Biophysical Bases of Electrotherapy

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CONTENTS

ELECTRICITY Chapter Chapter Chapter Chapter Chapter

1: Electronic Components and Circuits 2: Properties of Capacitors and Inductors 3: Electrical Properties of Skin 4: Electrical Stimulation of Nerve and Muscle 5: Rectification and Amplification

FIELDS Chapter 6: Electric and Magnetic Fields Chapter 7: Therapeutic Fields: Shortwave Diathermy Chapter 8: Non-Diathermic Fields

WAVES Chapter 9: Sound and Electromagnetic Waves Chapter 10: Therapeutic Waves: Ultrasound Chapter 11: Electromagnetic Waves for Therapy

GENERAL Chapter 12: Dosage and Safety Considerations Chapter 13: Electrical Safety Appendices: Quantities, Units and Prefixes

To open a chapter, click on the chapter name. To turn pages within a chapter, use the keyboard arrow keys.

To navigate within a chapter, use the navigation buttons on each page.

COMPONENTS AND CIRCUITS

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1 Electronic Components and Circuits Though the use of electricity in a medical context is no recent development. the last several decades have seen an unparalleled growth in electronic technology and its application in medicine. Sophisticated electronic apparatus is now as commonplace in the hospital and clinical setting as it previously was in the research laboratory. It is evident that all health professionals must become more and more accustomed to the use of electronic instrumentation and its proper place in the practice of their profession. The physiotherapist makes use of such apparatus for a variety of diagnostic procedures and treatments, collectively described by the terms electrodiagnosis and electrotherapy. The aim here is to provide a physical basis for electrodiagnosis and electrotherapy. We begin by discussing the 'nuts and bolts' of electronic apparatus, the basic components which go to make up an electronic circuit. Electronic circuits and electronic components owe their existence to the fact that the everyday materials with which we are familiar have quite dissimilar electrical properties. The vast majority of materials can be readily categorized as belonging to one of two groups: either conductors or insulators. First let us consider conductors and insulators and some additional concepts which are fundamental to an understanding of electricity.

SOME FUNDAMENTALS Conductors are familiar to us in everyday life: the filament of a light bulb, connected to car batteries and copper wire in the cables leading to appliances are a few examples. If we picture atoms as consisting charged nucleus together with electron 'shells' surrounding the nucleus of conductors have a characteristic feature.

the copper wire most electrical of a positively then the atoms

Although the central theme of this chapter is electronic components and hardware, most of the concepts are needed to understand the relevant physiology and electrotherapy covered in later chapters.

Apart from conductors and insulators there is a third, smaller group comprised of materials displaying intermediate conduction properties: this group includes the semiconductors which form the basis of modern electronics technology. We will talk more about semiconductors in chapter two.

Good conductors (iron, copper and other metals are examples) share the common feature that the outermost electrons of the atoms are only loosely bound to the nucleus. For this reason they can readily transmit electrons. A good way to picture this is to

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imagine a fairly rigid lattice of atomic nuclei with a sea of electrons contained within the boundaries of the piece of metal. An electron entering one end of a length of wire causes a 'ripple in the sea of electrons' and, a split second later, an electron is ejected from the other end. A flow of charges through any material is what we call an electric current and materials which readily transmit charges are termed conductors. The standard unit of current flow in the systeme internationale (Sl) is the ampere (abbreviated amp and given the symbol A). The current in amperes is a measure of the quantity of charge flowing through a conductor each second. The unit is named after the noted French physicist André Marie Ampère. Insulators have their electrons tightly bound in their shells so that the process of conduction described above can not take place under normal conditions. Plastics and ceramics are good insulators; thus we can use copper wires coated in plastic for the power leads to electric toasters, jugs, TV sets and so on. The telegraph poles in the street have the wires fastened not to the wood but around small white ceramic insulators. In order to cause a flow of current through a conductor we need a device which will produce an excess of electrons at one end of a wire and a deficiency at the other: such a device is a source of electrical potential energy and is said to produce a potential difference. A typical example of one of these devices is the battery found in a torch or transistor radio. The two terminals of a battery are at a different electrical potential - that is, there is a potential difference between the terminals. If a conductor is connected between the terminals electrons move from the terminal at a high potential to that at low potential. The Sl unit of potential difference is the volt (symbol V). Often the terms

a metal consists of a rigid and regular, crystalline array of metal ions ( ). A 'sea' or cloud of electrons ( ) fills the spaces between the ions.

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potential difference and voltage are used interchangeably. It is important to clearly understand the difference between voltage and current. Using the analogy with water in pipes we can liken voltage to the water pressure and current to the volume rate of flow of water. If we connect a piece of material between the terminals of a battery the amount of current flow will depend on the battery voltage and the resistance to current flow which the material offers. Good conductors offer little resistance to current flow while insulators offer substantial resistance. The Sl unit of resistance is the ohm (abbreviated Ω). We say something has a resistance of one ohm when a potential difference of one volt produces a current of one amp through it. A good conductor such as a metre length of household mains wire would have a resistance measured in milliohms while a good insulator, a block of ceramic, would have a resistance of several thousand megohms. Resistance values used in electronic circuits are typically measured in ohms (Ω), kilohms (kΩ) or megohms (MΩ). 1 kΩ = 103 Ω = 1000 Ω

Just as we can have a substantial pressure in the water mains without any flow of water through a tap, so we can have a substantial potential difference without any flow of current.

1 MΩ = 106 Ω = 1 000 000 Ω

COMPONENTS An electronic device such as a radio, TV set or CD player can be very complex but the complexity lies in the arrangement and total number of components. When we examine a typical circuit we find only a few different kinds of components. Resistors are the most common circuit components: they come in a variety of shapes and sizes and usually have their values coded in the form of three or four coloured stripes on the body. High power resistors are larger and usually have the resistance and power rating stamped on the body. Rheostats and potentiometers are variable resistors having two or three terminals respectively.

COMPONENTS AND CIRCUITS Capacitors come in a greater range of sizes and shapes than resistors but can readily be distinguished with a little practice.

fixed value capacitors They consist of two metal plates (usually aluminium) and an insulator. The insulator may be air (in the case of variable capacitors), mica or a plastic film. The different kinds of capacitor are named after the insulator used, thus we have mica capacitors, polyester capacitors, ceramic capacitors and so on. Typical examples of fixed value capacitors are shown, together with a variable capacitor such as might be used in the tuning section of a radio. For typical values of capacitance large areas of metal are involved and it is convenient to save space by making the plates of thin aluminium foil and rolling them into a cylinder. Thus a tubular capacitor has the internal construction shown. We will return to discuss what a capacitor does in the next chapter. The unit of capacitance is the farad (F) but the values found in typical circuits are measured in microfarads (µF),

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nanofarads (nF) or picofarads (pF). 1 µF =

1 F = 10-6 F 1 000 000

1 nF =

1 µF = 10-9 F 1 000

1 pF =

1 µF = 10-12 F 1 000 000

Electrolytic capacitors are polarized; that is, they have definite positive and negative terminals and can only be connected one way around in a circuit. Ordinary capacitors (mica, polyester, etc.) are non-polarized and can be connected either way around. Inductors and transformers form our third category of components. An inductor is simply a coil of wire: the wire may be wound on various kinds of core, depending on the specific role for which it is intended. An example is shown. The unit of inductance is the henry (H). You will also see the terms millihenry (mH) and microhenry (µH) used frequently. 1 mH =

1 H = 10-3 H 1000

1 µH =

1 mH = 10-6 H 1000

When two inductors are wound on the same core in close proximity or overlapping we have a transformer. Valves, Transistors and Diodes are the 'workhorses' of any electronic circuit, the devices which have permitted the development of long distance voice communication, radio and television, computers and space exploration, guided missiles and the reproduction of music to name a few of the more obvious applications. Today valves are seldom used except for special applications: they are bulky and inefficient and have largely been superseded by transistors, their semiconductor equivalents. The transistor performs much the same job as a valve but is physically much smaller.

COMPONENTS AND CIRCUITS Most electronic equipment today uses Integrated Circuits (IC's), small devices having 8 or more pins and which contain complete circuits having many transistors, diodes, resistors and capacitors fabricated directly in a single package. The pins are used for access to various points in the circuit so that the designer can tailor the circuit to specific requirements by connection to external components. Integrated circuits have permitted a further reduction in the physical size of complex circuits comparable to the reduction that was achieved by the replacement of valve circuitry with transistors. The pocket calculator, desktop computer and space satellites have all been made possible by the miniaturization permitted with integrated circuits.

ClRCUITS The electronic components we have met so far are typically found with a complex maze of interconnections between them. The particular way in which they are connected defines a circuit and the form that the circuit takes will depend on the job it is designed to do. If we consider, for example, a radio, the job it has to perform is exceedingly complex. It must pick up radio transmissions, convert the radio waves to electrical signals, amplify them and convert them to audible sound waves. Not only that but it must also be capable of selecting a particular frequency of radio wave (the one from the radio station you wish to tune-in on) and ignoring the remainder. Needless to say, the circuitry required to perform these tasks is very complex. In the next section we will consider extremely simple circuits in order to examine the characteristic behaviour of some of the components discussed so far. But first a few words on circuits themselves and their physical construction.

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COMPONENTS AND CIRCUITS

7

The earliest circuits, those found for example in a valve radio receiver, were made up of a large number of bulky components - the transformer, valves, etc., and the circuit interconnections reflected this bulkiness. Large components such as the transformer were bolted to a metal frame or chassis along with sockets for the valves, and the circuit was built-up by interconnecting with smaller components (resistors and capacitors) and lengths of wire. With the advent of the transistor and the consequent trend to miniaturization which this permitted, it was found to be more convenient and simpler to replace most of the wiring with copper strips firmly attached to an insulating board having holes drilled in appropriate places. All of the miniature components (transistors, resistors, capacitors) could then be mounted on the board and soldered to the copper conductors. Only a limited number of bulky components (transformer, volume control and speaker in a mains-powered radio) need then to be mounted on a chassis or casing. Circuits assembled on these printed circuit hoards can be rapidly mass produced and easily put together. Consequently they are used almost exclusively by manufacturers. When integrated circuits are used rather than transistors a further reduction in size is achieved. The printed circuit boards can be made smaller as the l.C.'s themselves are small and relatively few external components are required.

SIMPLE CIRCUITS WITH RESISTORS Perhaps the simplest kind of circuit which can be constructed comprises a battery (a source of electrical energy) and a single electrical component such as a resistor or a torch globe. First consider the simple circuit arrangement shown in figure 1.1. On the left is a small lamp (torch globe) connected by means of wires to a dry cell (battery). On the right is the circuit diagram of the arrangement: note the circuit symbols used for the lamp and dry cell. When connected this way current will flow from one terminal of the battery, through the lamp and back to the opposite terminal. It is common convention to say that current flows from the positive terminal of the battery to the negative terminal; although this is the opposite direction to the flow of

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electrons. The difference between reality and convention is not important as it has no effect on the magnitudes of currents and voltages in a circuit. Under normal circumstances the flow of current through the lamp will cause it to glow - the normal purpose of such a circuit. If a break occurs in the wire for example, or the wire is not secured to the battery, current will not flow and we have an open circuit. If the wire connected to one terminal of the lamp comes loose and touches the other wire, current can flow directly through the wires and bypass the lamp this condition is known as a short circuit. The switch on a torch is just a means of introducing an open circuit condition at will: preventing the flow of current and thus conserving the electrical energy stored in the battery. Going one step further than the circuit shown in figure 1.1 we may add a resistor to produce the circuits shown in figure 1.2. Note the circuit symbol (zig-zag line) for a resistor. Figure 1.2(a) shows the resistor in series with the lamp, i.e. the resistor and lamp are connected end to end so that current must flow through both the lamp and the resistor. In 1.2(b) the resistor is connected in parallel with the lamp so that some current can flow through the lamp and some through the resistor. The resistor is thus allowing part of the current to bypass the lamp and return to the opposite terminal of the battery.

Figure 1.1 A very simple circuit and its representation using a circuit diagram

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RESISTORS IN SERIES Two or more resistors may be connected in series or in parallel. When the resistors are connected in series as in figure 1.3 the current must pass through each resistance in turn and the total resistance offered by the circuit is equal to the sum of the individual resistances.

Figure 1.2 Two less simple circuits. (a) a series and (b) a parallel arrangement.

In general for a circuit having n resistors connected in series the total resistance of the circuit, R, is equal to the sum of the individual resistances i.e. R = R1 + R2 + R3 ........ + Rn

.... (1.1)

As mentioned previously, the current flowing is a measure of the rate of flow of charges around the circuit and is measured in amperes, or amps for short. Since there can be no accumulation of charges within a resistor the current flowing into a certain resistor is equal to the current flowing out. Thus the same current must flow through each resistor when they are connected in series. Although the current in each resistor is the same, the potential difference across each resistor in a series circuit is, in general, different. The sum of the potential differences across the resistors is equal to the potential difference generated by the battery (the battery voltage).

COMPONENTS AND CIRCUITS

Figure 1.3 Resistors in series

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RESISTORS IN PARALLEL When a number of resistors are connected in parallel as in figure 1.4 the current flowing in each resistor may be different. In this case the total resistance of the circuit, R, is given by the equation 1 1 1 1 = + + R1 R2 R3 R or more generally, where there are n resistors in parallel in a circuit 1 1 1 1 1 = + + ..... + R1 R2 R3 Rn R

.... (1.2) Figure 1.4 Resistors in parallel

Although the current in each resistor may be quite different, the voltage or potential difference across each resistor is the same and is equal to the battery voltage. This must be so since the ends of each resistor are connected directly to the terminals of the battery. The largest amount of current will flow through the smallest resistance when resistors are connected in parallel.

OHM'S LAW The relationship between potential difference (V), resistance (R) and current (I ) through a resistor is summarized by Ohm's law which states that the current flowing through a resistor is proportional to the potential difference across the resistor and is inversely proportional to the resistance. The equivalent mathematical expression of Ohm's law is

I =

V R

.... (1.3)

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where I is the current in amperes, V is the potential difference in volts and R is the resistance in ohms. Example 1. If, in figure 1.3, R1 is 50 ohms, R2 is 100 ohms and R3 is 150 ohms then the total resistance R is R = 50 + 100 + 150 = 300 Ω If these are connected to a 10 volt battery the current flow will be

I =

V 10 = = 0.033 amps or 33 milliamps. R 300

Here we have calculated the total current drawn from the battery which is equal to the current flowing through each resistor. The battery gives a total potential difference of 10 volts across the combination of resistors. If we wish to calculate the potential difference across each individual resistor we again use equation 1.3 but apply the equation to each resistor in turn. For example, for the 50 ohm resistor we know that

I = 0.033 amps and R = 50 Ω so I =

V V becomes = 0.033 and hence V = 50 x 0.033 = 1.7 volts. R R

For the 100 ohm resistor V = 100 x 0.033 = 3.3 volts and for the 150 ohm resistor V = 150 x 0.033 = 5.0 volts. The total potential difference across the combination of resistors is V = 1.7 + 3.3 + 5.0 = 10 volts as expected.

COMPONENTS AND CIRCUITS

Example 2. Now consider the same three resistors as in the previous example connected in parallel to a 10 volt battery as in figure 1.4. The total resistance, R, is now given by 1 1 1 6 3 2 11 1 = + + = + + = 50 100 150 300 300 300 300 R 300 = 27 ohms 11 V 10 = = 0.37 amps The total current is I = R 27 so R =

To calculate the current in each individual resistor we use the fact that the potential difference across each resistor is the full 10 volts. Then for the 50 ohm resistor we have

I =

V 10 = = 0.20 amps. R 50

For the 100 ohm resistor I =

10 = 0.10 amps 100

and for the 150 ohm resistor I =

V 10 = = 0.07 amps R 150

So the total current drawn from the battery is

I = 0.20 + 0.10 + 0.07 = 0.37 amps as calculated previously. The calculations are made relatively simple by remembering that when resistors are connected in series the current through each is the same and when resistors are connected in parallel the potential difference across each is the same.

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POWER DISSIPATION IN RESISTORS When electric current flows through a conductor, whether the conductor is a resistor, copper wire, the filament of a light bulb or whatever, electrical energy is converted to heat energy in the conductor. An extreme example of this is seen with a lamp - the heat generated is so great that the filament wire glows brightly. The same heating effect takes place, though to a lesser extent, in a resistor and clearly there will be a limit to how much heat a resistor can generate per second before it starts glowing and disintegrates or melts. The maximum rate at which a resistor can safely dissipate electrical energy is called the power rating of the resistor and this depends on the size and physical construction. The actual rate of dissipation of electrical energy at any particular time is the power dissipated. This depends on both the current flowing through the resistor and the potential difference across it. The power (in watts) is equal to the product of current (in amps) and potential difference (in volts). If we denote power by the symbol P the relationship may be written P = V.I

.... (1.4)

This formula can be usefully used with Ohm's law to calculate the maximum values for voltage or current for a particular resistor of known resistance and power rating. Alternatively if we know any two of the values of V, I or R for a given resistor, we can calculate the power dissipated in the resistor using equation 1.4 or the two expressions obtained by solving equations 1.3 and 1.4. These are: P =

V2 R

.... (1.5)

where we have eliminated I in solving the two expressions and P = I2.R

.... (1.6)

Starting with the expression P = V.I and substituting V = I.R we have P = (I.R).I or P = I 2R Starting with the expression P = V.I and substituting I = V/R 2 we have P = V.(V/R) or P = V R

where we have eliminated V.

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With these three alternative formulae for calculating power we would choose the equation which gives the required answer with a minimum of calculations. Consider the three resistors connected in series as in example 1 previously. If we wish to calculate the power dissipated in each resistor we would first calculate the current (which is the same for each resistor) and then use equation 1.6. In example 2 previously the resistors are in parallel and the potential difference is the same for each resistor. Equation 1.5 is thus the choice for power calculation.

RESISTORS IN A SERIES/PARALLEL COMBINATION The ideas we have considered previously can be applied to more complex circuits where resistors are combined in a series/parallel arrangement. Figure 1.5 shows one possible series/parallel combination. To calculate the total resistance in this circuit we would first need to calculate the resistance of R2 and R3 in parallel using equation 1.2. The total resistance is obtained by adding this to R 1 . This is because R 1 is in series with the parallel combination of R2 and R3 so equation 1.1 applies. To calculate voltages and currents in this circuit it is important to visualize current flowing from the positive terminal of the battery, through R1 and then through the combination of R2 and R3 . The current flowing through R1 splits. Some goes through R2 and some through R3. The current though R2 plus the current through R 3 must equal that through R1 since the moving charges cannot disappear nor can new ones appear. Another important idea is that current (charge) flows around the circuit, losing electrical potential energy in the process. Some energy is lost as charges flow through R1 . The remainder is lost as charges flow either through R2 or R3 . The potential difference across R 2 is one-and-the-same as the potential difference

Figure 1.5 Resistors in a series/parallel arrangement

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across R3. The potential difference across R1 plus that across the R2/R3 combination must equal the battery voltage. Example 3. If we have a resistor combination like that shown in figure 1.5, with the resistance values and battery voltage are as shown alongside: (i)

what is the current through each of the resistors?

(ii)

do the currents calculated add-up i.e. is the current through the 50 Ω resistor equal to the total current through the 100 Ω and 200 Ω combination?

(iii) what is the potential difference across the 50 Ω resistor? (iv) what is the power dissipated in the 100 Ω resistor? Answers: (i)

First calculate the total resistance. For the parallel combination,

1 1 3 200 1 = + = so R = = 67 Ω 100 200 200 3 R

The total resistance is thus 67 + 50 = 117 Ω The battery voltage is 10 V so the total current is I =

V 10 = = 0.085 A R 117

All of this current flows through the 50 Ω resistor so the current through it is 0.085 A or 85 mA. The potential difference across the 50 Ω resistor is V = I.R = 0.085 x 50 = 4.3 V, which answers question iii above. The potential difference across the 100 Ω/200 Ω parallel combination is 10 - 4.3 = 5.7 V.

COMPONENTS AND CIRCUITS

(ii)

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The current through the 100 Ω resistor is I =

V 5.7 = = 0.057 A R 100

The current through the 200 Ω resistor is I =

V 5.7 = = 0.029 A R 200

The total current through the 100 Ω/200 Ω parallel combination is 0.057 + 0.029 = 0.086 A which (allowing for rounding errors as we have used two-figure accuracy) agrees with the value calculated for the 50 Ω resistor.

(iii) See above. (iv) The power dissipated in the 100 Ω resistor can be calculated using either equation 1.4 or 1.5 or 1.6 as we know the potential difference, current and resistance. Using equation 1.4 we have P = V.I = 5.7 x 0.057 = 0.33 W.

AC AND THE MAINS SUPPLY DC and AC If we connect a battery to a lamp, as in figure 1.1 previously, current will flow from one terminal of the battery, through the lamp and return to the opposite terminal. Such a flow of current is termed direct current (DC) because the flow is always in one direction. A graph of voltage against time or current against time would look like figure 1.6. DC is directly available from a battery and it is DC that we need to power most electronic circuits. The voltage available at the power points in the laboratory or home is not, however, constant or direct but fluctuates rapidly and regularly between about + 340 and -340 volts. The repetition rate or frequency of the alternating voltage is 50 times per second. If we plotted a graph of voltage against time it would look like figure 1.7.

Figure 1.6 Direct current as from a battery

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The alternating voltage supplied by the mains gives rise to an alternating current (AC) in any device which is connected to a power point. During the positive half-cycles of the waveform shown in figure 1.7 the current flows in one direction through the appliance and during the negative half-cycles the current flows in the opposite direction. Because the waveform in figure 1.7 is symmetrical the current flow in one direction during the positive half-cycles will be exactly balanced by the current flow in the opposite direction during the negative half-cycles. Hence there is no net transfer of charge over a full cycle; electrons are simply moved back and forth. This is in contrast to direct current where electrons are moved in one direction only. Figure 1.7 Alternating current as from the mains supply

The frequency or number of times an AC waveform repeats itself per second is expressed in units of hertz (abbreviated Hz). 1 Hz is 1 cycle per second. It is easy to calculate the frequency of an AC waveform from a graph like figure 1.7. First we determine the period of the waveform. The period is simply the time taken for one complete cycle which is read from the horizontal axis. In this example, the alternating current is from the mains supply and the period, the time for one complete cycle or oscillation, is 0.02 second or I /50th of a second. The frequency (f) is the reciprocal of the period (T). In symbols: f =

1 T

.... (1.7)

Hence for AC supplied by the mains the frequency is 1/0.02 = 50 Hz. During the first hundredth of a second the voltage in figure 1.7 has gone from zero to +340 volts and back to zero again. The effective (average) voltage will be some figure between these limits. The figure of 340 volts (the maximum value) is termed the peak voltage but we normally specify AC by an average rather than peak value. The or 240 volts in this example. average during the first hundredth of a second is 340/

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Thus the appliances we plug into the mains must have a rating of 240 volts AC. Since the voltage is rapidly oscillating between + 340 and -340 volts the current through a light globe connected to the mains will flow alternately in one direction and then the other and the globe will flicker in brightness according to the voltage. It is only because the frequency of fluctuation is sufficiently rapid that this effect is not noticeable.

TRANSFORMERS AND THE MAINS SUPPLY Transformers are the devices we use to step-up or reduce AC voltage. If we apply a certain AC voltage to one winding of a transformer (the primary winding) an AC signal of the same shape but different amplitude (peak voltage) will appear on the other winding (the secondary winding). Power Transformers are specifically designed to convert 240 volts AC to some other voltage: for example a stereo tuner/amplifier would require only 20 to 60 volts to power the circuitry comprising the radio and amplifier, it would thus require a power transformer to step down the mains voltage to an appropriate value. An example of a transformer was shown in section 1.1. Figure 1.8 illustrates the circuit symbol and construction of a transformer. A transformer consists of two coils of wire which are insulated from each other but in close proximity. In practice the two coils of wire are often wound on a common core, made of iron (see figure 1.8(b)). This allows the transformer to work more efficiently. When AC is applied to one coil of wire (the 'primary' coil), an alternating current is induced in the other (the secondary) and the voltage will, in general, be different. If the transformer is constructed with more turns on the primary winding than on the secondary then it will act as a step-down transformer. For example if there are 200 turns on the primary winding and 50 on the secondary the voltage from the secondary will be one quarter of that applied to the primary. If 240 volts is applied to the primary 60 volts will be produced at the secondary.

Figure 1.8 (a) Circuit symbol and (b) construction of a transformer

COMPONENTS AND CIRCUITS If the transformer is constructed with fewer turns on the primary winding than on the secondary then it will act as a step-up transformer. For example if there are 20 turns on the primary winding and 100 on the secondary the voltage from the secondary will be five times that applied to the primary. If 240 volts is applied to the primary 1200 volts will be produced at the secondary. Because of their physical construction - two inductors in close proximity but insulated - they play a useful role in isolating the voltage applied to one winding (the primary) from that available at the other winding (the secondary). Most appliances are isolated from the mains voltage in this way - by their transformers. The isolation given by a transformer is important because the voltage from the secondary is earth-free.

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The principle that 'you can't get something for nothing' applies here. If the transformer steps-up the voltage, the available current is reduced. If the transformer steps-down the voltage, the current is increased.

The power from the mains comes from two cables, the active cable (colour coded brown) and the neutral cable (colour-coded blue). The neutral cable is earthed at the power station and at distribution substations. That is, the neutral cable is physically connected to the ground through a metal conductor (for example, a thick length of wire connected to a metal stake or water supply pipe buried in the ground). The neutral cable is thus kept at zero volts potential. The problem with this arrangement is that if a connection is inadvertently made between the active cable and earth, such as by a person accidentally touching the active cable, this will complete a circuit and current will flow from the active wire through the person to earth, giving an 'earth' shock. To obtain a shock from the 'earth free' power supplied at the transformer secondary it would be necessary for the person to touch both terminals simultaneously. As long as there is no connection between the primary and secondary windings, touching only one secondary terminal and thus making a connection between the secondary circuit and earth will not complete a circuit and no current can flow, hence no shock. Mains supply and the mechanisms of electric shock are described in more detail in a later chapter.

COMPONENTS AND CIRCUITS

EXERCISES 1

The lamp shown in figure 1.1 has a resistance of 3.0 Ω and is connected to a 1.5 V battery. Calculate the current flowing through the lamp and the power dissipated.

2

An electric jug draws a current of 1.2 amps when connected to the 240 volts mains supply. Calculate the resistance and power rating of the heating element.

3

The power dissipation of an electric heater is 1000 watts when connected to a 240 volt supply. Calculate its resistance, and the current it will draw.

4

Three resistors are connected in series as shown in figure 1.3. The battery voltage is 12 V. The resistors have values R1 = 100 Ω, R2 = 500 Ω and R3 = 600 Ω. Draw a diagram of the circuit and label the values of the components. What is: (a) the current through each resistor (b) the potential difference across each resistor (c) the power dissipated in each resistor Do the values calculated in (b) add to equal the battery voltage?

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Three resistors are connected in parallel as shown in figure 1.4. The battery voltage is 5 V and the resistors have values R1 = 50 Ω, R2 = 100 Ω and R3 = 75 Ω. Calculate: (a) the current through each resistor (b) the potential difference across each resistor (c) the power dissipated in each resistor (d) the total resistance of the circuit and hence (e) the total current drawn from the battery.

Figure 1.9 Isolation with a power transformer Current can only flow along the 'active' wire if it can simultaneously return via the 'neutral' wire. Because the neutral wire is earthed, anything touching the earth is provided with a return pathway via the neutral wire.

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COMPONENTS AND CIRCUITS (f) 6

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does the value calculated in part (e) agree with the sum of the values found in part (a) above?

Three resistors are connected in the circuit arrangement shown. The current through the 100 Ω resistor is 50 mA. Determine: (a) the battery voltage and (b) the power dissipated in the 30 Ω resistor.

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In the course of an experiment a student sets-up the circuit shown below using a torch globe, 2 resistors and a battery. A voltmeter connected across the 200 Ω resistor registers 8 volts.

(a) (b)

What is the resistance of the torch globe? What is the current through the globe?

COMPONENTS AND CIRCUITS 8

Consider the circuit shown. The current through the 50 kΩ resistor is measured as 67 µA. Determine the: (a) value of the unknown resistor, R. (b) total current drawn from the battery. (c) power dissipated in the 50 kΩ resistor. (d) potential difference across the resistor, R. (e) current through the 100 kΩ resistor.

9

Consider the circuit shown. The total current drawn from the battery is 0.2 A. What is the: (a) current through the 30 Ω resistor. (b) value of the unknown resistor, R. (c) power dissipated in the 50 Ω resistor. (d) potential difference across the 30 Ω resistor. (e) power dissipated in the resistor, R.

10

(a) (b)

(c)

What are the two important functions of a transformer? A transformer has 100 turns on the primary winding and 40 turns on the secondary. If an AC signal of amplitude 60 V is applied to the primary winding what voltage will appear at the secondary? A transformer has 200 turns on the primary winding. An AC signal of amplitude 240 V is applied to the primary winding and a voltage of 12 V is produced by the secondary. How many turns are there on the secondary?

22

CAPACITORS AND INDUCTORS

23

2 Properties of Capacitors and Inductors Figure 2.1 shows the circuit symbol for a capacitor and an inductor. Note the parallel between the circuit symbol and the physical construction of the component. A capacitor consists of two metal plates (often sheets of aluminium foil) separated by an insulator. An inductor is simply a coil of wire. Knowing the physical construction of each we can predict what resistance they offer when connected in a circuit to a battery. The capacitor, with an insulator between its plates, offers an extremely high (virtually infinite) resistance to DC. Current can not flow from one plate to the other. Inductors, being simply coils of conducting wire, have an extremely low resistance to DC (almost zero).

STORAGE CAPABILITY OF CAPACITORS A special property of a capacitor is its ability to store charge and thus electrical energy. If we connect a battery to the capacitor current will flow for a very short time and, since the current can not flow continuously through the insulator, the capacitor will accumulate charge on the plates. When the charge has built up until the voltage across the capacitor is equal to the battery voltage, current will no longer flow.

Figure 2.1 Circuit symbol for (a) a capacitor and (b) an inductor.

The capacitance tells us how much charge a capacitor will accumulate for a given battery voltage. Capacitance (C), charge (q) and voltage (V) are related by the formula C =

q V

.... (2.1)

Once charged, a capacitor can store the charge indefinitely. If the battery is disconnected, the charge on the capacitor and can only leak away if we connect an external resistance. The circuit shown in figure 2.2 can be used to observe the discharge of a capacitor.

CAPACITORS AND INDUCTORS

24

Suppose that the capacitor is first fully charged to 12 V, then connected into the circuit. Upon closing the switch, current flows through the 500 kΩ resistor, allowing the capacitor to discharge. The reading on the voltmeter progressively falls. Table 2.1 shows measured values of the voltage taken at 2 second intervals for 20 seconds using a 25 µF capacitor. The 25 µF capacitor was then replaced with a 50 µF capacitor and the second set of results was obtained. Time (s) 0 2 4 6 8 10 12 14 16 18 20

Voltage on the 25 µF capacitor 12 10.2 8.7 7.4 6.3 5.4 4.6 3.9 3.3 2.8 2.4

Voltage on the 50 µF capacitor 12 11.1 10.2 9.4 8.7 8.0 7.4 6.8 6.3 5.8 5.4

Figure 2.2 A circuit for measuring the discharge of a capacitor

Table 2.1 Measured voltages for the discharge of capacitors through a resistance of 500 kΩ

A graph of these voltages against time is shown in figure 2.3. Note that it takes about 8.5 seconds for the 25 µF capacitor to discharge to half of the original voltage and 17 seconds for the 50 µF capacitor. On this basis we predict that it would take about 85 seconds for a 250 µF capacitor to discharge to half the original voltage through the same resistance.

CAPACITORS AND INDUCTORS

25

What we observe is that the time to half discharge (τ1/2) is directly proportional to the capacitance (C) of the capacitor. In symbols this is written

τ1/2 α C Other experiments, where the resistance through which the capacitor discharges is varied, show that the time for half-discharge is also proportional to the resistance (R) through which the capacitor discharges. That is,

τ1/2 α R Combining these equations we have that the time for half discharge is directly proportional to the product of the resistance and the capacitance. In symbols

τ1/2 α R.C

.... (2.2)

This is a general relationship between resistance, capacitance and discharge time for a resistor/capacitor combination. Sometimes the term 'half-discharge time' is used, sometimes 'RC time-constant'. They are related, but are not the same thing. The RC timeconstant is simply the product R.C. The constant of proportionality in equation 2.2 is 0.693 (the natural logarithm of 2) so if we know the RC time constant, τ1/2 is easily calculated.

Figure 2.3 Discharge of capacitors through a resistance

Notice that for a short period of time, such as 1/50th of a second, the 25 µF capacitor in table 2.1 will not have discharged appreciably. If it discharges by 0.9 volt in the first second, in l/50th of a second it will discharge by less than 0.02 volt. The importance of this and the previous ideas will be considered later.

CAPACITORS AND INDUCTORS

26

INDUCTORS AND MAGNETIC FIELDS When current flows though an inductor, a magnetic field is produced. In fact, whenever current flows, there is an associated magnetic field so a straight piece of wire which is carrying current will have a magnetic field around it. The reason inductors are special is that the wire is wound in closely spaced loops and this results in a much more intense magnetic field. The field can be further intensified by wrapping the coil around a piece of iron or mild steel. Both the magnetic field intensity and the inductance are increased if the inductor is wound around a core of iron or other so-called ferromagnetic material.

Most of you would have made an electromagnet at school. Insulated copper wire is wound around, say, a nail to make an inductor. When the ends of the wire are connected to a battery, the nail becomes magnetized.

We will have more to say about magnetic fields (and their effects on tissues) in later chapters.

IMPEDANCE OF CAPACITORS AND INDUCTORS A major part of the field of electronics is concerned with processing alternating current of various frequencies and it is for this reason that capacitors and inductors are so important. Resistors are impartial as far as direct current and alternating current is concerned: they offer the same resistance to AC and DC. Capacitors and inductors display no such impartiality: they both react quite differently to AC and DC. Before describing the properties of capacitors it is useful to have some idea of the kind of frequencies we are dealing with. Sounds which are audible to humans lie in the range from 20 Hz to up to a maximum of about 20 kHz (1 kHz = 1000 Hz). Normal (AM) radio transmissions are made in the frequency range from 500 to 1500 kHz. FM radio transmissions are made in the range 88 to 108 MHz. (l MHz = 1000 kHz or 1000 000 Hz). Check-out the numbers on your tuner or pocket radio display panel. Television transmissions are made in the frequency range on either side of the FM range, i.e. between 40 and 80 MHz and above 108 MHz up to about 130 MHz. In each case the sound, radio or television frequency wave can be converted to alternating current of the same frequency which can be amplified, filtered and processed by an electronic circuit to produce the resulting sound or image.

Infants can hear sound frequencies of 20 kHz or more. With age, this maximum frequency drops so that most people over 35 can not hear frequencies above about 14 kHz.

CAPACITORS AND INDUCTORS

27

Capacitors and inductors are useful because their resistance to alternating current depends on the frequency of the AC. Next we examine this frequency dependence. Knowing the physical construction of capacitors and inductors we can predict what resistance they offer when connected in a circuit to a battery. The capacitor, with an insulator between its plates, offers an extremely high (virtually infinite) resistance to DC. Current can not flow from one plate to the other. Inductors, being simply coils of conducting wire, have an extremely low resistance to DC (almost zero). For alternating current the resistance of these components is not predicted quite so simply. Instead of attempting a prediction let us look at some experimental observations with AC. When we talk about resistance to the flow of alternating current we no longer use the term 'resistance' but impedance. The impedance is still measured in ohms but we reserve 'resistance' for use when talking about DC. The symbol Z is usually used for impedance. Impedance can be calculated from measurements using a circuit such as that shown in figure 2.4. The potential difference across R and that across C are found to depend on the AC frequency. The two values added equal the potential difference produced by the AC source. The voltage across the resistor can be used to calculate the current flowing around the circuit (using Ohm's Law, I = V/R). Since C and R are in series, this equals the current flowing to the capacitor. The impedance of the capacitor is then calculated using V Z = .... (2.3)

I

Which is Ohm's law in its more general form.

Figure 2.4 A circuit for measuring the impedance of a capacitor

If the capacitor in figure 2.4 is replaced by an inductor, impedance values for the inductor can be calculated in a similar manner. Table 2.2 gives the measured impedance of a 20 millihenry (mH) inductor and a 1 microfarad (µF) capacitor at various frequencies.

CAPACITORS AND INDUCTORS Table 2.2 Variation of impedance with frequency for a 1 µF capacitor and a 20 mH inductor

28

Frequency f (kHz)

Impedance of 1 µF capacitor (ohms)

Impedance of 20 mH inductor (ohms)

0.2 0.3 0.4 0.6 1.0 2.0 3.0 4.0 5.0

795 530 398 265 159 79.5 53.0 39.8 31.8

25.1 37.7 50.2 75.4 126 251 377 502 628

A graph of these values of impedance at each frequency is shown in figure 2.5. Notice that for the inductor the graph is a straight line passing through the origin. This tells us that the impedance of the inductor is directly proportional to the frequency of the AC. For the capacitor the curve suggests an inverse relationship between impedance and frequency: this is, in fact, the case. To check that the relationship is an inverse one rather than inverse square or inverse cube we would need to plot impedance versus 1/frequency and establish that this gives a straight line graph. This is left as an exercise for the interested reader. The mathematical expression for the dependence of impedance on frequency for each of these components is

and

Zαf

for the inductor

Z α 1/f

for the capacitor.

By substituting different values of inductance and capacitance, it is established that for inductors, Z α L and for capacitors, Z α 1/C.

CAPACITORS AND INDUCTORS

29

Since for inductors, impedance is proportional to the frequency and is also proportional to the inductance, we can combine these expressions and write Z α f.L. When the impedance is specified in Ohms, the inductance in Henrys and the frequency in Hz, the constant of proportionality is 2π. The relationship between Z and f for an inductor is thus Z = 2πfL

.... (2.4)

Similarly for the capacitor: since impedance is proportional to the reciprocal of frequency and the reciprocal of the capacitance we can combine the expressions and write Z a 1/f.C. When the impedance is specified in Ohms, the capacitance in Farads and the frequency in Hz, the constant of proportionality is 1/2p. The relationship between Z and f for a capacitor is thus Z =

1 2πfC

.... (2.5)

From equations 2.4 and 2.5 we can see that it is predicted that doubling the value of inductance would double the impedance at all frequencies. Doubling the value of capacitance would halve the impedance at all frequencies. These predictions are easily confirmed experimentally.

Figure 2.5 Impedance versus frequency graph for (a) a 1 µF capacitor and (b) a 20 mH inductor

Two important points to note from the equations (or the graphs) are that capacitors have an infinite impedance to direct current (zero frequency) whereas inductors have zero impedance to DC. This is in agreement with our previous predictions based on the physical construction of each.

CAPACITORS AND INDUCTORS

30

RESONANT CIRCUITS A traditional and simple way of producing alternating current relies on the properties of capacitors and inductors and their behaviour when connected together in a circuit. An inductor and capacitor combined in parallel as in figure 2.6 forms what is called a parallel resonant circuit. The impedance of the inductor increases with frequency while the impedance of the capacitor decreases with increasing frequency (figure 2.5). The parallel combination will have a low impedance at low frequencies (because of the inductor) and at high frequencies (because of the capacitor). At an intermediate frequency the combination has a maximum impedance. This is when the impedances of the components are equal. The peak in the graph shown in figure 2.6 occurs at this frequency. A formula for calculating the frequency at which the impedances of the capacitor and inductor are equal is obtained as follows. The impedance of the capacitor is given by equation 2.5, and the impedance of the inductor by equation 2.4. When the impedances of the components are equal 1 2πfC

= 2πfL

which rearranges to give f2 =

1 4π 2LC

Figure 2.6 A parallel resonant circuit and how its impedance varies with AC frequency

CAPACITORS AND INDUCTORS

or

f =

1 2π√  L C

31

.... (2.6)

This frequency is termed the resonant frequency of the combination, and for good reason. Imagine that the capacitor in figure 2.6 is disconnected from the inductor and connected to a battery to charge it. The battery is then removed and the capacitor connected to the inductor again. What subsequently happens? The capacitor immediately begins to discharge and current flows through the inductor. In so doing the current flow sets up a magnetic field around the inductor. Immediately the charge disappears the current ceases, causing the magnetic field to collapse. The collapsing magnetic field then induces a current in the inductor and the capacitor recharges, but with opposite polarity. The sequence of events is illustrated in figure 2.7 where the arrows show the direction of current flow. The capacitor next begins to discharge in the opposite direction (not shown in figure 2.7 - this would follow (d)) and the whole cycle is repeated again and again. What is happening is that the electrical energy stored in the capacitor is converted to magnetic field energy around the inductor. Because a magnetic field can only exist if current is flowing, the magnetic field must collapse as the capacitor becomes discharged. The magnetic field energy can not disappear, but is converted back into electrical energy which moves charge around the circuit and reverse-charges the capacitor. The reverse-charged capacitor (figure 2.7(d)) would then discharge through the inductor. Current would flow in the

CAPACITORS AND INDUCTORS

Figure 2.7 Initial current flow in a resonant circuit

32

reverse direction to that in figure 2.7, producing a magnetic field which would collapse and re-charge the capacitor to its original state (figure 2.7(a). The process would repeat indefinitely. If this were the whole story the circuit would resonate, generating a continuous alternating flow of current through the inductor. A graph of current versus time would be an undamped sine wave as shown in figure 2.8(a). A graph of voltage across the capacitor against time would also resemble figure 2.8(a). In practice, capacitors and inductors are not ideal. The inductor will have some DC resistance and the capacitor will allow some leakage of current so that energy is lost during each oscillation. The oscillations will be damped and must eventually come to an end. The net result will be a damped oscillation as shown in figure 2.8(b). The natural frequency of oscillation is the same frequency indicated by the peak in the graph shown in figure 2.6: the resonant frequency. This is also the frequency at which the impedances of the capacitor and inductor are equal (where the graphs intersect in figure 2.5). [A graph similar to that in figure 2.6 could be arrived at simply by adding impedances graphically, say by using figure 2.5. The method gives a correct qualitative result. It does not give a correct quantitative prediction as no consideration is given to the phase relationship between the current in each component. The 'resonant frequency' obtained is correct but the height of the impedance peak is not. A rigorous mathematical treatment shows that, for ideal components, the impedance is infinite at the resonant frequency. For real components the impedance will be finite because of the resistance of the inductor and leakage of the capacitor].

Figure 2.8 (a) undamped and (b) damped sinewaves

CAPACITORS AND INDUCTORS

33

Although resonant circuits are most commonly encountered in the parallel combination shown in figure 2.6, they can occur as a series combination (figure 2.9). Again there is a definite resonant frequency at which the impedances of the two components are equal. For a series resonant circuit the impedance has a minimum value at the resonant frequency. Because of the non-ideal nature of the components the impedance is not zero at the resonant frequency but when component losses are low the impedance can suddenly drop to a very low value as this frequency is approached.

COUPLING AND RESONANT CIRCUITS If an oscillator is designed to operate at a fixed frequency the usual way of coupling its signal to an amplifier or, more generally, between circuits is by use of a resonant circuit. This method is most commonly used at high frequencies (above 100 kHz). Figure 2.10 shows how a resonant circuit is used to transfer electrical energy from one circuit to another most efficiently. In other words to 'couple' the circuits most efficiently. The signal from the first circuit supplies energy for the resonant circuit to oscillate. The signal is induced in the second circuit by the transformer action of the two inductors.

Figure 2.9 A series resonant circuit and how its impedance varies with AC frequency

In order for transformer action to operate most efficiently the signal produced by circuit 1 should have the same frequency as the resonant circuit. This means either that the frequency of circuit 1 can be adjusted to match that of the resonant circuit or that the frequency of the resonant circuit can be adjusted to match that of circuit 1. The latter would require the inductor or capacitor in the resonant circuit to be variable so that the resonant frequency could be adjusted to suit.

CAPACITORS AND INDUCTORS

34

Figure 2.10 Coupling with a resonant circuit.

The combination of L and C will have a resonant frequency which is determined by their inductance and capacitance values according to equation 2.6. The current through the inductor connected to circuit 1 will induce a current of the same frequency in L. If the current induced in L does not have the same frequency as the resonant circuit, little energy will be transferred. If the resonant frequency of L and C in combination is the same as the frequency of the signal supplied by circuit 1, then the circuit will resonate and maximum electrical energy will be transferred.

SUSTAINED OSCILLATION By appropriate choice of the capacitor and inductor a resonant circuit can be made to generate any frequency of sine wave. A resonant circuit alone is not sufficient, however, to generate a sustained oscillation. To produce a continuous, steady supply of alternating current (as in figure 2.8a) we must arrange for the resonant circuit to be continuously supplied with energy to overcome the losses in the components and keep it oscillating. By use of an amplifying circuit, we can provide the energy to overcome the circuit losses and prevent the oscillations from dying. An amplifier provides positive feedback. Amplifiers and positive feedback are considered in chapter 5.

This is analogous to pushing a swing. The swing has its own, natural or resonant frequency. If you apply pushes at the same frequency, the swing oscillations are reinforced. Pushing at a different frequency would be counterproductive.

CAPACITORS AND INDUCTORS

35

PIEZOELECTRIC CRYSTALS So far our discussion of oscillators has been restricted to resonant circuits in the form of inductor and capacitor combinations. Unfortunately, such combinations tend to drift in frequency over a period of time and with changes in ambient temperature. The effect is only slight and not very important at low frequencies unless high accuracy is required. The effect is much more significant at radio frequencies (greater than 500 kHz) where it is very important to have good frequency stability - imagine switching on your radio and never knowing quite where to find your favourite radio station! The effect is really quite serious in TV and radio transmission and communications. The frequencies allocated to users are quite closely spaced and any drift in transmission frequency could result in overlap with adjacent transmitters. Similarly the shortwave diathermy equipment used in physiotherapy operates at radio frequencies and radiates a certain amount of energy as radio waves - for this reason only certain frequencies are permitted for their operation and little deviation or drift is permitted. Good frequency stability can be achieved by use of a piezoelectric crystal. These crystals have the special property that when squeezed or stretched, a potential difference is produced between each surface. This piezo-electric effect is illustrated in figure 2.11. The other side of the coin with piezoelectric crystals is that if a potential difference is applied to their opposite sides, they change in thickness. Thus in the example shown in figure 2.11, if a potential difference is applied to an unstressed crystal and the voltage is positive on top, the crystal will shrink in thickness. If the potential difference is applied negative on top in this example, the crystal will expand. An interesting thing happens if a potential difference is suddenly and briefly applied. The piezoelectric crystal reacts like a bell which is struck by a hammer and starts ringing. The thickness of the crystal changes when the voltage is applied but the molecules have a momentum which causes them to overshoot their equilibrium positions. They are pulled back and overshoot in the opposite direction. The cycle continues and the result is that the crystal vibrates continuously. Figure 2.12 illustrates the process and what would be observed in practice.

CAPACITORS AND INDUCTORS

Figure 2.11 A piezo-electric crystal compressed and stretched

36

The crystal resonates mechanically at a particular frequency. Its thickness oscillates about a mean which is its normal, unstressed thickness (the horizontal axis in figure 2.12a). As the circuit resonates mechanically, the surfaces become charged according to whether the crystal is stretched or compressed (figure 2.11). Hence, as the thickness oscillates, the charge on each surface will vary and a graph of charge versus time would resemble figure 2.12(a). A graph of potential difference versus time would look the same. Crystal resonators consist of a quartz wafer between two electrodes. The physical dimensions of the crystal determine the resonant frequency and if the crystal is maintained at a constant temperature a very high order of frequency stability can be obtained. When included in a circuit the crystal only permits current to flow when the frequency of the current is equal to the natural frequency of oscillation of the crystal. Quartz crystals can thus replace the resonant circuits of figure 2.6 and 2.9 and set the frequency of the oscillator more precisely. The frequency stability of quartz crystals is exploited in applications where precise timekeeping is required. Everyday applications include wristwatches and the clock which is the heart of every computer. We will meet quartz crystals again in a later chapter describing the production of ultrasound waves.

CAPACITORS AND DEPOLARISATION Earlier in this chapter we discussed the charge storing capability of capacitors and introduced the half-discharge time, τ 1/2 which characterizes the rate of discharge of a capacitor through a resistor (equation 2.2).

τ1/2 α R.C

.... (2.2)

Figure 2.12 (a) The change in thickness of a piezoelectric crystal in response to (b) a briefly applied voltage

CAPACITORS AND INDUCTORS

37

When a capacitor is charged through a resistor, equation 2.2 also applies to the charging behaviour. Hence in a circuit such as that shown in figure 2.13, when the switch is closed the voltage across the capacitor will increase as shown.

The voltage across the capacitor increases and approaches the battery voltage asymptotically. For the graph shown the battery voltage is 12 V. From the graph, τ1/2 is approximately 6 seconds. The reason that the graph has this shape is that the current flowing to charge the capacitor decreases as the capacitor charges. Initially, the capacitor is uncharged so when the switch is closed, the potential applied to the left side of the resistor is 12 V and the potential on the right side is zero. The potential difference across the resistor is maximum (in this circuit, 12 V) so the current flow is maximum. When the capacitor has charged to, say, 3 V, the potential difference across R is 9 volts, so the current

CAPACITORS AND INDUCTORS

38

flow (calculated using Ohm's law or measured directly) is less. When the capacitor is charged to 9 V, the potential difference across R is only 3 V and the current is onequarter of its original value. The closer the capacitor comes to fully charged, the smaller is the potential difference across R and the smaller is the charging current. Consequently, the rate of charging, determined by the current flow through the resistor, becomes smaller and smaller as the capacitor approaches fully charged and the capacitor voltage approaches the battery voltage but never quite gets there. Next consider what happens if a pulsed voltage is applied to the resistor-capacitor combination shown in figure 2.14. At the start of the pulse, the current flow through the resistor is high and the capacitor charges rapidly. As the capacitor charges, the rate of charge decreases and the potential difference across the capacitor (VC ) approaches a plateau (figure 2.14a). A graph of current flow through the resistor or potential difference across the resistor versus time mirrors the graph of V C (figure 2.14b). The initial charging current is high but reduces rapidly as the capacitor charges.

Figure 2.14 A pulsed voltage applied to a resistor-capacitor combination. (a) potential difference across the capacitor and (b) current through the resistor versus time.

Note that the current flow through the resistor is AC. Current flows through the resistor as the capacitor charges and an equal amount flows in the reverse (negative) direction as the capacitor discharges.

CAPACITORS AND INDUCTORS

39

The pulse duration in this example is significantly greater than the halfdischarge/half-charge time, τ 1/2 so the pulse is able to (almost) completely charge the capacitor before coming to an end and allowing the capacitor to discharge. If the RC time constant (and hence τ1/2) of the resistor-capacitor combination was comparable to the pulse width, the capacitor would not fully charge (figure 2.15b) and the current flow through the resistor would be more sustained. [The dashed lines in figure 2.15 indicate the voltage across the capacitor when it is fully charged (in this case, it is the pulse voltage).] If the RC time constant was greater than the pulse width, the capacitor would charge minimally (figure 2.15c) and the current flow through the resistor would be well sustained. The importance of the RC time constant will be apparent when we consider transcutaneous electrical nerve stimulation in subsequent chapters.

Figure 2.15 Response of the circuit shown in figure 2.14 when the pulse duration is (a) much greater than R.C, (b) comparable to R.C and (c) less than R.C.

CAPACITORS AND INDUCTORS

A final point which should be noted is that in each example in figure 2.15, the current through R is AC. The capacitor charges by a certain amount then discharges, so the net movement of charge is zero. Whatever charge flows through R during the pulse must flow back afterwards.

EXERCISES 1

The table below shows readings of voltage obtained at 2 second intervals as a capacitor discharged through a resistance. The circuit arrangement used is shown in figure 2.2. In this case the value of the capacitor was known to be 50 µF but the resistance was unknown. _______________________________________________________ Time Voltage on the 50 µF Time Voltage on the 50 µF (sec) capacitor (sec) capacitor _______________________________________________________ 0 12.0 12 1.8 2 8.7 14 1.3 4 6.3 16 0.9 6 4.6 18 0.7 8 3.3 20 0.5 10 2.4 _______________________________________________________ (a) (b) (c)

Plot a graph of voltage against time for comparison with figure 2.3. How long does it take for the 50 µF capacitor to discharge to half the original voltage? What would the unknown resistance need to be to give the results shown above?

40

CAPACITORS AND INDUCTORS 2

41

A capacitor is charged to a potential of 15 V then connected to a 250 kΩ resistor and allowed to discharge. The time taken to half discharge is 2.4 seconds. How long would the capacitor take to half discharge if: (a) the resistor was replaced by one of value 500 kΩ (b) the resistance was halved in value (c) the capacitance was doubled in value (d) the resistance was halved in value and the capacitance doubled (e) the resistance was doubled in value and the capacitance also doubled.

3

What is the impedance of: (a) (b) (c) (d)

a 10 µF capacitor a 0.047 µF capacitor a 10 µH inductor a 2 mH inductor

at a frequency of 1 kHz? How would the impedance change if the frequency was doubled? 4

A capacitor is connected directly to a 12 volt AC supply (of frequency 50 Hz). The current flowing through the capacitor is found to be 15 mA. What is the capacitance?

5

A capacitor has an impedance of 600 ohms at a frequency of 2 kHz. What is its impedance at a frequency of: (a) (b)

6

100 Hz 1 MHz.

Figure 2.5 includes a graph of impedance versus frequency for a 20 mH inductor. The graph is obtained using the values of impedance and frequency listed in table 2.2. Use the values listed to plot your own graph of impedance

CAPACITORS AND INDUCTORS versus frequency for a 20 mH inductor. On this graph draw lines showing the relationship between impedance and frequency for: (a) (b) 7

a 10 mH inductor a 50 mH inductor.

Use the values in table 2.2 to plot a graph of impedance versus frequency for a 1 µF capacitor. On the same graph draw curves relating impedance and frequency for: (a) (b)

a 0.5 µF capacitor a 2 µF capacitor.

8

Figures 2.6 and 2.9 show the variation of impedance with frequency for series and parallel resonant circuits. Explain the shape of each graph in terms of the variation of impedance with frequency of the individual inductors and capacitors.

9

A resonant circuit is made by combining a 0.1 µF capacitor with a 3.0 mH inductor. What is the resonant frequency of such a combination?

10

A 50 mH inductor in combination with a capacitor is found to have a resonant frequency of 1 kHz. What is the value of the capacitor?

11

When a pulse of current is applied to the circuit shown in figure 2.22 it 'resonates' generating an alternating voltage as shown in figure 2.24(b). (a) (b)

12

explain what is meant by resonance and why the circuit resonates explain why the AC is damped.

Two circuits are coupled together as shown in figure 2.10. The values of the components are C = 100 pF and L = 1 mH. (a)

Under what circumstances will power be transferred most efficiently between the circuits?

42

CAPACITORS AND INDUCTORS

13

(b)

At what frequency will maximum power transfer occur?

(c)

If C was increased to 500 pF, what would be the new frequency for maximum power transfer?

An AC signal of frequency 100 kHz is generated by circuit 1 in figure 2.10. The value of C is 0.01 µF. (a) What value of L is needed for maximum power transfer to circuit 2? (b)

14

If L is 3 mH, at what frequency will maximum power be transferred?

A DC pulse is applied to a resistor/capacitor combination. The result is a potential difference and flow of current through the resistor as shown.

(a)

why does the graph of current flow through the resistor droop so rapidly?

(b)

why must the current flow through R be purely AC?

(c)

if the capacitance was increased, how would this affect the shape of the graph of current through R? Why?

(d)

under what circumstances will the graph of current flow through the resistor resemble that of the applied voltage?

CAPACITORS AND INDUCTORS 15

43

DC pulses as shown below are applied to a resistor/capacitor combination.

(a) (b)

what can you conclude about the RC time constant for this circuit? if the RC time constant was reduced (by decreasing C or R) how would this affect the shape of the graph of potential difference or current flow through the resistor? Give an explanation of your reasoning.

44

ELECTRICAL PROPERTIES OF SKIN

45

3 Electrical Properties of Skin Physiologists study the bioelectric properties of nerve and muscle using invasive surgical techniques where the nerve trunk is exposed and so rendered accessible to direct electrical stimulation. Sometimes nerve fibres are teased-out and stimulated individually. In this way much has been learned about the electrical characteristics of nerve. This knowledge is used by the physiotherapist to choose electrical stimulus characteristics (such as the intensity, treatment time, pulse width and frequency) to invoke the desired physiological response during patient treatment. A complication is that electrical stimulation for therapy is almost invariably applied non-invasively. It is normally achieved by applying an electrical stimulus through the skin using surface mounted electrodes. The skin has complex electrical characteristics and these must be taken into account when transcutaneous electrical nerve stimulation (TENS) is used.

SKIN AND APPENDAGES The biological makeup of skin and its physiological state determine its electrical properties so it is worthwhile to consider some aspects of its structure which are relevant to electrical stimulation. Examined microscopically, the skin is found to have two distinct layers, the dermis and epidermis, as illustrated in figure 3.1. The dermis and epidermis together constitute the skin. The epidermis is punctured by the skin appendages: the sweat gland ducts and hair follicles. Beneath the skin is the subcutis, also referred to as the superficial fascia or simply subcutaneous tissue. Blood vessels, lymph vessels and nerves infiltrate the subcutis and dermis but not the epidermis. There are other fundamental differences between the dermis and epidermis. The dermis is well hydrated. It consists of a matrix of collagen and elastin fibres embedded in a 'ground substance' rich in proteoglycans and hyaluronic acid. Fibroblasts are the predominant cells in this layer. The high degree of hydration makes the dermis electrically conductive. The epidermis is less hydrated and consists of a matrix of keratin fibres. The cells present in this layer are predominantly keratinocytes which

ELECTRICAL PROPERTIES OF SKIN receive nutrients from capillaries in the underlying dermis. The basal layer of the epidermis is metabolically very active, with the cells regularly undergoing mitosis. Keratinocytes, formed and pushed upwards from this layer, synthesise keratin and store it within the cytoplasm. In their life cycle, the keratinocytes move towards the skin surface, becoming less metabolically active as diffusion limits the rate of nutrient supply. Near the surface the cells die and shrivel, turning into little sacs of (mostly) keratin. The dead cells form a scaly shell called the stratum corneum. The stratum corneum is thus the withered, dehydrated remains of keratinocytes packaged full of keratin. From formation in the basal layer to desquamation (flaking off as tiny scales from the surface) takes the keratinocyte approximately 40 days. The balance between desquamation and mitosis in the basal layer keeps the thickness of the epidermis constant. The stratum corneum is a dry, insulating but very thin shell which separates and isolates the highly hydrated soft tissues of the human body from the drier and far more changeable external environment. If the stratum corneum is removed, the body loses water and if too great an area of the stratum corneum is lost or damaged, the resulting water loss can be fatal. As noted above, the fibrous protein, keratin forms the bulk of this dead, insulating layer which is so essential for water homeostasis. From the point of view of transcutaneous electrical nerve stimulation, the presence of the stratum corneum is significant

In most areas of the body, the subcutis is predominantly adipose (fat storing) tissue. It is, nonetheless, quite conductive because of its water (and ion) content and the presence of an extensive network of blood vessels.

46

Figure 3.1 Some important features of skin.

ELECTRICAL PROPERTIES OF SKIN

47

because it is an electrical insulator. The skin appendages, the hair follicles and sweat gland ducts, are important both physiologically and electrically. The keratinous structures which we call hairs serve two important physiological roles; one sensory, one thermal. Hairs are relatively rigid structures which are anchored in the skin. Sensory receptors close to the hair root detect and respond to movement of the hair, so providing sensitive touch receptors. A sufficient drop in temperature will signal a reflex response from receptors which triggers contraction of smooth muscle fibres connected to the hair root. The myofibril contraction causes the hairs to 'stand on end', trapping a volume of warmer air which separates the living tissue from the external environment. This effect is pronounced in humans and is also seen in other species which have greater hair covering. An elevated temperature triggers an increased blood flow to the dermis, thus increasing the rate of heat transfer from deeper structures and also activates the sweat glands to release water on the skin surface and provide evaporative cooling. Electrically, the hair follicles and sweat gland ducts are conductive pathways through the insulating layer, the stratum corneum.

AN ELECTRICAL MODEL FOR TRANSCUTANEOUS STIMULATION When an electrode is placed on the skin surface we create a situation where two conductors are separated by an insulator, in other words, a capacitor. The electrode and the tissue beneath the stratum corneum are the two conductors. The stratum corneum is the insulator although, as we have seen, the stratum corneum is a 'leaky' insulator as it is punctured by conductive channels created by the skin appendages. The combination of electrodes and tissue can thus be modelled by a resistor capacitor combination. Figure 3.2 shows an electrical model for two electrodes placed on the skin surface.

