Modelling in Medicine and Biology VIII (Wit Transactions on Biomedicine and Health)

Modelling in Medicine and Biology VIII

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EIGHTH INTERNATIONAL CONFERENCE ON MODELLING IN MEDICINE AND BIOLOGY

BIOMEDICINE VIII

CONFERENCE CHAIRMAN C.A. Brebbia Wessex Institute of Technology, UK

INTERNATIONAL SCIENTIFIC ADVISORY COMMITTEE C. Bignardi A. Doi W.J. Federspiel M. Hyre W. Lakin A. Macpherson S. Mekaoui A. Peratta D. Poljak K. Shimano R.M. Shoucri M. Ursino

Sponsored by WIT Transactions on Biomedicine and Health. Organised by Wessex Institute of Technology, UK

WIT Transactions Transactions Editor Carlos Brebbia Wessex Institute of Technology Ashurst Lodge, Ashurst Southampton SO40 7AA, UK Email: [email protected]

Editorial Board B Abersek University of Maribor, Slovenia Y N Abousleiman University of Oklahoma, USA P L Aguilar University of Extremadura, Spain K S Al Jabri Sultan Qaboos University, Oman E Alarcon Universidad Politecnica de Madrid, Spain A Aldama IMTA, Mexico C Alessandri Universita di Ferrara, Italy D Almorza Gomar University of Cadiz, Spain B Alzahabi Kettering University, USA J A C Ambrosio IDMEC, Portugal A M Amer Cairo University, Egypt S A Anagnostopoulos University of Patras, Greece M Andretta Montecatini, Italy E Angelino A.R.P.A. Lombardia, Italy H Antes Technische Universitat Braunschweig, Germany M A Atherton South Bank University, UK A G Atkins University of Reading, UK D Aubry Ecole Centrale de Paris, France H Azegami Toyohashi University of Technology, Japan A F M Azevedo University of Porto, Portugal J Baish Bucknell University, USA J M Baldasano Universitat Politecnica de Catalunya, Spain J G Bartzis Institute of Nuclear Technology, Greece A Bejan Duke University, USA

M P Bekakos Democritus University of Thrace, Greece G Belingardi Politecnico di Torino, Italy R Belmans Katholieke Universiteit Leuven, Belgium C D Bertram The University of New South Wales, Australia D E Beskos University of Patras, Greece S K Bhattacharyya Indian Institute of Technology, India E Blums Latvian Academy of Sciences, Latvia J Boarder Cartref Consulting Systems, UK B Bobee Institut National de la Recherche Scientifique, Canada H Boileau ESIGEC, France J J Bommer Imperial College London, UK M Bonnet Ecole Polytechnique, France C A Borrego University of Aveiro, Portugal A R Bretones University of Granada, Spain J A Bryant University of Exeter, UK F-G Buchholz Universitat Gesanthochschule Paderborn, Germany M B Bush The University of Western Australia, Australia F Butera Politecnico di Milano, Italy J Byrne University of Portsmouth, UK W Cantwell Liverpool University, UK D J Cartwright Bucknell University, USA P G Carydis National Technical University of Athens, Greece J J Casares Long Universidad de Santiago de Compostela, Spain, M A Celia Princeton University, USA A Chakrabarti Indian Institute of Science, India

A H-D Cheng University of Mississippi, USA J Chilton University of Lincoln, UK C-L Chiu University of Pittsburgh, USA H Choi Kangnung National University, Korea A Cieslak Technical University of Lodz, Poland S Clement Transport System Centre, Australia M W Collins Brunel University, UK J J Connor Massachusetts Institute of Technology, USA M C Constantinou State University of New York at Buffalo, USA D E Cormack University of Toronto, Canada M Costantino Royal Bank of Scotland, UK D F Cutler Royal Botanic Gardens, UK W Czyczula Krakow University of Technology, Poland M da Conceicao Cunha University of Coimbra, Portugal A Davies University of Hertfordshire, UK M Davis Temple University, USA A B de Almeida Instituto Superior Tecnico, Portugal E R de Arantes e Oliveira Instituto Superior Tecnico, Portugal L De Biase University of Milan, Italy R de Borst Delft University of Technology, Netherlands G De Mey University of Ghent, Belgium A De Montis Universita di Cagliari, Italy A De Naeyer Universiteit Ghent, Belgium W P De Wilde Vrije Universiteit Brussel, Belgium L Debnath University of Texas-Pan American, USA N J Dedios Mimbela Universidad de Cordoba, Spain G Degrande Katholieke Universiteit Leuven, Belgium S del Giudice University of Udine, Italy G Deplano Universita di Cagliari, Italy I Doltsinis University of Stuttgart, Germany M Domaszewski Universite de Technologie de Belfort-Montbeliard, France J Dominguez University of Seville, Spain

K Dorow Pacific Northwest National Laboratory, USA W Dover University College London, UK C Dowlen South Bank University, UK J P du Plessis University of Stellenbosch, South Africa R Duffell University of Hertfordshire, UK A Ebel University of Cologne, Germany E E Edoutos Democritus University of Thrace, Greece G K Egan Monash University, Australia K M Elawadly Alexandria University, Egypt K-H Elmer Universitat Hannover, Germany D Elms University of Canterbury, New Zealand M E M El-Sayed Kettering University, USA D M Elsom Oxford Brookes University, UK A El-Zafrany Cranfield University, UK F Erdogan Lehigh University, USA F P Escrig University of Seville, Spain D J Evans Nottingham Trent University, UK J W Everett Rowan University, USA M Faghri University of Rhode Island, USA R A Falconer Cardiff University, UK M N Fardis University of Patras, Greece P Fedelinski Silesian Technical University, Poland H J S Fernando Arizona State University, USA S Finger Carnegie Mellon University, USA J I Frankel University of Tennessee, USA D M Fraser University of Cape Town, South Africa M J Fritzler University of Calgary, Canada U Gabbert Otto-von-Guericke Universitat Magdeburg, Germany G Gambolati Universita di Padova, Italy C J Gantes National Technical University of Athens, Greece L Gaul Universitat Stuttgart, Germany A Genco University of Palermo, Italy N Georgantzis Universitat Jaume I, Spain G S Gipson Oklahoma State University, USA P Giudici Universita di Pavia, Italy F Gomez Universidad Politecnica de Valencia, Spain R Gomez Martin University of Granada, Spain

D Goulias University of Maryland, USA K G Goulias Pennsylvania State University, USA F Grandori Politecnico di Milano, Italy W E Grant Texas A & M University, USA S Grilli University of Rhode Island, USA R H J Grimshaw, Loughborough University, UK D Gross Technische Hochschule Darmstadt, Germany R Grundmann Technische Universitat Dresden, Germany A Gualtierotti IDHEAP, Switzerland R C Gupta National University of Singapore, Singapore J M Hale University of Newcastle, UK K Hameyer Katholieke Universiteit Leuven, Belgium C Hanke Danish Technical University, Denmark K Hayami National Institute of Informatics, Japan Y Hayashi Nagoya University, Japan L Haydock Newage International Limited, UK A H Hendrickx Free University of Brussels, Belgium C Herman John Hopkins University, USA S Heslop University of Bristol, UK I Hideaki Nagoya University, Japan D A Hills University of Oxford, UK W F Huebner Southwest Research Institute, USA J A C Humphrey Bucknell University, USA M Y Hussaini Florida State University, USA W Hutchinson Edith Cowan University, Australia T H Hyde University of Nottingham, UK M Iguchi Science University of Tokyo, Japan D B Ingham University of Leeds, UK L Int Panis VITO Expertisecentrum IMS, Belgium N Ishikawa National Defence Academy, Japan J Jaafar UiTm, Malaysia W Jager Technical University of Dresden, Germany Y Jaluria Rutgers University, USA C M Jefferson University of the West of England, UK

P R Johnston Griffith University, Australia D R H Jones University of Cambridge, UK N Jones University of Liverpool, UK D Kaliampakos National Technical University of Athens, Greece N Kamiya Nagoya University, Japan D L Karabalis University of Patras, Greece M Karlsson Linkoping University, Sweden T Katayama Doshisha University, Japan K L Katsifarakis Aristotle University of Thessaloniki, Greece J T Katsikadelis National Technical University of Athens, Greece E Kausel Massachusetts Institute of Technology, USA H Kawashima The University of Tokyo, Japan B A Kazimee Washington State University, USA S Kim University of Wisconsin-Madison, USA D Kirkland Nicholas Grimshaw & Partners Ltd, UK E Kita Nagoya University, Japan A S Kobayashi University of Washington, USA T Kobayashi University of Tokyo, Japan D Koga Saga University, Japan A Konrad University of Toronto, Canada S Kotake University of Tokyo, Japan A N Kounadis National Technical University of Athens, Greece W B Kratzig Ruhr Universitat Bochum, Germany T Krauthammer Penn State University, USA C-H Lai University of Greenwich, UK M Langseth Norwegian University of Science and Technology, Norway B S Larsen Technical University of Denmark, Denmark F Lattarulo, Politecnico di Bari, Italy A Lebedev Moscow State University, Russia L J Leon University of Montreal, Canada D Lewis Mississippi State University, USA S lghobashi University of California Irvine, USA K-C Lin University of New Brunswick, Canada A A Liolios Democritus University of Thrace, Greece

S Lomov Katholieke Universiteit Leuven, Belgium J W S Longhurst University of the West of England, UK G Loo The University of Auckland, New Zealand J Lourenco Universidade do Minho, Portugal J E Luco University of California at San Diego, USA H Lui State Seismological Bureau Harbin, China C J Lumsden University of Toronto, Canada L Lundqvist Division of Transport and Location Analysis, Sweden T Lyons Murdoch University, Australia Y-W Mai University of Sydney, Australia M Majowiecki University of Bologna, Italy D Malerba Università degli Studi di Bari, Italy G Manara University of Pisa, Italy B N Mandal Indian Statistical Institute, India Ü Mander University of Tartu, Estonia H A Mang Technische Universitat Wien, Austria, G D, Manolis, Aristotle University of Thessaloniki, Greece W J Mansur COPPE/UFRJ, Brazil N Marchettini University of Siena, Italy J D M Marsh Griffith University, Australia J F Martin-Duque Universidad Complutense, Spain T Matsui Nagoya University, Japan G Mattrisch DaimlerChrysler AG, Germany F M Mazzolani University of Naples “Federico II”, Italy K McManis University of New Orleans, USA A C Mendes Universidade de Beira Interior, Portugal, R A Meric Research Institute for Basic Sciences, Turkey J Mikielewicz Polish Academy of Sciences, Poland N Milic-Frayling Microsoft Research Ltd, UK R A W Mines University of Liverpool, UK C A Mitchell University of Sydney, Australia

K Miura Kajima Corporation, Japan A Miyamoto Yamaguchi University, Japan T Miyoshi Kobe University, Japan G Molinari University of Genoa, Italy T B Moodie University of Alberta, Canada D B Murray Trinity College Dublin, Ireland G Nakhaeizadeh DaimlerChrysler AG, Germany M B Neace Mercer University, USA D Necsulescu University of Ottawa, Canada F Neumann University of Vienna, Austria S-I Nishida Saga University, Japan H Nisitani Kyushu Sangyo University, Japan B Notaros University of Massachusetts, USA P O’Donoghue University College Dublin, Ireland R O O’Neill Oak Ridge National Laboratory, USA M Ohkusu Kyushu University, Japan G Oliveto Universitá di Catania, Italy R Olsen Camp Dresser & McKee Inc., USA E Oñate Universitat Politecnica de Catalunya, Spain K Onishi Ibaraki University, Japan P H Oosthuizen Queens University, Canada E L Ortiz Imperial College London, UK E Outa Waseda University, Japan A S Papageorgiou Rensselaer Polytechnic Institute, USA J Park Seoul National University, Korea G Passerini Universita delle Marche, Italy B C Patten, University of Georgia, USA G Pelosi University of Florence, Italy G G Penelis, Aristotle University of Thessaloniki, Greece W Perrie Bedford Institute of Oceanography, Canada R Pietrabissa Politecnico di Milano, Italy H Pina Instituto Superior Tecnico, Portugal M F Platzer Naval Postgraduate School, USA D Poljak University of Split, Croatia V Popov Wessex Institute of Technology, UK H Power University of Nottingham, UK D Prandle Proudman Oceanographic Laboratory, UK

M Predeleanu University Paris VI, France M R I Purvis University of Portsmouth, UK I S Putra Institute of Technology Bandung, Indonesia Y A Pykh Russian Academy of Sciences, Russia F Rachidi EMC Group, Switzerland M Rahman Dalhousie University, Canada K R Rajagopal Texas A & M University, USA T Rang Tallinn Technical University, Estonia J Rao Case Western Reserve University, USA A M Reinhorn State University of New York at Buffalo, USA A D Rey McGill University, Canada D N Riahi University of Illinois at UrbanaChampaign, USA B Ribas Spanish National Centre for Environmental Health, Spain K Richter Graz University of Technology, Austria S Rinaldi Politecnico di Milano, Italy F Robuste Universitat Politecnica de Catalunya, Spain J Roddick Flinders University, Australia A C Rodrigues Universidade Nova de Lisboa, Portugal F Rodrigues Poly Institute of Porto, Portugal C W Roeder University of Washington, USA J M Roesset Texas A & M University, USA W Roetzel Universitaet der Bundeswehr Hamburg, Germany V Roje University of Split, Croatia R Rosset Laboratoire d’Aerologie, France J L Rubio Centro de Investigaciones sobre Desertificacion, Spain T J Rudolphi Iowa State University, USA S Russenchuck Magnet Group, Switzerland H Ryssel Fraunhofer Institut Integrierte Schaltungen, Germany S G Saad American University in Cairo, Egypt M Saiidi University of Nevada-Reno, USA R San Jose Technical University of Madrid, Spain F J Sanchez-Sesma Instituto Mexicano del Petroleo, Mexico

B Sarler Nova Gorica Polytechnic, Slovenia S A Savidis Technische Universitat Berlin, Germany A Savini Universita de Pavia, Italy G Schmid Ruhr-Universitat Bochum, Germany R Schmidt RWTH Aachen, Germany B Scholtes Universitaet of Kassel, Germany W Schreiber University of Alabama, USA A P S Selvadurai McGill University, Canada J J Sendra University of Seville, Spain J J Sharp Memorial University of Newfoundland, Canada Q Shen Massachusetts Institute of Technology, USA X Shixiong Fudan University, China G C Sih Lehigh University, USA L C Simoes University of Coimbra, Portugal A C Singhal Arizona State University, USA P Skerget University of Maribor, Slovenia J Sladek Slovak Academy of Sciences, Slovakia V Sladek Slovak Academy of Sciences, Slovakia A C M Sousa University of New Brunswick, Canada H Sozer Illinois Institute of Technology, USA D B Spalding CHAM, UK P D Spanos Rice University, USA T Speck Albert-Ludwigs-Universitaet Freiburg, Germany C C Spyrakos National Technical University of Athens, Greece I V Stangeeva St Petersburg University, Russia J Stasiek Technical University of Gdansk, Poland G E Swaters University of Alberta, Canada S Syngellakis University of Southampton, UK J Szmyd University of Mining and Metallurgy, Poland S T Tadano Hokkaido University, Japan H Takemiya Okayama University, Japan I Takewaki Kyoto University, Japan C-L Tan Carleton University, Canada M Tanaka Shinshu University, Japan E Taniguchi Kyoto University, Japan

S Tanimura Aichi University of Technology, Japan J L Tassoulas University of Texas at Austin, USA M A P Taylor University of South Australia, Australia A Terranova Politecnico di Milano, Italy E Tiezzi University of Siena, Italy A G Tijhuis Technische Universiteit Eindhoven, Netherlands T Tirabassi Institute FISBAT-CNR, Italy S Tkachenko Otto-von-GuerickeUniversity, Germany N Tosaka Nihon University, Japan T Tran-Cong University of Southern Queensland, Australia R Tremblay Ecole Polytechnique, Canada I Tsukrov University of New Hampshire, USA R Turra CINECA Interuniversity Computing Centre, Italy S G Tushinski Moscow State University, Russia J-L Uso Universitat Jaume I, Spain E Van den Bulck Katholieke Universiteit Leuven, Belgium D Van den Poel Ghent University, Belgium R van der Heijden Radboud University, Netherlands R van Duin Delft University of Technology, Netherlands P Vas University of Aberdeen, UK W S Venturini University of Sao Paulo, Brazil

R Verhoeven Ghent University, Belgium A Viguri Universitat Jaume I, Spain Y Villacampa Esteve Universidad de Alicante, Spain F F V Vincent University of Bath, UK S Walker Imperial College, UK G Walters University of Exeter, UK B Weiss University of Vienna, Austria H Westphal University of Magdeburg, Germany J R Whiteman Brunel University, UK Z-Y Yan Peking University, China S Yanniotis Agricultural University of Athens, Greece A Yeh University of Hong Kong, China J Yoon Old Dominion University, USA K Yoshizato Hiroshima University, Japan T X Yu Hong Kong University of Science & Technology, Hong Kong M Zador Technical University of Budapest, Hungary K Zakrzewski Politechnika Lodzka, Poland M Zamir University of Western Ontario, Canada R Zarnic University of Ljubljana, Slovenia G Zharkova Institute of Theoretical and Applied Mechanics, Russia N Zhong Maebashi Institute of Technology, Japan H G Zimmermann Siemens AG, Germany

Modelling in Medicine and Biology VIII EDITOR: C.A. Brebbia Wessex Institute of Technology, UK

Editor: C.A. Brebbia Wessex Institute of Technology, UK

Published by WIT Press Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK Tel: 44 (0) 238 029 3223; Fax: 44 (0) 238 029 2853 E-Mail: [email protected] http://www.witpress.com For USA, Canada and Mexico Computational Mechanics Inc 25 Bridge Street, Billerica, MA 01821, USA Tel: 978 667 5841; Fax: 978 667 7582 E-Mail: [email protected] http://www.witpress.com British Library Cataloguing-in-Publication Data A Catalogue record for this book is available from the British Library ISBN: 978-1-84564-183-2 ISSN: 1747-4485 (print) ISSN: 1743-3525 (on-line) The texts of the papers in this volume were set individually by the authors or under their supervision. Only minor corrections to the text may have been carried out by the publisher. No responsibility is assumed by the Publisher, the Editors and Authors for any injury and/ or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. The Publisher does not necessarily endorse the ideas held, or views expressed by the Editors or Authors of the material contained in its publications. © WIT Press 2009. Printed in Great Britain by MPG Book Group. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the Publisher.

Preface

Considerable advances have been made in computational modelling of complex problems in biomedicine, as demonstrated by the continuous success of this International Conference on Modelling in Medicine and Biology. The first meeting started in Southampton (1991), followed by one in Bath (1993), and others in Milano (1995), Acquasparta (1997), Ljubljana (2003), Bologna (2005), The New Forest (2007) and culminating in the 2009 meeting in Crete. This book contains most of the papers presented at that last meeting. Advances in medical and biological technology are due to the increasing interaction and collaboration between medical and engineering scientists. Computer models which have successfully been developed to represent a series of biomedical systems are now becoming increasingly used for a wide range of applications ranging from cardiovascular modelling to virtual reality and simulation in surgery. The books resulting from these Conferences, containing a series of outstanding papers, are produced by WIT Press, the academic publisher of Wessex Institute of Technology. Since 1993, all WIT conference papers have been archived in an electronic database where they are easily and rapidly available to the international scientific community (see http://library.witpress.com/). All the papers published in this book can also be found there. The contributions in this volume are divided into the following topics: • • • • • •

Cardiovascular system Computational fluid mechanics Biomechanics Physiological processes Data acquisition and analysis Virtual reality in medicine

The Editor is grateful to all authors for their excellent contributions and to members of the International Scientific Advisory Committee and other colleagues who helped to review the work published in this volume. The Editor Crete, 2009

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Contents Section 1: Cardiovascular system Equivalence of two approaches to study the stress-strain relation in the myocardium R. M. Shoucri ..........................................................................................................3 The variation of dobutamine induced heart stress with heart rate A. K. Macpherson, S. Neti, C. Chutakositkanon, M. Averbach & P. A. Macpherson .............................................................................................17 Prediction of stent endflare, arterial stresses and flow patterns in a stenotic artery M. R. Hyre, R. M. Pulliam & J. C. Squire ...........................................................27 Fractal behaviour of pathological heart rate variability dynamics G. D’Addio, A. Accardo, G. Corbi, F. Rengo & N. Ferrara ...............................39 Microvascular disorders induced by malaria infected red blood cells: a computational mechanical study using the biological particle method T. Yamaguchi, H. Kondo, Y. Imai & T. Ishikawa ................................................49 Mechanical characterization of deep vein thrombosis in a murine model using nanoindentation K. C. McGilvray, R. Sarkar & C. M. Puttlitz........................................................57 Section 2: Computational fluid dynamics Comparison of blood flow patterns in cerebral aneurysms K. Shimano, T. Kudo & Y. Enomoto.....................................................................71 Modelling of flow through the circle of Willis and cerebral vasculature I. D. Šutalo, A. Bui, S. Ahmed, K. Liffman & R. Manasseh.................................83

Computational hemodynamics analysis in realistic 3D geometries of the human coronary atherosclerosis S. I. Bernad, T. Bărbat, E. Bernad & R. Susan-Resiga .......................................93 Numerical investigation of the flow field in the upper human airways G. Eitel, W. Schröder & M. Meinke ...................................................................103 Numerical simulations of high frequency respiratory flows in a model bifurcating lung geometry N. Valleru, S. Smirnov, J. Tan, S. Parameswaran & R. Raj ..............................115 Numerical prediction of the focal sites of ozone-induced tissue injury in the respiratory tract B. Keshavarzi, J. Ultman & A. Borhan ..............................................................123 Computational hemodynamics coupled with mechanical behaviour of the surrounded materials, in the specific case of the brachial artery R. Paulus, S. Erpicum, B. J. Dewals, S. Cescotto & M. Pirotton......................133 Section 3: Biomechanics Biomechanical consideration for dorsal-lumbar and lumbar sagittal spine disorders C. Bignardi, A. Ramieri & G. Costanzo.............................................................149 A new brace for the treatment of scoliosis M. G. Antonelli, P. Beomonte Zobel, P. Raimondi, T. Raparelli & G. Costanzo.....................................................................................................159 Parameters of kinaesthesis during gaits derived from an ultrasound-based measuring system R. M. Kiss............................................................................................................171 Evaluating elbow joint kinematics with the Stewart Platform Mechanism M. Alrashidi, İ. Yıldız, K. Alrashdan & İ. Esat ..................................................181 Dynamics of the human stomach R. Miftahof & N. Akhmadeev..............................................................................191

Section 4: Physiological processes The human body exposed to a magnetotherapy device magnetic field D. Poljak, S. Sesnic, D. Cavka, M. Titlic & M. Mihalj......................................203 Electronic active model for saccadic eye movements O. Terán & E. Suaste..........................................................................................213 Vestibular apparatus: dynamic model of the semicircular canals L. Gastaldi, S. Pastorelli & M. Sorli..................................................................223 A mathematical model to predict the performance of advanced therapies in wound healing J. Ko, S. Dickman & V. W. Li.............................................................................235 Modeling of capacitance relaxation phenomena in a malignant membrane T. K. Basak, K. Bhattacharya, S. Halder, S. Murugappan, V. Cyril Raj, T. Ravi, G. Gunasekaran & P. Shaw ................................................247 Section 5: Data acquisition and analysis A novel 3D torso image reconstruction procedure using a pair of digital stereo back images A. Kumar & N. Durdle........................................................................................257 Computer simulations and modeling in oncology: methods and applications C. Guiot , P. Paolo Delsanto & A. S. Gliozzi.....................................................267 PKAIN: an artificial immune network for parameter optimization in pharmacokinetics L. Liu, C.-H. Lai, S.-d. Zhou, F. Xie & H.-w. Lu ...............................................277 Readability of reassigned scalograms and extraction of spectra features for signal analysis S. Mekaoui, A. Houacine & T. Gharbi...............................................................287

Section 6: Virtual reality in medicine An internal examination training system supporting abnormal labor conditions A. Doi, K. Noguchi, K. Katamachi, T. Ishii, H. Uno, S. Mega & K. Matsui.........................................................................................................303 Development of a training system for interventional radiology M. Ide, Y. Fujii, B. Fujioka, T. Komeda, H. Koyama, S. Yamamoto, M. Mohri & P. Beomonte Zobel .........................................................................313 Author Index .....................................................................................................323

Section 1 Cardiovascular system

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Modelling in Medicine and Biology VIII

3

Equivalence of two approaches to study the stress-strain relation in the myocardium R. M. Shoucri Department of Mathematics and Computer Science, Royal Military College of Canada, Kingston, Ontario, Canada

Abstract A method to study the mechanics of ventricular contraction was developed in previous publications by the author. In those studies, the active force of the myocardium is represented as force per unit volume of the myocardium. Other authors have developed studies in which the active force of the myocardium is included in the expression of the total stress derived from the constitutive relations. The purpose of the present study is to show how to make the connection between these two approaches. Derivation is done in a general way, expressions for the stress components are derived and application to experimental data is presented. The possibility of relating the pseudo strain energy function to the tension generated by the muscular fibre is also shown. Keywords: cardiac mechanics, mathematical modelling of ventricular contraction, pressure-volume relation, active force of the myocardium, pseudo strain energy function.

1

Introduction

In previous studies by the author a method to study the stress-strain relation in the myocardium was developed in which the active force generated by the myocardium was represented as force per unit volume of the myocardium [1-6]. This mathematical approach used a cylindrical model of the left ventricle and was successfully developed by using both large elastic deformation [1, 2] and linear elasticity [3, 4], the transition from large elastic deformation to linear elasticity was discussed in [6]. Most other studies that have been developed focus on the way the expression of the total stress can be derived from the constitutive relations [9-11]. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090011

4 Modelling in Medicine and Biology VIII In this study the myocardium is represented as a thick-walled incompressible cylinder in which the myocardial fibres are imbedded in a helical way in a soft incompressible medium (see fig. 1). The contraction of the cylinder is modelled is a way to take into consideration torsion and shear, it turns out that their effect is small for the purpose of the numerical calculation of this study. In the quasistatic approximation used in this study, inertia forces and viscous forces are neglected. The total stress induced in the passive medium of the myocardium tij is written in the form t ij = σ ij + q ij , where qij is the stress induced by the muscular fibre tension T and reflects the directional character of the stress; σij is the stress resulting from the deformation of the passive medium of the myocardium (passive medium assumed isotropic). A similar approach can de found in Spencer [11]. Nevo and Lanir [8] have introduced a quantity similar to qij as the derivative of a hydrostatic pressure, and Arts et al. [12] have used an approach where σij is replaced by a hydrostatic pressure. The purpose of this study is to show the equivalence of the formalism developed in [1-6] by the author with the formalism developed by Humphrey and Yin [9] in which the total stress tij induced in the myocardium is derived from a pseudo strain energy function W. It is also shown how W can be directly related to the muscular fibre tension T, and that the splitting of W = Wiso + Waniso into an isotropic and an anisotropic component [9, 13, 14] is equivalent to the aforementioned splitting of the total stress tij.