ELECTRICAL PROPERTIES OF SKIN

C is the capacitance of the stratum corneum. Rp , the resistor in parallel with C, represents the conductive pathway created by the skin appendages. Rs, the series resistance, represents the resistance of the tissue volume under the stratum corneum between the two electrodes. The resistance of the electrodes is assumed to be negligible. The actual values of Rp and C depend on factors such as the electrode size, ambient temperature and skin condition. The quantity Rs varies little. An important point which we will return to is that nerve fibres, whether sensory motor or pain, are located beneath the stratum corneum. In terms of the electrical model, this means that the potential difference across Rs determines the stimulation intensity experienced by nerve fibres. The potential difference across C and Rp represents lost electrical energy as far as nerve fibres are concerned.

RESPONSE OF THE MODEL TO STEADY DIRECT CURRENT If a source of direct current, such as a battery is connected to the circuit shown in figure 3.2, some current will flow through the three resistors in series and some will flow on to the capacitors, so charging them. The initial flow of current is large as it includes charging current flowing to the capacitors (figure 3.3a) which is limited only by Rs. After the capacitors have charged, the current flow is lower as there is only current flow through the resistors (figure 3.3b). If Rp , the parallel resistance of the stratum corneum, is high (as it is, under normal circumstances), the 'steady state' current flow (figure 3.3b) will be low - appreciably lower than the initial current flow which includes charging current for the capacitors. At the extreme, if the stratum corneum was not 'leaky' i.e. if the parallel resistance, Rp, in figure 2 was infinite, no steady direct current could flow. There would be a brief flow of current as the capacitors charged, then no further current flow. In reality, Rp allows current flow, so after C is charged the only current flow is through Rs and the two resistances Rp which are connected in series.

Figure 3.2 An electrical model for transcutaneous stimulation.

48

In this model, the resistance of the electrodes is assumed to be negligible, but could be included putting a resistor at each end of the model or including the electrode resistance In R s .

ELECTRICAL PROPERTIES OF SKIN

49

Question 1: How much steady direct current would flow in a typical 'real life' situation if the applied DC voltage is 50 V and electrodes with an area of 10 cm 2 are used? Information: Skin capacitance is proportional to the electrode area. A typical value for the capacitance per unit area is 0.05 µF.cm-2, so for a 10 cm2 electrode area, the capacitance, C, is 0.5 µF. Skin resistance is inversely proportional to the electrode area. The larger the area, the greater the number of conductive channels through the stratum corneum and the lower the total resistance. The parallel resistance x unit area might typically be 10 kΩ.cm 2 , so for a 10 cm2 electrode area, the parallel resistance, Rp , is about 10 kΩ.cm2/10 cm2 = 1 kΩ. The resistance of the tissue volume beneath the stratum corneum is typically about 200 Ω. Answer: Under steady-state conditions, the capacitors are charged so no current flows to the capacitors and the value of C is irrelevant. The combination of electrodes and tissue behaves as three resistors in series. The total resistance is (1 kΩ + 200 Ω + 1 kΩ) = 2200 Ω. The applied potential difference is 50 V so the resulting current flow is, from Ohm's Law, I = V/R = 50/2200 = 0.0227 A = 22.7 mA.

Figure 3.3 Response of the electrical model to direct current (a) initially and (b) after the capacitors have charged.

ELECTRICAL PROPERTIES OF SKIN

Question 2: How big is the initial current flow for the 'real life' situation described in question 1? Information: Initially, the capacitors act as a 'short circuit'. They are uncharged and offer no resistance to current flow. As each capacitor charges, the potential difference across the capacitor increases and this opposes further flow of current. When fully charged, an ideal capacitor has infinite resistance. Uncharged, the 'resistance' is zero. Answer: The initial flow of current is resisted only by Rs. The capacitors offer zero resistance when they are uncharged so the current flow is I = V/R = 50/200 = 0.25 A = 250 mA. This is more than ten times larger than the steady-state current (question 1). Question 3: What is the steady-state potential difference across Rp and Rs for an applied potential difference of 50 V? Answer: Since the three resistors are in series, the same current (22.7 mA) flows through each. The potential difference across each resistor can be calculated using Ohm's Law. The resistance Rs is 200 Ω so the potential difference across it is V = I.R s = 0.0227 x200 = 4.5 Volts. For each parallel resistance, Rp, V = I.Rp = 0.0227x1000 = 22.7 Volts.

THE USE OF STEADY DC The previous calculations demonstrate an important idea concerning the use of steady direct current for patient treatment. Namely, that steady DC is of little practical use for stimulation of nerve fibres. For steady, continuous flow of direct current

50

ELECTRICAL PROPERTIES OF SKIN

51

through the skin, the tissue impedance is high (Rs + 2Rp) and the potential difference across Rs is small. Nerve fibres are located in the tissues underlying the stratum corneum (represented by Rs) so the potential difference across them is small and their stimulation intensity is low. Most of the applied voltage drop occurs across the stratum corneum (Rp ). This is one reason why steady direct current is not used for transcutaneous nerve stimulation. A second, more important, reason is that nerves are relatively insensitive to steady DC because they accommodate. The firing threshold progressively increases with a constantly applied DC stimulus, so the nerve will cease firing once the threshold has risen above the applied DC stimulus. Accommodation is described in more detail in chapter 4.

Notice that in the answers to question 3, for an applied potential difference of 50 V, only 4.5 V is produced across the tissue beneath the stratum corneum.

IONTOPHORESIS It is for the reasons outlined above that steady direct current is not used for transcutaneous nerve stimulation. Steady DC does, however, have an important clinical role in iontophoresis, where drugs or other chemical agents are driven through the skin to the underlying tissue. Iontophoresis only works if the chemical to be driven is charged. It uses steady DC so that the chemical ions are driven continually in one direction while no nerve stimulation (in particular, no stimulation of pain or motor fibres) occurs. An example is acetate ion iontophoresis for decalcification of connective tissue. A soluble acetate, such as sodium acetate is dissolved in conductive gel. This is applied to the skin under the positive electrode (anode). Steady DC is applied at a current level of about 10 mA over a treatment period of 10 to 20 minutes. The negatively charged acetate ions are driven through the skin into deeper tissue. The calcium in calcified tissue is in the form of calcium phosphate crystals, which react with the acetate ions to form the soluble substance, calcium acetate. In this way the calcium phosphate crystals are removed. If the ion to be driven through the skin was positively charged, it would be applied under the negative electrode (the cathode).

ELECTRICAL PROPERTIES OF SKIN

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RESPONSE OF THE MODEL TO PULSED DIRECT CURRENT If a pulse of direct current is applied to the circuit shown in figure 3.2, the initial current flow through Rs will be large, but will quickly drop as the capacitors charge. The time constant for the charging process is proportional to Rs.C. Figure 3.4 shows the effect of stimulus pulse width and time constant on the current flow. In (a) the pulse duration is long compared to Rs .C so the capacitors charge during the pulse and the current quickly drops to resistive current flow only. This means that if R p is large, the current will drop to a very low value. At the end of the applied voltage pulse the capacitors discharge, producing a negativegoing current spike. Nerve fibres, located in deeper tissue beneath the stratum corneum (represented by Rs), would thus experience two shortduration pulses of stimulating current, one at the beginning and one at the end of the applied voltage pulse. Most of the applied voltage is ineffective as far as stimulation of nerve fibres is concerned. In (b) the pulse duration is short compared to Rs .C. The current decreases only fractionally during the pulse and most of the resulting current is due to the capacitors charging. The resulting current through R s is high and does not decrease much during the pulse. Most of the applied voltage pulse is effective.

Figure 3.4 Response of the electrical model to pulsed direct current (a) when the pulse duration is long compared to R s.C and (b) when the pulse duration is short compared to R s.C.

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Figure 3.4 shows examples close to the range of extremes for stimulation with rectangular pulsed current. Note that the horizontal (time) axis scale is different in each example. In figure 3.4(a) the time scale is relatively long compared to Rs.C and sharp spikes are seen in the graphs of current through Rs versus time. In figure 3.4(b) the time scale is relatively short compared to Rs.C and the current through Rs shows only a small decrease during the stimulus pulse. During the spikes in 3.4(a), appreciable current flows through Rs, meaning that electrical energy is dissipated in the tissue volume beneath the stratum corneum. Between spikes, little current flows through the deeper tissue (Rs), and most of the electrical energy is dissipated in the stratum corneum. The short duration pulse in 3.4(b) results in appreciable and sustained current flows through Rs, so most of the electrical energy is available for nerve stimulation. The above is one of the reasons why modern stimulators use short duration pulses for transcutaneous nerve stimulation. A second, perhaps more important reason is that better discrimination between sensory, motor and pain fibre stimulation is achieved with narrower stimulus pulses. This will be considered further in chapter 4.

RESPONSE OF THE MODEL TO ALTERNATING CURRENT The response of the model to alternating current is particularly interesting as alternating current (at kilohertz frequencies) is commonly used in physiotherapy for patient treatment, either in the form of 'interferential currents' or 'Russian stimulation'. Interferential currents are 'medium frequency' AC currents normally applied at a frequency of 4 kHz. Russian current has a frequency of 2.5 kHz. Part of the reason that these frequencies are used is that at kHz frequencies, the stratum corneum has a low electrical impedance. The impedance of the stratum corneum is the total impedance of Rp and C in the electrical model (figure 3.2). It is strongly dependent on the alternating current frequency because of the capacitance, C. The impedance of C is given by the formula:

ELECTRICAL PROPERTIES OF SKIN

Zc =

1 2πfC

54

.... (2.5)

where Zc is the impedance and f is the alternating current frequency. Question 3: What is the capacitative impedance of the stratum corneum, Zc, at frequencies of (a) 50 Hz, (b) 500 Hz and (c) 5 kHz? Information: A typical value for the skin capacitance per unit area is 0.05 µF.cm-2 . A typical electrode area is 10 cm2 so the capacitance, C, in the model is about 0.5 µF. Answer: (a) Using equation 2.5, the impedance at 50 Hz is Zc = (b)

(c)

1 1 10 6 = = = 6400 Ω 2πfC 2 x 3.142 x 50 x 0.5 x 10-6 50 x 3.142

Using equation 2.5, the impedance at 500 Hz is Zc =

1 1 10 6 = = = 640 Ω -6 2πfC 2 x 3.142 x 500 x 0.5 x 10 500 x 3.142

Using equation 2.5, the impedance at 5 kHz is Zc =

The term 'low frequency' currents refer to those in the 'biological' frequency range from 0 to approximately 100 Hz. 'Medium frequencies' are in the kHz to tens of kHz range. 'High frequencies' are in the 100 kHz and upward range.

1 1 10 3 = = = 64 Ω -6 2πfC 2 x 3.142 x 5000 x 0.5 x 10 5 x 3.142

If the stratum corneum was purely capacitative, its impedance would vary as calculated above. In fact the skin appendages provide a parallel resistive path for current flow (Rp in figure 3.2) so the total impedance of the stratum corneum is lower,

In most descriptions of interferential or Russian currents, the choice of a kilohertz frequency is justified as being due to the low skin impedance at kHz frequencies. The low skin impedance is due to the low impedance of both the stratum corneum and the underlying skin layers.

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significantly so at low frequencies. The parallel resistance x unit area of the stratum corneum is typically about 10 kΩ.cm 2 , so for a 10 cm2 electrode area, the parallel resistance, Rp, is about 10 kΩ.cm2/10 cm2 = 1 kΩ. At a frequency of 5 kHz, the capacitative impedance of the skin is about 64 Ω and placing a 1 kΩ resistance in parallel makes virtually no difference to the total impedance. At a frequency of 500 Hz, the capacitative impedance of the skin is about 640 Ω and placing a 1 kΩ resistance in parallel reduces the total impedance to appreciably less than 640 Ω. At a frequency of 50 Hz, the capacitative impedance of the skin is about 6400 Ω and placing a 1 kΩ resistance in parallel reduces the total impedance to less than 1 kΩ. Despite the effect of skin appendages offering a conductive pathway (Rp ) at low frequencies, the total impedance of the stratum corneum (C and Rp in parallel) still shows a dramatic decrease with increasing frequency. Thus, for a given stimulus voltage, greater current flows at higher frequencies. This means that the current through Rs, representing the underlying tissues, is higher at higher frequencies and the potential difference across Rs is correspondingly higher. Question 4: What is the impedance of the stratum corneum at frequencies of (a) 50 Hz, (b) 500 Hz and (c) 5 kHz? Use the model in figure 3.2 and values of C and Rp used previously assuming an electrode area of 10 cm2. Use the formula for two resistances in parallel to add the two impedances. Information: As indicated previously, a typical value for the skin capacitance for an electrode area of 10 cm2 is about 0.5 µF. The capacitative impedance is thus 6400 Ω, 640 Ω and 64 Ω at frequencies of 50 Hz, 500 Hz and 5 kHz respectively. The parallel resistance, Rp, is about 1 kΩ.

ELECTRICAL PROPERTIES OF SKIN

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Answer: The impedance of the stratum corneum at each frequency is the total impedance of C and Rp in parallel. Two resistances in parallel can be added using the formula 1 1 1 = + RT R1 R2 Adding the impedances of C and Rp in this way we use 1 1 1 = + Z Rp Zc where Zc is the impedance of C at the frequency concerned. (a)

Using the above formula, the impedance at 50 Hz is calculated as follows: 1 1 1000 + 6400 6500 1 = + = = 6400 1000 1000 x 6400 1000 x 6400 Z so Z =

(b)

6400 x 1000 = 985 Ω 6500

The impedance at 500 Hz is given by 1 1 1000 + 640 1640 1 = + = = 640 1000 1000 x 640 1000 x 640 Z

(c)

so Z =

640 x 1000 = 390 Ω 1640

The impedance at 5 kHz is given by 1 1 1000 + 64 1064 1 = + = = 64 1000 1000 x 64 1000 x 64 Z

Strictly speaking, capacitative and resistive impedances must be added using complex number algebra. If a resistor, R p and a capacitor, C, are connected in parallel, the total impedance, Z, is given by the formula: Rp Z = 21 (1+(2πfRp C) ) /2

so Z =

64 x 1000 = 60 Ω 1064

The impedance of the stratum corneum (C and Rp in parallel) thus shows a dramatic decrease with increasing frequency: from about 1 kΩ at 50 Hz to about 60 Ω at 5 kHz.

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It is the potential difference across Rs which determines the stimulation intensity experienced by nerve fibres, which are located under the stratum corneum. This depends on both Z and Rs. Rs is about 200 Ω and does not vary appreciably with frequency. Since Z decreases appreciably with increasing frequency, the potential difference across Rs will correspondingly increase. Question 5: What is the potential difference across Rs at frequencies of (a) 50 Hz, (b) 500 Hz and (c) 5 kHz for an applied potential difference of 50 V? Use the values of Z and Rs calculated previously assuming an electrode area of 10 cm2. Information: Accurate calculation of the potential difference across Rs requires complex number algebra, but a good first-approximation can be made by treating the impedances as simple resistances in series. Answer: (a) The impedance of the stratum corneum at 50 Hz is 985 Ω and R s is 200 Ω . Thus the total impedance is 985+200+985 = 2170 Ω. For an applied potential difference of 50 V, the current through the series combination is I = V/Z = 50/2170 = 0.023 A = 23 mA. This is also the current through Rs , so the potential difference across Rs is V = I.Rs = 0.023 x 200 = 4.6 volts. (b)

The impedance of the stratum corneum at 500 Hz is 390 Ω and Rs is 200 Ω. Thus the total impedance is 980 Ω. For an applied potential difference of 50 V, the current through the series combination is I = V/Z = 50/980 = 0.051 A = 51 mA. This is also the current through Rs, so the potential difference across Rs is V = I.Rs = 0.051 x 200 = 10.2 volts.

ELECTRICAL PROPERTIES OF SKIN (c)

The impedance of the stratum corneum at 5 kHz is 60 Ω and Rs is 200 Ω. Thus the total impedance is 320 Ω. For an applied potential difference of 50 V, the current through the series combination is I = V/Z = 50/320 = 0.156 A = 156 mA. Thus the potential difference across Rs is V = I.Rs = 0.156 x 200 = 31.2 volts.

Notice that at 5 kHz, the potential difference across Rs is 31.2 volts, which is more than 60% of the applied (50 V) stimulus. At 500 Hz, the potential difference across Rs is about 20% of the applied stimulus and at 50 Hz, about 9%. If the objective is to stimulate nerve fibres which are located beneath the stratum corneum, the higher the alternating current frequency, the less the energy wasted in the stratum corneum and the greater the current through, and voltage across, the deeper tissues. If skin impedance was the only factor, then the higher the alternating current frequency, the lower the skin impedance and the more efficiently nerves could be stimulated. Following this logic, if 5 kHz stimulation is more efficient than 500 Hz, then 50 kHz should be even more so. In fact this is not the case. The reason is that the impedance of the nerve fibre also varies with frequency.

NERVE FIBRE IMPEDANCE The electrical properties of nerve fibres are analogous to those of skin and underlying tissue. The nerve fibre membrane, like the membrane of all cells, is a phospholipid bilayer with an embedded patchwork or mosaic of protein molecules. This fluid-mosaic structure, illustrated alongside, explains the measured electrical properties. Orange shapes are the polar, high-water-content, protein molecules. Blue circles with two dangling chains represent the phospholipid molecules which have two long, non-polar 'tails'. The phospholipid bilayer is an insulator. The hydrocarbon tails of the phospholipid molecules separate the conductive intracellular fluid from the equally conductive extracellular fluid. Thus the nerve-fibre membrane acts as a capacitor. Protein molecules which span the width of the

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59

membrane provide conductive channels, allowing some ions to cross the membrane continuously. For this reason the membrane behaves electrically like a capacitor, C, with a parallel resistance Rp. Electrical stimulation of the nerve fibre involves current flow across the membrane, through the intracellular fluid and out across the membrane at some other point (usually the adjacent node of Ranvier). The internal contents of the nerve fibre, the intracellular fluid, is purely resistive so the nerve fibre can be electrically modelled as shown in figure 3.5. C in figure 3.5 is the capacitance of the nerve-fibre membrane: an insulating phospholipid bilayer. Rp is the resistance of conductive channels through the bilayer (the protein molecules which span the full width of the membrane). Rs is the resistance of the intracellular fluid.

Figure 3.5 An electrical model for a nerve fibre

To excite a nerve fibre, that is, to cause the nerve to fire and produce an action potential, the membrane potential must be changed from its resting value to the excitation threshold. For nerve excitation it is not the potential difference across the fibre which determines this but the potential difference across the fibre membrane (C and Rp in figure 3.5). Since the membrane acts as a capacitor, if alternating current is used for nerve stimulation, the potential difference across the fibre membrane will be less at higher alternating current frequencies. The decrease in membrane impedance with increasing frequency means that a greater potential difference will be produced across Rs and a correspondingly lower potential difference will be produced across C and Rp. At high frequencies, nerve fibres are less excitable because the potential difference produced across the fibre membrane is reduced. In practical terms this means that electrical stimulation with alternating current does not become more and more efficient at higher alternating current frequencies. the decrease in impedance of the stratum corneum, which results in a higher potential difference across the

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underlying tissues at kilohertz frequencies is countered by the decrease in impedance of the nerve fibre membrane. In theory, the decreased nerve fibre sensitivity could be compensated-for by increasing the stimulus intensity but in practice this would result in a high current flow through the skin and consequently a high electrical power dissipation and the risk of tissue damage. The result is that frequencies above about 10 kHz are of little use for eliciting a nerve response. A similar argument applies to stimulation with rectangular pulses. As indicated previously, long duration pulses are relatively ineffective for transcutaneous nerve stimulation as most of the electrical energy is dissipated in the stratum corneum and the current through the deeper tissues is in the form of two spikes, at the onset and cessation of the stimulus (figure 3.4(a)). With short duration pulses (figure 3.4(b)), little energy is dissipated in the stratum corneum and the current through the deeper tissues is sustained during the pulse. However, if the pulse duration is extremely short (tens of microseconds), there will be insufficient time during the pulse for the nerve membrane capacitance to charge. Figure 3.6 illustrates the response of the electrical model shown in figure 3.5 to short and longer duration current pulses. With a sufficiently long duration pulse, there is time for the membrane capacitance to charge (figure 3.6(a)) and the change in membrane potential is maximum. With a very short duration pulse, there is insufficient time for the membrane capacitance to fully charge and the change in membrane potential is small (figure 3.6(b)). The use of a

Figure 3.6 Change in potential difference across the nerve fibre membrane (voltage across C and Rp in the electrical model shown in figure 3.5) when the pulse duration is (a) much longer than R s.C and (b) shorter than R s.C.

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higher stimulus intensity can compensate for the short pulse duration, but if the intensity required is too large, there is the risk of skin damage. Specifically, at stimulus intensities of several hundred volts, there is the risk of skin electrical breakdown where tiny regions of skin under the electrodes suddenly become highly conductive, allowing an extremely high current to flow and causing tissue damage. The net effect is that for transcutaneous stimulation using rectangular pulses, there is a 'window' of pulse widths between tens of microseconds and about 1 ms, outside which stimuli are either ineffective or inefficient and potentially dangerous.

EXERCISES 1

Explain, in terms of the structure of skin, why the electrical model shown in figure 3.2 is an appropriate model for a description of transcutaneous electrical nerve stimulation.

2

When a constant DC stimulus is applied, via electrodes, to the skin surface, there is a large transient flow of current through the tissue. Explain why the initial flow of current is high and why the steady-state current is lower.

3

(a) (b) (c)

4

Figure 3.4, page 52, shows graphs of the current flow through tissue in response to stimulus pulses of different duration.

5

What is iontophoresis? Why must the active agent be charged (i.e. in the form of an ion)? What is the advantage of iontophoresis for application of medication?

(a)

explain why long duration pulses produce only short-duration current flows, as in figure 3.4(a).

(b)

Why is the current flow sustained when short duration pulses, as in figure 3.4(b), are applied transcutaneously?

For transcutaneous nerve stimulation, body tissue can be modelled by the resistor/ capacitor combination shown below. Skin capacitance per unit area is

ELECTRICAL PROPERTIES OF SKIN approximately 0.05 µF.cm-2 and the parallel skin resistance x unit area, approximately 10 kΩ.cm2. The subcutaneous resistance is typically around 200 Ω. Sinusoidal AC with an amplitude of 20V is applied between A and B using 1 cm2 electrodes.

6

(a)

Calculate the impedance of the capacitor at AC frequencies of: * 50 Hz * 500 Hz * 5 kHz.

(b)

Calculate the impedance of the stratum corneum; i.e. the impedance of the parallel (Rp and C) resistor capacitor combination, Z||, at each of the three frequencies.

(c)

What is the potential difference across the deep tissue, modelled as Rs, at each of the frequencies?

If the electrode area in question 5 above was 50 cm2, what would be: (a) (b)

the impedance of the capacitor, C, at the same AC frequencies? the impedance of the stratum corneum; i.e. the impedance of the (Rp and C) resistor-capacitor combination at each of the three frequencies?

62

ELECTRICAL PROPERTIES OF SKIN (c) 7

the potential difference across Rs (representing the deeper tissues) for the same AC frequencies?

A rectangular pulse of amplitude 40 V is applied to the circuit shown in question 5. What is: (a) (b)

the peak current through Rs the minimum current through Rs

when a long duration pulse is used? 8

(a)

(b) 9

10

Draw graphs of the voltage across Rs versus time when rectangular pulsed current is applied to the circuit in question 1. Consider three situations (i) the pulse width is small compared to Rs .C, (ii) the pulse width is comparable to Rs.C and (iii) the pulse width is large compared to Rs.C. explain, using graphs to illustrate, the effect of Rp on the voltage measured across Rs.

The skin acts as a capacitative barrier to the flow of current, meaning that the higher the AC frequency, the lower the skin impedance. For this reason kHz frequency sinusoidal AC is used for transcutaneous electrical nerve stimulation. (a)

why are higher frequencies (tens or hundreds of kHz) not used clinically to reduce skin impedance further?

(b)

what is the clinically useful frequency range for AC stimulation?

Pulsed current with widths between about 50 µs and 1 ms are normally used for transcutaneous electrical nerve stimulation. Why are pulses of duration (a) less than 50 µs and (b) greater than about 1 ms not normally used?

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4 Electrical Stimulation of Nerve and Muscle Both low frequency pulsed current and kHz frequency alternating current are used by the physiotherapist for the stimulation of nerve and muscle. Low frequency stimulation using short duration pulses has most often been used by physiologists for studying nerve and muscle. Consequently, the physiological basis for electrical stimulation with these currents is reasonably well understood. Less is known of the effects of kHz frequency alternating currents, particularly when applied transcutaneously. The aim of this chapter is to present some of the important observations that have been made concerning the response of nerve and muscle to electrical stimulation using both kinds of currents. When a nerve fibre is in its resting state there is a potential difference of some 70 millivolts between the interior and exterior of the fibre. This is called the 'resting membrane potential'. The inside of the fibre is negative with respect to the outside. The resting membrane potential originates from the difference in concentration of different ions inside and outside the cell and the permeability of the fibre membrane to particular ions. Potassium ions contribute most to the resting membrane potential. The intracellular and extracellular concentrations of potassium ions differ markedly, with the result that they diffuse down their concentration gradient, producing a difference in electrical potential across the nerve fibre membrane. The origin of the resting membrane potential is described in most physiology text books and so will not be elaborated here.

The resting membrane potential is also affected by the movement of other ions, including sodium ions, but because their permeability is much lower than that of potassium, their effect on the resting membrane potential is less.

If a current of sufficient intensity is passed through tissue containing a nerve fibre the potential difference set-up across the fibre may be sufficient to cause depolarization of the fibre membrane and the nerve is stimulated. The depolarization of the membrane, once induced, is transmitted along the length of the nerve fibre and is indistinguishable from a normal nerve impulse (sometimes referred to as an action potential). The important idea is that the potential difference across the membrane must be changed by a critical amount to produce the transient, but large, membrane depolarization which is known as an action potential.

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STIMULATION OF NERVE FIBRES In order to elicit an action potential the potential difference across a nerve fibre must be changed to more than some critical value known as the threshold potential. An electrical stimulus resulting in less than the threshold potential across the nerve fibre will not trigger any response. The process is illustrated in figure 4.1. The threshold potential of most excitable membranes is between 5 and 15 mV more positive than the resting potential. Thus if the resting potential is -70 mV the threshold potential may be -60 mV. To generate a nerve impulse the potential must be changed by more than 10 mV in this case. Once the potential is increased above threshold the nerve fibre is 'fired'. The response is 'all or none'. That is, any stimulus above the threshold value produces the same size of response. The membrane potential rapidly changes to around +30 mV then decreases to the resting value, typically in about one millisecond.

Refractory Periods After a nerve has been stimulated there is a short period of time, typically around 10 milliseconds for sensory and motor neurones, during which the sensitivity of the nerve to stimuli is decreased. During this time the nerve membrane is said to be refractory to a second stimulus. The threshold potential is increased above the normal value as shown in figure 4.2.

Figure 4.1 Response of a nerve fibre to stimuli of increasing intensity.

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For the first millisecond or so after initiation of a nerve impulse no stimulus, no matter how large, will produce a second impulse. This is the absolute refractory period. This is followed by a further period, the relative refractory period, during which only a larger than normal stimulus will produce a response. Full recovery is normally complete in about 10 to 15 milliseconds, when the threshold potential has returned to its original value. Termination of the action potential involves the membrane potential returning to its resting value. The membrane potential typically decreases towards resting, overshoots, and returns to baseline during the relative refractory period. During the overshoot, the magnitude of the membrane potential is somewhat greater than the normal 70 mV and the membrane is described as being hyperpolarized. Hyperpolarization and refractoriness following an action potential lasts for 10 to 15 milliseconds in large diameter sensory and motor neurones. For smaller diameter fibres, the refractory period is longer. This is associated with the observation that the firing rates of smaller diameter fibres are typically less than those of larger diameter fibres.

Figure 4.2 Refractory periods for a nerve fibre.

Accommodation Three characteristics of an electrical impulse influence its ability to stimulate nerve fibres: * the size or amplitude of the pulse, * the width or duration of the pulse, and * the rate of change (or rise) of the pulse.

ELECTRICAL STIMULATION OF NERVE AND MUSCLE The size or amplitude of the pulse is clearly important in that the larger the pulse, the more rapidly the nerve fibre will reach threshold. The width or duration of the pulse is also important in that the longer the pulse duration, the more time is available for the fibre to reach threshold.

Figure 4.3 The effect of stimulus rise-time on action potential production. (a) short stimulus rise-time, (b) lower rate of rise of the stimulus and (c) very low rise time. The rate of change (or rise) of the pulse is important because, in general, a stimulus pulse which rises slowly to its maximum value is less effective than a sudden sharp pulse, other things being equal. If a slow rising pulse is used then the minimum amplitude needed to elicit an action potential will be greater. This happens because the nerve fibre is able to accommodate to a

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slow change in potential. Indeed, if the pulse rises at a sufficiently low rate, no nerve impulse will be generated. The effect of accommodation is illustrated in figure 4.3. In figure 4.3(a) the pulse rise-time is short and threshold is reached before any appreciable change in threshold potential occurs. In 4.3(b) the pulse increases at a faster rate than the threshold potential so threshold is reached after several milliseconds. In 4.3(c) the pulse rises at the same rate as the threshold potential and so does not 'catch up' and cannot generate an action potential: the nerve fibre has 'accommodated' to the rising stimulus intensity by becoming insensitive to electrical stimulation. The refractory period and accommodation both stem from the same basic molecular process by which the nerve impulse, once generated, is terminated and the membrane potential returns to the resting value. An explanation involves voltagesensitive 'gates' which control opening and closing of conduction channels in the nerve fibre membrane.

STIMULATION OF NORMALLY INNERVATED MUSCLE When the nerve supply to a muscle or group is intact, transcutaneous electrical stimulation will normally evoke a motor response - not by a direct effect on the muscle fibres, but indirectly via excitation of the motor nerve fibres (α-motoneurons). The reasons for this are twofold. The first is that many muscle fibres are deeply located and so less likely to be stimulated than those closer to the stimulating electrodes. The second, and perhaps more important reason, is that the time constant for depolarization of a muscle fibre is much greater than that of a nerve fibre. If short duration pulses are used, muscle fibres do not have sufficient time to respond. Nerve fibres, which have a smaller time constant and depolarize more rapidly, are more likely to have sufficient time to reach threshold and fire.

Hille (Hille B. Ion Channels of Excitable Membranes. 2nd Edn. Sinauer Associates. 1992) provides a simple, yet detailed explanation of the molecular dynamics of an action potential.

Transcutaneous electrical stimulation then, particularly with short duration pulses (durations less than the time constant for muscle fibres), preferentially recruits nerve fibres.

Where to Stimulate? When the aim of transcutaneous stimulation is to produce a motor response, the electrodes are normally placed either over the nerve trunk or directly over the muscle to be stimulated. Stimulation of the nerve trunk is described as 'indirect' and will

ELECTRICAL STIMULATION OF NERVE AND MUSCLE evoke a response from all of the muscles innervated by fibres in the trunk. Stimulation over the muscle ('direct' stimulation) will preferentially activate just that muscle. For example, if the femoral nerve is stimulated at the level of the groin, the quadriceps femoris group will be activated. Alternatively, electrodes may be positioned over an individual muscle to activate just one member of the quadriceps group. The best response from an individual muscle is obtained if the stimulus is applied at a motor point. This is often the region of skin which is over the point where the main nerve enters the muscle. In the case of deeply placed muscles the motor point is usually where the muscle emerges from under cover of the more superficial ones. Note, however, that motor point locations are defined as points where a motor response is most easily produced and such points are determined experimentally. Thus a particular motor point may not fit either of the above descriptions - for example it may be simply an area where the nerve is located more superficially.

Electrode Orientation and Size To stimulate nerve, current must flow through tissue between two electrodes which are normally positioned so that current flows parallel to the nerve fibres. If the current flow is at right angles to the fibres, much higher stimulus intensities are required. The reason is that in order to produce an action potential, current must flow in across the fibre membrane at one Node of Ranvier, along the fibre and out at an adjacent node. The amount of current flow depends on the applied potential difference and the greatest potential difference will be produced between adjacent nodes if the current flow direction is parallel to the nerve fibre. A related idea is that action potentials will more readily be generated under the negative electrode (the cathode) than under the positive electrode (the anode). The reason for this is as follows. The nerve fibre membrane, in its resting state, is polarized. The outside is positively charged and the inside, negatively charged. A potential difference of about 70 mV exists across the membrane. If a stimulus is applied using two electrodes, the resulting current flow will depolarize the membrane at one Node of Ranvier while hyperpolarizing the membrane at its neighbour.

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Note that the terms 'direct' and 'indirect' stimulation refer to proximity to the muscle. Direct stimulation does not mean that muscle fibres are stimulated directly but rather that stimulation is via nerve fibres entering the muscle.

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Figure 4.4 illustrates the effect. The resting membrane is positively charged on the outside and negatively charged on the inside. The membrane also acts as a capacitor, which can store charge. The flow of current between anode and cathode increases the positive charge on the membrane closer to the anode, while reducing the charge on the membrane closer to the cathode. The result is that the potential difference across the membrane near the anode is increased and the membrane becomes hyperpolarized. The potential difference across the membrane near the cathode is decreased and the membrane becomes less polarized. When the reduced polarization (i.e. depolarization) is sufficient, an action potential is generated. To enhance the efficiency of stimulation under the cathode, a small electrode size can be used. A larger anode helps to ensure that the current density beneath the electrode is low while the smaller cathode concentrates the current flow and more specifically targets Nodes of Ranvier closest to this electrode. Under these conditions, the location of the anode is relatively unimportant as the cathode acts as the 'active' electrode. The anode acts as an 'indifferent' or 'dispersive' electrode ('dispersive' referring to spreading of the current over a larger area). The main criterion for the location of the anode is that it should not be located over an electrically sensitive area such as a motor point or muscle belly.

Figure 4.4 Current flow between anode and cathode produces depolarization under the cathode and hyperpolarization under the anode.

Sometimes equal size electrodes are used for transcutaneous electrical stimulation. This is often the case when electrodes are placed over a muscle belly. The cathode is positioned distally and the anode proximally. The reason for the 'cathode distal' arrangement is that an action potential generated near the cathode may not propagate through the region under the anode. The phenomenon is referred-to as anodal block. The idea is

ELECTRICAL STIMULATION OF NERVE AND MUSCLE that if an action potential is generated at a particular node of Ranvier, it will normally trigger an action potential at the nodes immediately adjacent. This means that when the nerve is stimulated electrically, action potentials can propagate in both directions along the fibre from the site of stimulation. Propagation in the direction towards the anode, however, will not occur if the adjacent node is kept hyperpolarized by the stimulus pulse. If the objective is to elicit a motor response, a cathode distal arrangement should be used, so that action potential propagation towards the muscle fibres is not blocked. Conversely, if the objective is stimulation of sensory fibres, where the aim is sensory input to the central nervous system, a cathode proximal arrangement would be appropriate.

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Two-way action potential propagation does not happen physiologically as action potentials are always initiated either at a synapse or a nerve ending.

Recruitment and Summation When muscle fibres are stimulated indirectly, via their nerve supply, the muscle fibres are activated synchronously because their motoneurons are activated simultan-eously. Each stimulus pulse activates a proportion of the fibres in the nerve trunk and the activated fibres evoke a twitch response in the muscle fibres which they innervate. The number of fibres recruited, and hence the force of the muscle contraction depends on the stimulus intensity. At low intensities, only a small proportion of the fibres are recruited. At higher intensities, a greater proportion of the nerve fibres are activated. Whatever the intensity, the muscle response to transcutaneous electrical nerve stimulation is critically dependent on the stimulus frequency. At low frequencies (a few Hz or less), isolated twitches are produced in response to each stimulus pulse. There is time for the muscle to relax before the next contraction. If the frequency is more than a few Hz, the muscle fibres do not have time to completely relax between pulses. Each successive contraction occurs on the tail of the previous one and the peak force is greater. With a further increase in the frequency it becomes more difficult to distinguish the effects of individual stimuli. The twitch responses fuse and the contraction becomes stronger still. With most human muscles, at a stimulus frequency of about 20 Hz, only small ripples are seen in the force record. This is described as partial tetany. Between 20 Hz and 50 Hz, the ripples disappear, the contractile force reaches a plateau and the contraction is described as tetanic.

The term 'tetanus' refers to a tetanic muscle contraction. It is also used to describe a pathalogical condition produced by the toxin of a bacillus which causes tetanic muscle contraction.

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Figure 4.5 illustrates the effect of a progressive increase in stimulus frequency on the evoked muscle response. In this diagram, the frequency has been ramped from 2 Hz to 50 Hz. Note that at about 20 Hz, the response is almost tetanic. Above this frequency complete tetany occurs. Note also that the tetanic force is about four times greater than the isolated twitch force. In the example considered, the fusion frequency is about 30 Hz. The fusion frequency varies between muscles and depends on the muscle fibre types present. In terms of twitch times, two groups of fibres are distinguished: fast- and slow-twitch. The contraction time, defined as the time from the start of the contraction to peak force, is about 40 ms for human fasttwitch muscle fibres and about 120 ms for slow-twitch fibres. Muscles such as soleus contain mostly (80%) slow twitch fibres. The twitch contraction time is long and consequently the fusion frequency is low. At the opposite extreme, orbicularis oculi, an eye movement muscle, contains mostly (85%) fast-twitch fibres and the fusion frequency is high. Fusion frequencies can thus vary from less than 20 Hz to close to 80 Hz. Many human skeletal muscles have roughly equal proportions of slow and fast-twitch fibres. For example, biceps and triceps brachii are comprised of about 60% fast-twitch fibres, wile the figure for quadriceps is close to 50%.

Figure 4.5 Muscular force produced in response to short duration rectangular pulsed current with frequencies in the range 2 Hz to 50 Hz.

Effect Of Pulse Duration Earlier we saw that the characteristics of an electrical impulse which determine its effectiveness in stimulating nerve fibres were pulse amplitude, duration and rate of rise. We now examine the first two factors in more detail. For the stimulation of normally innervated muscle it is customary to use rectangular pulses of short duration. The reasons for this are as follows:

ELECTRICAL STIMULATION OF NERVE AND MUSCLE *

the short rise-time of a rectangular pulse overcomes the problem of accommodation of the nerve fibre membrane.

*

some sensory nerves will invariably be stimulated. The sensation associated with pulses of short duration (less than 1 ms) is less unpleasant than that associated with a longer pulse duration (above 1 ms).

*

for long pulses, only the early part is effective in stimulating nerve. If the pulse duration is a few multiples of the skin RC time constant, significant subcutaneous current flow will only occur at the beginning and end of the pulse (chapter 3, figure 3.4).

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A graph of stimulus strength needed to produce a minimal muscle contraction against pulse duration is shown in figure 4.6. The method of obtaining these results is as follows. Pulses of long duration (usually 100 ms) are applied to a muscle and the stimulus intensity is increased until a minimal contraction is obtained. The stimulus intensity (voltage or current) is then recorded. The pulse duration is then decreased and the intensity needed for minimal contraction is again determined. The process is repeated until enough results are obtained to give a graph like that shown in figure 4.6. The graph (the strength-duration curve) is normally plotted using a logarithmic axis for pulse duration. This magnifies the region of the curve showing the effect of short pulse durations and makes interpretation easier. It is found that pulses of long duration (about 10 ms or more) produce a muscle contraction with the same voltage for all durations. When the duration is decreased below a certain point (in this example, a little over 1 ms) the

Figure 4.6 A strength-duration curve for normally innervated muscle

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stimulus intensity needed to produce contraction is increased. The increase in intensity with decreasing pulse duration can be explained in terms of the electrical properties of the nerve fibre: specifically the capacitance of the nerve fibre membrane (see chapter 3 previously). To generate an action potential, the membrane must be depolarized to threshold. That is, the potential difference across the membrane, the resting potential, must be changed by a certain amount by charging the capacitative membrane. As figure 3.6 shows, if the pulse duration is long, the capacitor charges fully. With short duration pulses, the membrane does not have time to fully charge. However, if a higher voltage is used, the overall current flow is higher and the membrane potential changes more rapidly. Threshold is approached more rapidly so a shorter charging period (short duration) is compensated by a greater charging force (higher voltage pulse). The plateau above 1 ms in this example relies on two factors: the membrane capacitance and the skin capacitance. When the pulse duration is long, the skin capacitance will fully charge and the flow of current through the deeper tissues will decrease to a negligible amount during the pulse (figure 3.4a). A long duration pulse also allows time for the membrane capacitor to fully charge, and any additional time does not produce further charging (figure 3.6a). Once the time has elapsed for both of these processes to reach a steady state, any further increase in pulse width has a negligible effect.

Chronaxie and Rheobase Two important quantities are obtained from the strength-duration curve, the chronaxie and the rheobase: * The rheobase is the minimum voltage (or current) which will produce a response if the stimulus is of infinite duration. In practice a pulse width of 100 ms duration is used, quite satisfactorily, to assess this. * The chronaxie is the minimum duration of impulse which will produce a response with a voltage (or current) of double the rheobase.

ELECTRICAL STIMULATION OF NERVE AND MUSCLE In figure 4.6 the rheobase is 25 volts so the chronaxie is the minimum duration required with a 50 volt stimulus. In this case the chronaxie is 0.03 ms (see alongside). Strength-duration curves and their chronaxie and rheobase values can be used clinically to assess and monitor muscle which may have suffered damage to its nerve supply. Strength-duration graphs for denervated muscle are quite different to those of normally innervated muscle, as are the chronaxie and rheobase values - but more of this later.

Effect of Pulse Frequency We have already described the response of typical skeletal muscles to nerve impulses of different frequencies. Single muscle twitches are produced with low frequency stimuli (less than about 5 per second) and as the frequency approaches 20 Hz, the twitches summate to produce partial tetany (see figure 4.5). At some frequency above 20 Hz, a tetanic contraction results. Once a fused, tetanic contraction is induced, any further increase in stimulus frequency does not induce any increase in muscle force. For typical human muscles (which have mixed fibre types in roughly equal proportions) the fusion frequency is around 40 Hz. For muscles with a high proportion of fast twitch fibres, the fusion frequency is higher. For muscles with a high proportion of slow twitch fibres, the fusion frequency is lower. What of stimulation at frequencies above the fusion frequency? At frequencies which are high enough that successive stimuli arrive within the refractory period, the nerve fibre response depends on the intensity of the stimulus. Just at threshold, one stimulus pulse will produce an action potential, the next will not, as the nerve fibre will

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be refractory. Well above threshold, when fibres are stimulated at multiples of their threshold intensities, firing will occur within the refractory period and the firing rate will equal the stimulus frequency. It has been demonstrated experimentally that nerve fibre firing rates up to the limit determined by the absolute refractory period can be produced by stimulus intensities of only a few times threshold. Thus if the absolute refractory period is 1 ms, the maximum firing rate would be every millisecond so the frequency would be 1 kHz. The experimentally determined maximum firing rate of αmotoneurons is 800 Hz, a value in close agreement with measured absolute refractory periods. Nerve fibre firing rates with electrical stimulation can thus be much higher than those produced physiologically. In a sustained, weak voluntary contraction, firing rates of 8 - 12 Hz are typical. Lower firing rates are found with repetitive weak contractions. For a steady, sustained forceful contraction, an upper limit to the firing rate seems to be about 30 Hz in human skeletal muscle. These are firing frequencies which result in a partially fused contraction. How, then, are smooth, controlled voluntary movements possible when low forces are involved and the firing rates are very low? Why is it that no twitching or fluttering is seen when the firing rates are below the fusion frequency? The answer is that the activity of different motor units is asynchronous. Although individual motor units may be firing at low frequency and producing a fluttering, partly fused contraction in individual muscle fibres, there is no synchronization between different motor units. At the level of the whole muscle, the total force is the sum of the contributions of all active motor units so the ripples in force output from each motor unit are smoothed i.e. lost in the total. By contrast, when muscles are activated electrically, all of the activated fibres are synchronously activated so smooth contractions are only possible when the induced firing frequencies are greater than, or equal to, the fusion frequency.

McComas (McComas AJ. Skeletal Muscle: Form and Function. Human Kinetics. 1996) provides a wealth of fascinating information on nerve firing frequencies in different activities.

The very large range of force output of which human muscles are capable is only partly due to variation in nerve firing rates. A second factor which is at least as important is recruitment. In a weak contraction only a few motor units may be active. In a stronger

ELECTRICAL STIMULATION OF NERVE AND MUSCLE contraction, more motor units are recruited. The gradation in force which all skeletal muscles exhibit is achieved by a combination of increase in firing rate and increase in number of motor units recruited. Different skeletal muscles rely to different extents on these two strategies. With rapid, forceful contractions, initial nerve fibre firing rates can be as high as 100 Hz or so, but this is never sustained. Such rates are only observed at the start of a contraction and drop to much lower 'steady' values within a few seconds. With prolonged effort and fatigue, the maximum steady firing rate might typically drop from 30 Hz to about half this figure. It is interesting to note that as muscle fibres fatigue, their twitch duration increases so the associated decrease in firing rate does not result in a partially fused contraction becoming unfused. Were this to occur, a very large drop in force would result (figure 4.5). Rather the decrease in firing rate seems to be balanced by the increase in contraction time.

Fatigue considerations An observation made very early in the history of electrically induced muscle contraction is that the rate of fatigue is much greater with electrically induced contractions than with voluntary contractions of the same magnitude. Two factors contribute to the difference: the firing rates of the motor units and the number and nature of the motor units which are recruited. As discussed previously, in order to produce a smooth, non-twitching motor response, the frequency of electrical stimuli must be higher than the fusion frequency of the excited muscle fibres. 50 Hz is a 'ball-park' figure for most skeletal muscles. A voluntary contraction of the same magnitude would involve lower firing frequencies and, to compensate, greater recruitment of motor units. The difference is that physiologically, the load is spread over more motor units which, individually, do not have to work as hard. The result is a lower rate of fatigue. The second difference involves different muscle fibre types. Muscle fibres are typed, as described previously, as fast- or slow-twitch. They are further categorized according to fatigue resistance where slow, fast-resistant and fast-fatigable fibres are

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Larger muscles such as biceps brachii and deltoid, which contain a large number of motor units, rely more on recruitment than smaller muscles, such as adductor pollicis and the first dorsal interosseous muscle.

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distinguished. Fatigue-resistance depends on cellular metabolism. Slow fibres have a long twitch-force duration and are fatigue-resistant. They rely on aerobic glycolysis for energy production. Fast-fatigable fibres are designed to produce very high peak forces for a very short time (from a fraction of a second to a few seconds). They rely on anaerobic metabolism for peak energy production: an oxygen supply from the bloodstream is not an immediate concern. Fast-resistant fibres are relatively fatigue resistant but also have a short twitch-force duration and moderately high peak force. In a steady or repetitive voluntary contraction it is the slow, highly fatigue-resistant motor units which are recruited first. For contractile forces up to about 20% of maximum, slow motor units dominate. Above this level, the contribution of fastresistant units increases. Fast-fatigable units are the last to be recruited. Slow motor units are the smallest in terms of the number of fibres innervated by an individual motoneuron and also in terms of the motoneuron diameter. Fast-resistant motor units are larger and the motoneuron diameters are larger. Fast-fatigable units are the largest on both counts. Clearly it is optimal to have small motor units with long twitch times and low fusion frequencies used for weak contractions. In this way, an unsteady twitching contraction is avoided. The order of recruitment described above applies to steady or repetitive contractions. In sudden movements, fast-fatigable units are activated at the beginning of the movement. These units fire very few action potentials in a single high-frequency burst. This produces a high peak force with a rapid initial rate of increase, such as would be needed to produce a sudden, brief acceleration of a limb segment.

Henemann et al, in 1965, proposed the size principle of motoneuron recruitment based upon their own experimental work and that of others. The principle states that with increasing contractile force, recruitment proceeds in an orderly fashion from smallest to largest motoneuron diameter.

With electrical stimulation, the pattern of recruitment is very different to that which occurs physiologically. Two factors determine the order of recruitment: proximity to the stimulating electrode and nerve fibre diameter. Fibres closer to the stimulating electrode will experience a higher stimulation intensity than those further away. This is because current spreads within the tissue, resulting in a decrease in intensity. Close to the electrodes, spreading is minimal and the current density is highest. With increasing distance, the current density decreases.

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The current density, i, is given by the formula i =

I A

...... (4.1)

where I is the current (in amperes) and A the area through which the current passes (in square metres). The units of current density are thus amperes per square metre (A.m-2). Close to the electrodes the current density will be greatest: approximately I/Ao, where Ao is the area of the electrode (figure 4.7). Further from the electrodes the area A through which the current passes is larger than Ao so the current density is less. Fibre diameter is important because the distance between adjacent nodes of Ranvier is greater for larger diameter fibres. Histological measurements show that the distance between nodes is directly proportional to nerve fibre diameter. As stated previously, initiation of an action potential relies on producing a potential difference between adjacent nodes. The greater the distance between the nodes, the greater will be the potential difference for a given stimulus intensity applied to the tissue. Larger diameter nerve fibres then, are more easily recruited than those of smaller diameter. This means that for nerve fibres at a certain distance from the stimulating electrode, the order of recruitment will be the reverse of that which occurs physiologically. The largest diameter fibres, which innervate fast-fatigable motor units which have the highest fusion frequencies, will be recruited first. The effect of electrode-to-nerve-fibre distance means that the reverse recruitment order will not be followed exactly, but fast-fatigable motor units will contribute

Figure 4.7 The spreading of current within a volume conductor

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disproportionately to an electrically induced contraction. A consequence is that a high stimulus frequency is needed to achieve a smooth, fused response and this inevitably induces a high rate of fatigue.

STIMULATION OF DENERVATED MUSCLE Even though a muscle may have lost its motor nerve supply it is possible to stimulate the fibres directly. The effect of electrical stimuli on muscle fibre and on nerve is similar: the potential difference across the muscle fibre membrane is reduced and this results in a wave of excitation which propagates along the fibre and is transmitted into the interior of the muscle fibre via the transverse tubule (T-tubule) system. Depolarization of the T-tubules triggers the calcium ion release which results in contraction of the fibre. There are three main differences between the response of innervated and denervated muscle to electrical stimulation. One difference is in the type of contraction produced and the others are in the effects of pulse shape and duration: *

The contraction and subsequent relaxation of denervated muscle is more sluggish than innervated muscle. This is mainly due to the absence of synchronization in stimulation of the muscle fibres.

*

Denervated muscle shows a much less marked accommodation effect than nerve. Thus it is not necessary to use short duration, rectangular pulses for stimulation. An impulse which rises slowly in intensity can depolarize the muscle fibre membrane. For this reason impulses having, for example, sawtooth, trapezoidal or triangular shape and long duration are effective in stimulating denervated muscle. Such pulses are termed selective because it is possible to adjust the pulse duration and intensity for adequate stimulation of denervated muscle with minimal stimulation of nearby intact nerve fibres.

*

Denervated muscle is relatively insensitive to short duration stimuli. important point is discussed next.

This

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Effect of Pulse Duration The much smaller accommodation effect with denervated muscle compared to nerve results in longer pulses being more effective for stimulating denervated muscle. However, short duration pulses are considerably less effective. These experimental observations are illustrated in figure 4.8, which shows a strength-duration curve typical of completely denervated muscle. The graph shows that painfully high voltages must be used to stimulate denervated muscle if the duration is short. In contrast, if long duration pulses are used only low voltages are needed. In this particular example, comparison with figure 4.6 shows a greater sensitivity for denervated muscle to pulses of duration above 50 ms. The increased sensitivity to long pulse duration stimuli is typical of recently denervated muscle. The lack of neural input apparently causes an increased sensitivity of the muscle fibre membrane. The increase may be the result of normal control mechanisms whereby the muscle fibre adapts-to or compensates-for changes in neural activity.

Effect of Pulse Frequency We saw previously that for frequencies above about 100 Hz, the higher the frequency the less efficient pulses are for direct stimulation of nerve fibres. Similar behaviour is observed when muscle is stimulated directly, though the frequencies concerned are different, as is the explanation of the effect. For stimulus frequencies above about 10 Hz the sensitivity of denervated muscle decreases with the effect becoming quite marked at frequencies above 50 Hz. A simple explanation for this behaviour is evident from the strength-duration graph of

Figure 4.8 A strength-duration curve for completely denervated muscle

ELECTRICAL STIMULATION OF NERVE AND MUSCLE denervated muscle. From figure 4.8 we see that for pulse durations below 100 ms the stimulus intensity needed for contraction begins to increase. For a pulse duration of 100 ms the pulse frequency cannot exceed 1/(100 ms) = 1/(0.1 s) = 10 Hz. This would allow no 'rest' time between stimuli. As the frequency is increased above 10 Hz the pulse duration must inevitably decrease if there is to be any separation between the pulses. One pulse must finish before the next one is applied. To charge the muscle fibre membrane enough to depolarize it, the stimulus intensity needs to be increased if the pulse duration is decreased. The membrane capacitor must be charged by a certain amount to trigger depolarization. This can be achieved by a long duration pulse of relatively low intensity or a shorter duration pulse of higher intensity. Charge movement is the critical factor, as with nerve fibres, but the timeframes are very different.

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The effect of stimulus frequency then is somewhat simpler to explain than for nerve fibre stimulation, where the effects of accommodation and pulses applied within the relative refractory period are needed to account for the observed variation in sensitivity.

SENSORY, MOTOR AND PAIN RESPONSES So far the focus has been on the motor response to electrical stimulation. In reality, the forcefulness of the motor response will be limited by pain. Pain can be a direct result of the muscle contraction or can be due to stimulation of pain fibres (nocioceptive afferent nerve fibres) by the electrical stimulus. Clearly, if pain is produced as a result of the forcefulness of the muscle contraction, more efficient electrical stimulation will not enhance the motor response. On the other hand, if noxious electrical stimulation is the limiting factor, stimuli which preferentially recruit motor (A-α) fibres ahead of pain (A-δ and C) fibres will be more effective. Fortunately, a degree of selectivity can be achieved by appropriate choice of pulse duration. The reason is that α-motoneurons and pain fibres have a different range of diameters and different strength-duration behaviour. α-motoneurons have the largest diameters (range 12-20 µm), the largest internodal spacing (distance between adjacent nodes of Ranvier) and consequently, the lowest thresholds for electrical stimulation. The range of diameters of sensory fibres (A-α afferents, diameters 6-17 µm) overlaps with that of α-motoneurons, making it virtually impossible to elicit a motor response without also electrically activating sensory fibres. When stimulation is applied transcutaneously, a sensory response is, more

ELECTRICAL STIMULATION OF NERVE AND MUSCLE often than not, elicited before a motor response. The reason is that although motor fibres have, on average, larger diameters, they are located more deeply. Sensory fibres are in abundance near the skin surface and so will inevitably be closer to the electrodes. In other words, the effect of current spreading with depth tips the balance to favour sensory fibre activation before motoneuron activation in most individuals. Pain fibres are also found in abundance near the skin surface. If this was the whole story then one would expect a painful sensation before a motor response. That this seldom occurs is due to the fact that A-δ and C fibres are less sensitive to electrical stimulation than the larger diameter sensory and motor nerve fibres. Suppose then that skin surface electrodes are attached to a subject and stimuli of increasing intensity are applied. Three distinct responses may be obtained, each response having a different threshold for its onset. As the stimulus intensity is increased the first response normally noticed is sensory. The subject perceives the electrical stimulation before any muscular response is elicited. A further increase in intensity is needed for the onset of a motor response. This is followed, at higher intensities, by the subject reporting a sensation of pain. The sequence of responses and their separation in terms of the intensity required, depend on four factors: the placement of electrodes, the electrode size, the stimulus pulse width and the stimulus frequency. *

Electrode placement is important in that to obtain a pronounced motor response without pain the electrodes should be over a motor point or region where the motor nerve is located superficially. Conversely, if the aim is to produce sensory fibre stimulation with no motor response (as would be appropriate for pain control), motor points and nerve trunks should be avoided.

*

The electrode size should be as large as possible to avoid concentrating the current in a small superficial region. Current spreads as it enters tissue and the greatest spreading is produced near the edges of the electrodes. Near the electrode centre the spreading is less. By using a large electrode the central part, where current spreading is least, covers a larger deep tissue area (see figure 4.9).

α -motoneurons are intrinsically more sensitive to electrical stimulation than the smaller diameter A-δ and C fibres.

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A motor response is elicited first in some individuals. This can happen when the person has a low skinfold thickness i.e. a very thin layer of subcutaneous adipose tissue. The motor response inevitably produces a sensory response, but this is a result of the muscle activity rather than electrical activation of sensory fibres.

ELECTRICAL STIMULATION OF NERVE AND MUSCLE

*

The stimulus pulse width should be sufficiently small. Experimentally it has been shown that best discrimination between sensory, motor and pain responses is achieved using relatively narrow stimulus pulses. This point is discussed further below.

*

The stimulus frequency would be expected to influence discrimination because the nerve fibres associated with sensory, motor and pain responses have different refractory periods. To date, no studies of discrimination as a function of frequency appear to have been published.

Effect of Pulse Duration The differences in electrical characteristics of nerve fibre types and their different depths of location in tissue results in separate strength-duration curves for sensory, motor and pain responses. Consider first the effect of fibre type and consequently fibre diameter. Other things being equal, the observation is that the strength-duration curve is shifted to the right (to longer pulse durations) for smaller diameter fibres. The smaller the fibre diameter, the larger is the chronaxie. This is another way of saying that the smaller the fibre diameter, the larger is the associated RC time-constant. Figure 4.10(a) illustrates the differences in strength-duration curves of different diameter nerve fibres. The results apply to nerve which is stimulated directly using surgical intervention. On this basis we would predict a recruitment order of motor then sensory then pain fibres, with the sensory fibres recruited almost as soon as the motor fibres.

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Figure 4.9 The effect of electrode size on current density in tissue. (a) small electrodes, greater spreading of current, (b) larger electrodes, more uniform current density

As noted previously, with transcutaneous stimulation, depth in tissue also affects

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The graphs shown in figure 4.10(a) are based on animal studies. There are greater ethical problems associated with human experimentation using the same design, so such studies have not been carried-out.

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the threshold for nerve excitation. This is because of current spreading and a consequent reduction in the local stimulus intensity. Superficially located fibres are therefore recruited at lower stimulus intensities. Figure 4.10(b) shows measurements obtained with human subjects and transcutaneous electrical stimulation.

Note the horizontal axis (time) scale. In this figure the pulse widths are measured in microseconds (µs) and not milliseconds as have been previously used to describe action potentials and the subsequent refractory period. Here we are dealing with pulse widths which are small compared to the time-course of an action potential.

Figure 4.10 Strength-duration curves for (a) different nerve fibre types, with the nerve trunk exposed and stimulated directly and (b) sensory, motor and pain thresholds measured using transcutaneous stimulation.

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Two things are apparent from figure 4.10(b). First, that in reality the order of recruitment is usually sensory, then motor, then pain at all pulse durations when current is applied transcutaneously. Second, that as we go to shorter pulse durations the separation between the curves increases. The separation due to fibre diameter is most marked in figure 4.10(a) and indicates that with direct nerve stimulation, by using sufficiently short pulse durations (around 500 µs), the small diameter C fibres will not be stimulated at intensities which very effectively recruit the larger A-δ, A-β and A-α fibres. A shorter pulse durations (around 50 µs), neither C nor A-δ fibres will be stimulated at intensities which efficiently recruit A-β and A-α fibres. This indicates that as one goes to smaller pulse widths, the ease of discrimination between sensory and motor responses on the one hand, and pain responses on the other, is increased. The extent of discrimination evident with transcutaneous stimulation is less. As figure 4.10(b) shows, the sensory, motor and pain threshold graphs are more overlapped and the variation occurs at smaller pulse widths. This is because the measured response depends not just on the fibre type (and the associated diameter) but also two other factors: the depth of the fibres within tissue and the electrical characteristics of the skin and underlying tissues. The capacitative nature of the stratum corneum means that longer duration pulses are not more effective for nerve stimulation (whatever the fibre type) as the current flow in tissue beneath the stratum corneum is transient (figure 3.4). Spikes in the current flow are produced at the start and end of long duration pulses and increasing the pulse width does not result in a longer duration flow of current in tissue. Thus C fibres are not as more readily recruited at longer pulse durations as would be expected from figure 4.10(a). Nor are A-δ fibres, though the effect is less. The result is a plateau in the transcutaneous sensory, motor and pain threshold graphs at a pulse width much less than in figure 4.10(a).

The results shown in figure 4.10(a) indicate that, with surgically implanted electrodes, very good discrimination between nerve fibre types can be achieved by choice of an optimal pulse width.

The pain threshold graph in figure 4.10(b) should not be identified with the A-δ and C fibre graphs. Very little contribution would be made by C fibres. Rather, A-δ fibre activity and the pain associated with a forceful muscle contraction would determine pain thresholds.

The observations regarding the effect of pulse width have important practical implication for therapy, and the results shown in figure 4.10(b) are most relevant. If long duration pulse widths are used then only small changes in intensity will be needed to change from a sensory response to a motor or pain response. By contrast,

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if short duration pulses are used, much larger changes in intensity will be needed to recruit motor and pain fibres. If the objective is to produce a sensory response with minimal motor or pain responses then short duration pulses are preferred (less than 50 µs from figure 4.10a or perhaps 'the shorter, the better' from figure 4.10b). Short duration pulses will also be capable of producing an effective motor response with minimum pain sensation. It is for this reason that modern electronic stimulators produce higher voltage, shorter duration pulses than their predecessors. A question arising from the foregoing discussion is whether very short pulses, around 2 to 10 µs duration, will give better discrimination with transcutaneous stimulation than, say, 20 µs pulses. The evidence certainly indicates that pulses of duration in the range 20 to 50 µs will more effectively discriminate than pulses with duration greater than 100 µs. It is not known whether this trend continues to very short pulse durations. Further research is needed before any firm conclusions can be drawn.

STIMULATION USING SINUSOIDAL AC Sinusoidal alternating current has been used for patient treatment almost since devices for producing AC were first marketed in the late 1800s. It was soon established that low frequency AC produced noxious stimulation while AC in the kHz frequency range could produce strong muscle contractions without the degree of noxious stimulation associated with lower frequencies. The French scientist Arsène d'Arsonval studied the effect of AC stimulation on nerve and muscle, both using dissected animals and by transcutaneous stimulation of human subjects. He used an alternator, the first device built for generating AC and the one which is used in every modern-day motor car. An alternator works on the principle that if a coil of wire is made to spin in a stationary magnetic field, alternating current is produced in the coil. d'Arsonval reported in 1891 that with increasing frequency, the neuromuscular response to sinusoidal AC becomes stronger up to about 1400 Hz, is constant between 1500 and 2500 Hz and decreases to 5,000 Hz. He also reported that a current of 1500 Hz is more painful than 5,000 Hz but much less painful than currents of 75 and 20 Hz.

On the basis of his work, d'Arsonval described the electric chair, recently adopted in New York state for criminal executions, as 'barbarous and unholy' as the voltage chosen was too low (1500 V) and death is slow.

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d'Arsonval's observations of transcutaneous stimulation using human subjects and sinusoidal AC laid the foundations for the use of kHz frequency AC in clinical practice. Sinusoidal AC stimulation has figured in clinical practice since the 1950s when a German physician, Hans Nemec, began advocating 'Interferential Currents' as a means of producing comfortable, pain-free, muscle contractions and 'promoting tissue healing'. More recently, the use of interferential currents for pain control has been advocated. Interferential currents used clinically are sinusoidal AC with frequencies around 4 or 5 kHz.

The Nemectrodyne interferential stimulator was the first on the market and the company continues to sucessfully market interferential units.

More than two decades after the introduction of interferential currents, 'Russian currents' became popular, principally due to the claims made by a Russian physician, Yakov Kots, in the late 1970s. Kots claimed that kHz frequency AC, modulated at 50 Hz with a 1:1 duty cycle, could produce large strength gains in stimulated muscle. He based his claims on studies made with young Russian athletes as subjects: athletes who were hoping to qualify for the Olympic games. Russia's success in the Olympics and the intense competitiveness which existed at the time seems to have given weight to Kots' claims. Kots argued that an optimal AC frequency for muscle strengthening, one which produced maximal force at the pain-tolerance threshold, was 2.5 kHz if the muscle was stimulated directly (with the active electrode over the muscle) or 1 kHz if the muscle was stimulated indirectly (with the active electrode over the nerve trunk supplying the muscle). Both Russian currents and Interferential currents continue to be used in clinical practice. Interferential currents are popular in England, Europe and Australia. Russian currents are, somewhat paradoxically in the light of political relations post world war two, more popular in the USA. Stimulation with low frequency AC is seldom used nowadays. It is particularly painful. Nonetheless, it did experience some popularity in Europe in earlier decades. A particular form of low-frequency AC stimulation, called 'Diadynamic current' was popularized in Europe. The argument seems to have been that the discomfort associated with the stimulation had therapeutic benefits resulting from a counterirritant effect.

ELECTRICAL STIMULATION OF NERVE AND MUSCLE Low Frequency Alternating Current The term low frequency AC as applied in therapy relates to frequencies between about 1 Hz and 100 Hz. A sinusoidal current is, in effect, a continuous train of current pulses. For example, 50 Hz AC has one complete cycle every 1/50th of a second or 20 milliseconds. The 20 ms sinewave has a rounded 10 ms positive pulse followed by a rounded 10 ms negative pulse. The stimulus is therefore a series of 10 ms pulses. As noted previously, a pulse width of 10 ms results in little discrimination between sensory, motor and pain thresholds. Smaller diameter pain fibres are recruited at thresholds not much above those of the larger sensory and motor fibres (figure 4.10b). As the waveform does not have an abrupt rise, some nerve fibre accommodation will occur meaning that a greater current intensity will be needed to produce the same response as a 10 millisecond rectangular pulse. The effect of accommodation is greater in large diameter nerve fibres so there is dropout of their contribution if sinewaves rather than square waves are used. This means that there will be less stimulation of large (A-α and A-β) fibres with low frequency AC and, relatively, more contribution of smaller (A-δ) fibres. The pulse duration will result in stimulation which is both superficial and relatively non-discriminatory between sensory, motor and pain responses. Thus if the aim of therapy is to stimulate superficially and to produce, say, modest muscle contraction together with stimulation of pain fibres, or simply painful stimulation, then 50 Hz sinusoidal AC or one of its variants is a logical choice. Unsurged 50 Hz AC is sometimes used for a counter irritant effect. Counter-irritation has been dismissed as treating a patient with a sore right thumb by hitting the opposite, left thumb and producing more pain. Suddenly, the patient finds the right thumb more comfortable! The point which is ignored in this simplistic argument is

Diadynamic currents have all but disappeared from clinical practice and will not be discussed here. Details can be found in older texts e.g. the 2nd (1986) edition of Electricity, Fields and Waves in Therapy'.

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whether the pain relief persists, in which case counter-irritation stimulation is vindicated. Unfortunately, no properly documented studies seem to have addressed this question. Low frequency AC might thus have some potential in clinical practice. The evidence base has yet to be established.

Medium Frequency Alternating Current Medium frequency alternating currents are defined as currents in the frequency range 1 kHz to 100 kHz. Above 100 kHz, alternating current is not able to excite nerve fibres and the only effect is one of tissue heating. Currents above 100 kHz are classified as 'high frequency'. In clinical practice, currents with frequencies between 1 kHz and 10 kHz are commonly used. Frequencies above 10 kHz are not. The reason is that above 10 kHz or so, the nerve fibre response diminishes while the power dissipated in tissue increases. At frequencies above 10 kHz, nerve sensitivity becomes lower while the electrical energy dissipated in tissue, and consequently the heating rate, increases. 10 kHz to 100 kHz is evidently the transition zone between direct electrical stimulation and tissue heating.

Alternating current at high (>100 kHz) frequencies does have a part to play in physiotherapy, but not transcutaneous electrical nerve stimulation. Rather the tissue-heating effects are exploited. This is discussed in the later topic 'Fields'.

As noted previously, nerve-fibre firing rates are well below 100 Hz during most voluntary activities, including strenuous exercise and generally less than a few tens of Hz on a sustained basis. With electrical stimulation at higher frequencies and sufficiently high intensity, firing rates approaching 1 kHz can be produced. The absolute refractory period places the limit on the maximum firing rate. If nerve is stimulated with AC at frequencies above 1 kHz, action potentials are produced with every second, third or fourth succeeding AC pulse. The fibre firing rate will thus be a sub-multiple of the AC frequency. If, for example, 4 kHz AC is used, the induced firing rate might be 100 Hz at intensities just above threshold. In this case the firing rate is determined by the relative refractory period. At higher intensities, higher firing rates are induced as action potentials are produced during the relative refractory period. At the highest intensities the firing rate might approach 1 kHz i.e. fibres firing immediately after the absolute refractory period.

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With AC stimulus frequencies above 10 kHz or so, the physiological response of nerve fibres become less and less while the power dissipated, and heating rate, become larger and larger. The decreased nerve fibre response is because the membrane capacitor has less and less time to charge during a pulse, so less depolarization is produced. The higher tissue heating rate is because skin impedance decreases with increasing frequency (chapter 3 previously) so the current flow is higher for a given stimulus voltage and the power dissipation is correspondingly higher.

Higher frequency AC (above 10 kHz) thus has less direct effect on the nerve-fibre membrane and more effect on the sensory receptors which detect heat.

INTERFERENTIAL CURRENTS Hans Nemec popularized interferential currents in the 1950s. Although Nemec published a number of articles describing and reporting on the effect of interferential currents, these were in German. Only one English language article exists. It was translated from German and published in the British Journal of Physiotherapy in 1959. In it, Nemec described interferential currents and made claims of therapeutic benefits. The claims, judged in terms of modern criteria, were inappropriately speculative i.e. were not adequately documented. They are, however, intriguing and not without some credence. Here we focus on the less speculative aspects. An interferential stimulator has two separate, electrically isolated circuits for applying current to the patient. The currents are applied using two diagonally opposed pairs of electrodes as shown in figure 4.11. The idea is that the two currents 'interfere' within the tissue volume, reinforcing each other and producing a greater effect at depth than would be possible using a single circuit. In the region of intersection (the crosshatched area in figure 4.11), the resultant intensity is

Figure 4.11 Application of interferential currents

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high as it is the sum of the contributions of each current. Each circuit (A and B) supplies an AC signal of constant amplitude to the patient. If current spreading is not great, as is assumed in figure 4.11, the region of maximum stimulation is the cross-hatched area (the region of diamond shapes) in figure 4.11. This contrasts with the regions of maximum stimulation when only one circuit is used. In this case maximum stimulation is produced immediately under the electrodes. Figure 4.12 illustrates the difference. In practice, current spreading will make the difference shown in figure 4.12 less marked. The superimposition of the two currents will, however, help to counteract the reduction in stimulus intensity with depth, thus increasing the depth efficiency of stimulation. The original interferential machines produced a sinusoidal waveform with a frequency around 4 kHz: thus the stimulus pulse width was 1/8000 sec or 125 µs. Some modern machines offer a choice of AC frequencies and use a rectangular pulsed AC waveform, rather than a sinewave. There is some evidence that a rectangular pulsed waveform is more comfortable than its sinusoidal counterpart and also evidence that optimal comfortable stimulation is achieved at a frequency of about 9 kHz.

Figure 4.12 Depth efficiency of (a) bipolar and (b) quadripolar stimulation.

Beat or Modulation Frequencies A key feature of interferential currents is that the two circuits produce current of slightly different frequency. The difference is normally between 1 Hz and 150 Hz. When

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applied to the patient the two currents interfere and produce a 'beating' effect in the patient's tissue. The interference or 'beat frequency' effect is illustrated in figure 4.13. Figures 4.13(a) and (b) show two sinusoidal waveforms applied to the patient via diagonally opposing pairs of electrodes as shown in figure 4.11. The total current at a particular point in the patient's tissue is the sum of the currents from each pair of electrodes. At points where the two currents are of equal amplitude the sum of the two signals will be an AC waveform which is amplitude modulated as shown in 4.13(c).

The surge or modulation frequency is equal to the difference in frequency of the two currents. The frequency, f, of waveform (a) might be 4000 Hz and the frequency (f-δ) of waveform (b) might be 4000-10 = 3990 Hz. In this case the value of δ, the modulation frequency is 10 Hz.

Figure 4.13 Interference of two sinusoidal currents of different frequency.

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To make clear which frequency we are talking about, the terms 'carrier frequency' and 'beat frequency' are used. In this example, the carrier frequency is 3995 Hz and the beat frequency is 10 Hz. In figure 4.13, the currents are assumed to be of equal amplitude. In regions of tissue where the two currents are not the same size, an interference effect will still be produced, but the resulting waveform will not drop to zero midway between the maxima. Figure 4.14 shows the effect of adding two currents of slightly different frequency when one current is twice as big as the other. An interference effect is still produced but the depth of modulation of the waveform is less. Depth Efficiency and Localization As noted earlier in this chapter, for maximum stimulation efficiency, current should flow parallel to the nerve fibres when there is a single current flow through tissue. When there are two intersecting currents of equal amplitude, maximum stimulation occurs along lines midway between the current paths. The reason is that the net current flow is the vector sum of the two currents.

Figure 4.14 Interference of two sinusoidal currents of different frequency and different amplitude.

Consider first the situation where two current pathways are at right angles and the currents are equal. Nerve fibres aligned parallel to one of the current pathways will experience an unmodulated AC stimulus as shown in figure 4.13(a) or (b). Fibres aligned along lines midway between the current paths will experience a modulated stimulus (figure 4.13(c)) of higher intensity. Those fibres aligned in other directions will experience a partially modulated stimulus, similar to figure 4.14, with a depth of modulation which depends on the fibre orientation. Figure 4.15 shows the net current flow in different directions for the simple configuration in figure 4.11. The length of the black arrows is proportional to the

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current intensity. In the horizontal and vertical directions, the net current is maximum and the modulation is 100%. In directions at 45 o , there is no modulation and the intensity is some 30% lower. The pattern of stimulation is clearly more complex with interferential currents than with current applied using a single pair of electrodes. We can, however, draw some important conclusions: *

nerve fibres aligned in directions which bisect the angle between the current pathways (horizontally and vertically in figure 4.15) will experience the greatest stimulation intensity and the stimulus will be a modulated AC signal.

*

fibres aligned parallel to the direction of the individual current flows will experience a lower, but still relatively high, stimulation intensity. The stimulating current will not be modulated.

*

nerve fibre firing rates will be much higher than with stimulation using single pulses applied at low frequency. Fibres aligned parallel to the direction of the individual current flows will fire at a rate determined by how far above threshold is the local stimulation intensity.

*

Fibres aligned in directions which bisect the angle between the current pathways will fire in bursts. The bursts of activity will be at the beat frequency and the number of action potentials per burst will depend on how far above threshold is the local stimulation intensity.

Figure 4.15 The variation in current intensity and amount of modulation with direction when using interferential currents.

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A widespread misconception is that with interferential currents, the nerve fibre firing frequency is equal to the beat frequency. This would only be the case for fibres stimulated at, or just above, their threshold. As noted previously, for stimulation intensities above threshold, nerve fibres will fire at much higher rates. When the stimulus intensity is modulated at low frequency, nerve fibres will fire in bursts, with each 'beat' of the current intensity. The beat frequency only determines the burst frequency of the action potentials. The number of action potentials per burst depends on how far the stimulus intensity is above threshold. Thus if a beat frequency of 50 Hz is chosen to produce repetitive, forceful muscle contractions, the rate of fatigue will be higher than if 50 Hz single-pulses were used as the average firing rate will be much higher. Another widespread misconception about interferential currents is that the pattern of stimulation is in the shape of a clover-leaf (a four-leafed clover) rather than the rounded-diamond shape shown in figures 4.11, 4.12 and 4.15. The misconception seems to have originated from the idea that nerve fibres are insensitive to an unmodulated AC stimulus i.e. that modulation at low ('biological') frequencies is necessary to produce a physiological response. Were this true, then fibres aligned parallel to the current paths (figure 4.15) would not be excited while those aligned along lines bisecting the angle between the current paths would be excited maximally. The pattern of stimulation would have four lobes, each lobe pointing to a corner of the rounded diamond shape. In fact, the clover-leaf pattern shows the areas of maximum interference, not maximum stimulation. The pattern applies to every small diamond shaped segment in the region of interference. It indicates the direction in which the stimulus intensity is greatest. It does not, in any way, represent the area of maximum stimulation. Within each diamond-shaped segment, a clover-leaf pattern can be drawn, showing the directions of maximum interference: in other words, the directions in which nerve fibres must be aligned to experience maximum stimulation. A misleading implication of the pattern is that no stimulation is produced if the nerve fibres are aligned along either of the current paths.

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Premodulated Interferential Current Most interferential machines make provision for stimulation using either two pairs or a single pair of electrodes. Two pairs are needed for true interferential stimulation. The term 'premodulated interferential current' refers to a current waveform as shown in figure 4.13(c), which is produced inside the interferential machine and applied to the patient using a single pair of electrodes. 'Premodulated interferential' is thus something of a misnomer, as there are no currents interfering in tissue. Premodulated current has the advantage that it is easier to apply, as only two electrodes are needed. The disadvantage is that there is no reinforcing at depth so maximum stimulation is produced immediately beneath the electrodes (figure 4.12(a)). RUSSIAN CURRENTS Russian currents are a particular form of electrical stimulation which became popular as a result of a talk given by Dr Y M Kots of the Central Institute of Physical Culture, Moscow, at a conference hosted by Concordia University, Montreal in 1977. He claimed strength gains of up to 40% in elite athletes as a result of this form of electrical stimulation. The term 'Russian currents' refers to sinusoidal AC of frequency 2.5 kHz which is burst-modulated at 50 Hz. The waveform is shown in figure 4.16. It consists of 10 ms bursts of AC separated by 10 ms 'off' periods. The waveform repeats every 20 ms (1/50th sec) so the burst or modulation frequency is 50 Hz.

Figure 4.16 Russian currents: 2.5 kHz sinusoidal AC, burst modulated at 50 Hz i.e. 10 ms 'on' and 10 ms 'off'.

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Kots and co-workers measured the maximum force which could be elicited using AC in the frequency range 100 Hz to 5 kHz. Current was applied either using either two equal-sized electrodes placed over the muscle belly (referred-to as 'direct' stimulation) or using a small 'active' electrode over the nerve trunk supplying the muscle and a larger 'indifferent' electrode placed elsewhere, so as to avoid excitable tissue (referred-to as 'indirect' stimulation). They established that maximal force at the paintolerance threshold was obtained at 2.5 kHz if the muscle was stimulated directly or 1 kHz if the muscle was stimulated indirectly. Kots also advocated a '10/50/10' treatment regime i.e. 10 seconds of stimulation followed by a 50 second rest period, repeated 10 times. His argument was that to produce strengthening, the electrical stimulation should be non-fatiguing. He reported that with intense stimulation for periods over 10 sec, fatigue is evident, whereas no force decline is seen if the duration is 10 sec or less. To avoid a force decline from one 10 sec stimulation period to the next, a rest period of 50 sec is needed. If this rest period is allowed, no force decline is seen over the 10 repeats. The validity of Kots' argument for the '10/50/10' treatment regime is questionable. The quoted findings were obtained using low frequency monophasic pulsed current, not kHz frequency AC. With AC bursts, the nerve firing rates would be expected to be higher and, as a consequence, the rate of fatigue would be higher. The strength gains reported by Kots are supportive, but whether the '10/50/10' treatment regime is optimal with AC burst stimulation remains open to question.

EXERCISES 1

Briefly explain what is meant by the following terms: (a) threshold potential (b) absolute refractory period (c) relative refractory period (d) hyperpolarization

ELECTRICAL STIMULATION OF NERVE AND MUSCLE 2

What is meant by the term 'accommodation' as applied to nerve fibres? Explain the significance of accommodation as far as stimulation by very low frequency AC is concerned.

3

Consider the strength-duration curve shown in figure 4.6. Explain why: (a) greater stimulus amplitudes are needed to elicit a muscle contraction when very short pulses (less than about 0.1 ms) are used. (b) a muscle contraction is produced with the same stimulus voltage for all values of pulse duration above a few milliseconds.

4

It is observed that the sensitivity of nerve fibres to stimuli changes as the stimulus frequency is increased above about 100 Hz. (a) Describe the change and briefly explain why it occurs. (b) Why does the effect become more pronounced at frequencies above about 1 kHz?

5

(a) (b)

6

Compare figures 4.6 and 4.8 and explain why the chronaxie for denervated muscle is much greater than that of typical nerve fibres. Does this also account for the rheobase of denervated muscle being lower than that of nerve? Explain.

Consider the strength-duration curves for sensory, motor and pain responses shown in figure 4.10(b). What range of pulse duration should be used for producing: (a) A maximum sensory response with minimum motor involvement? (b) A maximum motor response with minimum physical discomfort? (c) A pain response with a minimum motor response? What is a motor point and what relevance has it to (a), (b) and (c) above?

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8

9

10

Consider DC pulses used for stimulation of denervated muscle. (a) (b)

What range of pulse width and frequency is most useful? What is meant by the term 'selective' stimulus? Give an example and explain why the waveform you have nominated is selective.

(c)

Apart from pulse width and frequency, what other characteristic of a 'selective' waveform should be adjustable by the therapist?

Consider rectangular pulsed waveforms used for stimulating nerve fibres. (a) (b)

What is the most useful range of pulse width and frequency? How does the depth efficiency of stimulation vary with pulse width and why?

(c)

When is surging of a pulse train useful? Why?

(a)

What is the principal disadvantage of 50 Hz sinusoidal AC when the objective is to stimulate muscle?

(b)

Describe one useful practical application of 50 Hz sinusoidal AC in therapy.

(a)

Describe the result of combining two medium frequency (interferential) currents in tissue. List the characteristics (carrier frequency, beat frequency, period and shape) of the resulting waveform. What are the differences in depth efficiency of short duration pulsed current and interferential currents? Draw diagrams to illustrate. Why do interferential currents have greater depth efficiency?

(b)

11

Consider interferential currents flowing through tissue as in figure 4.15. (a) Why is the stimulation intensity greater in the regions between lines connecting electrode pairs rather than along lines connecting the electrodes? (b) Assuming that the stimulus intensity is high enough to excite nerve fibres regardless of their orientation, what differences in fibre firing rates would

ELECTRICAL STIMULATION OF NERVE AND MUSCLE exist between fibres aligned parallel to the current pathways and those aligned along lines bisecting the angles between the current paths? 12

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Describe the similarities and differences between Russian currents and premodulated interferential currents.

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5 Rectification and Amplification DIODES, TRANSISTORS AND VALVES The humble electric light bulb is such a commonplace item that we take it for granted. Yet its development marked a turning point in human societal development. No longer were we constrained by daylight hours. Although fire, the candle and later, gas lighting had enabled people to extend their daytime activities, the light bulb (and the establishment of a statewide system for providing electricity) virtually eliminated human dependence on daylight. The development of the light bulb also signalled the birth of electronics, an area of science which has transformed the world in which we live. The radio, TV, mobile phone and desktop computer are but a few examples of developments in electronics. Electronic technology led to the development of the neuromuscular stimulators used by physiotherapists and also enabled scientists to better study the workings of the human body, in particular the nervous and neuromuscular systems. So what is so special about an electric light bulb, that it can lay claim to the birth of electronics? A common electric light bulb consists of a filament mounted inside an evacuated glass envelope. The filament is a coil of resistance wire, usually tungsten. The glass bulb has virtually no air inside as oxygen in the air would react with the tungsten at high temperatures, forming tungsten oxide, which is an insulator so the filament would no longer conduct. Oxidation is prevented by the vacuum and when current from the mains or a suitable power source is passed through the tungsten filament It heats up and glows. Heat and light energy are produced at the expense of electrical energy. Heat and light production by a metal are fundamentally related to the movement of electrons. The greater the movement energy (principally of the electrons: the atoms are more locked in place in the crystal structure), the greater the heat energy and the greater the emission of light. Light is emitted whenever electrons are accelerated and, in the case of a light globe, electrons are continually accelerated when a current

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Electric light bulbs are very inefficient. More than 80% of the electrical energy is dissipated in the form of heat. Less than 20% of the energy is released as light.

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flows through the filament. Acceleration of electrons at red-hot temperatures and above, results in emission of electrons from the surface of the hot filament. The phenomenon of electron emission accompanying heating is given the name thermionic emission. The relationship between heat, light and movement will be considered further in later chapters. For the moment consider the electrons emitted by a glowing filament. In the normal run of things these electrons, having left behind positively charged ions from the parent metal, will be attracted back to the filament. The filament of a light globe is thus continually emitting and recapturing electrons in the process of emitting light. If heat and light production were the only physical phenomena associated with light bulbs then this, in itself, would be justification for their importance. the history of electronics would, however, would be quite different to the one we know. Electronics has its origin in a device based upon a simple light globe and invented by the English physicist J. A. Fleming in 1904. The device is a valve diode. Fleming's diode resembled an electric light bulb with an extra part, a metal electrode called the plate, included inside the glass envelope. The construction is illustrated in figure 5.1(a). By including the plate Fleming was able to capture some of' the electrons emitted by the filament. This is achieved by making the plate positively charged so that electrons are attracted to the plate more strongly than to the filament. Thus the diode allows a flow of current with electrons moving from filament to plate, providing the plate is sufficiently positive with respect to the filament. Notice, however, that a diode is asymmetric. Electrons can move from filament to plate but not from plate to filament even when the filament is made positive with respect to the plate. This is because the plate is not heated and so does not spontaneously emit electrons. The diode allows current to flow in one direction but not the other. Valve diodes require the plate to be at a potential some tens of volts higher than the filament for conduction to occur. Semiconductor diodes (figure 5.1(b)), their modern day counterparts, require potential differences of only a few tenths of a volt tor

Some electrons are given so much energy as a result of collision that they can escape the confines of the metal surface.

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conduction to occur and their energy efficiency is much higher i.e. less electrical energy is dissipated as heat. Fleming's diode was an extremely important development but the next step was even more important. By adding a 'grid' (actually a fine wire grid or mesh) between the filament and metal plate of a diode a device which could amplify electrical signals was invented. The development of this device (the triode valve) is generally credited to an American, Lee de Forest. This was in 1906. The construction of a triode valve is shown in figure 5.2. The mechanism by which it amplifies electrical signals will be described shortly. The triode valve, because of its ability to amplify very weak signals from a microphone and then apply them to a transmitter, started a revolution in science and technology. Although transmissions of signals across the Atlantic Ocean had been made by Marconi in 1901 these transmissions strained to the limit the detection facilities available. With amplifying valves it became possible to transmit and receive over much greater distances and even to amplify the signals so that they could be clearly heard without headphones, through a loudspeaker! By 1920 the valve had transformed wireless transmission from an extension of electrical and telegraph practice into a new and fascinating technology. In the 1940's when valves were a familiar part of the fields of home entertainment, communications and science the growing pressures of developments in science and technology had pointed up the limitations of the valve - high power consumption, large size and excessive heat generation. These factors became more and more critical as scientists and engineers attempted to apply electronics to more sophisticated tasks. The first computers were constructed using hundreds of valves, fully occupying large rooms and requiring elaborate ventilation and cooling systems in addition to enormous quantities of electrical power.

RECTIFICATION AND AMPLIFICATION Fortunately science again came to the rescue with the development in 1947 of the first functional transistor. Developed by physicists Shockley, Brittain and Bardeen the device revolutionized future developments in technology. The computers could be drastically reduced in size and power consumption. Printed circuit boards became a possibility so resistors, capacitors and inductors were miniaturized to suit. The computer occupying a large room could now be assembled as a device the size of a filing cabinet. More was still to come - the pressure was on for smaller and lighter components to build the apparatus used in navigation, guidance and communication in aeroplanes, missiles and satellites - to perform increasingly complex scientific tasks. Engineers quickly realized that transistors could be scaled down in size to fit many devices in the same volume which one previously occupied - they could even be interconnected in the one device in the kind of circuit arrangement equipment engineers might require. The same fabrication techniques could be adapted to produce resistive or capacitive interconnections between the transistors microscopic versions of the familiar resistors and capacitors used in everyday circuits. Thus was the first integrated circuit (IC) produced. Early IC's, introduced in 1964 contained up to about 10 components in one tiny package. By 1968 this figure had risen to about 500. Today it numbers in the millions. Nowadays, large scale integration (LSI) permits many hundreds of transistors and diodes together with the necessary interconnections of components to fit into a package not much bigger than a postage stamp. A good example of LSI technology is the pocket calculator. Because they are based upon one or at most a few large scale integrated circuits, they can be cheaply mass produced. Most of the fabrication, testing and assembly is performed by automated apparatus under the control of minicomputers, themselves the product of IC technology. Many more examples of the impact of modern electronics are to be found in everyday life - digital wrist watches and clocks, the desktop computer and video games are a few which spring to mind. Watch for others - they are evident in virtually all aspects of life, art and science.

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DIODES AND RECTIFICATION Valve diodes are rarely used nowadays. They have been superseded by semiconductor diodes, their solid-state equivalent. Semiconductor diodes are smaller, more efficient and generate less heat. The first semiconductors were fabricated from crystals of germanium. Nowadays silicon is the semiconductor of choice because of its superior physical properties. Both germanium and silicon are elements which fall midway between the good conductors and the good insulators. We now consider the properties of a silicon diode but you should bear in mind that other kinds of diode (selenium or germanium diodes, vacuum tube diodes) have similar though not identical properties. In addition there are diodes specifically tailored for somewhat different roles to those we will examine in this section (zener diodes and light emitting diodes are examples). These have distinct circuit symbols of their own which are variations on the normal diode symbol. Figure 5.3 shows the circuit symbol of a simple diode. Note the naming of the different sides of the diode. The circuit shown in Figure 5.4 can be used to demonstrate the conduction properties of a diode. With the circuit arrangement shown we would find that the potential difference across the diode is about 0.7 volt and the current flowing about 110 mA (0.11 amp). The resistance of the diode, calculated using Ohm's law, is thus 0.7 V = = 6.4 ohms R= 0.11 I Figure 5.4 Circuit for demonstrating the conduction properties of a diode

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This arrangement, with the anode (the arrow in the circuit symbol) connected to the positive terminal of the power supply and the cathode (the straight line) to the negative terminal, is called forward biasing of the diode. When forward biased, the diode has a very low resistance. With the diode reversed in the circuit, that is anode and cathode reversed, the diode is reverse biased. The voltage across the diode would be measured as 12 volts and the current about 0.5 microamp or 5 x 10-7 amp - too small to register on most ammeters. For all intents and purposes, the reverse current flow is negligible, meaning that the resistance is close to infinite. The diode resistance calculated from the measurements quoted is R=

12 V = = 24 x 106 Ω = 2.4 x 107 Ω or 24 MΩ. 5 x 10-7 I

An ideal diode has zero resistance in the forward direction (when forward biased) and an infinite resistance in the reverse direction (when reverse biased). Silicon diodes come reasonably close to his ideal.

Half Wave Rectification This unique conduction property of diodes makes them well suited to the task of' rectifying alternating current, that is converting alternating current to direct current. The circuit shown in figure 5.5 illustrates one way of doing this. Figure 5.5 Rectification by a diode

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The AC source produces a sinusoidal alternating voltage, as shown in figure 5.6(a). This means that the diode will be forward biased when the voltage is positive i.e. during the positive half-cycles then reverse biased during the negative half-cycles. Only when the diode is forward biased will current flow through the resistor so a graph of current in the resistor will resemble figure 5.6(b). A graph of potential difference across the resistor would also resemble 5.6(b).

Figure 5.6 (a) sinusoidal alternating voltage produced by the AC source in figure 5.5 and (b) the resulting current flow through the resistor, R. The current through the resistor is DC: not like the DC produced by a battery to be sure, but DC nonetheless because the current flow is only in one direction. A more accurate description of this half-wave rectified current would be pulsed DC. By placing a capacitor in parallel with the resistor as shown in figure 5.7 we can 'smooth' the pulsed DC to provide a more even flow of current. Figure 5.8 shows the waveforms which are obtained when different size capacitors are connected in parallel with the resistor, R.

Figure 5.7 Smoothing rectified AC with a capacitor

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As the capacitor is made larger the waveform becomes more like the straight line graph obtained with a battery. The voltage becomes more nearly constant. The reason that the capacitor has this effect is that the capacitor stores electrical energy. While the diode is conducting, current will flow through the resistor and at the same time the capacitor will charge to the peak voltage of the waveform. When the diode ceases to conduct the capacitor will discharge through the resistor (we discussed capacitor discharging in chapter 2). The rate at which the capacitor discharges depends on the size of the capacitor and also the resistance through which it discharges. The larger the capacitance and the larger the resistance, the slower the rate of discharge will be and the smoother will be the waveform. If we replaced the resistor in figure 5.7 with a higher value resistor then the waveforms shown in figure 5.8 would be obtained with capacitors of lower value. Conversely, if the capacitors remained the same, smoother waveforms would result. The amount of smoothing is determined by the RC time constant (chapter 2). For mains supplied electricity, where the AC frequency is 50 Hz, the time between pulses in figure 5.8(a) is 1/50th sec or 20 ms. For efficient smoothing, the RC time constant should be greater than 20 ms. Thus if R is large, meaning that the amount of current

Figure 5.8 Smoothing using a capacitor. The effect of different size capacitors.

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drawn from the supply is small, C can be relatively small. If R is small and a lot of current is drawn from the supply, C would need to be relatively large to produce adequate smoothing.

Full Wave Rectification The half wave rectifier circuit is very simple but rather inefficient. Only half of' the original AC waveform is being used. The power supply is virtually switched off for half the time as no current flows during negative half cycles. The full wave rectifier circuit shown in figure 5.9 is a more efficient rectifier. The arrangement is called a bridge rectifier. The positioning of diodes ensures that current will always flow through the resistor in one direction and both halves of the AC waveform are used. Consider what happens when terminal A of the AC supply is positive - that is, on the positive half-cycle of the waveform. Current flows through diode 2 but not diode 1 (because of their polarity). It then flows through the resistor through diode 3 and back to terminal B of the transformer. It can not flow through diodes 1 or 4 because their opposite terminals are at a higher potential - remember there is a relatively large potential difference across the resistor. On the negative half cycle of the waveform, terminal A is at a lower potential than terminal B. Current flows from B through diode 4 through the resistor in the same direction as before, through diode 1 to terminal A. Current can not flow through diodes 2 or 3 as the voltage is higher at the opposite terminal. The net result is that the current through the resistor or potential difference across the resistor resembles figure 5.10(a).

RECTIFICATION AND AMPLIFICATION

Full-wave rectified AC can be smoothed with capacitors in the same way as the half-wave rectified AC. A smooth waveform is more easily obtained with the full-wave rectifier because the capacitor is recharged 100 times per second rather than 50 times per second with half-wave rectified AC. Compare figure 5.10(b) with the half-wave rectified waveform of figure 5.6(c). The R and C values are the same but the waveform in 5.10(b) is smoother because the pulses of rectified current are closer together.

Figure 5.9 full-wave rectification using a diode bridge.

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Figure 5.10 Current flow through the resistor in figure 5.9 (a) without a capacitor and (b) with a capacitor as in figure 5.9(c)

AMPLIFICATION The Triode Valve The triode valve (figure 5.2) was the first electronic device capable of amplification. If a relatively high voltage is applied to the plate, electrons emitted by the filament will be attracted to, and accelerated towards, the plate, so a current will flow. The grid can control the flow of current. If a (relatively small) negative voltage is applied to the grid, electrons will be repelled and the flow of current will be reduced. If a small positive voltage is applied to the grid, electrons will be attracted and the flow of current will be increased. The grid of the valve is placed closer to the filament than to the plate, with the result that very small changes in grid voltage produce very large changes in the current through the valve. The valve thus functions as an amplifier. If a small alternating voltage is applied to the grid, the result is a large fluctuation in the current flowing through the valve. Again the small voltage applied to the grid

Athough the grid, when positive, attracts electrons, most shoot straigh through the empty spaces in the grid and add to the current flowing between filament and plate.

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results in a large change in the current flowing through the triode. If a resistor is connected in series with the triode, the large fluctuations in current through the triode produce a large change in the potential difference across the resistor. A small alternating voltage applied to the grid produces large fluctuations in the voltage across the resistor. Thus the small signal is amplified.

The Transistor The transistor is the semiconductor equivalent of the triode valve. It is used today in preference to valves in almost all electrical equipment, either in the form of a discrete component or as a part of an integrated circuit. A transistor has no filament and hence no heating requirements, is much smaller than a valve and consumes less power. It is more suited to low power applications, and so is used almost exclusively in electronic stimulators and many other pieces of apparatus. Figure 5.11 shows a transistor and its circuit symbol. The transistor (like the triode valve) has three terminals - called the collector, base and emitter (abbreviated c, b and e in figure 5.11). The names were given to indicate that the emitter 'emits' electrons which are 'received' or collected by the collector. The base controls the flow of current between emitter and collector. The arrow in the circuit symbol for the transistor is used in the same way as for a diode (figure 5.3). It points in the direction of easy current flow. The base-emitter junction in fact behaves just like a diode: the resistance to current flow in the direction of the arrow is very low, the resistance in the opposite direction is extremely high. The resistance between collector and emitter can vary from very low to very high depending on the current flowing between base and emitter. This 'variable resistance' property gives the transistor its name. Transistor is an abbreviation of trans-resistor.

Figure 5.11 (a) a transistor and (b) its circuit symbol

The current flowing between the collector and the emitter is directly proportional to the base-emitter current. Thus if the base to emitter current is zero, the collector to emitter current is also zero. If a small current flows from base to emitter, a larger current can flow between the collector and the emitter. The collector current is always

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many multiples of the base current. The ratio (collector current/base current) is called the current gain (or amplification) of the transistor. The amplification of the transistor depends on how the transistor has been made, its size and other factors. Typical values of current gain lie in the range 50 to 500. In other words the resistance of' the transistor between collector and emitter decreases in proportion to the base current. As the base current is made greater the collector to emitter resistance decreases so that the collector current increases in proportion to the base current. The transistor is thus a very good current amplifier - if we pass a certain amount of current through the base-emitter junction a much larger current will flow from collector to emitter.

Operational Amplifiers It would be unusual nowadays to find a piece of electronic equipment built entirely from discrete components. Integrated circuits are now produced in huge numbers using automated fabrication techniques and this has reduced their cost to a point where it is, more often than not, cheaper to use one integrated circuit in applications where previously several individual transistors were used. One of the most common integrated circuits is the operational amplifier or op-amp for short. The operational amplifier is comprised of many transistors, resistors and capacitors fabricated in one tiny package with the components interconnected to produce an amplifier of very high gain. By adding a few external components the op-amp can be adapted to suit a variety of particular applications. Figure 5.12 shows an integrated circuit which contains four, independent operational amplifiers alongside the circuit symbol for a single operational amplifier.

Figure 5.12 (a) an integrated circuit containing four operational amplifiers and (b) the circuit symbol for an operational amplifier.

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All operational amplifiers require a power supply. For simplicity, this is not shown in figure 5.12. Of the 14 pins on the IC shown in figure 5.12, twelve are used for connection to the four op-amps and the remaining two are used for connection to a power supply. The operational amplifier has two inputs, the inverting input (labelled -) and the noninverting input (labelled +). When a signal is applied to the inverting input the output is a much amplified and inverted version of the input signal. Signals applied to the non-inverting input are amplified without being inverted. If the same signal is applied to both inputs, the output is zero.

An op-amp amplifies the potential difference between the inverting and non-inverting inputs.

The voltage amplification or gain of an op-amp is very high, typically in the range 106 to 1014. More often than not such high gains are not required in practical electronic circuits. The gain is easily reduced by adding a few external resistors. Figure 5.13 shows a practical op-amp circuit which acts as an inverting amplifier. Notice that we have again omitted the power supply connections for simplicity. The circuit is arranged so that signals are applied to the inverting input via resistor R1. The non-inverting input is connected to ground (earthed). A resistor (R2 ) connects from the output, back to the inverting input. This resistor will allow some of the output signal to feed back into the input. Note that the output is inverted with respect to the input. This means that the signal fed back will tend to cancel the input and so reduce both the input and output of the op-amp. The principle being used here is that of negative feedback. The gain (G) of the amplifier shown in figure 5.13 is given by the formula G =

R2 R1

For example, if R2 is 20 kΩ and R1 is 4 kΩ , the gain would be 20/4 = 5. Negative feedback can be used to reduce the gain of an amplifier to any desired value. For the circuit shown in figure 5.13, if the feedback resistor (R2) is one hundred times as big as the input resistor (R1 ) the gain is set at 100 times. If, instead, the feedback resistor was ten times as big as the input

RECTIFICATION AND AMPLIFICATION

Figure 5.13 An inverting amplifier.

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resistor (R1) the gain would be 10 times. The advantage of using external resistors and negative feedback is that the op-amp is then a general purpose device. Instead of having to produce a multiplicity of different op-amps, each with a particular gain, manufacturers need only produce one device which can be tailored to suit any particular application. An alternative amplifier arrangement is shown in figure 5.14. This shows a non-inverting amplifier. In this circuit negative feedback is once again used to set the gain. The input signal is applied to the non-inverting input and the inverting input is connected to ground by a resistor (R1 ). The inverting input cannot be connected directly to ground otherwise all the feedback current would flow to ground and not into the inverting input: there would be no feedback. The gain (G) of this circuit is given by the formula G = 1 +

R2 R1

Thus if R2 is 50 kΩ and R1 is 2 kΩ, the gain is 1 + 50/2 = 26. If the 50 kΩ resistor was decreased to 10 kΩ the gain would be reduced to 1 + 10/2 = 6. Notice that although different inputs are used for the signal to be amplified with inverting and non-inverting amplifiers, feedback is always applied to the inverting input to produce negative feedback and reduce the gain to a value determined by the ratio of two resistors. If the feedback was applied to the non-inverting input we would have positive feedback and the amplifier would be unstable. Most of us have experienced the effect of positive feedback when a microphone is moved too close to the loudspeaker of a public address system or the volume control is advanced too far. The circuit becomes unstable and an unpleasant howl is generated which can quickly damage the loudspeaker and/or amplifier and/or listener's ears! Used properly and carefully positive feedback can be of advantage in electronic

Figure 5.14 A non-inverting amplifier.

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circuits. We will consider an example later. As a general rule, however, positive feedback is not used in circuits whose sole purpose is to amplify.

PRODUCTION OF ALTERNATING CURRENT In chapter 2 we saw that a resonant circuit, consisting of an inductor and a capacitor connected in parallel, has a natural (resonant) frequency which can be calculated using the formula

If electrical energy (i.e. a pulse of current) is applied to the circuit, it resonates. That is, current flows backwards and forwards around the circuit and this alternating current has a particular frequency, the resonant frequency. As noted in chapter 2, by appropriate choice of the capacitor and inductor a resonant circuit can be made to generate any frequency of sine wave. A resonant circuit alone is not sufficient, however, to generate a sustained oscillation. To produce a continuous, steady alternating current we must arrange for the resonant circuit to be continuously supplied with energy to overcome the losses in the components and keep it oscillating. By use of an amplifier and positive feedback we can provide this energy. The circuit alongside shows one way of generating sustained oscillations using an operational amplifier with positive feedback.

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L1 and C1 form the resonant circuit and L2 is an extra inductor in close proximity to L1. The combination of L1 and L2 is, of course, a transformer. The oscillating current in L 1 will induce a current in L2 . The current induced in L2 produces an AC potential difference between the two input terminals of the op-amp. The output of the op-amp, which is an AC signal which is in synchronization with the AC in the resonant circuit, is fed back to the resonant circuit through R and this compensates for the natural energy loss and so keeps the resonant circuit oscillating. A problem with this circuit is that it is unstable. If the amount of feedback is too small, the oscillations will die-out. If the amount of feedback is too large, the oscillations will increase out of control. In practice it is impossible to have precisely the right amount of feedback to generate a steady, sustained oscillation. The problem is overcome by using a voltage controlled amplifier whose gain is controlled by negative feedback. Figure 5.15 shows how this is achieved. The AC potential difference across the resonant circuit is rectified and smoothed to produce a DC voltage which is directly proportional to the AC signal. This DC voltage is used to control the gain of the op-amp. If the AC signal increases, the DC voltage applied to the op-amp increases and its gain is reduced. This reduces the amount of feedback and the AC signal is reduced. If the AC signal decreases, the DC voltage applied to the op-amp decreases and its gain is increased. This increases the amount of feedback and the AC signal is increased. In this

Figure 5.15 A circuit for producing steady continuous AC

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way the AC signal is prevented from either decreasing or increasing appreciably.

Piezoelectric Crystal Oscillators As indicated in chapter 2, LC resonant circuits tend to drift slightly in frequency due to factors such as ageing and temperature change. When extreme frequency stability is needed, piezoelectric crystals are used. These crystals have the property that when a potential difference is applied to their opposite sides, the crystal resonates mechanically. When included in the circuit the crystal only permits current to flow when the frequency of the current is equal to the natural frequency of oscillation of the crystal.

positive feedback

In practice crystal resonators consist of a quartz wafer between two electrodes. The physical dimensions of the crystal determine the resonant frequency and if the crystal is maintained at a constant temperature a very high order of frequency stability can be obtained.

R

piezoelectric crystal

voltage controlled amplifier

rectifier with smoothing

negative feedback

Figure 5.16 shows a suitable circuit for the production of stable, high frequency AC using a piezoelectric crystal. The LC resonant circuit of figure 5.15 is replaced by a piezoelectric crystal which is connected directly to the voltage controlled amplifier.

Figure 5.16 A circuit for producing stable high frequency AC using a piezoelectric crystal

A circuit using a piezoelectric crystal can also be made to produce rectangular pulses over an extremely wide range of pulse durations and repetition rates.

RECTIFICATION AND AMPLIFICATION Extremely short pulse durations are required for computing and other applications. For reaction testing in electrotherapy, the shortest duration in use would not normally be less than 10 microseconds. For muscle therapy the pulse widths in use might lie in the range 20 microseconds up to a few seconds.

EXERCISES 1

(a) (b)

2

Draw the circuit symbol for a semiconductor diode. Include arrows to show the directions of high current flow (low resistance) and low current flow (high resistance). What is meant by the terms 'forward bias' and 'reverse bias' of a diode? State typical values for the resistance of a semiconductor diode when forward biased and reverse biased.

The circuit below is used to convert AC from the mains to DC of lower voltage.

(a) (b) (c) (d)

output

Draw a graph of the potential difference across the 1 kΩ resistor versus time. Why does this graph represent DC and not AC? How would the graph be changed if the diode was connected into the circuit with its terminals reversed? Describe (with the aid of graphs) the effect of connecting different size capacitors in parallel with the resistor.

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3

4

The circuit shown below is used to convert AC from the mains to DC of lower voltage.

(a)

Draw a graph of potential difference across the I kΩ resistor versus time.

(b)

In what way is this circuit more efficient than that shown in question 2?

Consider the circuit shown in question 3 above. (a) (b) (c)

5

Draw graphs to show the effect of different size capacitors connected in parallel to the 1 kΩ resistor. What is the advantage of full-wave rectification compared to half wave rectification as regards the size of capacitor needed to smooth the rectified waveform? Draw graphs to show the effect of removing one of the diodes from the circuit.

Consider an operational amplifier integrated circuit. (a) Draw a circuit diagram for an inverting amplifier with a gain of 20. (b)

describe two ways by which the gain of the amplifier in (a) above could be increased to 50.

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6

Figure 5.14 shows a circuit diagram for a non-inverting amplifier. (a) What would be the gain of this circuit if resistor R2 is 50 kΩ and R1 is 10 kΩ? (b) What value resistor would need to be used instead of the 50 kΩ resistor tor a gain of 8?

7

What is meant by the term 'positive feedback'? Why does positive feedback produce instability?

8

Consider the following amplifier circuits.

9

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(a)

Which is the inverting amplifier? Which is the non-inverting amplifier?

(b)

What is the gain (amplification) of each circuit?

(c)

If the 4 kΩ resistor was replaced by a 1 kΩ resistor, what would be the new value of the gain of each circuit?

(d)

Explain, in terms of positive and negative feedback, the arrangement of resistors in each circuit.

The circuit shown in figure 5.15 can be used to produce continuous sinusoidal

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RECTIFICATION AND AMPLIFICATION alternating current.

10

(a)

Explain why positive feedback is needed in this circuit. How is positive feedback produced?

(b)

Why is negative feedback needed? Explain how negative feedback is achieved.

Under what circumstances would the circuit shown in figure 5.16 be preferred to that shown in figure 5.15?

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6 Electric and Magnetic Fields Pulses or bursts of electric current, delivered at relatively low frequency, have a major effect on nerve and muscle because the current causes depolarization of the nervefibre membrane and production of action potentials. When sinusoidal alternating current (AC) is used, the physiological effects depend on the AC frequency. This is, at least in part, because each AC sinewave produces a positive and negative pulse of current. At higher frequencies (several tens or hundreds of kHz), the pulses of current in each sinewave are of short duration and the nerve fibre does not have time to respond to the positive pulse before the negative pulse cancels out its effects. In other words, the pulse duration is too short to produce an appreciable change in the nerve fibre membrane potential before the effects are reversed. Thus high frequency alternating current flow has no direct excitatory effect on nerve fibres. Current flow does, however, result in heating of tissue because electrical energy is converted to heat energy according to Joule's Law: P = V.I. Any flow of current will result in heat production. In this and the following chapter we consider how AC current flow induced in tissue by high-frequency alternating electric and magnetic fields produces heating of tissue. We also ask why certain tissues are heated more rapidly by high-frequency alternating electric and magnetic fields.

The French scientist, Arsenne d'Arsonval described, in 1893, passing a current of 1 Amp AC at a frequency of 500 kHz, through two human volunteers and an electric light bulb connected in series. The light glowed brilliantly but the volunteers felt nothing. Only when the current was increased to 3 Amperes did the subjects complain of a disagreeable sensation of heat.

Chapters 6 and 7 are both important for an understanding of the principles of diathermy. Diathermy (which literally translates as 'through-heating') refers to the heating of deeply located tissues; that is, the heating of tissues lying below both the skin and superficial fatty tissue. The practical difficulty involved in this form of therapy is that of minimizing the heating of superficial tissue while raising the temperature of the deeper structures by a therapeutically useful amount. The objective, then, is to produce a greater increase in temperature in the tissue layer to be treated. By varying the dosage, the desired amount of tissue heating can be chosen. We begin our coverage of diathermy using electric and magnetic fields with some basic physical principles associated with these fields.

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STATIC ELECTRIC FIELDS It is common knowledge that there are two kinds of electric charge: the negative charge associated with electrons and the positive charge associated with the nucleus of an atom. We also know that like charges repel each other and unlike charges attract. The quantitative relationship between the forces between charges and their magnitude and distance apart was first determined by Charles Augustin de Coulomb in 1785. The relationship is summarized in what we now call Coulomb's law which states that the force between two charges is directly proportional to the magnitude of each charge and inversely proportional to the square of their separation. Coulomb's law is expressed mathematically as: F α

q1.q2 r2

.... (6.1)

where F is the force experienced by two charges q1 and q2 and r is their distance apart. If we wish to calculate the force on a charge q due to a large number of charges q1, .... qn, we could achieve this by using equation 6.1 and calculating the force on q due to q 1 , the force on q due to q2 and so on and then summing the forces to obtain the resultant force. Since forces are vector quantities the summation must be by vector addition. In practice we do not often come across situations which approximate to two point charges nor even to simple distributions of charges. More often we encounter charges spread over surfaces of different shapes where the actual charge distribution

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is unknown (and in practice difficult to measure). Under these circumstances it is difficult to calculate the force on another charged object. It is usually more convenient and easier experimentally to measure the magnitude and direction of the force on a 'test charge' placed in the field of influence of the charged object. The concept of a field of influence, the electric field, is an extremely useful one because it is possible to measure forces in the field without any prior knowledge of the arrangement of charges causing the field.

By moving a small positive test charge around in an electric field and measuring the force at numerous points, the electric field can be mapped-out.

We define the intensity at any point in an electric field, E, to be the force per unit positive charge at that point. This defines the magnitude of the field and its direction. If a small test charge is placed in a field the magnitude of the force it experiences together with the direction of the force determine the electric field intensity at that point. The field intensity is thus a vector quantity given by the relationship E =

F q'

.... (6.2)

Where q' is the magnitude of the test charge placed in the field. Since the units of force are Newtons and the units of charge are Coulombs, the field intensity has units of Newtons per Coulomb (written N.C-1). The field surrounding a single positive charge q can be obtained from equations 6.1 and 6.2 as E =

q F = 2 r q'

.... (6.3)

Where E is the magnitude of the field and its direction is the same as that of the force (in a straight line between the charges in this case).

ELECTRIC AND MAGNETIC FIELDS Figure 6.1 shows the pattern of the electric field around point charges and pairs of charges. Figure 6.1 Fields of point charges (a) single positive charge (b) single negative charge (c) unlike pair of charges (d) like pair of charges

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The lines drawn in figure 6.1 indicate the direction of the field and are called lines of force because they indicate the direction along which the electric force acts at any point. An important point to note is that electric fields are always produced by separation of charges. Electrons are removed from their parent atoms to produce the charge separation so that for every positive charge that is produced, there is a corresponding negative charge somewhere. The electric field lines thus have their origin on charges and can only terminate on opposite charges.

Positive charge will move along the lines of force in the same direction as the field, negative charge along lines in the opposite direction.

By convention we indicate the relative strengths of electric fields by the number of lines of force going through a unit area perpendicular to them. In figure 6.1 the field lines are closest together near the point charges so the field intensity is greatest there. In figures 6.1(a) and (b) it is apparent that at greater distances from the point charges the field line density is reduced meaning that the field intensity is correspondingly decreased. Another point worthy of note is that lines of force represent vector directions. It follows that no two lines of force can ever cross each other - the fields at the point would simply add by vector addition. This is just another way of saying that a point charge placed in a field will only move in one direction it cannot be pushed in either of two directions.

THE FIELD BETWEEN CAPACITOR PLATES Consider the case of two large, flat metal surfaces each carrying opposite charges and parallel to each other. The metal plates are good conductors so charges are able to move freely in response to electric forces. On one plate we have an accumulation of positive charges and on the other an accumulation of negative charges.

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In this situation electric field lines will originate on the positive plate and terminate on the negative plate. The field lines so produced will originate perpendicular to the conductor surfaces (this is illustrated in figure 6.2). It is generally true that at the surface of a conductor through which no charge is moving, the field lines are perpendicular to the conductor surface. If this were not so the lines of force would have a component tangential to the conductor surface. The effect of the tangential component would be to move charges, so changing the field. The charges would continue to move and readjust the field until there were no longer any field line components tangential to the surface causing them to move. In insulators, where the charges are not free to move, this does not apply and the field lines may be at any angle to the insulator surface. Figure 6.2 shows the pattern of electric field lines between two capacitor plates. The field lines originate perpendicular to one plate and terminate perpendicular to the opposite plate. If the surfaces of each metal plate were infinitely large, the field lines would all be straight and uniformly spaced - in other words the field intensity would be the same at all points, not growing weaker at large distances from the centre of the plates. In practice the plates cannot be infinitely large and the field lines curve away from each plate at the edges. When the plates are close together as in figure 6.2(a) the field is relatively uniform - only the outermost areas (near the edges of each plate) show any curvature, or weakening, of the field pattern. In 6.2(b) the plates are more widely spaced and more weakening of the field is evident. Only in the central region is the field uniform. Clearly, to obtain the most uniform field intensity it is desirable to have large capacitor plates separated by a relatively small distance.

Figure 6.2 Electric field between parallel, equal size capacitor plates.

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Figure 6.3 shows the field patterns expected for different sizes and arrangements of electrodes.

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When interpreting electric field patterns it is important to remember that not only is the field direction shown by the lines of force but that the field intensity is by convention indicated as the number of lines per unit cross section area. Imagine now a small positive charge placed in the field between two capacitor plates. The charge will experience a force in the direction along the field lines and hence will accelerate in this direction, thus gaining kinetic energy. The gain in kinetic energy is offset by a loss of (electrical) potential energy. In other words the charge will lose potential energy as it moves along field lines. The change in potential energy per unit positive charge is called the electric potential difference between the points. This quantity is the same potential difference we have already met in chapter 1 and the unit is, of course, the volt. A region of positive charge is a region of high electric potential and a negative charge region is at a low potential so that going from a positive to a negative region involves a potential drop, i.e. a negative potential difference. The electric field also goes from positive to negative so that moving along a field line in the direction of the field involves travelling across a potential drop. The idea of charges losing potential energy as they move in an electric field gives us another way of talking about electric field intensity. Clearly an intense electric field will be associated with a large potential difference between two points a certain distance apart. A weak field has a lower potential difference for the same distance. The field intensity can thus be specified in units of potential difference per unit distance: in other words in volts per metre (V.m-1). The units of volts per metre are identical to the units of field intensity mentioned earlier, namely newtons per coulomb.

STATIC MAGNETIC FIELDS The basic groundwork for our understanding of magnetic phenomena was laid around 1820 by André Marie Ampère when he first studied the forces of attraction and repulsion between wires which are carrying current. Before these pioneering

Saying that the electric field intensity is 100 newtons per coulomb is the same as saying the intensity is 100 volts per metre.

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experiments magnetism was regarded as an isolated and puzzling phenomena associated with a few naturally occurring (magnetic) materials. What Ampère did was to establish the first links between electricity and magnetism: essential groundwork which culminated in the development of modern electromagnetic theory. Ampère observed that if two loops of wire were mounted so that each had one side in close proximity to a side of the other loop, a force was produced when a current was made to flow in each loop. The arrangement is shown in figure 6.4. Ampère found that when current was passed through each loop in the same direction the loops were attracted to each other - this could be seen by movement cf the pivoted loop. If the current in one loop was reversed the force was also reversed - a repulsive force was produced. He also noted that if current was made to flow in only one loop, no force was experienced by either loop. Experiments can be performed to ascertain what effect the size of the current in each loop has on the force and what effect have the loop separation, r, and the length of wire, L. The results are summarized in what we now know as Ampere's law for the force between two parallel conductors (equation 6.4).

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I1.I2.L

.... (6.4)

r

In comparing this with Coulomb's law of electrostatics, (equation 6.1) we can see some similarities and some differences. The table below summarizes the corresponding features of each kind of force. Electrostatic Force * * * * *

between two charges. repulsion of Iike charges. depends on size of charges. is proportional to 1/r2 . force in a straight line between charges.

The two most striking properties of the magnetic force which contrast with the electrostatic case are the attraction of 'like' currents, i.e. currents in the same direction (figure 6.5) and the line of action of the magnetic force perpendicular to the direction of current flow.

Figure 6.5 Forces between two currentcarrying wires.

Magnetic Force * * * * *

between two currents attraction of 'like' currents. depends on size of currents. is proportional to 1/r. force at right angles to direction of current.

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Inspection of figure 6.5 leads to the prediction that two current carrying conductors placed at right angles will experience no force. This is borne out experimentally. Remember that the electric field near charged bodies could be mapped by placing a test charge in the field and observing the direction of the force on the charge - let us now consider the relationship between magnetic fields and forces. The use of a compass or iron filings to visualize the magnetic field around a bar magnet is an experiment which most of us have carried-out. The compass or iron filings line up in the direction of the magnetic field and the field lines are readily apparent. The same technique can be used to map the field lines around a wire carrying an electric current. Figure 6.6 shows the sort of pattern which is obtained. It was Ampère himself who first thought of obtaining a more intense magnetic field by passing current through a wire wound in a closely spaced spiral. Figure 6.7 shows the magnetic field expected for a single loop of wire and a number of loops wound as a solenoid.

Figure 6.6 Magnetic field lines around a wire carrying current.

Figure 6.7 Magnetic field lines around a loop and a solenoid.

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Now let us return to consider what is happening to a small element of current in a wire placed near to the straight wire in figure 6.6. Figure 6.8 shows the main features of such an arrangement. Here we have the most striking feature of magnetism: that the force on a moving charge is perpendicular to both the current and magnetic field directions. This observation stands in stark contrast to our experience with gravitational and electrostatic fields, where the field direction is also the direction of the force experienced by the mass or charge. The scope of this book precludes our delving into theories of the origin of magnetic force: its explanation was a triumph for relativity theory in modern physics, when the magnetic force was shown to be simply a result of the movement of charges with their associated electric field. The theory indicates that the force between charges in motion is slightly larger than that between stationary charges. This slight increase in the electric force is, in fact, the 'magnetic' influence of moving charges on one another. Although we have, so far, only discussed electric current in the form of charges flowing through wires, what we have described also applies to isolated charges moving in a vacuum. A good practical example of this is in the television set. Here a stream of electrons emitted from a glowing filament (the cathode) is accelerated by an electric field and impinges on the screen. The phosphor coating on the screen is utilized to convert the kinetic energy of the electrons to light energy and a visible spot is produced on the screen. The electron beam is deflected rapidly both horizontally and vertically by magnetic fields produced by coils or solenoids attached to the neck of the picture tube. By applying rapidly alternating pulses of current to the coil, impulsive forces are applied to the stream of electrons and the spot is moved accordingly.

Figure 6.8 Forces on a current element in a magnetic field (B).

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In summary it must be remembered that the movement of charges is central to our understanding of magnetic phenomena. The movement of charges results in a magnetic field and a static magnetic field can only have an influence on a charge when that charge is in motion. Motion at right angles to the field results in a maximum force. Movement of charges along a magnetic field line will result in no force being exerted.

The size of the magnetic force depends on sin(θ ) where θ is the angle between the field line and the direction of charge movement.

Since we know that movement of charge in a static magnetic field is necessary for a force to be exerted on the charge, we might logically ask such questions as whether the charges in a conductor will experience a force if the conductor is moved in a magnetic field, or whether a force is exerted on a stationary charge by a moving or changing magnetic field. We will discuss the answers to these questions later in this chapter.