2

Mathematical formalism: first approach

2.1 Equilibrium conditions This is the approach used in [9, 10], in which the calculation is carried out by using the total stress tij. By assuming symmetry around the z-axis (solution independent of θ), the conditions of local equilibrium the myocardium (div t = 0) can be written as follows in cylindrical coordinates

∂t rr t rr − tθθ ∂t zr + + =0 r ∂z ∂r

(1a)

2 1 ∂ (r t rθ ) ∂t zθ + =0 ∂r ∂z r2

(1b)

1 ∂ (rt rz ) ∂t zz + =0 r ∂r ∂z

(1c)

The stress can be dependent on the z variable, but we shall simplify the mathematical formalism by assuming that tzr, tzθ and tzz are independent of z as in [9]. In this case eqns (1b) and (1c) give WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

r 2 t rθ = const = H 1

rt rz = const = H 2

5

(2)

The radial stress boundary conditions on the surface of the cylinder are given by (see fig. 1).

t rr (ri ) = − Pi

t rr (ro ) = − Po

(3)

2.2 Deformation gradient The contraction of the myocardium is assumed to change the stress free configuration (R,Θ, Z) to the end-diastolic configuration (red, θed, zed) and finally to (r, θ, z) during the systolic phase according to the relations

red = red ( R),

θ ed = α 1 Θ,

z ed = k1 Z

(4)

r = r (red ), θ = α 2θ ed + ψ 2 z ed + χ (red ),

(5)

z = k 2 z z ed + k 2θ θ ed + ω (red ) which combined together give

r = r ( R), θ = αΘ + ψZ + χ ( R), z = k z Z + kθ Θ + ω ( R)

(6)

The deformation gradient F1 for the transformation from the stress free configuration (R,Θ, Z) to the end-diastolic configuration (red, θed, zed) in cylindrical coordinates is given by [10]

∂red 1 ∂red  ∂red  ∂R R ∂Θ ∂Z    ∂θ r ∂θ ed ∂θ F1 = red ed ed red ed ∂R R ∂Θ ∂Z     ∂z ed ∂z ed 1 ∂z ed  R ∂Θ ∂Z  ∂R

  dred   dR        = 0 α 1        0    

WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

0

red R 0

0

     0     k1   

(7)

6 Modelling in Medicine and Biology VIII The deformation gradient F2 for the transformation from the end-diastolic configuration (red, θed, zed) to the final configuration (r, θ, z) is given by

 ∂r  ∂r  re   ∂θ F2 = r  ∂red    ∂z   ∂red by

1 ∂r red ∂θ ed r ∂θ red ∂θ ed 1 ∂z red ∂θ ed

 ∂r   dr 0 0    dr ∂z ed    ed       r ∂θ   dχ rψ 2  (8) r α2 = r    dred red ∂z ed          k 2θ ∂z dω   k2z  ∂z ed   dred red 

The deformation gradient F = F2.F1 of the combined transformation is given

 ∂r  ∂R    ∂θ F = r  ∂R    ∂z  ∂Z

 ∂r   dr 0 0   ∂Z   dR        r r ∂θ ∂θ   dχ = r rψ  r α  R R ∂Θ ∂Z   dR         1 ∂z ∂z  kθ dω   k z R ∂Θ ∂Z   dR R 

1 ∂r R ∂Θ

(9)

where α = α2α1, ψ = ψ2k1, kθ = k2θα1, kz = k2zk1. By assuming that the transformations take place at constant volume, the incompressibility constraint can be written as follows

I 3 = (det F ) 2 = (det F2 ) 2 (det F1 ) 2 = 1

(10)

where I3 is the third strain invariant. By calculating the determinants in eqn (10) one obtains

dr R = , dR Kr

Kr 2 − R 2 = Kri 2 − Ri2

WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

(11)

Modelling in Medicine and Biology VIII

dred R = , dR K 1 red r dr = ed , dred K 2 r

2 K 1 red2 − R 2 = K 1 ried − Ri2

2 K 2 r 2 − red2 = K 2 ri 2 − ried

7

(12)

(13)

We have K1 = α1k1 = det(F1), K2 = α2k2z - ψ2k2θ = det(F2) and K = K1K2 = det(F) = α1k1α2k2z - α1k2θk1ψ2 = αkz - kθψ, with α = α1α2, kz = k1k2z, ψ = ψ2k1, kθ = α1k2θ. The inner radii are respectively Ri and ri in the stress free configuration and during the systolic phase. A muscular fibre in the myocardium is supposed to have a helical form on a cylindrical surface. In the undeformed configuration a unit vector N with fibre angle Γ(R) is transformed into a vector n in the deformed configuration with fibre angle γ(r) calculated with respect to the circumferential direction. With λN representing the stretch ratio in the direction of the muscular fibre, we have

n = [0, cos(γ (r ), sin(γ (r )]T ,

N = [0, cos(Γ( R), sin(Γ( R)]T (14)

with

n=

1

λN

F .N

(15)

By using eqns (9) and (15) we get

cos(γ ( r )) =

1 αr [ cos(Γ( R)) + ( rψ ) sin(Γ( R))] λN R

(16)

sin(γ (r )) =

1 kθ [ cos(Γ( R)) + k z sin(Γ( R))] λN R

(17)

2.3 Constitutive relations Relations between the components of the stress and deformation are known as constitutive relations. By assuming transverse isotropy with respect to the z-axis of the cylinder, Humphrey and Yin [9] have focused on a subclass of transverse isotropic material with pseudo strain energy function W given by the expression

W = W (I1 , λ N )

(18)

where I1 is the first strain invariant I1 = tr(B), and B = F.FT is the left CauchyGreen deformation tensor. Written explicitly we have WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

8 Modelling in Medicine and Biology VIII

Figure 1:

Cross-section of a cylinder representing the left ventricle. The dotted circle represents the projection of a helical fibre on the cross-section. Dr(r) is the radial active force/unit volume of the myocardium. Pi is the ventricular pressure, Po is the outer pressure, ri is the inner radius, ro is the outer radius, h = b – a is the thickness of the myocardium.

 dr 2 ( )  dR   dr dχ B = r  dR dR    dr dω   dR dR

r

dr dχ dR dR

(r

dχ 2 α 2 r 2 ) + 2 + ( rψ 2 ) dR R

r

dχ dω αrkθ + 2 + rψk z dR dR R

     dχ dω αrkθ r + 2 + rψk z  dR dR R   2  k dω ( ) 2 + θ2 + k z2  dR R  dr dω dR dR

(19) The Cauchy stress t (force/current area) can be expressed by using eqn (18) in the form [9]

t ij = − pδ ij + 2W1 Bij + WλN λ N n i n j where W1 =

(20)

∂W ∂W and W λN = , p is a Lagrange multiplier introduced to ∂I 1 ∂λ N

express the incompressibility condition for the myocardium. The first two terms in eqn (20) have an isotropic symmetry and the third term (the components of ni are shown in eqn (14), part one) has a directional character corresponding to the WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

9

direction of the muscular fibre in the myocardium. We are now in a position to make the junction with the second approach developed in [1-6].

3 Mathematical formalism: second approach 3.1 Equilibrium conditions We shall now see that it can be more instructive to work with the components σij and qij of tii = σij + qij. T(r,z) is the stress in the direction of the muscular fibre in the myocardium. By assuming that a muscular fibre in the myocardium has a helical cylindrical shape as in the previous section, one can derive for the components of the stress qij the following relations as in [2, 15]

q rr = 0,

qθθ = T (cos γ (r )) 2 ,

q zθ = T sin γ (r ) cos γ (r ),

q zz = T (sin γ (r )) 2 q rθ = q rz = 0

(21)

It is also possible to write the following relations

Dr =

T (cos γ (r )) 2 , r

q zθ = rDr tan γ ,

q zz = rDr tan 2 γ (22)

where Dr is obtained by substituting tij = σij + qij in eqns (1) and by writing the terms containing qij in the following way

Dr =

qθθ , r

Dθ = −

∂q zθ , ∂z

Dz = −

∂q zz ∂z

(23)

The quantities Dr, Dθ and Dz have the units of force/unit volume of the myocardium expressed in the three orthogonal directions of a cylindrical coordinate system. By using this notation, eqns (1) can be written in the form

∂σ rr σ rr − σ θθ ∂ ∂σ rr σ rr − σ θθ ∂σ zr + + − Dr = + + (σ zr −∫ D r dz ) ∂r r ∂z ∂r r ∂z z (24a)

1 ∂ (r σ rθ ) ∂σ zθ 1 ∂ (r σ rθ ) ∂ + − Dθ = 2 + (σ zθ − ∫ Dθ dz ) (24b) 2 ∂r ∂z ∂r ∂z r r z 2

2

1 ∂ (rσ rz ) ∂σ zz 1 ∂ (rσ rz ) ∂ + − Dz = + (σ zz − ∫ D z dz ) r ∂r ∂z r ∂r ∂z z WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

(24c)

10 Modelling in Medicine and Biology VIII Because trθ = σr θ (qrθ = 0) and trz = σrz (qrz = 0) we can write in a way similar to eqns (2)

r 2σ rθ = const = H 1

rσ rz = const = H 2

(25)

We assume that no external moment is applied to the myocardium, consequently the moment of forces M around the z-axis is zero ro

M = 2π ∫ (σ zθ +q zθ ) r 2 dr = 0

(26)

ri

which gives

σ zθ = −q zθ

(27)

Equilibrium of forces in the longitudinal direction gives

σ zz + q zz + τ av = 0

(28)

where τav is the average traction on the cross-section and is given by [8]

τ av =

Pi ri 2 − Po ro2 ro2 − ri 2

(29)

It is assumed that the average stress τav is independent of r and z. 3.2 Constitutive relations It is assumed that the muscular fibre tension T(r,z), and consequently qij, is uniformly distributed throughout the myocardium. By writing tij = σij + qij and by comparing eqns (21) with the last term of eqn (20) we can write

q ij = WλN λ N ni n j

(30)

with WλN appropriately chosen such that

T = WλN λ N

(31)

From eqn (20), the stress σij can be expressed as follows

σ ij = − pδ ij + 2W1 Bij WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

(32)

Modelling in Medicine and Biology VIII

11

This last equation can be written with the help of eqn (19) in explicit form as follows

σ rθ = 2W1 r

dr dχ H 1 = dR dR r 2

(33)

σ rz = 2W1

dr dω H 2 = dR dR r

(34)

dχ dω αrkθ + 2 + rψk z ] dr dR R

σ zθ = 2W1 [r

σ zz

2 dω 2 k θ = − p + 2W1 [( ) + 2 + k z2 ] dr R

σ θθ = − p + 2W1 [(r σ rr = − p + 2W1 (

dχ 2 αr 2 ) + ( ) + ( rψ ) 2 ] R dr

dr 2 R ) = − p + 2W1 ( ) 2 dR Kr

(35)

(36)

(37)

(38)

These equations are used in the experimental application described in what follows. The term Dr appearing in eqn (24a) is similar, but not identical, to the introduction of a derivative of a hydrostatic pressure in eqn (26) of [8].

4 Application and results The fibre angle γ(r) (referred to the circumferential direction) is supposed to be constant with respect to the axial and circumferential directions. The radial variation of the fibre angle is supposed to be linear and given by

γ = γ end (

ro − r r − ri ) + γ epi ( ) ro − ri ro − ri

(39)

where γend = 45o is the fibre angle at the endocardium, and γepi = - 45o is the fibre angle at the epicardium. The dimensions of the left ventricle in the diastolic configuration are outer radius ro = 3.38 cm, inner radius ri = 1.02 cm and length l = 3.06 cm as taken from experiment on dog reported in Feit [7]. The corresponding radii in the reference stress free configuration were estimated from eqn (11), part two, as Ro = 3.4474 cm and Ri = 1.1 cm. The tension T developed by the muscular by the fibre near the end-diastolic phase is taken from fig. 7a of Feit [7] and is reproduced in fig. 2 (left) of this study, fig. 2 (right) WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

12 Modelling in Medicine and Biology VIII

Figure 2:

Radial variation from endocardium to epicardium of the fibre tension T reproduced from [7] (left), and of the radial active force/unit volume of the myocardium Dr (right).

shows the variation of Dr calculated from eqn (22), part one and (39). Similarly the active stress components qzθ and qzz are calculated respectively from eqn (22), parts two and three, σzθ and σzz are calculated respectively from eqns (27) and (28) and shown in fig. 3. We took the ventricular pressure Pi = 10 mmHg and the epicardial pressure Po = 0 mmHg in the calculation of τav from eqn (29). The two quantities dχ/dr and dω/dR are small and have be neglected in the calculation that follows. Consequently from eqns (35) – (38) one can derive the following equation to calculate σrr

σ zθ (k αr ) / R 2 + k zψr = 2 θ2 σ zz − σ rr kθ / R + k z2 − R 2 /( Kr ) 2 and the following equation to calculate σ θθ σ zθ (kθ αr ) / R 2 + k zψr = σ θθ − σ rr (αr ) 2 / R 2 + (ψr ) 2 − R 2 /( Kr ) 2

(40)

(41)

The radial variation of σrr and σθθ is shown respectively in the left and right side of fig. 4. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 3:

Radial variation from endocardium to epicardium of the stress σzθ (left), and of the axial stress σzz (right).

Figure 4:

Radial variation from endocardium to epicardium of the radial stress σrr (left), and of the circumferential stress σθθ (right).

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14 Modelling in Medicine and Biology VIII

Figure 5:

Radial variation of the total circumferential stress tθθ from endocardium to epicardium.

In the calculation of σrr and σθθ we have made use of the two following conditions, first the incompressibility condition

αk z − ψ kθ = K

(42)

The second condition is that numerator of eqn (41) for σzθ is zero for rzero = 2.2 cm from fig. 3 (R2zero = 5.1160 from eqn (11), part two), which gives 2 αkθ + ψk z R zero =0

(43)

Eqns (42) and (43) are solved to express α and ψ in terms of kz and kθ as follows

ψ =−

kθ K 2 kθ + k z2 R zero

(44)

α=

2 k z KR zero 2 kθ2 + k z2 R zero

(45)

2

These values of ψ and α are substituted into eqn (40) evaluated at ri (σrr = Pi = 10 mmHg) and at ro (σrr = -Po = 0) with σzz calculated from eqn (28). The two equations obtained in this way are solved by using the Newton iteration WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

15

algorithm to calculate the two roots kθ and kz of a system of two coupled equations. For ri = 1.02 cm, ro = 3.38 cm, K = 1.028, Ri = 1.1 cm, Ro = 3.4474 cm, we have calculated kθ = -0.1472, kz = 1.0043, eqns (44) and (45) give α = 1.0228 and ψ = 0.0057. These are the values used to calculate the results of fig. 4. From the results shown in figs 3 to 6, the radial variation of each of the stress components appears to be similar to that reported in ref. [9] with a difference of sign probably due to the fact that we use the convention that a negative stress represents compression, a positive stress represents tension. It is also important to note the difference between the stress components σij and tij as is clear for instance by comparing fig. 4 (right) and fig. 5, and also how the three quantities W1, WλN and Dr can be expressed directly in terms of the muscular fibre T.

Figure 6:

5

Radial variation from endocardium to epicardium of WλN (eqn (31)) (left), and of W1 (eqn (38)) (right).

Conclusion

By introducing the concept of radial active force/unit volume of the myocardium, we have shown that it is possible to calculate the stress induced in the passive medium of the myocardium and the components of the active stress generated by the muscular fibre. One should note that all the calculations have been carried out without having to assume a model for the pseudo strain energy function; instead knowledge of the muscular fibre stress generated in the direction of the muscular fibre was necessary for our calculation. It is also WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

16 Modelling in Medicine and Biology VIII evident that the trend in the literature to split the pseudo strain energy function into an isotropic and a directional component is equivalent to the splitting of the total stress into two components as explained in this study.

References [1] Shoucri, R.M., The pressure-volume relation and the mechanics of left ventricular contraction, Japanese Heart Journal, 31, pp. 713-729, 1990. [2] Shoucri, R.M., Theoretical study of pressure-volume relation in left ventricle, American Journal of Physiology, 260, pp. H282-H291, 1991. [3] Shoucri, R.M., Active and passive stresses in the myocardium, American Journal of Physiology, 279, pp. H2519-H2528, 2000. [4] Shoucri, R.M., The calculation of the intramyocardial stress, Technology and health Care, 10, pp. 11-22, 2002. [5] Shoucri, R.M., Studying the mechanics of left ventricular contraction, IEEE engineering in Medicine and Biology Magazine, 17, pp. 95-101, May/June 1998. [6] Shoucri, R.M., Comparison between linear elasticity and large elastic deformation in the study of the contraction of the myocardium, Modelling in Medicine and Biology VII, ed. C.A. Brebbia, WIT Press: Southampton & Boston, pp. 3-13, 2007. [7] Feit, T.S., Diastolic pressure-volume relations and distribution of pressure and fiber extension across the wall of a model left ventricle, Biophysical Journal, 28, pp. 143-166, 1979. [8] Nevo, E. & Lanir, Y., Structural finite deformation model of the left ventricle during diastole and systole, Journal of Biomechanical Engineering, 111, pp. 342-349, 1989. [9] Humphrey, J.D. & Yin, F.C.P., Constitutive relations and finite deformations of passive cardiac tissue II: stress analysis in the left ventricle, Circulation Research, 65, pp. 805-817, 1989. [10] Guccione, J.M., McCulloch, A.D. & Waldman, L.K., Passive material properties of intact ventricular myocardium determined from a cylindrical model, Journal of Biomechanical Engineering, 113, pp. 42-45, 1991. [11] Spencer, A.J.M., Deformation of fiber-reinforced Materials, Clarendon Press: Oxford, U.K., p. 82, 1972. [12] Arts, T., Bovendeerd, P.H.M., Prinzen, F.W. & Reneman, R.S., Relation between left ventricular cavity pressure and volume and systolic fiber stress and strain in the wall, Biophysical Journal, 59, 93-102, 1991. [13] Driessen N.J.B., Bouten C.V.C. & Baaijens F.P.T., A structural constitutive model for collagenous cardiovascular tissues incorporating the angular fiber distribution, Journal of Biomechanical Engineering, 127, 494-503, 2005. [14] Zulliger M.A., Fridez P., Hayashi K. & Stergiopoulos N., A strain energy function for arteries accounting for wall composition and structure, Journal of Biomechanics, 37, 989-1000, 2004. [15] Chadwick, R.S., Mechanics of the left ventricle, Biophysical Journal, 39, 279-288, 1982. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

17

The variation of dobutamine induced heart stress with heart rate A. K. Macpherson1, S. Neti1, C. Chutakositkanon1, M. Averbach2 & P. A. Macpherson3 1

Institute of Biomedical Engineering and Mathematical Biology, Lehigh University, USA 2 Radiology Department, St Luke’s Hospital, USA 3 Department of Applied Technology, Rogers State University, USA

Abstract Dobutamine stress echocardiography is a common test to provoke myocardial ischemia in patients unable to undergo routine exercise stress testing. Heart rate elevation, achieved by staged increases in dobutamine doses, acts as a surrogate for exercise. The physicians monitor the ECG of the patient and echocardiographic images to evaluate for myocardial ischemia. However, the actual mechanical stress on the heart is not readily available to the physician. The motivation for the present preliminary study is to both investigate the feasibility of producing such information for clinicians as well as to investigate the variation between different patients as the heart rate varies. Echocardiograms were obtained from three patients undergoing dobutamine stress tests. Using standard equations of motion, the surface shear stress at the surface of the left ventricle was calculated. The average shear stress around the left ventricle is shown, as well as the peak stresses at selected locations as a function of time. It was found that generally the surface shear stress increased with heart rate around most of the left ventricle. While the time averaged shear stress may be important for diagnosis, the maximum shear stress is probably the limiting factor in terminating testing. Keywords: heart stress, dobutamine testing, heart rate, heart diagnosis.

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18 Modelling in Medicine and Biology VIII

1

Introduction

Dobutamine stress echocardiography is a common test to provoke myocardial ischemia in patients unable to undergo routine exercise stress testing. Heart rate elevation, achieved by staged increases in dobutamine doses, acts as a surrogate for exercise. A study is currently underway relating isolated myocardial stress to neurologic activity, without other confounding factors. The effects of shear stress on the heart are very extensive. Two surveys [1,2] showed that stress can cause changes in the genetic structure of the heart. In patient review it would be useful for a clinician to have details of the stress being applied to the heart. The dobutamine stress test is stopped if a patient develops concerning symptoms or demonstrates evidence of significant myocardial ischemia. In making a decision to terminate a test, information on the level of stress being experienced by the patient would be useful information for the clinician. As the present preliminary study was both a feasibility study as well as a preliminary investigation for a neural cardiovascular study, available echocardiogram results were used. Three-dimensional MRI results will be used in subsequent studies.

2

Method of calculation

The general method of calculation used here has been described previously [3,4]. In the solution the bloodflow into the left atrium is simulated by a source distributed throughout the atrium. In order to conserve mass sinks are distribute around the periphery of the integration domain. The change in shape is obtained from the echocardiograms and used as boundary conditions for the flow. The source strength has to match the change in volume of the ventricle. The valves have to be modelled as thicker than in reality as Lagranian integration must go around both sides of the valve. The Navier-Stokes equations are then solved with a predictor corrector scheme [4]. The Navier Stokes equations defined on an x-y Cartesian co-ordinate system for an incompressible fluid are  ∂uˆ  + uˆ • ∇uˆ  + ∇p = µ∇ 2 uˆ + Fˆ  ∂t 

ρ

∇uˆ = 0

(1) (2)

where uˆ is the velocity vector, ρ is the density, t is the time, p is the pressure and the viscosity is µ . The boundary force Fˆ arising from the heart muscles is L

(

)

Fˆ (xˆ , t ) = ∫ fˆ (s, t )δ xˆ − Xˆ (s, t ) ds 0

WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

(3)

Modelling in Medicine and Biology VIII

19

Here fˆ is the force on the boundary element at the point s defined on a Lagrangian system where xˆ is defined on the Cartesian system and Xˆ n is the nth point on the Lagrangian system The flow velocities and pressures can be used to calculate the stresses on the surface of the heart walls. These forces can then be used to examine the microscopic interaction with the cells in the heart wall (endocardium). The first step in the solution involves obtaining the shape of the ventricle at various times. This is often difficult as echocardiogram images are sometimes indistinct. Following a method often used by echocardiographers only five images in a cardiac cycle were selected. One image when the valves were closed, a second image when the valves were fully open, a third just before the atrium starts to contract, one at the end of the ventricle filling (diastole) stage and a final one as the aortic valve opens. A linear variation was assumed between each image, time frame. It was assumed that the motion of the wall would be normal to the surface. As described below the times required for valve opening and atrium contraction can be obtained from Doppler measurements of the velocity through the mitral valve and the shape was obtained from the echocardiogram contained many irregularities. The echocardiogram tracing was obtained as a digital image. If the source were allowed to start while the valves were closed then the program would fail due to excessive pressure. Similarly the wall could not be allowed to start moving until the source started. Thus an initial short period was required without source or wall motion to allow the valves to start opening (these events are independent of fluid motion are dependent on cardiac electrical signals). The second step required the simulation of the atrium. The atrium changes shape during the diastole stage and thus changes the pressure. However the use of a source in place of the correct inflow pattern to the atrium was an artifice that made the actual atrium shape unimportant. The atrium shape was fixed at near hemispherical shape with valves in the closed and early open positions. After some time the atrium contracts for a period before the mitral valve closed. The shape was expanded and contracted as required for the different sized mitral valves. The source strength was increased slowly as the valves opened in accordance with the increase in volume of the ventricle. Once the calculation of the flow velocities and pressures were completed the stresses at the walls were calculated. In accordance with the aim of the research, evaluation of wall stresses, the boundary layer had to be modelled properly. Two points were chosen as close to the wall as possible along a line normal to the surface. A finite difference method was used to obtain the derivative of the velocity along this line. Similarly the velocity normal to the wall was calculated along the same line. As only pressure gradients are used in the calculations, an arbitrary constant was added to the pressure to make it relative to atmospheric pressure. The method of the microscopic calculation of the blood, to obtain details on the affinity involved the effects of dobutamine contained in the blood, on the cells of the myocardium will not be discussed here. It is necessary to have a length scale in between the continuum calculation and the above microscopic WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

20 Modelling in Medicine and Biology VIII scale. This is undertaken using a Monte Carlo method. These details are presented elsewhere [4]. The basic process of describing the effects of dobutamine starts with the Landau equation which for the test particle takes the form below and has been described as a generalized diffusion equation in velocity space by Chandrasekhar [5]. Expressed in a non-dimensional form it becomes [6]

∂φτ = ∂ vr ( − Fr + 0.5∂ sTrs )φ

(4)

where φ is the velocity distribution, the v r differentiation is with respect to nondimensional velocity v/2kT, subscript τ is differentiation with respect to the nondimensional time defined below. The solution is obtained in terms of the drag force Fr and a random force Trs .

Fr = −8v −1G ( v )vr Trs = 2v −1H ( v )δ rs + 2v −3 E ( v )vr vs

(5) (6)

The movement of the blood components assumes they are sufficiently far apart so that collisions between the components will not occur. This is the usual assumption made for the application of the Landau equation. Under these circumstances the force on an ion will consist of a drag due to G(v) and a random force due to H(v). The docking mechanism for dobutamine with the receptor is unknown. It is useful in considering the present results to have an estimate of the fraction of dobutamine, which will dock with the receptor. Both dobutamine and Losartan dock with a G-protein so in order to make an estimate of the fraction of dobutamine absorbed, the docking of Losartan was calculated under the various conditions simulated in the present paper. The density of dobutamine receptors was used to estimate the affinity of the dobutamine to the receptor based on the affinity values calculated for Losartan. This is only presented in order to provide an indication of the possible outcome.

3

Results

The m-mode tracings were not available so that the opening and closing of the valves had to be obtained from the one EKG recording and the heart rate provided. The EKG recording is shown in the bottom left of figure 1. The heart rate is shown as 67 on the bottom right. The results of the stress calculations are presented for four regions of the ventricle. The regions are the apex (A), the mid endocardium (ME), across the mitral valves (MV) and in the middle of the septum (SE) as shown in figure 2. Typical variation of maximum and average shear stress over the whole ventricle is shown as a variation of time in figure 3. It can be seen that the peak maximum stress occurs when the aortic valve opens. The average of the shear stress over the whole ventricle is shown as a dashed line. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

Figure 1:

Figure 2:

21

Typical echo recording showing EKG recording and heart rate.

Areas over which the averaged stress was calculated.

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22 Modelling in Medicine and Biology VIII

Figure 3:

The variation of shear stress over the whole ventricle at a resting heart rate of 63 BPM for patient 1.

Figure 4:

The variation of shear stress over the whole ventricle at the maximum heart rate of 154 BPM for patient 1.

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Modelling in Medicine and Biology VIII

Figure 5:

23

The variation of shear stress over the septum at the heart rate of 150 BPM for patient 1.