ELECTRIC FIELDS AND DIELECTRICS In talking about the electric field between charged bodies and the magnetic field associated with moving charges we have not considered what effect, if any, the surrounding medium has on the fields in each case. Let us consider the electrostatic case first. When a material is given an electric charge, the charge may remain localized in the region of generation for some considerable time or it may spread over the surface almost instantaneously. In the first case the material is called an insulator or dielectric, and in the latter case a conductor. An ideal insulator would retain a charge indefinitely and materials such as glass, plastics and waxes come very close to this ideal. Pure metals such as gold and copper are close to being ideal conductors. Of course the line of demarcation between conductors and insulators is not sharp and many substances, particularly

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those of biological origin, are neither good conductors nor good insulators.

Dielectric Constant It was found by Henry Cavendish, and later independently by Michael Faraday, that the capacitance of a capacitor - its ability to store charge - can be increased by placing a dielectric material between the plates. If Co is the capacitance when measured in a vacuum and C the capacitance when the region between the plates is filled with a dielectric, the ratio of C to Co, is found to be independent of the shape and size of the capacitor. This ratio is dependent only on the dielectric medium itself and is called the dielectric constant (ε) of the material. We thus have C

ε = C o

.... (6.5)

Capacitance is defined as the amount of charge the capacitor will acquire (and store) for each volt of applied potential difference. Mathematically this is written: C =

q V

.... (6.6)

where the symbols have their usual meaning. This equation tells us that if the capacitance is changed by inserting a dielectric between two capacitor plates either the charge will be increased or the potential will be decreased. If the capacitor plates are connected to a battery or other power source capable of maintaining a constant potential difference then the charge on the plates will be increased. The increase in capacitance is very small in the case of gases at normal pressure, but for materials such as oil the capacitance is doubled. Values of ε determined in this way range from very close to 1 for gases up to 81 for pure water. Table 6.1 gives values of ε for a representative range of materials.

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Notice that the dielectric constant of petroleum oil is similar to that of polythene. Although these materials are quite different in some properties, most notably physical appearance, they have the same basic molecular composition. Each is made up of a long hydrocarbon chain. The chains in polythene are longer and the subunits have many chain crosslinks but the elemental compositions of the two materials are very similar. Compare these with ethyl alcohol and then water. The dielectric constant is closely correlated with an intrinsic property of the molecules making up a substance - the polarity. Of the substances in table 6.1 the most polar is water, then ethyl alcohol. The principal constituents of air, namely oxygen and nitrogen, are non-polar. The remaining substances in the table are of intermediate polarity. We consider the effect of dielectrics on an electric field next but it is worthwhile to pause and note at this point the distinction between insulating properties and dielectric properties. The terms insulator and dielectric are used almost synonymously to describe a particular group of materials - those with poor conduction properties. The terms 'good insulator' and 'good dielectric', however, mean two different things.

Table 6.1 Dielectric constants of materials

*

A good insulator is a substance which offers a very high resistance to the flow of current. This refers to how easily charges can move through a material and says nothing about the dielectric constant.

*

A good dielectric is a substance with a high dielectric constant. This depends on the polarity of the molecules in the substance, not on its conduction properties.

Though good dielectrics must be reasonably good insulators the best dielectrics are not necessarily the best insulators. For example water in table 6.1 is the substance

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with the highest dielectric constant. Although pure water is a reasonably good insulator the other substances in this table are all better insulators.

Dielectrics In An Electric Field Consider what happens when we put a material made up of polar molecules between two charged capacitor plates; that is, in an electric field. For simplicity we assume that the polar molecules are dipoles having a single 'slightly negative' region and a corresponding 'slightly positive' region. In the absence of a field the molecules will be randomly aligned as shown in figure 6.9(a). In an electric field the dipolar molecules will-align themselves as shown in figure 6.9(b) with the positive and negative ends pointing in directions along the field lines. The material is then said to be polarized. The alignment of the molecules has an effect on the applied electric field. This is because each dipole has its own field comprised of field lines which spread from the positive end of the molecule to its negative end. The field of a single dipole is shown in figure 6.10. Although the field around a single dipolar molecule is extremely weak, when a dipolar material is polarized and the molecules are aligned as in figure 6.9(b) the net effect of all of the dipoles can have a significant effect on the externally applied field.

Figure 6.9 (a) Unpolarized dielectric and (b) polarized dielectric

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Consider first the effect of a single dipole on the external field. The central field line in figure 6.10 points in the direction of the externally applied field (from left to right). The remaining field lines, however, curve around above and below the dipole and the arrows point in a direction opposing the external field (from right to left). The net result is that these field lines cancel part of the externally applied field and so weaken it. When we take into account the fields surrounding all of the dipoles in a polarized material the net result is a weakening of the externally applied field within a dielectric. The field inside a dielectric will be less than the externally applied field: this is because each of the dipoles aligns to produce a field opposing the externally applied field. Why then should there be any effect with 'non-polar' molecules such as oil or bakelite? The origins of the effect with these materials can best be understood by considering a single non-polar molecule such as the one shown in figure 6.11. In the absence of a field the electron 'cloud' around the molecule is symmetrical: the molecule is non-polar. If now an electric field is applied the electron cloud can distort giving rise to an induced dipole. The molecule becomes polarized and will remain so as long as the external electric field is maintained. Non-polar molecules will polarize and so have the same effect on an externally applied field as polar molecules. The dipole field produced will oppose the external field and so weaken it. Generally the effect of non-polar molecules on an electric field is not as great as those of polar molecules. This is because the induced dipoles are not as strong as those of naturally polar substances. We thus have the general result that the field within any dielectric is less than the field that would exist without the dielectric being present. The next question we must ask is how the pattern of field lines is affected.

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Figure 6.10 Electric field of a dipole

Figure 6.11 Polarization of a molecule in an electric field

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For simplicity we take the single dipole shown in figure 6.10 and consider what happens when it is placed in a uniform field. The original field must be added (vectorially) to that of the dipole. The result is the field pattern shown in figure 6.12. The most significant thing to note from figure 6.12 is that the field lines converge towards the dipole. The field is made more intense immediately in front of the faces perpendicular to the field direction. Above and below the dipole the field lines are more spread out; that is, the field here is weaker. If now we wish to consider a large object of high dielectric constant in an electric field we can apply the same ideas.

Figure 6.12 The effect of a dipole on a uniform electric field

Take the case of a simple cube of dielectric. We can add the contribution to the field from each dipole using the arrangement shown in figure 6.9(b). The result will be similar to figure 6.12 but now we have many dipoles joined head to tail and stacked above and below each other. The net result is shown in figure 6.13. Again we observe the field lines converging towards the dielectric, so increasing the field intensity near faces perpendicular to the field. The field intensity near the remaining faces of the cube is correspondingly decreased. Figure 6.13 also shows why, even though field lines converge on the cube, the field within the dielectric is weak. It is because some field lines terminate on charges on the surface of the dielectric. The origin of the surface charge can be seen by inspection of figure 6.9(b). Within the polarized dielectric, dipoles are aligned in a head to tail pattern so that for each positive 'head' there is an

Figure 6.13 Effect of a dielectric cube on a uniform electric field

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adjacent negative 'tail'. The opposite charges so cancel each other. On the faces perpendicular to the field, however, there is a charge imbalance. One surface has an excess positive charge and the opposite surface has an excess negative charge. It is on this surface charge of polarization that field lines will terminate. The amount of surface charge produced in an electric field will depend on the dielectric constant of the material. For a material with a high dielectric constant the surface charge of polarization will be high and a large proportion of field lines will terminate on the dielectric surface. When the dielectric constant is low the proportion of field lines which terminate will be small. This means that the field within a dielectric is reduced in proportion to the dielectric constant. The higher the dielectric constant, the lower is the field within the material. In the limiting case where the dielectric constant is extremely high the field within the material is close to zero. Figure 6.14 shows what happens when either a spherical or cylindrical dielectric is placed in an electric field. Note the regions in which the field is intensified (more field lines per unit area) and where the field is reduced.

Figure 6.14 Effect of a dielectric sphere on a uniform electric field

REFRACTION OF FIELD LINES Inspection of figures 6.13 and 6.14 reveals certain discontinuities. Two things are apparent. First we notice that some field lines terminate at the dielectric interface. Second we notice that the lines which do not terminate impinge on the interface at one angle and leave it at a different angle. The change in direction at an interface is called refraction. Field lines are refracted at a dielectric interface. The phenomenon is not restricted to dielectrics but also applies to conductors. In this section we consider how the dielectric constant and conductivity of materials determines the extent of refraction at an interface.

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Refraction and Dielectrics First let us consider the refraction of field lines as they pass from one dielectric medium to another. We ask the question 'what are the constraints on field lines crossing a dielectric boundary?' There are two constrains, or boundary conditions, on the electric field at an interface. The first is that the tangential component of the field (the component parallel to the boundary) should be continuous. That is, the tangential component of the field must be the same on each side of the boundary. This is illustrated in figure 6.15. The tangential component of E 1 is E1 .sinθ 1 and the tangential component of E2 is E2.sinθ2. These are equal so we have E1.sinθ1 = E2.sinθ2

.... (6.7)

The second boundary condition is that the normal component of the field in each dielectric (the component at right-angles to the boundary) is decreased in proportion to the dielectric constant. We have seen previously that the field within a dielectric is decreased. The second boundary condition states this precisely. Since the normal component of the field is decreased in proportion to the dielectric constant the product of dielectric constant and the normal component of the field is the same in each medium. Mathematically this is written

ε1.E1.cosθ1 = ε2.E2.cosθ2

.... (6.8)

Figure 6.15 Refraction of an electric field line at a boundary between two dielectrics

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Dividing equation 6.8 by equation 6.7 and cancelling the E1s and E2s, we obtain cosθ1

ε1. sinθ or

1

cosθ2 = ε 2. sinθ 2

ε1.cotθ1 = ε2.cotθ2

.... (6.9)

An Example. Suppose that in figure 6.15, medium 1 is air with a dielectric constant of 1.0 and medium 2 is water with a dielectric constant of 81 (table 6.1). The field line shown for E1 strikes the boundary at an angle θ 1 of 20o . We wish to calculate the angle θ2 at which the field line leaves the boundary. Rearranging equation 6.9 we have

ε1 1.0 1.0 .cot20o = .2.75 = 0.0399 cotθ2 = ε .cotθ1 = 81 81 2 which gives

A similar calculation for an incident angle θ1 of 10 o gives an angle of refraction θ2 of 86 o.

θ2 = 88o For an angle of incidence θ1 of 20o, the angle of refraction, θ2, is predicted to be 88o. In this example the angle of refraction is always considerably greater than the angle of incidence. This is because medium 2 has a much higher dielectric constant than medium 1. The implication is that field lines are refracted greatly on entering a medium of high dielectric constant. For a moderate angle of refraction the angle of incidence must be very small: in other words the field lines must enter the medium of high dielectric constant almost at right angles.

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Dielectrics Between Capacitor Plates Now consider what happens when we place a material of high dielectric constant between two capacitor plates. The result is shown in figure 6.16. To explain the effects shown here we must remember the main points of the preceding discussion. Two important effects are evident in figure 6. 16. An increase in field strength between each plate and the dielectric surface. This is predicted by equation 6.6. The capacitance is increased in the presence of a dielectric. This means that the charge on the plates must increase and consequently the field strength between the plates and the dielectric must increase if the potential difference between the plates is kept constant. A decrease in field strength within the dielectric. Two factors contribute to this effect. The first is the surface charge of polarization which allows field lines to terminate on the dielectric surface, so reducing the field intensity within. The second is the refraction of field lines at the dielectric interface (equation 6.9). Notice that the field lines in figure 6.16 enter the dielectric with a small angle of incidence and are refracted greatly. Refraction spreads the field lines and so reduces the field in the dielectric.

Figure 6.16 Change in the electric field pattern in the presence of a dielectric.

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Refraction and Conductors So far we have talked about dielectrics in an electric field. We dealt with dielectric properties alone and made no mention of the effect of conductivity on the electric field pattern. In the case of good insulators such as plastics, glass and oil it is reasonable to treat them as ideal, non-conducting dielectrics. In the case of water and materials containing water, this approximation cannot be made. Hence for biological materials which generally have a high water (and ion) content we must also consider the effect of conductivity on the electric field pattern. Here then we consider the effect of conductors on an electric field. We will not be so concerned with the ideal or perfect conductors, which must have field lines perpendicular to the conductor surface, but with non-ideal conductors which are much more commonplace in biological and in everyday situations. For ideal conductors two conditions apply: *

the field lines must be perpendicular to the conductor surface.

*

all field lines must terminate on the surface of the conductor. In this sense an ideal conductor can be considered to be 'infinitely polarizable'. This means that the field within a perfect conductor is zero.

For non-ideal conductors these constraints do not apply. What then are the properties of a non-ideal conductor and what are the laws governing its behaviour? We have already met one of the properties of a conducting medium in a previous chapter. This is Ohm's law which states that the current in a conductor is proportional to the potential difference between the conductor ends. The constant of proportionality is 1/R where R is the resistance. Ohm's law was covered in chapter 1.

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The field within a non- ideal conductor is not zero, nor do all the field lines terminate on the conductor surface.

146

Mathematically, Ohm's Law is written: V I = .... (1.3) R When we are interested in the bulk flow of current within a material it is more convenient to use a different expression for Ohm's law. The alternative expression is E i = ρ or i = σ.E .... (6.10) Where E is the electric field intensity in volts per metre and i is the current density in amps per square metre passing through a surface perpendicular to E (figure 6.17). For an area A, with a current I flowing through it, the current density is I/A amperes per square metre (A.m-2). The resistivity, ρ, is defined as the resistance of a one metre length of conductor with a cross sectional area of one square metre. The unit of resistivity is the ohm.metre. The conductivity, σ, is simply the reciprocal of resistivity and thus has units of (ohm.m-1) which are called siemens per metre (S.m-1). We now ask what happens to the electric field lines (and hence the direction of current flow) on passing from one conductive medium to another. Figure 6.18 shows the refraction which occurs when a field line crosses to a medium of higher conductivity. The angle of refraction is greater than the angle of incidence in this case. The conditions applying to the electric field lines are analogous to those which apply in dielectric media. When a field line crosses into a medium of higher conductivity, the angle of refraction is greater than the angle of incidence. Conversely, when a field line crosses into a medium of lower conductivity, the angle of refraction is less than the angle of incidence.

Figure 6.17 Electric field and current in a conductor

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Compare figure 6.18 with figure 6.15. The first condition on the electric field is that the tangential components of the field be equal on both sides of the boundary i.e. E1.sinθ1 = E2.sinθ2

.... (6.11)

The second condition is that the flow of current normal to one side of the boundary should be equal to that normal to the other side i.e. i1.cosθ1 = i2cosθ2 This makes sense since the net current flow into the boundary from side 1 must equal that entering side 2 from the boundary. Using equation 6.10 to eliminate i1 and i2 from this last equation we have: σ1.E1.cosθ1 = σ2.E2.cosθ2

.... (6.12)

Figure 6.18 Refraction of electric field line at a boundary between two conductors.

Combining 6.11 and 6.12 we obtain: σ1.cotθ1 = σ2.cotθ2

.... (6.13)

Compare these equations with 6.7, 6.8 and 6.9. The close analogy between current flow in conductors and polarization in a dielectric is quite evident from these equations. The implications of equation 6.13 for conductors are similar to those we found using equation 6.9 which applies to dielectrics. Electric field lines are refracted significantly on passing from a poor conductor to a better conductor.

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Equation 6.13 predicts that when the conductivity of medium 2 is extremely high (when medium 2 comes close to being an ideal conductor) the field lines in medium 1 will enter the boundary between the two conductors almost normally (almost at right angles). In other words, if material 2 is an ideal conductor the field lines in medium 1 will impinge on the boundary at an angle of 90o. The similarity of equations 6.9 and 6.13 lead to the general conclusion that field lines are refracted greatly when entering a medium of high dielectric constant or conductivity. To obtain an equation which takes account of both factors simultaneously the dielectric constant and conductivity of the media must be combined to calculate their electrical impedances. The amount of refraction then depends on the impedance of each medium and the refraction equation has a similar form to equations 6.9 and 6.13. It turns out that for biological materials there is a good correlation between the insulating properties (hence the conductivity) and the dielectric properties (dielectric constant). Tissues with high dielectric constant are poor insulators. In general, the higher the value of ε, the higher the value of σ. Hence if both of ε and σ increase by a factor of 10, say, on going from one medium to another equations 6.9 and 6.14 predict the same relationship between angle of incidence and angle of refraction. In this case both equations correctly predict the angle of refraction and the complexity of combining ε and σ in a single equation is avoided.

Electric Fields in Tissue Table 6.2 gives values of dielectric constant and conductivity for some body tissues measured at 37oC. Also included for comparison are data for water, oil and metals. The fields used in therapy are not static but alternate at very high frequencies (normally 27.12 MHz). At high frequencies the dielectric constant is lower and the conductivity higher than in the electrostatic case.

The reason that high σ values accompany high ε values is that tissues of high water content have a high ion content. A high water content results in a high ε, a high ion content results in a high σ.

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The results in table 6.2 were obtained with an alternating electric field of frequency 50 MHz the values of conductivity would be about 20% lower at 27 MHz (the frequency normally used for therapy) and the dielectric constant some 20% higher. Both quantities depend quite strongly on the frequency of alternation of the field. From table 6.2 it is apparent that at this frequency tissues with high water content (muscle, spleen and kidney) have very similar dielectric and conduction properties to saline solution. This is not surprising as the tissues contain about 80% by weight of water with the remaining 20% being tissue protein. The lower water content of fatty tissue and bone marrow is also reflected in their dielectric constant and conductivity values. Now let us apply what we have found so far to a more practical situation and ask what the pattern of field lines would look like for a patient's arm or leg placed between circular capacitor plates. Figure 6.19 shows a highly simplified representation of the arm or leg in (a) longitudinal and (b) perpendicular cross-section.

Table 6.2 Dielectric constants and conductivities at 37o C and 50 MHz.

It can be seen that the field intensity is highest in the air space and decreases markedly on entering the fatty tissue. A further reduction occurs when the field lines enter the muscle.

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In figure 6.19(a) the reduction in field intensity within the arm or leg is due to two factors: (a) refraction of field lines at the air/fatty tissue and fatty tissue/muscle boundaries which spreads the lines apart, so reducing the field intensity and (b) termination of some of the field lines on surface charge of polarization of the fat and muscle.

150

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Note that in figure 6.19(a), in bone there are more field lines than in muscle. This is because bone has a lower dielectric constant and conductivity than muscle and so does not polarize to the same extent. Its electrical properties are similar to fatty tissue. Figure 6.19(b) also shows the effects of refraction and termination of field lines. We will return to consider the implications of these field patterns. For the moment it is sufficient to note the diminished field intensity in materials of high dielectric constant due to the polarization of the material and consequent termination of some field lines. Note also the refraction of field lines on entering a material of high dielectric constant or conductivity which can result in a focussing of the field (figure 6.19(b)) or defocussing (figure 6.19(a)).

MAGNETIC FIELDS IN MATERIALS We saw previously that the dielectric constant of a material could be measured by the increase in capacitance which occurs when the material is placed between the plates of a parallel plate capacitor. An analogous situation is found with magnetism. When different materials are placed along the axis of a solenoid or inductor (figure 6.7), the inductance changes. There is also a proportionate change in the strength of the magnetic field around the inductor. The effect can be seen in the dramatic increase in magnetic field strength which occurs when a bar of iron is placed in a solenoid to make an electromagnet.

Iron-cored electromagnets can generate fields up to several thousand times as strong as would be produced without the iron core present.

Iron is, however, a member of a small group of elements which show such an effect on the magnetic field. Nickel and cobalt are two others. The materials in this group are said to be ferromagnetic. An extremely small effect on the magnetic field is found with other substances.

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By analogy with the electrostatic case we define the permeability of a material by its ability to change the inductance of a solenoid. The permeability, µ, is the ratio of the inductance L, to the inductance without the material being present, Lo, i.e. µ =

L Lo

.... (6.14)

The permeability of most materials is found to be extremely close to unity. Unlike the electrostatic case (where ε is always greater than unity) µ may be less than or greater than unity.

Substances with µ less than one are termed diamagnetic while substances with µ greater than one are called paramagnetic.

Table 6.3 lists values of µ for a number of different materials. Certainly for biological materials we can consider µ to be sufficiently close to one to make no difference. In other words the magnetic field lines will be virtually unchanged on passing through biological materials: tissues are 'transparent' as far as the magnetic field is concerned. Ferromagnetic materials will influence a magnetic field in the same way that dielectrics influence an electric field. While an electric field will polarize a dielectric a magnetic field will magnetise a ferromagnetic material. The analogy could be carried further with discussion of the field in and around a ferromagnetic material. Figures 6.13 and 6.14 would be appropriate if we replaced the electric field E with the magnetic field B and substituted µ for ε. However since the effects are negligible for biological materials we will not pursue this topic further.

Magnetic Fields and Induced EMF We now return to the questions left unanswered earlier and further examine the effect of magnetic fields on charges. We have already considered electric charge moving in a magnetic field and found that the

Table 6.3 Permeability of various materials.

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charge experiences a force in a direction perpendicular to both the magnetic field and the current direction (figure 6.8). This applies both to current flowing along a wire, as in figure 6.8 and to current flow in a vacuum or near vacuum, as in a television picture tube. Consider the following experiments which can be performed with two loops of wire. Figure 6.20 shows the arrangement of the apparatus which is needed. Loop 2 is connected, through a switch, to a battery. Loop 1 is connected to a sensitive current meter or galvanometer. If the switch were in the closed position, current is flowing through loop 2 and a magnetic field exists. Since the electrons in loop 1 are not moving, we know from Ampere's law that there is no force on them and hence no force on the loop. What happens if now we move loop 1 away from loop 2? In moving the loop upwards we find that the galvanometer deflects indicating a flow of current in loop 1. As soon as we stop moving loop 1, the current flow ceases. If we use the three dimensional axes of figure 6.8 and regard the direction of movement of the loop as the current direction we predict a force on the electrons in loop 1 in a direction along the wire. This then is the explanation for the induced current in loop 1.

Figure 6.20 An experiment with moving wire loops.

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If now we fix loop 1 in figure 6.20 and instead move loop 2 downwards what is the effect? In this instance instead of moving electrons through a magnetic field we have left the electrons alone and moved the field. The net result is the same as we found in the first example: the galvanometer deflects indicating a flow of current in loop 1. Clearly it is only the relative motion of the conductor and field which is important in determining whether current is induced. When moving the conductor in a fixed field or moving the field with the conductor fixed, the essential process occurring is that charges are crossing magnetic field lines. Whenever this happens the charges experience a force. What happens then if we keep both loops fixed and suddenly switch off the current in loop 2? As far as loop 1 is concerned one of two things could have happened. Either the magnetic field disappeared because the current causing it was stopped or the loop responsible for the field was suddenly accelerated away from the vicinity. The net effect is the same - current is induced in loop 1 because of the changing magnetic field.

v in the diagram above is the direction of movement either of the field, the wire or electrons (current).

The observation can be explained by picturing the magnetic field collapsing on loop 2. With current flowing in loop 2 a magnetic field, represented by concentric circles around the wire, is present (figure 6.7). When the current is switched off the circular field lines can be visualized as shrinking; converging on the wire and disappearing into it. Thus the field direction is in concentric circles but the direction of movement of the field is radially inwards towards the wire of loop 2. This is illustrated in figure 6.21.

Figure 6.21 Magnetic force acting on charges when current in in loop 2 is switched off.

ELECTRIC AND MAGNETIC FIELDS Closing the switch in figure 6.20 causes current to flow in loop 2 and a field builds up around the wire. The increasing field is represented by a series of expanding concentric rings emanating from the wire. The direction of increase of the field is radially outwards (opposite to v in figure 6.21). Experimentally we observe a flow of current in loop 1, in the opposite direction to that when the switch was opened (i.e. F is reversed when v is reversed in figure 6.21) Once the current flow is steady, the magnetic field is constant and current is no longer induced in loop 1. Now consider what happens if, instead of switching the current in loop 2 of figure 6.20 on or off, we pass an alternating current through it. By the principles outlined above we would expect to find an alternating current produced in loop 1. This process, as you may have realized, forms the basis of transformer action: a process described in chapter 1.

155 The experiments described demonstrate the principle of electromagnetic induction. A current is induced in loop 1 either by moving the loops or by switching the current and so causing the field to change. In each instance the charges in loop 1 are crossing magnetic field lines. This results in a force on the charges and hence charge movement. The direction of the induced current is at right angles to both the field direction and the direction of movement.

When an alternating current flows in the primary winding of a transformer, an alternating magnetic field is produced around the primary and an alternating current is thus induced in the secondary. The secondary circuit need not be closed. If no ammeter or other components are connected to the secondary, current will still be able to flow to the ends of the secondary winding. This will result in a difference in charge between each end of the winding and thus a potential difference between the endings. A major point that should be noted here is that although we have talked about the current induced in a conducting wire loop, we need not have restricted the discussion to conductors. Although insulators have their electrons tightly bound to the molecule and current will not flow, the charges will still experience a force and this force will polarize the molecules.

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Since electric field strength is defined as the force per unit charge we could include both insulators and conductors in our discussion by referring to the induced electric field or electromotive force (EMF) arising as a result of changing magnetic field. The principle of electromagnetic induction applies to any material placed in a changing magnetic field. An electric field is always produced as a result of the change in a magnetic field. If a conductor is in the changing magnetic field, a current will be induced whereas if an insulator is in the changing magnetic field only polarization will result. When an alternating current is induced in a slab of conducting material rather than a wire the currents are given the special name 'eddy currents'. The term arises because the most common geometry, a conducting cylinder placed in a solenoid as in figure 6.22, gives rise to circular current paths at right angles to the magnetic field. Provided that the magnetic field of the solenoid is changing i.e. the field lines are moving, force will be produced on charges in the conductor, resulting in current flow. An alternating current in the solenoid will result in an induced alternating current flow in the conductor. To understand why the induced current follows circular pathways we need to think about the direction of the magnetic field lines and their direction of movement. A force will be produced with a direction at right-angles to each of these. Figure 6.7 shows the magnetic field pattern around a solenoid. Magnetic field lines inside the solenoid run parallel along the central axis. If alternating current flows though the solenoid, the magnetic field will build-up then collapse, build-up in the reverse direction then collapse in repetitive cycles. As the field builds-up, the field line loops in figure 6.7 will grow larger as new loops form. This is illustrated in figure 6.23.

Figure 6.22 Current induced in a conductive material placed in a solenoid.

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Note the movement of the field lines. One field line is highlighted in red to show how the line moves towards the central axis as the current flow (and field intensity) increases. For field lines inside the solenoid, the field line movement is always radially inwards as the field increases and radially outward as the field decreases.

Figure 6.23 Movement of field lines as the current through a solenoid increases.

Figure 6.24 shows the conductive cylindrical object in figure 6.22, viewed endon. The magnetic field lie (B), direction of movement (v) and resulting force (F) vectors are shown at different points. The B arrows point out of the page, directly towards you and are shown as blue circles. Because the field, B, is always pointing out of the page (along the cylinder axis) and v is always radially inwards, the resulting force (EMF) always acts around the circumference of a circle. This is why the induced current follows circular pathways.

Figure 6.24 Direction of force and induced current as a result of an increasing magnetic field intensity.

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The force, F, acts clockwise when v points inwards i.e. when the current and, consequently the magnetic field is increasing, When the current decreases, the magnetic field collapses and loops shrink towards the coil (the reverse of that shown in figure 6.23) so the direction of v in figure 6.24 is reversed. The consequence is that the direction of F is reversed and the induced current flow reverses direction. Thus an alternating current is induced in the conductor as a result of the alternating magnetic field. The important conclusion to draw from figures 6.22 to 6.24 is that an alternating current in the solenoid gives rise to an alternating magnetic field. This, in turn, gives rise to an alternating EMF in the material within the solenoid. If the material is a conductor a current will be induced which follows a circular path parallel to the current in the solenoid. If the material is an insulator the molecules will polarize in alternating directions along arcs parallel to the solenoid loops. In either case an induced electric field is produced with the field direction parallel to the wires in the solenoid. The actual amount of induced current flow will depend on the dielectric constant and conductivity of the material.

Magnetic Fields in Tissue From the previous discussion it should now be apparent that three factors determine the effect of an alternating magnetic field on a material: the permeability, µ, which is a measure of the 'magnetizability' of a material. This determines the magnetic field strength around and within a material placed in the field. The permeability is very close to unity for most biological materials: only ferromagnetic substances have a significant effect on the field strength. the conductivity, σ , which determines the amount of current flow in response to the applied (alternating) magnetic field. The higher the conductivity the greater will be the

158

When the current through the coil is increasing, the induced current flows in an anticlockwise direction. When the current flow is decreasing, the induced current flows in a clockwise direction.

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induced current. the dielectric constant, ε , which measures the 'polarizability' of a material. The higher the dielectric constant the greater will be the amount of charge movement as a result of polarization of the material. A significant difference between electric and magnetic fields in tissue is that biological tissue is 'transparent' in a magnetic field. The permeability, µ, is close to 1.000 meaning that the magnetic field is virtually unaffected by the presence of biological tissue. This contrasts with biological tissue in an electric field, where the field intensity varies according to the electrical properties of different tissues. While the electrical properties of fat, muscle and bone are quite different, the magnetic properties are almost identical. This means that for a body segment in an electric field, the field within tissue will vary according to tissue type. The field in muscle is lower than in the fatty tissue or bone (figure 6.19). In a magnetic field, no such variation occurs. The field intensities in fat, muscle and bone are virtually identical. Thus if, for example, a limb segment is exposed to a magnetic field by a surrounding coil as in figure 6.22, the magnetic field intensity (and consequently the induced EMF) within fatty tissue, muscle and bone will be the same. The differences between current induced by an alternating electric field and that induced by an alternating magnetic field will be discussed further in chapter 7.

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EXERCISES 1

Two oppositely charged spheres are separated by a distance d as shown. The force of attraction between the spheres is measured as 50 N. What would this force be if: (a) the distance d was halved. (b) the distance d was doubled. (c) the charge q1 was reduced by one half. (d) both charges were halved. (e) both charges were halved and the distance doubled.

2

The quantitative relationship between charge and electrostatic force is: q .q F = k. 12 2 r

.... (6.1)

where k has the value 9 x 109 N.m2.C-2. Calculate the magnitude of the force of attraction between: (a)

one coulomb of negative charge and one coulomb of positive charge separated by a distance of 1 m.

(b)

two opposite charges, both of magnitude 1 microcoulomb separated by a distance of 1 mm.

(c)

a sodium ion (Na+ ) and a chloride ion (Cl- ) in crystalline NaCI. The distance between ions in the crystal is 0.3 nm and the charge of each ion is 1.6 x 10-19 coulomb.

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3

161

The magnitude of the electric field around a single positive charge q is: q E = k. 2 r

.... (6.3)

where k has the value 9 x 109 N.m2.C-2 (see equation 6.3). Calculate the field of: (a) one coulomb of positive charge at a distance of 1 m. (b) one coulomb of positive charge at a distance of 1 km. (c) one microcoulomb of positive charge at a distance of I m. (d) a sodium ion (charge 1.6 x 10-19 C) at a distance of 0.3 nm. 4

The electric field of a single positive charge is measured as 1.5 V.m-1 at a distance of 0.3 m. Use equations 6.1 and 6.3 to calculate: (a) the magnitude of the charge (b) the force which would be experienced by a charge of 2 µC placed at a distance 0.3 m from the first charge. The constant of proportionality in equations 6.1 and 6.3 is 9 x 109 N.m2.C-2.

5

Consider the electric field patterns shown in figure 6.3. Draw diagrams to show the effect on the field when: (a) the small plate in figure 6.3(a) is made smaller (b) the angle between the plates in 6.3(b) is made greater (c) the plates in 6.3(c) are offset further.

6

Two current carrying wires are separated by a distance, d, as shown. The force of attraction between the wires is 50 N.

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162

What would be the force if: (a) the distance d was halved (b) the distance d was doubled (c) the current, I1, was reduced by one half (d) both currents were halved (e) both currents were halved and the distance doubled I1 was reduced to zero. (f) (g) I1 was reversed in direction. 7

The quantitative relationship between current and magnetic force is:

I .I .L F α k. 1 2 r

.... (6.4)

where k has the value of 10-7 N.C -2.s2 (see equation 6.4). Calculate the force between two parallel wires of length 0.8 m. The wires each carry a current of 0.5 ampere and are separated by a distance of 2 cm. 8

9

Two metal plates separated by a thin air space are found to have a capacitance of 15 pF. The plates are connected to a power supply and charged to a potential difference of 200 V. (a)

Use equation 6.6 to calculate the resulting charge on each plate.

(b)

The power supply remains connected and the space between the plates is filled with petroleum oil. What is the charge on each plate? What is the potential difference between the plates?

If the power supply in question 8(b) above was disconnected before the oil was introduced what would be: (a) the charge on each plate

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10

(b)

the potential difference between the plates after introduction of the oil?

(a)

Define the terms 'good insulator' and 'good dielectric'.

(b)

Which of the substances listed in table 6.2 are good insulators and which are good dielectrics?

11

Briefly describe how the dielectric constant and conductivity of a material will change if there is a change in proportion of: (a) polar molecules (b) non- polar molecules (c) ions.

12

(a)

Draw a diagram similar to figure 6.13 to show the effect of a rectangular dielectric on a uniform electric field. Label your diagram to show clearly regions where the field intensity is increased and where it is decreased.

(b)

Explain the origin of the charges shown on the dielectric surface in figure 6.13. What effect do the charges have on the field intensity within the dielectric?

13

163

An electric field line crosses from medium I (air) to medium 2 (water) as shown in figure 4.15. Use equation 4.9 and values of s from table 4.1 to calculate the angle of refraction for an incident angle of: (a) 1o (b) 5o (c) 15o What general conclusion can be drawn about field lines crossing into a medium of high dielectric constant?

ELECTRIC AND MAGNETIC FIELDS 14

15

Draw a diagram similar to figure 6.16 to show the effect of a dielectric on the field between capacitor plates. (a)

Label the diagram to show clearly regions where the field intensity is increased or decreased.

(b)

Briefly explain the origin of the changes in field intensity.

A cylindrical block of material of conductivity 0.8 S.m-1 carries a current of 0.2 amps along its length. The block has faces each of area 10 cm2 and a length of 25 cm. (a)

16

(b)

Use equation 6.11 to calculate the electric field intensity in the material.

(c)

What is the potential difference between the two faces?

A rectangular length of material of resistivity 25 Ω.m has a potential difference of 5 V between its ends. The material has a length of 30 cm and a cross-sectional area of 2 cm2. (a)

17

What is the current density in the material?

What is the field intensity within the material?

(b)

What is the current density within the material?

(c)

What is the total current in the material?

An electric field line crosses from medium 1 (conductivity 0.05 S.m-1) to medium 2 (conductivity 0.8 S.m-1) as shown in figure 6.18. Use equation 6.14 to calculate the angle of incidence for a refraction angle of: (a) (b) (c)

40 o 80 o 89 o

164

ELECTRIC AND MAGNETIC FIELDS 18

Figure 6.19 shows the pattern of field lines in a model for an arm or leg (longitudinal section) placed between capacitor plates. Explain the pronounced difference in field intensity in fat and muscle tissue in terms of refraction of field lines and surface charge of polarization in each tissue.

19

Figure 6.20 shows the pattern of field lines in a model for an arm or leg (perpendicular cross-section) between two capacitor plates. Explain why: (a)

the field intensity in air close to the fatty tissue is greater than that without the limb present.

(b)

the field in muscle tissue is lower than without the limb present and lower than that in fatty tissue.

20

Consider a length of wire wound as a solenoid as shown in figure 6.7. Draw a diagram of the solenoid and the associated magnetic field when a current flows through the wire. Is the magnetic field changed by the introduction of biological materials (for example, an arm or leg) within the coil? Explain.

21

(a)

What is meant by the term 'electromagnetic induction'?

(b)

Describe the effects of electromagnetic induction on conductors placed in a magnetic field.

22

Consider a material placed within a coil of wire wound as a solenoid as shown in figure 6.24(a). Describe the effect of an alternating magnetic field on the material if the material is: (a) a good conductor (b) a good dielectric.

23

Figure 6.22 shows the pathway of induced current when a conductive material is placed in a solenoid through which alternating current is flowing. Explain why the

ELECTRIC AND MAGNETIC FIELDS induced current follows circular paths, parallel to the coil. 24

Consider a loop of wire positioned parallel to the surface of a conductive material as shown. (a)

Draw a diagram showing the magnetic field produced by the loop.

(b)

Suppose the current through the loop is alternating so that current is induced in the conductor. Draw a diagram showing the pathways of the induced current. Explain why the current flows follows these particular pathways.

(c)

where, in the conductive material, is the induced current greatest? Why?

165

166

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7 Therapeutic Fields: Shortwave Diathermy 'Shortwave diathermy' refers to heating of deeply located tissue using electric or magnetic fields which alternate at high frequency. The term 'shortwave diathermy', is something of a misnomer as the contribution of waves, as such, to the treatment is negligible. The physiological effects are a result of electric and magnetic fields generated by the shortwave diathermy apparatus. Shortwave radiation plays little or no role in the therapy. The apparatus used by physiotherapists generates alternating electric and magnetic fields with a frequency of 27.12 MHz. Since radio waves with frequencies in the range 10 MHz to 100 MHz are termed short waves the term has been, rather inappropriately, applied to this therapeutic modality.

While shortwave diathermy units do radiate waves with a frequency of 27.12 MHz, this is a side-effect. The physiological effects are due to the powerful electric or magnetic fields generated by the apparatus.

PRODUCTION OF THE FIELD Shortwave diathermy apparatus consists of a sinewave generator circuit which produces alternating current with a frequency of 27.12 MHz and a resonant circuit which can be tuned to exactly the same frequency. The sinewave generator supplies energy to the resonant circuit by transformer action. Figure 7.1 illustrates the arrangement.

Figure 7.1 Shortwave diathermy apparatus (schematic).

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The sinewave generator consists of a power supply (chapter 5), an oscillator with good frequency stability (chapters 2 and 5) and a power amplifier (chapter 5). The power supply converts AC from the mains (of frequency 50 Hz) to DC which is needed to power the equipment. It consists of a transformer (to convert the 240 V AC from the mains to the voltage needed by the rest of the circuitry), and a rectifier to convert the AC to DC. The DC is used to power a sinewave generator; a resonant circuit which oscillates at 27.12 MHz and an amplifier, which boosts the current produced by the resonant circuit to higher levels, as needed for patient treatment. Electrical energy produced by the sinewave generator is coupled to the patient tuning circuit by transformer action (figure 7.1). Two inductors are placed close together so that energy produced by the power amplifier is transferred to the patient circuit. This method of coupling ensures that DC in the apparatus is unable to reach the patient and the risk of electric shock is minimized. A variable capacitor, C, is included in the patient circuit so that the resonant frequency of the patient circuit can be made equal to the frequency of the oscillator. This ensures maximum efficiency of energy transfer (chapter 2) and reliable operation of the apparatus. A power meter or indicator lamp shows when resonance is achieved and maximum power is transferred. In older machines, the variable capacitor, C, was manually adjusted with the operator adjusting a knob while observing the power meter and adjusting for maximum power. Modern machines use electronic control of the variable capacitor and are described as 'auto-tuning'. The principal advantage of automatic tuning is that if the patient should move during treatment the machine will adjust to keep the patient circuit in resonance. With manual tuning machines, movement of the patient or electrodes can result in de-tuning and a drop in output of the machine. The output of the apparatus is coupled to the patient via electrodes (in the capacitor field technique represented in figure 7.1) or via an induction coil. The coil or electrodes are connected directly to the output of the machine and the part of the

Any mains-frequency AC produced by the apparatus is also not conducted appreciably to the patient circuit as the resonant frequency (27.12 MHz) is vastly different to the mains frequency (50 Hz).

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patient to be treated is positioned in the electric or magnetic field. In figure 7.1, the area highlighted in yellow is circuitry inside the machine. The part of the patient to be treated would be positioned between the external capacitor plates shown in figure 7.1. The plates are normally in the form of two metal disks, each inside a clear plastic container or envelope. The electrical characteristics of the patient's tissue affects the capacitance of the patient circuit, as does the electrode size and spacing. For this reason it is necessary that the apparatus be tuned (by adjusting C in figure 7.1) with the patient positioned in the field. Similarly, if an induction coil is used rather than capacitor plates, tuning will be necessary. This is because when the coil is wrapped around the part of the patient to be treated, the inductance of the coil will depend on the number of turns of the coil and their radius.

When an induction coil is used, the presence of biological tissue in the field is irrelevant but the tissue volume to be treated will influence the number of turns of the coil and their radius.

MOLECULES IN AN ELECTRIC FIELD In shortwave diathermy treatment a high frequency AC electrical signal is produced and applied to the patient via an induction coil or electrodes. The high frequency signal will produce a corresponding high frequency alternating electric or magnetic field in the patient's tissue. We now consider what effect this has on the tissue. Since an alternating magnetic field gives rise to an induced alternating electrical field (as described in chapter 6) we first examine the effects of an alternating electric field on the different molecules found in human tissue.

Charged Molecules The conductivity of tissue is determined by the number of free ions in the tissue fluid. In the presence of an electric field these ions will migrate along field lines and so constitute an electric current. The process is not unlike electrical conduction in metals. Metallic conduction results from the movement of free electrons. In electrolytes the charge carriers are not electrons but ions; these are tens of

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thousands of times more massive than electrons. Under the influence of the electric field ions will be accelerated along field lines - but they will not travel far before colliding with other molecules and losing their acquired kinetic energy. The repeated sequence of accelerations and collisions is the way in which electrical energy is converted to heat energy, which is the random-motion energy of the molecules. At the frequencies associated with shortwave diathermy the field alternations are so rapid that the ions oscillate about a mean position rather than undergoing any large scale movement, but the alternations are not so rapid that movement is prevented and heat generation is not impaired.

Dipolar Molecules Dipolar molecules such as water will orient themselves in an electrical field and if the field is alternating this will result in backwards and forwards rotation of the dipoles. In a liquid the molecules are continually in motion (due to their thermal energy) and are loosely associated with each other (coupled); thus some of the rotational energy of the molecules will be converted to heat energy by what can be thought of as a frictional drag between adjacent molecules.

Non Polar Molecules Though not normally polar these molecules will undergo a distortion of their electron 'clouds'; that is, they will polarize in an electric field. In an alternating field the electron clouds will

Figure 7.2 Response of molecules to a high frequency alternating electric field.

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oscillate back and forth to each end of the molecule. Since this kind of motion does not involve transport or rotation of the molecule as a whole it can only be coupled indirectly with the gross molecular movement associated with heat energy. Figure 7.2 summarizes, by illustration, the response of ions, polar molecules and non-polar molecules to a high frequency alternating electric field. In each case there is a net back and forth movement of charge: in other words, an alternating flow of current.

REAL AND DISPLACEMENT CURRENT From the previous discussion it is apparent that the different kinds of molecule in a material will each respond differently to an applied electric field. The back and forth movement of ions and the consequent collisions will result in a very efficient conversion of electrical energy into heat energy. The rotational movement of polar molecules provides a less efficient mechanism of energy conversion. The electron cloud distortion of non-polar molecules represents the least efficient means of heat production. Nonetheless each kind of molecule responds to an alternating electric field in a way which results in movement of charges and hence an alternating current. The difference is in the proportion of electrical energy converted to heat energy when the alternating current is produced. With this in mind we distinguish real and displacement current. *

Real current is that associated with heat production. When real current flows through a material the rate at which electrical energy is converted to heat energy is given by Joule's law: P = V.I

.... (1.4)

where V is the potential difference and I is the real current flowing through the material. P is the power dissipated (in watts), in other words the amount of electrical energy dissipated per second (1 watt (W) = 1 joule per second (J.s-1)).

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*

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Displacement current is current flow which does not produce any heating. In this case the power dissipated, and hence the heat generated, is zero.

Ionic materials are associated principally with real current and hence substantial heat production. Polar substances are associated with both real and displacement current and hence less heat production. Non-polar materials are principally associated with displacement current and hence minimal heat production. An example which serves to illustrate the distinction between real and displacement current is given in figure 7.3. Here we have a resistor and a capacitor connected in series to a source of alternating current. In this case we suppose that the capacitor is ideal - it comprises two metal plates separated by a perfect insulator which can polarize and depolarize with no loss of electrical energy to heat energy. The magnitude of the current flowing in this circuit will depend on the voltage of the AC source and the total impedance of the resistor/capacitor combination. The actual impedance of the capacitor is calculated using equation 2.5. The real current (Ir) flowing through the resistor will result in power dissipation according to equation 1.4 and hence heat production in the resistor. The displacement current (Id ) flowing through the capacitor (assumed ideal) gives no power dissipation and hence no heat production as the material between the plates is able to polarize and depolarize with no energy loss. In this case, then, the current flowing from the AC source appears as real current in the resistor R and displacement current in the (ideal) capacitor C. Charges move and heat is produced in the resistor while the charge movement (displacement current) in the capacitor produces no heating. The two currents, which are different forms of the same thing, are necessarily the same size.

Figure 7.3 Real and displacement current in an AC circuit.

SHORTWAVE DIATHERMY For a capacitor to be ideal the material between the plates must be an ideal dielectric - a substance capable of polarizing in an electric field and depolarizing on its removal without any dielectric absorption. In other words, with no conversion of electrical energy to heat energy. Biological materials, particularly those with high water and ion content are far from being ideal dielectrics. When placed in an electric field the induced current will be a combination of real and displacement current. The proportions of each kind of current will depend on the proportions of ionic, polar and non-polar molecules. We now consider biological tissue exposed to an electric or magnetic field which alternates at a frequency of 27.12 MHz, the frequency licensed for use in shortwave diathermy. As we have seen, shortwave diathermy may be applied using capacitor plates (which produce an electric field) or an inductive coil (which generates a magnetic field).

173

Most gases come close to being ideal dielectrics, as do some oils. Water being a highly polar molecule, falls short of this ideal and dielectric absorption results in significant heating at any frequency below about 1010 Hz.

CAPACITOR FIELD TREATMENT Consider first the situation depicted in figure 6.19(a), where an arm or leg is positioned between two capacitor plates. Figure 6.19(a) shows the electric field pattern, which is affected by refraction and termination of field lines. The total current flowing through the tissue will be determined by the total impedance of the tissue plus the air space between tissue and capacitor plates. Current will flow in the direction of the field lines and the proportions of real and displacement current will depend on the electrical properties of the particular tissue. The amount of heating in any tissue layer will be determined by two factors: the field intensity within the layer and the amount of real, rather than displacement, current. Calculation of the proportions of real and displacement current in a particular tissue is not difficult. Measured values of dielectric constant and conductivity are all that are

SHORTWAVE DIATHERMY needed. Calculation of the field pattern is much more difficult and has only been done using simplified models: even simpler than the somewhat idealized geometries shown in figure 6.19. Useful qualitative pictures are nonetheless obtained by combining diagrams such as those shown in figure 6.19, with calculated values of real and displacement current in each tissue layer. At a frequency of 27.12 MHz the current flow in fatty tissue and bone is approximately 50% displacement. In muscle and tissues of high water content the proportions are approximately 80% real current to 20% displacement current. Figure 7.4 shows a revised view of figure 6.19(a) which takes into account the two kinds of current flow which occur. In the air spaces the current flow is entirely displacement current. In fatty tissue and bone the current is assumed to be one half real current and one half displacement current. For simplicity, muscle is shown as having entirely real current.

Figure 7.4 Current type and directions in a model for an arm or leg.

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The conductivity determines the amount of real current flow, the dielectric constant determines the amount of displacement current.

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When viewing diagrams such as these, bear in mind the simplifications made. The pictures can be misleading if interpreted too literally. You should also bear in mind that even a single tissue layer may be inhomogeneous at both the microscopic and macroscopic level. An example of the complications introduced by tissue inhomogeneity is seen with fatty tissue in the shortwave field.

Fatty Tissue A practical limitation on the amount of heat which can be produced in deeply located tissue is the heat production in fatty tissue. When using capacitor plates the rate of heating of fatty tissue is always greater than that of the underlying muscle tissue. Part of the reason is that fatty tissue is inhomogeneous. The tissue is not a uniform distribution of cells but a complex structure incorporating regions of high conductivity and dielectric constant: the lymphatic and blood vessels. The high conductivity and dielectric constant of the vessels will result in field lines being focussed or channelled into them with a resulting high local field intensity and corresponding high rate of heating in and near the vessels. The phenomenon is illustrated in figure 7.5. The localized high heat production will result in greater temperature elevation of the vessels than the fatty tissue as a whole and a greater sensation of heat than would be expected if the tissue layer was homogenous.

Figure 7.5 Focussing of electric field lines in blood and lymphatic vessels in fatty tissue.

INDUCTIVE COIL TREATMENT We now consider application of the shortwave field with an induction coil. The objective is to induce an electric field and hence a flow of current as a result of the alternating magnetic field produced by the coil. In the example illustrated in figure 7.6 a cable carrying the shortwave frequency current is wrapped around a patient's lower

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limb. Figure 7.6(a) shows the inductive coil wound as a solenoid around the patient's lower limb and figure 7.6(b) shows the current pathways in the different tissues. The current pathways shown are predicted assuming that the alternating magnetic field gives rise to an induced EMF in the patient's tissue. In this case the current will follow circular paths parallel to the turns of the coil in figure 7.6(a). Note that in figure 7.6(b) the current through the fatty tissue is shown as half displacement current and half real current while muscle is assumed to have real current only. As indicated previously, this is only an approximation: while the proportion of real current in fatty tissue is about 50%. in muscle it is about 80%.

If the coil in figure 7.6 had a large number of closely spaced turns and the coil diameter was small compared to its length, then the magnetic field inside the coil would be uniform and the induced EMF would be the same throughout the tissue volume. Were this the case, the relative amounts of current flow in each tissue would simply depend on the tissue impedance (which is determined by the dielectric constant and conductivity).

Figure 7.6 Current flow induced in a limb by inductive coil treatment.

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A complication is that with more widely spaced turns and a relatively large diameter, the magnetic field inside the induction coil will be non-uniform. In an arrangement like that shown in figure 7.6(a), the magnetic field would be strongest close to the coil and decreasing in intensity towards the centre. The highest field intensity is thus in the superficial tissues of the limb.

Another Kind of Coil

This means thaqt a greater EMF will be induced in the superficial tissues and consequently there will be a greater current flow.

Most manufacturers of shortwave diathermy apparatus offer accessories which include a compact coil mounted in a plastic housing. This device is called a monode. The monode is pointed at the part of the patient to be treated so that the coil is in a plane parallel to the skin surface. With this arrangement (figure 7.7), currents are induced which flow in circular paths parallel to the skin surface. The cable supplied with the shortwave machine can, of course, also be wound into a spiral and positioned to produce a similar distribution of induced current. The spiral coil placed parallel to the skin produces more superficial heating than the solenoidal coil (figure 7.6). This is because the magnetic field intensity decreases rapidly with distance from the coil, as the field lines diverge, spreading apart and looping round to the opposite side of the coil. The field spreading is similar to that which occurs at the ends of the coils in figures 6.7 and 7.6. Magnetic field lines become more separated, indicating a weaker magnetic field further from the coil and consequently less induced EMF and less induced current. Hence although the current induced in muscle is mostly real current, the amount of current at depth is much less than with a solenoid (figure 7.6).

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Figure 7.7 Induced currents with a spiral coil mounted parallel to the skin surface.

178

Capacitative Effects A practical complication which occurs with inductive coil treatment, whether with a solenoid or a spiral coil (monode), is that in addition to the currents induced by the magnetic field there is also a pronounced electrostatic effect. There is a certain capacitance between the loops of the coil. In fact whenever a cable or wire is folded back on itself or coiled we have produced a situation where there are two conductors separated by a space; thus we have produced a capacitor. Although in the case of a cable wound as a coil the capacitance is very small, the effect is quite significant at MHz frequencies. The inductive coil behaves as an inductor in parallel with a capacitor. At the high frequencies used for shortwave diathermy the inductance of the coil results in a high impedance to current flow in the cable (equation 2.4). The capacitance associated with the coil presents a lower impedance pathway for current to take (equation 2.5). In consequence the induced current patterns are not as simple as those shown in figure 7.6(b). The electric field between adjacent turns (Figure 7.8(a)) results in current flow along the field lines shown in blue. Because the electric field is stronger closer to the coils, greater current flows and this adds to the current induced by the magnetic field. The consequence is greater current flow in, and greater heating of, superficial tissue (figure 7.8(b)). The electric field between adjacent loops is similar to that between two small electrodes (figure 6.1(c)). The field is most intense close to the cable. A consequence is that there is a risk of burning the superficial tissues with the electric field of the coil rather than

Figure 7.8 Electric field pattern (blue lines) between adjacent turns of an inductive coil.

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heating deeper tissue with current induced by the alternating magnetic field. A similar argument applies for a spiral coil. An electric field is produced between adjacent turns within the loop. Close to the coil, the electric field is intense and greater current flows. This adds to the current induced by the magnetic field so there is greater current flow in, and greater heating of, superficial tissue. Superficial heating due to the electric field can be minimized in three ways: (a) by winding the turns of the coil close together, (b) by keeping the cable away from the patient's skin using towelling and/or rubber spacer designed for this purpose and (c) by using an electrostatic shield. Electric field heating effects can also be minimized, in the case of a solenoid, by positioning an earthed metal cylinder between the coil and the patient's limb. If a monode is used, a flat metal plate between the monode and the patient's tissue would be needed. The plate will screen-out the electric field while having little effect on the magnetic field of the coil. The electric field inside the metal cylinder or behind the metal plate would be almost nil because the metal is a good conductor and field lines will terminate on its surface. Most metals are, however, transparent as far as magnetic fields are concerned so the magnetic field is virtually unchanged. Some, but not all, inductive coil applicators are supplied with an inbuilt electric field screen. Screening is an important feature when depth efficient heating is required.

When the adjacent turns are closer together, the electric field is actually greater, but it is also more localized to the space between the turns of the coil.

In summary, the options with inductive coil treatment are a coil wound around the part of the patient to be treated or a flat coil (monode) positioned over the body part. The difference is the depth efficiency of treatment. A solenoidal coil (figure 7.6) has greater depth efficiency as far as tissue within the coil s concerned. A flat spiral coil (figure 7.7) has greater effect on superficial tissues. With either method of application, there is the risk of excessive superficial heating due to the electric field between adjacent turns of the coil or spiral. the risk is minimized by spacing the coil or spiral away from the patient's superficial tissues.

SHORTWAVE DIATHERMY

ELECTRODE PLACEMENT - CAPACITOR FIELDS With capacitor field treatment, the therapist has more control over the field intensity in different areas than with inductive coil treatment. We have discussed previously how the combination of tissue layers in the part of the patient being treated alters the shape of the electric field. The other factors influencing the field pattern involve the placement of the electrodes. Each factor listed below must be taken into account in the practical application of shortwave diathermy using capacitor field treatment. *

The shape of the part of patient in the field. Compare Figure 6.19(a) with 6.19(b). In addition, if the electrodes are placed over any prominence an undesirable concentration of the field can result.

*

The arrangement of tissues layers in the treated structure. As discussed previously, this factor plays a significant role in determining the final shape of the field.

*

The size, spacing and orientation of the electrodes. Some examples of the electric field in the absence of any object were shown in figures 6.2 and 6.3. We consider below the effect when the patient is in the field.

Electrode Size In general, it is preferable to use electrodes which are somewhat larger than the structure to be treated. This results in the central, more uniform, part of the field being used (figure 6.2). The dielectric constant and conductivity of tissue are much higher than those of air (table 4.2). Thus, with large electrodes, the field lines are bent towards the limb and spreading of the field is minimised. The effect is illustrated in figure 7.9 where the effect of the different tissue layers is ignored for simplicity.

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Compare these with figures 4.19 and 4.20. The use of small electrodes results in an undesirably high field intensity in the superficial tissues. Unequal size electrodes (figure 5.10(c)) can be used to selectively heat tissue located closer to one surface of a limb. Large differences in electrode size, however, can sometimes lead to difficulty in tuning or instability in machine operation.

Electrode Spacing The electrode spacing should normally be as wide as possible. In this way the problems associated with a non-uniform field pattern are minimised. The machine itself, however, sets the limit on the maximum spacing which can be used. As the electrodes are moved further apart the capacitance of the two plates decreases. In addition the field intensity (and consequently the rate of heating) will decrease. A point will be reached where the machine can no longer be tuned or insufficient power is available for adequate heating: this sets the limit on the separation of the electrodes.

Figure 7.9 Effect of electrode size: (a) correct electrode size (b) electrodes too small (c) arrangement for selective heating.

By use of a wide spacing the electrical properties of the tissue have a smaller effect on the overall field pattern and the electrical properties of air play a greater role. Thus the field pattern is more uniform and less subject to variation with movement of the patient in the field. Figure 7.10 illustrates the effect of electrode spacing. In 7.10(a) the electrode to surface distance varies considerably resulting in a local high field intensity

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in the limb. In 7.10(b) the electrode to surface distance does not vary greatly and the field within the limb is more uniform. Clearly, if a relatively uniform field pattern is required the arrangement shown in 7.10(b) is to be preferred. The arrangements shown in 7.10(c) and 7.11(c) are both suitable if we wish to selectively heat one surface of a limb. They would also be suitable for heating a structure which is located close to one surface of a limb or trunk - for example, the hip joint.

Electrode Orientation In the examples considered previously the electrodes were placed parallel to each other in order to obtain a relatively uniform heating pattern. However if one part of the surface of a structure is closer to an electrode, the field lines will be concentrated in that region. Figure 7.11 shows electrodes applied to the shoulder. Compare this with figure 6.16. Electrodes which are parallel to each other as in figure 7.11(a) do not give a uniform field because the air spacing varies considerably. The dielectric constant and conductivity of each field-line pathway varies considerably, resulting in variation in the field intensity. In figure 7.11(b) the distance between the plates varies but the electrical characteristics of each pathway are similar: thus the field is relatively uniform. Clearly the arrangement shown in figure 7.11(b) is preferred when uniform heating is the objective.

Figure 7.10 Effect of electrode spacing: (a) narrow spacing, (b) wide spacing and (c) unequal spacing.

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In all of the examples discussed previously the arrangement of electrodes is contraplanar: that is, electrodes are placed over opposite sides of a structure. Such an arrangement is needed if deeply located tissue is to be heated. In some circumstances it is preferable to use a coplanar electrode arrangement. For example superficial structures, such as the spine, which are too extensive for contraplanar treatment may be treated in this way. Figure 7.12 shows a coplanar arrangement of electrodes. When using a coplanar arrangement it is very important to ensure that the spacing between electrodes is greater than double the skin to electrode distance. This results in the majority of field lines passing through tissue rather than the air space between the electrodes. Figure 7.11 Effect of electrode orientation. (a) and (c): incorrect orientation (b) correct orientation.

Figure 7.12 A coplanar arrangement of electrodes.

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Even when using a contraplanar arrangement of electrodes considerable heating occurs in the superficial tissues closest to the electrodes. This effect can be minimized by using the cross-fire technique of treatment shown in figure 7.13. Half of the treatment is given with electrodes in one position (figure 7.13(a)). The electrodes are then moved so that the new electric field is at right angles to the old one (figure 7.13(b)) and the treatment is continued. In this way deeply located tissue receives treatment for twice as long as the skin. The cross-fire treatment may be used, for example, on the knee joint or pelvic organs. It is also particularly useful for treating the walls of cavities within a structure, for example the sinuses. Figure 7.14 shows the field pattern obtained with an object of high dielectric constant which has an air-filled hollow at its centre.

Figure 7.13 The cross-fire technique.

The field lines are concentrated in the dielectric resulting in uneven heating of the walls of the cavity. Cross-fire treatment ensures that all of the cavity wall area is treated.

HEATING OF TISSUE Earlier we discussed qualitatively and in molecular terms, the heating effect of a high frequency alternating electric field. We now consider heat production and temperature rise and take a larger scale view of matter: a view at the level of tissue rather than molecules. We saw in chapter 1 that the power dissipated by a resistor, the rate at which electrical energy is converted to heat energy, is given by equation 1.4: P = V.I

.... (1.4)

Figure 7.14 A hollow dielectric between capacitor plates.

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This expression relates the current, I, flowing through a resistor to the total power, P, dissipated in the resistor. For resistors the current, I, is entirely real current and thus produces heat. When we consider biological tissues we must distinguish between real current and displacement current since only the real current results in heat production. In additional, we are usually more interested in the rate of heating at a particular point in the tissue rather than in the tissue as a whole. In this case a more useful expression of equation 1.4 is equation 7.1. Pv = E.ir

.... (7.1)

Here Pv is the power dissipated per unit volume of tissue at a particular point. The units of Pv are thus watts per cubic metre. E is the field strength (in volts per metre) and ir is the real component of current density (in amps per square metre) at that point. The power dissipated, Pv is equal to the rate of heat production. Hence, in order to determine the rate of heating at a particular point in tissue we need to know the electric field strength and the real current density. We begin by considering fields and currents produced using capacitor field treatment.