It can be seen that the peak stress is more than 50% higher at 154 BP than at 63 BPM over the whole ventricle. Over the septum the peak stress is shifted to the opening of the aortic valve and is less than the peak stress over the endocardium. The variation of maximum shear stress around the whole ventricle is shown in figure 6 for the three patients. It can be seen that it increases very rapidly with heart rate. Patient 1 was a 55 year old woman, Patient 2 was a 75 year old woman and Patient 3 was a 75 year old woman. An unresolved problem is whether the maximum stress causes the onset of hypertrophy or the average stress applied over an extended time. In [4] it was shown that there is a finite length of time required for the angiotensin II to dock with the AT receptors on the G-protein. Thus it appears reasonable to assume it is the average sustained shear stress that is the more important stress for the onset of hypertrophy. The time averaged shear stress over one heart beat in the ME region, as a function of heat rate is shown for the three patients in figures 7. It can be seen in all patients that the effect of a moderate increased heart rate is to increase the stress very rapidly. However with further increase in heart rate there is comparatively small increase in the time-averaged heart rate. In the case of Patient 3 the time averaged shear stress decreases with high heart rate. This may be a valid result or it could be due to the interpretation of the echocardium. Further study using MRI output will resolve this result.

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24 Modelling in Medicine and Biology VIII

(a)

(b)

(c) Figure 6:

4

The variation of the maximum shear stress on left ventricle with heart rate. (a) Patient 1, (b) patient 2, (c) patient 3.

Conclusions and future work

As the time averaged stress only increases slowly with high heart rate then it appears that the time averaged stress is not the best criteria for terminating the testing. As seen in Figure 6 the maximum shear stress around the ventricle increases very rapidly with the heart rate. Therefore the best termination criterion probably is the maximum stress at any point around the ventricle. Values of the average shear stress would be useful for the physician in determining treatment. Only limited results have been presented for conditions along the septum. However hypertrophy of the septum can occur and will been addressed in future work. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

(a)

25

(b)

(c) Figure 7:

Variation of time averaged shear stress over on heartbeat as a function of heart rate. (a) Patient 1, (b) patient 2, (c) patient 3. The location is in the ME region as in figure 2.

This preliminary study has shown that data useful to a clinician monitoring dobutamine testing can be obtained. In addition diagnostic results can be extracted from the data. Future studies will use MRI data the output of which can be automated. This will be used in a study of a heart-brain study presently underway.

References [1] Sadoshima J. and Izumo S. The cellular and molecular response of cardiac myocytes to mechanical stress Ann. Revs of Physiology, 59, 551–571, 1997. [2] Ruwhof C, van der Laarse A. Mechanical stress-induced cardiac hypertrophy: mechanisms and signal transduction pathways Cardiovascular Research, 47, 23–37, 2000. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

26 Modelling in Medicine and Biology VIII [3] Macpherson AK and Neti S, “The effect of Angiotensin II on heart blood flow and hypertension”, Advances Fluid Mechanics IV, eds M. Rahman, R. Verhoeven, C.A. Brebbla, WIT press, Southampton, 1–12, 2002.1–12, 2002. [4] Macpherson AK, Neti S, Macpherson PA, Houser SR, Hari M and Marzillier J. Mechanical Stress and Hypertrophy, Modelling in Medicine and Biology VI, eds M. Ursino, C.A. Brebbia, G. Pontrelli, E. Magasso, WIT Press, Southampton, 171–179, 2005. [5] Chandrasekhar, S. Principles of Stellar Dynamics, Uni. Of Chicago Press, Chicago, 1942. [6] Balesu, R. Equilibrium and Nonequilibrium Statistical Mechanics, Wiley, New York, 1975.co.

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Prediction of stent endflare, arterial stresses and flow patterns in a stenotic artery M. R. Hyre1, R. M. Pulliam2 & J. C. Squire1 1

Department of Mechanical Engineering, Virginia Military Institute, USA Department of Mechanical Engineering, Villanova University, USA

2

Abstract Restenosis remains a significant problem in coronary intervention. While stent migrations, collapses, and positioning difficulties remain serious issues, it is the problem of restenosis that is the most common long term problem in treating atherosclerotic coronary arteries with stents. While much attention has focused on biocompatibility, thrombosis and neointimal pathology, less attention has been given to matching stents to the inflation balloon, artery and occlusion size. Results from this study indicate a 100% increase in balloon overhang results in a 4% increase in maximum endflare and a 39% change in peak arterial stress. At the end of expansion, which is of the most clinical importance, the increase in maximum endflare is 2% and the increase in maximum arterial stress is 93% at the balloon point of contact and 45% at the point of contact with the far proximal and distal ends of the stent. When comparing the results of calcified and cellular plaque, a maximum endflare of about 55% was observed for both the calcified and cellular plaque cases during expansion. At the end of expansion the increase in maximum endflare was 10% for the cellular plaque and 40% for the calcified plaque. The peak equivalent stress seen by the artery was about 100% larger in the cellular case than in the calcified plaque case. Keywords: stent, vascular injury, balloon, restenosis, finite element analysis.

1

Introduction

Atherosclerotic stenosis and its ischemic complications necessitate arterial reconstruction. Current strategies to restore normal blood flow in stenotic coronary arteries include angioplasty, intracoronary stents, and coronary artery WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090031

28 Modelling in Medicine and Biology VIII bypass surgery. Depending on the method of treatment, the incidence of restenosis is high: up to 40% within six months after angioplasty [1], 25% after stenting [2], and 20% after bypass surgery [3]. Therefore, restenosis remains a significant problem in coronary intervention. Advances in prosthetic science and engineering have spurred the rapid development of many new permanent implants such as arterial reinforcement grafts, venous filters, myocardial perforation-sealing clamshells, and stents that strengthen and scaffold the biliary duct, urethra, veins, and arteries. These devices are typically attached to a delivery catheter and threaded to the site of interest where they are expanded. The very nature of the remote delivery systems make the mechanical details of implantation difficult to ascertain, yet this is important to quantify since there may be a link between how the devices are emplaced and the body’s acute and chronic response. Endovascular stents in particular are ideal devices to quantify these relationships because of the extreme levels of stress they impose and because of their ubiquity; more than one million are annually implanted in the U.S. alone [4]. These studies suggest an upper limit exists to the success of purely biomedical approaches for managing post-device implantation, and a return to examining the mechanical initiators of vascular injury that occur during implantation. A complete understanding of the manner that stents expand may thus lead to both a new understanding of the processes of vascular adaptation to implants and possibly to the design and development of less-injurious devices. Experimental data are indirect; stents are too thin to be fully radioopaque, and methods of bringing a camera to the stent, such as intravascular ultrasound, are blocked by the balloon that expands the stent during the critical moments of implantation. Post mortem examinations indicate that restenosis is paradoxically more severe in the parts of the artery immediately outside the stented region, and animal studies have shown an unusual pattern of endothelial cell denudation occurring at a regular pattern at the center of stent struts, a superficial injury that may be a marker for deep vascular injury [8]. These data are not explained by current finite element analyses of arterial stresses in a stent-expanded artery [912] because no finite element models included the expansion of a balloon catheter in the model of a plastically-deformed stent. The inclusion of the balloon catheter in the stent expansion model is not trivial. The problem is highly non-linear and includes complex contact problems between the stent, balloon, and arterial wall. Additionally, the balloon properties change dramatically depending on whether it is fully or partially inflated.

2

Geometry

The finite element stress analysis was performed on a three-dimensional stent/balloon/plaque/artery geometry. In addition to the usual difficulties in modeling the mechanical behavior of soft tissue, the overall system response is highly nonlinear due to the large plastic/multilinear-elastic/hyperelastic deformations of the individual components. The component geometries and constitutive material models are described below. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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2.1 Artery The coronary artery modeled was 30mm in length, with an inside diameter of 2.8mm and thickness of 0.3mm. Average element size was about 0.5mm long with a thickness of 0.15mm. This configuration yielded a total of 7,680 elements. The artery elements were defined by eight nodes capable of large deflections and hyperelasticity 2.2 Plaque The plaque has a semi-parabolic profile and corresponds to percent blockage data presented by Lally et al. [12]. The plaque was 16mm in length with a maximum thickness of 0.48 mm. This configuration corresponds to a maximum percent blockage of about 60%. The characteristic shape of plaque can be seen in Figure 1.

Figure 1:

Figure 2:

Characteristic plaque curve.

Balloon geometry, shown with mounted slotted tube stent.

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30 Modelling in Medicine and Biology VIII 2.3 Balloon The balloon was modeled in its unfolded state, and already assumed to be in contact with the stent. The balloon dimensions are given at 0 Pa, before stent expansion occurs. Depending on the amount of balloon overhang, the overall length of the balloon can range from 22-23 mm. For 1mm balloon overhang, the total length of end balloon is 1mm, or 0.5 mm on each side of the inside balloon, yielding a total balloon length of 22 mm. For 2 mm balloon overhang, the total length of balloon overhang is 2 mm, or 1 mm on each side, yielding a total balloon length of 23 mm (see fig. 2). The balloon was meshed using triangular shell elements with an average base size of 0.05 mm and an average side length of 0.05 mm. This yielded 54,456 elements for the 2 mm overhang case and 51,616 elements for the 1 mm overhang case. For finite element analysis, elements capable of modeling shell structures, large deflections and plasticity were used. 2.4 Stent A three dimensional model of the slotted tube geometry intravascular stent was created. The stent is 16 mm in length (L), with an inside diameter (ID) of 1 mm, and a thickness (t) of 0.1 mm. The diamond-shaped stent consists of 5 slots in the longitudinal direction and 12 slots in the circumferential direction with a length of 2.88 mm. The slots were cut such that in a cross-section, the angle describing the slot was approximately 23 degrees, and the angle describing the metal between slots was 6.9 degrees (see figs. 3 and 4). These dimensions refer to the model in an unexpanded state [10, 12]. The stent was meshed using hexahedral elements. There are two elements through the thickness of the stent yielding a total of 12,036 elements. The stent was assigned an element type of solid45 for finite element analysis in ANSYS. These elements are defined by eight nodes and are capable of large deflections and plasticity.

Figure 3:

Medial slice of modeled slotted tube stent.

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Figure 4:

Side view: stent geometry.

Figure 5:

31

Final model geometry.

A cross-section of the final model geometry with the stent/balloon/artery and plaque is shown in fig. 5.

3

Materials

3.1 Artery The material properties of the artery are based on a previous study by Lally et al. [12]. This model describes the behavior of the artery using a five parameter, third-order, Mooney-Rivlin hyperelastic constitutive equation. This model has been found to be suitable for modeling an incompressible isotropic material [13]. Prendergast et al. developed the constants for this model by fitting the fiveparameter Mooney-Rivlin expression to uniaxial and equibiaxial tension tests of human femoral arterial tissue data [14]. See Table 1. 3.2 Plaque The material properties of the plaque are based on a previous study by Loree et al. [15]. Two histological classifications of plaques were modeled: cellular and calcified. The cellular and calcified specimen results were chosen to provide models of stent expansion dynamics with plaques whose stress-strain slopes differed significantly. This model describes the behavior of the plaque using a five parameter, third-order Mooney-Rivlin hyperelastic constitutive equation. This model for plaque behavior neglects the artery laminate compositions, tissue anisotropy as well as the residual strain and active smooth muscle stresses [6]. The final form of the strain density function used to model the artery is given in eqn. (1). W = a10(I1 – 3) + a01(I2 – 3) + a20(I1 – 3)2 + a11(I1 – 3)(I2 – 3) + a30(I1 – 3)3 (1) The constants were developed for this model by fitting the five-parameter Mooney-Rivlin expression to uniaxial tension tests of human aortic atherosclerotic tissue data [15]. The hyperelastic constants for the plaques are given in Table 2. 3.3 Balloon To model the mechanical properties of the balloon without evaluating the balloon’s behavior during unfolding, empirical data was used. The stress-strain WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

32 Modelling in Medicine and Biology VIII Table 1:

Hyperelastic constants for the artery.

Hyperelastic Constants [Pa] a10 = 0.01890 a01 = 0.00275 a20 = 0.08572 a11 = 0.59043 a02 = 0

Table 2:

Hyperelastic constants for plaque.

Hyperelastic Constants [Pa] Cellular Plaque Calcified Plaque a10 = -0.088314 a10 = -3.0254 a01 = 0.10619 a01 = 3.1073 a20 = 0.11373 a20 = 107.39 a11 = 0.89382 a11 = -234.7 a02 = -0.96676

a02 = 137.22

curve for the full expansion of the balloon produced a linear piecewise function. The first segment of the piecewise function is representative of the unfolding balloon, while the second is of the balloon expansion after unfolding. 3.4 Stent The stent was modeled after the slotted tube geometry given by Migliavacca et al. [10]. This model assumes the stent to be made of 316LN stainless steel. The Poisson ratio is 0.3 and the Young Modulus is 200 GPa.

4

Boundary conditions

The artery, balloon, plaque, and stent were all constrained in the rotational directions allowing no rotation. The artery was constrained axially at the distal ends. The artery was at a minimum, 7mm longer than the stent on each side and 3.5mm longer than the end of the balloon on each side. This constraint on the artery did not affect the behavior of the artery at the point of contact with the stent or balloon because of the extra length of the artery on both sides. The same axial constraint was placed on the balloon. To model the expansion of the balloon the balloon was assigned a ramped internal pressure load.

5

Deformed geometry export and solution

In order to examine the blood flow patterns thorough the stented artery, a conversion module was written to export the final expanded stent/plaque/artery system for meshing and use within a finite volume computational fluid dynamics code (FLUENT/UNS). The deformed geometry was exported to STL format and the internal geometry (solid mirror of the stent/plaque/artery) was meshed using a hex dominant grid. Holes and seams between the various geometry parts were automatically filled within the meshing algorithms. The inlet boundary condition was specified as a velocity inlet using the time dependent flow rate equation of Womersley found in [16]. The flow was WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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assumed to enter the simulation with a constant cross-sectional velocity. A noslip condition where the stent/plaque/artery surfaces were in contact with the blood was assumed. The Casson model was used to evaluate the viscosity of the blood since the artery was relatively large and inhomogeneities associated with Fahraeus effects were negligible.

6

Results and discussion

Figure 6 shows the endflare during stent expansion for the 1mm and 2mm balloon overhang cases when there is no plaque included in the models. Endflare is defined as the ratio of the stent diameter at the distal ends to the diameter at the stent centerline. There is a significant difference in the endflare both at the point of peak endflare and at the end of expansion, indicating that the amount of balloon overhang can have a significant impact on vascular injury. Figure 7 shows the endflare during stent expansion for the 2mm balloon overhang case when calcified and cellular plaque are included in the models. The endflare with plaque present in the model is significantly higher than when it was not included. This is because the distal ends of the stent were located such that they did not contact the plaque. Therefore, the effective diameter against which the stent was expanding was significantly larger in this area leading to a lower expansion resistance. There is a significant difference in the endflare both at the point of peak endflare and at the end of expansion, indicating that the amount of balloon overhang can have a significant impact on vascular injury. Figure 8 shows the arterial stresses at the end of stent expansion when calcified and cellular plaque was included in the model. At the end of expansion, the increase in maximum endflare for the cellular plaque geometry over the calcified plaque geometry is about 200%. The increase in maximum arterial stress is 200% at the point of stent contact at the proximal and distal ends.

Figure 6:

Endflare without plaque for 2mm (upper line) and 1mm (lower line) overhang.

Figure 7:

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Endflare with plaque for calcified (upper line) and cellular (lower line) plaque.

34 Modelling in Medicine and Biology VIII

Figure 8:

Arterial/plaque stresses for calcified (top) and cellular (lower) plaque.

Figure 9:

Max arterial stresses for calcified (lower line) and cellular (upper line) plaque.

Figure 10:

Arterial stresses (radial, hoop, and axial) for calcified plaque.

Figure 11:

Arterial stresses (radial, hoop, and axial) for cellular plaque.

Figure 9 shows the max arterial stress during the expansion process for the cellular and calcified plaque cases. The cellular plaque case results in much higher stresses at a given balloon expansion pressure than the calcified plaque case. Figures 10 and 11 show the directional stresses when calcified and cellular plaque are included in the model. As expected, the arterial stresses were significantly lower for the calcified plaque cases when compared to the cellular case. This is the result of the much higher rigidity of the cacified plaque layer over the celluar plaque. This is further supported by the higher stresses seen by the plaque in the calcified case as compared to the cellular case. It should be mentioned that no plaque breakup model was included in the simulations. It would be expected that the calcified plaque would break up before the high stresses at the end of expansion predicted by the model.

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As one would expect, the most significant compenent of stress is the hoop. The smallest compenent is the axial. The clinical significance of the directional stresses on vascular injury is unclear. Future experimental studies will be aimed at determining the most appropriate stress to analyze when evaluating vascular injury (von Misses, maximum principal, directional, etc.). The current model does not include the anisotropic behavior of the artery and plaque materials. Additionally, no distinction was made between the passive arterial medium and the active fibers, the orentations of which may vary from inner to the outer layers of the arterial wall. This is currently being included in a more sophisticated model which also includes arterial prestresses. Finally, Figure 12 shows the preliminary results of the flow pattern within the final stented geometry. The development of Poiseuille flow is apparent as the blood moves from the model entrance along the artery and plaque surface (not show) and through the stented portion of the artery. The strain rates are quite interesting and show a decrease after the initial entry into the arterial section. However, they rapidly increase as the blood travels through the stented portion of the artery. This CFD model is now being used to evaluate the diffusion process associated with drug eluting stents, as well as modeling neointima formation, thrombus formation mechanics, and blood flow patterns. It has been found that standing vortices and regions of stagnation are responsible for the rises in concentrations of platelet-activating agents within those regions. Platelets accumulate preferentially in the regions of large platelet-activating agent concentrations and low fluid velocities. Figure 13 shows a contour plot at 0.05 m/s. Regions of the domain near where the stent contacts the plaque exhibit

Figure 12:

Velocity magnitudes and strain rates in stented artery.

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36 Modelling in Medicine and Biology VIII

Figure 13:

Iso-contour at 0.05 m/s on region near plaque/stent interface.

zones of low velocity. These regions are potential locations for thrombus formation. Additionally, the wall shear rates and blood strain rates at the wall can be evaluated to determine if they exceed a critical embolizing limit.

7

Clinical significance of report

Concerns that drug-eluting stents interfere with the process of reendothelization and thus may encourage long-term thrombosis have spurred interest in understanding the mechanisms causing acute deendothelization during the stenting procedure. This model aids in the prediction of regions of endothelial cell (EC) denudation during stent implantation. This is an important phenomenon since regions of EC denudation profoundly impact drug absorption/loading profiles of anti-proliferative agents in drug-eluting stents (DES). Additionally, anti-proliferative drugs are hypothesized to inhibit EC regrowth causing increased rates of long-term thrombosis, so predictive capability of regions of EC denudation during implantation provides the tool to reduce thrombosis rates of DES. The model developed also helps in the prediction of regions of high arterial stresses, which may cause vascular injury. Acute superficial and deep vascular injury has been found to be a strong predictor of chronic restenosis. This method provides a predictive tool to evaluate the degree of acute vascular injury of new stent geometries prior to in-vivo studies. Finally, the ability to examine blood flow patterns in stented arteries allows for the prediction of standing vortices and high platelet-activating substances capable of trapping and stimulating platelets for aggregation. It also allows for the evaluation of embolizing stresses acting on a thrombus. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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References [1] Schillinger, M., Exner, M., Mlekusch, W., Haumer, M., Sabeti, S., Ahmadi, R., Schwarzinger, I., Wagner, O., Minar, E., Restenosis after femoropopliteal PTA and elective stent implantation: predictive value of monocyte counts. Journal of Endovascular Theory, 10, pp. 557-565, 2003. [2] Antoniucci, D., Valenti, R., Santoro, G., Bolognese, L., Trapani, M., Cerisano, G., Fazzini, P. Restenosis after coronary stenting in current clinical practice, American Heart Journal, 135(3), pp. 510-518, 1998. [3] Griffiths, H., Bakhai, A., West, D., Petrou, M., De Souza, T., Moat, N., Pepper, J., Di Mario, C., Feasibility and cost of treatment with drug eluting stents of surgical candidates with multi-vessel coronary disease. European Journal of Cardiothroacic Surgery, 26, pp. 528-534, 2004. [4] Feder, B.J., Panel Urges Caution on Coated Stents, New York Times Health p. 1, Dec. 9, 2006 [5] Kuchulakanti, P.K., Chu, W.W., Torguson, R., Ohlmann, P., et al. Correlates and long-term outcomes of angiographically proven stent thrombosis with sirolimus- and paclitaxel-eluting stents. Circulation. 113, pp. 1108 –1113, 2006. [6] McFadden, E.P., Stabile, E., Regar, E., Cheneau, E., et al. Late thrombosis in drug-eluting coronary stents after discontinuation of antiplatelet therapy. The Lancet, 364, pp. 1519-1521, 2004. [7] Ong, A.T., McFadden, E.P., Regar, E., de Jaegere, P.P., van Domburg, R.T., Serruys, P.W. Late angiographic stent thrombosis events with drug eluting stents. Journal of the American College of Cardiology, 45, pp. 2088-2092, 2005. [8] Rogers, C., Parikh, S., Seifert, P., Edelman, E.R. Endogenous cell seeding: Remnant endothelium after stenting enhances vascular repair. Circulation, 11, pp. 2909-2914, 1996. [9] Auricchio, F., Di Loreto, M., Sacco, E., Finite element analysis of a stenotic artery revascularization through stent insertion, Computer Methods in Biomechanics and Biomedical Engineering, 4, pp. 249-263, 2001. [10] Migliavacca, F., Petrini, L., Colombo, M. et al. Mechanical behavior of coronary stents investigated through the finite element method. Journal of Biomechanics, 35, pp. 803-811, 2002. [11] Petrini L, Migliavacca F, Dubini G, Auricchio F. Evaluation of intravascular stent flexibility by means of numerical analysis. Proc. Of the 2003 Summer Bioengineering Conference, June 25-29, Key Biscayne, FL, pp. 251-252, 2003. [12] Lally, C., Dolan, F, and Prendergast, P.J., “Cardiovascular stent design and vessel stresses: a finite element analysis, Journal of Biomechanics, 38, pp. 1574-1581, 2005. [13] Lally, C, Prendergast, P.J., An investigation into the applicability of a Mooney–Rivlin constitutive equation for modeling vascular tissue in cardiovascular stenting procedures. Proceedings of the International

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38 Modelling in Medicine and Biology VIII Congress on Computational Biomechanics. Zaragoza, Spain, pp. 542-550, 2003. [14] Prendergast, P.J., Lally, C., Daly, S., Reid, A.J., Lee, T.C., Quinn, D., Dolan, F., Analysis of prolapse in cardiovascular stents: a constitutive equation for vascular tissue and finite element modeling. ASME Journal of Biomechanical Engineering, 125, pp. 692-699, 2003. [15] Loree, H.M., Grodzinsky, A.J., Park, S.Y., Gibson, L.J., Lee, R.T., Static circumferential tangential modulus of human artherosclerotic tissue. Journal of Biomechanics, 27, pp. 195-204, 1994. [16] Nichols, W.W., O'Rourke, M.F. McDonald's Blood Flow in Arteries Theoretical, Experimental and Clinical Principles. Oxford University Press: New York, 1998.

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Fractal behaviour of pathological heart rate variability dynamics G. D’Addio1, A. Accardo2, G. Corbi3, F. Rengo1,4 & N. Ferrara1,3 1

S. Maugeri Foundation, Italy DEEI University of Trieste, Italy 3 University of Molise, Italy 4 University Federico II of Naples, Italy 2

Abstract Heart rate variability analysis (HRV) is a well recognized tool in the autonomic control assessment. It has been suggested that nonlinear analysis of HRV might provide more valuable information than traditional linear methods. Several non linear fractal techniques recently gained wide interest: that based on indirect fractal dimension (FD) estimation from the 1/f spectral power relationship, and that based on a direct FD estimation from HRV time sequences. Aim of the study was to assess whether FD discriminates pathological HRV dynamics, comparing results with normal subjects and traditional linear indexes. We studied 7 groups of 10 ECG 24h-Holter recordings in normal and different pathologies: obstructive pulmonary disease, stroke, hypertension, post myocardial infarction, heart failure, heart transplanted. HRV was assessed by spectral power in very low, low and high frequency bands and standard deviation between normal beats. FD was estimated directly from the HRV sequences by Higuchi method (HM) and from the 1/f slope of spectral power relationship (beta). Results showed differences in the autonomic control impairments better described by FD than by traditional linear methods. Although HM and beta tried to measure the same FD property, the latter seemed to be rather insensitive to changes in autonomic control. These preliminary results clearly suggest that FD, estimated by HM, contains relevant information related to different HRV pathological dynamics. Keywords: HRV, fractal analysis, nonlinear dynamics.

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40 Modelling in Medicine and Biology VIII

1

Introduction

Heart rate variability (HRV) is a well known noninvasive tool in the investigation of the heart autonomic control. Although most studies on HRV have been performed using time- and frequency-domain linear methods, it has been suggested that HRV nonlinear analysis might provide more valuable information for physiological interpretation of heart rate fluctuations and for cardiac risk assessment [1]. Fractal analysis is an emerging nonlinear technique and, among several methods proposed so far to measure the fractal behaviour of the HRV signal, that based on spectral power-law relationship [2–7], and that based on iterative algorithms directly from RR time series, [8,9] have gained wide interest in the last years. The first way has traditionally been approached following the chaostheory, with the aim of modelling the attractor extracted from HRV sequences [6], estimating the fractal dimension from the slope of the 1/f-like relationship [7]. Alternatively a fractal dimension value can be directly estimated from HRV sequences by means of Higuchi algorithm [9]. All two the approaches were followed in this study, estimating fractal features by beta exponent of the 1/f (beta) and by fractal dimension of the Higuchi algorithm (HM). The latter method, whose good reproducibility has been already studied in congestive heart failure [10], allows a better fractal estimation, eliminating the errors due to the indirect estimation of FD from the spectral power. HRV has been usually investigated in cardiac patients, where abnormalities of the autonomic control to the heart have a common diagnostic and prognostic use [11,12]. Evidences of clinically significant impairment of the autonomic nervous system are known in two others widely diffuse pathologies like stroke and chronic obstructive pulmonary disease, although only limited data are available on the use of HRV in the assessment in these not strictly cardiac patients. Impaired cardiovascular autonomic regulation has been described in stroke patients with abnormalities hypothesized to be mediated by the central nervous system as a result of the cerebrovascular event, whereas the mechanism of this phenomenon is not fully understood [13,14]. Respiratory arrhythmia in chronic obstructive pulmonary disease, represents the most recognizable evidence of a functional link between neural cardiac and respiratory controls. Changes in respiratory patterns and lung volumes in these patients influence the autonomic outflows by complex reflex adjustments, mediated by both vagal and sympathetic efferent activity [15,16]. Aim of the study was to assess whether FD discriminates pathological HRV dynamics, comparing results with normal subjects and traditional linear indexes.

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41

Study population

All enrolled patients were admitted to S. Maugeri Foundation Rehabilitation Institute of Telese Terme, Italy. We studied 7 groups of ECG 24h-Holter recordings in normal (NR) and different pathologies: hypertension (HY), post myocardial infarction (MI), heart failure (HF), heart transplanted (TR), obstructive pulmonary disease (COPD), stroke (SP). Hypertension diagnosis was defined as systolic blood pressure 140 mm Hg and/or diastolic blood pressure 90 mm Hg. A prior diagnosis or ECG evidence of Q waves was used to define MY patients. The diagnosis of chronic systolic heart failure was based on a HF history of at least 6 months and previous echocardiographic and/or scintigraphic evidence of an ejection fraction of <40%. COPD selected patients had a positive medical history for obstructive pulmonary disease, without coronary artery disease. Patients were considered to be affected by COPD if they fulfilled either of the following criteria: 1) they had an FEV1/FVC of <70% and no change or an FEV1 increase of >12%, but not FEV1 normalisation after 100 mg fenoterol; or 2) they nor reported history of wheeze in the last year, had an FEV1/FVC of <70%, an FEV1 of <80% and an FEV1 increase of <12% after 100 mg fenoterol. SP selected patients had a positive past medical history for previous first-ever stroke (ischemic and/or hemorrhagic), without coronary artery disease, presence of neuromotor monolateral deficit at physical examination, a CT finding of medium cerebral artery multiple lesions, and FIM score between 40 and 60. The control group (N) consisted of 10 healthy subjects. See Table 1 for details. Table 1:

Descriptive statistics of studied populations.

Population

Code

#

Age

N

10

42 ± 6

Hypertension

HY

10

41 ± 1

Post-myocardial infarction

MI

10

50 ± 10

Heart failure

HF

10

54 ± 11

Heart transplanted

TR

10

45 ± 15

COPD

16

68 ± 07

SP

17

63 ± 05

Normal

Obstructive pulmonary disease Post-stroke

3

Holter analysis

Twenty-four-hours Holter ECG recordings were assessed by a portable threechannel tape recorder, processed by a Marquette 8000 T system with a sampling frequency of 128 Hz. In order to be considered eligible for the study, each WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

42 Modelling in Medicine and Biology VIII recording had to have at least 12 hours of analyzable RR intervals in sinus rhythm. Moreover, this period had to include at least half of the nighttime (from 00:00 AM through to 5:00 AM) and half of the daytime (from 7:30 AM through to 11:30 AM) [17]. Before analysis, identified RR time series were preprocessed according to the following criteria: 1) RR intervals associated with single or multiple ectopic beats or artefacts were automatically replaced by means of an interpolating algorithm, 2) RR values differing from the preceding one more than a prefixed threshold were replaced in the same way as for artefacts (Table 2). The RR time series were finally interpolated by piecewise cubic spline and resampled at 2 Hz. The signal was divided in one hour tracts; for each tract the linear and non linear parameters were calculated. Table 2:

4

Beats correction summary.

Population

# beats

# corrections

%

NR

102115

433

0.4

HY

107755

568

0.5

MI

98664

446

0.5

HF

107145

557

0.5

TR

116043

60

0.1

COPD

97658

4645

4.8

SP

93202

8857

8.7

Fractal dimension analysis

Fractal dimension was calculated by using the Higuchi's algorithm [18]. From a given time series X(1), X(2), ... X(N), the algorithm constructs k new time series; each of them, Xmk, is defined as Xmk:X(m),X(m+k),X(m+2*k),..., X(m+int((N-m)/k)*k)

(1)

where m=1,2,...,k and k are integers indicating the initial time and the interval time, respectively. Then the length, Lm(k), of each curve Xmk is calculated and the length of the original curve for the time interval k, L(k), is estimated as the mean of the k values Lm(k) for m=1, 2, ..., k. In our analysis a k value of 6 was used. If the L(k) value is proportional to k-D, the curve is fractal-like with the dimension D. Then, if L(k) is plotted against k, for k ranging from 1 to kmax, on a double logarithmic scale, the data should fall on a straight line with a slope equal to -D. Thus, by means of a least-square linear best-fitting procedure applied to the series of pairs (k, L(k)), obtained by increasing the k value, the angular coefficient of the linear regression of the graph ln(L(k)) vs. ln(1/k), which constitutes the D estimation, is calculated (Fig. 1).

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Modelling in Medicine and Biology VIII

Figure 1:

5

43

Example of an hi sequence determination on a curve for the length calculation. hi = |X(m+i*k) - X(m+(i-1)*k)| k = 3, m= 2.

1/f analysis

Power law beta exponent was calculated from the power spectral density function estimated by the Blackman-Tukey method after linear trend removal. The beta index represents the slope of the linear regression analysis between log(power) and log(frequency) per-formed on the portion of the power spectrum between 10-4 and 10-2 Hz.

Figure 2:

6

Example of the beta exponent evaluation by means of the slope of the linear best fitting (dashed line) of the power spectrum for frequencies < 0.05Hz.

Linear analysis

Spectral analysis was performed by homemade software [18] on 5-minute RR sequences extracted from 24-hours holter recordings. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

44 Modelling in Medicine and Biology VIII Power spectral density was estimated by the Blackman-Tukey method in all accepted segments after linear trend removal. The total power and the power in the very low frequency band (VLF, 0.01-0.04 Hz), low frequency band (LF, 0.04-0.15 Hz) and high frequency bands (HF, 0.15-0.45 Hz) were then computed by numerical integration of the spectral density function. Standard deviation between normal-to-normal RR values (SDNN) was also evaluated for all RR time series. Mean ± SD of HRV indexes in all the studied groups.

Table 3: N

COPD

SP

HY

MI

HF

TR

HM

1.35±0.06

1.68±0.11

1.86±0.10

1.58±0.06

1.67±0.08

1.76±0.13

1.96±0.12

Beta

-1.02±0.14 -1.10±0.13

1.17±0.13

1.26±0.06

1.42±0.27

1.50±0.30

1.84±0.30

VLF

1048±362

260±139

1355±672

784±264

643±260

519±512

45±50

LF

1188±531

188±85

453±344

878±383

404±188

190±151

20±19

HF

366±254

266±187

169±116

433±314

205±147

112±108

28±24

SDNN

133±21

136±49

56±12

135±38

102±18

87±19

70±28

HM

2.00

1.75

1.50

1.25 Figure 3:

N COPD SP

HY

MI

HF

TR

HM values in the seven studied populations.

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45

Results

Descriptive statistics for HM, beta exponent, spectral and time-domain HRV indexes in all the studied groups are reported in Table 3. Kolmogorov-Smirnov (KS) test was used to assess the normality of the distribution of all variables (P>0.1 for all variables). According to the severity of the autonomic control impairment of the six patients' populations studied, the fractal dimension index, gradually increases from normals to heart transplanted subjects (Fig. 3). A one-way ANOVA, with Tukey's multiple comparison test, was performed to assess statistical differences between each couple of studied populations (Tab.4). Results showed that differences in the autonomic control impairments seem to be better described by HM and LF than by other HRV indexes. Although several indexes were able to discriminate between some groups, only HM reached a significant p-value (p<0.001) in all the populations. Table 4:

8

Tukey's Multiple Comparison Test P values between N and pathological studied populations. HM

Beta

VLF

N vs COPD

P < 0.001

P > 0.05

N vs SP

P < 0.001

P > 0.05

P > 0.05

N vs HY

P < 0.001

P > 0.05

N vs MI

P < 0.001

LF

HF

SDNN

P > 0.05

P > 0.05

P < 0.001

P > 0.05

P < 0.001

P > 0.05

P > 0.05

P > 0.05

P > 0.05

P < 0.01

P > 0.05

P < 0.001

P > 0.05

P > 0.05

N vs HF

P < 0.001 P < 0.001

P < 0.05

P < 0.001

P > 0.05

P < 0.05

N vs TR

P < 0.001 P < 0.001 P < 0.001 P < 0.001

P < 0.01

P < 0.001

P < 0.001 P < 0.001

Discussion

These preliminary results allow to discuss the following three findings. First of all, only FD by Higuchi method and LF parameters showed very significant differences between Normal and pathological studied groups, while beta and other linear indexes were not so able to detect significant differences. The second novel finding is that the sensitivity of the HM and beta exponent parameters in regard to the severity of the central nervous system damage appears to be different. Indeed, the Higuchi's index strongly changes passing from normal to pathological subjects. The beta exponent, on the contrary, seems rather insensitive to changes in autonomic cardiovascular regulation. These considerations suggest that, although the two algorithms try to measure the same fractal property of HRV, they provide non superimposable results. This could be WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

46 Modelling in Medicine and Biology VIII due to the fact that the beta exponent is usually calculated considering only the low band of the signal (<0.05 Hz). Probably the changes in autonomic cardiovascular regulation much more affect a band with higher frequency. The third consideration is that the difference in the mean HM values between N and HF subjects, ranging about the 23% is very interesting because very much higher than the standard error of measurement of just about the 3.4% that we found in a reproducibility study of the same parameter in a HF population [10]. Although a major limitation of this study is the low sample size of the groups, nevertheless these preliminary results clearly suggest that HRV fractal dimension parameters, obtained from morphologic quantification of HRV directly in the time series sequence, contains relevant information related to different HRV dynamics and can be candidates for future risk assessment studies as relevant measures of the overall physiologic and functional status of these patients.

References [1] Task Force of the European Society of Cardiology. Heart Rate Variability – Standard of Measurement, Physiological Interpretation and Clinical Use. Circulation 1996;93:1043-65 [2] Saul JP, Albrecht P, Berger RD, Cohen RJ. Analysis of long term HRV: methods, 1/f scaling and implications. In: Computers in Cardiology 1987. IEEE Computer Society Press, 1987:419-22. [3] Bigger T, Steinman R, Rolnitzky L, Fleiss J, Albrecht P, Cohen R. Power law behavior of RR-Interval Variability in healthy middle-aged persons, patients with recent acute myocardial infarction and patient with hearth transplants. Circulation 1996;93:2142-51. [4] Makikallio TH, Hoiber S, Kober L, Torp-Pedersen C, Peng CK, Goldberger AL, Huikuri HV. Fractal analysis of heart rate dynamics as a predictor of mortality in patients with depressed left ventricular function after acute myocardial infarction. Am J Cardiol 1999;83(6):836-9. [5] Makikallio TH, Huikuri H, Hintze U, Videbaek J, Mitrani RD, Castellanos A, Myerburg R, Moller M. Fractal analysis and time- and frequencydomain measures of heart rate variability as predictors of mortality in patients with heart failure. Am J Cardiol 2001;87(2):178-82. [6] Cerutti S, Carrault G, Cluitmans PJ, Kinie A, Lipping T, Nikolaidis N, Pitas I, Signorini MG. Non-linear algorithms for processing biological signals. Comp Met Prog Biomed 1996;1:51-73. [7] Butler GC, Yamamoto Y, Xing HC, Northey DR, Hughson RL. Heart rate variability and fractal dimension during orthostatic challenges. J Appl Physiol 1993;75(6):2602-12. [8] Goldberger AL. Fractal mechanisms in the electrophysiology of the heart. IEEE Eng Med Biol 1992;11:47-52. [9] Higuchi T. Approach to an irregular time series on the basis of the fractal theory. Physica D 1988;31:277-83.

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[10] D'Addio G, Accardo A., Maestri R, Picone C., Furgi G, Rengo F. Reproducibility of nonlinear indexes of HRV in chronic heart failure. Pace, February 2003, Volume 26, No.2, Part II p. S171. [11] Brouwer J, Veldhuisen D et al. Prognostic value of heart rate variability during long-term follow-up in patients with mild to moderate heart failure. J Am Coll Cardiol 1996;28:1183-9 [12] D'Addio G., Pinna GD., La Rovere MT., Maestri R., Furgi G., Rengo F. Prognostic Value Of Poincarè Plot Indexes In Chronic Heart Failure Patients Computers in Cardiology 2001; 28: 57-60. IEEE Computer Society Press [13] Meglic B., Kobal J., Osredkar J. And Pogacnik T. Autonomic Nervous System Function in Patients with Acute Brainstem Stroke. Cerebrovasc. Dis. 2001:11, pp. 2-8. [14] Korpelainen J.T., Sotaniemi K.A., Makikallio A., Huikuri H.V., and Myllyla V.V. Dynamic behavior of heart rate in ischemic stroke. Stroke 1999: 30, pp. 1008-13 [15] Volterrani M, Scalvini, et al. Decreased heart rate variability in patients with chronic obstructive pulmonary disease. Chest 1994;106:1432-37 [16] Pagani M, Lucini D, Pizzinelli P, Sergi M, Mela GS, Malliani A. Effects of aging and of chronic obstructive pulmonary disease on RR interval variability J Auton Nervous System l996;59:125-132 [17] Bigger T, Fleiss J, Rolnitzky L, Steinman R. Stability over time of heart period variability in patients with previous myocardial infarction and ventricular arrhythmias. Am J Cardiology 1992;69:718-23. [18] Maestri R. and Pinna G.D. POLIANN: a computer program for poliparametric analysis of cardio-respiratory variability signals.’ Computer Methods and Programs in Biomedicine, 1998: 56, pp. 37-48.

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Microvascular disorders induced by malaria infected red blood cells: a computational mechanical study using the biological particle method T. Yamaguchi1, H. Kondo2, Y. Imai2 & T. Ishikawa2 1

Department of Biomedical Engineering, Graduate School of Biomedical Engineering, Tohoku University, Sendai, Japan 2 Department of Bioengineering and Robotics, Graduate School of Engineering, Tohoku University, Sendai, Japan

Abstract We simulated malarial microvascular blood flow disturbances by using a new particle method of biological solid-fluid interaction analysis especially developed for the analysis of malaria infection. Particle based spatial discretization and the sub time step time integration could provide us with stable computations for micro scale blood flow involving interaction with many cells. We performed numerical simulation of the stretching of infected red blood cells and the results agreed well with experimental results. Our model successfully simulated the flow of infected red blood cells into narrow channels. Keywords: malaria, computational fluid dynamics, particle method, red blood cell, microcirculation, mechanical properties of cell membrane.

1

Introduction

Malaria is one of the most serious infectious diseases on earth. There are 500 million patients with 2 million deaths arising from malaria infection. When a parasite invades and matures inside a red blood cell (RBC), the infected RBC (IRBC) becomes stiffer and cytoadherent. These changes are postulated to link to microvascular blockage [1]. Several researchers have investigated cell mechanics of IRBCs using recent experimental techniques. Methods to quantify WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090051

50 Modelling in Medicine and Biology VIII the stiffness of the IRBCs include the micropipette aspiration [2-4] and optical tweezers [5]. Suresh et al. [5] used optical tweezers to investigate the change of mechanical response of IRBCs at different development stages. They clarified that the shear modules can increase by ten-fold in the final schizont stage. Microfluidics has also been used to investigate the pathology of malaria. Shelby et al. [9] investigated the effect of IRBCs on the capillary obstruction and demonstrated that IRBCs in the late stages of the infection cannot pass through micro channels that have diameters smaller than those of the IRBCs. These studies are reviewed in [6-8]. These experimental studies are still limited to the effect of the single infected cell. However, microvascular blockage may be a hemodynamics problem, involving the interactions between IRBCs, healthy RBCs and endothelial cells. This is due to the limitation of the current experimental techniques. Firstly, it is still difficult to observe the RBC behaviour interacting with many other cells, even with the recent confocal microscope. Secondly, three-dimensional information on the flow field is hard to obtain. Thirdly, capillaries in the human body are circular channels with complex geometry, but such complex channels cannot be created in micro scale. Instead, numerical modelling can be a strong tool for further understanding the pathology of malaria. Finite element modelling was performed for the stretching of an IRBC by optical tweezers [5]. Dupin et al. [10] also modelled the stretching of the IRBC using the lattice Boltzmann method. We have also developed a hemodynamic model involving adhesive interactions [11]. In this report, we present our methodology and preliminary numerical results.

2

Method

2.1 Particle method Blood is a suspension of RBCs, white blood cells, and platelets in plasma. An RBC consists of cytoplasm enclosed by a thin membrane. Assuming that plasma and cytoplasm are incompressible and Newtonian fluids, the governing equations are described as Dρ (2) = 0, Dt 1 Du (3) = − ∇p + ν∇2u + f , Dt ρ where the notation t refers to the time, ρ is the density, u is the velocity vector, p is the pressure, ν is the dynamic viscosity, f is the external force per unit mass, and D/Dt is the Lagrangian derivative. Our model is based on a particle method. All the components of blood are represented by particles (Fig. 1). Note that each particle is not a real fluid particle but a discrete point for computation. Fluid variables are calculated at the computational point and it is moved by the calculated advection velocity every time step. In conventional mesh methods, each computational point requires the WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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51

connection to neighbouring points for the discretization of Eqs. (2) and (3) and thus the computational meshes are generated. When a red blood cell approaches another other cell, however, the computational meshes can be distorted and destroyed easily. In contrast to mesh methods, the particle method does not require such computational meshes and the computation is stable even when many cells are interacting with each other. Another advantage of the particle method is the coupling with the front-tracking. The particle method tracks the front of the membrane using the membrane particles and the no-slip condition on the membrane is directly imposed to Eq. (3) by using the position and velocity of membrane particles. We use the moving particle semi-implicit (MPS) method [12] for solving Eqs. (2) and (3). In the MPS method, differential operators in the governing equations are approximated using the weight function, for example, φ −φ d (4) ∇φi = 0 ∑ j 2 i rij w rij , n j ≠i rij

( )

( )

2d (5) ∑ (φ − φ )w rij , λ n 0 j ≠i j i where φ is a fluid variable, d is the space dimension number, n0 is the reference particle number density, λ is the constant, r is the position of particle, rij = rj - ri, and w is the weight function. ∇2φi =

Plasma Membrane of RBC Cytoplasm of RBC Membrane of IRBC Cytoplasm of IRBC Malaria parasite Endothelial Cell

Figure 1:

Scheme of free mesh particle model of blood. All the components of blood are represented by the finite number of particles (computational points). Velocity and pressure are calculated at the position of each particle and it is moved by the calculated advection velocity every time step.

2.2 IRBC model The membrane of IRBCs is represented by the two-dimensional network consisting of the finite number of particles (Fig. 2). Particle j is connected to particle i by linear spring, giving a force,

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52 Modelling in Medicine and Biology VIII

(

Fjs = k s rij − l0

) rr

ij

,

(6)

ij

where ks is the spring constant and l0 is the equilibrium distance. A bending force is also considered in our model: Fb + Flb , (7) Fbj = Fkb = i 2 θ  (8) Fib = kb tan nijk , 2 θ  (9) Flb = kb tan  n jkl , 2 where kb is the spring constant, θ is the angle between the triangles ∆ijk and ∆jkl, and nijk is the normal vector to the triangle ∆ijk. The external force per mass is given as F s + Fbj , (10) fi = j ρV0 where V0 is the reference volume V0 = r03 and r0 is the averaged particle distance at the initial time step. Note that the spring constants ks and kb should be model parameters to control the deformation of IRBCs. As described in the next section, we adjust these model parameters through numerical experiments. A malaria parasite inside IRBCs is modelled by a rigid object constructed by some particles. The treatment of the rigid object in the MPS method was proposed by Koshizuka et al. [13].

a

k l

i

j

b

‰

j l

i Figure 2:

Spring network model of IRBC membrane. (a) Each membrane particle is connected to the neighbouring membrane particles by a spring with linear elasticity. (b) To express the deformation of the thin membrane, the bending force is also introduced.

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Figure 3:

53

Stretching of IRBCs. An IRBC is replaced at the centre of the computational domain at the initial time step. The IRBC is stretched horizontally by a constant force. Governing equations are time integrated until the steady state solution is obtained.

Normalized diameter [-]

1.6 1.4

Axial (Numerical) Axial (Experiment) Transverse (Numerical) Transverse (Experiment)

1.2 1.0 0.8 0.0

Figure 4:

3

50.0 100.0 Stretching force [pN]

150.0

Comparison between the numerical results and the experimental results [11] for stretching of IRBCs in the schizont stage. Symbols indicate the experimental results and lines are the numerical results. The upper line represents the results for axial diameter and the lower line represents that for transverse diameter. The numerical results agree well with the experimental ones.

Results

Suresh et al. [5] performed stretching of IRBCs by optical tweezers to quantify the mechanical response of IRBCs. They measured the axial and transverse diameters of IRBCs at different development stages with several stretching WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

54 Modelling in Medicine and Biology VIII forces. We examine this stretching test numerically to validate our model. In the three-dimensional computational domain, we put an IRBC at initial time step. Then we stretch the IRBC horizontally with a stretching force as shown in Fig. 3. The governing equations are time-integrated until the axial and transverse diameters are well converged. If the obtained results do not follow the experimental results, the values of model parameters ks and kb are changed and the computation is carried out again. Finally we can get the appropriate values of the model parameters. Final results of IRBCs in the schizont stage are presented in Fig. 4. The numerical results agree well with the experimental results by Suresh et al. [5].

4

Discussion

We have proposed a numerical model of three-dimensional hemodynamics arising from malaria infection. A particle based spatial discretization and sub time step time integration method are employed to stably simulate many cell interactions in micro scale blood flow. To validate our model, we examined stretching of IRBCs. When we used appropriate values for the spring coefficients ks and kb, the numerical results agreed well with the experimental results. Small differences can be found in the axial diameter for high stretching force. It may be improved using a non-linear spring model for membrane. However, compared with previously presented numerical models, such as a finite element model [5] and a Lattice Boltzmann model [10], our model has similar accuracy for the stretching of IRBCs. Our model also successfully simulated flow into narrow channels, mimicking microcirculation in human body. We tested two sizes of narrow channels: a 6mm-square channel and a 4-mm-square channel. Both HRBCs and IRBCs in the schizont stage passed through a 6-mm-square channel. While HRBCs also flowed into the 4-mm-square channel, schizont IRBCs occluded the flow. These results follow the experimental observation by Shelby et al [9], where they revealed that IRBCs in the late trophozoite and schizont stages occluded flow into the channel with 4mm width.

5

Conclusion

In this report, we proposed a numerical model of three-dimensional hemodynamics arising from malaria infection. Particle based spatial discretization and the sub time step time integration can provide us stable computations for the micro scale blood flow involving the interaction with many cells. We performed the stretching of IRBCs and the numerical results agreed well with experimental results. Our model successfully simulated flow of IRBCs into narrow channels. Here, we have not considered the adhesion property of IRBCs. We have already developed an adhesion model [9] and this adhesion model can be applied easily to the current hemodynamic model.

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References [1] Cooke BM, Mohandas N, and Coppell RL. The malaria-infected red blood cell: structural and functional changes. Advances in Parasitology 26, 1–86, 2001. [2] Nash GB, O’Brien E, Goldon-Smith EC, and Dormandy JA. Abnormalities in the mechanical properties of red blood cells caused by plasmodium falciparum. Blood 74, 855–861, 1989. [3] Paulitschke M, and Nash GB. Membrane rigidity of red blood cells parasitized by different strains of Plasmodium falciparum. J Lab Clin Med 122, 581–589, 1993. [4] Lim CT, Zhou EH, and Quek ST. Mechanical models for living cells–a review. J Biomech 39, 195–216, 2006. [5] Suresh S, Spatz J, Mills JP, Micoulet A, Dao M, Lim CT, Beil M, and Seufferlein T. Connection between single-cell biomechanics and human disease states: gastrointestinal cancer and malaria. Acta Biomater 1, 15–30, 2005. [6] Lim CT, Zhou EH, Li A, Vedula SRK, and Fu HX. Experimental techniques for single cell and single molecule biomechanics. Mater Sci Eng C 26, 1278–1288, 2006. [7] Lim CT. Single cell mechanics study of the human disease malaria. J. Biomech Sci Eng 1, 82–92, 2006. [8] Lee GYH, and Lim CT. Biomechanics approaches to studying human diseases. Trends Biotech 25, 112–118, 2007. [9] Shelby JP, White J, Ganesan K, Rathod PK, and Chiu DT. A microfluidic model for single-cell capillary obstruction by Plasmodium falciparuminfected erythrocytes. PNAS 100, 14618–14622, 2003. [10] Dupin MM, Halliday I, Care CM, and Munn LL. Lattice Boltzmann modelling of blood cell dynamics. Int J CFD 22, 481–492, 2008. [11] Kondo H, Imai Y, Ishikawa T, Tsubota K, and Yamaguchi T. Hemodynamic analysis of microcirculation in malaria infection. Ann Biomed Eng, accepted. [12] Koshizuka S, and Oka Y, Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123, 421–434, 1996. [13] Koshizuka S, Nobe A, and Oka Y. Numerical analysis of breaking waves using the moving particle semi-implicit method. Int J numer meth Fluids 26, 751–769, 1998.

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Mechanical characterization of deep vein thrombosis in a murine model using nanoindentation K. C. McGilvray1, R. Sarkar 2 & C. M. Puttlitz1 1

Orthopaedic Bioengineering Research Laboratory, Department of Mechanical Engineering and School of Biomedical Engineering, Colorado State University, Fort Collins, CO, USA 2 Northern California Institute for Research and Education, University of California San Francisco, San Francisco, CA, USA

Abstract Seventy percent or more of the blood volume is contained by the venous return, and this system of vessels represents a highly distensible, low-pressure configuration. Disease states associated with the venous system have a high impact on society; complications from acute and chronic deep vein thrombosis (DVT), the precursor to post-phlebitic veins (PPV), contribute to more deaths each year than AIDS and breast cancer combined, affecting between 1 and 2% of hospitalized patients in the United States. To our knowledge, there are no published studies with regard to the biomechanical dispensability of either clinical or experimental PPV. To examine the effects of DVT induction on vessel wall biomechanics, a distinct DVT-induced murine model was evaluated. The DVT model was generated using a well-established model via partial ligation of the murine vena cava (MVC). The anisotropic biomechanical properties of each variant were determined along the primary loading axes: circumferential (perpendicular to the surface created by transecting the vein wall longitudinally), longitudinal (perpendicular to the beginning/end surfaces of the isolated intact vessel segment), and luminal (perpendicular to inner surface of vessel) of healthy and diseased vascular tissue via nanoindentation. Three indentation parameters were determined to describe the inherent tissue mechanics: unloading stiffness and two reduced elastic moduli (Oliver-Pharr and JKR formulations). Nanoindentation testing indicated that MVC induced with DVT demonstrated mean deviations (as compared to the normal, baseline tissue) in elastic moduli for the longitudinal and luminal directions. The data also indicate that the unloading stiffness in the longitudinal direction has the largest average magnitude among the three physiologically-relevant planes. Keywords: nanoindentation, deep vein thrombosis, elastic modulus. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090061

58 Modelling in Medicine and Biology VIII

1

Introduction

In order to better understand the resultant outcome of deep vein thrombosis (DVT), a technique that has significant spatial resolution must be employed to examine the biomechanical changes of the intra-wall constituents. Therefore, the purpose of this study was to quantitatively compare the structural/material changes through the use of nanoindentation between healthy and DVT-induced veins “diseased”, in the three primary axes of loading: in the circumferential (perpendicular to the surface created by transecting the vein wall longitudinally), longitudinal (perpendicular to beginning/end surfaces of the isolated intact vessel segment), and luminal (perpendicular to inner surface of vessel) directions (Figure 1).