Capacitor Field Treatment Whether electrodes are positioned in a coplanar arrangement (figure 7.12) or in a contraplanar arrangement (figures 7.9 to 7.11) the current flow in muscle will be determined by the total impedance of the tissue combination plus the air space between the tissue and capacitor plates. Figure 7.15 shows electrical equivalent circuits for the two electrode/tissue arrangements. The quantities Za, Zf, and Zm refer to the electrical impedances of air, fat and muscle respectively.

SHORTWAVE DIATHERMY

Figure 7.15 Electrode/tissue configurations and their electrical equivalent circuits. (a) coplanar arrangement, (b) contraplanar arrangement. In figure 7.15(a) we ignore (displacement) current flow through the air directly between the electrodes. We also ignore current flowing directly through the fatty tissue and bypassing the muscle. If the electrode spacing is at least twice the electrode to tissue spacing this will be a reasonable approximation. The impedance presented by each alternate pathway will be sufficiently high to make these currents negligible. In figure 7.15(b) we ignore current flow through the bone, directly around the fatty tissue or through the air around the tissue. Again this is because these pathways have very high impedance compared to the ones shown. With these approximations the electrical equivalent circuits in 5.16(a) and (b) are the

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same. Each of the electrode/tissue arrangements are equivalent to a series combination of impedances. Just as with resistive circuits (described in chapter 1), when impedances are connected in series the current in each impedance is the same. We thus have the following relationship: displacement current in air

=

displacement + real current in fatty tissue

=

displacement + real current in muscle

As mentioned earlier, the proportion of real current in fatty tissue is approximately 50% while in muscle the proportion is about 80%. Thus the amount of real current flow in muscle is 80/50 or about one and one half times greater than in fatty tissue. Let us take the simple case where current spreading is minimal and estimate the relative rate of heating in fatty tissue and muscle. We need to know both the real current density and field strength in each tissue. The field strength is estimated below.

The real current density in muscle may be increased or decreased if the electric field lines converge or diverge. This depends on the tissue/electrode geometry see figure 6.19 for example.

When resistors are connected in series the current flow in each is the same but the voltage across each resistor will, in general, be different. The largest resistor will have across it the greatest potential difference. The equivalent statement for tissues of different impedance is as follows: When tissues are arranged in series the field intensity will be greatest in the tissue with highest impedance. Inspection of table 6.2 shows that muscle has a higher conductivity and dielectric constant than fatty tissue: both figures are several times higher. Now a high conductivity and dielectric constant means a low impedance. Combining the two

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figures from table 6.2 we calculate that fatty tissue has an electrical impedance some ten times larger than muscle. The rate of heating of each tissue is given by equation 7.1: Pv = E.ir

.... (7.1)

The real current density in muscle is as we have seen, about one and a half times greater than in fatty tissue, however the field intensity in fatty tissue is approximately ten times higher. Hence the rate of heating of fatty tissue is predicted to be approximately 10/1.5 times higher than muscle. We thus have the general conclusion that if spreading or converging of the field is minimal the rate of heat production in fatty tissue will be about seven times higher than in muscle. If the electrode/tissue configuration permits spreading of the field in muscle the current density will be reduced and the rate of heating of muscle correspondingly reduced. Conversely if the geometry produces convergence of the field lines in muscle the current density will be increased and the relative rate of heating will be increased accordingly.

Inductive Coil Treatment With capacitor field treatment tissues are effectively in a series electrical arrangement. The current flow in muscle is thus limited by the impedance of the fatty tissue layers. When inductive coil treatment is used such is not the case. Consider the coil and tissue arrangement shown in figure 7.7. Currents are induced in the plane of the fatty tissue and in the plane of the muscle. The current loops are complete electrical pathways in the one tissue. For this reason the current flow in

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muscle is not limited by the fatty tissue but depends only on the strength of the induced electric field and the electrical characteristics of the muscle tissue. In other words the induced currents flowing in each tissue layer are independent of each other. The real component of the current density, the current density which determines heat production, is given by equation 6.10, which can be written: ir = σ.E

.... (6.10)

Substituting this formula into equation 7.1 we obtain an alternate expression for the power dissipated per unit volume: Pv = σ.E2

.... (7.2)

Table 6.2 shows that the conductivity, σ, of muscle is some sixteen times greater than that of fatty tissue. Hence, for the same induced electric field strength, both the real current density and the power dissipated in muscle will be sixteen times greater than in fatty tissue. How large is the magnetically induced electric field? The intensity of the induced field is determined by the rate of change of the magnetic field and the permeability, µ, of the material. The permeability is close to one for biological materials (see table 6.3) so fatty tissue and muscle are alike in this regard. For the same strength of alternating magnetic field then, both fatty tissue and muscle will have the same strength of induced electric field. Thus the rate of heating of muscle in this situation will be about sixteen times greater than that of fatty tissue.

The intensity of the induced electric field is determined by the rate of change of the magnetic field and the permeability, µ, of the material. It does not depend on the electrical properties, σ and ε, of the tissue.

In practice such a degree of selective heating is difficult to achieve. This is for two reasons:

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*

Muscle is located beneath fatty tissue and so is further from the induction coil. Thus the magnetic field is weaker in muscle and the strength of the induced electric field is correspondingly smaller.

*

Fatty tissue, being closer to the induction coil may also experience an appreciable electric field due to the capacitance between adjacent turns of the coil. This effect was described earlier (see figure 7.8).

These two factors combine to increase the heating of fatty tissue relative to muscle so that a sixteen to one advantage is rarely obtained. Nonetheless efficient selective heating is achieved with close spacing of the turns of the coil and a sufficiently large coil to patient distance. One would also expect good discrimination with applicators which incorporate an electric field screen in front of the inductive coil.

HEAT AND TEMPERATURE RISE Having described the factors determining the rate of heating of tissue we now consider the relationship between rate of heating and rate of increase of temperature. The rate of heating per unit volume is given in terms of electric field intensity and real current density by equation 7.1. Hence the amount of heat produced per unit volume, ∆Qv, in a time interval ∆t is given by equation 7.3. ∆Qv = E.ir.∆t

.... (7.3)

∆Qv has units of joules per cubic meter (J.m-3). In considering the therapeutic effects of diathermy it is not the heat produced, as such, which determines the physiological response but the resulting temperature rise. Temperature is a key factor in determining the rates of chemical reactions and hence

For more information on relative heating rates see the book 'Therapeutic Heat and Cold' J F Lehmann (ed) (1982) chapters 6 and 10.

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physiological processes. The SI unit of temperature is the kelvin (symbol K). It is related to the perhaps more familiar degree Celsius (oC) by the expression oC = K - 273.15

Notice that from this definition the size of the degree Celsius is the same as the kelvin. In other words a change in temperature of five degrees Celsius is precisely the same as a change of five Kelvin's. When we are talking about increases in temperature brought about by diathermy treatment the terms kelvin and degrees Celsius can be used interchangeably to describe the increase.

To convert from degrees Celsius to Kelvin's, simply add 273.15 to the Celsius temperature.

When a fixed amount of heat is supplied to different substances the increase in temperature of each will, in general, be quite different. The factor which determines the resulting temperature increase is the specific heat capacity of the substance. The specific heat capacity is defined as the amount of heat required to raise 1 kg of a substance through one kelvin. The units of specific heat capacity are thus joules per kilogram per kelvin. This can be measured experimentally by supplying a certain amount of heat (∆Q) to a known mass (m) of the substance an measuring the resulting temperature increase (∆T). The experiment must be arranged so that all of the heat supplied is used to increase the temperature of the substance. If the loss of heat is negligible then the specific heat capacity (c) can be calculated using equation 7.4: ∆Q c= .... (7.4) m.∆T When we consider the heating of tissues by diathermy, heat transfer between tissues and to the bloodstream will have a large effect on the temperature distribution during treatment.

SHORTWAVE DIATHERMY

Alternatively, when the specific heat capacity of a substance is known, equation 7.4 can be used to calculate the temperature increase resulting from the heat supplied.

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Prior to the start of treatment the body tissues are in a state of dynamic equilibrium. Cellular activity, metabolism and muscle contraction result in the steady production of heat and the circulation of blood and tissue fluids provide an efficient means of heat transfer. The net production of heat is balanced by net transfer of heat from the tissue and a stable temperature is maintained. Once treatment is started heat is produced in the tissue according to equation 7.3 and the temperature starts to increase. An expression for the initial rate of increase in temperature is obtained below. Rearranging 7.4 we have ∆Q = m.c.∆T Dividing this expression by volume we obtain: ∆Qv = ρ.C.∆T

.... (7.5)

where ρ is the mass per unit volume or density of the tissue. Dividing 7.5 by ∆t gives: ∆T ∆Qv = ρ.c. ∆t ∆t

.... (7.6)

where ∆Qv/∆t is the volume rate of heating (in Joules per cubic metre per second) and ∆T/∆t is the rate of increase in temperature (in Kelvin's per second). This equation can be used to compare the initial rate of temperature increase in fatty tissue with that of muscle. The densities of the two tissues are similar but the heat capacity of muscle is some 50% greater than that of fatty tissue. Thus if the rate of heating of each tissue is the same, the initial rate of temperature increase in muscle will be only two thirds of that of fatty tissue. To produce the same initial rate of increase in temperature in each tissue the rate at which heat energy is produced in muscle must be 50% greater.

Note that this conclusion is a general one. It applies not just to shortwave diathermy but to any diathermic modality.

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An equation specifically applicable to shortwave diathermy is obtained by solving equations 7.5 and 7.3. We then have: ρ.c.∆T = E.ir.∆t, which on rearranging gives: ∆T E.ir = ∆t ρc

.... (7.7)

Equation 7.7 shows that the initial rate of increase in temperature (∆T/∆t) in shortwave diathermy depends on four factors: * * * *

E, the field intensity at the point ir, the magnitude of the real current density at the point ρ, the density of the tissue c, the specific heat capacity of the tissue

Once the temperature of any tissue has increased appreciably two things happen: *

The body's temperature regulation mechanism responds. Blood vessels dilate, circulation is increased and more heat is transferred from the tissue.

*

Heat is transferred by the blood and tissue fluids to adjacent cooler tissues.

Both of these effects lower the rate of increase in temperature. Eventually, the stage is reached where the temperature ceases to increase. A new dynamic equilibrium is achieved where the net production of heat is once again balanced by the net transfer from the tissue. Figure 7.17 illustrates the temperature variation during treatment. There is a transient period during which the tissue temperature increases, followed by a steady state where a constant (elevated) temperature is produced. The transient period for tissue volumes of interest in physiotherapy is typically of the order of twenty to thirty minutes

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(see Lehmann (1982), chapter 10). Thus for treatment times of up to several minutes, equation 7.7 gives a reasonable approximation to the real physical situation. Application of equations 7.1 and 7.7 to quantitative prediction of the rate of heating and rate of temperature increase in different parts of tissue is difficult. The difficulty arises in the calculation of the field intensity in a particular area. For a review of results obtained using various approximations see A. W. Guy in J F Lehmann (1982). In patient treatment, shortwave diathermy remains something of an art as well as a science. The physiotherapist must use a knowledge of anatomy together with knowledge of the electrical properties of tissues to determine the optimum placement of electrodes or coil to give the required field pattern. Once the field pattern is selected, the physiotherapist uses a knowledge of the relative heating of the tissues and the patient's report of a sensation of warmth to adjust the intensity of the applied field to an appropriate level. With this procedure it is not possible to accurately monitor dose or dose rate for the individual tissues. Since this is a problem common to all diathermic modalities we will defer further discussion of dosage until chapter eleven.

Physiological Effects The therapeutic value of shortwave diathermy arises from the physiological response of tissues to an increase in temperature. A number of physiological responses are found: *

Figure 7.16 A simple model for tissue temperature variation during treatment.

at the cellular level an increase in temperature increases the rate of biochemical reactions. Thus cellular metabolism is increased - there is an increased demand for oxygen and nutrients and the output of waste products is increased.

SHORTWAVE DIATHERMY

*

blood supply is increased. A number of factors determine this response. The increased output of cellular waste products triggers dilation of the capillaries and arterioles. The temperature increase itself causes some dilation, mainly in the superficial tissues where heating is greatest. In addition, stimulation of sensory nerve endings (again mainly in the superficial tissues) can cause a reflex dilation.

*

a rise in temperature can induce relaxation of muscles. If there is abnormal muscle activity caused by pain, for example, repeated treatment with shortwave diathermy can be beneficial. The treatment helps to interrupt the vicious circle of pain producing muscle activity which in turn produces more pain and so on. A number of factors may contribute to relaxation: the direct effect of heat on muscle tissue, the removal of any accumulated metabolites due to increased circulation and the sedative effect of heat on sensory nerves.

*

the response of sensory nerves to heat is useful for the relief of pain generally. Mild heating appears to inhibit the transmission of sensory impulses via nerve fibres. In addition, when pain results from inflammation of tissue an increase in the rate of absorption of exudate with increase in temperature can result in a secondary pain-relief effect.

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Some claims have been made that additional non-thermal effects can be produced under the conditions used for therapy. As yet there is no clinical evidence for these claims. Non-thermal effects seem to have been demonstrated using pulsed shortwave treatment when the peak power level is significantly higher than used for diathermy. The few published comparative studies indicate little or no nonthermal effect at the low continuous power levels of conventional shortwave field treatment. These points are considered further in chapter 8 following.

SHORTWAVE DIATHERMY EXERCISES 1

2

3

4

5

Figure 7.1 shows a schematic diagram of shortwave diathermy apparatus. (a)

Briefly explain how the apparatus produces a high-frequency alternating electric or magnetic field.

(b)

What is the function of inductors L1 and L2?

c)

Why is the capacitor in the patient tuning circuit a variable one?

(a)

Why is it necessary to tune shortwave diathermy apparatus with the patient coupled to the machine?

(b)

What is the advantage of automatic versus manual tuning of shortwave diathermy apparatus?

Figure 7.2 illustrates the response of ions, polar molecules and non-polar molecules to a high-frequency alternating electric field. (a)

Briefly describe the movement of each kind of molecule in the field.

(b)

How is the movement related to heat production in a material?

(c)

Which kind of movement is associated with greatest heat production and which with least heat production?

(a)

What is meant by the terms 'real current' and 'displacement current'?

(b)

Consider the movement of ions, polar molecules and non-polar molecules in an alternating electric field. Describe the relationship between each kind of movement and real and displacement current.

(a)

Consider each of fatty tissue, muscle and bone in the shortwave diathermy field. Is current flow in each tissue best described as real or displacement current?

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SHORTWAVE DIATHERMY (b)

On the basis of your classification which tissue would be associated with maximum heat production?

(c)

Describe the complications (as far as prediction of heat production is concerned) caused by the presence of Iymphatic and blood vessels in fatty tissue.

6

Figure 7.4 shows current pathways in a model for an arm or leg. Describe the principal factors determining the relative rate of heating of each tissue layer.

7

A patient's lower limb is enclosed in a solenoidally wound coil as shown in figure 7.6. (a)

Describe the motion of polar, non-polar and ionic molecules when a high frequency alternating current flows through the coil.

(b)

Indicate (with a diagram) the direction of movement of molecules in the limb.

8

If the solenoidally wound coil in question 7 was replaced by a pair of capacitor plates (one above the knee, one below the sole of the foot), what would be the new directions of molecular motion? Draw a diagram to illustrate.

9

Figure 7.8 shows the electric field associated with two adjacent turns of an induction coil

10

(a)

what is the practical significance of this electric field in patient treatment?

(b)

how can the effects of this electric field be minimized?

For shortwave diathermy it is common practice to use electrodes which are somewhat larger than the structure to be treated (figure 7.9). Explain in terms of: (a)

the field pattern produced

SHORTWAVE DIATHERMY (b)

the pattern of heating of tissue.

What are the advantages and disadvantages of using unequal size electrode (figure 7.9(c) )? 11

(a)

It is normal practice to space electrodes as far apart as possible (figure 7.10(b)) in shortwave diathermy treatment. Why is this the case?

(b)

What is the practical limitation on the electrode spacing which can be used?

(c)

Is there any advantage in positioning one electrode close to the patient's tissue as shown in figure 7.10(c)?

12

Consider the electrode arrangements shown in figure 7.11. Explain why the field intensity is non-uniform in diagrams (a) and (c). Under what circumstances will the field intensity be uniform, as in (b)?

13

(a)

Draw a diagram showing a coplanar arrangement of electrodes over tissue and the resulting field pattern.

(b) What are the advantages and disadvantages of coplanar electrode arrangements? (c) 14

197

What practical limit is there on the spacing of coplanar electrodes?

Consider the hollow dielectric between capacitor plates which is shown in figure 7.14. (a)

Explain where heat production is greatest and why.

b)

What technique can be used to produce more uniform heating of the dielectric? Explain.

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SHORTWAVE DIATHERMY 15

16

17

18

When coplanar electrodes are used for patient treatment the tissues can be considered to be in series electrically (see figure 7.15(a)). (a)

what approximations are implicit in this statement?

(b)

draw an electrical equivalent circuit similar to that in figure 7.15(a) for the situation where the electrodes are close together.

(c)

how would bringing the electrodes closer together affect the relative heating rate of muscle and fatty tissue? Justify your answer.

(a)

Explain why, in principle, it is easier to produce selective heating of muscle with an inductive coil rather than capacitor field electrodes.

(b)

what practical constraints limit the selective heating of muscle with an induction coil?

(a)

Explain the meaning of each of the terms in equation 7.6.

(b)

The initial rate of temperature increase in fatty tissue in an experiment is found to be double that of muscle. Assume that the densities of each tissue are the same and that muscle has a 50% greater heat capacity and calculate the relative rate of heating of the tissues.

The relationship between heat production (∆Q) and current flow (I) in a conductor is given by Joule's law: ∆Q = V.I.∆t where V is the potential difference across the conductor and ∆t is the time interval for which current I flows. (a)

Show how equation 7.3 can be obtained as an alternative form of Joule's law.

(b)

An electric field intensity of 100 V.m-1 in a conductor results in a current density of 50 A.m-2 . Use equation 7.2 to calculate the amount of heat produced in a 30 second time interval. You may assume that the current is entirely real.

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(a)

Explain the meaning of each of the terms in equation 7.7.

(b)

An electric field of intensity 200 V.m-1 in a material results in a real current density of 50 A.m-2. The mass density of the material is 900 kg.m-3 and its specific heat capacity is 4.0 kJ.kg-1.K-1. Calculate the initial rate of increase in temperature (∆T/∆t) of the material.

20

A block of conducting material is placed in an electric field. The field intensity in the material is 300 V.m-1 and the resulting real current density is 120 A.m-2. If the material has a density of 1000 kg.m-3 and it has a specific heat capacity of 3.8 kJ.kg -1 .K -1 , calculate the initial rate of increase in temperature of the material (using equation 7.7).

21

Equation 7.7 describes the initial rate of increase in temperature of tissue in shortwave diathermy treatment. Describe the physiological response to the initial temperature rise and the effect this has on the subsequent rate of increase of temperature (figure 7.16).

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8 Non-Diathermic Fields PULSED SHORTWAVE APPARATUS Most shortwave diathermy machines offer the option of pulsed or continuous output. With continuous output, tissue heating is maximized as energy is transferred continually from the apparatus to the tissue. With pulsed output, energy is delivered in brief bursts with a long off-time between the bursts, so the average energy transferred is low. Pulsed shortwave is classed as non-diathermic in that the average power dissipated in the patient's tissue is too low to produce the appreciable temperature rises associated with traditional (continuous) shortwave treatment. For this reason it is described here rather than in the previous chapter where the emphasis was on the use of electric and magnetic fields to produce deep heating. Consider an example. If a burst of high frequency AC with a duration of 1 ms is generated at a burst frequency of 50 Hz, the period of each repetition ('on' time + 'off' time) is 1000/50 = 20 ms so 'on' time is 1 ms and the 'off' time is 19 ms and consequently the average energy is 1/20th of the peak energy (figure 8.1). Figure 8.1 (a) continuous and (b) pulsed output from shortwave machines.

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It is asserted, though it has not been demonstrated, that pulsed shortwave is clinically beneficial because there are physiological effects of a 'non-thermal' nature which are produced by the bursts of electromagnetic energy. Figure 8.2 shows the essential features of pulsed shortwave apparatus. It is the similar to figure 7.1 but with the addition of a gating circuit to control the output of the 27.12 MHz sinewave generator. The gating circuit switches the sinewave generator on and off at the operator-chosen frequency (50 Hz in the previous example). It also controls the burst duration (1 ms in the previous example). Some machines have a predetermined burst duration, others allow operator selection. Components and subsections within the yellow rectangle in figure 8.2 are inside the apparatus. The functions of each subsection are as follows: *

The sinewave generator produces a sinusoidal AC signal at the internationally approved frequency of 27.12 MHz [see chapter 5 and chapter 7].

*

The gating circuit generates rectangular pulses to control the gain of the sinewave generator. The pulse frequency can normally be adjusted in the range approximately 1 Hz to 200 Hz.

*

A power amplifier in the sinewave generator circuit amplifies the signal to a level suitable for driving the patient circuit. The pulse power is high but the average power is low.

*

The resonant circuit (the patient tuning circuit)

Figure 8.2 Pulsed shortwave diathermy apparatus (schematic).

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couples energy generated by the apparatus to the patient. Its operation was described in chapter 7. Output from the apparatus is applied to the patient using electrodes or an induction coil, in the same way that conventional (continuousmode) shortwave diathermy is applied. *

A power supply is needed to convert mains supplied 50 Hz AC to DC of whatever voltage is required to power the gating circuit, sinewave generator and amplifier.

Pulsed shortwave is described as 'non-diathermic', meaning that it does not produce deep heating. The rationale is as follows. Consider and compare two pulsed shortwave machines. A difference between them is in the pulse frequencies which can be selected and the pulse width. Both have a peak power output of approximately 1000 watts. Machine 1 has a pulse width of 65 microseconds and a pulse frequency selectable between 80 Hz and 600 Hz. Machine 2 has a pulse width of 400 microseconds and a frequency range from 15 Hz to 200 Hz. Suppose machine 1 was set to a frequency of 80 Hz. The pulse width is fixed, in the machine, at 65 microseconds so the output is on for a total of 65 x 80 = 5200 microseconds each second. When the intensity control is set to deliver a maximum output of 1000 W pulses the average power output is only: 5200 x 10-6 x 1000 W = 5.2 W 1 This is a tiny fraction, approximately 0.5%, of the peak power.

5.2 W is not a high power level. Imagine shining an ordinary battery-operated torch at your skin from a short distance. The heat and light energy produced by the torch has little impact on your tissues.

At a frequency of 600 Hz the average power at maximum output rises to: 65 x 600 x 10-6 x 1000 W = 39 W 1 This is 39/1000 W which is still only 3.9% of the peak power.

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The average power is low and is only a small fraction, between 0.5% and 4%, of the peak power per pulse. Machine 2, with a pulse width of 400 microseconds, has a frequency range of 15 Hz to 200 Hz and a peak power output of 1000 W. Calculations similar to the previous examples show that the maximum average power varies between 6 watts (at 15 Hz) and 80 watts (at 200 Hz). Again, the average power is low and is only a small fraction, between 0.6% and 8%, of the peak power per pulse. The low power levels of pulsed shortwave ensure that gross heating effects, due to an appreciable increase in tissue temperature, do not occur.

The calculation of average power is left as an exercise for the reader. It is the same as the previous calculation, but with different frequencies and burst durations.

Effects of Pulsed Shortwave Fields Pulsed shortwave treatment is advocated as therapeutically beneficial due to nonthermal effects. Unfortunately, the advocates seem, in the main, to be the manufacturers of the equipment, rather than independent researchers who have carried-out proper studies. The few studies which have been undertaken include human and laboratory animal observations at the tissue level. Healing of experimentally produced skin wounds and haematomas in laboratory animals has been shown, in one study, to be promoted by pulsed shortwave treatment. Another study showed that human soft tissue injuries responded more rapidly in comparison to control (untreated) groups and groups of patients receiving a similar dose (but not dose rate) of continuous shortwave treatment. Some promising results have also been obtained in studies of rate and extent of nerve regeneration in laboratory animals. These results are indicative of therapeutic benefit but, due to their small number, are by no means conclusive. Proponents of pulsed shortwave have argued that another diathermic modality, ultrasound (chapter 10, following), has been shown to be useful when applied in

See chapter 10 of JF Lehmann (Ed) 'Therapeutic Heat and Cold'. Williams & Wilkins (1982) for more details of these studies.

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pulsed mode for the treatment of a number of conditions where heating as such is either contraindicated or of dubious value. The argument is that if ultrasound can used to advantage in pulsed mode, where non-thermal effects are the explanation for any therapeutic benefits, then pulsed shortwave should also be beneficial. This arguments is based on two questionable premises. First that 'non-thermal' benefits of ultrasound treatment actually exist and second (perhaps more importantly) that non-thermal effects will also be produced by pulsed shortwave treatment.

Biophysical Mechanisms Although the evidence base for pulsed shortwave treatment is small, a biophysical argument can be made for possible non-thermal effects of pulsed shortwave. The response of ions, polar molecules and non-polar molecules to an applied electric or magnetic field is well understood (chapter 7) and heat production in the applied field is readily explained. What is not known is how these same molecules responding in a biological environment can produce non-thermal cellular effects of therapeutic value. When electromagnetic energy is applied in brief bursts, the ions, polar molecules and non-polar molecules will respond equally briefly, with vigorous movement during the burst and a dying-down of activity between bursts. During the bursts, we would expect considerable molecular movement which would not increase the average temperature appreciably but which would markedly increase the instantaneous temperature.

In the absence of such effects, pulsing the output so that tissue heating is a minimum would be of little therapeutic value.

With pulsed shortwave, the instantaneous temperature increase is high but the average temperature increase is low.

One could reasonably speculate that the transient excitation might affect concentration gradients, movement of molecules across the cell membrane and changes in membrane permeability in either or both of excitable cells and non-excitable cells. There may also be transient thermal effects on the synaptic junctions of nerve cells. These ideas remain speculative in the absence of appropriate experimental studies.

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Optimum Treatment Parameters The operating frequency of pulsed shortwave apparatus is 27.12 MHz. This particular frequency is used because there is international agreement that it be allocated for use in diathermy and it is known to produce heating at depth. There is no evidence to suggest that 27.12 MHz is the most appropriate frequency for clinical use. Another frequency might be optimum in terms of depth efficiency of heating. As yet, only limited experimental work has been conducted on the effect of different frequencies in the high frequency range. The burst frequency range from 10 Hz to 100 Hz or more may well turn out to be a clinically useful range as it overlaps the 'biological' frequency range. That is, the range of frequencies associated with best response from excitable cells. Unfortunately, comparative studies of the effect of different pulse frequencies (and burst durations) have yet to be conducted. The value of pulsed shortwave is thus questionable. Despite the long history of commercial availability and use, the question of therapeutic benefit remains to be answered. Therapeutic benefits seem possible but have not been adequately demonstrated.

LOW FREQUENCY PULSED MAGNETIC FIELDS During the 1980s, low frequency pulsed magnetic field (PMF) apparatus gained some popularity as a therapeutic modality. The apparatus consists of a signal generating circuit (a pulse or sinewave generator), an amplifier circuit, a patient circuit and a power supply (figure 8.3). Subsections within the apparatus are shown in the yellow rectangle. *

The signal generator circuit produces either low frequency sinusoidal AC or DC pulses of low frequency. The selectable frequencies are in the 'biological' frequency range up to about 100 Hz.

This limitation, of course, applies to both pulsed and continuous shortwave.

NON-DIATHERMIC FIELDS

*

Current from the signal generator circuit is amplified and delivered to an induction coil, creating an intense, pulsed magnetic field.

*

A power supply converts mains-frequency AC to the DC which is needed to power the signal generator and the amplifier circuit.

The part of the patient to be treated is placed within the induction coil. Large coils are used to enclose the trunk for treatment of, for example, low back pain. Smaller coils are used to treat smaller regions such as parts of the limbs As with pulsed shortwave, the advocacy and the recommendations for treatment seemed to come from the manufacturers of the equipment, rather than independent researchers. An argument has been presented that pulsed magnetic field (PMF) therapy is of benefit for the treatment of musculo-skeletal disorders in general and bone-healing in particular. The evidence of value for the treatment of musculo-skeletal disorders is lacking. The evidence for promotion of bone-healing is more convincing.

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Figure 8.3 Low-frequency pulsed magnetic field apparatus (schematic).

Promotion of Bone Healing In the mid 1960's it was found that hydrated living bone is a piezo-electric material: when stressed the bone becomes charged. The phenomenon is illustrated in figure 8.4. When a bending load is applied, one surface of the bone is subject to a compressive stress (the top surface in figure 8.4) and the opposite surface is subject to a stretching or tensile stress. The bone responds by becoming

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negatively charged on the surface under compression and positively charged on the surface under tension. This observation provides the basis for an explanation of the biological phenomenon of stress remodelling; the mechanism by which bones or areas of bone respond to stress by growing in size and load bearing capability. It was reasoned that tiny current produced as a result of the piezoelectrically induced potentials can stimulate bone growth or resorption. Subsequent experiments demonstrated that small (microampere range) currents applied by implanted electrodes could also promote bone formation. Direct current promoted bone formation near the cathode, alternating current promoted bone formation near both electrodes. Two problems associated with the use of implanted electrodes are the risk of infection and the localization of bone formation in the vicinity of the electrodes. These problems are overcome by inducing current flow in tissue using a pulsed magnetic field. As the magnetic field increases and decreases, eddy currents (chapter 6) are produced in the tissue.

Induced Current in Tissue When considering the effects of low frequency pulsed magnetic fields it is important to make the distinction between the voltage waveform produced by the apparatus, the current waveform in the induction coil and the current induced in the patient's tissue. If a rectangular voltage waveform is applied to an induction coil the current waveform will not be perfectly rectangular, but more rounded. This is because when the voltage suddenly changes it takes a finite time for the current through the coil and the magnetic field around the coil to change correspondingly. The larger is the inductance of the coil, the longer it takes for the change to be complete. Figure 8.5 shows the relationship between voltage and current in the coil for different values of

Figure 8.4 The piezoelectric effect in bone.

Extensive case studies have established the effectiveness of pulsed magnetic fields for the treatment of non-united fractures.

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inductance. The larger is the inductance, the longer it takes for the current to increase to maximum. The rate of increase in current intensity in the coil is important because this determines the induced current in tissue. Eddy currents are produced in tissue as a result of a changing magnetic field (chapter 6). When the magnetic field is constant, no current will be induced. Thus when the current in the coil is changing, and only when it is changing, a current will be induced in tissue. Figure 8.6 shows the relationship between induced current and current in the coil for the two waveforms shown in figure 8.5. When the coil current suddenly starts to increase, the rapid rate of increase (figure 8.6a) results in a high induced current. The magnetic field around the coil builds-up rapidly so the induced current is large. The rate of increase then drops rapidly and the induced current drops accordingly. A current spike is induced in tissue. When the coil current suddenly decreases, a current spike of the opposite polarity is induced due to the decreasing magnetic field intensity. The more rapidly changing coil current in 8.6(a) induces large current spikes but these are of short duration as the coil current rapidly reaches a steady value. The more slowly changing coil current in 8.6(b) induces current spikes which are smaller in amplitude but of longer duration.

Figure 8.5 Voltage and current waveforms for (a) small and (b) large value inductances.

Figure 8.6 Current in an induction coil and resulting current induced in tissue for (a) small and (b) large value inductances.

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The size of the induced current depends on the rate of change of the magnetic field and thus on the rate of change of current in the coil. If a sinusoidal current is applied to an induction coil the induced current will have the same shape but be shifted in phase. This is because the rate of change of a sine waveform is another sine waveform phaseshifted by one quarter of a wavelength; in other words a cosine waveform. This is shown in figure 8.7(a). Figure 8.7(b) shows the rectangular current waveform induced when triangular waveform is passed through an induction coil. A rectangular waveform is induced because the triangular waveform is alternately increasing at a constant rate then decreasing at a constant rate. The induced current is alternately constant and positive then constant and negative.

Treatment Parameters No definite statements can yet be made regarding the most appropriate current waveform for magnetic field therapy. Even in the case of bone repair, where the clinical evidence of effectiveness is substantial, there is uncertainty as to the best waveshape and

Figure 8.7 Induced current waveforms for (a) sinusoidal and (b) triangular currents in an induction coil.

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frequency. Excellent results seem to have been obtained using a burst of highfrequency pulses (frequency approximately 4 kHz) in 5 millisecond bursts repeated at a frequency of 15 Hz. Similar success has been achieved using single pulses and more intense fields wit a pulse frequency of 1 Hz. Optimum treatment parameters are yet to be established. Since heating appears to play no role (the energies involved are too low) one cannot predict effectiveness on the simple basis of total energy transfer. Nor can optimum frequencies be deduced without adequate knowledge of the cellular mechanisms involved. A conclusion is that chronic non-union of fractured bone can be successfully treated with low-frequency pulsed magnetic fields but that its value for the treatment of soft tissue injury remains open to question.

EXERCISES 1

2

Figure 8.2 shows a schematic diagram of pulsed shortwave diathermy apparatus. (a)

Briefly explain the function of each subsection.

(b)

What range of output pulse widths and pulse frequencies are normally provided?

Suppose the peak power output of a pulsed shortwave machine is 900 W and the pulse width is 300 ms. (a)

What is the maximum frequency which can be used if the average power output is not to exceed 50 W?

(b)

For a pulse frequency of 200 Hz, what is the average power output?

NON-DIATHERMIC FIELDS 3

4

5

One justification which is often invoked for pulsed shortwave treatment is that certain biological responses are 'non-linear' and exhibit a 'threshold effect'. (a)

What is meant by the terms 'non-linear' and 'threshold effect'?

(b)

The average power output may be increased either by increasing the pulse frequency or by increasing the peak power output. If threshold effects are important, which of these adjustments would have the greatest biological effect?

Suppose it was established that for pulsed shortwave therapy an optimum pulse frequency was 100 Hz and a desirable peak to average power ratio was 50 to 1. What pulse width would be necessary? (a)

What is a piezo-electric material?

(b)

It is known that direct current promotes bone formation near the cathode. If bone, in vivo, is loaded as in figure 8.4, in what region will bone formation be promoted?

6

Figure 8.3 shows a schematic diagram of low frequency pulsed magnetic field apparatus. Briefly explain the function of each subsection.

7

Consider figures 8.5 and 8.6.

8

(a)

Explain why the current waveforms in figure 8.5 are rounded versions of the voltage waveforms.

(b)

Explain why the induced current waveforms in tissue have the shapes shown in figure 8.6.

The diagrams below show current waveforms in an induction coil placed near tissue.

212

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Draw diagrams to show the waveforms of the induced current in tissue for each of waveforms (a), (b), (c) and (d). 9

Draw diagrams showing the pathways of the induced current for biological tissue placed (a) adjacent to an induction coil (b) within an induction coil.

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9 Sound and Electromagnetic Waves Many kinds of wave motion are found in nature - in this chapter we examine only two sound waves and electromagnetic waves. The two are very different in character but share a number of common properties and it is these common properties which we first consider. We look at how waves are produced in more detail in later chapters.

DIFFERENT KINDS OF WAVES Any kind of wave motion, be it the ripples on a pond, sound or light has four characteristics which are fundamentally associated with the wave. These are: the wavelength, frequency, velocity of propagation and amplitude (or size). Figure 9.1 shows one kind of wave motion; a sinusoidal oscillation travelling in a finely coiled spring. As the spring is shaken up and down, transverse oscillations are produced which travel along the spring at a characteristic velocity, v, determined by the physical properties of the spring. The wavelength, λ, is the distance between peaks of the waves, that is the distance over which the wave repeats itself. The (peak) amplitude is the maximum displacement of the spring from the mean position. Sometimes the peak-to-peak amplitude (which is double the peak amplitude) is specified. In order to sustain the oscillations, one end of the spring must be moved up an down with an appropriate frequency, f. The velocity,

Figure 9.1 Transverse oscillations in a spring.

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wavelength and frequency are related by the wave equation: v = f.λ

.... (9.1)

The wave shown in figure 9.1 is a transverse sine wave - so called because the displacement of the spring is perpendicular or transverse to the direction of propagation. When the oscillations are along the direction of propagation the wave is called longitudinal. Figure 9.2 shows a longitudinal wave generated in a spring. For a longitudinal wave, the wavelength and velocity are easy to determine. The wavelength is the distance between two regions of compression. The velocity is determined by measuring how far a region of compression moves along the spring (∆x) in a known time interval (∆t). One region of compression, moving to the right, is coloured in figure 9.2. The velocity is calculated using the formula v = ∆x/∆t. The amplitude is less apparent but, in the case of a spring, it can be determined by attaching a marker to a point on the spring and measuring how far the marker oscillates back and forth from its mean position. Sound waves are longitudinal compressional waves. By their very nature they require a material medium for their existence as they are displacements of the material medium - solid, liquid or gas - about some mean position. The human ear can detect only a restricted range of sound frequencies, from the lowest tones of an organ, around 16 Hz, up to some 12 to 20 kHz. The upper frequency limit of audibility diminishes with age. Frequencies greater than 20 kHz are termed ultrasonic, although some animals can hear frequencies up to 100 kHz.

Figure 9.2 Longitudinal waves in a spring.

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Figure 9.3 is a pictorial representation of a sound wave. Lines drawn close together represent regions of high pressure. Widely spaced lines represent regions of low pressure (rarefaction). These regions move through any particular medium at a fixed velocity. For example, all sound waves travel at 340 m.s-1 in air, regardless of the sound frequency. In water, the velocity of sound waves is close to 1100 m.s-1. Electromagnetic waves are a very special kind of transverse wave. They consist of a transverse sinusoidal electric field together with a transverse magnetic field. Light, radio waves, microwaves and X-rays are all electromagnetic waves with different frequencies, but the same velocity. The frequency and wavelength can vary but, in a particular medium, the velocity is constant. The speed of propagation of electromagnetic waves in empty space is a universal constant on which much of the structure of modern physics is based.

Figure 9.3 Diagramatic representation of a sound wave.

A convenient representation of an electromagnetic wave is shown in figure 9.4. The sinusoidal electric field E, is transverse to the direction of propagation (arrow labelled v) and also perpendicular to the magnetic field, B. The existence of electromagnetic waves was not suspected until 1864 when the Scottish scientist, James Clerk Maxwell published a theoretical paper in which their existence was predicted. The velocity predicted for these waves turned out to be extremely close to that measured experimentally for light, which led Maxwell to conclude that light itself was an electromagnetic wave. Prior to this scientist since Newton's day had puzzled over the nature

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Figure 9.4 Diagramatic representation of an electromagnetic wave.

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of light: whether it was corpuscular or a wave motion, and if a wave, a wave in what? In many way Maxwell's work formed the keystone of 19th century physics. Maxwell had set himself the task of generalizing all of the accumulated knowledge of electrostatics, electric current, magnetism and electromagnetism: to write a few simple laws from which everything else could be derived. He summarized his findings in a set of four equations which expressed the relationship between electric and magnetic fields. In writing the equations he noticed they had a certain symmetry about them, but that the symmetry could only be made complete by assuming the existence of a hitherto unobserved experimental result: that a changing electric field gives rise to a changing magnetic field. This assumption, together with other known facts of electricity and magnetism gave rise to the four equations which bear Maxwell's name. Not only did Maxwell's equations account for all that was known of electricity and magnetism, they also made one startling prediction: whenever charges are accelerated, an electromagnetic wave is produced. It was previously known that a moving charge produces a magnetic field which disappears when the charge stop moving. The equations predict that in addition an electromagnetic wave is produced if the charge accelerates and once the wave is produced its continued existence and propagation is independent of what subsequently happens to the charge. It is not a great step from this to the conclusion that all electromagnetic waves have their origin in the accelerated motion of charges. Since Maxwell's time electromagnetic waves with frequencies ranging from 5 Hz to 1024 Hz have been produced and used. Although they are produced an detected by seemingly different means and given different names, they all have essentially the same nature. A range of frequencies of electromagnetic waves is referred-to as an electromagnetic spectrum. Figure 9.5 shows such a spectrum and its most familiar regions. Electromagnetic waves with frequencies up to 1012 Hz can be generated electrically. For example the normal AM or FM waves received by a radio are produced by

The mathematics of Maxwell's equations can be found in most textbooks on electromagnetism and will not be discussed here. Rather the focus is on the implications of his equations.

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generating an oscillating electrical current in the transmitting aerial. Electrons are accelerated back and forth along the wire and the result is that electromagnetic waves are produced, radiating from the wire.

Figure 9.5 The electromagnetic spectrum. Production of current by electronic circuitry becomes increasingly difficult at higher frequencies and above 10 12 Hz it is necessary to use alternative methods for accelerating charges and producing the waves. Infrared radiation, sometimes referred-to as 'radiant heat' is emitted by all matter. This is because the atoms and molecules are continually moving. In a solid, for example, the atoms are constrained but are able to vibrate about their mean position. It is this movement energy which we call the heat energy of an object. The atomic jiggling means that charges (negative electrons and positive nuclei) are continually accelerating, so they radiate electromagnetic waves. At normal temperatures, most of the electromagnetic radiation has frequencies in the infrared portion of the spectrum.

There are limits to the speed with which charges can move through electronic (silicon chip) circuits. This limits the frequency of electromagnetic waves which can be produced by such means.

When something is heated, the molecules within it are given more energy and they move or jiggle more vigorously. A consequence is that the electromagnetic radiation produced has a higher average frequency. For example, if a piece of metal is heated from room temperature it first emits only infrared radiation, but as the temperature is increased, the metal becomes red-hot, then white, then blue hot. This is because the

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average frequency of the emitted radiation increases. Higher frequency (visible light) radiation can also be produced by movement the outer-shell electrons of an atom - hence the different colours produced when, for example, different chemicals are introduced into a bunsen flame. If atoms are bombarded with high energy electrons, inner shell electrons can be knocked from their orbitals, producing electromagnetic waves in the ultraviolet and Xray parts of the spectrum. Higher frequency gamma and hard-X radiation can not be produced by knocking electrons from their orbitals, but are produced when atomic nuclei are split, as in a nuclear reactor or nuclear explosion. The spontaneous decay of naturally occurring radioisotopes also results in production of gamma and hard-X radiation, due to the massive acceleration of the fragments when the nucleus is split.

Infrared, visible and ultraviolet light can be produced by heating of materials and temperature elevation is the major factor in determining the frequency distribution of the waves.

Electromagnetic Waves and Safety Any form of electromagnetic radiation can pose a biological hazard but higher frequencies are more dangerous. A distinction is made between ionizing and nonionizing radiation. Electromagnetic waves with frequencies somewhat higher than those of visible light constitute ionizing radiation. When these higher frequency waves are absorbed by matter, electrons are knocked from their orbitals producing ions. If the displaced electrons are involved in bonding atoms together in a molecule, the bond will be broken and the molecule will be damaged, sometimes split. Thus higher frequency electromagnetic waves can cause molecular disruption. In most instances, molecular disruption will not harm cells or tissues as the damaged molecules can be removed and replaced. But if the disrupted molecule is DNA, a mutation can result. This is why exposure to ionizing radiation is associated with cell mutation and tissue tumours (cancer). The medical use of ionizing radiation involves a risk/benefit analysis. X-radiation is very useful for diagnostic imaging but there is no 'safe' level of exposure - rather the risk is proportional to the dose. Ultraviolet radiation is useful for treating certain conditions (see chapter 11), but again there is no 'safe' level of exposure. In

Mutation may result either in cell death, cells with suboptimal function or daughter cells which are cancerous (replicate uncontrollably).

SOUND AND ELECTROMAGNETIC WAVES assessing what is an acceptable level of exposure or dosage, normal environmental exposure is a consideration. If the treatment does not add appreciably to the natural burden then it may well be considered acceptable. The question is,. what is an 'appreciable' increase and how does this weigh against the benefits of treatment? In contrast to ionizing radiation, non-ionizing radiation does have safe levels of exposure. At frequencies less than those of visible light, the principal effect of wave absorption is heating, so the risks are simply those associated with temperature elevation. Thus provided the temperature increase is within the physiological range, no harm will normally occur. In this sense, exposure to non-ionizing radiation is no more harmful than any other form of heating. A potential hazard with exposure to non-ionizing radiation is that it could stimulate cell proliferation in malignant tissue, simply as a result of heating. It is for this reason that any form of therapeutic heating is contraindicated when tissue malignancy is known or suspected. A second potential hazard is focussing of a beam of electromagnetic waves, which will result in concentration of the wave energy in a particular region, producing a local 'hot-spot'. This is discussed further in chapter 11. A general conclusion is that, provided the temperature increase in any region is below a physiologically harmful level and the tissue is non-malignant, treatment with non-ionizing radiation is quite safe.

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Low levels of naturally occuring ionizing radiation are ever present. Sources include sunlight, cosmic radiation and naturally occuring radioisotopes. It is no coincidence that visible light is close to the boundary between ionizing and non-ionizing radiation. Vision relies on light sensitive molecules (opsins) which split in a particular way when they absorb light. The 'damage' is, however, reversible.

WAVE TRANSMISSION AND ABSORPTION Having talked a little about sound and electromagnetic waves and considered some (but by no means all) hazards, we now address some fundamental questions about how these waves propagate and how they interact with and are absorbed by matter. Here the emphasis is on wave transmission and absorption. Hazards associated with particular frequencies of sound and electromagnetic waves are considered in more detail in later chapters. One striking difference between sound and electromagnetic waves is that a sound

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wave, being a periodic vibration of atoms or molecules, relies on a material medium for its existence and propagation. Electromagnetic waves on the other hand, require no material medium for their transmission. Thus we can see our sun, the stars and distant galaxies but cannot (even in principle) hear them! In applications to therapy this difference need not concern us. We examine what happens to both kinds of wave in a material medium. From everyday experience we know that sound and light are absorbed as they pass through materials. Ordinary window glass absorbs very little visible light - though it certainly absorbs some - but absorbs ultraviolet and infrared radiation quite strongly. Sound is absorbed by brick walls. In this case we find that low frequency sound is not absorbed as readily as higher frequencies. Absorption is related to the amount of absorbing material so it must be related to the density of the absorber. But it is also related to some other property of the material - otherwise why does glass transmit light while cardboard or paper does not?

Molecular Motion in Matter To gain some insight into the absorption process we consider the motion of molecules making up a material. At any temperature above absolute zero the molecules will be in a state of agitation oscillating back and forth and rotating. In addition, for molecules of more than one atom, vibrations of atoms relative to each other is possible. Figure 9.6 illustrates some of these modes of movement. Each of these three kinds of motion has a certain average frequency associated with it. For example if we consider a rotating molecule in a liquid, then as a result of its motion and the motion of other molecules it will suffer frequent collisions. In many instances the collisions will result in a change in the frequency of rotation of the molecule. Thus when we specify an average frequency of rotation we know that at any one instant some of the molecules will be rotating with frequencies much higher than the average, and some with frequencies much lower. A molecule may have a high

Figure 9.6 (a) translational oscillation (b) rotation and (c) Internal vibration of a diatomic molecule.

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frequency of rotation at one instant, suffer a collision and lose some rotational energy to the other molecule, thus changing to a lower rotation frequency. The same is true for vibration of the molecules - there is continual transfer of the vibrational energy back and forth between molecules. The motion of molecules within a material is, of course, what we measure as the heat energy of an object. As we heat up a material the energy we put in results in greater agitation and thus greater kinetic energy of the molecules. The extent to which energy is shared between the different modes of movement will depend on whether the material is a solid, liquid or gas and how many atoms make up molecule. For example in the case of a large protein molecule with many atom and many bonds, a significant proportion of the heat energy will appear as internal vibrations of the molecule.

Sound Waves in Matter What happens then to a sound (or ultrasound) wave as it travels through a medium? In generating a sound wave we are producing mechanical vibrations - an oscillating displacement of the molecules - with a specific frequency. Consider what happens when the sound frequency is the same as that of some of the molecules. The sound wave is oscillating the molecules in a particular direction in the medium (the direction of propagation) while the molecules are naturally oscillating in all directions and these directions are continually and randomly changing as a result of collisions. The tendency is for the collisions to randomize the direction of sound vibrations and so convert sound energy into heat energy. If any natural oscillation of the molecule corresponds in frequency to the sound wave then the sound will be rapidly absorbed in the medium.

This is analogous to the situation with resonant circuits (chapter 2) where if two circuits have the same resonant frequency, energy can be transferred very efficiently between them.

Even if the sound frequency differs somewhat from any average frequency of molecular movement the natural spread of oscillation frequency of the molecules will enable some energy to be absorbed. In addition if the difference in frequency of two natural modes of molecular oscillation is equal to the sound frequency, energy can be absorbed in converting one frequency of oscillation to the other.

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Electromagnetic Waves in Matter What happens as an electromagnetic wave travels through a medium? Since the wave consists of an alternating electric and magnetic field as in figure 9.4 we would expect the effect of the wave on the medium to be similar to the field effects discussed in chapter 7. Non-polar molecules will polarize in alternate directions in the alternating field, polar molecules will rotate back and forth and ions will try to move in the field direction. The energy losses in these processes - discussed previously in chapter 7 - will result in electromagnetic energy being converted into heat energy. The absorption of electromagnetic energy as a wave travels through a medium will thus depend on the frequency of the wave and the electric and magnetic properties of the material - the dielectric constant, conductivity and permeability. Using this (somewhat simplified) model we predict that biological tissues with low dielectric constant and conductivity, such as fatty tissue will absorb electromagnetic energy to a lesser extent than substances such as muscle and other tissues with a high dielectric constant and conductivity.

Figure 7.2 shows the response of ions, polar molecules and non-polar molecules to a high frequency alternating electric field.

PENETRATION DEPTH In general, for any kind of wave of a certain frequency, we find that the wave energy decreases exponentially with distance. Mathematically this is written: E = Eoe-x/δ

.... (9.2)

Where Eo is the original energy and E is the energy remaining after the waves have travelled a distance x through the medium. The quantity δ is called the penetration depth of the waves in the medium. It depends on the frequency of the wave and the properties of the medium through which the wave travels. The quantity e is a constant which crops-up in any mathematical description of exponential increases or decreases, in the same way that π crops-up when we are dealing with circular geometry.

e, like π is an irrational number, it cannot be expressed as a whole number or a simple fraction. Its value, to an accuracy of four significant figures, is 2.718.

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Figure 9.7 shows a graph of E against x for an exponential decrease. To see what is meant by equation 9.2 and the term 'penetration depth', try substituting different values for x into equation 9.2. *

when the distance x is zero, e-x/δ is eo = 1 since any number raised to the power zero is one. Thus E is equal to Eo the original energy, as we might expect.

*

when x = δ, e-x/δ is e-1 = 1/e = 1/2.718 = 0.37. Equation 9.2 then becomes E = Eo x 0.37. In other words the wave energy is reduced to 37% of the incident energy at a distance x equal to δ, the penetration depth.

*

when x = 2δ, e-x/δ is e-2 = 1/(2.718)2 = (0.37)2 . Equation 9.2 then becomes E = Eo x (0.37)2. In other words the wave energy is reduced to 14% (37% of 37%) of the incident energy.

The calculations show that as the wave travels through a material the energy is progressively absorbed. At a distance δ (the 'penetration depth') the wave energy is decreased to 37% of the original energy. At a distance 2δ the wave energy is reduced to 37% of 37% of the incident energy and so on. In other words the wave energy is reduced by 63% every time the wave travels a distance δ in the medium. The wave energy is never completely absorbed but is reduced by a certain fraction with every centimetre it travels through the material. Clearly we cannot specify 'depth for complete absorption' of the wave energy as this will never occur. Instead we specify the penetration depth as the depth required to absorb 63% of the incident wave energy.

Figure 9.7 Graph showing an exponential drop in energy, E, with distance, x.

An example. The penetration depth, δ, of 2000 MHz microwaves in fatty tissue is

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5.3 cm. Use equation 9.2 to calculate the energy remaining after travelling a distance of (a) 2 cm and (b) 10 cm through fatty tissue. (a)

For a distance x of 2 cm the wave energy, E, is given by equation 9.2 as E = 0.69 E = Eoe-2/5.3 = Eoe-0.38 = 0.69Eo so Eo The energy remaining after travelling a distance of 2 cm in fatty tissue is 69% of the incident energy.

(b)

For a distance x of 10 cm the wave energy E is given by E E = Eoe-10/5.3 = Eoe-1.9 = 0.15Eo so = 0.15 Eo The energy remaining after travelling a distance of 10 cm in fatty tissue is 15% of the incident energy.

Some authors prefer to specify a 'half-value depth' rather than a penetration depth to describe the rate of absorption of wave energy. The relationship between half-value depth and penetration depth can be calculated from equation 9.2 as follows. The half-value depth, the thickness required to reduce the wave energy by 50%, is d1/2 where Eo = Eoe-d1/2/δ 2 In other words we have substituted E = Eo/2 (50% of Eo) when x = d1/2 into equation 9.2. 1 = e-d1/2/δ Cancelling the Eo on each side gives 2 1 d d d Taking logarithms to the base e we have In = - δ1/2 i.e. In 2 = δ1/2 so δ = 1/2 2 ln2 hence δ = 1.44 d1/2

The half-value depth, d1/2, is the thickness of material required to absorb 50% of the incident wave energy.

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That is, the penetration depth is obtained from the half-value depth simply by multiplying by 1.44.

Ultrasound and Microwaves In the frequency range of therapeutic interest, microwave and ultrasound radiation share two common features: *

their penetration depths in fatty tissue are much higher than in muscle (or other tissues with high water and ion content).

*

as the frequency increases the penetration depth decreases. In other words as the wavelength decreases so does the penetration depth.

Table 9.1 shows values of the penetration depth, δ, for different frequencies of ultrasound and microwave radiation in different body tissues. It is clear from the table that microwaves and ultrasound are true diathermic modalities; that is, the waves are able to penetrate deeply into tissue. A significant proportion of the wave energy will be available for heating of muscle and other tissues Iying beneath the subcutaneous fat. In considering which frequencies are most useful for diathermy we would choose a frequency which gives adequate penetration of the waves. We would not, however, aim for a maximum penetration depth since if δ is too large the waves will penetrate right through the tissue with little absorption and thus little heating. The choice of 1 MHz for therapeutic application of ultrasound is a good compromise between adequate penetration

radiation

δ (cm) in fat

δ (cm) in muscle

Ultrasound 1 MHz Ultrasound 2 MHz Ultrasound 3 MHz Microwave 1000 MHz Microwave 2000 MHz Microwave 4000 MHz

7.2 4.8 2.4 7.0 5.3 4.0

1.7 1.2 0.6 1.6 1.2 0.6

δ (cm) in bone 0.22 0.15 0.07 ) similar ) to ) fat

Table 9.1 Penetration depth, δ, for microwaves and ultrasound in body tissues.

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and adequate heating of underlying tissue. The pattern of heating does not, however, depend solely on penetration depth - reflection of the waves plays an important role. We will discuss reflection shortly.

Infrared and Higher Frequency Radiation At higher frequencies than microwaves we have infrared, visible and ultraviolet radiation (see figure 9.5). As we go to these 'optical' frequencies the penetration depth becomes dependent on frequency in a complicated way; there is no longer a smooth increase or decrease in δ with frequency. Let us consider, first, infrared radiation. The infrared region of the spectrum extends from about 3 x 10 11 Hz up to 4 x 10 14 Hz. Traditionally we refer to the wavelength of these radiations rather than the frequency and the unit in popular usage is the nanometre which is abbreviated nm. One nanometre is 10-9 metre. We can convert from frequency in Hz to wavelength in nanometres by using equation 9.1. Since the speed of electromagnetic waves is close to 3 x 10 8 metres per second in most materials the wavelength, λ, in nanometres is related to the frequency, f, by λ =

v 3 x 1017 = f f

.... (9.3)

The infrared region of the spectrum extends from 700 nm wavelength up to about 400 000 nm. For therapeutic application, sources of infrared radiation are used which put out most of their radiation at the end of the spectrum close to visible light: from about 700 nm to about 15 000 nm. This includes both the so called 'near' infrared region, from about 700 nm to 4000 nm and part of the 'far' infrared region. The far infrared region extends from 4000 nm to about 400 000 nm. The penetration depth of near infrared radiation is very small. A maximum penetration depth of a few mm is obtained at about 1200 nm wavelength, and this decreases to a fraction of a millimetre at longer wavelengths. Wavelengths longer than 3000 nm are absorbed by the moisture on the surface of the skin. You may have noticed that the red end of the visible spectrum can be transmitted through the full thickness of your

To obtain this equation we have substituted v = 3 x 108 m.s-1 into equation 7.1 and multiplied by 109 to convert from metres to nm.

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hand: this property does not extend to the infrared region of the spectrum. Visible and ultraviolet radiation have frequencies corresponding to natural frequencies associated with electrons in the outer shells of atoms. Since these electrons are the ones involved in bonding between atoms it is possible for light and ultraviolet radiation to cause breaking of chemical bonds. We may summarize the absorption mechanisms for infrared, visible and ultraviolet radiations as follows: *

Infrared radiation has frequencies corresponding to molecular and atomic motion and to differences in vibration frequency between two modes of motion. It can thus produce heating directly (hence the term 'radiant heat') but has a very small depth of penetration.

*

Visible and near ultraviolet radiation have frequencies corresponding to the difference in natural frequency between two energy states of bonding electrons in atoms. Such radiation can initiate chemical reactions and is only indirectly associated with the production of heat.

*

Far ultraviolet radiation, at higher frequencies than visible and near ultraviolet light, can separate electrons completely from an atom thus producing an ion. For this reason there is some risk of causing irreversible damage to biological molecules.

The absorption mechanism for ultraviolet and visible light means that absorption and hence penetration depth, depends critically on frequency. Certain frequencies will be rapidly absorbed and have small penetration depths while others will not be absorbed so readily and hence have large penetration depths.

There is no sharp dividing line, but the boundary between ionizing and nonionizing radiation is between the near and far ultraviolet regions of the electromagnetic spectrum.

Clearly ultraviolet therapy is of more value in initiating chemical change than in heating as such. Infrared radiation would be indicated when heating of superficial tissue is required.

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WAVES AT BOUNDARIES So far we have discussed the absorption of a wave as it is transmitted through a medium. A knowledge of the rate of absorption of a wave in different tissues is not, however, sufficient to predict the amount of heating in a given tissue layer. Not all of the radiation striking a tissue interface will be transmitted, some will be reflected. In this section we consider the factors determining the relative proportions of reflection and transmission which occur in tissues.

Energy and Impedance First let us ask what determines the energy carried by a wave. The example of a transverse wave produced in a spring (figure 9.1) is a useful one. If the human oscillator in this figure were to shake the spring at a higher frequency this would result in more work being done and thus a greater energy in the wave. If the spring is displaced over a larger distance, resulting in a greater amplitude more energy is also produced in the wave motion. Wave energy depends on both the amplitude and frequency of the oscillations. The energy also depends on the properties of the spring itself. A very heavy spring will require more energy to move it: thus the energy depends on the mass of the spring, or for waves generally on the inertia of the medium. In the case of solids and liquids carrying sound waves the property which specifies the inertia of the medium is the density. Another property which determines the energy needed to produce oscillations in the spring is the elasticity. If waves are produced in a spring the energy needed will depend on its 'stretchiness' or elasticity. If the spring has high elastic compliance it will stretch easily and the restoring force which returns the spring to its original length is small. The two factors of elasticity and inertia together specify the impedance, Z, of a medium. In the case of sound waves in a solid or liquid the impedance is determined by the density and elasticity of the medium.

The energy E of a wave is proportional to the square of the wave amplitude, a. This is written: E α a2 It is also proportional to the square of the wave frequency, f, so we can also write: E α f2

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In the case of electromagnetic waves the properties determining the impedance are the dielectric constant and conductivity. Consider, for example, an ideal dielectric. The molecules will polarize in the electric field. The electron cloud will alternate about the atomic nucleus and be drawn back to the normal position by the electrostatic attraction of negative electrons for the positive nucleus. The polarizing of the atom is analogous to stretching of a spring, and the polarizability (elasticity) is determined by the dielectric constant. For any kind of wave, the relationship between wave energy and the three quantities amplitude (a), frequency (f) and impedance (Z) is E α a2.f2.Z

.... (9.4)

Impedance and Reflection We now consider what happens when a wave strikes a boundary between two media. The example of two different springs connected together is a useful one. If the first spring is made to oscillate with a certain frequency then if any energy is transferred to the second spring, the frequency of the waves in each spring must be identical. This must be so since the springs are fastened together so that the oscillations in the joined ends of each spring are the same. How would we arrange things so as to transfer all of the wave energy from one medium to another? For maximum energy transfer the wave amplitude must be a maximum in spring 2. It cannot be larger than in spring 1 as the springs are joined. So for maximum energy transfer the wave amplitudes must be equal. The frequency is always the same in each medium, so for maximum energy transfer we require both equal amplitudes and frequencies. The wave energy, however, depends not only on frequency and amplitude but also on the impedance of the medium (equation 9.4). It follows that complete energy transfer can only occur when the impedances of each medium are the same.