Figure 1:

Schematic of IVC sample preparation for nanoindentation testing. The respective biomechanical/physiological loading directions of the vessel wall have been labeled.

Biomechanical properties of vascular tissue on the cellular scale are difficult to evaluate through traditional mechanical testing techniques [4]. Nanoindentation has recently gained popularity for testing soft (elastic modulus ≈ 1MPa) biologic materials on the micrometer length scale [4-7]. This technique utilizes small-diameter spherical tips (approximate 100µm) and micro-scale loads (approximate 100μN) and displacements (1000nm) to obtain accurate measurements in small tissue regions [6]. A number of studies have demonstrated the feasibility of using nanoindentation to elicit biomechanical properties of vascular tissue, yet, to date, few have reported quantitative values comparing healthy and diseased vascular tissue. To our knowledge, no study has examined the biomechanical properties of each of the primary loading axes WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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59

(circumferential, longitudinal, and luminal) of healthy and diseased vascular tissue. An understanding of the properties of soft tissues, on all scales, may aid researchers in quantitatively characterizing local mechanics on the cell-to-tissue scale and could clarify relationships between tissue mechanics and pathologic response [8]. Nanoindentation involves application of a controlled load to a surface, with force applications typically ranging between 100nN and 500µN and spatial resolutions between 1nm and 30µm through the use of a nanoindenter [5]. Loaddisplacement data obtained during one cycle of loading and unloading can be analyzed using equations based on elastic contact theory (Hertzian contact) to quantify structural and material properties such as stiffness and elastic modulus. Currently, the standard technique for analyzing nanoindentation data has been developed and validated for elastic and elastic-plastic materials. The form most often used is that presented by Oliver and Pharr [9] and are based on the classical elastic solutions by Sneddon [10], who derived general relationships between the load, displacement, and contact area for any indenter shape that can be described as a solid revolution of a smooth function [9, 10]. Sneddon demonstrated that the applied load (P) is related to the shear modulus of the material (µ), the radius of contact (a), and the penetration depth (h) through the following relationship: P

µ

E√A

  or P

√

(1)

where ν is the Poisson ratio of the material. The elastic modulus (E) of a material is defined as the ratio of normal engineering stress (σ) over engineering strain (ε) in the portion of the stress-strain relationship that obeys Hooke’s law. By taking the first derivative of the applied load with respect to the indenter penetration depth, the unloading stiffness (S) of the substrate can be expressed as a function of the contact area (A) and steady state elastic modulus (E). The compliance of the indenter is considered to be several orders of magnitude less than the compliance of the substrate, and thus can be ignored in the calculation of the elastic modulus. Under these assumptions, the elastic modulus, termed the “reduced modulus” (Er), can be recast as a function of unloading stiffness and contact area (Ac): √

(2)

The contact area (Ac) between a spherical indenter tip and the substrate can be calculated as follows. A

π 2Rh

h

(3)

where hc is the contact depth at the maximum load, and R is the nominal radius of the spherical indenter. It has been found empirically that both elastic and plastic deformation may occur during the loading phase, but that the immediate unloading phase is dominated by the elastic response of the substrate material. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

60 Modelling in Medicine and Biology VIII That is, even if plastic or viscoelastic deformation occurs in the material, the instantaneous elastic response dominates the initial portion of the unloading curve [5]. Thus, calculating the reduced elastic modulus from nanoindentation data simply involves measuring the unloading stiffness and the contact depth at maximum load (Figure 2). Since Oliver and Pharr first proposed the formulation for reduced modulus (eqn. (2)) several modification and improvements have been proposed in order to account for deviations from the idealized Hertzian contact conditions. Nanoindentation of soft substrates usually involves a significant adhesive effect between the indenter tip and the sample. The two most common models used to correct for the effects of adhesion on contact mechanics behavior are the Johnson, Kendall, Roberts (JKR) and Deraguin-Muller-Toporov (DMT) models [7]. For low modulus materials, the JKR model, which accounts for adhesion forces only within the expanded area of contact, is considered to give a better representation of the elastic modulus. Specifically, work by Gupta et al demonstrated that adhesion plays a significant role in soft tissue contact mechanics [7]. Their investigation concluded that the work of adhesion must be included in the experimental protocol and resulting calculations for determining the mechanical properties with nanoindentation. According to the JKR model, the work of adhesion is related to the magnitude of the maximum tensile pull-off adhesive force, measured during the unloading of the substrate. The authors of this model postulated that the thermodynamic work of adhesion per unit of contact area (Wa) is related to the pull-off adhesion force as follows: F

(4)

RW

Explicitly, the thermodynamic work of adhesion per unit of contact area (Wa) is the work required to separate two surfaces from finite to infinite contact. This parameter can be directly calculated from the load-displacement profile generated during nanoindentation testing (Figure 2). Using the JKR formulation, the elastic modulus (EJKR) for a spherical indenter can be expressed as: S

EJKR R P

F

   F

(5)

P F

where S is the unloading stiffness, R is the nominal radius of the indenter tip, and FPull-off is a measure of the adhesion force generated during tip-substrate separation. The JKR model (eqn. (5)), which accounts for interfacial forces outside the Hertzian contact area, is the most applicable adhesion model for compliant materials indented with spherical probes with a large radius of curvature (approximate 100µm) [7].

2

Materials and methods

Nanoindentation tests were performed on both healthy and DVT induced (“diseased”) murine inferior vena cava tissue (IVC) (n = 8 IVC/group) [3]. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

Figure 2:

61

Typical load-displacement curve observed during nanoindentation testing of murine vena cava. Biomechanical indices used in the formulation of equations (2) and (5) are indicated.

Diseased murine vena cava tissue samples were created by causing a partial stenosis for seven days via tightening a suture around the IVC of a mouse until blood flow was reduced by 80-90%, invariably producing occlusive, laminar thrombi [3, 11]. Harvested IVC were transected longitudinally along the flow direction, creating a planar sheet. The thrombus was then separated from the vessel wall for the diseased group. Each planar murine sample was sectioned circumferentially into three equal samples sections. Samples were stored in cool (approximately 20ºC) isotonic saline (0.9%w/v sodium chloride) and biomechanically tested within forty-eight hours post sacrifice. Care was taken to ensure that the primary directions of physiological loading (circumferential, longitudinal, and luminal) were “tracked” through all processing procedures for each piece of tissue. Prior to nanoindentation testing, the samples were mounted using previously reported techniques [4-6], which have been shown to maintain sample hydration for up to eight hours and provide adequate mechanical substrate support for testing [4]. To insure complete equilibrium hydration testing, samples were submerged in saline at forty-five minute intervals for a minimum of twenty minutes. Using these dissection and hydration techniques, it was possible to mount hydrated samples such that one of the six sides from each planar sheet was aligned in one of the primary directions of interest. Nanoindentation measurements were performed using a Hysitron TriboIndenter (Hysitron Inc., Minneapolis, MN). A 100µm, 90º cone angle fluid cell nonporous diamond tip was used for all experiments. Ebenstein et al showed a conospherical diamond probe, with a 100µm radius of curvature was found to be suitable for testing a variety of soft hydrated materials [4]. Their work demonstrated repeatable measurement on all of the materials they tested, exhibited minimal approach problems, and had reasonable projected contact area WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

62 Modelling in Medicine and Biology VIII to measure local tissue properties rather than individual cells or globally averaged tissue properties [4]. A trapezoidal loading profile was selected; once the tip was brought into contact with the sample, the load was applied at a rate of 100 μN/s, held for 10 seconds at the maximum load (400 μN) to permit viscoelastic dissipation, and subsequently withdrawn at a rate of 100 μN/s [4, 6, 7]. For these experiments much of the applied load was associated with moving the tip at large displacements, the actual peak load imparted on the samples ranged from 10–80 µN (Figure 2). A trapezoidal load function was utilized to allow creep in the substrate to dissipate prior to unloading. Load–displacement curves were corrected for large displacements and 8 µN force offsets were imposed prior to the analysis. Brisceo et al. demonstrated that this loading profile, while not long enough for creep to fully dissipate, was sufficient for the unloading behavior to dominant the inherent viscoelastic effects [12]. Load and displacement were recorded simultaneously during indentation at 228 Hz. Three parameters are reported here as measures of tissue mechanical properties: the unloading stiffness, reduced elastic modulus [9] (Oliver-Pharr formulation, eqn. (2)), and JKR elastic modulus (eqn. (5)) [13]. Because an identical load function was applied to each indenter site, the changes in the indentation response of the tissue, as quantified by these parameters, provide a measure of relative functional properties in the different tissue specimens [4]. The unloading stiffness, (S), was calculated by fitting a linear function to the initial 10% portion of the unloading curve. Statistical significance in the aforementioned biomechanical parameters between groups was performed using a student’s t-test (SigmaStat, Systat Software Inc. Richmond, California, USA), p-values less than 0.05 were considered statistically significant.

3

Results

Using the JKR formulation of elastic modulus, which is considered to be the most accurate for “soft” biomaterials, the data indicate that the mean elastic moduli of healthy murine IVC ranges from 238.1 kPa in the luminal direction to 362.6 kPa in the circumferential direction. The mean elastic modulus in the longitudinal direction was 270.5 kPa. The DVT-induced samples had mean elastic moduli of 340.5 kPa in the luminal direction, 397.3 kPa in the circumferential direction, and 381.1 kPa in the longitudinal direction. No statistically significant differences between the healthy and diseased tissue was found for the unloading stiffness parameters in the three orthogonal directions. However, statistical differences were observed intra-group (Table 1). The longitudinal unloading stiffness of the healthy tissue demonstrated a statistical increase of 43.6% and 62.5% as compared to the circumferential and luminal directions, respectively (Figure 3). Similarly the diseased tissue showed a statistically significant increase in the unloading stiffness in the longitudinal direction of 40.5% and 62.2% when compared to the circumferential and luminal direction, respectively (Figure 3).

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Table 1:

63

Unloading stiffness formulations in the circumferential, longitudinal, and luminal directions based on nanoindentation data for healthy and diseased (induced with deep vein thrombosis) groups. Calculated means are shown with (±) one standard deviation. Similar letters correspond to the given p-values. Circumferential

Unloading Stiffness (μN/nm) Longitudinal

A Healthy 0.0275 ± 0.0120    A  (A) p = 0.340 Diseased 0.0319 ± 0.0024   

Luminal

0.0440 ± 0.0127   B 0.0192 ± 0.0080   C B    (B) p = 0.117 C  (C) p = 0.736 0.0513 ± 0.0099    0.0208 ± 0.0049   

In Heathly:                                                                In Diseased:                                              Radial > Longitudinal      p = 0.002                       Radial > Longitudinal      p < 0.001       Luminal > Longitudinal  p < 0.001                        Luminal > Longitudinal  p < 0.001       

Figure 3:

Unloading stiffness calculated from nanoindentation data in the three primary axes of loading. Averages are shown with one standard deviation error bars, with similar letters indicating statistical differences. Statistically significant p-values are as follows: (θ) p =0.002, (ψ) p < 0.001, (δ) p < 0.001, (φ) p < 0.001.

All reduced elastic moduli calculations were statistically different (p<0.05) from the corresponding JKR elastic moduli calculations, for the three primary directions of interest. No statistically significant findings were noted between the healthy and diseased tissue in the circumferential direction (Table 2). The circumferential direction did demonstrate a non-significant increase of 9.51% between the healthy and diseased tissue for the JKR elastic modulus formulation. The longitudinal reduced elastic modulus was statically greater (111.8%) for the diseased tissue as compared to the healthy tissue (Table 2, Figure 4). The longitudinal JKR elastic modulus for the diseased tissue was not statistically different than that of the healthy tissue; however, an increase of 40.9% was calculated for the diseased tissue. The luminal JKR elastic modulus was statistically greater (43%) for the diseased tissue compared to the healthy tissue, (Table 2, Figure 5). WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

64 Modelling in Medicine and Biology VIII Table 2:

Elastic modulus formulations (reduced and JKR) in the circumferential, longitudinal, and luminal directions based on nanoindentation data for healthy and diseased (induced with deep vein thrombosis) groups. Calculated means are shown with (±) one standard deviation. Similar letters correspond to the given pvalues. CIRCUMFERENTIAL PARAMATERS Reduced Elastic Modulus (MPa)

Healthy Diseased

0.6108 ± 0.1819 A 0.6455 ± 0.1132 A

 (A) p = 0.583

LONGITUDINAL PARAMATERS Reduced Elastic Modulus (MPa) Healthy Diseased

0.4558 ± 0.0222 C 0.9654 ± 0.3243 C

 (C) p < 0.001

LUMINAL PARAMATERS Reduced Elastic Modulus (MPa)

Elastic Modulus JKR (MPa)

0.3626 ± 0.1207 B 0.3973 ± 0.0424 B

 (B) p = 0.579

Elastic Modulus JKR (MPa)

0.2705 ± 0.0153 D 0.3811 ± 0.0902 D

 (D) p = 0.201

Elastic Modulus JKR (MPa)

0.2381 ± 0.0461 F 0.4651 ± 0.1007 E Healthy  (E) p = 0.086  (F) p = 0.024 0.5412 ± 0.0985 E 0.3406 ± 0.0854 F Diseased NOTE:  ALL Reduced Elastic Modulus calculations where statically different from the corresponding JKR Elastic Modulus  calculations p < 0.05.

Figure 4:

Elastic moduli (reduced and JKR formulations) calculated from nanoindentation data in the longitudinal axes of loading. Averages are shown with one standard deviation error bar, with similar letters indicating statistical differences. Statistically significant pvalues are as follows: (C) p < 0.001.

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Figure 5:

4

65

Elastic moduli (reduced and JKR formulations) calculated from nanoindentation data in the luminal axis of loading. Averages are shown with one standard deviation error bar, with similar letters indicating statistical differences. Statistically significant p-values are as follows: (E) p = 0.024.

Discussion

Nanoindentation is a compressive testing process. This type of testing does not explicitly mimic the physiologic in vivo loads expected. However, nanoindentation does give insight into general comparisons of the biomechanical response of the tissue. The assumption used for this analysis is that relationships between the compressive load histories can be applied to the more physiologically-relevant tensile loading history. This is to say that the compressive-tensile anisotropy is known to exist in vivo, however in order to make conclusions based on the presented nanoindentation data the underlying assumption is that the in vivo tensile response of the tissue is altered in a manner that is similar to our indentation data. The results presented above further emphasis that the formulation used to calculate the elastic modulus of the material based on nanoindentation data is crucial in determining accurate biomechanical response parameters. This is apparent when considering that all of the reduced elastic moduli calculations were statistically different from the corresponding JKR formulation of the elastic moduli. Interestingly, the longitudinal unloading stiffness was statistically greater than the circumferential or luminal responses. This could possibly be correlated to the physiologic in vivo response of the tissue, were the tissue is generally under circumferential stress rather than longitudinal or luminal stresses. It is hypothesized by our group that increased stiffness in the longitudinal direction in a 3D loading case is important in maintaining physiologic hemostasis and can be WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

66 Modelling in Medicine and Biology VIII directly correlated to collagen and elastin fiber alignment within the tissue. Future finite element and histological work by our lab will address this hypothesis. The only statistical difference from normal, healthy baseline data based on the JKR elastic modulus formulation was noted in the luminal direction. This correlates with previously published data examining only the luminal surface of vascular tissue [4, 5]. Surprisingly, no statistical changes where observed in the circumferential and longitudinal direction. Clinically, increases in intra-luminal pressure following DVT are associated with a reduced capability of the vessel to distend [14]. The DVT model used here has previously demonstrated strong histologic correlation to clinically recovered DVT veins [3]. The data presented within seems to indicate that even non-statistical changes in the circumferential and longitudinal response parameter coupled with statistical changes only in the luminal direction can have a large affect on the response of the tissue (and the entire venous system). In order to address these issues a more comprehensive battery of biomechanical tests are currently being performed by our lab, including bi-axial stretch experiments, strain energy function fitting, and finite element modeling.

References [1] Azuma T, Hasegawa M. Distensibility of the vein: from the architectural point of view. Biorheology 1973; 10: 469-79. [2] Anderson FA, Jr., Wheeler HB. Physician practices in the management of venous thromboembolism: a community-wide survey. J Vasc Surg 1992; 16: 707-14. [3] Humphries J, McGuinness CL, Smith A, Waltham M, Poston R, KG B. Monocyte chemotactic protein-1 (MCP-1) accelerates the organization and resolution of venous thrombi. J Vasc Surg 1999; 30: 894-9. [4] Ebenstein L, Pruitt L. Nanoindentation of soft hydrated materials for application to vascular tissues. J Biomed Mater Res A. 2004; 69: 222-232. [5] Ebenstein D. Biomechanical Characterization of Atherosclerotic Plaques: A Combined Nanoindentation and FTIR Approach. In Bioengineering: Dissertation, University of California, Berkeley; 2002. [6] Ebenstein D, Kuo A, Rodrigo J, Reddi A, Ries M, Pruitt L. A nanoindentation technique for functional evaluation of cartilage repair tissue. J. Mater. Res. 2003; 19: 273-281. [7] Gupta S, Fernando C, Cheng L, Pruitt L, Puttlitz C. Adhesive forces significantly affect elastic modulus determination of soft polymeric materials in nanoindentation. Materials Letters 2007; 61: 448-451. [8] Jacot J, Dianis S, Schnall J, Wong J. A simple microindentation technique for mapping the microscale compliance of soft hydrated materials and tissues. Journal of Biomedical Materials Research 2006: 485-494. [9] Oliver WC, Pharr GM. An Improved Technique for Determining Hardness and Elastic Modus Using Load and Displacement Sensing Indentation Experiments. Journal of Material Research 1992; 7: 1564-1584. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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[10] Sneddon IN. The relation between load and penetration in the axisymmetric boussinesq problem for a punch of arbitrary profile. Int J Engng Sci 1965; 3: 47-57. [11] McGuinness CL, Humphries J, Smith A, Burnand KG. A new model of venous thrombosis. Cardiovasc Surg 1997; 5. [12] Briscoe B, Fiori L, Pelillo E. Nano-indentation of polymeric surfaces. Journal of Physics D-Applied Physics 1998; 31: 2395-2405. [13] K.L. Johnson K. Kendall, A.D. Roberts. Surface energy and the contact of elastic solids. Proc R. Soc. Lond. 1971; A324: 310-313. [14] Dobrin PB. Mechanics of normal and diseased blood vessels. Ann Vasc Surg 1988; 2: 283-94.

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Section 2 Computational fluid dynamics

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Comparison of blood flow patterns in cerebral aneurysms K. Shimano, T. Kudo & Y. Enomoto Department of Mechanical Systems Engineering, Musashi Institute of Technology, Japan

Abstract Thrombogenesis is said to play an important role in the rupture of cerebral artery aneurysms and it was reported that the degree of platelet aggregation in an aneurysm had a significant correlation with the flow pattern in the aneurysmal dome. In this study, flows in three different models of cerebral saccular aneurysms at artery bifurcations were numerically investigated to compare flow patterns from a viewpoint of likelihood of platelet aggregation. It was shown that the relative size of the aneurysmal dome had a greater influence on the formation of a low-speed region responsible for active platelet aggregation than geometric features such as the aspect ratio of the aneurysm and the angle of the bifurcation. Keywords: cerebral saccular aneurysm, computational fluid dynamics, platelet aggregation, anterior communicating artery, artery bifurcation.

1

Introduction

Rupture of a cerebral artery aneurysm causes a life-threatening subarachnoid haemorrhage. Although not all aneurysms rupture, it is difficult to predict which aneurysms are likely to do so. Ujiie et al. [1] reported a significant correlation between the probability of rupture and aspect ratio of the aneurysmal dome depth to the neck width: almost 80% of ruptured aneurysms had aspect ratio larger than 1.6. According to a theory of Ujiie et al. [2], thrombogenesis is likely in an aneurysm of high aspect ratio due to the extremely slow blood flow in it. Ensuing fibrinolysis in which thrombi are dissolved is considered to cause damage to the endothelium. Takahashi et al. [3] showed evidence supporting the hypothesis: a canine blood flow in endothelialised model aneurysms made of glass was visualised and thrombogenesis was observed for a large aspect ratio. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090071

72 Modelling in Medicine and Biology VIII For further progress in the area of the rupture prediction, more understanding of the flow and the thrombogenesis in aneurysms is crucial. Approaches based on numerical techniques such as computational fluid dynamics (CFD) are promising for this purpose. In the previous study [4], the authors’ group applied an unsteady Navier-Stokes solver to the flow in one of model aneurysms tested by Takahashi et al. Furthermore, the authors proposed a platelet aggregation model which mathematically describes a series of physiological reactions of platelets to adenosine diphosphate (ADP). When the aggregation model was combined with CFD, the computational results showed that a lot of platelets were aggregated along the aneurysmal wall where the blood flow was considerably slow. It was also demonstrated that the flow pattern in the aneurysmal dome was closely correlated with the location of platelet aggregation. Therefore, it is important to classify blood flow patterns into some groups by investigation of a number of different aneurysms. Although a lot of CFD applications to blood flows in cerebral aneurysms have been reported (for example, References [5] and [6]), blood flow patterns were not discussed from a viewpoint of platelet aggregation. In this study, unsteady blood flow simulations with unstructured grids were conducted for three different model aneurysms in order to compare flow patterns in terms of likelihood of platelet aggregation.

2

Model aneurysms

Three aneurysms at artery bifurcations were modelled for the flow simulation because it is known that approximately 50% of cerebral aneurysms are found at bifurcations where anterior communicating arteries separate from anterior cerebral arteries. Each model had an aspect ratio larger than 1.6 so that aneurysms with potential risk of rupture could be investigated. Model 1 is shown in fig.1.The aneurysmal dome was located at a Y-shaped bifurcation and the whole geometry was symmetrical about the centre plane, which is, in fig.1, parallel to the surface of the page. Model 1 was based on one of the glass aneurysms made and experimentally tested by Takahashi et al [3]. Also Shimano et al. [4] used this glass model for application of the aggregation model. The geometric data was obtained manually from several digital images of the glass aneurysm cut in half because the original glass model was a handcraft and no CAD data was available. The depth of the aneurysmal dome and the neck width measured 6.5mm and 2.8mm, respectively, with 2.3 aspect ratio. The centre line of each tube lay on the same plane identical to the plane of symmetry. A thick arrow in fig.1 indicates the direction of the inlet flow. The diameter of the inlet tube was 2mm. A curve of the cyclically changing flow rate, which had been recorded during the experiment by Takahashi et al., was imposed as the inlet boundary conditions. The period of the cyclic change was 0.276sec. and the Womersley parameter was 2.39. The Reynolds number at the inlet ranged between 244 and 485. At the bifurcation, 40% of the blood proceeded to the anterior communicating artery and the rest continued flowing in the anterior cerebral artery. This ratio of the flow rates remained the same regardless of time. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Model 2 was a scale-up model originally designed by Shimamura [7] who had visualised the flow pattern in the dome. As shown in fig.2, the model had an aneurysmal dome at a T-shaped cerebral bifurcation where two arteries met at the right angle. The whole model geometry was symmetrical about the centre plane in the same manner as Model 1. The main pipe (anterior cerebral artery) and the branch (anterior communicating artery) were 5mm and 3mm in diameter, respectively. Anterior communicating artery

2.8 2

Anterior cerebral artery

Figure 1:

6.5

The geometry of Model 1 (all dimensions in mm).

In Model 2, the dome was an ellipsoid with the major and minor axes of 8 and 4 mm and inclined by 45 degrees. It is difficult to define the depth of the dome or the neck width because the dome was partly embedded into the arteries. However, the length of the major axis, 8mm, seems a good approximation to the dome depth and, similarly, the minor axis can be used to evaluate the neck width. Using these approximations, the aspect ratio was determined to be 2. As a sinusoidal curve had been used for the inlet flow rate in Shimamura’s experiment, the same curve was adopted as the inlet boundary conditions. The Womersley parameter, min. and max. Reynolds numbers were 2.4, 399 and 797, respectively, which represented a flow in a real cerebrovascular. Throughout the cycle, 26% of the incoming blood moved to the branch at the bifurcation. Influences of the bifurcation angle could be investigated by comparing Models 1 and 2. However, in these two models, the centre line of each pipe lay on the same plane and this is uncommon for real cerebral bifurcations. Therefore, the authors used another model with more realistic features. Model 3, shown in fig.3, consisted of 3-dimensionally curved pipes and an ellipsoidal aneurysm. The direction of protrusion of the dome was determined according to the most likely features of anterior communicating artery aneurysms mentioned in Reference [8]. In Model 3, the main pipe (anterior cerebral artery) shrank a little: the diameter measured 3mm near the inlet and 2.7mm after the bifurcation. On WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

74 Modelling in Medicine and Biology VIII the other hand, the branch (anterior communicating artery) was 1.8mm in diameter. The dome was an ellipsoid with the major and minor axes of 4 and 2 mm. The aspect ratio was determined to be 2 in the same manner as in Model 2. A flow rate curve similar to that used for Model 1 was imposed as the inlet boundary conditions. The Womersley parameter was 2.25. The maximum and minimum Reynolds numbers at the inlet were 366 and 728, respectively. 30% of the blood flowed into the anterior communicating artery.

3

Computational techniques

The unsteady incompressible Navier-Stokes equations were numerically solved. Blood was assumed to be Newtonian fluid and flows were treated as laminar. The cell-centred finite volume approach was used for spatial discretisation of the governing equations. Unstructured grid systems were employed so that the complicated geometry of each model could be properly expressed. The number of computational cells depended on the model: 886893 for Model 1, 1127077 for Model 2 and 1220816 for Model 3. Most computational cells were tetrahedra. Also hexahedra and trigonal prisms were allocated along the wall surfaces so that deterioration in accuracy in the boundary layers could be avoided. The SIMPLE algorithm was used for coupling of pressure and velocity. The Navier-Stokes solver was parallelised by domain decomposition and implemented on a PC cluster with sixteen Pentium-4 processors. 3

Anterior communicating artery 4 Anterior cerebral artery 8 5

Figure 2:

The geometry of Model 2 (all dimensions in mm).

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Anterior cerebral artery Anterior communicating artery (a) Top view Anterior cerebral artery

(b) Front view

Figure 3:

(c) Side view

Views of Model 3.