Although we have used the example of two joined springs, the rule that waves do not change frequency when passing into a new medium applies to any kind of wave motion.

If a wave arrives at a boundary between two media of different impedance only part of

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the wave energy can be transmitted: the rest must be reflected. Figure 9.8 shows two springs of different impedance connected together. Spring 2 is of lower impedance (more elastically compliant and/or lighter) than spring 1. A pulse travels along spring 1 until meeting the low impedance boundary. When the pulse hits the low impedance boundary the end of spring 2 is lifted to the same height as the incoming pulse. The energy transferred to spring 2 is given by equation 9.4 as E α a 2 f2 Z so if the impedance, Z of spring 2 is lower than spring 1 but a and f are the same, a2 f2 Z is less so the transmitted energy is lower than the incident energy. The energy that is not transmitted is reflected, producing the reflected pulse in figure 9.8. Part of the original pulse is reflected and part continues in the original direction. Note that the displacement of the reflected pulse is in the same direction as the original. Figure 9.9 shows the opposite scenario, reflection at a high impedance boundary. In this case spring 2 is heavier and less compliant (has a higher impedance) than spring 1. In this case the stiffness of spring 2 prevents the spring junction from moving as high as the crest of the incoming wave. The effect is that a net downward force is exerted on spring 1 when the pulse reaches the junction. A reflected pulse is generated with the displacement downward rather than upward as in figure 9.8.

Figure 9.8 Reflection from a low impedance boundary.

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The relationship between the mismatch in impedance and the amplitude of the reflected pulse (or wave) is given by equation 9.5. ρ =

ar Z -Z = 1 2 ai Z1 + Z2

.... (9.5)

The reflection coefficient, ρ, is defined as the ratio of the reflected wave amplitude (a r ) to the incident wave amplitude (a i ) and this depends on the difference in impedance of the two media. Only when the impedances are equal (Z1 = Z2 ) will the reflection coefficient be zero and the amplitude of the reflected wave be zero. If there is a mismatch in impedance some wave energy will be reflected. An example. Suppose that two springs are connected together as in figure 9.8 and the impedance of spring 1 is three times the impedance of spring 2. Calculate the proportion of energy reflected at the junction. Substituting Z 1 = 3Z2 into equation 9.5 the reflection coefficient is ρ =

3Z2 - Z2 = 0.50 3Z2 + Z2

This means that the reflected wave has an amplitude one half of the incident wave. Wave energy is proportional to the square of the amplitude, thus the fraction of energy reflected is one quarter.

Figure 9.9 Reflection from a high impedance boundary.

Although we have talked in terms of pulses or waves in a spring to illustrate the application of equation 9.5, the equation holds true for any kind of wave motion

SOUND AND ELECTROMAGNETIC WAVES including sound and electromagnetic waves. In the case of sound waves, Z refers to the acoustic impedance of the medium. In the case of electromagnetic waves, Z is the electrical impedance.

STANDING WAVES Consider what happens if a transverse wave rather than a pulse strikes the boundary between two media. Unless the impedances of both media are identical a reflected wave will be produced travelling in the opposite direction. The two waves will add together, sometimes reinforcing, sometimes cancelling and the result is a standing wave pattern. Figure 9.10 shows the resultant waveform (in red) when two waves of equal amplitude and frequency are travelling in opposite directions The incident wave (blue) travels to the right and strikes a boundary (not shown). The wave is fully reflected, generating a wave (green) travelling in the opposite direction. The waves add together, so that what is actually observed is no longer two separate waves travelling in opposite directions but a single resultant. The resultant is a stationary wave pattern (hence the term 'standing wave'). The wave amplitude varies from instant to instant, changing from zero to maximum and back again, but the wave crests do not change position. At certain points (called nodes) the wave amplitude is always zero while at other points (the antinodes) the wave amplitude alternates rapidly between extreme values.

Figure 9.10 A standing wave produced by interference of two equal size waves travelling in opposite directions.

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In figure 9.10(a) the incident and reflected waves are out of phase by one half of a wavelength. In this case the two waves exactly cancel and the resultant has zero amplitude. An instant later (figure 9.10(b)) the incident wave has moved 1/8th wavelength to the right and the reflected wave 1/8th wavelength to the left. Now the waves are only 1/4 of a wavelength out of phase and the resultant is non-zero. In figure 9.10(c) the waves have moved further: now they are in phase and the resultant has a maximum amplitude. See if you can construct the resultant waveform at two later times when the incident and reflected waves have progressed a further 1/8th wavelength then 1/4 wavelength. For waves travelling at high velocity, the variation from (a) to (d) in figure 9.10 would occur in a tiny fraction of a second and the resulting variation in amplitude would be so fast as to be seen as a blur. This is illustrated in figure 9.11. Notice that in figure 9.11 the nodes and antinodes are readily discerned.

You should find that the resultant is the same as figures 9.10(b) and 9.10(a) respectively.

The nodes are one half of a wavelength apart (as are the antinodes). One wavelength is one sinewave cycle, which is two of the 'beats' in figure 9.11. So half a wavelength is the distance between two antinodes or two nodes. An everyday example of standing wave production is seen with stretched wires or strings (for example guitar strings) which, when plucked, resonate and produce standing waves at any frequency for which the string length is a multiple of half a wavelength. The mismatch in impedance at each end of the string results in almost complete reflection and superposition of the waves results in a standing wave. If a wave is not fully reflected at a boundary (ρ < 1 in equation 9.5) the incident and reflected waves have different amplitudes and the resultant will be a combination of a standing wave and a travelling wave. This is the more

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Figure 9.11 The (blurred) standing wave pattern which would be seen when the incident and reflected waves travel at high velocity.

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usual case with reflection. It is a useful exercise to draw two waves as in figure 9.10 but with unequal amplitude and see what effect this has on the resultant. What is produced is an amplitude modulated oscillation with maxima and minima but no true nodes (figure 9.12).

REFLECTION AND REFRACTION

Figure 9.12 Effect on the standing wave pattern of unequal size incident and reflected waves.

We have seen that a mismatch in impedance results in reflection of waves at a boundary. A difference in impedance also results in the phenomenon of refraction. When a beam of waves is incident on a boundary at a certain angle (i in figure 9.13) the reflected wave will leave the boundary at the same angle. i' in figure 9.13 is the same size as i. The transmitted wave will be refracted: that is, its direction of propagation will change. The angle of refraction, r, will not be equal to the angle of incidence, i. The laws of reflection and refraction arise in most discussions of how light behaves, but these laws are not restricted to optics: they apply equally to any kind of wave motion. All that is required for refraction to occur is that the wave have a different velocity in the two media. The wave velocity is in turn determined by the impedance of the medium. Thus refraction of light occurs when a beam passes from air to glass because of the different velocity of

Figure 9.13 Reflected and refracted waves at a boundary.

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light (and, more fundamentally, electrical impedance) of the two media. To see how the laws arise, consider a beam incident on a boundary as shown in figure 9.14. First we look at the reflected wave. For simplicity, we consider a beam of width AB chosen so that AC is exactly one wavelength and we assume that the wavecrests are synchronized (the results are perfectly general, but the maths is more complicated when the waves are not synchronized and the distances do not match). Waves will be reflected at B while those at A on the same wavefront still have to travel a distance AC before being reflected. This will take a time t where AC = v1t During this time waves reflected at B will have travelled a distance BD where BD = v2t and the new wavefront is DC. Clearly distance AC is equal to distance BD: this is because the velocities of the incident and reflected waves are equal. If the angle of incidence is i then angle ACB is (90-i) - from simple geometric considerations - thus angle ABC is i and AC .... (9.6) sin i = BC In triangle DBC angle DBC is (90-i') and angle DCB is thus i' and i' is given by BD .... (9.7) sin i' = BC Since we know that AC and BD are equal, equations 9.6 and 9.7 together give sin i = sin i' thus i = i' and we have the 'Law of Reflection':

Figure 9.14 Reflection of a beam.

angle of incidence = angle of reflection

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Now consider figure 9.15 where the refracted beam is shown. Waves entering medium 2 at B will travel a distance BD where .... (9.8) BD = v2t in the same time it takes for waves at A to travel the distance AC where .... (9.9) AC = v1t In this case v1 is not necessarily equal to v2 so distance BD is not equal to distance AC. Angle ABC is equal to i (as in the previous example with reflected waves). Hence AC .... (9.10) sin i = BC Similarly, angle BCD is equal to r and BD .... (9.11) sin r = BC Dividing equation 9.10 by equation 9.11 gives: AC sin i = BD sin r and substituting equations 9.8 and 9.9 this becomes v sin i = 1 v2 sin r

..... (9.12)

Which is the 'Law of Refraction' for waves at a boundary.

Figure 9.15 Refraction of a beam.

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Since v1 is not necessarily equal to v2 , sin i is not equal to sin r and so the angle of incidence is not equal to the angle of refraction. The angles of incidence and refraction depend on the relative velocity of the waves in each medium. Equation 9.12 is a less familiar form of the law of refraction. It is more common in the case of light to define an 'index of refraction'. This is simply the ratio of the velocity of light in a vacuum to its velocity in the medium. Equation 9.12 then has v1 and v2 replaced by n1 and n2, the refractive indices of each medium. The refractive index is dictated by the wave velocity in the medium.

The law of refraction is often written: n1 sin i = n2 sin r

For light waves in air their velocity, v1 , is always greater than the velocity, v2 , in a denser medium (glass or whatever). Consequently the angle of incidence is always greater than the angle of refraction.

where n1 and n2 are the socalled refractive indices of each medium.

An example. The velocity of sound in air is 340 m.s-1 and in water is close to 1500 m.s-1 (see table 10.1 in the next chapter). Use equation 9.12 to calculate the angle of refraction when sound waves in air are incident upon water at an angle of 6o. Substituting v1 = 340 m.s-1 and v2 = 1500 m.s-1 into equation 9.12 we have 340 sin i = = 0.23 ..... (9.13) 1500 sin r that is sln i sin r = 0.23

When sound waves enter a denser medium, they travel faster. Electromagnetic waves are slowed in a denser medium.

In this example i = 6o so we have 0.1045 sin 6o = = 0 4548 0.23 0.23 and the angle of refraction, r, is calculated to be 27o. sin r =

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Critical Angle From the previous discussion it is apparent that waves are refracted at a boundary when the wave velocity is different in each medium. The relationship between incident and refracted angle is given by equation 9.12. Consider again the example of sound waves in air incident upon a boundary with water. Equation 9.13 relates the incident and refracted angle in this case. If this equation is used to calculate r for different values of i, a table similar to table 9.2 is produced. The results show a smooth increase in r as i increases in the range 0o to 13o . The value i = 13o is called the critical angle for the air/water system. At this angle of incidence the angle of refraction, r, is 90o. In other words the refracted wave travels along the air/water boundary. For angles of incidence greater than 13o there is no real solution to equation 9.13. Experimentally what we observe is that total reflection occurs; that is, no refracted wave is produced. The critical angle is the largest incident angle for which a refracted wave exists. Although we have used the air/water system as an example, the general conclusions apply to any pair of materials where the wave velocity in medium 2 is greater than in medium 1. In this circumstance, the angle of refraction is greater than the angle of incidence and at a critical angle of incidence the refracted angle will be 90o . For angles of incidence greater than the critical angle, total reflection occurs. The actual value of the critical angle for a given pair of materials is calculated using equation 9.12. An example. The velocity of sound in muscle tissue is 1550 m.s-1 and in bone is 2800 m.s-1 (table 10.1 following). Calculate the critical angle for sound waves incident upon a muscle/bone boundary.

angle of incidence i

angle of refraction r

3o 6o 9o 12o 13o

13o 27o 44o 67o 90o

Table 9.2 Angle of incidence and refraction for sound waves at an air-water interface.

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Using equation 9.12 we have 1550 sin i = = 0.55 2800 sin r At the critical angle the refracted angle is 90o so sin r = 1.00. We thus have sin i = 0.55 which gives i = 34o . Hence the critical angle for the muscle/bone boundary is 34o.

EXERCISES 1

(a)

(b)

2

Figure 9.1 shows transverse oscillations induced in a spring. The oscillations are observed to travel along the spring at a velocity of 2.5 m.s-1. If the end of the spring is vibrated with a frequency of 2 Hz what will be the wavelength of the oscillations? When the spring shown in figure 9.2 is vibrated back and forth at a frequency of 3 Hz regions of compression separated by a distance of 48 cm are produced. What is the velocity of longitudinal waves in the spring?

The frequencies of some of the waves used in therapy are: ultrasound: 1 MHz, microwave: 2450 MHz, infrared: 3 x 10 11 to 4 x 10 14 Hz, ultraviolet :0.8 x 10 15 to 1.6 x 1015 Hz (a) Given that the speed of light is 3 x 10 8 m.s-1 and that of sound 340 m.s-1, calculate the wavelength (or wavelength range) of these waves. (b) For each of these waves, indicate whether the wavelength concerned is closest in size to a house. a human limb. a tissue layer. a cell a protein molecule. (c)

What would be the wavelength of electromagnetic radiation of the same frequency as the ultrasound?

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(a) (b) (c)

Describe the similarities and differences between sound and electromagnetic waves. Give four examples of electromagnetic waves. What are their differences and similarities? On what basis do we distinguished sound and ultrasound?

4

The number of possible modes of vibration of a molecule depends on the number of atoms in the molecule. For example the diatomic molecule in figure 9.6 has three possible modes of vibration. How many modes of vibration has: (a) a single atom? (b) a triatomic molecule? What are the implications of this for absorption of, say, sound energy in liquids made up of diatomic compared with polyatomic molecules?

5

Sound waves in liquid A are absorbed more rapidly than in liquid B. What conclusions can you draw regarding the frequencies of molecular oscillation in each liquid?

6

Consider an electromagnetic wave travelling through a material. (a) Describe the effect of the wave on each of polar molecules, non-polar molecules and ions in the material. (b) If the proportion of ions in the material was increased what effect would this have on the rate of absorption of wave energy? Explain. (c) If the proportion of non-polar molecules was increased what effect would this have on the rate of absorption of wave energy? Explain.

7

The penetration depth of 1 MHz ultrasound in muscle tissue is 1.7 cm. Using equation 9.2 construct a table showing fraction of energy remaining (E/Eo ) at different depths in the muscle. Use a range of values of depth from 0 to 5 cm.

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Plot a graph of E/Eo vs depth and determine the depth at which the energy is reduced to: (a) (b) (c) (d) (e) 8

The penetration depth of 1 MHz ultrasound in fatty tissue is 7.2 cm. Use equation 9.2 to construct a table showing the fraction of energy remaining (E/Eo ) at different depths in the tissue. A suitable range of values of depth is from 0 to 5 cm. Plot a graph of E/Eo versus depth. (a) Compare your graph with that obtained in question 7. What conclusions can you draw about the relative 'absorbing power' of fatty tissue compared with muscle for 1 MHz ultrasound? (b) (c)

9

75% 50% 37% 25% 10%

Use your graph to determine the thickness of fatty tissue required to reduce the wave energy to 75% of the original value. What thickness of fatty tissue is required to absorb 10% of the incident energy?

The penetration depth of 4000 MHz microwaves in muscle tissue is 0.6 cm (table 9.1). Use equation 9.2 to calculate the thickness of muscle tissue required to absorb: (a) 10% (b) 50% (c) 90% of the incident wave energy.

SOUND AND ELECTROMAGNETIC WAVES 10

(a) (b)

Electromagnetic waves with wavelengths in the range 700 nm to 400 nm are termed 'near infrared'. What is the frequency range of near infrared radiation? (See equation 9.3). The portion of the far infrared spectrum used in therapy ranges from 4000 nm to 15 000 nm in wavelength. What is the frequency range of these waves?

11

What are the factors which determine: (a) the impedance of a spring? (b) the acoustic impedance of a material? (c) the impedance of a material to electromagnetic waves (the electrical impedance)?

12

Two springs are connected together as in figure 9.8. A pulse travels along spring 1. After reflection the reflected pulse has an amplitude one fifth of the incident amplitude. (a) What is the reflection coefficient of the boundary? (b) If the impedance of spring 1 is 3 x 10 3 kg.m-2.s-1 what is the impedance of spring 2?

13

Two springs are connected together as in figure 9.8. The impedance of spring 1 is one quarter of the impedance of spring 2. (a) Calculate the reflection coefficient of the boundary. (b) What is the significance of the negative value for the reflection coefficient? (c) What is the fraction of energy reflected at the spring junction? (d) What is the fraction of energy transmitted from spring 1 to spring 2?

14

Calculate the reflection coefficient for two media in contact when: (a) the impedances differ by a factor of 100.

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(b)

the impedances differ by a factor of 10.

(c) (d)

the impedances differ by a factor of 2. the impedances are equal.

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For each of the cases considered in question 14 above calculate the percentage of the original energy which will be: (a) reflected (b) transmitted at the boundary between the media.

16

Figure 9.10 shows the standing wave pattern produced by two waves of equal amplitude travelling in opposite directions. The time interval between each diagram is the same. Construct (graphically) the standing wave pattern at four successive (equal) time intervals.

17

Figure 9.10 shows the standing wave pattern produced when incident and reflected waves are of equal amplitude. Construct (graphically) the corresponding pattern produced when the reflected wave is only 2/3 the amplitude of the incident wave.

18

Ultrasound waves travelling through water strikes fatty tissue at an angle of incidence of 45o. The velocity of sound in water is 1500 m.s-1 and in fatty tissue is 1450 m.s-1 . Use equation 9.12 to calculate the angle of refraction of the waves.

19

Ultrasound waves travelling through muscle strike an interface with bone. The transmitted wave has an angle of refraction of 80o . Given that the velocity of sound in muscle is 1550 m.s-1 and in bone is 2800 m.s-1, calculate the angle of incidence of the ultrasound waves.

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Microwaves strike a tissue with an angle of incidence of 20o. If the ratio: speed of light in air speed of light in tissue is 1.3, calculate the angle of refraction of the waves.

21

Light incident on tissue at an angle of 40o has an angle of refraction of 29o. If the speed of light in air is 3.0 x 10 8 m.s-1 , calculate the speed of light in the tissue (using equation 9.12).

22

(a) (b)

What is meant by the term 'critical angle' for a wave at a boundary? Draw diagrams showing the incident, reflected and transmitted waves when the angle of incidence is (i) zero, (ii) less than the critical angle, (iii) equal to the critical angle and (iv) greater than the critical angle.

23

(a)

The velocity of sound in fatty tissue is 1450 m.s-1 and in muscle, 1550 m.s1. Calculate the critical angle for the fat/muscle interface. The critical angle for an air/water interface is 13o. Given that the velocity of sound in air is 340 m.s-1, calculate the velocity of sound in water.

(b)

24

Consider three parallel layers, fat/muscle/bone, on top of each other. A sound wave travelling through fatty tissue is incident upon the fat/muscle interface at an angle of 20o. Given: vfatty tissue = 1450 m.s-1 vmuscle = 1550 m.s-1 vbone = 2800 m.s-1

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SOUND AND ELECTROMAGNETIC WAVES Calculate: (a) (b)

the angle of refraction at the fat/muscle interface, the angle of incidence at the muscle/bone interface.

Would any wave energy be transmitted at the muscle/bone interface?

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10 Therapeutic Waves: Ultrasound Of the diathermic modalities commonly used in therapy, ultrasound is the most popular. This is not because ultrasound is necessarily the most depth-effective. While ultrasound is a deep-heating modality and more depth-efficient than superficial modalities such as hot packs, infrared lamps or lasers, the depth effectiveness is strictly limited.

Their popularity may be because ultrasound units are relatively cheap, simple to use, compact and portable.

The ultrasound frequencies most commonly used are 1 MHz and 3 MHz. The reasons for these being popular operating frequencies will become apparent in later sections of this chapter. In water and tissues of high water content the velocity of sound is close to 1500 m.s-1 thus the wavelength of 1 MHz ultrasound is (from equation 9.1) about 1.5 mm and that of 3 MHz ultrasound is about 0.5 mm.

PRODUCTION OF THE WAVES The apparatus used to generate ultrasound waves consists of a high frequency oscillator, a power amplifier and a piezo-electric crystal which is mounted in a hand-held probe. A gating circuit is usually interposed between the oscillator and the power amplifier to provide pulsing of the ultrasound output. Figure 10.1 illustrates the arrangement.

Figure 10.1 An ultrasound machine (schematic).

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A power supply is needed to convert mains supplied 50 Hz AC into DC to power the subsections shown. The heart of the circuit is the oscillator which produces high frequency sinusoidal AC (see chapter 5). This current is amplified and applied to a piezo-electric crystal, causing it to vibrate (change in thickness) at the same frequency. The piezo-electric effect was discovered in 1880 by the brothers Pierre and Paul-Jaques Curie. They found that certain crystals display the remarkable property of producing a small potential difference between their faces when subjected to mechanical pressure. The reverse of this effect, namely that when a voltage is applied to a piezo-electric crystal it changes in thickness, was discovered a short time afterwards.

Various crystals display piezo-electric properties but two, quartz and barium titanate (both ceramics) are most useful for practical applications.

All piezo-electric crystals are found to exhibit a resonance effect - that is, they vibrate most efficiently at a certain (resonant) frequency. This natural frequency depends on the dimensions, most importantly on the thickness, of the crystal. The resonant frequency of the oscillator (see chapter 5) is normally adjusted during manufacture to correspond to the crystal's resonant frequency. In continuous mode the gating circuit is not used and the piezo-electric crystal is supplied with high frequency AC continuously. In pulsed mode the AC is applied to the crystal in bursts. The burst frequency is normally 100 Hz; thus the time from the start of one burst to the start of the next is one-hundredth of a second or 10 milliseconds. The duty cycle is the ratio of 'on' time to total time ('on' plus 'off') for the output. In other words the duty cycle is the fraction of time for which ultrasound is being produced. Typical values of duty cycle for apparatus used in therapy are in the range 1:2 to 1:10. An alternative to specifying the duty cycle of pulsed ultrasound is to specify the markspace ratio. The mark-space ratio is the ratio of 'on' time to 'off' time for the output. The rationale for the use of pulsed ultrasound will be discussed in a later section of this chapter.

For example a pulsed ultrasound signal which is on for 5 ms then off for 5 ms has a duty cycle of 1:2 and a mark-space ratio of 1:1. A signal which is on for 1 ms and off for 9 ms has a duty cycle of 1:10 and a markspace ratio of 1:9.

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PATTERN OF THE ULTRASOUND FIELD The ultrasound generator produces a beam of ultrasonic waves by vibration of the metal end plate of the treatment head (the transducer) shown in figure 10.1. The plate is typically a few centimetres in diameter - perhaps twenty or thirty wavelengths. When the diameter of the transducer is many multiples of the wavelength, the sound beam is cylindrical in shape and the beam divergence is low. At a frequency of 1 MHz, an ultrasound beam in water, produced by a typical size transducer (2.8 cm diameter) has a divergence of about 4o. This figure increases to about 40o at a frequency of 100 kHz and 90o at 65 kHz. Thus at frequencies of 65 kHz or less, there is no beam: sound waves radiate in all directions. At MHz frequencies, the sound beam is pencilshaped and almost the same diameter as the transducer. Although the beam has a relatively uniform, cylindrical shape, the relatively small size of the treatment head of typical ultrasound machines results in a marked variation in ultrasound intensity across the width of the beam. To see how this occurs, consider a particular point in front of the transducer such as point A in figure 10.2.

The angle of divergence of a sound beam produced by a flat vibrating disk is given by: 0.61 v sinθ = f.r where v is the wave velocity, f is the frequency and r is the transducer radius.

Every point on the transducer surface will act as a source of sound waves. The total wave amplitude, and hence total wave energy, at point A will depend on the contribution from all points on the transducer surface. Waves from some points will arrive in phase and reinforce each other; others will arrive out of phase and cancel. Figure 10.3 shows waves originating at only two points on the transducer surface: in this case the waves are out-of-phase and cancel. By adding (vectorially) the waves originating from all points on the transducer surface we can calculate the resulting intensity at any particular point. The calculations are made complex by the fact that the surface of the transducer does not remain planar, but flexes and undulates as it oscillates.

Figure 10.2 Interference of sound waves from a radiating source.

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The net result is that the ultrasound field is not uniform. Near the transducer a distinctive pattern of maxima and minima of intensity are produced. Beyond this region (called the near or interference field) the distant field is more homogenous and decreases smoothly in intensity with distance from the transducer. The effect is illustrated in figure 10.3 where the undulations of the transducer surface have been ignored for simplicity. The pattern of intensity was calculated for a frequency of 1 MHz and a transducer of diameter 2.8 cm (area 6.2 cm2 ): with larger diameters the pattern is qualitatively similar with the interference effects extending to greater distances. At an ultrasound frequency of 3 MHz the pattern would again be qualitatively similar but with the interference effects extending over approximately three times the distance shown in figure 10.3. Figure 10.3 shows the variation in intensity of an ultrasound beam at points along the central axis. In the near field, local 'hot-spots' or regions of maximum intensity are separated by 'cold-spots' or regions of minimum intensity.

Figure 10.3 Intensity along the axis of a sound beam for a transducer of diameter 2.8 cm, operated at 1 MHz frequency in water.

Off-axis, patterns of hot-spots and cold-spots are also observed. The location of their maxima and minima are, however, different. Averaged across the beam, the intensity is relatively constant, only decreasing slowly with distance. So at any particular distance, hot-spots and cold spots are produced in different locations across the beam, while the average energy is constant. Figure 10.4 shows another view of the energy distribution in an ultrasound beam. This time a two-dimensional view showing the high intensity regions off the central axis. The shaded areas indicate regions of high local ultrasound intensity. Note that regions of low intensity on the central axis have, alongside, regions of high intensity

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and vice-versa. Most of the ultrasound energy is confined within the area defined by the brown lines. There is a slight convergence of the beam in the near (interference) field and a small divergence in the far field. The complex interference pattern makes it essential that in therapeutic application of ultrasound the transducer be moved around over the area to be treated. If the sound-head (ultrasound transducer) were kept stationary, localized 'hot spots' would be produced in tissue which could result in excessive local heating. By moving the sound-head in circular paths, production of local areas of high temperature rise is avoided.

Figure 10.4 Variation in intensity within the ultrasound beam described in figure 10.3.

If the sound-head is moved in a circular path so as to produce a treated area of at least twice the diameter of the head, hot-spot production will be avoided.

Beam Nonuniformity Ratio (BNR) A quantity of interest is the beam nonuniformity ratio or BNR. This is the ratio of the peak intensity to average intensity of the beam. Because there are always local regions of high intensity, the BNR is always greater than 1. In figure 10.4, the intensity pattern is that which would be produced by a piezo-electric crystal which was about the same size as the ultrasound treatment head. If the crystal were appreciably smaller than the metal end-plate of the ultrasound treatment head, the metal end-plate would vibrate differently and the pattern shown in figure 10.4 would be different. The high-intensity regions would be in different positions and, more importantly, the peaks would be higher. So the BNR would be higher. A low BNR is clearly an advantage but movement of the treatment head is of much

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If a hollow, doughnut-shaped crystal were used, the intensity pattern would again be different to figure 10.4. In this case the highest peak (and the BNR) would be lower.

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more crucial importance in clinical practice. In the following sections we ignore beam nonuniformity and assume that the treatment head is moved to produce the effect of a uniform ultrasound beam. For a more accurate analysis we would have to consider the exact shape of the beam and its movement and also include the effect of tissue inhomogeneity in the calculations (for example the effect of blood vessels in fatty tissues). Much useful insight can, however, be gained with the simplifying assumptions used.

TRANSFER OF ENERGY TO TISSUE: COUPLING Before examining the effect of ultrasound on tissue we must first consider the transfer of energy from the transducer to tissue. Previously we saw that reflection of waves at an interface depends on the difference in impedance of the two media. Table 10.1 (following) lists the acoustic impedances of air, water, steel and body tissues. Notice that the impedance of air and metal differ significantly from the rest with air showing by far the largest deviation. The acoustic impedance of air is only a tiny fraction of the impedance of body tissues. For this reason the ultrasound transducer must be in intimate contact with the skin for appreciable transfer of energy to the tissues. If the transducer is separated from the skin by even a tiny air gap most of the ultrasonic energy will be reflected back into the transducer from the air/tissue boundary. It is a useful exercise to calculate the amount of energy transmitted at an air/tissue interface. Using the values in table 10.1 and equation 9.5 ρ =

ar Z -Z = 1 2 ai Z1 + Z2

.... (9.5)

we predict that the amplitude of the reflected wave will be 0.9997 times the amplitude of the incident wave for an air/tissue interface. Hence (0.9997)2 x 100 = 99.94 per cent of the incident energy is reflected! Clearly the amount of energy transmitted (0.06%) is negligible. Almost all of the wave energy is reflected back into the air.

When the two impedances are the same, no reflection occurs while if the impedances differ greatly most of the wave energy is reflected.

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Only by having a coupling medium between the transducer and tissue can efficient transfer of energy be ensured. The coupling medium is spread on the surface of the skin so that the ultrasound transducer contacts the skin via the coupling medium. No air/tissue boundary is present. Many different coupling media can be used. The desirable characteristics of the coupling medium are: *

It should be fluid, so as to completely fill the gap between skin and treatment head and exclude air bubbles.

*

It should be viscous so that it stays on the skin rather than rapidly flowing and spreading.

*

It should not inhibit heat loss from the skin otherwise high temperatures may be produced in skin and subcutaneous tissue.

*

It should have an impedance similar to that of steel and tissue, so as to minimize reflection.

*

It should absorb a negligible amount of the ultrasound energy: in other words, should have a high penetration depth.

In practice the first two criteria listed above are the most important. The principal function of a coupling medium is to eliminate air gaps and provide contact between treatment head and tissue. Criterion three is also very important and water or water based gels are best in this regard. Criterion four is important but most liquids have similar values of acoustic impedance. Water meets all the above criteria with the exception of the second (viscosity). For this reason water is most often used either in a coupling cushion (a polythene or rubber bag filled with water) or in a bath - when the part to be treated can be immersed.

It is also important that there be no tiny air bubbles within the coupling medium as these too would produce reflection and prevent efficient transmission of the ultrasound energy.

Criterion five is least important since a very thin layer of couplant is used in practice so that energy absorption by virtue of the penetration depth in the coupling medium is relatively insignificant.

Oils and liquid medicinal paraffin have appropriate viscosities and so can be used as

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couplants, however they inhibit heat loss from the skin and produce greater superficial heating than water, water based gels or glycerol. Glycerol is viscous and has similar acoustic properties to water. It makes a very good coupling medium. Thixotropic couplants are solids at room temperature which liquefy when ultrasound is applied. They are ideally suited to treatment of a vertical surface as they will not run down the skin. A number of thixotropic couplants are available.

THERMAL EFFECTS OF ULTRASOUND At the intensity levels used for therapy the major effect of ultrasound waves on tissue is, as with all diathermic modalities, the production of heat. The amount of heat generated in a particular tissue depends on two factors: *

the rate at which energy is absorbed by the tissue - which is determined by the penetration depth, δ.

*

the extent to which the waves are reflected back into the tissue on striking a tissue interface: determined by the difference in impedance between the two media.

We considered in chapter 9 the rate of absorption of energy in different tissues. Values of the penetration depth are shown in table 9.1. The figures are not very accurate and vary by up to a factor of two between different tissue samples. They do, however, give a clear indication of the relative absorbing power of different tissues. The amount of reflection at a tissue interface is determined by the difference in impedance of the two materials. The acoustic impedance (Z) depends on the elasticity and density of a medium according to equation 10.1: .... (10.1)

In this section we ignore the effects of different coupling media and focus attention on what happens to the ultrasound energy which is transmitted into tissue.

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where ρ is the density and Y the modulus of elasticity (stiffness) of the medium. The velocity of sound in the medium, v, also depends on elasticity and density according to equation 10.2: .... (10.2) Combining these two equations we obtain a simple expression for the impedance in terms of velocity and density: Z = ρ.v

.... (10.3)

Table 10.1 lists the acoustic properties of air, water, various tissues and steel. As noted previously the acoustic impedance of air differs considerably from the remaining materials. The table also shows that there is little difference in the acoustic impedance of muscle, fatty tissue and water. For this reason we expect little reflection at a fat/muscle interface. The reflection coefficient calculated using equation 9.5 is 0.10, thus the amount of energy reflected is 0.1 squared or 1%. The impedance of bone is higher than that of muscle hence we expect significant reflection at a muscle/bone interface. The reflection coefficient is 0.50 so we expect about 25% of the energy to be reflected.

Material Air Fatty Tissue Muscle Bone Water Steel

velocity (m.s-1) 340 1450 1550 2800 1500 5850

density (kg.m-3)

impedance (kg.m.s-1)

0.625 940 1100 1800 1000 8000

213 1.4 x 106 1.7 x 106 5.1 x 106 1.5 x 106 47.0 x 106

Table 10.1 Acoustic properties of materials.

Heating Rate of Boneless Tissue Consider first an ultrasound beam travelling through fatty tissue and muscle with no bone present. Equation 9.5 tells us that reflection at the fat/muscle interface is

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negligible and equation 9.12 indicates that refraction is minimal for incident angles up to about 50o . Thus we can consider the waves to be travelling in one direction in a straight line through the tissue. The wave intensity at a point is the energy per unit area per unit time; the area being taken perpendicular to the wave direction. Since energy varies with distance according to equation 9.2, the wave intensity I (in watts per square metre) is given by equation 10.4: I = Io e-x/δ .... (10.4) where x is the distance in the tissue and δ is the penetration depth. The rate of heating is equal to the rate of decrease of intensity with distance. It depends on two factors, the wave intensity at a particular point and the rate of absorption of energy (specified, indirectly, by the penetration depth). The rate of decrease of intensity with distance is obtained by differentiating equation 10.4 to give: .... (10.5) where Pv is the heat developed per unit volume per second. We can use equations 10.4 and 10.5 together with values for δ (from table 9.1) to calculate the wave intensity and heat development in different parts of a fatty tissue/muscle combination once we know the thickness of the fat and muscle layers. For example, suppose that we have a fat layer of uniform thickness (1 cm) on top of a thick muscle layer and that ultrasound of frequency 1 MHz is incident upon this tissue combination. The penetration depth in fatty tissue at this frequency is 7.2 cm (table 9.1) thus the wave intensity (equation 10.4) will be reduced by a factor of e-1/7.2 or 0.87 on traversing the fatty tissue - a decrease of only 13%. After travelling a distance of one centimetre in the muscle the intensity would be reduced by a factor of e-1/1.7 or

The lower the penetration depth, the greater the rate of heating.

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0.56 so the intensity would be 56% of 87% or 49% of the original energy. Figure 10.5 shows the overall reduction in wave intensity with distance in the tissue and also the relative rate of heating of the tissue (equation 10.5). Calculated heating rates are scaled to a value of 100% at the muscle surface (because this is where maximum heating occurs). Even though we have made a number of simplifying assumptions (to be discussed shortly) the general implications of figure 10.5 are valid. It is clear that only modest heating is produced in the fatty tissue. Greatest heating is produced in the few centimetres of muscle tissue adjacent to the fat/muscle interface. Using our simplified model, even after penetrating 2 cm of muscle tissue the ultrasound is predicted to produce a higher rate of heating than at any point in the fatty tissue.

In 2 cm of muscle the intensity would be reduced by a factor of e-2/1.7 = e-1.2 = 0.31 so the intensity would be 31% of 87% or 27% of the original value.

This validates ultrasound as being classified as a diathermic modality. A greater rate of heating is produced at depth. [The depth efficiency of MHz frequency ultrasound is not, however, as great as implied by figure 10.5. Depth efficiency is best assessed by the rate of temperature increase, which is not the same as the rate of heating. A graph of temperature increase resulting from this heating pattern does not show as great a difference between fatty tissue and muscle. This is because temperature elevation in tissue depends on a number of factors other than rate of energy input. Temperature elevation will be considered later in this chapter].

Heating Rate of Tissues With Bone The same kind of calculation as made above can be performed for the more complex tissue combination of fat/muscle/bone. In this case we can not ignore reflection. Equation 9.5 indicates that close to 25% of the wave energy incident upon the muscle/bone interface

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Figure 10.5 Wave intensity and relative rate of heating in fat and muscle tissue with ultrasound of frequency 1 MHz.

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is reflected. Figure 10.6 shows the relative rate of heating which is predicted for a combination of 1 cm fatty tissue and 1 cm muscle overlying bone. The reflection has two effects: * a greater proportion of the total wave energy is absorbed in the fat and muscle tissue. * the reflected and incident waves will interfere and produce a standing wave pattern. The first of these effects is quite significant. Energy will be absorbed both as the wave travels through fat and muscle to the boundary with bone and as the reflected wave travels back through muscle then fatty tissue. Hence the total rate of heating of fat and muscle tissue at any depth is greater than without the bone (compare figures 10.6 and 10.5). The effect on fatty tissue is small. As might be expected, in muscle the effect is larger because the reflected wave, and thus the reflected wave energy, is larger. The second effect is of less practical importance. Certainly an interference pattern will be produced but consider the distance between nodes and antinodes (figure 9.11). The wavelength of the standing wave pattern is the same as that of the incident and reflected waves with nodes and also antinodes separated by one half of a wavelength. For ultrasound of frequency 1 MHz the wavelength is 1.5 mm so the antinodes will be separated by 0.75 mm. The antinodes represent points of maximum wave energy and hence maximum heat production. We have, then, that points of maximum heat production are only 0.75 mm apart. This is too close to be of practical significance, particularly if the treatment head is kept

Figure 10.6 Wave intensity and relative rate of heating in fat, muscle and bone with ultrasound of frequency 1 MHz.

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moving. We have already seen why the treatment head can not be kept stationary: movement is necessary to smooth out the effects of variations in ultrasound intensity with depth shown in figure 10.4. This same movement will produce variations in tissue thickness well in excess of 0.75 mm. The net result will be an averaging of any standing wave pattern as the treatment head is moved: so much so that no evidence for standing waves would be detected. In addition factors such as the pulsatile nature of blood flow through tissue and muscle contraction will result in variations in the thickness of tissue layers: the standing wave pattern will hence shift back and forth, further smoothing the pattern of heat production. The most significant feature of figure 10.6 is the high rate of heating of the bone surface. Most of the wave energy transmitted into the bone is absorbed in the first few millimetres. This is predicted from the value of penetration depth given in table 9.1. The result is substantial heating. As can be seen, the heating rate is predicted to be about three times greater at the bone surface than anywhere in the muscle tissue. Heat development is confined to the first few millimetres of bone but is quite substantial. In practice, heat production at the bone surface is often the factor which limits the intensity which can be used in therapeutic application of 1 MHz ultrasound. Too great an intensity or too prolonged a treatment can result in periosteal pain and significant tissue damage (a periosteal burn). The risk of periosteal burns is reduced by movement of the ultrasound transducer (treatment head). Movement distributes the ultrasound energy over a larger area of the bone surface, thus reducing the average energy in a specific location. The pattern of heat production shown in figures 10.5 and 10.6 indicate the value of 1 MHz ultrasound for heating of deeply located tissue. Figure 10.6 also highlights the risk when the soft tissue layers are thin and underlying bone is exposed to the ultrasound beam.

In summary, in figure 10.6 we have included the effect of reflection which occurs at the muscle/bone interface. This results in a less rapid dropoff in heat production with depth in the fat and muscle tissue when compared with figure 10.5. The effect of standing waves has been ignored for the reasons described.

There have been reports of deep tissue burns being inflicted by unqualified practitioners who neglected to move the ultrasound head during treatment.

If a frequency of 3 MHz is used rather than 1 MHz, values of penetration depth are smaller (table 9.1). The ultrasound intensity decreases more rapidly so heat production is greater in the superficial tissues. A less pronounced 'deep heating' effect results but there is less energy remaining at depth to heat underlying bone.

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Figure 10.7 shows the wave intensity and relative rate of heating calculated for ultrasound of frequency 3 MHz in a tissue combination with the same dimensions as assumed in figure 10.6. Note that with the assumptions made, the peak heating rate at the bone surface does not exceed that at the muscle surface. Comparison of figures 10.6 and 10.7 bears out the qualitative observation made earlier: if a maximum depth efficiency of heating is required then 1 MHz ultrasound is the modality of choice. For less deeply located structures, 3 MHz ultrasound may be preferred to avoid excessive heating of the bone. Figures 10.6 and 10.7 indicate the great usefulness of ultrasound for heating of joints, particularly those located under thick tissue layers. Heat developed at the bone surface will be transferred to heat the adjoining tissue. Experimental work in which the temperature elevation of the hip joint was measured directly confirms that ultrasonic therapy is very useful in this regard. Let us now briefly summarize the approximations made in calculating the results shown in figures 10.6 and 10.7: *

*

We have assumed that the ultrasound beam is uniform: that is, we have neglected the original beam shape. This is a reasonable approximation to make if the treatment head is moving and the transfer of heat between and within tissues is taken into account. We have neglected reflection and refraction at the fat/muscle interface. A reasonable approximation as only about 1% of the energy is reflected and the refraction effect is very small (equation

Figure 10.7 Wave intensity and relative rate of heating in fat, muscle and bone with ultrasound of frequency 3 MHz.

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9.12 and table 10.1). Reflection at the bone surface was taken into account refraction in bone is unimportant as the penetration depth is so small. *

We have assumed the tissue layers to be homogeneous. A valid assumption when we are considering heating of the tissue as a whole. We return to this point shortly in considering the mechanical effects of ultrasound.

*

We have neglected heat losses to the bloodstream and heat transfer to adjacent tissues. These effects are considered next.

Temperature Distribution in Tissue The results shown in Figures 10.5 to 10.7 correctly describe the rate of heating due to the absorption of wave energy. They do not, however, describe the resulting rate of temperature increase. The rate of temperature increase depends on the heat capacity of the tissue. The heat capacity is measured by how much heat must be provided to increase the temperature by one degree (see chapter 7). A tissue with a high heat capacity will require more heat to increase the temperature by 1o than a tissue with a low heat capacity. Muscle tissue (principally because of its high water content) has a higher heat capacity than fatty tissue or bone. Consequently, for the same amount of heat energy, the temperature increase in muscle will be less than that of fatty tissue or bone. At the commencement of treatment, the relative rate of heating will be more or less as indicated in the figures. Graphs of rate of temperature increase would, however, show an overall lesser rate of temperature increase in muscle than fat or bone, because of the heat capacity effect. A second factor affecting the rate of temperature increase in tissue is heat loss to the surrounding tissue and, more importantly, the blood vessels. Muscle has a much higher vascularity and therefore volume rate of blood flow, than fatty tissue or bone. We would thus expect convective cooling of muscle (i.e. heat carried away by the bloodstream) to reduce the rate of temperature increase still further.

The rate of temperature increase depends not just on the rate of energy absorption, but also on heat capacity of the tissue and the rate of heat transfer to adjacent tissue.

Water has a much higher heat capacity than most other substances. This is illustrated by the fact that a cup of lukewarm tea has a higher heat content than a (large) red-hot nail.

A third factor is that any temperature increase in muscle would also be expected to trigger reflex dilation, whereby arterioles dilate to increase the blood flow in response

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to an increase in temperature. This would further decrease the rate of temperature increase. The main points relating temperature increase to heating rate are as follows: *

the low specific heat capacity of fatty tissue and poor thermal conductivity will result in a greater temperature rise than indicated by the graphs. In addition the thermal conductivity of fatty tissue is low and its vascularity is not as good as muscle; consequently heat can not be removed as rapidly. This adds to the temperature elevation of fatty tissue as compared to muscle.

*

efficient heat transfer through muscle tissue and to blood vessels will result in more uniform heating of muscle and less temperature rise than might otherwise be expected. At the same time heat transfer to the adjacent fatty tissue will reduce the temperature elevation of muscle near the fat/muscle interface.

*

bone is a relatively good conductor of heat. The heat will be rapidly distributed in the bone and also transferred to the periosteum. The higher thermal conductivity partially compensates for the rapid absorption of energy near the bone surface and reduces the selective heating. It is still possible, however, to produce a maximum temperature elevation in the periosteum when the intervening tissue layers are not very thick. This gives rise to the periosteal pain mentioned previously.

Despite these limitations, some of which also apply to other diathermic modalities, ultrasound is an effective deep-heating modality. The principal factor limiting the temperature elevation which can be produced at depth is heating of the periosteum.

Mechanical Effects The predominant physiological effects of ultrasound therapy are due to a rise in temperature of the treated tissues. Certain effects are, however, produced which are a direct result of the mechanical vibration of tissue.

Equation 7.6 tells us that if the specific heat capacity is low the rate of temperature elevation will be greater for a fixed rate of heat energy supplied to the tissue.

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As an ultrasound wave propagates the particles in the medium experience rapidly alternating compressions and rarefactions. The pressure varies with distance as shown in figure 10.8. Regions of high pressure (dark shading) are separated by one wavelength - about 1.5 mm for 1 MHz ultrasound. For a wave intensity of 2 to 4 watts per square centimetre (near the upper limit for therapeutic application) the pressure amplitude of the waves is about 2 to 4 atmospheres (20 to 40 newtons per Figure 10.8 square centimetre). This means that the pressure extremes - a Pressure variation in an ultrasound wave. difference of 4 to 8 atmospheres - are separated in tissue by only one half of a wavelength (0.75 mm). Any tissue component or substructure with dimensions of about one wavelength will be subject to substantial mechanical stresses, alternating at a frequency of 1 MHz. Smaller structures such as tissue cells will experience lesser, though still substantial stresses and will be vibrated back and forth by the pressure changes. Figures 10.4 and 10.8 are two important views of the ultrasound intensity variation in a beam. Figure 10.4 is at a gross, large-scale level and shows the variation in intensity within a beam. Regions of high intensity (shown by the deepest blue colouration) are separated by distances measured in centimetres and the positions are stationary in the beam. Viewed on a smaller scale, figure 10.8 shows a sound wave within the beam. At any point within the beam shown in figure 10.4, the pressure varies from maximum to minimum over a distance of half of one wavelength and regions of high pressure move through the medium (tissue, air or water) at a high velocity. The sound velocity, v, is about 1500 m.s-2 in water and soft tissue.

Therapeutic ultrasound produces large stresses in biological tissues, acting over distances of a fraction of a millimetre. The stresses are greatest in the regions shown in figure 10.4.

Listed below are some examples of the effects of ultrasound where mechanical stresses are thought to play a significant role. It should be emphasized that in all instances heating contributes to the observed results: in most cases it is difficult to ascertain the relative contribution of thermal and mechanical effects. *

Experiments have shown that the extensibility of connective tissue can be increased by exposure to ultrasound. Since connective tissue fibres are key

THERAPEUTIC WAVES: ULTRASOUND constituents of tendons, scar tissue, joint capsules and muscle the results are major significance for therapy. Part of the effect is attributed to the increase temperature on exposure to ultrasound: the separation of fibres and loosening the structure as a result of the mechanical stresses would also be expected contribute.

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*

The rate of diffusion of ions across cell membranes is found to increase on exposure to ultrasound. An effect is observed over and above that due to heating alone. A possible explanation is that a stirring effect is produced which increases the concentration gradient of ions and other materials. In any diffusion process there will be a narrow region on either side of the membrane where the ion concentrations are not the same as in the bulk of the fluid. Another possible contributing factor involves the fluidity of the cell membrane. If we picture the cell membrane as a thixotropic barrier we predict that the fluidity, and hence the permeability, of the cell membrane will increase in response to the mechanical agitation of the ultrasound waves.

*

Ultrasound is useful in relieving pain and muscle spasm. While any form of heating is useful in this regard, it appears that ultrasound can have an effect other than via direct heating. The mechanism of this action has not been conclusively established but it is interesting to note that an optimum effect appears to be produced using pulsed ultrasound beams. The pulse frequency normally available is 100 Hz - the same frequency used to produce analgesia by electrical stimulation.

PULSED ULTRASOUND Most ultrasound apparatus makes provision for either pulsed or continuous output. In pulsed mode the ultrasound is produced in bursts, normally with a frequency of 50 or 100 Hz. If the duty cycle ('on' time to 'on + off' time) is 1:5 then the apparatus is 'on' for only one fifth of the time: consequently the rate of transfer of energy is one fifth of that obtained using the continuous mode at the same intensity. If the dose required (continuous mode) necessitates treatment for 20 minutes then to obtain a similar thermal effect using pulsed ultrasound we would have to extend the treatment time or

For example, the diffusion of sodium ions across a membrane will deplete the concentration close to one side of the membrane and increase it on the other side: this will result in a decrease in the rate of diffusion. If the fluids are agitated, mixing will proceed more rapidly and the concentration gradient will be more efficiently maintained.

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increase the intensity to compensate. An increase in the treatment time alone will not compensate adequately. Suppose the duty cycle is 1:5 then a 20 minute (continuous) treatment could be increased to 100 minutes (pulsed). Although the total energy (the dose) supplied to the tissue is the same in both cases, spreading the treatment over 100 minutes will considerably reduce the temperature elevation produced. Increasing the intensity by a factor of five will result in the same rate of energy transfer to tissue (dose rate) but the much higher peak intensities could result in tissue damage through gaseous cavitation - the rapid formation and collapse of tiny gas bubbles in the tissue fluid. The cavitation effect will be described more fully in chapter 12 along with other potentially harmful effects.

Temperature elevation depends not just on the dose but also on the dose rate.

Proponents of the use of pulsed ultrasound argue that heat production is rarely the sole objective of therapy and that in some applications it may even be undesirable. By use of pulsed ultrasound, at low to moderate intensities, mechanical effects are produced while heat production is kept to a minimum. Of course the same (low) rate of heat production could be achieved using the continuous mode at one fifth of the peak intensity. We would, however, expect some differences in the mechanical effects produced: continuous mild mechanical agitation does not necessarily produce the same effect as brief vigorous mechanical agitation. The idea is that mechanical effects do not depend linearly on intensity: that there is a threshold intensity level below which the mechanical effects are negligible. Pulsed ultrasound would ensure that intensities above threshold are achieved while keeping heat production to a minimum. One study which indicates the possibility of therapeutically significant mechanical effects was carried out by Dyson et al. (1968). These authors examined the rate of tissue repair using continuous output treatment compared with pulsed mode treatment using different duty cycles. The frequency used was 3 MHz and the output in pulsed mode was adjusted to keep the average power the same in each experiment. Tissue growth rate was increased using a duty cycle of 1:5 but retarded when a duty cycle of 1:80 was used. It seems that modest duty cycles may promote repair activity but that (for the same average power) too small a duty cycle involves peak power levels which are damaging to tissue. The results of this and other relevant studies

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The referances here are to two old but reliable papers: Dyson M et al. Clinical Science 35, 273-285 (1968) and Dyson M & Suckling J. Physiotherapy 64, 105-108 (1978).

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are summarized by Dyson and Suckling (1978). To date, insufficient research has been done to quantitatively determine the relative contributions of heating and mechanical effects in different therapeutic applications of ultrasound. When such information is available the therapist will be in a better position to select between pulsed and continuous mode and, when pulsed mode is indicated, to chose the appropriate duty cycle. At this stage it is possible, at least, to say that if ultrasound is chosen (rather than another diathermic modality) on the basis of the predicted pattern of heat production, then a continuous output is indicated. Where mechanical effects are a desirable part of the therapy, continuous or pulsed mode might be selected.

EXERCISES 1

Consider the schematic diagram of an ultrasound generator shown in figure 10.1. (a) Briefly explain the function of each subsection. (b) What is the function of the piezoelectric crystal shown in the probe? Explain what happens when an AC signal is applied to opposite faces of the crystal.

2

(a) (b)

3

Explain why the ultrasound beam produced by apparatus used in therapy is non-uniform. What is meant by the terms 'near field' and 'far field' as applied to therapeutic ultrasound?

Figure 10.3 shows the distribution of ultrasound energy with distance along the central axis of an ultrasound transducer. In this case the interference pattern is generated by a 2.8 cm diameter, 1 MHz frequency, ultrasound source in water.

There is no clinical evidence that pulsed has greater therapeutic benefit than continuous ultrasound. Biophysical evidence shows that there are different effects, but whether this translates into clinical practice is unknown at present.

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Describe how the interference pattern would change if:

4

5

(a) (b)

the transducer diameter was made smaller the operating frequency was increased (to 3 MHz, say)

(c)

fatty tissue rather than water was used. What is the effect of the different wave velocity in fatty tissue?

Use equation 9.5 and the impedance values given in table 10.1 to calculate the reflection coefficient of a metal/air boundary and an air/fatty tissue boundary. (a)

What percentage of the incident ultrasound energy would be transmitted at these boundaries?

(b)

What are the implications of your results for ultrasound therapy?

(a)

What is a coupling medium and why is one needed in ultrasound therapy?

(b)

List the characteristics of a 'good' coupling medium and describe the relative importance of each.

(c)

State the relative advantages and disadvantages of water, oils, glycerol and thixotropic fluids as coupling media.

6

Ultrasound of frequency 1 MHz travels through water and strikes fatty tissue. Use the data in table 10.1 to calculate the reflection coefficient of the water/fat boundary. What percentage of the incident wave energy is transmitted?

7

Use equation 9.5 and the figures in table 10.1 to calculate the reflection coefficient of ultrasound at the following boundaries: (a) fatty tissue/muscle (b) muscle/bone What are the practical implications of these figures?

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(a) (b)

Refer to figure 10.5 and explain how the graph of wave intensity vs distance can be used to obtain the graph of relative rate of heating versus distance. What is the relationship between penetration depth (table 9.1) and relative rate of heating (figure 10.5)?

9

For ultrasound of frequency 2 MHz (table 9.1) calculate the fraction of energy remaining after travelling through: (a) 2 cm fat (b) 2 cm muscle (c) 2 cm bone

10

Ultrasound of frequencies 1, 2 and 3 MHz travels through 3 cm of fatty tissue and strikes a fat/muscle interface. Assuming no reflection at the interface, what fraction of the original energy reaches the muscle? Use the data in table 9.1 and equation 9.2.

11

Consider 3 MHz ultrasound travelling through a tissue combination of 2 cm fatty tissue over muscle. (a) Use equation 10.4 and the data in table 9.1 to calculate the wave energy at depths of 1, 2, 3 .... 10 cm in the tissue combination. You may assume no reflection at the fat/muscle interface. (b) Use equation 10.5 to calculate the relative rate of heating from the intensity figures obtained in part (a). (c) Plot a graph of wave intensity versus depth and relative rate of heating versus depth for comparison with figure 10.5. (d) What similarities and differences are evident in comparing your graphs with figure 10.5?

12

Compare figures 10.5 and 10.6 and explain why the wave intensity appears to diminish less rapidly with distance in fat and muscle in figure 10.6.

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Consider figure 10.6, which shows the wave intensity at different depths in a muscle/fat/bone tissue combination. The graph is obtained assuming that 25% of the incident energy is reflected at the muscle/bone interface. (a) How would the graph of intensity versus depth differ if reflection at the muscle/bone interface was negligible? (b)

Reflection of waves at an interface results in the production of a standing wave pattern. Why is this not shown in figure 10.6?

(c)

Under what conditions will the production of standing waves be of practical significance for 1 MHz ultrasound in complex tissue combinations?

14

Compare figures 10.6 and 10.7 and summarize the advantages and disadvantages of 1 MHz ultrasound compared with 3 MHz ultrasound for different thicknesses of tissue over bone.

15

The results shown in figures 10.5 to 10.7 take no account of heat losses to the air, the bloodstream and between adjacent tissues. Redraw figure 10.7 to show, qualitatively, the relative rate of heating when heat loss and heat transfer are taken into account.

16

Figures 10.5 to 10.7 show the relative rate of heating of different tissue exposed to ultrasound. (a) What additional factor must be taken into account to predict the initial rate of temperature increase in each tissue? (b)

17

Draw a diagram, based upon figure 10.7 to show (qualitatively) the initial rate of temperature increase in each tissue. You may assume that the specific heat capacity of muscle is twice that of fatty tissue and bone.

Explain why graphs such as those shown in figures 10.5 to 10.7 can be used to accurately predict the initial rate of temperature increase in tissue but not the final temperature increase in therapy. What additional factors must be taken into account to predict the final temperature elevation?

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(a) (b)

19

Briefly explain why ultrasound would be expected to have mechanical effects on tissue in addition to thermal effects. List and briefly describe the effects of ultrasound where mechanical vibration is thought to play a direct role.

Compare the following two methods of treating a patient with ultrasound: (i) Exposure at a level of 2 W.cm-2 for 10 minutes in continuous mode (ii) Exposure at a level of 2 W.cm-2 peak for 20 minutes using a duty cycle of 1:2 Would you expect: (a) the same mechanical effect? (b) the same total heat production? (c) the same temperature elevation? How do the dose and dose rate compare in each case?

20

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Compare the following two methods of treating a patient with ultrasound: (i) exposure at a level of 1 W.cm-2 for 15 minutes in continuous mode. (ii) exposure at a level of 2 W.cm-2 peak for 15 minutes in pulsed mode (duty cycle 1:2). Would you expect: (a) the same mechanical effects? (b) the same total heat production? (c) the same temperature elevation? How do the dose and dose rate compare in each case?

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11 Electromagnetic Waves for Therapy We saw, in chapter 9, that Maxwell's equations predict that whenever charges are accelerated electromagnetic waves are produced. In this chapter we consider the electromagnetic waves used in therapy: how they are produced and why they are useful to physiotherapists. Three main kinds of electromagnetic wave are used in therapy: microwaves, infrared and ultraviolet radiation. Of these, only microwaves are able to penetrate tissue significantly and so be classed as diathermic. The different therapeutic applications of these radiations arise from their differing effect on tissue. These effects, in turn, are determined by the wavelength (or frequency) of the waves. Before considering the effect on tissue we examine the way in which each kind of electromagnetic wave is produced: this gives a first insight into their physical and physiological effects.

Ultraviolet and infrared radiation have low penetration depth but are useful for therapy in applications other than diathermy.

PRODUCTION OF WAVES AROUND OPTICAL FREQUENCIES In what follows we consider the way in which infrared and ultraviolet radiation are produced for therapy. Both kinds of radiation are normally produced by similar apparatus: more fundamental are the similarities in the molecular processes involved.

Production Of Ultraviolet Radiation Electromagnetic waves with frequencies from 0.75 x 10 15 Hz to 3.00 x 10 15 Hz are classified as ultraviolet radiation (see figure 9.5). Their frequencies are above those of visible light and below those of X-rays. Ultraviolet radiation has wavelengths between 400 nm and 100 nm. The wavelengths used in therapy are restricted to the high end of this range: 190 nm to 400 nm, as wavelengths less than 190 nm are strongly absorbed in air.

When discussing ultraviolet radiation it is a common convention to talk in terms of wavelength rather than frequency.

By international convention the ultraviolet spectrum is divided into three regions.

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These are: * * *

UV-A: wavelengths between 400 nm and 315 nm UV-B: wavelengths between 315 nm and 280 nm UV-C: wavelengths between 280 nm and 100 nm.

UV-C radiation is used to sterilize things when you don't want to boil them. This is because UV-C, at sufficiently high intensities, destroys bacteria. It does this by damaging the bacterial DNA. UV-C exposure will also damage human cells in the same way and can produce malignancies (cancer). UV-C and, in fact, UV-B and -A have an extremely low penetration depth, so most of the absorption of UV is by the skin. The low penetration depth of UV is the reason that UV exposure (in particular, exposure to UV-C) is associated with skin cancer. The usual means of producing ultraviolet light is by the passage of an electric current through an ionized gas or vapour. Gases at normal temperature and pressure are very poor conductors. They can, however, be made to conduct at high temperature or low pressure in the presence of a sufficiently strong electric field. Ultraviolet radiation for therapeutic application is usually produced by current flow through mercury vapour. Mercury under reduced pressure is contained in a sealed envelope of quartz or special glass with an electrode inserted in each end. The device is similar to the strip-lights (fluorescent lights) commonly found in the kitchen at home and the office or tutorial room. The difference is that UV lights operate at lower pressures than household or business lights. This means that more energy is required to initiate conduction and charges are accelerated over greater distances so that when they collide, the energy release is larger and, as a result of the higher energies, UV rather than visible light is produced. The arrangement used with a mercury vapour lamp is shown in figure 11.1. Figure 11.1 Schematic diagram of a mercury vapour lamp.

When deciding whether to use UV for therapy, the clinician must, as with all interventions, weigh the benefits against the risks.

Indeed, we have evidence for gaseous conduction at normal temperature and pressure with every lightning flash in a thunderstorm.

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The reduced pressure in the lamp ensures that mercury vapour is present, but in order for current to flow the vapour must be ionized. This means that electrons must be separated from the parent atoms. Cosmic rays and gamma-rays are high frequency and high energy and can 'kick' electrons from their orbitals, so producing positive ions and free electrons. Under normal circumstances the electron returns to its parent atom, because of the attraction between positive and negative charges. However, in a sufficiently strong electric field (as in the lamp) the excited electron can accelerate and collide with other atoms. If the electric field is strong enough, the electron can gain enough energy to cause further ionization and produce an 'avalanche' effect: one electron is accelerated and collides, producing more metal ions and free electrons which in turn accelerate, collide and cause further ionization.