4 Results and discussions Figure 4 shows velocity vectors on the central cross-sectional plane in Model 1 at diastolic and systolic phases. It is clearly observed that the blood flow hit the edge of the aneurismal neck. Not only at the timings shown in fig.4, the blood flow was always impinging upon the neck at the anterior cerebral artery. This impingement is considered to cause haemolysis in which ADP is released from RBCs to the plasma. After the impingement upon the neck, some blood entered the dome with changing its direction and, again, hit the internal wall of the aneurysm. The spot of the second impingement depended on the time. At the higher flow rate (fig.4(b)), stronger inertia made the flow proceed deeper. After the second impingement, the flow spread 3-dimensionally with reducing its speed as clearly seen in fig.5. The shape of these curved path lines agreed with the vortex reported by Takahashi et al [3]. Consequently, a low-speed region of the blood was formed and occupied a large part in the dome (see the lower half of the dome in fig.4). Importantly, the blood motion in the dome after WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

76 Modelling in Medicine and Biology VIII the second impingement was so slow that there was a fairly long time for platelets to aggregate. According to Shimano et al. [4], aggregation occurred most actively in this region. The blood moving slowly in the dome changed its direction to the exit and returned to the arteries. The most blood proceeded to the anterior communicating artery after discharged from the dome. (a) Diastolic phase

(b) Systolic phase

Figure 4:

Velocity vectors on the central cross section in Model 1.

Figure 6 shows velocity vectors on the central plane in Model 2. The blood flow hit the right edge of the aneurismal neck at both timings. This feature is common to Models 1 and 2. However, in Model 2, the flow entered the dome along the aneurysmal wall and the second impingement, which was clearly seen in Model 1, was not present. The contrast between Models 1 and 2 can be highlighted if figs 5 and 7, in which particle paths in the domes are drawn, are compared. The fluid moving along the endothelium in Model 2 reached the depths of the dome, turned round and was discharged. This flow pattern in Model 2 agreed well with a video recorded in Shimamura’s experiment [7]. The flow speed in the dome was so high that aggregation seems less likely than in Model 1. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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2nd impingement

1st impingement

Figure 5:

Particle paths after the first impingement to the aneurysmal neck (Model 1 around a diastolic phase).

(a) Diastolic phase

Figure 6:

(b) Systolic phase

Velocity vectors on the central cross section in Model 2.

Calculated velocity vectors on two central planes of the aneurysmal dome in Model 3 are drawn in figs 8 and 9, of which directions of the view correspond to fig.1(a) and fig.1(b), respectively. In addition, the 3-dimensional structure of the flow in the aneurysmal dome is schematically illustrated in fig.10.

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78 Modelling in Medicine and Biology VIII A strong impingement of the mainstream upon the aneurysmal neck was clearly observed in fig.8. As similar impingements were observed in all the models regardless of the bifurcation angles, impingements of this kind and resulting haemolysis are presumably common to most aneurysms at artery bifurcations.

Figure 7:

Particle paths in Model 2 around a systolic phase.

(a) Diastolic phase

Figure 8:

(b) Systolic phase

Velocity vectors on the central plane of the aneurysm in Model 3 parallel to fig.3(a).

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Modelling in Medicine and Biology VIII

Anterior communicating artery

Anterior communicating artery

Anterior cerebral artery

Anterior cerebral artery (a) Diastolic phase Figure 9:

79

(b) Systolic phase

Velocity vectors on the cross-sectional plane parallel to fig.3(b) (Model 3).

Spot of the impingement

Mainflow to impinge upon the neck

Figure 10:

Schematic illustration of flow structure in the aneurysmal dome in Model 3.

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80 Modelling in Medicine and Biology VIII When Model 3 was used, the flow pattern in the aneurysmal dome was similar to that in Model 2 because, after the impingement upon the neck, the flow entered the dome along the endothelium and turned round. Unlike Model l, neither impingement upon the internal aneurysmal wall nor extremely low speed region was observed in Model 3. According to Figure 9, the fluid was mainly discharged from the dome through a lower part of the neck and moved towards the anterior cerebral artery. As a whole, the flow structure in Model 3 was similar to that in Model 2 although the flow in Model 3 was more 3-dimensional. The flow pattern in Model 1 was characterised by the second impingement within the aneurysm and the low-speed region. On the other hand, impingement upon the internal aneurysmal wall and an extremely low speed region were absent in Models 2 and 3. It can be concluded that platelet aggregation would have occurred less actively in Models 2 and 3 because ADP arising from the flow impingement upon the neck would have been swept out more quickly. The Y-shaped bifurcation or the angle of the aneurysmal dome does not seem responsible for the unique flow pattern in Model 1 because Models 2 and 3 showed similar flow features in spite of different angles of the bifurcation and dome protrusion. The aspect ratio (2.39 for Model 1 and approx. 2 for Models 2 and 3) might have had an influence on the different flow patterns to some extent. However, the larger volume of the aneurysmal dome in Model 1 seems a more reasonable explanation for the low speed region. The amount of fluid filling the dome was so large in Model 1 that the mainstream could not provide the fluid with enough energy to allow rapid motion in the dome. Therefore, the degree of platelet aggregation is presumably influenced not only by the aspect ratio but also the relative size of the aneurysmal dome to the artery.

5

Conclusions

Unsteady flow simulation was conducted for three different model aneurysms: Model 1 at a Y-shaped bifurcation, Model 2 at a T-shaped bifurcation and Model 3 which represented more realistic anatomical features of arteries. Impingements of the main flow upon the aneurysmal neck were observed in all the models. This means that haemolysis is likely to occur at the entries to most cerebral aneurysms. A large region filled with low speed fluid was found in Model 1 while fluid entering the domes was discharged quickly in Models 2 and 3. Therefore, it can be concluded that platelet aggregation would have been less active in Models 2 and 3. This difference in flow patterns in the aneurysms was presumably due to the relative size of the aneurysmal dome to the artery diameter. The authors suggest that, in addition to the aspect ratio, the relative size of the aneurysm should be taken into consideration when the likelihood of platelet aggregation and thrombogenesis and the probability of rupture are discussed.

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References [1] Ujiie, H., Tamano, Y., Sasaki, K. & Hori, T., Is the aspect ratio a reliable index for predicting the rupture of a saccular aneurysm? Neurosurgery, 48(3), pp.495-503, 2001. [2] Ujiie, H., Tachibana, H., Hiramatsu, O., Hazel, A.L., Matsumoto, T., Ogasawara, Y., Nakajima, H., Hori, T., Takakura, K. & Kajiya, F., Effects of size and shape (aspect ratio) on the hemodynamics of saccular aneurysms: a possible index for surgical treatment of intracranial aneurysms, Neurosurgery, 45(1), pp.110-130, 1999. [3] Takahashi, N., Ujiie, H., Yotoriyama, T., Suzuki, Y., Hori, T. & Kaibara, M., Flow visualization study of the endothelialized glass aneurysm model implanting canine carotid artery (in Japanese with English abs.), J. Jpn. Soc. Biorheol., 18(4), pp.143-148, 2004. [4] Shimano, K., Hayashi, T., Ujiie, H., Ono, T. & Enomoto, Y., Modelling of platelet aggregation in aneurysm, Proc. of 7th Int. Conf. On Modelling In Medicine and Biology, WIT Press, Southampton, pp.43-52, 2007. [5] Torii, R., Oshima, M., Kobayashi, T., Takagi, K. & Tezduyar, T.E., Influence of Wall Elasticity on Image-Based Blood Flow Simulations (in Japanese with English abs.), Transactions of Japan Society of Mechanical Engineers, Series A, 70(697), pp.1224-1231, 2004. [6] Funazaki, K., Higashi, M., Yamada, K., Taniguchi, H. & Tomura, N., FlowStructure Coupled Analysis of Cerebrovascular Artery with an Aneurysm of Realistic Geometry (in Japanese with English abs.), Transactions of Japan Society of Mechanical Engineers, Series B, 73(731), pp.1472-1479, 2007. [7] Shimamura, T., A study on flow in aneurysms (in Japanese), Graduation thesis, Tokyo Science University, Tokyo, 2006. [8] Satoh, A., Nakamura, H., Kobayashi, S., Miyata, A., Tokunaga, H., Wada, M. & Watanabe, Y., Surgical approaches and techniques for anterior communicating artery aneurysms: from angioanatomical point of view (in Japanese with English abs.), Surgery for Cerebral Stroke, 30, pp.240-246, 2002.

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Modelling of flow through the circle of Willis and cerebral vasculature I. D. Šutalo1,2, A. Bui1, S. Ahmed1, K. Liffman1,2 & R. Manasseh1 1

Materials Science and Engineering, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Australia 2 School of Public Health, Curtin University of Technology, Australia

Abstract The blood flow through the circle of Willis was modelled by coupling a Computational Fluid Dynamics (CFD) model of the circle of Willis with a branching tree model of the cerebral vasculature. The cerebral small vascular networks, which often cannot be accurately obtained by medical imaging, were modelled using a branching tree fractal model that accurately simulated the cerebral vasculature geometries and flow. This provided realistic mass flow boundary conditions for the outlet arteries of the circle of Willis. CFD was used to model the three-dimensional transient flow through a simplified and a patient-specific circle of Willis. The patient specific geometry was obtained directly from Computed Tomography (CT) images. A pipe network model was also used to predict the flow through the simplified circle of Willis and the predictions for the flow rates were within 4% of the CFD predictions. The coupled CFD and branching tree model provided useful insight into the variation of the flow through the circle of Willis. Keywords: circle of Willis, cerebral hemodynamics, computational fluid dynamics, cerebral circulation, carotid arteries, numerical models.

1

Introduction

Blood flows into the circle of Willis (CoW) system via the vertebral (VA) and internal carotid arteries (ICA) and flows out via the posterior (PCA), middle (MCA) and anterior (ACA) cerebral arteries. At the CoW periphery the outflowing blood enters a network of cerebral arterioles and capillaries. The resistance within the brain cerebral small vasculature is non-homogeneous and may change the flows through the ICA and VA. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090081

84 Modelling in Medicine and Biology VIII Most previous numerical studies on the CoW have investigated occlusions in the ICA or VA, or anatomical variations within the CoW. There have been several simplified and one-dimensional (1D) models of the blood flow through the CoW. Kufahl and Clark [1] developed a 1D finite difference model with elastic walls. Hillen et al [2] developed a 1D non-linear model. The peripheral resistances were determined from mass fluxes and anatomical structure. Linear models were also developed [3–5]. Other simplified and 1D models were compared to experimental data or observations [6–13]. Of these models, the Alastruey et al [10] model with compliant vessels compared anatomical variations and occlusions for common CoW geometries. Computational Fluid Dynamics (CFD) models of the CoW include a 2D nonlinear model with time varying resistances [14, 15]. There have been several CFD studies on 3D patient-specific models of the CoW that considered anatomical variations. Cebral et al [16] modelled the flow through 3D patientspecific models of the CoW. They compared flow resistances calculated from flow simulations with those obtained using a vascular bed model based on parallel resistors. Moore et al [17, 18] modelled three anatomical variations and showed the flow redistribution with these anatomical variations. Kim et al [19, 20] modelled three patient-specific geometries and found communicating arteries play an important role in the cerebral autoregulation mechanism. Alnaes et al [21] modelled three geometric variants of the posterior part of the CoW and showed that differences in vessel radii and bifurcation angle influence the wall shear stress. Most of these CFD models included autoregulation [14, 15, 17–20] and only Kim et al [19] and Kim [20] modelled flexible walls. In the CFD simulations of the CoW the vascular bed resistance models used for the outlet boundary conditions include: network of resistors [16, 19, 20], porous media as terminating blocks [14, 15, 17, 18], constant pressure [21] and calculating flow resistance of branching tree models [16]. Cebral et al [16] recommended using arterial models to specify outflow conditions and include arterial compliance. To describe the outflow boundary conditions in the CFD CoW modelling, previous work usually used network resistances or porous media models for the peripheral resistances rather than calculating them from a branching tree model of the cerebral vasculature. This study investigates the effect of changes in peripheral resistances of the CoW on the flow through the ICA and VA without occlusions. A coupled CFD and branching tree model was developed where a 3D CFD model of the CoW was coupled with a branching tree fractal model of the cerebral vasculature.

2

Methods

2.1 Branching tree model for cerebral vasculature Fractal branching tree models were created for all territories of cerebral vasculature, namely ACA, MCA, and PCA, using the Constrained Constructive Optimisation (CCO) method [22]. The fractal scaling parameter of the branching WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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tree models was chosen so that the generated fractal models are physiologically correct for the description of the human cerebral vasculature. With the systemic and venous pressures prescribed, the transient flow rates at the six outlets of the CoW were obtained using the pulsatile flow model developed for a fractal vascular network, which is described in more detail in a previous work [22]. Coupling of the CFD and fractal tree models was carried out by applying these flow rates as boundary conditions in the CFD simulations. The branching tree predicted peripheral resistance ratios for the ACA, MCA, and PCA for a normal person were: 6.906, 3.453, and 4.615, respectively. This fractal tree model more accurately simulates the cerebral vasculature geometry and peripheral resistances of the CoW than previous models, and includes wall compliance and viscosity change due to artery size [22]. 2.2 Pipe network model A commercial pipe network code Design Flow Solutions (ABZ Incorporated, Chantilly, Virginia USA) was used to investigate the steady-state flow through the CoW. The dimensions for the simplified complete CoW were taken from Hillen et al [2]. The network of arterioles and capillaries produces a pressure drop in the cerebral arterial system from an average inlet pressure of around 100 mmHg to pressures approaching venous pressures of around 10 mmHg [2]. The outlet boundary conditions at the periphery of the CoW for the CFD model were obtained from the branching tree model. 2.3 Computational fluid dynamics model The CFD model simulated the 3D transient incompressible laminar pulsatile flow fields through the CoW. The numerical modelling was performed using the commercial CFD package ANSYS CFX-11, which has a coupled solver and uses an unstructured mesh based on the finite element method. The inlet boundary condition was set by specifying a pressure pulse ranging 80–125 mmHg at the inlet for a period of 0.7 s. The outlet boundary conditions were pulsatile flow rate obtained from branching tree model. The density of the blood was assumed to be 1050 kg/m3, and the blood flow was assumed to be Newtonian. The mesh consisted of tetrahedral elements, and the solutions were mesh-independent when the total number of elements was 267,440 for the simplified geometry and 74,250 for the patient-specific geometry. The dimensions of simplified CoW geometry were the same as used in the pipe network model. The average Reynolds number based on the ICA diameter for the simplified and patientspecific CoW geometries were about 400. The patient-specific geometry data was acquired from CT scans and converted to surface (STL) and then finite element models using Slicer-3D, meshlab and ICEMCFD. The patient-specific geometry was more complex than the symmetrical simplified CoW geometry. This patient-specific geometry was missing the left posterior communicating artery (PCoA) and had a penal anterior communicating artery 1 (ACA1). This anatomical configuration with missing PCoA occurs in 9% of the population [10]. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

86 Modelling in Medicine and Biology VIII

3

Results and discussion

3.1 CFD predictions of carotid and vertebral artery flow rates for the simplified CoW geometry Modelling runs with the coupled CFD and vascular tree fractal model investigated the effect on the VA and ICA flow rates of reducing by 10% the diameters of the peripheral vasculature of the ACA, MCA and PCA, respectively. For a normal person with a complete CoW, under the peripheral resistance conditions stated earlier, the CFD model predicted that 69% of the blood flow enters the simplified CoW geometry via the ICA and 31% enters via the VA. When the diameters of the ACA peripheral vasculature was reduced by 10% then 66% of the blood flow entered the CoW from the ICA, and 34% entered via the VA. A 10% decrease in the diameters of the MCA peripheral vasculature reduced the flow through the ICA further so that 64% of the blood flow entered the CoW from the ICA and 36% entered from the VA. However, a 10% decrease in the diameters of the PCA peripheral vasculature decreased the flow through the VA so that 76% of the flow entering the CoW entered via the ICA, and 25% entered via the VA. A summary of results showing the ICA and VA flow rates for the different peripheral resistances are given in table 1. Table 1:

Percentage blood flow rates through the ICA and VA as obtained from the CFD simulations for the simplified and patient-specific CoW geometries compared to the pipe-flow code simulations.

Resistance Ratio (ACA: MCA: PCA)

% ICA

%VA

pipe

CFD

CFD

pipe

CFD

CFD

simple

simple

patient specific

simple

simple

patient specific

normal 6.906:3.453:4.615

70

68.6

74.4

30

31.4

25.5

increased ACA resistance

68

66.3

72.3

32

33.7

27.7

66

63.7

69.7

34

36.3

30.3

77

75.5

81.0

23

24.5

19.0

10.526:3.453:4.615 increased MCA resistance

6.906:5.263:4.615 increased PCA resistance

6.906:3.453:7.034

Fig. 1 shows the pressure distribution at peak systole in the simplified CoW geometry where the pressure is highest at the VA and ICA inlets and then decreases along the CoW. Fig. 2 shows that there is negligible change in VA flow rate magnitude for a 10% decrease in diameters of the peripheral vasculature of the ACA and MCA, most of the decrease occurs in the ICA flow rate. While for a 10% decrease in the diameters of the PCA vasculature most of the flow rate decrease occurs in the VA. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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RMCA RACA LACA

RPCA

LMCA BA LPCA

LICA

RICA RVA

LVA Figure 1:

Instantaneous pressure on walls of simplified CoW at peak systole. The two VA merge into the basilar artery (BA).

Figure 2:

CFD predictions of inlet flow rates in the simplified (left) and patient-specific (right) CoW.

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88 Modelling in Medicine and Biology VIII 3.2 Comparisons between CFD and pipe network model predictions The pipe network model predictions for the inlet flow rates to the CoW were within 4% of the CFD predictions with the simplified CoW geometry as shown in table 1. A major difference between the two models was the run time: 3 hours for the CFD code compared to a few seconds for the pipe flow model. However, the pipe network model results were for steady-state flow, while the CFD results were for pulsatile flow and showed the whole pulsatile flow rate profile. 3.3 Patient-specific CoW geometry model predictions of carotid and vertebral artery flow rates The patient-specific CoW geometry simulations show significant changes of the VA and ICA flow rates when compared to the symmetric simplified CoW geometry. The patient-specific CoW geometry percentages of the flows in VA and ICA became 26% and 74%, respectively. Patient-specific CoW had a missing PCoA and different dimensions compared to the simplified case. Table 1 and fig. 2 show that when the peripheral resistances were reduced then the same trend in the VA and ICA flow rates movements was observed for the patientspecific case as for the simplified CoW geometry. In fig. 3 the right ACA had a lower pressure due to the higher pressure drop in the remaining PCoA. The missing PCoA resulted in higher velocity in ACoA and ACA1 to deliver blood to the ACA2 (fig. 4). This result supports the predictions by Kim [20], which included an auto-regulation mechanism for a patient-specific CoW with a missing PCoA. 3.4 Current models and future model improvements We used a number of different strategies in an effort to accurately simulate cerebral vascular flow. CFD can compute flow through patient-specific vascular geometries, but is a computationally intensive method that has limitations in terms of memory and processing speed. For the pipe network model the approach is to neglect much of the patient-specific spatial structure of the cerebral vascular system and to treat sections of the system as a network of pipes, where only the lengths and diameters of the pipes are derived from a particular patient. The advantage of using a pipe network is that for simple CoW geometries it may give nearly the same result as CFD, but with a much faster processing time. The difference in VA and ICA flow rate predictions between the patientspecific and simplified CoW geometry highlighted the importance of using patient-specific geometry. In the future we plan to simulate further patientspecific geometries for the main CoW anatomical variations, to comprehensively investigate the effect of changes in the peripheral resistances of the CoW on the flow through the ICA and VA. In our modelling we took into consideration the compliance of the blood vessel walls in the branching tree fractal model (which acts as an autoregulation mechanism), but did not include the active feedback control loop. Autoregulation WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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ACA2 MCA

ACA1 MCA ICA

ICA

Missing PCoA PCA BA

PCA

Figure 3:

Instantaneous pressure distribution on the wall in the patientspecific CoW at peak systole. The two VA merge into the basilar artery (BA).

ACA2 MCA

ACA1 MCA

ICA

ICA

Missing PCoA PCA

PCA Figure 4:

BA

Instantaneous blood velocity in the patient-specific CoW at peak systole.

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90 Modelling in Medicine and Biology VIII has been better modelled in previous CFD studies of CoW [14, 15, 17, 18, 19, 20] and we plan to incorporate such a method in our model. Currently, our model has only one-way coupling between 3D CFD and the vascular tree fractal models. We are in the process of implementing a two-way coupling of the models that can more accurately predict the pressure variation within the CoW. The cerebral small vascular networks that often cannot be accurately obtained by medical imaging was modelled using a branching tree fractal model that accurately simulated the cerebral vasculature geometries and flow. This provided realistic pulsatile flow rate for the outlet arteries of the CoW. This fractal tree model also includes wall compliance and viscosity change due to artery size. Cebral et al [16] showed that blood simulated using a Newtonian model and a Casson non-Newtonian model in the CoW compared quite well, although the Newtonian model had lower wall shear stresses. In future we also plan to include non-Newtonian blood properties in the CFD model.

4

Conclusion

A coupled CFD model of the CoW and branching tree fractal model of the cerebral vasculature was used to investigate the effect of changes in the peripheral resistances of the CoW on the flow through the ICA and VA without occlusions. For the simplified CoW geometry the coupled model showed that a 10% decrease in diameters of the PCA peripheral vasculature predominately decreases the VA flow rate, while a 10% decrease in diameters of the ACA and MCA periphery vasculature predominately decreases the ICA flow rate. A simple pipe network model predicted flows through the simplified CoW geometry were within 4% of the CFD predictions, but took much less time to compute. Thus, the simple network model can be useful tool to obtain general information about the flows in simple CoW geometries. The coupled numerical model was also used to model the three-dimensional transient flow through a patient-specific CoW. It showed the same trends for changes in the peripheral resistances of the CoW on the flow through the ICA and VA, but the flow rate magnitudes differed due to the more complex geometry. In future we will include: • Two-way coupling between the 3D CFD and branching tree fractal models. • An improved autoregulation model that includes an active feedback control loop. • More patient-specific CoW geometries to simulate the main CoW anatomical variations when the CoW peripheral vascular resistances are changed.

References [1] Kufahl, R.H. & Clark, M.E., A circle of Willis simulation using distensible vessels and pulsatile flow. Journal of Biomechanical EngineeringTransactions of the ASME, 107(2), pp. 112-122, 1985. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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[2] Hillen, B., Hoogstraten, H.W. & Post, L. A mathematical model of the flow in the circle of Willis. Journal of Biomechanics, 19(3), pp. 187-194, 1986. [3] Hillen, B., Drinkenburg, B.A.H., Hoogstraten, H.W. & Post, L., Analysis of flow and vascular-resistance in a model of the circle of Willis. Journal of Biomechanics, 21(10), pp. 807-814, 1988. [4] Cassot, F., Zagzoule, M. & Marc-Vergnes, J.P., Hemodynamic role of the circle of Willis in stenoses of internal carotid arteries. An analytical solution of a linear model. Journal of Biomechanics; 33(4), 395-405, 2000. [5] Moorhead, K.T., Doran, C.V., Chase, J.G. & David, T., Lumped parameter and feedback control models of the auto-regulatory response in the circle of Willis, Comput Methods Biome, 7(3), pp. 121-130, 2004. [6] Dickey, P.S., Kailasnath, P., Bloomgarden, G., Goodrich, I. & Chaloupka, J., Computer modeling of cerebral blood flow following internal carotid artery occlusion. Neurological Research, 18(3), pp. 259-266, 1996. [7] Kailasnath, P., Dickey, P.S., Gahbauer, H., Nunes, J., Beckman, C. & Chaloupka, J.C., Intracarotid pressure measurements in the evaluation of a computer model of the cerebral circulation. Surgical Neurology, 50(3), pp. 257-263, 1998. [8] Cieslicki, K. & Ciesla, D., Investigations of flow and pressure distributions in physical model of the circle of Willis. Journal of Biomechanics, 38(11), pp. 2302-2310. 2005. [9] Roessler, F.C., Reith, W. & Siegel, G., Simulation of cerebral hemodynamics for preoperative risk assessment. Brain Research, 1118, pp. 183-191, 2006. [10] Alastruey, J., Parker, K.H., Peiro, J., Byrd, S.M. & Sherwin, S.J., Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. Journal of Biomechanics, 40(8), pp. 1794-1805, 2007. [11] Matthys, K.S., Alastruey, J., Peiro, .J, Khir, A.W., Segers, P., Verdonck, P.R., Parker, K.H. & Sherwin, S.J., Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. Journal of Biomechanics, 40(15), pp. 3476-3486, 2007. [12] Ursino, M., Lodi, C.A. & Russo, G., Cerebral hemodynamic response to CO2 tests in patients with internal carotid artery occlusion: Modeling study and in vivo validation. J Vasc Res, 37(2), pp. 123-133, 2000. [13] Viedma, A., Jimenez-Ortiz, C. & Marco, V., Extended Willis circle model to explain clinical observations in periorbital arterial flow. Journal of Biomechanics, 30(3), 265-272, 1997. [14] Fernandez, A., David, T. & Brown, M.D., Numerical models of autoregulation and blood flow in the cerebral circulation. Comp Meth Biomech En, 5(1), pp. 7-19, 2002; [15] David, T., Brown, M. & Ferrandez, A., Auto-regulation and blood flow in the cerebral circulation. Int. J. Numer. Meth. Fluids, 43, 701-173, 2003.

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92 Modelling in Medicine and Biology VIII [16] Cebral, J.R., Castro, M.A., Soto, O., Lohner, R. & Alperin, N., Blood-flow models of the circle of Willis from magnetic resonance data. Journal of Engineering Mathematics, 47(3-4), pp. 369-386, 2003. [17] Moore, S., David, T., Chase, J.G., Arnold, J. & Fink, J., 3D models of blood flow in the cerebral vasculature. Journal of Biomechanics, 39(8), 1454-1463, 2006. [18] Moore, S.M., Moorhead, K.T., Chase, J.G., David, T., & Fink, J., Onedimensional and three-dimensional models of cerebrovascular flow. Journal of Biomechanical Engineering-Transactions of the ASME, 127(3), pp. 440-449, 2005. [19] Kim, C.S., Kiris, C., Kwak, D. & David, T., Numerical simulation of local blood flow in the carotid and cerebral arteries under altered gravity. Journal of Biomechanical Engineering-Transactions of the ASME, 128(2), pp. 194202, 2006. [20] Kim, C.S., Numerical simulation of auto-regulation and collateral circulation in the human brain. Journal of Mechanical Science and Technology, 21(3), pp. 525-535, 2007. [21] Alnaes, M.S., Isaksen, J., Mardal, K.A., Romner, B., Morgan, M.K. & Ingebrigtsen, T., Computation of hemodynamics in the circle of Willis. Stroke 38(9), pp. 2500-2505, 2007. [22] Bui, A., Šutalo, I.D., Manasseh, R. & Liffman, K, Dynamics of pulsatile flow in fractal models of vascular branching networks. Med. & Biol. Eng. & Comp. 2009.