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Ionization occurs continuously in all materials: it is brought about by various radiations always present at low levels; cosmic rays and natural radioactivity are examples.

The glow of a mercury-vapour lamp is a consequence of the avalanche of ionization. After participating briefly in the avalanche, electrons reattach to ions, dropping into a particular orbital and releasing energy in the form of electromagnetic waves. In the discharge process ions are being continually formed and are continually recombining with electrons. As ions and electrons recombine energy is released in the form of electromagnetic radiation which has frequencies characteristic of the parent atom. The range of frequencies put out by the lamp is modified by the pressure within the lamp and further modified in passing through the glass envelope which contains the vapour. Figure 11.2 compares the range of wavelengths over which various lamps put out appreciable energy. The spectral range for an ordinary incandescent (tungsten filament) lamp, a fluorescent tube (strip-light) and for sunlight is also included for comparison.

It is generally necessary to help initiate the avalanche, or discharge, by pulsing the lamp with a high voltage. Once the discharge is started the current must be regulated to limit and control the output of light from the lamp.

Also indicated in the figure are approximate proportions of ultraviolet, visible and infrared radiation expressed as a percentage of the total energy output. The proportions vary with the pressure of mercury vapour in the lamp or tube and with the thickness and composition of the lamp envelope. Percentages are not shown for fluorescent tubes ('strip lights') or incandescent lamps (normal globes) as the

ELECTROMAGNETIC WAVES FOR THERAPY proportions vary depending on the construction of the device, the power rating and whether filters are used to block-out certain wavelengths. Figure 11.2 Spectral range of various lamps and sunlight

Mercury vapour lamps operating at lower pressure put out more radiation in the high frequency region of the spectrum (towards the far ultraviolet region). Even so, all ultraviolet lamps produce a considerable amount of energy in the infrared and visible region. Both the visible and far ultraviolet radiation can be removed by the use of filters. If water cooling of the lamp is used - as with the Kromayer lamp - the water serves the dual role of keeping the lamp cool and absorbing the infrared radiation.

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Within the ultraviolet region of the spectrum there are significant differences in the output of mercury vapour lamps and tubes: *

Low pressure mercury vapour lamps, otherwise known as cold quartz lamps when the envelope material is quartz, emit most of their ultraviolet radiation in the UV-C region, at a wavelength of 253.7 nm. The operating temperature of the lamp envelope rarely exceeds about 60oC.

*

High pressure mercury vapour lamps, known as hot quartz lamps when the envelope material is quartz, put out a proportion of their ultraviolet energy at a wavelength of 366.0 nm (in the UV-A region). There is also significant output at specific wavelengths in the UV-B and UV-C regions. The amount of energy in each region depends on the construction of the lamp. The normal operating temperature of these lamps is several hundred degrees Celsius: if they are to be used close to, or in contact with the patient they must be cooled by a water jacket (Kromayer lamps) or an air blower.

*

Fluorescent ultraviolet tubes are usually low pressure mercury lamps in the form of a long tube. The tube is coated on the inside with fluorescent substances (phosphors). The purpose of the phosphor is to absorb the original ultraviolet radiation and re-emit it at longer wavelengths. Different phosphors have different wavelengths for re-emission of radiation. The commonly used ultraviolet tubes put out most of their energy in the UV-A region. Special tubes are available which produce a maximum output in the UV-B region. A negligible amount of UV-C radiation is emitted from any of these light sources.

In the past carbon arcs were used extensively for the production of ultraviolet radiation. Two carbon rods are brought into contact with each other and a current is passed through them. With a small point of contact the high current density heats and vapourises the carbon. The rods are then separated and the presence of carbon vapour enables a current to flow in the form of an arc discharge between the ends of the rod. The spectrum produced by carbon arcs has a range close to that of sunlight (figure 11.2): the proportions of ultraviolet, visible and infrared radiation are also similar.

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Quartz or special composition glass must be used for the envelope of all ultraviolet light sources as normal glass absorbs strongly over almost the whole of the ultraviolet spectrum.

The presence of a vapour, even at atmospheric pressure, enables a discharge to be sustained with a gap up to around 5 cm when about 80 volts potential difference is applied.

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Carbon arcs are rarely used today in physiotherapy departments: they have been largely superseded by mercury vapour lamps which are cleaner and easier to operate.

Production Of Infrared Radiation Infrared radiation - sometimes referred to as radiant heat - is produced (and absorbed) by all materials at temperatures above absolute zero. Absorption of infrared radiation results in changes in molecular and atomic motion of a material; the continuous agitation and changes in the motion of molecules, and within molecules also results in the emission of infrared radiation. For example, chemical bonds in molecules can absorb energy and 'stretch', changing the bond length and thus the energy of the bonding electrons. When the bond reverts to its original size, infrared radiation is produced at a frequency characteristic of the bond. Any molecule may, as a result of absorption of radiation or collision, change its state of rotation or vibration, or both simultaneously. On changing to a rotation or vibration state of lower energy, infrared radiation is produced. A particular kind of molecule has very many possible states of rotation and vibration and therefore many options for going from one state to some other. At a given temperature a body will emit a continuous spectrum of radiation - the maximum intensity occurring at a particular frequency but with significant intensities extending over a wide range of frequencies. The frequency of maximum production of radiation is directly proportional to the absolute temperature of the source. Since wavelength and frequency are inversely related (by equation 9.1, v = f.λ), it follows that the wavelength of maximum production of radiation is inversely proportional to the absolute temperature of the source. (This is called Wien's Law). As the source of radiation becomes progressively warmer, the wavelength of maximum emission becomes progressively shorter: thus an iron bar turns from black to 'red-hot' to 'white hot' to 'blue hot' as its temperature increases. In the black to redhot temperature range both near infrared (770 to about 4000 nm) and far infrared

Any object will be emitting and absorbing infrared radiation on an ongoing basis. Whether emission outweighs absorption depends on the temperature of the object relative to its surroundings.

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(4000 to 15 000 nm) radiation is produced in appreciable amounts. A suitable device for producing such radiation consists of a coil of wire through which a current is passed. If the coil is wound on an insulator such as a ceramic rod, both the wire and the ceramic will emit radiation. The ceramic, being at a lower temperature will produce more far infrared radiation. The common household electric heater is usually of this kind. A way of producing most radiation in the far infrared region of the spectrum is to encase the heating element inside a ceramic rod or mount it behind a plate so that the major source of radiation is the rod or plate. If a reflector is used, the reflector will absorb some radiation and re-emit it at higher wavelengths thus adding to the far infrared component. These devices are often used for therapy. Use is also made of incandescent infrared lamps which produce a significantly greater proportion of near infrared radiation. Incandescent infrared lamps (similar to household lamps - consisting of a tungsten filament mounted in a glass envelope) have maximum emission at a wavelength around 1000 nm: some visible and ultraviolet light is produced but the ultraviolet is absorbed by the glass envelope and not transmitted. Use may be made of specially shaped lamps with internal reflectors. The reflectors may be shaped to give a floodlight beam - suitable for treating large areas - or a spotlight beam for treatment of localized areas. Some lamps have a clear glass lens while others have a red lens: there is little difference in the therapeutic effects of each.

For an ordinary household light bulb the tungsten filament is at about 3000 K and the wavelength of maximum emission is about 960 nm - that is, in the near infrared. For skin at about 300 K it would be 9600 nm, in the far infrared.

EFFECTS OF ULTRAVIOLET AND INFRARED RADIATION ON TISSUE Infrared and ultraviolet radiation share the common feature that their effects are produced in the surface layers of the skin. This was mentioned briefly in chapter 9(link to p227). Figure 11.3 summarizes the penetrating properties of these radiations. The penetration depth of waves of these frequencies clearly distinguishes them from waves used for diathermy. Considering first the infrared radiation (wavelengths of 770 nm and above) the figure indicates that shorter infrared waves (770 to 1200 nm) penetrate to the deeper parts of the dermis while the longer wavelengths are absorbed in the superficial epidermis.

ELECTROMAGNETIC WAVES FOR THERAPY From a penetration depth of a few millimetres at 1200 nm there is a decrease to about 0.1 mm at 3000 nm. Wavelengths above 3000 nm are absorbed by moisture on the surface of the skin. The trend does not continue indefinitely and we find that in the far infrared region from 10 000 to 40 000 nm, the penetration depth increases to several centimetres. In effect, the tissues become much more transparent. Figure 11.3 Penetration of radiation into skin in the infrared to ultraviolet region of the electromagnetic spectrum

Over the whole of the near infrared spectrum and up to about 20 000 nm in the far infrared, reflection is minimal. Close to 95 per cent of energy incident perpendicular to the skin is absorbed only about 5 per cent is reflected. To a reasonable approximation then, we can consider infrared radiation to be wholly absorbed by tissue. The region of the ultraviolet spectrum of interest in therapy extends from about 180 nm to 390 nm. From figure 11.3 we can see that most of this radiation is absorbed in the epidermis. In the region from 220 nm to 300 nm about 5 to 8 per cent of incident radiation is reflected. The reflectance increases to about 20 per cent at 390 nm. Within the range there are regions of very low reflectance corresponding to specific absorption by particular molecules in the skin - for example,

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nucleic acids absorb strongly at frequencies between 250 and 260 nm and at 280 nm.

Heating by Infrared Radiation From the foregoing discussion it is clear that the major effect of infrared radiation is thermal: to increase the temperature of cutaneous tissue. The penetration depth is very small but some heat will be transferred to the subcutaneous tissues via the capillaries. The main effects of treatment are: *

An increase in metabolic rate in the superficial tissues. This is the direct effect of temperature on the rate of chemical reactions generally. As a result there will be an increased demand for oxygen and an increased output of waste products.

*

Dilatation of capillaries and arterioles due directly to the heating and also as a reflex reaction to the presence of increased concentrations of metabolites. The flow of blood to the superficial tissues is thus increased producing a reddening of the skin (erythema) and an increased supply of oxygen and nutrients. The erythema produced by infrared therapy, unlike that resulting from ultraviolet treatment, appears quite rapidly and begins to fade soon after treatment ceases.

*

Sensory sedation. Mild heating has a 'sedatory' effect on sensory nerves and is thus useful for the relief of pain.

*

Muscle spasm relief. This results from both the effect of heat on nerve fibres and the direct effect of heat which is transferred to muscle from the superficial tissues.

The physiological effects of infrared radiation differ from those of other forms of heating (e.g. shortwave diathermy) only in the location of heat production.

The effects of infrared radiation are not damaging unless the temperature elevation is too high.

Effects of Ultraviolet Radiation The effects of ultraviolet radiation are mainly non-thermal and due to cellular damage and protective responses. While damage might seem an undesirable consequence, there are therapeutic benefits of treatment. Five principal effects of therapeutic significance are found to result from treatment with ultraviolet radiation:

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*

An increased blood supply to the skin results from dilation of the capillaries and arterioles. Dilation does not result from heating of the tissue but as a reflex response to destruction of cells. Cells are destroyed as a result of chemical changes caused by the absorption of radiation, and reddening of the skin (erythema) results. The effects are similar to the changes observed in inflammation. Two groups of waves produce this reaction, one with wavelengths in the UV-C region around 250 nm and one with wavelength close to 300 nm (UV-B).

*

Production of vitamin D. Ultraviolet radiation in the range 250 to 300 nm initiates a sequence of chemical reactions by which vitamins of the D group are synthesized. The effect has been used in the past for the treatment of rickets and tetany, but is not used any longer.

*

Pigmentation. The amino-acid tyrosine is converted, via a sequence of reactions, to the pigment melanin. The accumulation of melanin in the epidermis is triggered by the same wavelengths of ultraviolet radiation responsible for erythema production - in addition UV-A wavelengths around 340 nm in low doses can produce tanning without erythema.

*

Sterilization. Shorter wavelength ultraviolet radiation (UV-C, around 250 nm) is effective in destroying bacteria. In therapy this effect finds application in the treatment of indolent ulcers: ultraviolet treatment is found to promote and accelerate the healing process. It is not clear to what extent the sterilization contributes as compared to erythema production. The increased blood supply evidenced by erythema will increase the number of white blood cells and antibodies in the area, hence reinforcing the body's defence mechanism.

*

Desquamation occurs some time after exposure to ultraviolet rays - it is a casting-off of dead cells from the surface of the body. The amount of peeling varies with the strength of the dose: it ranges from virtually imperceptible through powdery peeling to free peeling of epidermal layers. This can be of value in the treatment of skin diseases such as acne and psoriasis.

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Nowadays, vitamin tablets provide a means of achieving results more quickly and economically when treating vitamin D deficiency.

In laboratories and pharmaceutical preparation areas, contamination by bacteria must be avoided, so lamps producing UV-C are used to irradiate the areas.

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The degree of erythema production is used to characterize the dose in ultra-violet therapy using UV-B fluorescent tubes or mercury vapour lamps. The reaction is graded into four levels: *

A first-degree erythema is a slight reddening of the skin which takes from six to eight hours to develop. The erythema has faded in about twenty four hours leaving the skin apparently unchanged. A minimum erythema dose (MED) is also a slight reddening which takes from six to eight hours to develop but in this case the erythema is still just visible at twenty four hours.

*

A second-degree erythema is a more marked reddening of the skin (resembling mild sunburn). There is a slight soreness. The reaction fades in about two days and is followed by pigmentation. After one or two weeks desquamation (peeling, usually powdery) occurs.

*

A third-degree erythema resembles severe sunburn. The skin may begin to show the effects as soon as two hours after treatment. The reaction is severe and the skin becomes hot, sore and oedematous. Effects subside gradually over several days and the skin often peels off in sheets or flakes.

*

A fourth-degree erythema is similar to a third-degree reaction but exudation and oedema are so marked that blisters form. Production of a third or fourth degree erythema in a small localized area results in a counter-irritation effect.

Dose characterization in this way is appropriate for sources which produce an appreciable proportion of UV-B radiation. When using UV-A fluorescent tubes, dosage can not be assessed in this way as erythema production is minimal except at extremely high dose levels. In practice this is not a problem as the principal use of UV-A is in conjunction with a photosensitizing drug, 8-methoxy-psoralen, for the treatment of psoriasis. For psoralen - UV-A, therapy a special procedure is used for dose characterization. The procedure is described in chapter 12.

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The counter-irritation effect of a fourth degree erythema has been used in the past as a quick and effective method of relieving pain from joints and other deep structures in degenerative arthritis and rheumatic conditions.

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PRODUCTION OF MICROWAVES Having considered the low penetration electromagnetic waves - infrared, visible and ultraviolet - we now turn to lower frequency waves used in therapy; microwaves. Microwaves occupy the region of the electromagnetic spectrum between radio waves and infrared radiation: their wavelengths are in the range from about a centimetre to a meter - corresponding to frequencies in the range 300 MHz to 30 000 MHz. Three main frequencies are used for physiotherapy, 2450 MHz (wavelength 12 cm), 915 MHz (wavelength 33 cm) and 433.9 MHz (wavelength 69 cm). Note that the wavelengths quoted are in air. In biological tissues the wavelength is significantly lower because the wave velocity is lower. Radio waves can be produced by first generating a very high frequency AC signal in an ordinary electronic circuit and then applying this signal to a suitable antenna. The high frequency alternating current in the antenna results in radio frequency waves being produced and radiated. The limit to the frequencies that can be produced by standard electronic circuits is determined by the time it takes for an electron to travel through a transistor. If the transit-time, the time taken, becomes comparable to the time of oscillation or period of the wave we wish to produce, then the transistor can no longer function at this frequency. Microwave frequencies are extremely high, by electronic standards, and are at the limit of those which can be produced by transistors. Although vacuum tubes (valves) are an older design and are generally more inefficient than transistors, two vacuum tube devices which can operate at microwave frequencies were developed many years ago: these are the magnetron and the klystron. The magnetron valve, first described by Hull in 1921, was developed for radar use during the second world war. It is more useful for high power applications than the klystron. After the war, apparatus operating at a frequency of 2450 MHz (the standard radar frequency) was made available to physiotherapists. Microwave apparatus (figure 11.4) consists of a device (a magnetron or klystron), powered by an electronic circuit. The high frequency alternating current which is produced is fed to an antenna. The current flowing in the antenna results in the

Electromagnetic waves travel more slowly in biological tissues than air. The higher the dielectric constant and conductivity, the lower the wave velocity.

As we will see, a frequency of 2450 MHz is not the best choice for therapeutic applications and for some years the use of lower frequencies has been advocated.

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production of electromagnetic waves (chapter 9) which are beamed by the reflector. Figure 11.4 Schematic diagram: microwave apparatus

The frequency of the microwaves is equal to the frequency of the AC produced by the magnetron. This is determined by the physical construction of the magnetron and is fixed during manufacture. A number of differently shaped antennas and reflectors may be used for directing the beam. Each gives a different beam shape though none gives a perfectly uniform beam. To obtain a collimated uniform beam (like a searchlight) would require a parabolic reflector with a point source of radiation as shown in figure 11.5(a). If a point source of radiation is placed at the focus of the parabola the beam emerges with a uniform cylindrical shape as shown. In the case of microwaves used by physiotherapists, the most common frequency is 2450 MHz and the wavelength in air is 12 cm. The source of radiation is normally a half-wave antenna; a rod shaped conductor about 6 cm long. Placed in a small parabolic reflector the antenna would produce a highly non-uniform beam (figure 11.5b). To produce a reasonably uniform beam the antenna would need to be placed in a reflector very much larger than its 6 cm length. A reflector with a focal length of a

ELECTROMAGNETIC WAVES FOR THERAPY metre or more and a diameter of several metres would be needed - producing a beam which is metres in diameter. Figure 11.5 (a) a uniform beam from a parabolic reflector and point source, (b) a non-uniform beam from a parabolic reflector and extended source

For therapeutic application, a microwave beam only 10 to 20 cm in diameter is desirable, in order to localize the microwave energy. Reflectors 10 or 20 cm in diameter with antennas about 6 cm in length cannot produce a uniform beam but can be designed to produce a diverging beam. The beams obtained from reflectors presently used in therapy diverge considerably - the wave intensity decreasing rapidly with distance from the reflector. The reflectors must be designed this way: if a less divergent beam is produced part of the beam will be divergent, part will be parallel and part focussed at some point in front of the reflector as in figure 11.5(b). This has the obvious risk of producing a local hot-spot in the patient's tissue and causing tissue damage. Microwave applicators are available to produce a number of beam patterns. The pattern is not obvious from inspection of the shape of the reflector but the manufacturers do supply this information.

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EFFECT OF MICROWAVES ON TISSUE The penetration depth of microwaves (table 9.1) indicates that the waves are useful for diathermy. The three factors determining the depth efficiency of waves generally are (chapter 9) the penetration depth (δ) of the waves in a particular tissue and the extent of reflection and refraction at tissue interfaces. Considering first penetration depth, we make the observation that tissues with high values of dielectric constant (ε) and conductivity (σ) absorb electromagnetic radiation more rapidly than tissues with low values of ε and σ. The reasons were given in chapter 9 previously. Values of ε and σ are significantly different at microwave frequencies to those appropriate to shortwave diathermy at 27 MHz (table 6.2). Table 11.1 lists the values applicable at microwave frequencies. Notice that fatty tissue and bone marrow have quite similar values of ε and σ - this explains why the penetration depth of microwaves (table 9.1) is almost the same in both tissues. The relatively high values of ε and σ for muscle result in a greater rate of microwave absorption and hence a lower value for the penetration depth in this tissue. The extent of reflection at an interface is calculated from equation 9.5: it is determined by the mismatch in impedance of the tissues. Since we are talking about electromagnetic waves the imped-

Table 11.1 Dielectric constant and conductivity of tissue at microwave frequencies.

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ance of interest is the electrical impedance - determined by the dielectric constant (ε) and conductivity (σ ) of the tissue. Reflection of microwaves at the fat/muscle and muscle/bone interfaces will be pronounced due to the difference in electrical properties (changed by a factor of 10) on either side of the boundary concerned. The amount of refraction at an interface is calculated from equation 9.12: it is determined by the mismatch in wave velocity of the tissues. The wave velocity in turn is determined by the dielectric constant and conductivity of the tissue. Because of the large difference in the electrical properties (ε and σ) of air, fatty tissue, muscle and bone, refraction effects will be significant unless the microwave beam strikes each boundary at a right angle (zero angle of incidence).

The Fraction of Total Energy Absorbed A knowledge of the dielectric constant and conductivity of each tissue enables us to calculate the relative rate of heating of each tissue. This information alone does not allow us to predict the actual amount of heat produced since much of the microwave energy is reflected at the air/skin interface.

Clearly if we wish to calculate the pattern of heating in tissue we must take account of both the penetration depth in each tissue and the amount of reflection and refraction at each tissue interface. Each factor will have a significant effect on the heating pattern.

The significant difference in the electrical properties of air (for which ε ≈ 1 and σ ≈ 0) and soft tissue will result in a considerable amount of the energy incident upon the skin being reflected. The total percentage of microwave energy absorbed deeper in the body tissues and hence converted into heat also depends on the thickness of the skin/fatty tissue layer. This is because a proportion of the wave energy reflected from the fat/muscle interface will penetrate the skin and be re-radiated into the air. Some decades ago, H. P. Schwan (see Licht (1968)) calculated the percentage of total energy reflected at different frequencies and various thicknesses of skin and fat. His results show that: *

At frequencies less than 1000 MHz, 60 to 70% of the energy is reflected this almost independently of skin and fat thickness.

See the chapter by Schwan in: Licht, S H, Therapeutic Heat and Cold, (2nd Edition), New Haven (1968).

ELECTROMAGNETIC WAVES FOR THERAPY *

Between 1000 and 3000 MHz reflection depends critically and in a complex way on tissue thickness. Between 0 and 80% of the energy is reflected.

*

Above 3000 MHz around 60% of the energy is reflected - again almost independently of tissue thickness.

One major implication of the above results is that at a frequency of 2450 MHz the effective dosage is virtually impossible to determine in a clinical situation, due to the practical difficulty in establishing skin and fat thickness which may vary considerably in the treated area. Clearly a frequency above or below the range 1000 to 3000 MHz is to be preferred on these grounds. As we will see in what follows, a lower frequency is preferable.

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Another practical implication of the large amount of reflection is the need to avoid unintentional exposure of body parts (including those of the therapist).

The Distribution of Absorbed Energy We examine now the absorption of the proportion of microwave energy which is not reflected by the skin or re-radiated. Consider a 2 cm fatty tissue layer adjoining muscle tissue. For simplicity we begin by making two assumptions: *

that no bone is present. We will take bone into account in subsequent examples.

*

that refraction can be ignored. In other words the angle of incidence is assumed to be zero. Refraction effects will be described separately.

The relative rate of heating can be calculated from the dielectric constant and conductivity of each tissue: the two factors which determine the amount of reflection and the penetration depth. The method of calculation is described by Schwan (see Licht (1968)). Figure 11.6 shows the pattern of heat production for microwaves at the relatively high frequency of 8500 MHz (wavelength 3.5 cm in air). A standing-wave pattern (see chapter 9) is produced in the fatty tissue: this is because of reflection at the fat/muscle interface.

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Figure 11.6 Heating pattern predicted for microwaves of frequency 8500 MHz in a specimen of 2 cm fatty tissue over muscle.

The standing-wave pattern in the fatty tissue is not ideal since reflection is not 100% and the wave is progressively absorbed in its travel. The actual pattern is a combination of an exponential decrease (determined by the penetration depth, δ) and interference of unequal size waves (figure 9.12). At this frequency, most heat is produced in the fatty tissue close to the skin and in the superficial region of the muscle. A reasonable heating rate is obtained at the muscle surface but the effect extends to only a fraction of a centimetre into the muscle tissue. The total amount of heat produced in each tissue is indicated by the area under the curves in figure 11.6. It is evident that there is greater overall heat production in the fatty tissue. This problem is typical of higher microwave frequencies. The peaks in the heating pattern in the fatty tissue are separated by one half of a wavelength (see chapter 9 - this is close to 1 cm in figure 11.6) so the wavelength of the microwaves in fatty tissue is about 2 cm.

When unequal size waves interfere, the standing-wave effect produces peaks and troughs in the heating-rate pattern, but there are no true nodes (points where the intensity is zero).

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At a frequency of 2450 MHz, the frequency most commonly used in therapy, the relative rate of heating is as shown in figure 11.7. Figure 11.7 Heating pattern predicted for microwaves of frequency 2450 MHz in a specimen of 2 cm fatty tissue over muscle.

Again a standing wave pattern is found in the fatty tissue where most heat is produced. At this lower frequency, the wavelength is greater (since equation 9.1 holds: v = f.λ) so only a single peak is seen in the heating pattern in fatty tissue. Heat production in the muscle tissue is improved over the 8500 MHz results but is still limited to the first centimetre or so. Evidently the lower frequency is preferable from a 'deep heating' point of view - but we saw earlier that frequencies in the range 1000-3000 MHz result in uncertain dosage. What of frequencies below 2450 MHz? Figure 11.8 shows the relative rate of heating predicted for a microwave frequency of 915 MHz in a tissue specimen with the same dimensions as used previously.

Figure 11.8 Heating pattern predicted for microwaves of frequency 915 MHz in a specimen of 2 cm fatty tissue over muscle.

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At 915 MHz, a standing wave pattern is still produced in the fatty tissue but the wavelength is so large that no peaks are evident. Figure 11.9 shows the relative rate of heating predicted for a microwave frequency of 434 MHz in the same tissue specimen.

Figure 11.9 Heating pattern predicted for microwaves of frequency 434 MHz in a specimen consisting of 2 cm fatty tissue over muscle. The depth efficiency of lower frequency microwaves is apparent from figures 11.8 and 11.9. Both frequencies give maximum heating in the muscle with much the same decrease in heating rate with distance into the tissue. The lowest frequency (434 MHz) produces least heating of fatty tissue; the difference being most noticeable near the tissue surface. Both frequencies give a heating pattern which is suitable for diathermy and dosage is reasonably predictable. The heating of the fatty tissue surface with 915 MHz microwaves can be compensated for by using a contact applicator with surface cooling. The microwave director (applicator) is designed to be used in direct contact with the patient. Cooling air is blown through the applicator and on to the patients' skin during treatment in order to minimize the temperature elevation of superficial tissues.

The wavelength in fatty tissue at 915 Mhz is about 18 cm so a peak and a trough would be separated by 4.5 cm (one quarter of a wavelength).

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When we consider the three-layer system of fat/muscle/bone we predict reflection at both the fat/muscle interface and at the muscle/bone interface. In consequence a complex heating pattern is produced in both the fat and muscle tissue. Figures 11.10, 11.11 and 11.12 show the patterns predicted for frequencies of 2450 MHz, 915 MHz and 434 MHz respectively. Tissue dimensions are the same as those chosen to illustrate the heating pattern for ultrasound (figures 10.6 and 10.7).

Figure 11.10 Heating pattern predicted for a microwave frequency of 2450 MHz in a tissue combination of 2 cm fat, 2 cm muscle and 2 cm bone.

Figure 11.11 Heating pattern predicted for a microwave frequency of 915 MHz in a tissue combination of 2 cm fat, 2 cm muscle and 2 cm bone.

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For each frequency, heat production in bone is minimal. Both 915 and 434 MHz microwaves produce maximum heating in the muscle layer: the lower frequency having greater depth efficiency. Not too much significance can be attributed to the actual positions of maxima and minima of heat production as these vary with the tissue dimensions and electrical properties assumed. The general implications of the figures are, however, clear: frequencies below 1000 MHz are needed if tissues located beneath a few centimetres of fat are to be effectively heated. For treating structures located closer to the skin surface for example a knee or elbow joint which is not covered by a thick layer of fat - a frequency of 2450 MHz is adequate, though the dose will be somewhat unpredictable. More deeply located structures - for example, the hip joint - are not heated appreciably at this frequency.

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Figure 11.12 Heating pattern predicted for a microwave frequency of 434 MHz in a tissue combination of 2 cm fat, 2 cm muscle and 2 cm bone.

TISSUE GEOMETRY AND REFRACTION EFFECTS The heating patterns shown in figures 11.6 to 11.12 were calculated ignoring refraction effects. This is appropriate for a uniform microwave beam incident upon a plane surface with tissues of constant thickness beneath. When microwaves are incident upon a curved surface then, even if the beam is uniform, refraction will occur. This is illustrated in figure 11.13, where reflected waves are omitted for clarity. The amount of refraction depends on the curvature of the tissue surfaces an the electrical characteristics of the tissues. When the curvature of the tissues is pronounced, as for example with an arm or leg, the amount of refraction is considerable. The smaller is the radius of the limb, the greater is the refraction effect.

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Tissues of high dielectric constant and conductivity have a low electrical impedance and consequently a low wave velocity. Thus the velocity decreases as the wave progresses from air to fatty tissue and then to muscle. This means that waves will be refracted to produce convergence of the beam. A focussing effect is produced in the tissues. What effect will the beam convergence have on the heating pattern? As the beam travels through fatty tissue and muscle it is progressively absorbed. The wave energy is converted into heat energy and the wave intensity (energy per unit area) decreases. Convergence of the beam will result in the energy remaining any particular depth being concentrated in a smaller area. This tends to increase the beam intensity. Thus with curved tissue surfaces as shown in figure 9.12 the beam intensity does not diminish as rapidly as would occur with plane surface. Consequently the depth efficiency for heat production is greater. H. S. Ho (1976) has calculated the relative rate of heating for cylindric models with dimensions approximating to adult human arms and legs. His results are qualitatively similar to those shown in figures 11.10 to 11.12 but the relative rate of heating of muscle is significantly higher. Nonetheless the conclusions to be drawn from Ho's work are those described earlier. For patient treatment, better heating patterns are produced with frequencies lower than the 2450 MHz currently used. Ho's results indicate an optimum frequency of around 750 MHz for efficient and relatively uniform heating of muscle tissue. For a description of depth efficiency calculations using different frequencies and other geometric shapes resembling parts of the human body see A. W. Guy in Lehmann (1982), chapter 6. In summary we may conclude that 2450 MHz microwaves have low depth efficiency. This frequency is best suited to heat production in fatty tissue and the superficial region of muscle. 915 MHz and 434 MHz microwaves produce greater depth heating of muscle and less heating of fatty tissue. The optimum frequency for selective and uniform heating of muscle tissue being around 750 MHz.

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Figure 11.13 Refraction of a microwave beam at tissue interfaces.

Ho HS. Health Physics, 31, 97-108 (1976). Lehmann, J F, Therapeutic Heat and Cold, (3rd Edition), Williams and Wilkins (1982).

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Microwaves are intrinsically unsuited to heating of bone (see figures 11.10 to 11.12) because of its electrical characteristics: for this reason joints can only be heated when the overlying tissue layers are very thin. For heating of deeply located joints, ultrasound or shortwave diathermy would be more effective. As a final point it should be stressed that the graphs shown in figures 11.6 to 11.12 show where heat is produced but not the temperature increase in each tissue. The temperature increase depends on such factors as the specific heat capacity of the tissue and heat transfer within and between tissues and to the bloodstream (see chapter 7).

LASERS

The difference between heating rate and rate of temperature increase, and the relationship between these quantities, was discussed in chapter 7.

The acronym 'laser' stands for 'light amplification by stimulated emission of radiation'. Lasers are electromagnetic wave amplifiers which can produce beams of electromagnetic waves with two special properties: *

the beam has very little divergence. It has a pencil-like shape.

*

the beam is coherent. That is, all the waves in the beam are of exactly the same frequency and wavelength and are synchronized with each other.

The pencil-like beam of the laser means that the wave energy is always concentrated on the same area: the intensity (which is the energy per unit area) does not decreased appreciably with distance due to beam-spreading.

Production of a laser beam Visible light can be produced by excitation of atoms. For example if crystals of a copper salt, such as copper sulphate, are heated in a flame, the flame turns blue. When strontium salts are heated, the flame turns violet. Sodium salts produce a yellow colouration. This is because electrons in the copper, strontium or sodium atoms are kicked from their 'ground state' orbitals by the heat energy of the flame and

The divergence of a laser beam is so small that a beam pointed at the moon could illuminate a target less than a metre across.

ELECTROMAGNETIC WAVES FOR THERAPY when they fall back into their original orbitals, the energy released is radiated as light of a particular frequency. Because electrons may be kicked out of, and fall back to, different orbitals, the light emitted is a mixture of several specific frequencies. By contrast, laser radiation has but a single frequency. The light emitted by a burning salts is also incoherent, meaning that electrons drop back into their ground-state orbitals randomly so there is no synchronization of the radiated electromagnetic waves. By contrast, lasers are devices which force electrons to drop back into one particular orbital in an avalanche effect, i.e. almost simultaneously. The result is that the emitted waves are all synchronized (coherent) and have the same frequency. The avalanche effect and resulting coherence of a laser beam is achieved by bouncing waves back and forth between two reflectors. For example, a helium-neon laser consists of a cylindrical tube containing helium and neon gas. Each end of the tube has a reflector, one is fully reflecting and the other is partially reflecting so as to allow some light (the laser beam) to escape. The back-and-forth reflection triggers a resonance effect where electrons to drop back into a specific ground-state orbital synchronously and a coherent, monochromatic beam of waves is produced. each wave having the same frequency. To keep the laser operating it is necessary to bump electrons out of their ground-state orbitals and into a higher-energy orbital, ready to drop. For this reason a power supply (a source of energy) is required. Sometimes the energy is provided by an electric current, sometimes by a by a burst of light energy. In the case of a heliumneon laser, a power supply is used to energise a flashlight (rather like a camera flash) which provides rapid-fire bursts of light energy to push electrons into an excited state. In the case of diode lasers, current flow through the diode provides the necessary energy.

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The light emitted by a burning salt is usually at a mixture of waves of different frequency. The different frequencies correspond to electrons returning to different 'ground state' orbitals.

The term 'monochromatic' literally means 'one colour'. In most contexts this means that each wave has the same frequency.

We can summarize the differences between laser light and light from a common, incandescent light bulb as follows. Light from a normal incandescent source has a spectrum of frequencies and the waves are incoherent. Lasers are beams of coherent waves of identical frequency. There is some clinical evidence that laser

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beams can be therapeutically beneficial. What has not been established is whether laser beams have any advantage over simpler (and cheaper) torch beams. No comparisons have yet been reported.

Beam Intensity The output of a laser can vary from tens of milliwatts to tens of kilowatts, depending on the type and the physical construction. Lasers used therapeutically have power levels between these two extremes. They are typically of relatively low power and intensity. Intensities are normally in the range 1 mW.cm-2 to 50 mW.cm-2. The beam diameter of the low power lasers used clinically is about 3 mm (an area of about 7 mm2). Thus if the output intensity is, for example, 20 mW.cm-2 and the area is 7 mm2 = 0.07 cm2, the power of the beam is 20/0.07 mW ≈ 300 mW or 0.3 W.

High power lasers are used to cut steel sheets several centimetres thick. Much lower powers are used in microsurgery, where focused beams are used to cut tiny regions of tissue.

By way of comparison, a torch might have a beam 8 cm in diameter (an area about 50 cm2) and use a 12 W light bulb. As far as visible light output is concerned, the bulb is about 25% efficient (75% of the energy is emitted at infrared frequencies). Hence the power of the visible light-beam is approximately 3 W. The visible-light beam intensity is 3/50 = 0.06 W.cm-2 or 60 mW.cm-2 . The intensity of the infrared component is approximately 180 mW.cm-2. A torch beam thus has a similar and, if anything, a higher power and intensity than a clinical laser but is polychromatic. The wave energy is spread over a range of frequencies. Any clinical significance of the polychromatic/monochromatic difference has yet to be established.

Beam Divergence Light from a light bulb can be formed into a pencil-like beam (as in a searchlight) by using a parabolic reflector but the beam divergence is larger than that of a laser because of the practical difficulty of producing a perfectly shaped reflector. This difference would be of no clinical significance for beams between a light source and the patient, a distance of only a few centimetres or tens of centimetres.

Lasers are often applied with only a thin film of plastic separating the laser from the skin surface, so beam divergence is not important.

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Beam Diameter The beam diameter of the low power lasers used clinically (commonly referred-to as 'low level lasers') is about 3 mm (an area of about 7 mm 2 = 0.07 cm2 ). A consequence is that if the area of the skin surface which is to be treated is several cm 2 , the beam must be scanned over the area. This means that both the average intensity and the energy delivered per unit area are reduced. For example, if the area to be treated is 5 cm x 5 cm (25 cm2), the reduction in average intensity and energy delivered per unit area is 25/0.07 = 3500 times. By contrast, a torch beam would illuminate the same area with no reduction in intensity or energy delivered.

Coherence The light from a light-globe is incoherent. The radiated waves have different frequencies (a spread of frequencies about some mean) and the waves are not 'in synch' with each other. Synchronization is impossible because the wavelengths are different. The coherence of a laser beam is not likely to be of practical significance as biological tissues are quite inhomogeneous at a microscopic level. This means that waves will be scattered and slowed to varying extents so coherence will be lost. A coherent beam striking the skin surface will be incoherent after traversing a distance through tissue of only a few cell diameters. Although coherence is rapidly lost in biological tissue, the beam remains monochromatic i.e. the waves still have identical frequencies.

Coherence is only possible if waves have identical frequencies. If the frequencies (and thus, the wavelengths) are different, they cannot stay in-phase.

Producing a coherent beam using diode lasers is technically difficult. Superluminous diodes are easier to manufacture. These are devices which produce monochromatic, laser-like beams which are non-coherent. It should be noted that some diode 'lasers' used in physiotherapy produce relatively incoherent beams and should more correctly be described as 'superluminous diodes'. The lack of coherence in the beam of radiation produced would appear to be of no clinical significance.

Since coherence is lost when lasers are beamed through tissue, whether the light source is a laser or superluminous diode appears irrelevant.

Laser Light Wavelengths The particular wavelength of radiation emitted by a laser is determined by the physical

ELECTROMAGNETIC WAVES FOR THERAPY design; in particular its chemical composition. Thus helium-neon lasers emit red light with a wavelength of 632.8 nm. Ruby lasers, which consist of a cylindrical rod of synthetic ruby (a gemstone made of aluminium oxide) emit red light with a wavelength of 694.3 nm. Gallium aluminium arsenide (GaAlAs) diodes emit radiation at a frequency determined by the ratio of gallium to aluminium. The particular wavelength can be between 650 nm (in the visible, red part of the spectrum) and 1300 nm (in the near infrared).

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The range of wavelengths which can be produced by laser action is quite large, from the microwave region of the spectrum to the X-ray region.

Two types of lasers are commonly used in physiotherapy: helium-neon lasers, which, as noted above, produce red light of wavelength 632.8 nm and gallium aluminium arsenide diode lasers, operating at near-infrared wavelengths (normally between 810 and 850 nm).

Penetration Depth The penetration depth of laser radiation is the same as ordinary electromagnetic radiation of the same frequency. The wave coherence and the monochromatic nature of the laser beam make no difference. Thus the penetration depth of visible light from a helium-neon laser is a mm or so and most of the wave energy is absorbed in the epidermis (figure 11.3). The infrared radiation produced by commercial GaAlAs diodes has greater penetration depth but most of the wave energy is absorbed in the epidermis and dermis. This perhaps explains why laser irradiation has been shown to be of value for treating ulcers and other skin conditions. What has not been shown, and is not likely to be shown, is that laser treatment is any better than shining a torch beam on the area. Similar considerations indicate that laser irradiation is not likely to be of value for treating deeper tissue injuries. The therapeutic benefit and relative cost effectiveness of laser therapy must thus be questioned.

Consideration of beam area and average intensity indicates that torch-beam therapy might be a cheaper and more effective treatment than laser therapy.

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EXERCISES 1

(a) (b)

2

Figure 11.1 shows a schematic diagram of a mercury vapour lamp. (a)

3

4

5

What are the similarities and differences between infrared, ultraviolet and microwave radiation? State the wavelength range and frequency range of each kind of radiation. Describe the mechanism whereby ultraviolet radiation is produced in the lamp.

(b)

Why must the power supply used for the lamp be current limiting?

(c)

Why must special glass be used for the lamp envelope?

Compare the output of UV, visible and infrared radiation of air and water cooled UV lamps and fluorescent tubes (figure 11.2). (a)

Why do water-cooled lamps put out a negligible proportion of infrared radiation?

(b)

Why do fluorescent tubes put out a negligible amount of radiation at wavelengths less than 280 nm?

(a)

Describe the process of production of infrared radiation by lamps and electric heaters.

(b)

What effect does the use of a reflector have on the directionality and wavelength of the radiation produced?

The filament of a light bulb is at a temperature of 3000 K and its wavelength of maximum emission is 960 nm. If the filament temperature was lowered to 1000 K by decreasing the current what would be the new wavelength of maximum emission? In what part of the electromagnetic spectrum is this wavelength?

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6

(a) (b)

Use figure 11.3 to describe the variation with frequency of the penetration depth of near infrared radiation. Describe the ways in which heat produced by near infrared radiation is transferred to subcutaneous tissue. Which would you expect to be the most efficient transfer mechanism?

7

Compare and contrast the principal effects of infrared and ultraviolet radiation on tissue. How are the differences related to the wavelength of the radiation?

8

(a) (b)

What is meant by the term erythema as related to dosage in ultraviolet therapy? Briefly list the characteristics of a first, second, third and fourth-degree erythema reaction to ultraviolet radiation.

9

Figure 11.4 shows a schematic diagram of apparatus used for the production of microwaves. (a) Briefly describe the function of each subsection. (b) Why is a magnetron valve rather than conventional electronic circuitry used in microwave apparatus? (c) What is the relationship between the size of the antenna in figure 11.4 and the wavelength of the microwaves produced? (d) What determines the frequency of the microwave radiation produced by the apparatus?

10

Figure 11.5 shows the beam produced by a point source of radiation positioned at the focus of a parabolic reflector. Draw diagrams to show the effect on the beam shape of: (a) mounting the point source between the focus and the reflector surface (still on the central axis)

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11

(b)

mounting the point source on the central axis but further from the reflector than the focus.

(a)

Explain why parabolic reflectors are not used with microwave diathermy apparatus. What are the most important factors determining the size and shape of the reflectors used with microwave diathermy apparatus?

(b) 12

(a) (b)

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Give a brief explanation (in molecular terms) of why tissues with high dielectric constant and conductivity have low values of penetration depth for microwave radiation Refer to the figures given in table 11.1 and comment on the relative values of penetration depth for microwaves in fat, muscle and bone. Which tissues would be expected to have similar values of penetration depth and why?

13

It has been said that a frequency of 2450 MHz represents a very poor choice for microwave radiation used in therapy because of the unpredictability of dosage. Explain.

14

Using data in table 11.1 determine the thickness of fat required to absorb 50% of the transmitted microwave energy at a frequency of: (a) 1000 MHz (b) 2000 MHz (c) 4000 MHz

15

For microwaves of frequency 2000 MHz (table 9.1) calculate the fraction of energy remaining after travelling through (a) 2 cm fat (b) 2 cm muscle (c) 2 cm bone. In which tissue is the energy absorbed most rapidly?

ELECTROMAGNETIC WAVES FOR THERAPY

16

Refer to figure 11.6 and explain the origin of the peaks and troughs (maxima and minima) in the heating pattern.

17

Refer to figure 11.6. Draw the corresponding graph of relative rate of heating which would be expected if the reflection coefficient of the fat/muscle interface was: (a) 0.0 (b) 1.0

18

Compare figures 11.7, 11.8 and 11.9 and explain how the microwave wavelength is related to the differences in heat production in each case.

19

H. P. Schwann has shown that from the point of view of reliable dose prediction microwaves with frequencies either below 1000 MHz or above 3000 MHz are preferred. Compare figures 11.6 and 11.9 and say whether high frequencies or low frequencies would be preferred from the point of view of the pattern of heating produced.

20

Compare figures 10.6 and 10.7 with figures 11.10 to 11.13 and explain the differences in heat production in terms of: (a) (b)

the wavelength associated with the modality the penetration depth in each tissue.

21

Figure 11.13 shows a beam of microwaves striking an arm or leg. Briefly explain why the beam converges in fatty tissue and muscle.

22

The diagram below shows a uniform microwave beam striking a tissue surface. The fat and muscle layers have only slight curvature. The bone surface is markedly curved. Fatty tissue and bone have low values of dielectric constant and conductivity. The corresponding values for muscle are high.

302

ELECTROMAGNETIC WAVES FOR THERAPY

Complete the diagram to show the refraction effects. Briefly explain what happens to waves at each boundary and why.

303

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12 Dosage and Safety Considerations Having examined the effects of electric current, electric fields, ultrasound and electromagnetic waves on tissue we conclude by looking at two factors of great importance in the clinical situation. These are the assessment of dose and some specific safety considerations. The two are firmly linked. Under the heading 'safety considerations' we include hazards associated with diathermy and exposure to electromagnetic radiation. Under the heading 'dosage' we concern ourselves with how to establish safe but therapeutically effective dosage both with diathermic modalities and those which are more superficial (infrared, ultraviolet and visible radiation, including laser). When considering electrical stimulation of nerve and muscle for therapy a satisfactory statement of 'dosage' includes specification of the waveform used, duration of treatment, position and size of the electrodes used and a description of the response obtained. Since a response is produced immediately, the therapist can adjust the machine controls to obtain precisely the required effect. In this way there is the potential for optimum conditions to be achieved and for the patient to obtain maximum therapeutic benefit at each treatment. With other forms of treatment, using electric or magnetic fields, electromagnetic radiation or ultrasound, the therapist does not have such reliable feedback. For these modalities there is too long a delay between the commencement of treatment and the effects produced. For example, in the case of diathermy the subjective response of the patient - a feeling of warmth - gives only a poor guide to dosage and effect. With efficient diathermic modalities it is possible that by the time a sensation of heat is felt, the deeper tissue temperatures are high enough to produce irreversible tissue damage. This is because temperature receptors are located superficially, where they are needed to detect the kind of damaging temperature elevations which are experienced normally. For this reason particular consideration needs to be given to the question of dosage as applied to diathermy.

Electrical safety including hazards associated with electrical stimulation and the use of mains powered apparatus are treated separately in chapter 13.

There have been reports of deep tissue injury requiring surgical intervention following incorrect application of 1 MHz ultrasound.

It is convenient to consider dosage in two parts: the first as applied to infrared, visible

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and ultraviolet radiation, the second as applied to the diathermic modalities. Before discussing how we might reliably and reproducibly estimate dosage we need a clear definition of three important quantities. These are: *

The dose: in other words the total energy supplied to the patient - normally expressed in joules (J).

*

The dose rate: the rate at which energy is supplied. This has units of joules per second. One joule per second (J.s-1) is one watt (W).

*

The irradiance: the dose rate per unit area of body surface. Normally in units of joules per square centimetre per second (J.cm-2 .s -1 ) i.e. watts per square centimetre (W.cm-2). When talking about radiation (sound or electromagnetic waves) this quantity is what we call the intensity.

The general requirement in specifying dosage is that all three of these quantities be stated, either directly or indirectly. Each gives important facts about the treatment. For example, consider the heating effect of ultrasound. The total amount of heat developed is determined solely by the dose. The temperature increase, however, depends on the dose rate, the time of treatment and the area treated: that is, on all three factors listed above. Since it is the temperature rise rather than heat production as such which determines the physiological response, a knowledge of dose alone is insufficient.

DOSAGE: INFRARED AND ULTRAVIOLET RADIATION When we consider the question of dosage with infrared or ultraviolet radiation two problems arise: *

The therapeutic effects depend not just on the energy output of the lamp but also on the frequency of the radiation (chapter 11). This is most noticeable with ultraviolet radiation where only narrow ranges of frequency produce the desired reactions.

In this book, the more familiar term 'intensity' is used rather than the more technically correct 'irradiance'.

A dose of 10 J administered in a second or so would evoke a marked physiological response. The same dose applied over a 10 minute period would have little effect.

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*

306

A given dosage from the same lamp will produce a greater response in some patients than others. Again greater variation is found with ultraviolet radiation.

For infrared treatment, specifying the particular type of lamp, the reflector used, the patient-to-lamp distance and the time of exposure is an adequate statement of dosage. Dose, dose rate and intensity are thus specified indirectly. Generally the intensity used is that which produces a mild, comfortable warmth after 5 minutes. If this does not come about, the lamp-to-patient distance can be adjusted during the treatment. With ultraviolet therapy the maximum effects are not produced until long after treatment is complete. For this reason no adjustment of the dose can be made during treatment. A close estimate of the dose requirement is needed beforehand. How can this be achieved? A measurement of the total power output of the lamps is insufficient. Even if the output was measured at different frequencies this would take no account of variation in sensitivity of individual patients. A more useful and direct method is to test lamps in terms of the amount of radiation needed to produce a specific biological response in each particular patient.

With infrared exposure, the intensity used is normally that which produces a mild, comfortable warmth after 5 minutes. If this does not come about, the lamp-topatient distance can be adjusted during treatment.

Ultraviolet Therapy and Erythema Dosage For lamps which produce an appreciable output of UV-B radiation (see chapter 11) tests are carried out to determine the amount of radiation needed to produce a first degree erythema. This will vary from patient to patient and even between different skin areas on a particular patient, but an average figure for the lamp will provide a useful starting point in determining the test dose requirements of an individual. Once known, the dose requirements of a particular patient can be specified as multiples of the first degree erythema dose (erythema dosages were described in chapter 11 previously).

As a UV lamp ages the ultraviolet output diminishes: for this reason the average dose figure must be redetermined periodically.

The dose required to produce a first-degree erythema is determined by exposing small parts of an area similar to that to be treated (usually a few square centimetres) for varying lengths of time. The patient-to-lamp distance is kept constant. Inspection of the exposed areas after 24 hours enables the dosage to be determined.

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Specification of the dosage in this case requires a statement of the particular lamp used, the exposure time and the patient-to-lamp distance. Once the time and distance required for a particular lamp are known the dosage needed to produce any other degree of erythema can be established from table 12.1. E 1 refers to a first-degree erythema, E2 to one of seconddegree and so on. The values quoted are experimentally determined and represent a consensus of agreement amongst physiotherapists. To obtain the exposure time required for a second, third or fourth degree erythema the time for first-degree erythema production is multiplied by the appropriate conversion factor. For example if a first-degree erythema is produced after 6 seconds exposure, table 12.1 indicates that 5 x 6 s = 30 s exposure is required to produce a third-degree erythema. In table 12.1 the lamp-to-patient distance is assumed to be the same. For different distances an inverse-square law is applied (see below) to correct the conversion factor.

Erythema reaction

E1

E2

E3

E4

Conversion factor

1

2.5

5

10

Table 12.1 Conversion factors for different degrees of erythema.

After a first exposure to ultraviolet radiation, there is thickening of the epidermis. Consequently an increase in exposure time is required to produce the same effect on subsequent occasions. Table 12.2 lists the increase required. Again the conversion factors quoted are experimental results, not theoretical values (which would be extremely difficult to calculate).

Table 12.2 Conversion factors for repeated exposure to ultraviolet radiation.

Erythema reaction

E1

E2

E3

E4

Conversion factor

1.25

1.5

1.75

*

*

not normally progressed

DOSAGE AND SAFETY CONSIDERATIONS Thus a second exposure to ultraviolet radiation would require a 50% increase in exposure time in order to reproduce a second degree erythema. The figures shown are only approximate and may need modification in many cases.

PUVA Therapy and Dosage As indicated in chapter 11 previously, UV-A radiation alone does not produce erythema except at extremely high dosages. The common use of UV-A, however, is in combination with a photosensitizing drug, 8-methoxy-psoralen, for the treatment of psoriasis.

308

Following marked desquamation it is common practice to reduce the exposure time to its original (first exposure) value.

In psoralen-UVA (PUVA) therapy the drug is administered two hours before UV-A exposure. The drug renders the patient UV-A sensitive and an erythema response is readily evoked. The dosage required to produce a minimal erythema 72 hours after exposure is determined. This is called the minimal phototoxicity dosage (MPD). It is found by exposing test areas of the patient's skin to predetermined dosages of UV-A (for example 0.5, 1, 2, 3 and 4 J.cm-2) and inspecting the test areas 72 hours later. Once the MPD has been determined, treatment can be given with the dosage specified in J.cm-2. The present practice is to use the MPD for the first treatment and to progress the dosage by 0.5 J.cm-2 or 1 J.cm-2 (depending on skin type) on each subsequent treatment. UV-A fluorescent tubes display a significant drop in output intensity, particularly over the first 200 hours of use. For this reason it is essential that the output intensity of the UV-A source be regularly measured. Special meters, calibrated in W.cm-2 , are available for this purpose.

Patients who always burn in the sun are progressed by 0.5 J.cm-2. Those who never or rarely burn are progressed by 1 J.cm-2.

If the output intensity of the source is known the dosage in J.cm-2 can be calculated using the relationship: Dosage (in J.cm-2) = Intensity (in W.cm-2) x time (in s)

DOSAGE AND SAFETY CONSIDERATIONS THE INVERSE SQUARE LAW AND DOSAGE Turning now to the more general aspects of dosage we consider the effect of the distance from the source of radiation to the patient. These considerations apply to both infrared, visible and ultraviolet radiation - and also to microwaves to a more limited extent. Consider a screen with a small square aperture, behind which is placed a point-source of radiation. The arrangement is shown in figure 12.1. Figure 12.1 The law of inverse squares.

Radiation produced by the point source spreads uniformly in all directions. The aperture in the screen allows a beam of square cross-section through. The distance from source to screen is s and the sides of the screen aperture have length x. The beam area at the screen is thus x 2 . At a distance 2s from the source (a further distance s from the screen) the beam area is (2x)2 = 4x2. At a distance 3s from the source the area is (3x)2 = 9x 2 . In other words, as we progress 1,2,3,4 units of distance from the source the beam area increases to 1,4,9,16 times the original area: it increases in proportion to the square of the distance.

309

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310

What happens to the wave energy which passes through the square aperture? For infrared, ultraviolet and microwave energy there is very little absorption in air over a distance of a few metres. In other words their penetration depths in air are large. This means that the wave energy in the rectangular beam in figure 12.1 is virtually constant. If the beam of radiation has an energy E at the aperture then the intensity - the energy per unit area - is E/x2. At a distance 2s from the source this energy is spread over an area 4x2 , so the intensity is E/4x2 or one quarter of its value at the screen. At a distance 3s the intensity is one-ninth of the value at the screen. An inevitable conclusion is the law of inverse squares which states that the intensity of radiation from a point source varies inversely with the square of the distance from the source.

If the energy of the beam is constant but is spread over a larger and larger area with distance from the source then the intensity, which is the energy per unit area, must decrease.

Mathematically this is written:

I

o .... (12.1) d2 where I in the intensity at a distance d from the source and Io is the intensity at unit distance.

I =

How is the law of inverse squares applied to dosage? Strictly speaking, the inverse square law only holds for point sources of radiation. Sources of infrared and ultraviolet are extended sources, usually mounted in reflectors. The effect of the reflector is to reduce the divergence of the beam, but for the lamps used in physiotherapy departments the effect is not too great and the law provides a rough, rule-of-thumb, but satisfactory basis for calculations.

For example if the intensity is 0.1 W.cm-2 at a distance of 1 m from the source, then at a distance of 2 m the intensity will be 0.1/4 = 0.025 W.cm.

An Example: Consider treatment with a high pressure mercury vapour lamp (a hot quartz lamp; chapter 11). Suppose we know that the minimal erythema dose with a particular lamp is 18 seconds at a distance of 1 metre and we wish to use the lamp at a distance of

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1.5 metres. What exposure time is required at this new distance? The production of the erythema reaction depends on the intensity of radiation and the exposure time. At a distance of 1.5 metres the intensity is reduced by a factor of (1.5)2: thus the exposure time needs to be increased by a factor of (1.5)2 to produce the same effect. The new exposure time is then 18 x (1.5)2 = 41 seconds. This kind of calculation shows how the law of inverse squares, which nominally relates intensity to distance is adapted to relate exposure time to distance. t =

to.d2 do2

.... (12.2)

Here t o and do refer to the original exposure time and the original distance respectively. t is the new exposure time at the new distance d.

The Effect of the Angle of Incidence What happens if the beam of radiation does not strike the surface of the patient's skin at right-angles? This happens in the circumstances illustrated in figure 12.2. In each case some or all of the radiation has an angle of incidence, θ, which is not zero. When this happens the amount of reflection is increased and the beam is spread over a larger area.

Figure 12.2 (a) effect of beam divergence on the angle of incidence, θ. (b) effect of both beam divergence and surface curvature. (continued overleaf)

Using equation 12.1 and the fact that dose per unit area is the product of intensity and exposure time it is a simple matter to derive equation 12.2 which relates exposure time at one distance to exposure time of any other distance.

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Figure 12.2 (c) effect of beam divergence and angulation of the reflector on the angle of incidence, θ.

Reflection is minimal at an angle of incidence of 0o. As the angle is increased, the amount of reflection increases. The relationship between reflectance and angle is a complex one (at the critical angle, reflection is 100%). In each of figures (a), (b) and (c), reflection is increased because the angle of incidence is not always zero. The beam (or part of the beam) is spread over a greater area in figures 12,2(b) and (c). This results in a decrease in the intensity of radiation at the surface. In figure 12.2(b) the area illuminated by the beam is larger because the surface is curved. In figure 12.2(c) the beam is spread over a wider area because of the angulation of the reflector. The effect of angulation is further illustrated in figure 12.3 where, for simplicity, we consider a tiny portion of the beam with width x and square cross-section.

Figure 12.3 The effect of angle of incidence on intensity.

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When the beam is incident at right angles the energy is spread over an area x2 so the intensity is E/x2. With the angle of incidence θ the area irradiated is x2/cosθ and the intensity if Ecosθ/x2: that is, the intensity is reduced by a factor of cosθ. For example, for an angle of incidence of 15o, cosθ = 0.97 and the intensity is reduced by 3%. Table 12.3 shows the relative intensity (as a fraction of the intensity for θ = 0) for different values of θ. This effect is quite noticeable when using a torch to see one's way on a dark night. Pointing the torch downwards gives a circular beam. Shining the torch ahead gives a larger area of illumination with an egg shape. Here the area depends on both the distance (through the inverse square law) and the angle of incidence. In the application of infrared and ultraviolet radiation, the therapist should be aware of this effect. It is normal practice to keep the beam as near to perpendicular to the treated surface as possible: thus the situation shown in figure 12.2(c) should be avoided. Even if a perpendicular arrangement is used, parts of the treated area near the periphery may receive a lower dosage (figure 12.2 (a and b)). If necessary the lamp should be moved to give additional exposure to these areas.

DOSAGE AND DIATHERMIC MODALITIES In the previous chapters we examined the relative rate of heating of different tissues in combination. We saw that the resulting temperature increase depends on a number of factors including the dose rate and time of exposure. In any treatment there will be threshold values of dose and dose rate which cannot be exceeded without risk of harm to the patient. There will also be minimum values of useful dosage. Below this, heat development in the tissues will be within the range which the body's temperature regulating mechanism can cope with and there will be an insignificant local rise in temperature. The maximum safe energy dose exceeds the minimum required to produce an appreciable effect by less than a factor of ten: thus a knowledge of dosage is of great

angle of incidence (degrees)

relative intensity

0 15 30 45 60 75 90

1.00 0.97 0.87 0.71 0.50 0.26 0.00

Table 12.3 Relative intensity for different values of θ, the angle of incidence.

See the chapter by Schwan in: Licht, S H, Therapeutic Heat and Cold, (2nd Edition), Williams & Wilkins (1968).

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importance. In order to specify dose and dose rate we need to know first the energy produced by the apparatus and second, the fraction of that energy which is absorbed by the body. Unfortunately both of these quantities are not always known, as we will see.

Shortwave Diathermy This is the modality which has been in use for the greatest length of time. It is also the one for which the dosage is least predictable. In the case of the capacitor field technique the energy produced by the apparatus varies with the position and size of the electrodes and the amount and type of tissue in the field. It is possible to simulate the conditions of therapy by placing a 'dummy load' between the electrodes. The load must have the right electrical properties and be correctly positioned to simulate the conditions of therapy. In this way the energy produced can be measured, though not the energy absorbed by the patient. Scott (see Licht (1968)) describes how a series of subjects were tested with apparatus adjusted for a predetermined rate of energy production. The extreme variation in the responses obtained indicates that a knowledge of energy production alone is of little value in establishing dosage.

The meter on the front panel of most shortwave diathermy machines only indicates a relative value of power - it gives no indication of the actual dosage.

A further complication is that the field spreads as it passes through the body. This results in the area treated being much larger than the electrodes used, and varying with depth. This makes it impossible to predict accurately the heat developed in a particular part of the tissue. With the inductive coil technique of application the situation is just as complex - due to the difficulty in establishing the pattern of induced electric field intensity with the geometries used. For the present, the most reliable estimate of correct dosage is obtained by adjusting the intensity until the patient feels a mild, comfortable warmth in the treated area. This is a relatively safe method of assessing dose as greatest temperature elevation is

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produced in tissues where pain and temperature receptors are abundant. Since temperature elevation is less in the more deeply located structures there is little risk of overheating them without first producing pain and damage to superficial tissues. The need to rely on physical sensations indicates why shortwave diathermy (or indeed any diathermic modality) is contra-indicated for areas where sensory impairment is suspected.