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Computational hemodynamics analysis in realistic 3D geometries of the human coronary atherosclerosis S. I. Bernad1, T. Bărbat1, E. Bernad2 & R. Susan-Resiga3 1

Romanian Academy – Timisoara Branch, Timisoara, Romania University of Medicine and Pharmacy Timisoara, Timisoara, Romania 3 “Politehnica” University of Timisoara, Timisoara, Romania 2

Abstract The present investigation concentrates on one particular problem related to fluid mechanics: namely, the description of disturbed flow fields in the coronary atherosclerosis. Atherosclerosis creating a constriction can significantly alter the local blood flow dynamics. From a biological aspect, the changes that take place in the flow have a profound effect on the structure and function of the arterial wall and the development of the disease. The purpose of this paper was to non-invasively assess hemodynamic parameters, such as mass flow, wall shear stress, and wall pressure, with computational fluid dynamics (CFD) in coronary arteriosclerosis using patient-specific data from computed tomography (CT) angiography. Keywords: blood flow, coronary artery, hemodynamics, numerical simulation.

1

Introduction

Statistics show that atherosclerosis is the leading cause of death in European countries. The vessels most commonly affected are the abdominal aorta, femoral, carotid, and coronary arteries. A number of reports indicate that arterial stenosis has a significant effect on the character of blood flow in blood vessels. For example, stenosis induces formation of local vortices, increases energy losses, changes the pressure on the blood vessel walls, and modifies flow parameters in the branching distal portions of the vessels [4, 7–9]. Blood flowing through a vascular segment exerts a WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090091

94 Modelling in Medicine and Biology VIII tangentially directed shear stress on the luminal surface of endothelial cells, the inner layer of the vascular wall. Wall shear stress is the product of wall shear rate and local blood viscosity. Wall shear stress has been shown to be an important determinant of endothelial cell function [3, 4, 6]. Atherosclerosis involves the development of plaque in the inner lining of large and medium sized arteries. Eventually the plaque matures into a structure consisting of two main parts: a soft “inner core” that consists of cholesterol, waste products, inflammatory cells and calcium, and a thin outer shell called the “fibrous cap” [6]. As plaque builds up, it can begin to block the flow of blood through an artery. Hemodynamic flow assessment in coronary arteries is usually performed with intravascular Doppler ultrasound by measuring local velocities [1, 5]. Even if these data enable a hemodynamic characterization of stenosis severity, the introduction of an ultrasound-catheter into the lumen is an invasive procedure, leads to flow disturbances, and the results of such measurements are, therefore, often difficult to interpret [5]. An alternative means for invasive flow measurements is presented by the calculation of models in which blood flow can be virtually simulated, a method that is called computational fluid dynamics (CFD). In fact, several in vitro studies [3, 6] and some in vivo investigations [11] have shown that CFD allows reliable physiologic blood flow simulation and measurements of WSS, wall pressure, and mass flow. A requisite for obtaining reliable results from coronary CFD is to use exact anatomical models [7, 9], which today are provided by multi-detector row computed tomography (CT). The three-dimensional reconstruction of coronary arteries from CT datasets has been shown to be more accurate than corresponding 3D reconstructions obtained from conventional angiography combining two simultaneously captured twodimensional views [1, 2, 6]. The purpose of this study was to simulate pulsatile blood flow in coronary arteries using CFD based on geometric models from CT datasets and to measure the WSS, wall pressure, and mass flow and visualize flow patterns.

2

Materials and method

For the case presented in this paper, spiral CT (computed tomography) was performed 4 days following the CA (coronary angiography), (44 year old, patient by typical angina symptoms is investigated). A Somatom Sensation 64 Scanner (Siemens Medical Systems, Erlangen, Germany) was used in non-enhanced spiral scan technique with a slice thickness of 2 mm, a table feed of 3 mm/s, and an increment of 2 mm. Data corresponding to the investigated patient is presented in Table 1. The Vascular Model: the coronary artery model simulating the flow field in the RCA (right coronary artery) is illustrated in fig. 1 and fig. 2 respectively. The RCA is modelled to be of length 52 mm with variable diameters, depending of the stenosis severity. Model assumptions, data input, and boundary conditions: the blood is assumed to be incompressible, with a Newtonian behaviour having dynamic viscosity (µ) of 0.00408 Pa and a density (ρ) of 1050 kg/m3. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Table 1:

95

Preoperative patient characteristics.

Variables Value Mean age 44 Gender (M/F) male History of MI NO Previous PTCA NO Renal insufficiency NO Cardiovascular risk factor Hypertension Yes Diabetes NO Smoke Yes Obesity moderate Angiographic data RCA stenosis multiple/severe LCX stenosis intermediate MI = myocardial infarction, PTCA = percutaneous transluminal coronary angioplasty. RCA = right coronary artery, LCX = left circumflex artery.

Figure 1:

Invasive coronary angiography (CA). Multiple stenosis of right coronary artery (arrows).

Figure 2:

Axial tomographic images of heart after intravenous injection of contrast agent (64slice CT).

The blood vessel walls are assumed to be rigid and impermeable. The distributions of velocity and wall shear stress are obtained by computationally solving the Navier-Stokes equations. 2.1 Flow parameters To complement the geometric factors we must also consider the physiological flow parameters that describe our problem. A wide variety of pulsatile flow WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

96 Modelling in Medicine and Biology VIII waveforms are observed in arteries, these vary with position in the arterial system, in response to exertion, and from individual to individual. We assume that u (t ) is described by the mean and two harmonic terms in the form [10]:

  2πt   4πt  u (t ) = u m 1 + a1 sin   + a 2 cos   T   T  

(1)

The pulsatile flow waveforms presented in eqn. (1), is more realistic approximation of physiologic waveforms since it has a higher peak-to-mean ratio, is obtained using two harmonics and with a1 = −a2 =0.75. We note that if the magnitude of the two harmonics is fixed at 0.75, fig.3.

Figure 3:

Waveforms of u (t ) in the pulsatile flow cases considered, eqn. (1).

Fluid dynamics simulation setup: the fluid dynamics simulations are performed by using a control-volume-based technique, implemented in the computational fluid dynamics code, FLUENT [11]. The computation procedure of the commercial code consists of (i) construction of the geometry using a preprocessor, GAMBIT [11], (ii) meshing the computation domain, (iii) assigning boundary conditions in terms of velocities and flow-rate weightings, (iv) assigning fluid properties, and (v) the solution algorithm. The geometry of the stenosed coronary artery is constructed using the dimensions provided by the CT measurements. In figures 4c the reconstructed coronary artery is shown to have a circular cross-section, larger distal to the stenosed area. In order to carry out the mesh sensitivity analysis, numerical simulations were carried out by varying the number of mesh elements in the computational domain. The accuracy of the simulation results was then improved by employing a finer mesh that contained 1598752 elements.

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97

Results and discussion

In order to generate the geometry for the present study, a series of stenosed right coronary artery (RCA) slices were acquired in vivo using CT scan imaging. A profile of the reconstructed RCA is shown in fig. 4A. A very important aspect of the analysis of three-dimensional flows in the RCA is the graphical presentation of the flow field. The vortical type velocity field has been induced by the curved entrance geometry, and it is not a result of the inlet condition, fig. 5. RCA curvature gradually changes the flow in sections d0, d1, d2, d3. Far downstream of the section d3, the flow is highly three-dimensional with little resemblance to classical steady flow in a pipe, being sharply skewed toward the outer wall of the RCA.

Figure 4:

Axial tomographic images of the RCA (right coronary artery), 64slice CT (A); multiple stenosis of right coronary artery (arrows), (B); 3-D geometry reconstruction (C).

The natural curvature of the RCA is expected to induce significant secondary flow motion or crossflow motion within the artery, and the presence of the successive curvature has an impact on the structure of the induced secondary flows. Figure 5 depicts the distribution of velocity-vectors in the flow field for the different time step. In section d0, the inner wall of the RCA curvature creates a recirculation region. In constricted regions of the RCA between the sections d1 and d6 a strong region of recirculation is observed near to the inner wall, which forces the flow to move distal in the bypass graft, figures 5 and 7. The average velocity and pressure distribution in the different sections of the bypass graft is presented in Table 2. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

98 Modelling in Medicine and Biology VIII Table 2:

Hemodynamic data in different sections of the venous bypass graft.

Section

Time T1 *

**

0.15 0.69 0.66 0.15 0.32 0.12 0.76 0.68 0.21 0.84

28.2 25.4 23.8 24.4 24.2 24.4 21.1 20.2 21.1 17.5

VT 1

D0 D1 D2 D3 D4 D5 D6 D7 D8 D9

PT 1

Time T2 *

VT 2

0.0096 0.0447 0.0429 0.0125 0.0209 0.0111 0.0496 0.0441 0.0147 0.0547

**

PT 2

1.40 1.36 1.29 1.28 1.27 1.26 1.22 1.19 1.17 1.12

Time T3 *

VT 3

0.06 0.277 0.266 0.055 0.128 0.049 0.307 0.273 0.069 0.330

**

PT 3

8.78 8.20 7.75 7.85 7.79 7.80 7.13 6.86 7.05 6.26

Time T4 *

VT 4

**

0.0096 0.0447 0.0429 0.0140 0.0209 0.0083 0.0496 0.0441 0.0151 0.0547

PT 4

0.88 0.85 0.82 0.81 0.82 0.82 0.82 0.79 0.77 0.77

*Area Weighted Average Velocity Magnitude [m/s] for time steps T1, T2, T3, T4. **Area Weighted Average Pressure [mmHg] for time steps T1, T2, T3, T4.

For the patient of this study, the wall pressure decreased towards the periphery of the coronary artery tree with elevated pressure drops in stenotic segments. The increased pressure drop in stenoses reflects the elevated energy needed to drive the flow through these regions. As shown in atherosclerotic coronary arteries, regions of flow acceleration were associated with high WSS. Relating our CFD results to coronary artery morphology, the location of atherosclerotic plaques correlated well with the regions of high WSS. The flow pattern variations give us insight into the wall shear distribution. The high velocity gradients in the anastomosis give rise to large spatial variations in the resulting wall shear stress. In the spacing between successive constrictions (sections d1 and d6) the flow repeatedly undergoes an expansion after each constriction where separation zone formed distal to the section d2, d4, and d6, and re-attaches on the wall between the two constrictions. This separation zone subsequently covers the whole space between the two constrictions and leads to vortex shedding. Secondary flows subtract energy from the forward axial flow motion. Thus their location and extent can affect severely the fluid dynamic performances in stenosed artery. Wall shear stress and pressure are likely the most relevant parameters from the standpoint of the fluid mechanical involvement in the development of atherosclerosis. The wall shear stress distribution patterns shown in fig. 6 suggest that the maximum shear stresses occur in the vicinity of the stenosed regions. Parallel to the changes of WSS, the flow pattern was more variable and inconsistent in vicinity of the constriction. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 5:

(a)

(b)

(c)

(d)

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Velocity field and vector computed in the stenosed RCA. Parabolic profiles of the velocity vectors are seen inside the RCA. Maximum flow velocity approaching the artery curvature in sections d6 (maximum constriction) is around 1 m/s. The flow exiting the RCA curvature in section d2 with a higher velocity results in a stronger impingement on the wall of the artery. The maximum flow velocity magnitude (1 m/s in section d6 and 0.8 m/s in section d2), seen close to the exterior wall in section d2, and d6.

Artery section constriction and post-constriction dilatations have led to acceleration and rapid deceleration, respectively, including a distortion of flow. Large recirculation regions found in the vicinity of the each constricted section. An effective way to investigate these highly three-dimensional fluid motions is to numerically inject a passive or non-interactive tracer into the flow, and we have carried out such an investigation in the present work. We have introduced a tracer, which serves as a visualization tool to analyze the flow characteristics. Flow separation occurs on the inner walls. However, the separation zone on the inner wall downstream curvature of the section d2 is larger that separation zone downstream of the section d4, fig. 7. Prominent vortex shedding with 3D features is seen at the end of the flow deceleration zone in the spacing between the constrictions. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

100 Modelling in Medicine and Biology VIII

Figure 6:

(a)

(b)

(c)

(d)

Wall shear stress distribution on the RCA wall for different time steps (fig. 3). There are many regions of elevated wall shear stress correlating either to artery curvature or to irregular vessel geometry (stenosed regions).

Figure 7 present the tracer distribution inside to the stenosed right coronary artery for different time steps. The associated velocity profiles are shown in fig. 5. However, in the low-velocity regions, i.e., distal to the sections d2, d4, and d6, the relatively long residence times for particles help them to establish a relatively large mass-transfer zone. The role of secondary flows in particle transport is not as pronounced as it is for micro particles. Specifically, in some cases micro particles can be swept out of the strong secondary vortices due to their inertia, and preferential ring areas of high concentrations are generated outside the secondary vortices.

4

Conclusions

The problem of flow disturbances in the arterial system is generally complex from the fluid dynamics point of view. For physicians, three-dimensional WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 7:

(a)

(b)

(c)

(d)

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Particles motion inside the stenosed coronary artery (RCA).

modelling becomes particularly helpful when dealing with complex vascular geometries as seen with large stenosis. Highly accurate anatomy for the generation of geometric models is a principal requirement to perform reliable flow simulations and to make assumptions about mass flow, WSS, and wall pressure. In summary, this paper describes a method for the construction of a flow model based on two-dimensional in vivo images acquired noninvasively. In order to create a real physical artery model, the method uses commercially available software, both to create an anatomy model and reproduce the blood hemodynamics. In conclusion, our study demonstrates that the simulation of pulsatile blood flow is feasible in-vivo in coronary arteries of patients with geometric data obtained from multi-detector row CT.

Acknowledgements The present research has been supported by the Romanian National Authority for Scientific Research through the CNCSIS 798/2008 project. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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References [1] Achenbach S., Current and future status on cardiac computed tomography imaging for diagnosis and risk stratification, Journal of Nuclear Cardiology, 12(6), pp: 703-713, 2005. [2] Bernad S.I., Barbat T., Bernad E.S., Susan-Resiga R., Cardio vascular surgery – simulation based medical intervention, Proceedings of the 9th WSEAS Int. Conf. on MATHEMATICS & COMPUTERS IN BIOLOGY & CHEMISTRY (MCBC '08), eds. L Vladareanu, V. Chiroiu, P. Bratu, I. Magheti, WSEAS Press, pp: 100-106, 2008. [3] Gardhagen R., Renner J., Lanne T., Karlsson M., Subject Specific Wall Shear Stress in the Human Thoracic Aorta, WSEAS TRANSACTIONS on BIOLOGY and BIOMEDICINE, 10(3), pp: 609-614, 2006. [4] Hornero F., Cervera V., Estornell J., Rodirguez I., Buendia J.A., Esteban J.M., Montero J.A., Virtual Vascular Endoscopy for Acute Aortic Dissection, Ann Thorac Surg, 80, pp: 708-710, 2005. [5] Ruengsakulrach P., Joshi A.K., Fremes S., Foster S., Wall Shear Stress and Atherosclerosis: Numerical Blood Flow Simulations in the Mouse Aortic Arch, WSEAS TRANSACTIONS on FLUID MECHANICS, 2(3), pp: 90-100, 2008. [6] Marsahall I., Zhao S., Papathanasopoulou P., Hoskins P., Yun Xu X., MRI and CFD studies of pulsatile flow in healthy and stenosed carotid bifurcation models, Journal of Biomechanics, 37, pp: 679-687, 2004. [7] Rathish Kumar B.V., Yamaghuchi T., Liu H., Himeno R., A numerical study of an unsteady laminar flow in a doubly constricted 3D vessel, Int. J. Numer. Meth. Fluids, 38, pp: 1159-1176, 2002. [8] Saa A.A., An experimental investigation of pulsatile flow through a smooth constriction, Experimental Thermal and Fluid Science, 17, pp: 309-318, 1998. [9] Shahcheraghi N., Dwyer H.A., Cheer A.Y., Barakat A.I., Rutaganira T., Unsteady and Tree-Dimensional Simulation of Blood Flow in the Human Aortic Arch, Journal of Biomechanical Engineering, 124, pp: 378-387, 2002. [10] Sherwin S.J., Blackburn H.M., Three-dimensional instabilities and transition of steady and pulsatile axisymmetric stenotic flows, J. Fluid Mech., 533, pp: 297-327. [11] FLUENT 6.3 User’s Guide, Fluent Incorporated, 2006.

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Numerical investigation of the flow field in the upper human airways G. Eitel, W. Schr¨oder & M. Meinke Institute of Aerodynamics, RWTH Aachen University, W¨ullnerstrasse 5a, 52062 Aachen, Germany

Abstract The flow in a realistic model of the human lung is numerically simulated at steady and unsteady inspiration and expiration. A model of a human lung ranging from the trachea down to the sixth generation of the bronchial tree is used for the simulation. The numerical analysis is based on the Lattice-Boltzmann method, which is particularly suited for flows in extremely intricate geometries such as the upper human airways. The results for steady air flow at inspiration and expiration for a diameter based Reynolds number of ReD = 1250 evidence secondary vortex structures and air exchange mechanisms. It is shown that the asymmetric geometry of the human lung plays a significant role for the development of the flow field in the respiratory system. Secondary vortex structures observed in former studies are reproduced and described in detail. The solutions for unsteady respiration allow a detailed analysis of the temporal formation of secondary flow structures whereas the time dependence is much more pronounced at inspiration than at expiration. Keywords: computational fluid dynamics, Lattice-Boltzmann method, respiratory system, human lung, unsteady flow.

1 Introduction The human lung is a complex respiratory system consisting of a repeatedly bifurcating network of tubes with progressively decreasing diameters. The understanding of the flow processes in the upper human airways is of great importance in developing aerosol drug delivery systems and to improve the efficiency and usability of artificial respiration. Numerous experimental and numerical investigations of WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090101

104 Modelling in Medicine and Biology VIII lung flow have been conducted so far [1–5]. However, due to the high geometric intricacy of the human lung, there is still a considerable amount of uncertainty concerning the very complex flow field and results for realistic lung models are still rare. Most of the investigations are based on simplified models of the lung structure. The most popular of these models is the so-called Weibel model [6], which describes the bronchi as a symmetric tree structure of subsequent bifurcating tubes with 23 generations. However, in most of the studies only the first three to five generations are considered and a planar representation is often favored for simplicity. In [7], detailed experimental studies were performed for a planar Weibel model where fundamental flow phenomena, such as m-shaped velocity profiles and counter-rotating vortices, were described. It was shown in numerical studies [1,8], however, that the flow field for the non-planar configuration differs significantly from the planar case. Studies considering asymmetric bifurcations [9], non-smooth surfaces [10] and CT-based models [2, 5, 11] show that the inspiratory flow in the upper human airways is asymmetric and swirling and the results emphasize the importance of realistic airway models. Generally, the aforementioned results show that an accurate lung geometry, i.e., CT data or a real human lung cast, is required to obtain physically relevant results. However, when numerical simulations are considered the accuracy does not only depend on the geometry, but also on the quality of the numerical method, the computational resolution, and the boundary conditions. Therefore, it is necessary to validate the numerical solutions either by experimental results or by a detailed comparison with existing numerical data from the literature or by proving the quality of the method through an analysis of a similar, well established flow problem. The present work focuses on the detailed investigation of the three-dimensional flow in a realistic model of the human based on an actual lung cast. The geometry covers the trachea and the bronchial tree down to the sixth generation. A silicon model of the same geometry has been experimentally investigated in [3]. The flow field is simulated via the Lattice-Boltzmann method (LBM) [12]. Unlike former numerical and experimental investigations, in which a simplified geometry was used, the present method can be efficiently applied to variable, realistic airway geometries. For instance the flow field downstream of the laryngeal region has recently been investigated in [13] by an LBM. Since the numerical method is capable of reproducing small-scale features of lung flow, the results serve to fundamentally understand respiratory mechanisms. Thus, the numerical results allow an extended analysis of the three-dimensional flow structures observed in [3]. The steady flow field at inspiration and expiration has been simulated for a constant Reynolds number based on the hydraulic diameter of the trachea D of ReD = 1250. Furthermore, the flow field at time dependent inhalation and exhalation has been computed with a peak Reynolds number of ReD = 1050 and a Womersley √ number of α = 3.27, where the Womersley number is defined as α = 0.5D 2πf/ν and f is the frequency of the respiratory cycle and ν is the kinematic viscosity of air. The results mainly serve to fundamentally understand WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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the three-dimensional flow structures within the upper bifurcations of the human lung under normal breathing conditions and the time-dependent development of the flow field. The structure of this paper is as follows. First, the numerical method is briefly described. Then, the results for steady air flow are presented and compared with experimental data. Subsequently, the temporal development of secondary flow structures are discussed. Finally, the findings are summarized and some conclusions are drawn.

2 The Lattice-Boltzmann method In the following, a concise description of the Lattice-Boltzmann Method (LBM) using the Bhatnagar, Gross, and Krook (BGK) [14] approximation will be given. A detailed derivation of the LBM and an extensive discussion can be found in [12]. The BGK approximation uses a simplified collision term for the Boltzmann equation leading to the so-called BGK equation without external forcing ∂f ∂f = ω(f eq − f ). + ξi · ∂t ∂xi

(1)

The quantity ω represents the collision frequency, f eq is the Maxwell equilibrium distribution function, f is the particle distribution function, and ξi is the i-th component of the molecular velocity vector. That is, the left-hand side of Eq. 1 contains the temporal change and the propagation term, whereas the right-hand side describes molecular collisions. The corresponding algorithm is based on the iterative computation of propagation and collision processes for each cell of the computational grid. The macroscopic flow variables are determined by summation over the base moments of the distribution function f . Since the LBM formulation is based on a uniform Cartesian grid, it is highly adapted for parallel computation and it offers an efficient boundary treatment for fixed walls. The computational grid is automatically generated from arbitrary surface data by an in-house grid generator [15]. The ability of reproducing variable organic geometries makes this method well suited for biomedical applications. The standard LBM describes weakly compressible flows and it has been shown in the literature [16] that the LBM yields indeed solutions to the Navier-Stokes equations. All results presented in this study for the incompressible flow in the upper human airways have been obtained by the Lattice Boltzmann method using the BGK equation (LBGK) method. More details on the numerical approach can be found in [17, 18].

3 Results 3.1 Steady flow field To understand the global structure of the flow field, simulations have been performed for steady inspiration and expiration at a constant Reynolds number WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

106 Modelling in Medicine and Biology VIII

(a)

(b)

Figure 1: Reference plane to compare experimental and numerical data (left) and definition of the location of the cross sections for the investigation of secondary flow structures in the left primary bronchus (right).

of ReD = 1250. The value for ReD is based on the hydraulic diameter of the throat and corresponds to a mass flux of 240 ml/s. A Dirichlet condition, i.e., the velocity distribution, has been imposed at the tracheal cross section at inspiration and expiration. To validate the numerical solutions, the results are compared with experimental data obtained by Particle Imaging Velocimetry (PIV) measurements, which have been performed using a silicon model of the same lung geometry [3]. The results are investigated at a reference plane the location of which is indicated in Fig. 1(a). The comparison of the velocity magnitude contours at inspiration in Fig. 2 shows good agreement between experiment and simulation. The numerical bulk velocity distribution in the tracheal section is nearly identical with the experimental profile and the decomposition of the mass flux in the left and right primary bronchus is very similar. The velocity contours at the upper left primary bronchus, which indicate a recirculation region with counterrotating vortices whose axes lie in the stream-wise direction, are also in good agreement. At expiration (Fig. 3) the numerical and experimental data for the velocity contours are also alike, although the velocity magnitude in the numerical solution is smaller than that in the measurements. The velocity distribution in the tracheal region appears to be more developed in the experimental case. This slight difference is likely to be due to the boundary WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 2: Velocity contours and distributions at inspiration at ReD = 1250; PIV (left), LBM (right).

Figure 3: Velocity contours and distributions at expiration at ReD = 1250; PIV (left), LBM (right).

conditions at the outlet cross sections of the highest generations of the bronchial tree. In the experiment these cross sections were accessed by drilling holes in the silicon body for mass feeding or discharge, respectively. In the simulation a vanishing velocity gradient has been prescribed at inspiration and expiration. The current numerical method allows a detailed investigation of secondary flow structures in the primary bronchi. These flow structures have been observed in other numerical studies [5, 8, 9] and in experimental findings [3, 19]. The velocity distributions and contours in the left primary bronchus at four cross sections described in Fig. 1(b) are shown in Fig. 4. At inspiration Fig. 4 shows that the main mass flux is located near the lower wall indicated by a high stream-wise velocity. Downstream of the first bifurcation a pair of counter-rotating vortices develops in which air is transported away from the high speed region along the outer walls. The left vortex has a center of rotation near to the upper wall and fills up nearly the upper half of the cross section. The right vortex is much smaller and is located very close to the right wall. The vortex pair has also been observed in the experiment [3]. Additionally, the analysis of WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

108 Modelling in Medicine and Biology VIII

Figure 4: In-plane velocity distributions (arrows) and axial velocity contours (shades of gray) in several cross sections defined in Fig. 1(b) in the left principal bronchus at steady inspiration (left) and steady expiration (right). the numerical data emphasizes the strong asymmetry of the vortical structures. When the next bifurcation is reached, the vortices do separate and each one enters a branch of the next bronchial generation as shown in Fig. 4(d) at inspiration. At expiration the stream-wise velocity is fully distributed and the indicated vortical structures clearly possess a much smaller ratio of azimuthal momentum to stream-wise momentum than at expiration. The mixing of two streams coming from the higher generation bronchi generates a shear layer which develops into a slightly swirling region in Fig. 4(c). In Fig. 4(a) an in-plane velocity, pointing upwards, is evidenced near the walls. In conclusion, the findings of the steady flow WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 5: Modeling of the unsteady flow by a sinusoidal distribution of the mass flux.

field confirm that much more secondary flow structures are generated at inspiration than at expiration. 3.2 Unsteady flow field In order to investigate the temporal development of the secondary flow structures, extensive simulations concerning the unsteady behavior of the flow field during oscillating respiratory ventilation have been performed. The results evidence that the temporal rate of change of the velocity has a strong impact on the development of the flow field in the bifurcations. Simulations of oscillating flows have been conducted for a ReD = 1050 and a α = 3.27 corresponding to a time period of T = 3.7 s which describes a normal respiration at rest. The temporal change of the mass flux has been prescribed by a sinusoidal curve shown in Fig. 5. The data acquisition has been started after a transient time of two respiration cycles. The velocity distributions for a whole respiration cycle are shown in Fig. 6 for the trachea and the first bifurcation. When the mass flux peaks, the flow field is almost identical with the steady case distribution. At a phase angle of 45◦ the overall flow structures are already fully developed. However, the high speed region in the left primary bronchus still becomes more narrow at increasing phase angle. At expiration the overall flow structure does not appear to change noticeably in time. The comparison with the time dependent PIV findings from [3] shows the temporal behavior of the numerically determined flow field to be in good agreement with the experimental results. In order to obtain insight into the development of secondary flow structures at inspiration, the flow at the very beginning of inhalation, i.e., at very low Reynolds WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 6: Velocity distributions at unsteady simulation. The numbers indicate the phase angles depicted in Fig. 5.