Ultrasound In the case of ultrasound therapy, virtually all the energy produced by the generator is transferred to the patient, provided that intimate contact is maintained between the transducer and body surface (chapter 10). Generator-produced power can be read directly from the meter on the front of the apparatus: thus the dose is obtained simply by multiplying the power (in watts: 1 W = 1 J.s-1) by the treatment time (seconds). The irradiance or average intensity (in W.cm2 ) is not so reliably known when the usual massage technique of application is used and the treatment head is moved in small circles over the area to be treated. The average intensity is calculated by dividing the total power by the area treated: it will only be a reliable figure if the therapist is able to expose all parts of the treated area for the same length of time.

Since even minute amounts of air can interrupt the flow of energy, the space between transducer and skin is filled by a coupling medium.

Ultrasound ranks highest of the diathermic modalities in terms of reliability and reproducibility of dosage. Even so it is difficult to assess the dosage applied to a particular structure or tissue layer within the treated part. This is partly because of the difficulty in estimating the thickness of different tissue layers and, more significantly, because the ultrasound beam intensity (figure 10.4) is nonuniform.

Microwaves The power produced by a microwave source can be quite accurately measured. It can usually be read directly from a meter on the front of the apparatus. Unfortunately, at a frequency of 2450 MHz, the proportion of energy actually absorbed depends on a complex way on the thickness of skin and subcutaneous fatty tissue (chapter 11).

Up to 80% of the microwave energy may be reflected at a frequency of 2450 MHz (see chapter 11).

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In view of the practical difficulty in estimating tissue thickness, microwave therapy at 2450 MHz does not permit accurate dosage measurement. As with shortwave diathermy, the dosage, and consequently the heat development can only be estimated roughly. The physiotherapist must be guided by a knowledge of the pattern of heat production, a knowledge of 'normal' dose rates and the subjective reports of the patient. A more quantitative assessment of dose can, at least, be made for lower frequency microwaves, though the proximity of the patient to the source of radiation (less than one wavelength in normal applications) makes the distribution of energy difficult to calculate.

The difficulty in establishing microwave dosage is discussed in Lehmann, J F, Therapeutic Heat and Cold (3rd Edition) Williams and Wilkins (1982): chapter 6).

IMPLANTS AND CAVITIES In this and the following sections we consider some of the safety aspects of the treatments we have discussed. When using any of the diathermic modalities in the region of a metallic implant, air or fluid-filled cavity, particular consideration needs to be given to the likely effects of the cavity or implant on the pattern of heat production in nearby tissue. The effects are different for each modality so we will consider each in turn.

Shortwave Diathermy

The scope of this book precludes consideration of the indications and contraindications for each modality. Rather we restrict ourselves to the effect of implants and cavities, and to some specific safety hazards.

We considered in chapters 6 and 7 the way in which different tissues (fat, muscle and bone) modify the field pattern and determine the magnitude of real and displacement current. This in turn determines the pattern of heat production in tissue combinations. The two quantities determining these effects are, as we saw, the dielectric constant and conductivity of the tissues. To determine the effect of a cavity or implant we need to know its depth, shape and size and, most importantly, its electrical properties. *

Metals have extremely high conductivities - several thousand times higher than muscle.

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317

*

Air, for all practical purposes, can be considered a perfect insulator. conductivity can be taken as zero and the dielectric constant as 1.

*

Body Fluids can be considered equivalent to muscle and other tissues of high water content. The differences in electrical properties are negligible as far as shortwave diathermy is concerned.

Figure 12.4 shows the effect of a cylindrical object of high dielectric constant or conductivity on a uniform electric field. This illustrates the focussing effect on the field lines of a metallic implant in tissue. The effect of the orientation of the object on the field is apparent. A greater focussing effect is produced when the long axis of the object is aligned with the field (figure 12.4a). When the length along the field direction is short, as in figure 12.4(b) and (c), the field distortion is less. Figure 12.4 A material of high dielectric constant or conductivity in a uniform electric field. The effect of different orientations.

The

DOSAGE AND SAFETY CONSIDERATIONS The heat production within a metal implant is very low because the field intensity within the metal is very low - due to the rapid, free movement of charge which results in accumulation of charge on the surface and termination of field lines. The field intensity is greatest near the surfaces of the metal perpendicular to the field lines. It is here that maximum heat is produced. The risk, then, is of overheating tissue adjacent to a metallic implant. For this reason shortwave diathermy is often contraindicated when a metallic implant is present. The effect of metallic implants is discussed more fully by B. O. Scott in Licht, (1968). Figure 12.4 is also applicable to fluid-filled cavities in fatty tissue or bone. This is because the dielectric constant and conductivity of body fluids are considerably higher than those of bone or fat. We saw an example of this with blood vessels in fatty tissue in chapter 7. Field lines will be focussed resulting in maximum heat production near the cavity. The field intensity within the cavity will be reduced by charge accumulation at the interface, but this may not be sufficient to prevent overheating.

318

If the metal was a perfect conductor the resistance would be zero, the field intensity zero and the heat production zero. See the chapter by Scott in: Licht, S H, Therapeutic Heat and Cold, (2nd Edition), Williams & Wilkins (1968).

Fluid-filled cavities in muscle or other tissues of high water content will not affect the electric field pattern appreciably. The temperature rise in the cavity will however be greater than in muscle because heat is not transferred efficiently to adjacent tissues or the bloodstream. The effect of an air-filled hollow in tissue was discussed in chapter 7 (see figure 7.14). The field lines bend around the hollow. This results in an increased intensity in the tissue adjacent to the sides of the hollow which are parallel to the field. The effect proves useful when it is desired to selectively heat the surfaces of hollows, such as the sinuses.

Ultrasound We saw in chapter 9 that reflection of ultrasound occurs when there is a mismatch of acoustic impedance between two adjacent tissue layers. The impedances of muscle and fatty tissue are similar but that of bone is much higher. There is thus an appreciable reflection of ultrasound at the muscle/bone interface.

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In order to determine the effects of implants or cavities we need to know the acoustic impedances of metals, air and body fluids. *

Metals have an acoustic impedance about thirty times higher than fat or muscle so there will be significant reflection at a tissue/metal interface. Using the figures in table 10.1 and equation 9.5 we find that the reflection coefficient for a fat/metal or muscle/metal interface is about 0.94. Thus about (0.94)2 x 100 or 90% of the ultrasound energy will be reflected.

*

Air has an acoustic impedance which is only a tiny fraction of that of tissue so virtually 100% of the energy incident upon a tissue/air interface will be reflected.

*

Body fluids have an acoustic impedance closer to that of water, muscle and fatty tissue. Fluid-filled cavities will not pose any problems as regards reflection of the ultrasound beam.

Whenever reflection occurs there will be an increase in the ultrasound intensity adjacent to the reflecting surface and greater heat production in this region (see chapter 10). For a metallic implant in tissue, reflection will be significant and almost all the heat will be produced between the implant and the tissue surface. The excess heating of the tissue layer above the implant may or may not be advantageous depending on the actual location of the tissue to be heated. The additional factor of the effect of shape of the implant on the ultrasound field pattern is discussed by Lehmann in Licht, (1968). The presence of an air-filled cavity in tissue will have a substantial effect on the pattern of heat production. Almost total reflection will occur at the interface and almost all of the heat will be produced in the intervening tissues. This has particular implications for treatment of the chest wall or throat. As we saw, the presence of fluid-filled cavities has little effect on the pattern of heat production with ultrasound. The only factors to be assessed are the likelihood of

A selective build-up of heat could result, however, if the heat cannot be transferred to adjacent tissues or the bloodstream.

Little heat will be produced in a metal as little energy is transmitted at the tissue/ metal interface and ultrasound is not absorbed very rapidly in metals (the penetration depth is high).

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selective heating within the cavity due to poor heat dissipation, and whether this is desirable.

Microwaves As we saw in chapter 11, the reflection of microwaves and the rate of absorption are determined by the electrical properties (dielectric constant and conductivity) of tissues. Since metals have a much higher conductivity than any biological tissue, reflection at a tissue/metallic-implant boundary will be pronounced. The high conductivity of metals also results in rapid absorption of microwaves - penetration depths are extremely small. The result is that pronounced reflection occurs at a tissue/metal boundary and the transmitted wave is absorbed over a very short distance. Microwaves penetrating metallic implants will be absorbed in a fraction of a millimetre with significant heat production. However, metals are good conductors of heat and the energy will be rapidly conducted throughout the metal and spread into the adjacent tissues. Reflection of microwaves at a tissue/metal interface will result in the production of standing waves. The energy reflected and the resulting standing wave pattern will produce a concentration of energy in the tissues adjacent to the metal. There is also the risk of focussing the waves with a curved metal surface (chapter 11) which can result in 'hot spots' being produced in the patient's tissues. The rather poor penetration depth of 2450 MHz microwaves suggests, however, that metallic implants located well below the surface of the body will have little effect on heat production.

The electrical properties of metals, air and body fluids were considered earlier in this chapter.

The phenomenon of rapid absorption (and consequently great heat production) can be demonstrated quite convincingly by igniting a piece of steel wool in a microwave oven.

The effect of an air-filled cavity is similar to that of a metal implant: reflection occurs at the boundary and a standing wave pattern is produced. The implications of this were discussed above. Fluid-filled cavities within muscle and other tissues of high water content will not affect the pattern of heat production, but may undergo a selective rise in temperature if heat is not conducted away efficiently.

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SOME SPECIFIC HAZARDS Electromagnetic Waves and Safety It is known from various studies that certain body tissues are more susceptible than others to damage from electromagnetic waves. The eyes and reproductive organs are most frequently mentioned in this regard. The eyes, not having a covering of skin, are susceptible to damage by ultraviolet radiation. In ultraviolet therapy the eyes should always be protected from direct irradiation by use of glasses which reduce the visible light intensity and absorb most of the ultraviolet radiation. Sunglasses perform this role quite adequately. The only risk is of UV exposure through the areas not covered by the sunglasses. For this reason, protective goggles, which cover the eyes completely, are preferred.

Sunglasses or goggles made of plastic or glass and painted with a filter (often coloured blue) can effectively block ultraviolet transmission.

Exposure of the eyes to a sufficiently high dose of ultraviolet radiation produces photopthalmia - acute inflammatory reactions of the superficial parts of the eye. This is commonly known as snow-blindness (snow reflects a large part of the UV radiation in sunlight). It can be produced by sunlight, electric welding arcs or any other source of ultraviolet radiation. The reactions cause acute pain, beginning after a latency period of a few hours and reaching a maximum in about 48 hours. The effects subside over a period of days. Only very large doses produce permanent damage. It should also be borne in mind when considering prolonged or repeated courses of treatment with ultraviolet radiation that such radiation is carcinogenic. Certain forms of cancer are known to occur more frequently in people exposed to higher levels of ultraviolet radiation. Microwave therapy is contraindicated for treatment of eye conditions. Generally, extreme caution should be exercised when treating nearby structures. The susceptibility of the eyes to damage by microwave radiation is due to two factors (a) reflection and refraction producing 'hot-spots' within the eye cavities and (b) a relatively poor blood supply which limits the eye's ability to conduct heat away.

As with all forms of therapy the risks associated with UV exposure must be calculated and weighed against the therapeutic benefit in deciding a course of treatment.

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It has been known for some time that sufficiently high intensities of microwave radiation can bring about the formation of cataracts in the eye. Experimental work using laboratory animals indicates a threshold intensity level for cataract formation a little in excess of 100 mW.cm-2 for prolonged exposure. It is common practice to avoid exposing the reproductive organs to microwave radiation. The testes are particularly susceptible to stray radiation in therapy. For a detailed description of the hazards of microwave exposure and references to relevant experimental work see S. M. Michaelson in Lehmann (1982). For a more general description of the hazards of both radio-frequency and microwave radiation see publications by the World Health Organization (search their website at www.who.int). These documents discuss the known biological effects of such radiation and summarize exposure safety limits proposed or in use in different countries.

The hazards with microwave exposure are discussed in Lehmann, J F, Therapeutic Heat and Cold (3rd Edition) Williams and Wilkins (1982): chapter 7).

All practicing physiotherapists should be familiar with the relevant safety standards and their implementation. It should be noted, however, that the exposure limits stipulated apply to the general public but not to the patient receiving treatment, nor the therapist. For example, the maximum exposure level for a therapist using 27 MHz shortwave diathermy apparatus is 1.2 mW.cm-2 . For non-occupationally exposed individuals such as secretarial staff and members of the general public the stipulated levels are one fifth of these values. For patient exposure, there is no prescribed limit. It is assumed that the therapist has weighed the therapeutic benefits against the potential hazards and on this basis has prescribed treatment.

Australian standard AS2772.1 (1998) stipulates the maximum exposure levels allowed for radio-frequency and microwave frequency radiation.

Ultrasound And Boundary Effects A fundamental characteristic which distinguishes ultrasound from other diathermic modalities is that the ultrasound wave is a mechanical disturbance in a material medium. Particles within the medium oscillate back and forth, undergoing large changes in velocity and acceleration. This gives rise to two phenomena which can result in selective heating at or near a boundary. The processes are called velocity gradient heat production and shear wave production.

DOSAGE AND SAFETY CONSIDERATIONS

We consider first velocity gradient heat production. Suppose an ultrasound beam is directed so as to strike a tissue boundary at a grazing angle. In other words the waves travel almost parallel to the interface. At this angle of incidence little or no wave energy will be transmitted. There will be a thin layer, just each side of the boundary where the velocity changes from maximum to zero. Thus there is a velocity gradient in this narrow region. If the boundary region is very narrow a very high velocity gradient exists and the adjacent region are subject to greater stresses than those outside. The higher oscillatory stresses give rise to greater heat production than occurs in the medium in which the waves are travelling. Hence boundary layer heat production can be significantly greater than heat production due to normal wave energy absorption in a medium.

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This means that particles on one side of the boundary are oscillating back and forth with the wave motion while particles on the other side of the boundary are stationary.

The thickness of the boundary layer determines the velocity gradient and this in turn depends on the rigidity of each medium. For tissues of similar stiffness, such as muscle and fatty tissue the boundary layer is wide, the velocity gradient is small and boundary layer heat production is minimal. For tissues of quite different stiffness, such as muscle and bone, the velocity gradient is high and the rate of heat production at the interface is much higher than in the bulk of the tissues. Shear waves can be produced when an ultrasound beam strikes a boundary. They are not produced when the wave strikes the boundary at a right angle (zero angle of incidence), nor is production appreciable at grazing angles. Maximum production occurs near the middle of the range. While normal sound waves are a longitudinal wave motion, shear waves are transverse. In other words the particle displacement is at right angles to the direction of propagation. A further point which should be noted is that shear waves can only exist in solids or very viscous liquids and they are absorbed more rapidly than transverse waves. Shear waves are produced when the wave velocity is different in two adjacent tissues. The wave frequencies must be identical so the difference in wave velocity results in a

In practical terms this means that shear wave production is relatively unimportant at soft tissue interfaces but is important at the muscle/bone interface.

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different wavelength in each tissue (by equation 9.1, v = f.λ). This means that regions of compression and rarefaction are separated by different distances in the adjoining tissues. So at the boundary, a region of compression on one side will periodically be aligned with a region of rarefaction on the other. The resulting pressure differential will cause particles near the boundary to oscillate in a direction transverse to the direction of the reflected and transmitted waves. The high stresses produced at the interface result in greater heating than in the bulk of each tissue. When an ultrasound beam in muscle strikes the muscle/bone interface the amount of energy taken by the shear wave can be large. As the rate of absorption of shear wave energy is much higher than that of longitudinal waves, heating of the surface region of the bone is accentuated. In chapter 10 we saw that in the fat/muscle/bone system, 1 MHz ultrasound produces the greatest rate of heating in the first few mm of bone. The resulting temperature rise in the periosteum places a limit on the rate at which energy can be supplied to the patient. This happens when ultrasound is incident on each tissue boundary at a right angle and shear wave production and velocity gradient effects are negligible. For ultrasound incident upon the bone at other than a right angle, the heating rate of tissue adjacent to the bone will be even greater so the rate at which energy can be supplied to the patient will be further limited. The implication is that in therapy, the risk of producing periosteal pain and tissue damage is enhanced if the ultrasound beam does not strike the bone surface at a right angle. The therapist should position the treatment head as close to parallel to the surface of the underlying bone as possible so as to minimize velocity gradient and shear wave effects.

Dunn and Frizzel (see Lehmann, 1982, chapter 9) calculate that for 1 MHz ultrasound, up to 80% of the energy transmitted at the muscle/bone interface may go into production of shear waves.

Ultrasound and Cavitation We discussed in chapter 10, the mechanical stresses produced when an ultrasound wave travels through tissue. With 1 MHz ultrasound at an intensity of 2 W.cm-2 , regions separated by 0.75 mm differ in pressure by about 20 N.cm-2 . The large pressure gradient can result in gaseous cavitation: the formation of tiny bubbles from

DOSAGE AND SAFETY CONSIDERATIONS

A pressure of 20 N.cm-2 is 20 x 104 N.m-2 or 20 kPa). This is about 1/5 of atmospheric pressure.

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gas dissolved in the tissue fluid in a region of rarefaction (low pressure). An upper limit to the therapeutic dose rate of ultrasound is set by the threshold intensity for cavitation. Treatment at intensities above this threshold could only produce the therapeutically undesirable outcome of mechanical tissue damage. In practice this threshold is unlikely to be achieved clinically. An intensity of 2 W.cm-2 would produce sufficient heating for temperature elevation to be the limiting factor. Thus although cavitation is a risk and a therapeutic hazard, it is not likely to be of practical significance. Cavitation effects are described more fully by Coakley (Physiotherapy. 64, 166-169, 1978). Treatment of the eyes with ultrasound is generally avoided because of the risk of damage in any tissue which has a poor or restricted blood supply.

Cardiac Pacemakers Cardiac pacemakers present a special hazard as far as diathermy is concerned. Two kinds of pacemaker are used: the fixed rate unit which provides a constant frequency train of stimuli to the heart and the more popular noncompetitive units which provide a stimulus frequency based on feedback signals from the heart. Noncompetitive units are more satisfactory medically as the heart rate is adjusted by the oxygen demand of the patient. There are two risks in the application of diathermy: *

the risk of selective heating of the unit and tissues in contact with the unit and its wires. Each diathermic modality presents this hazard when used close to the unit.

*

more importantly, the risk of interfering with pacemaker action. Microwave and shortwave diathermy present the greatest hazard in this regard. The fixed

The formation of a gas bubble can cause damage, tearing apart the tissue. Also, the bubble may collapse during the subsequent compression phase, creating a minute but intense shock wave in the immediate area.

DOSAGE AND SAFETY CONSIDERATIONS frequency pacemaker is less susceptible as it does not require any feedback signal. Noncompetitive units can change their frequency or cease to function completely as a result of currents induced by microwave radiation or the shortwave field. The limited amount of research in this area to date indicates that shortwave and microwave radiation are contraindicated when a pacemaker is present. Some units are found to cease functioning when brought within a few metres of microwave apparatus. Ultrasound therapy seems safer in this regard but extreme care should be exercised, when using this modality, to avoid the area containing the pacemaker and wires.

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See J. F. Lehman in Krusen's Handbook of Physical Medicine and Rehabilitation (3rd Edition), Williams and Wilkins (1982) and J. Heath Modern Medicine (Abstract) Vol. 43, June 15, 1975

EXERCISES 1

(a) (b)

2

3

Define the terms dose, dose rate and irradiance. What are the units of each quantity? Use infrared therapy as an example and explain how each of the above quantities are specified in describing the treatment conditions.

(a)

List the treatment modalities for which the response of the patient can be used as an immediate guide as to dosage requirements.

(b)

For each treatment modality not listed in (a) above, explain why the immediate response of the patient can not be used to assess dosage requirements. Distinguish those cases where the immediate response is a poor guide and where an immediate response is not normally produced.

Describe the way in which dose requirements are established for (i) ultraviolet therapy and (ii) PUVA treatment in terms of: (a) calibration of the lamp (b)

testing an individual patient

DOSAGE AND SAFETY CONSIDERATIONS 4

A patient about to receive ultraviolet therapy is tested with a particular lamp and found to require 25 seconds exposure at a distance of 0.6 m to produce a firstdegree erythema reaction (E1). (a) What exposure time would be required to produce a second-degree erythema reaction (E2) with the same lamp-to-patient distance? (See table 12.1). (b) What exposure time would be required to again produce a second degree erythema reaction with a subsequent treatment? (See table 12.2). (c) What exposure time would be required to produce an E2 reaction at the second treatment if the patient-to-lamp distance is decreased to 0.4 m?

5

A patient is found to require a 35 second exposure from a particular lamp to produce a first-degree erythema reaction. The lamp-to-patient distance is 0.5 m. (a) What exposure time would be required to produce an E1 reaction if the lamp-to-patient distance was changed to 1 m? (b) What exposure time would be required to produce an E3 reaction with the lamp-to-patient distance kept at 0.5 m? (c) What exposure time would be required to produce an E3 reaction with the lamp-to-patient distance reduced to 0.25 m?

6

Equation 12.1 was derived by considering light from a point source passing through a rectangular aperture in a screen (figure 12.1). Show that the same relationship is obtained when we consider light passing through a screen with a circular aperture of diameter x.

7

Using the fact that dosage depends on the product of intensity and exposure time, derive equation 12.2 from equation 12.1.

8

What are the two factors which contribute to lower the dosage when radiation is not incident at right angles to a tissue surface?

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DOSAGE AND SAFETY CONSIDERATIONS

9

(a) (b) (c)

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Draw a diagram similar to figure 12.3(b), showing a light beam of square cross-section striking a surface at an angle θ. Use simple trigonometry to show that the square beam consequently illuminates a rectangular area with sides of length x and x/cosθ. Hence show that the light intensity at the surface is Ecosθ/x2, where E is the light energy in the beam.

10

A beam of light strikes a surface at right angles as shown in figure 12.3. The light intensity at the surface is measured as 50 J.m-2. Calculate the light intensity if the surface is tilted so that the angle of incidence is: (a) 15 o (b) 30 o (c) 45 o

11

Briefly summarize the particular problems of assessing dosage in: (a) shortwave diathermy (b) ultrasound diathermy (c) microwave diathermy For which modality is the dosage most reproducible and for which is it least reproducible?

12

(a) (b) (c)

Draw a diagram similar to figure 6.19(a) but including a metal implant (say a metal plate on the surface of the bone). Draw the pattern of electric field lines and indicate clearly where the field intensity is increased and where it is decreased. Describe the effect of the implant on heat production in different areas of the tissue. Why would heat production within the metal be low?

DOSAGE AND SAFETY CONSIDERATIONS

13

Assume that the electric field within a tissue layer is originally uniform and draw diagrams to show the effect of a spherical implant or cavity for the following cases: (a) an air-filled cavity in muscle (b) an air-filled cavity in fatty tissue (c) a fluid-filled cavity in muscle (d) a fluid-filled cavity in fatty tissue (e) a metal implant in muscle In each case state reasons for the effects on the field pattern which you have shown.

14

Briefly describe the effects of shape and orientation of a metal implant in tissue subjected to an electric field (figure 12.4). What are the implications for shortwave diathermy treatment using parallel plate electrodes?

15

Consider the effect of implants and cavities on the distribution of ultrasound energy in tissue. (a) What factors determine the amount of energy reflected at a tissue/implant or tissue/cavity interface? (b) Would you expect significant reflection at the following interfaces? (i) tissue/metal (ii) tissue/air cavity (iii) tissue/body fluid cavity Justify your answers.

16

The pattern of heat production for 1 MHz ultrasound in a fat/muscle tissue combination is shown in figure 10.5. (a) Draw diagrams similar to figure 10.5 to show qualitatively the effect on the

329

DOSAGE AND SAFETY CONSIDERATIONS

17

(b)

pattern of heat production of a metal implant located 4 cm below the tissue surface. Would you predict any difference in the pattern of heat production if an airfilled cavity rather than a metal implant was included in the tissue? Justify your answer.

(c)

What differences would be expected in the temperature elevation of tissue adjacent to a metal implant as compared to an air-filled cavity? Explain.

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Briefly describe the effects of the following on the pattern of heat production in tissue exposed to microwaves: (a) an air-filled cavity (b) a metal implant (c) a fluid-filled cavity What differences in temperature elevation would occur with (a) as compared to (b)? Explain.

18

(a)

Briefly describe why the eyes must be protected from exposure to (i) microwaves and (ii) ultraviolet radiation in therapy.

(b)

For ultraviolet therapy adequate shielding of the eyes is achieved by wearing protective glasses. Would similar glasses provide adequate protection from microwaves? Explain. List other specific hazards associated with microwave and ultraviolet radiation.

(c) 19

(a) (b) (c)

What is meant by the term 'velocity gradient heat production' and under what circumstances is this effect significant? What is a shear wave and under what circumstances is shear wave production significant? What effect will shear wave production and velocity gradient heat production have on the temperature distribution in tissue?

DOSAGE AND SAFETY CONSIDERATIONS

20

(a) (b)

21

What is meant by the term 'gaseous cavitation'. Does it pose a hazard in ultrasound therapy? Why is gaseous cavitation associated with ultrasound waves and not microwaves or ultraviolet radiation?

Cardiac pacemakers present a special hazard for diathermy. What are the risks and how do shortwave, microwave and ultrasound diathermy differ in this regard?

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13 Electrical Safety Most of the apparatus used in diathermy and electrotherapy is plugged into the mains supply - 240 volts AC with a frequency of 50 Hz. Any apparatus of this kind represents a potential hazard: the risk of electric shock. In this chapter we consider how a shock can occur, its likely effect and methods of shock protection. It is convenient to distinguish two kinds of shock mechanism; these are macroshock and microshock.

In Australia, the UK and many other countries the mains supply is 240 volts AC at a frequency of 50 Hz. Other countries (including the USA) use a 120 volts, 60 Hz supply.

* Macroshock: The familiar mechanism which has posed a risk since the advent of commercially supplied electricity. Here current flows from the body surface, through the skin and into the body. In order to produce harmful effects a relatively large voltage and current are needed. A high voltage is needed to produce a sufficiently high current as the skin offers a high electrical impedance. A high current is needed as current spreads as it flows through deeper tissues and it is the current density (in A.m2 or mA.cm-2) which determines the physiological effects. * Microshock: As a result of increasing sophistication in medical technology the patient, in a hospital setting, may be connected to a number of pieces of apparatus some of which provide a direct electrical pathway to the heart (for example a myocardial electrode or a transvenous catheter). A very small current applied directly to the heart via this pathway can be fatal. Only a low voltage is needed as the subcutaneous tissues have a low electrical impedance and the current is localized, resulting in a high current density.

For the clinician in an intensive-care ward, the risk of microshock is an important consideration. In private practice, the potential hazards are less likely to exist.

How to get a shock To avoid the risk of electric shock, it is necessary to understand how it can occur. In order to produce an electric shock two conditions must be satisfied. Firstly the victim must complete a circuit and secondly the current levels involved must be high enough to produce an adverse reaction.

ELECTRICAL SAFETY Suppose a person inadvertently contacts one terminal of a battery. In this situation no shock can occur. A shock current can only flow when the person completes a circuit and current is able to flow from one terminal through the person and ultimately to the opposite terminal of the battery. In order for a current to flow the person must simultaneously contact both terminals of the battery. This is illustrated in figure 13.1.

Figure 13.1 A person must 'complete a circuit' for shock to occur. In figure 13.1(a) a shock can not occur, regardless of the size of the battery voltage, as there is no continuous pathway for the current to travel. In figure 13.1(b) current is able to flow from one terminal of the battery, through the person, to the opposite terminal: the circuit is complete and a shock can result if the current flow is large enough.

SIZES OF SHOCK CURRENT For shock to occur the current flowing through the person must exceed a certain level. Currents below about 10 mA (0.01 amps) when applied to the whole body via the skin are unlikely to cause an electric shock. This is because the current is distributed through the body so that the amount of electrical energy applied to a particular organ is small. Macroshock only poses a significant risk if the current level exceeds 10 mA.

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ELECTRICAL SAFETY

By the same token a current in excess of about 100 µA (0.0001 amps) applied directly to the heart (for example via a myocardial electrode) may be fatal. The microshock risk threshold is more than 100 times lower than that of macroshock. Table 13.1 shows the effect of macroshock, i.e. when current passes through the skin and through the body: that is when the shock is not given directly to vital organs. The values quoted refer to mains frequency (50 Hz) AC, since shock via the mains supply is the greatest hazard in most situations which the physiotherapist will encounter. For figures appropriate to DC and other frequencies see Standards Association publication AS/NZS 60479:2002.

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Shocks are described in terms of current flow and not voltage. A shock's effect is determined by the amount of current which flows through a particular organ, not the voltage which produces it.

Table 13.1 Effects of shock current through body. While any amount of current over 10 mA is capable of producing painful to severe shock, currents between 50 and 250 mA are potentially lethal. At values as low as 20 mA breathing becomes laboured, finally ceasing completely even at values below 75 mA: the victim can suffocate due to uncontrollable contraction of the muscles of the thorax and abdomen.

ELECTRICAL SAFETY

If the current exceeds about 50 mA, ventricular fibrillation of the heart is likely to occur an uncoordinated twitching of the walls of the heart's ventricles. Once ventricular fibrillation is induced the heart will not spontaneously revert to its normal pattern of beating. Normal cardiac rhythm can only be restored by administering a massive current pulse from a cardiac defibrillator. The machine, which should only be operated by qualified personnel, supplies a short (3-4 ms) current pulse with an instantaneous amplitude of up to 40 to 80 amperes. Such high currents forcibly clamp the heart. When the clamping action ceases the heart is more likely to revert to its normal pattern of contraction.

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A fibrillating heart is unable to pump blood so the victim will die unless first aid (cardiac massage and artificial respiration) is administered until medical help arrives.

For shock currents above about 250 milliamps, the muscular contractions are so severe that the heart is forcibly clamped during the shock. This clamping protects the heart from going into ventricular fibrillation and the chances of survival are improved. From a practical viewpoint, after a person is knocked out by an electrical shock it is impossible to tell how much current passed through the vital organs of his body. Artificial respiration must be applied immediately if breathing has stopped: if no pulse is detectable external cardiac massage should also be applied. An important question is 'how much current will flow if a particular voltage is applied externally i.e. to the skin surface'. This depends more on the skin impedance than on the impedance of deeper tissues. The impedance of deeper tissues depends on their shape and volume, but does not vary a lot. Between the ears, for example, the internal resistance at low frequencies (less the skin resistance) is 100 ohms, while from hand to foot it is close to 500 ohms. The skin impedance varies much more than that of the underlying tissue. For 50 Hz AC it can be lower than 1000 ohms for moist skin to higher than 0.5 megohms for dry skin. The body current flowing when a person contacts the mains supply (240 volts) is calculated from Ohm's law to vary between 0.5 mA when the skin is dry and 240 mA when the skin is moist. If the victim is startled from an initial mild shock, sweating can

A useful rule-of-thumb is that in most practical situations, the deep-tissue impedance is around 200 Ω.

ELECTRICAL SAFETY result in a lowering of skin resistance and a rise in current from sub-lethal to lethal levels in a short space of time. This is one reason why it is essential, in an electric shock situation, to terminate the shock current as quickly as is safely possible.

HOW SHOCK CAN OCCUR: MACROSHOCK To understand the hazards associated with the use of mains powered apparatus we need a clear picture of the way in which mains electricity is supplied.

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Perspiration is an unfortunate accompaniment to pain and fright. This lowers the skin resistance and increases the shock current.

The very high voltage electricity which is generated at power stations is distributed by cables to electricity substations where step-down transformers reduce the voltage to a lower value. A single, large step-down transformer may be used to supply the 240 volts to many buildings in a residential neighbourhood. Large buildings in a city (for example a hospital) may have their own step-down transformers. Figure 13.2 shows the essential features of the power supply to a building.

Figure 13.2 Mains supply to a building (schematic)

ELECTRICAL SAFETY One terminal of the stepped-down supply is earthed at the electricity substation. This is called the neutral line. When the substation serves several buildings the neutral line is normally also earthed at the fuse box in each building. 240 volts AC is thus supplied to the fuse box in a building using two wires, the active wire and the neutral wire. The neutral wire is nominally at earth potential (zero volts) and the active wire is at a high potential. The active line connects through a power meter to a switch and fuse or to a circuit breaker. From the fuse box, power wires run to light switches and power outlets. Power outlets have three terminals; an active, a neutral and an earth terminal. The earth terminal is connected to a wire which is physically connected to earth at the building. Figure 13.3 shows the connections of the active, neutral and earth wires to a power outlet socket. Figure 13.3 Wiring convention for an Australian power outlet (viewed from the front).

Both the neutral and earth terminals of a power point are normally at earth or ground potential. However, it should not be assumed that the active terminal (on the left in figure 13.3) is the only hazardous one. For example it is quite possible for the active and neutral connections to be inadvertently interchanged when the power point is installed. Mainspowered equipment will still function normally when plugged in to the power point: the fault can only be determined by a specific test. Even when the power point is correctly wired it is possible for the neutral terminal to be above ground potential. This happens when appliances which draw a high current

337

As we will see later, earthing the supply affords a simple but efficient means of primary protection against shock hazard situations.

ELECTRICAL SAFETY are connected to the same circuit. A high current flowing in the neutral line will result in a potential difference between the power point neutral terminal and the connection to earth at the fuse box. This is because the resistance of the neutral cable, while small, is not zero. If the neutral wire has a resistance R and carries a current I, the potential difference produced is given by Ohm's law as V = I.R. In what follows we assume that the power point is correctly wired and consider other hazards associated with the mains supply.

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For these reasons both the active and neutral terminals of a power point should be treated with equal respect when considering potential shock hazards.

In normal operation, when an appliance is plugged into the mains outlet, current flows between the active and neutral terminals. The earth wires does not normally carry any current. The earth connection is only provided as a safety measure. The advantages of a three-terminal mains supply can be seen by inspecting figure 13.4.

Figure 13.4 Earthing of mains-powered apparatus casing. The circuitry within the apparatus (represented by an equivalent resistance Re in figure 13.4) is powered from the active and neutral wires. The earth wire is connected to the casing of the apparatus to ensure that there is never any voltage on the casing. The idea is that if the active wire within the apparatus makes accidental contact with the casing a very high current will flow through the earth wire to ground. The low

ELECTRICAL SAFETY resistance of the earth wire ensures that the current flow will be large enough to blow the fuse, thus cutting off the active supply. In this way, the casing of the apparatus can not become 'live' and present an electric shock hazard to anyone touching it. As long as the earth wire and connections remain intact there is no risk of shock from touching the apparatus. Some apparatus - electric shavers and hair dryers are examples - is 'double insulated'. The casing is usually made of a non-conducting plastic and special precautions are taken to ensure that an electric shock is virtually impossible. The advantage here is that no reliance is placed on an earth wire which could come loose or break. In fact, no provision at all is made for an earth connection to the apparatus. The use of the double insulation principle is restricted to small and easily insulated apparatus. Any exposed metal on double insulated apparatus is not connected to earth but is doubly isolated from the internal electrical circuitry. All apparatus which plugs into the mains, then, is macroshock protected, either by double insulation or by earthing. Nevertheless hazards remain in the form of faulty or worn equipment or careless workmanship. Figure 13.5 illustrates how an electric shock can result when apparatus is not earthed - because the earth wire is damaged or disconnected. The shock hazard in figure 13.5 arises when the active terminal short-circuits to the casing of the apparatus. In this case two faults have occurred a break in the earth connection and a short circuit of the active wire to the casing.

Figure 13.5 A person contacts apparatus which is not earthed and has the active wire touching the casing.

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For this reason a fuse should never be replaced with a conductor other than a fuse of the same rating. Australian Standard AS3100 requires that double insulated apparatus has two distinct and independent layers of insulation. Failure of both insulating layers is almost impossible without complete fracture of the apparatus' casing.

ELECTRICAL SAFETY

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Since the neutral line is earthed at the fuse box and power sub-station, a person standing on the ground is effectively connected to the neutral terminal of the mains supply. To complete the circuit and receive a shock, the person need only touch the active terminal or something connected to the active terminal. Current flows from the active terminal through the person to ground and hence to the neutral connection at the fuse box or power substation. Two important things should be noted about the situation illustrated in figure 13.5 *

A shock has occurred because the earth wire is damaged. If the earth connection was intact the fuse in the active line (figure 13.2) would blow and isolate the apparatus from the mains supply.

*

The fuse in the active line will not protect the person from receiving an electric shock. The fuses used for normal apparatus have a rating of several amperes. The person can receive a lethal shock (see table 13.1) without blowing the fuse.

A question which might occur to you is 'do both faults shown in figure 13.5 have to exist in order for a shock to result?' The answer is no. A shock can result when the apparatus is not earthed even though there is no direct physical contact between the active terminal and the casing. This is because the active wire and the case must have a small capacitance associated with them and insulation will not be perfect. Thus it is possible for small currents to leak via the insulation to the casing. With new and well looked-after apparatus the insulation impedance will be high and the maximum leakage current will be very small. Bad design or deteriorating insulation can, however, increase leakage currents to hazardous levels. Only by earthing the casing and providing an extremely low resistance pathway to ground can the risk of shock be minimized.

The fuses used in typical apparatus must have high ratings so they will not blow in normal operation (where the current flowing between active and neutral may be measured in amperes).

MACROSHOCK PROTECTION From the previous discussion it should be apparent that the fuses in the mains supply serve a protective role only when currents of several amperes are involved. For this to

ELECTRICAL SAFETY happen the active wire must short-circuit to the earthed casing. How then can we protect against shock involving much lower currents? There are two commonly used ways - by using a core balance relay or a protected earth-free supply.

Core Balance Relays Under normal circumstances the currents flowing in the active and neutral wires are equal. When an electric shock occurs the current in the active wire will be slightly greater than that in the neutral wire. This is because some current flows from the active wire through the victim to ground and through the ground to the neutral connection at the fuse box. Core balance relays are used to detect any imbalance and disconnect the power supply when the imbalance exceeds a predetermined value. The arrangement is shown in figure 13.6. Figure 13.6 Core-balance relay protection.

The active and neutral wires both pass through a magnetic core around which a sensing coil is wound. The currents in these wires are in opposite directions and when they are equal no current is induced in the sensing coil. If the currents are unequal a current proportional to the difference in active and neutral current is induced in the sensing coil. The induced current is amplified and used to operate a magnetic relay which disconnects both the active and neutral supply lines.

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Once the core balance relay has been 'tripped', the supply remains disconnected until the circuit breaker is manually reset. The response time of core balance relays is quite short (less than 100 ms) and typical units can be adjusted to trigger on an imbalance of as little as 5 mA. They are available for permanent installation (usually inside the fuse box) and are also supplied as portable units suitable for connecting between power points and appliances. From the foregoing description it should be apparent that these units protect against the 'normal' situation where a shock current flows through a person's body to earth. They will not protect against the more unusual situation where a person inadvertently contacts both the active and neutral lines simultaneously.

Earth Free Supplies In the situation shown in figure 13.5 the person receives an electric shock because his hand makes contact with the active line and his feet are in contact with the ground to which the neutral is connected. A question which might occur to you is 'would it be safer if the supply neutral was not earthed?' In this case the earthed person could not complete a circuit by touching the active line and so would not receive a shock. The answer to the question is a qualified 'yes'. Figure 13.7 shows how the normal mains supply can be rendered earthfree by using an isolating transformer. If neither side of the transformer secondary is earthed a person can

Figure 13.7 Isolation with a transformer.

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touch both earth and one transformer terminal without receiving a shock. At first glance it would seem that a person can only receive a shock if both transformer secondary terminals are contacted simultaneously. Unfortunately this is not the case in practice and the reasons are twofold: *

If a piece of apparatus plugged into the power point should develop a short circuit to earth no fuses will blow. The fault can remain unnoticed indefinitely. In the meantime the earth free supply has been converted to an earthed supply and we have no knowledge of which side of the transformer has become 'active' and which 'neutral'.

*

If faulty or poorly designed apparatus is plugged into the power point the insulation impedance between either supply terminal and earth can be reduced to the extent that the supply is effectively earthed: again with no knowledge of which terminal is at earth potential.

The system can be rendered safe by adding an earth leakage detector between the mains earth and the two transformer secondary wires as shown in figure 13.8. In normal operation a negligible amount of current flows through the leakage detector. If, however, apparatus with a short circuit or defective insulation is plugged into

Figure 13.8 Isolation with earth leakage detection.

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the power point a current will flow through the detector and activate the alarm. Of the two systems the protected earth free supply is somewhat safer than an earthed supply fitted with a core balance relay. Unfortunately the isolation transformers and leakage detection circuitry needed are both bulky and expensive. For this reason protected earth free supplies are only found in areas of high shock hazard. Core balance relays which are relatively cheap and easy to install are considered adequate for more general use such as in physiotherapy clinics and the physiotherapy departments of hospitals. Whichever method of protection is used it is important that the system be checked at regular intervals to ensure that the protection mechanisms are operating correctly.

When earth free supplies are used an earth leakage detector system is mandatory. The combination provides a high degree of electrical safety.

MICROSHOCK The use of electronic monitoring and measuring devices in the hospital setting has proved of immense value for patient monitoring and assessment. It has, however, also introduced some special risks of which the modern member of the health care team must be aware. Consider the patient in an intensive care unit. In some cases the patient may have apparatus connected by a direct electrical pathway to the heart. One such situation is illustrated in figure 13.9. Here a very special hazard exists because of the low current needed to cause ventricular fibrillation. Even if all the equipment is earthed the patient can still be electrocuted unless adequate precautions are taken. Figure 13.9 A microshock hazard situation.

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The patient, in this situation, is connected to two pieces of apparatus: an electrocardiograph (ECG) machine and a blood pressure monitor. For simplicity only the earth wires are shown. The patient is connected to earth by two pathways: the electrode connected to the right leg is earthed via the ECG machine and the fluid filled catheter is connected to a pressure transducer which is earthed via the blood pressure monitor. The risk of shock arises when a potential difference exists between the earth terminals on outlets 2 and 3. If a current I flows along the earth wire connecting the two outlets a potential difference V will result. V is given by Ohm's law V = I.R where R is the resistance of the earth wire between the outlets. Although R is very small it is not zero. If I is large enough the potential difference produced will be sufficient to electrocute the patient - remember that currents in excess of 100 microamperes or so flowing through the patient's heart may be fatal. Normally, of course, little or no current flows in the earth wire - it is only there to carry leakage current from the apparatus plugged-in. If, however, an appliance with a high leakage current, such as a vacuum cleaner, is plugged into outlet 1 a dangerous situation can result. Vacuum cleaners are notorious for producing large leakage currents (particularly at switch-on) because the motor is continually exposed to dust and moisture which lower the insulation impedance. Visualize the situation where the patient in figure 13.9 is connected as shown and a cleaner, working his way down the corridor, plugs a vacuum cleaner into outlet I (on the corridor outside) and switches it on. The instantaneous leakage current flowing in the earth wire could raise the potential at the earth terminal of outlet 2 to a sufficiently high value (relative to outlet 3) to electrocute the patient. The solution, in this case, is to plug all apparatus around the patient into a single power outlet or to interconnect the earth terminal of each outlet with heavy gauge copper wire. It is also necessary to ensure that the wiring for the power outlets in the patient's room does not connect to the power outlets in adjacent rooms or corridors.

For example, if the resistance of the earth wire is 0.1 Ω, and a spike of leakage current of 100 mA flows, a potential difference of 10 mV is produced. If the resistance of the tissue is 100 Ω, a current of 100 µA will flow.

ELECTRICAL SAFETY A further precaution which must be taken is to ensure that any apparatus which is used in the patient's room has been tested for earth leakage and meets the appropriate safety standards.

PATIENT TREATMENT AND ELECTRICAL SAFETY From the foregoing considerations of shock and shock protection it is apparent that there are three levels of risk associated with patient treatment. The greatest risk is to patients coupled to apparatus which may have a direct electrical connection to the heart. A lower level of risk exists when there are no invasive electrical connections; however we should distinguish the patient who is coupled to electromedical apparatus by surface electrodes from the patient who is not electrically connected to any piece of apparatus. The reason is that if a patient is connected by electrodes to, say, an electrocardiograph the potential for a shock to occur is increased by the deliberate electrical connection. In addition the skin resistance has been minimized by cleaning and application of a conductive electrode gel. In this case the voltage needed to produce a fatal shock current is reduced.

346

This last criterion would exclude most domestic and industrial vacuum cleaners and many domestic appliances.

Protection is afforded at two levels: * *

by using apparatus which meets appropriate safety standards and by appropriate protection built into the mains supply.

We consider each factor in turn.

Protection and the Mains Supply Patient treatment areas in hospitals are distinguished according to the kind of procedures or treatment being used and different safety standards apply to the mains supply in each case. Three types of treatment area are distinguished: *

Cardiac protected electrical areas. These are areas which are suitable for carrying out procedures which involve direct electrical connection to the heart. The safety requirements for both the electrical supply and apparatus to be used in such areas are stringent (see SAA Standards AS 3200 and AS 3003). These

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347

are described as 'Type CF' or simply 'cardiac protected' areas. In Australia and some other countries, these used to be described as 'Class A' treatment areas. *

Body protected electrical areas. These are areas which are suitable for carrying out procedures which do not involve direct electrical connection to the heart but which do involve the patient being in direct electrical contact with electromedical apparatus. Safety requirements are more stringent than those applying to areas where no electrical connection between patient and apparatus is necessary. Such areas are described as 'Type BF' or simply 'body protected' areas. They used to be known as 'class B' treatment areas.

*

Other patient areas. These are areas which are not specifically suited to 'cardiac type' or 'body type' procedures. Apparatus which is not electrically connected to the patient can be used. Apparatus which is intended to connect electrically to the patient can be used in these areas, but only if the apparatus itself meets stringent safety requirements (equivalent to those of a cardiac protected or body protected treatment area).

When direct electrical connection is made to the heart, shock currents as low as 100 µA can be fatal. For this reason cardiac-protected treatment areas are designed to minimize this risk. The earth wiring in these areas is constructed from heavy gauge copper wire so that even when substantial currents (up to 1 ampere) flow in the earth wire the potential difference between different earth terminals is kept below 100 mV. An area which meets this and other requirements (see SAA Standard AS3003) is described as an equipotential earth (EP) area. In addition to the requirement for equipotential earth wiring, cardiac protected areas must also have core-balance relay protection or have a protected earth free supply. Body protected areas are those designed to protect patients who may be connected directly to electromedical apparatus from macroshock currents. It is not necessary for the area to have an equipotential earth system but the supply must have core-balance relay protection or a protected earth free supply.

If the maximum potential difference is kept below 100 mV then the maximum patient current will be below 100 µA (assuming a minimum patient resistance of 1000 ohms).

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348

Best protection is afforded by a protected earth free supply but such installations are expensive. Core-balance relay protection can be provided economically and gives an adequate level of safety. Body protected areas which have appropriate core-balance protection will have the mains supply disconnected within 60 milliseconds of the active and neutral current imbalance exceeding 10 mA (SAA Standard AS3003). Class CF (cardiac protected) and BF (body protected) treatment areas are normally identified by signs displayed in, or on the doors of, the area. The signs have an identifying symbol and the words 'CARDIAC PROTECTED ELECTRICAL AREA' or 'BODY PROTECTED ELECTRICAL AREA' printed in white letters on a green background. The symbols for these areas are shown in figure 13.10. Figure 13.10 Symbols used to identify different classes of equipment or treatment area. (a) class CF (microshock protected) (b) class BF (body protected).

Patient areas which are not designated class A or B have no 'special' safety requirements other than those which apply to commercial, industrial and domestic supplies (SAA Standard AS3000). This means that the area does not provide protection if contact is made (either directly or indirectly) between the active supply wire and earth. It is recommended, though not mandatory, that such areas be provided with corebalance relay protection. The recommendation should be considered seriously since normal protective devices (fuses or circuit breakers) can allow currents of up to 150 times the macroshock hazard level without operating to cut-off the supply.

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349

Protection and Electromedical Apparatus Electromedical apparatus used for patient treatment falls into one of two categories. In the first category we have apparatus which does not have a deliberate and direct contact with the patient, such as an infrared or ultraviolet lamp. In the second category we have apparatus which requires deliberate electrical connection with the patient; for example, apparatus for delivering interferential or conventional TENS currents. In this case the apparatus has a patient circuit. Consider first electromedical apparatus which does not have a patient circuit. In this case the significant risk to the patient is if the patient inadvertently contacts the apparatus casing. If the maximum contact current which can flow is below a specified value (10 µA through a 1000 ohm load) and the earth leakage current is less than 100 µA then the apparatus is designated class CF. This is the safest kind of electromedical apparatus. Other electromedical apparatus must have a maximum casing-contact current below 100 µA: this is considered to offer adequate protection when the patient has no possibility of direct electrical connection to the heart. Class CF equipment can easily be recognized by the 'heart in the square' symbol (figure 13.10a). This is normally displayed on the rear panel of the equipment, near where the power cord enters. When a piece of electromedical apparatus has a patient circuit then the patient circuit itself can be either class CF, BF or B. *

A class CF patient circuit is the most safe. If the leakage current to the patient circuit is normally below 10 µA and below 50 µA even when a fault condition exists (when the earth lead is broken or the patient inadvertently contacts the active terminal of the mains supply) the patient circuit is designated class CF. A class CF patient circuit affords microshock protection.

*

A class BF patient circuit is macroshock protected. The normal leakage current is below 100 µA and the current which can flow when the earth lead is broken is below 500 µA. To comply with class BF specifications the fault current which can

Both kinds of apparatus must meet certain safety standards specified in terms of the amount of leakage current which can flow under different conditions. Apparatus which has a patient circuit must meet additional safety standards.

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350

flow from the patient circuit through the patient to the active terminal of the mains supply (in the event of the patient accidentally contacting the mains active lead, either directly or indirectly) must be below 5 mA. In other words a class B patient circuit has adequate isolation from the mains supply to minimize the risk of macroshock. *

A class B patient circuit affords a minimum level of macroshock protection. This kind of patient circuit may have one terminal earthed. Such a circuit must have leakage currents below those needed to represent a macroshock hazard when the apparatus is operating normally or when the earth lead is broken. However, no protection is offered against the situation where the patient inadvertently contacts the mains active lead.

Class CF and BF patient circuits are identified by the symbols shown in figure 13.10. The appropriate symbol is prominently displayed immediately adjacent to the patient circuit output sockets of the machine. If no symbol is found, the patient circuit should be assumed to be class B.

Protection in Summary It should be apparent, from the foregoing description, that electrical safety is only ensured if: *

the equipment meets appropriate safety standards for the treatment procedures involved;

*

the electrical supply meets appropriate safety standards for the treatment procedures involved.

Figure 13.11 shows a flowchart summarizing the requirements for earthed mainspowered apparatus and the class of area in which it can be used. The flowchart is based upon those of Australian Standard AS3200 and figure 5.3 of AS2500.

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351

START Procedure must be carried YES Is there a possibility of an intra-cardiac conductor out in a class CF area. (class CF procedure)?

NO

Does the equipment have a patient circuit?

NO

YES All mains powered equipment must: * be connected in the same equipotential area. * meet relevant safety standards.

NO

YES Use class CF or BF patient circuit.

Will individual patient circuits (if any) have possibility of intra-cardiac connection?

Does the mains supply have class CF or class BF protection?

Use class CF, BF or B patient circuit.

NO

YES Patient circuit must be class CF.

Use class CF, BF or B patient circuit.

Figure 13.11 Flowchart for the safe application and use of electromedical equipment.

No special precautions

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352

The strictest safety standards are mandatory when the patient has apparatus connected directly to the heart. In this case the mains supply should be that provided in a Class CF area and electromedical apparatus with a patient circuit should not be used unless either the patient circuit is class CF or there is no possibility of a direct electrical connection with the heart. In this way the risk of microshock is minimized. When there is no direct electrical connection to the heart it is sufficient to protect against the risk of macroshock. This can be achieved either by using equipment with a class CF or BF patient circuit or by treating the patient in a class CF or BF area. If the electrical wiring in a patient treatment area is class CF or BF then patients can be safety treated with apparatus which has a class CF, BF or B patient circuit. If the electrical wiring in a patient treatment area is not class CF or BF then the patient circuit must be class CF or BF. In other words if the mains supply is of the normal household variety then electromedical apparatus should have either a class CF or BF patient circuit. When there is no patient circuit and no possibility of intra-cardiac connection, electromedical equipment may be used on a normal earthed (but unprotected) mains supply.

EXERCISES 1

2

(a)

What is meant by the terms 'macroshock' and 'microshock'?

(b)

Electric shocks are always described in terms of shock current rather than voltage. Why is this so?

(a)

Consider macroshock and list the factors which will determine the size of shock current when a person accidentally contacts the mains supply.

(b)

Table 13.1 shows the effect of different sizes of shock current. Explain why shock currents in the range 50-250 mA represent a greater hazard than shock currents above 250 mA.

ELECTRICAL SAFETY

3

A person accidentally contacts the mains supply (240 volts). Given that the skin resistance may vary from 1000 Ω to 0.5 MΩ calculate the possible range of shock currents which may result. What is the practical significance of this in terms of terminating the shock as quickly as possible?

4

Figure 13.3 shows the wiring convention for a power outlet. Given that the neutral and earth terminals are grounded (earthed) at the fuse box in a building according to figure 13.2, does this mean that the neutral and earth terminals are 'safe'? Explain.

5

(a) (b)

Figure 13.2 is a schematic diagram of the mains supply to a building. The neutral line is earthed at the electricity sub-station. Why is it an advantage to also earth the neutral line at the fuse box? The neutral line of the mains supply is normally earthed at the fuse box. Even so, this does not mean that the neutral terminal of a correctly wired power point is at earth potential. Explain.

6

Consider figure 13.4 and explain what would happen if: (a) the active line accidentally makes contact with the apparatus casing (b) the neutral line accidentally makes contact with the apparatus casing. Would this blow the fuse? Could a short-circuit of the neutral line to the casing pose a shock hazard? Explain.

7

(a) (b) (c)

What is meant by the term 'double insulated' as applied to electrical apparatus? What is the principal advantage of double insulation for electrical safety? Explain the relative merits of double insulated apparatus and earthed apparatus as far as leakage currents are concerned.

353

ELECTRICAL SAFETY 8

(a) (b)

Figure 13.5 shows a person receiving an electric shock because two faults have occurred. What are they? Is it possible for a person to receive a shock from apparatus in which only one of these faults has occurred? Explain.

9

It has been said that fuses are included in the mains supply line only to protect the apparatus. Is it possible for a fatal shock to be delivered without blowing the fuse in the following two cases: (a) when the earth wiring is damaged? (b) when the earth wiring is undamaged? Explain.

10

(a) (b)

(c)

11

Describe the principles of operation of a core balance relay (as shown in figure 13.6). The fuses included in the mains supply must have a rating of several amperes and so can not protect against macroshock, yet a core balance relay can have a 'rating' of a few milliamperes but will not disconnect the mains supply in normal operation. Explain. Under what circumstances is a core balance relay unable to protect against macroshock (even involving shock currents of several hundred milliamperes)?

Figure 13.7 shows an isolating transformer used to generate an earth-free supply. (a) Could a person receive a shock by contacting any one of the terminals of the power outlet? (b) Under what circumstances could a person receive a shock from an earthfree supply (assuming no faults in the wiring)? (c) What is the principal disadvantage of the simple earth-free supply shown in figure 13.7?

ELECTRICAL SAFETY 12

354

(a) (b)

Explain why earth-free supplies can only be regarded as safer than a normal earthed supply if an earth leakage detector (figure 13.8) is included in the circuitry. Why is a protected earth-free supply preferable to a core balance relay protected supply for areas of high shock hazard? Are there any disadvantages associated with the installation of protected earth-free supplies?

13

Consider figure 13.9 where the patient is connected to (i) the supply earth of a blood pressure monitor via a transvenous catheter and (ii) the supply earth of an ECG machine via an electrode attached to the right leg. Explain how a shock hazard situation arises as a result of the ECG machine and blood pressure monitor being connected to separate power outlets.

14

A microshock of only 200 µA flowing directly through the heart can be fatal. (a) Given that the resistance of the patient's tissues between the catheter and the electrode applied to the right leg in figure 13.9 is about 1000 Ω, calculate the potential difference needed to produce a fatal shock current. (b) The resistance of the earth wire connecting mains outlets 2 and 3 in figure 13.9 is 4.0 Ω . Calculate the current flowing through the earth wire connecting the outlets which would be sufficient to cause electrocution of the patient. (c) A cleaner plugs a vacuum cleaner into mains outlet 2 (see figure 13.9). The leakage current of the vacuum cleaner is 70 mA. Does this represent a microshock hazard?

15

Consider the microshock hazard situation shown in figure 13.9. (a) How might the microshock hazards be minimized? (b) What are the implications of this situation for the use of electrotherapy apparatus on or near the patient?

355

ELECTRICAL SAFETY

16 17

18

(a)

What is the difference between a class CF and a class BF treatment area?

(b)

How are class CF and class BF treatment areas identified?

(a)

How would you recognize equipment with a class CF or class BF patient circuit?

(b)

A piece of equipment has a class Z patient circuit. Under what circumstances should the equipment be used for patient treatment?

(c)

What class of patient circuit should apparatus have if it is to be used in a patient's home?

Suppose you have (a) an ultrasound machine and (b) an interferential therapy machine which are to be used with an unprotected mains supply. What electrical safety standards apply to each machine?

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APPENDICES

357

APPENDIX 1 Prefixes Used to Specify Multiples and Submultiples of Units. The following table lists some important prefixes used to specify multiples and submultiples of units in the système internationale (SI). Prefix

Symbol

Multiple

giga

G

109

mega

M

106

kilo

k

103

deci

d

10-1

centi

c

10-2

milli

m

10-3

micro

µ

10-6

nano

n

10-9

pico

p

10-12

SOME EXAMPLES 109 hertz (Hz) = 1 gigahertz (GHz) 106 ohms (Ω) = 1 megohm (MΩ) 103 joules (J) = 1 kilojoule (kJ) 10-2 metre (m) = 1 centimetre (cm) 10-3 watt (W) = 1 milliwatt (mW) 10-6 henry (H) = 1 microhenry (µH) 10-12 farad (F) = 1 picofarad (pF) Note that the prefixes deci- and centi- are considered acceptable (due to their common usage) but are not recommended in the SI.

APPENDICES

358

APPENDIX 2 Quantities and Units The following tables list the quantities used in this text, their units and their symbols. Table 1 lists quantities which are measured directly in SI base units. The quantities listed in Table 2 have units derived from the base SI units. For a comprehensive listing of quantities, their SI units and their definitions see Quantities and Units in Science by O. Ogrim and A. E. Vaughan, Science Press (1977). TABLE 1: QUANTITIES MEASURED IN Sl BASE UNITS quantity length mass time current temperature amount of substance angle

symbol for quantity* l m s

unit

T n

metre kilogram# second ampere kelvin## mole

m kg s A K mol

θ

radian###

rad

I

Some quantities have several alternative acceptable symbols. The ones shown are SI recommended and are used in this book.

symbol for unit #

notice that the base unit of mass is the kilogram, not the gram. The kilogram is the only Sl base unit to include a prefix in the name.

##

Although the base Sl unit of temperature is the kelvin, the degree Celsius (symbol °C) is an acceptable alternative. Temperatures in degrees Celsius are converted to kelvins by adding 273.15. The size of the degree Celsius is equal to the size of the kelvin.

### Although the base Sl unit of angle is the radian, the degree (symbol o ) is an acceptable alternative. One radian is 180/2π degrees (approximately 57o).

APPENDICES

359

TABLE 2: SOME QUANTITIES MEASURED IN Sl DERIVED UNITS name capacitance charge conductivity current density density (mass density) dielectric constant# electric field strength energy force frequency heat energy intensity (wave) impedance (electrical) impedance (acoustic) magnetic field strength penetration depth permeability## potential difference power resistance resistivity specific heat capacity velocity volume rate of heating wavelength

symbol C q σ i ρ

ε

E E F f Q

I

Z Z B δ µ V P R ρ c v Pv λ

unit farad F (= C.V-1) coulomb C (= A.s) S.m-1 (= Ω-1.m-1) A.m-2 kg.m-3 V.m-1 (= N.C-1) joule J (= N.m) newton N (= kg.m.s-2) hertz Hz (= s-1) joule J (= N.m) W.m-2 (= J m-2s -1) ohm Ω kg m-2 s-1 A.m-1 m volt V (= W.A-1) watt W (= J.s-1) ohm Ω Ω.m J .kg-1.K-1 m.s-1 W.m-3 ( = J m-3.s-1) m

#

This quantity is more properly termed the relative permittivity. The term dielectric constant is used in this text to conform with common usage in the literature.

##

This quantity is more properly termed the relative permeability to distinguish it from the absolute permeability. Since absolute permeability is not used in this text the term permeability is used for simplicity.