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Figure 7: In-plane velocity distributions (arrows) and axial velocity contours (shades of gray) for three cross sections depicted in Fig. 1(b). The numbers indicate three subsequent time steps as shown in Fig. 5.

numbers, has been investigated. In Fig. 7 the temporal evolution of the flow field in the left primary bronchus is shown for three time values indicated in Fig. 5. The observed sequence starts 173 ms after the bulk velocity at the inlet cross section has reversed. In Fig. 7(I) the distribution of the axial velocity shows a rather flat pattern and no vortical structures are visible. The flow field is very similar in all three cross sections. The second time step depicted in Fig. 7(II) shows an increasing axial velocity and an incipient asymmetry of the axial velocity distribution, i.e., the location of the peak value occurs close to the lower wall. This development is enhanced in the stream-wise direction. The in-plane velocity distribution evidences the formation of counter rotating vortical structures throughout the cross sections. In the last stage of the analysis shown in Fig. 7(III) both the counter-rotating vortices and the axial velocity distribution are very similar to those observed at maximum inspiration in all three cross sections. The numerical results show that the elementary vortical structures and the high speed region which have been observed at maximum inspiration are already encountered at a phase angle of 34◦ corresponding to a Reynolds number of WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

112 Modelling in Medicine and Biology VIII ReD = 590. This is agreement with the experiment where the size of the counterrotating vortices has been found to be Reynolds number independent as long as the Reynolds number is above a critical level.

4 Conclusions The flow field in a realistic model of the human lung at steady inspiration and expiration and at unsteady respiration has been simulated via the LBM. The LBM has proved to be an efficient tool to simulate flows through highly intricate geometries which have been resolved by an automatically generated Cartesian mesh. The steady flow field at inspiration and expiration has been analyzed for a constant flow rate of 240 ml/s resulting in a Reynolds number based on the hydraulic diameter of the throat region of ReD = 1250. The visualization has evidenced the intricate three-dimensional character of the flow field. A pair of counter-rotating vortices and a region of high speed flow have been observed downstream of the first bifurcation in the left bronchus. The results have been compared with PIV measurements and showed to be in very good agreement with the experimental findings. Furthermore, the separation of the vortices in the next bifurcation and a strong asymmetry have been observed. In order to evidence the impact of an oscillating mass flux the unsteady flow field has been analyzed at a Womersley number of α = 3.27 and a maximum Reynolds number of ReD = 1050. In a preliminary analysis the overall flow structure has been shown to be similar to that of the steady case. The growth of the vortical structures in the left branch of the first bifurcation has been investigated in detail for initiating inspiration. The numerical data evidences a strong dependence of the shape and size of the secondary flow structures on the instantaneous mass flux. At Reynolds numbers greater than ReD = 600 the overall shape of the flow field does not change. At expiration the steady and unsteady flow solutions are found to be very similar since hardly any secondary flow structures have been observed. The obtained results reveal insight into the overall structure of the flow field in a realistic lung geometry and emphasize the unsteady character of the flow field when the flow conditions reverse. This knowledge is essential for the improvement of artificial respiration devices and for the development of aerosol drug delivery systems.

Acknowledgement The support of the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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References [1] Comer, J.K., Kleinstreuer, C. & Kim, C.S., Flow structures and particle desposition patterns in double-bifurcation airway models. Part 1. Air flow fields. 435, pp. 25 – 54, 2001. [2] Nowak, N., Kakade, P.P. & Annapragada, A.V., Computational Fluid Dynamics Simulation of Airfoil and Aerosol Deposition in Human lungs. Annals of Biomedical Eng, 31, pp. 374 – 390, 2003. [3] Große, S., Schr¨oder, W., Klaas, M., Kl¨ockner, A. & Roggenkamp, J., Time resolved analysis of steady and oscillating flow in the upper human airways. Experiments in Fluids, 42, pp. 955–970, 2007. [4] Große, S., Schr¨oder, W. & Klaas, M., Time-Resolved PIV Measurements of Vortical Structures in the Upper Human Airways. Particle Image Velocimetry, Springer Berlin/Heidelberg, volume 112/2008 of Topics in Applied Physics, pp. 35–53, 2008. [5] van Ertbruggen, C., Hirsch, C. & Paiva, M., Anatomically based threedimensional model of airways to simulate flow and particle transport using computational fluid dynamics. J Appl Physiol, 98, pp. 970–980, 2005. [6] Weibel, E., Morphometry of the human Lung. Springer Berlin, 1963. [7] Zhao, Y. & Lieber, B.B., Steady Expiratory Flow in a Model Symmetric Bifurcation. Journal of Biomechanical Engineering, 116(3), pp. 318–323, 1994. [8] Zhang, Z., Kleinstreuer, C. & Kim, C.S., Gas-solid two-phase flow in a triple bifurcation lung airway model. International Journal of Multiphase Flow, 28(6), pp. 1021 – 1046, 2002. [9] Liu, Y., So, R.M.C. & Zhang, C.H., Modelling the bifurcating flow in a human lung airway. Journal of Biomechanics, 35(4), pp. 465–473, 2003. [10] Li, Z., Kleinstreuer, C. & Zhang, Z., Particle deposition in the human tracheobronchial airways due to transient inspiratory flow patterns. Journal of Aerosol Science, 38(6), pp. 625 – 644, 2007. [11] Lin, C.L., Tawhai, M.H., McLennan, G. & Hoffman, E.A., Characteristics of the turbulent laryngeal jet and its effect on airflow in the human intra-thoracic airways. Respiratory Physiology & Neurobiology, 157(2-3), pp. 295 – 309, 2007. [12] Benzi, R., Succi, S. & Vergassola, M., The Lattice Boltzmann Equation: Theory and Applications. Physics Reports, 222(No. 3), pp. 145–197, 1992. [13] Ball, C.G., Uddin, M. & Pollard, A., Mean flow structures inside the human upper airway. Flow, Turbulence and Combustion, 81, pp. 155–188, 2008. [14] Bhatnagar, P.L., Gross, E.P. & Krook, M., A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral OneComponent Systems. Phys Rev, 94(3), pp. 511–525, 1954. [15] Hartmann, D., Meinke, M. & Schr¨oder, W., An adaptive multilevel multigrid formulation for Cartesian hierarchical grid methods. Comput Fluids, 37, pp. 1103–1125, 2008. [16] H¨anel, D., Molekulare Gasdynamik. Springer: Berlin, 2004. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

114 Modelling in Medicine and Biology VIII [17] Freitas, R.K. & Schr¨oder, W., Numerical investigation of the threedimensional flow in a human lung model. Journal of Biomechanics, 2008. [18] Freitas, R.K., Meinke, M. & Schr¨oder, W., Investigation of Wall-Bounded Turbulent Flow using Lattice Boltzmann Methods. submitted to Computers & Fluids, 2008. [19] Adler, K. & Br¨ucker, C., Dynamic flow in a realistic model of the upper human lung airways. Experiments in Fluids, 43(2), pp. 411–423, 2007.

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Numerical simulations of high frequency respiratory flows in a model bifurcating lung geometry N. Valleru1, S. Smirnov1, J. Tan1, S. Parameswaran1 & R. Raj2 1

Department of Mechanical Engineering, Texas Tech University, USA Department of Internal Medicine, Texas Tech University Health Sciences Center, USA

2

Abstract Numerical studies of 2D cases are conducted using a CFD code FLUENT to analyze the flow patterns and gas transport at high oscillatory frequencies through a two-generation bifurcating lung model. The geometry corresponds to fifth to seventh generations of airways with the dimensions based on the Wiebel’s symmetric pulmonary model. Computations are carried out for the Reynolds numbers Re = 400 and Re = 1000, while the Womersley number is Wo = 4.0 and Wo = 16.0. The average mass distribution in the entire lung model is also investigated to analyze the influence of flow frequency on the mass diffusion efficiency. The numerical results of the current study pointed to: i) the numerical model successfully reproduces many results observed in the experiments; and ii) there is practically no net effect of the high frequency on the increased mass diffusion in the bifurcation geometry. The developed numerical model may be further used in more complicated 3D geometries and for determining the optimal conditions for artificial lung ventilation. Keywords: pulmonary airways, oscillatory flow, diffusion, numerical simulation, CFD, fluid mechanics.

1

Introduction

To better understand the human pulmonary system, the knowledge of fluid mechanics is essential. When air passes through human lungs, it constantly changes its direction and speed, which leads to complex flow phenomena such as WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090111

116 Modelling in Medicine and Biology VIII flow separation, recirculation, and mixing across airway cross-sections. Therefore, a physical description of the flow characteristics and gas transport in the pulmonary airways is essential. Although experimental studies can improve the insight, they are expensive and restricted in many ways due to the complex structure and small dimensions of the pulmonary airways. However, the rapid growth of the electronic computer technology enables us to utilize numerical simulation as a complement to the experimental studies. Moreover, once a numerical model is validated versus experimental measurements, it may be further used in studying the flow behavior in complex geometries where experimental studies are impossible.

2

An experiment model

The airways of human lungs are an extremely complex network of small bifurcations. This nature of the airways makes the study of flow and the transport of gases infeasible without reasonable simplifications. One of the famous and most commonly used simplified experimental models is provided by Wiebel and Gomez [1]. He conducted detailed measurements of human lungs and constructed a symmetric model by assuming the airways of any generation are identical and the surfaces are rigid, straight and smooth.

Figure 1:

Wiebel’s model.

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Table 1:

117

Parameters based on Wiebel’s model.

Generation D n (mm) 5 3.50 6 2.80 7 2.30

L (mm) 10.7 9.0 7.6

Re

α

341.6 213.5 130.0

0.54 0.43 0.35

The process of breathing can be split into two phases: the inspiratory phase and the expiratory phase. Because of the oscillatory nature, the lung flow behaves in an unsteady manner and thus cannot be characterized by the Reynolds number alone. Instead, the Womersley number is widely accepted to describe the oscillatory flow. It is defined as:

Wo 

D  2 

where Wo = the Womersley number, D = the airway diameter,  = the frequency of oscillation,  = the dynamic viscosity. In the experiment, steady-state Taylor-type dispersion is conducted as shown in Figure 2. At the beginning, the straight tube is filled with fluid A. When t = to, another fluid B is introduced into the tube at the inlet boundary with a parabolic velocity profile. After a certain period of time, when t = t1, the dispersion of fluid B is clear enough to be observed and measured. The apparent diffusion coefficient is hereby introduced to quantify the diffusion and it is defined as:

Dapp 

 2U 2 48Dmol

where Dmol = the molecular diffusion coefficient,

 = the radius of tube, U =

the average velocity of fluid B at the inlet boundary.

t

=

t

=

Fluid B

Fluid A

Fluid B

Figure 2:

Taylor’s dispersion.

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Fluid A

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3

Numerical techniques

In numerical simulations we modeled a double bifurcation (5th to 7th generation) of the lung airway network. Finite volume method was utilized to discretize the general transportation equation:

 (  )  ( U )  (  )  S t The second order UPWIND scheme and SIMPLE algorithm were performed to solve for the velocity and pressure distributions. All calculations were processed with double precision to ensure the accuracy of the results. GAMBIT software package was utilized to model and mesh the airways’ geometry, and FLUENT software package was used to solve the fluid governing equations to obtain the data on the flow field and fluid dispersion. The mesh generation for the straight tube is straight forward. Quadrilateral mesh was generated for the whole fluid domain, and the mesh density was increased near the tube wall surface due to the boundary layer effects. For the double bifurcation model, the whole fluid domain was divided into different sections to accommodate the requirements of different flow regions (Figure 3). Quadrilateral mesh is applied for the straight tube sections and triangular mesh is generated to fit the geometry of the junction area. Also mesh density is increased near the surfaces of the wall boundaries to take care of the boundary layer effects.

Figure 3:

Mesh for the double bifurcation model.

Two different numerical models were constructed and analyzed in the current study. Namely, 1) 2D double bifurcation scaled up model, and 2) 2D double bifurcation actual size model. For each model, we conduct research on both WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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steady-state and unsteady cases, change the inlet flow frequency and compare the results in order to uncover how the oscillatory frequency affects the flow diffusion. Generally the inlet velocity profile is specified as:

u  u0 sin(t ) ， where u0 

p r 2  a 2 x 4 

u = the axial velocity for unsteady case, u0 = the axial velocity for steady case, p / x = the axial pressure gradient, a = the radius of tube, r = the radial position,  = the dynamic viscosity, t = the time and  = the oscillatory frequency. The transport equation is solved using the following boundary conditions: (i) at the inlet boundary, the mass fraction of fluid B is fixed to 1.0 and the mass fraction of fluid A is fixed to 0.0, (ii) at the outlet boundary, the mass fraction of fluid B is fixed to 0.0 and the mass fraction of fluid B is fixed to 1.0. Moreover, for all unsteady cases, the initial conditions are always the same: at t = 0, the whole fluid domain is filled with fluid A, which means that the mass fraction of fluid B is equal to zero at t = 0 everywhere in the fluid domain, except at the inlet boundary.

4

Simulation results and discussions

4.1 2D scaled-up double bifurcation model The simulations were carried out for the inspiratory flow with the Reynolds number Re = 410 and the average velocity of 0.00137 m/s. Water (at the room temperature) was used as the working medium. We calculated the average non-dimensional velocities at different cross sections along the model (such as cross section 2B, 3B 2C, 3C, see Figure 4 below), and compared these data with the experimental results from literature [2]. In Figure 4, it is clearly indicated that numerical and experimental results reach a satisfactory agreement, which provides a solid validation for our numerical models. 4.2 2D double bifurcation model with actual size Since we have enough confidence in the 2D scaled-up double-bifurcation model, the next step was to apply it to the actual size model of a very small scale. Simulations are carried out for two cases with different Reynolds numbers Re = 400 and Re = 1000. The validation of this actual size model is performed by comparing the flow velocity distribution between the numerical results from simulations and the numerical results from literature [3]. As shown in Figure 5, the numerical results are well matched. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

120 Modelling in Medicine and Biology VIII

Figure 4:

Scaled-up model, non-dimensional velocity (vertical axis) vs. reduced distance (horizontal axis), the numerical results compared with experimental results.

Figure 5:

Actual size model, velocity distribution comparison, numerical results vs. results from literature [3], Re=400 and Re=1000.

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After validating our model, we studied two cases with the same Reynolds number Re = 1000, but different Womersley numbers, Wo = 4.0 and Wo = 16.0. Different Womersley numbers correspond to different flow frequencies. We were curious about whether the inlet flow frequency has any effect on the fluid diffusion or not. In other words, the purpose of this simulation was to study whether increasing the inlet flow frequency causes an increase of the mass fraction of fluid B at all four outlets of our model. The results of simulation have to be compared at exactly the same phases of oscillation due to the periodic nature of the flow at inlet boundary. The results also have to be compared at the same absolute flow times, because the molecular diffusion depends on the absolute time. So we had to follow two criteria: 1) The results are chosen at the same absolute flow time, and 2) they are chosen at the same phase of a cycle, e.g., peak of the inspiratory phase, peak of the expiratory phase, etc. In this simulation, we compute both cases under exactly the same boundary conditions and numerical set-ups (e.g., grid size, time step, etc.). Furthermore, the absolute time is chosen to be large enough (five cycles for Wo = 4) to guarantee that the periodic flow becomes well established inside the bifurcation model. The results are compared at the same phase of the last cycle for both cases. Tables 2 and 3 present the results of such computations. The mass fraction of oxygen (fluid B) is computed at four outlet boundaries. The flow time is 0.4009 seconds, which equals to five cycles for Wo = 4.0 case, and eighty cycles for Wo = 16.0 case.

5

Conclusion

The data in Tables 2 and 3 reveal the fact that at the end of the last cycle, the mass fraction is essentially zero for both values of the Womersley number. At the peak of the inspiratory phase the values of mass fraction of oxygen at the outlets are much larger for Wo= 4 than for Wo = 16. Based on this fact, our conclusion is that there is practically no net effect of the high frequency (or Womersley number) on the improved mass transfer in the bifurcation model. Table 2:

Mass fraction of oxygen at four outlet boundaries at the peak of the inspiratory phase (flow time 0.4009s). Wo=4.0

Wo=16.0

1st outlet (7C)

0.330531

0.003020

2nd outlet (6C)

0.529589

0.002776

3rd outlet (5C)

0.529775

0.002781

4th outlet (4C)

0.330554

0.003019

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122 Modelling in Medicine and Biology VIII Table 3:

Mass fraction of oxygen at the outlet boundaries at the end of the cycle (flow time 0.4009s). Wo=4.0

Wo=16.0

1st outlet (7C)

6.917803E-19

3.058330E-5

2nd outlet (6C)

6.801282E-16

2.807442E-5

3rd outlet (5C)

6.972378E-16

2.813371E-5

4th outlet (4C)

7.056232E-19

3.057900E-5

In other words, increasing oscillatory frequency does not cause any significant increase in the oxygen diffusion. We hypothesize that the situation might be different in 3D geometry, which opens the possibility for transverse flow instabilities, and as a result increased mixing and mass transfer along the bifurcation model. We intend to explore this possibility in the near future.

References [1] [2] [3]

[4] [5]

Wiebel, E.R. and Gomez, D.M., Architecture of Human Lung, Science, Volume 137 No. 3530, pp. 577–585, 1962. Theunissen, R. and Riethmuller, M.L., Particle Image Velocimetry in Lung Bifurcation Models, Springer Berlin/Heidelberg, Volume 112, 2008. Wilquem, F. and Degrez, G., Numerical Modeling of Steady Inspiratory Airflow Through a Three-Generation Model of the Human Central Airways, Journal of Biomechanical Engineering, Volume 119 Issue 1, pp. 59-66, 1997 Chang, H.K., Mechanisms of gas transport during ventilation by highfrequency oscillation, Journal of Applied Physiology, Volume 56, Issue 3, pp. 553-563, 1984 Pedley, T.J., Pulmonary Fluid Dynamics, Annual Review of Fluid Mechanics, Volume 9, pp. 229-274, 1977

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Numerical prediction of the focal sites of ozone-induced tissue injury in the respiratory tract B. Keshavarzi, J. Ultman & A. Borhan Department of Chemical Engineering, The Pennsylvania State University, USA

Abstract Numerical simulations of ozone transport and uptake in an anatomically-accurate model of the respiratory tract of a Rhesus monkey were performed. The model geometry was created using three-dimensional reconstruction of MRI images of the respiratory tract, including the larynx and the first thirteen generations of the tracheobronchial tree. An unstructured mesh was generated for the resulting structure, and three-dimensional flow and concentration distributions were obtained through numerical solution of the Navier-Stokes, continuity, and species convection-diffusion equations. A quasi-steady diffusion-reaction model was used to account for the interaction between O3 and endogenous substrates in the respiratory tract lining fluid. Hotspots of O3 flux on the walls were identified for steady inspiratory flow under quiet breathing conditions. Keywords: ozone, uptake, respiratory tract, larynx, conducting airways.

1 Introduction Ozone (O3 ) is a highly reactive gas and a harmful air pollutant. Ground level ozone is formed primarily by the action of sunlight on hydrocarbon vapors and nitrogen oxides emitted by combustion of fossil fuels. It can irritate the respiratory tract, reduce lung function, and increase the frequency of asthma attacks in populations suffering from asthma. Recent observations have shown that exposure to ozone can also produce intense remodeling in the developing lungs of infant primates, resulting in the loss of conducting airways [1]. The national ambient air quality standard for (O3 ) level is 0.075 ppm averaged over eight hours. However, tests WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090121

124 Modelling in Medicine and Biology VIII carried out on a group of healthy adults and children undergoing moderate exercise have shown that even lower levels of ozone can cause a decrease in breathing ability [2]. The toxicity of oxidant gases such as ozone is believed to be due to peroxidation of membrane lipids, leading to destruction of membrane integrity and eventual cell death. Ozone may also oxidize reduced sulfhydryl groups of proteins and peptides such as Glutathione [3]. Absorption of ozone occurs in all regions of the respiratory tract, and the pattern of lung injury induced by the inhalation of ozone appears to depend on the dose delivered to different tissues in the airways. For example, Castleman et al. [1] performed controlled O3 exposures of bonnet monkeys, and observed epithelial cell hyperplasia and macrophage accumulation that were primarily focused in the proximal alveolar region. They also reported less severe epithelial damage in the trachea and large bronchi, and little or no damage in other lung regions. Similarly, Chang et al. [4] found that the degree of focal injury occurring in the proximal alveolar region of Fisher-344 rats was directly dependent on the integrated O3 concentration-time profile of the inhaled gas. Thus, regional differences in cell injury are probably related to corresponding differences in the rates of O3 delivery. Although local dose cannot be easily measured, especially in a site-specific manner, mathematical dosimetry models can provide predictions of the uptake and distribution of absorbed gases such as O3 in the respiratory tract [5, 6]. Dosimetry refers to the estimation or measurement of the amount of a compound (or its toxic metabolites or reaction products) that reaches specific target sites after exposure to a given concentration of the compound. While it is critical to understand interspecies differences in the respiratory tract in order to assess the implications of toxicological results from animal studies to humans, dosimetry modeling can also play an extremely important role. Previous analysis of transport and removal of O3 in the lungs of guinea pigs, rabbits, and humans indicates the existence of a general similarity among these species in the shapes of the dose curves [7]. Thus, accurate dosimetry models that incorporate physical, biological and chemical properties of the respiratory tract, as well as the nature of gas transport in the lumen and air spaces, can serve as invaluable predictive tools in the extrapolation of animal toxicological results to humans [8]. The development of O3 dosimetry models for laboratory animals and humans has mainly focused on the effects of O3 in the lower respiratory tract (LRT) comprising the tracheobronchial tree which includes the trachea and a series of branching tubes that become narrower, shorter, and more numerous before ending at the terminal bronchioles (cf. Recent advances in the development of reliable algorithms for three-dimensional reconstruction of complex geometries, combined with well-established computational fluid dynamics (CFD) strategies for the computation of flows in such geometries, have paved the way for the development of more sophisticated dosimetry models based on anatomicallyaccurate geometries of the respiratory tract. In this paper, we present the results of three-dimensional simulations of ozone transport and uptake during inspiratory

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flow in an anatomically-accurate model of the LRT of a rhesus monkey that also includes the larynx.

2 Methods 2.1 Airway geometry Based on the structural similarity between the human and rhesus monkey respiratory tracts, the respiratory tract of a rhesus monkey was selected as the platform for this computational study. In addition, experimental assessment of ozoneinduced injury in the conducting airways of rhesus monkeys is currently being conducted, thereby allowing direct comparison between theoretical predictions and experimental observations in the near future. The anatomically-accurate airway geometry was created from 3-D reconstructions of the larynx and tracheobronchial tree using MRI data for the casts of the respective parts of a rhesus monkey. The MRI data consisted of 256 transverse slices depicting square fields of view 10 cm on a side for the tracheobronchial tree and 3.5 cm on a side for the larynx. The thickness of each slice and the resolution of its sides were both 391 microns for the tracheobronchial tree, and 137 microns for the larynx. All slices were stored as 8-bit TIFF (Tag Image File Format) files containing 256 × 256 pixels in DDV (Digital Data Viewer) format. The MRI images were imported as raw data into the three-dimensional visualization and volume modeling software, AMIRA (Mercury Computer Systems, Chelmsford, MA). Open-ended triangulated surfaces for the larynx and the tracheobronchial tree were created in AMIRA and exported to the commercial meshing software GAMBIT (ANSYS, Canonsburgh, PA). The extraneous lowresolution parts of small branches were cut off from the geometry, and the geometry was cleaned up by removing hard and short edges, as well as small faces. Closed volumes for the larynx and the tracheobronchial tree were then created from their respective cleaned-up surfaces, and the two volumes were subsequently attached together. The resulting structure included thirteen airway generations beyond the larynx. Accurate reconstruction beyond the thirteenth generation would require a better resolution of the MRI images. The entire airway geometry was meshed using tetrahedral elements of dimensionless size 0.05 (made dimensionless with the hydraulic radius of the trachea). Zone types such as inflow, outflow and wall were assigned to the bounding surfaces, and the meshed geometry was exported to the commercial CFD software FLUENT 6.3 (ANSYS, Canonsburgh, PA) for the simulations. 2.2 Mathematical formulation Simulations were performed for steady incompressible laminar flow of a binary gas-mixture of air and ozone. The flow field is governed by dimensionless continuity and Navier-Stokes equations given by ∇ ·u=0 WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

(1)

126 Modelling in Medicine and Biology VIII ∂u 1 2 (2) + u · ∇u = −∇p + ∇ u ∂t Re where u and p are the dimensionless velocity and dynamic pressure, respectively, and ∇ is the dimensionless gradient operator. All lengths are made dimensionless with the hydraulic radius of the trachea, Rh , velocities with U0 = Q0 /πRh2 , time with Rh /U0 , and pressure with ρU02 , where ρ denotes the density of the gas mixture, and Q0 denotes the volumetric flow rate through the trachea. The Reynolds number, Re = U0 Rh /ν, appearing in Eq. 2 represents the tracheal Reynolds number, with ν denoting the kinematic viscosity of the gas mixture. Equations (1) and (2) are solved subject to the following boundary conditions: • Uniform velocity normal to the inflow plane of the larynx for inspiratory flow. • Zero velocity (u = 0) at the airway walls (no-slip condition). • Zero viscous normal stress and prescribed volumetric flow rates at the outflow boundaries, with the flow splits among outflow boundaries specified to be proportional to their cross-sectional areas. The O3 concentration distribution in the gas mixture is governed by the dimensionless convection-diffusion equation, which can be written as 1 2 ∂C + u · ∇C = ∇ C, ∂t Pe

(3)

where the concentration C is made dimensionless with the inlet O3 concentration, C0 , P e = U0 Rh /Dg is the tracheal Peclet number, and Dg denotes the gas phase diffusivity of O3 . The boundary conditions for this equation include: • Uniform O3 concentration, C = 1, at the inflow boundary. • Zero diffusive flux, n · ∇C = 0, at the outflow boundaries • An appropriate diffusion-reaction condition at the airway wall, based on the specific model used for the reaction of O3 within the respiratory tract lining fluid (RTLF) The last boundary condition considers either an infinitely fast or a slow firstorder reaction of O3 with substrates in the RTLF. Hence, several different reactions can take place between O3 and the RTLF components. In the absence of reliable data for the rates of these reactions, we have to rely on simplified models for the reactions between O3 and RTLF substrates. The simplest model that is considered in our simulations is that of an infinitely fast reaction. In this model, the rate of reaction is so fast (compared to the rate of O3 transport to the RTLF) that the O3 concentration vanishes at the gas-RTLF interface, leading to the boundary condition C = 0 at the airway wall. Although using an infinite reaction rate overestimates the rate of O3 uptake, it also magnifies the non-uniformity in the wall flux distribution, and facilitates the identification of hot spots of O3 flux (which is one of the main objectives of this study). In reality, the rate of reaction of O3 in the RTLF is finite. Hence, a second model for the reaction between O3 and RTLF substrates can be formulated based on the assumption that the substrate concentrations in the RTLF are much higher than O3 concentrations that can result from any reasonable O3 exposure. In that WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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case, the reaction can be viewed as a pseudo-first order reaction with respect to O3 concentration in the RTLF. Neglecting convection in the RTLF, the quasisteady diffusion and reaction of O3 within the RTLF is described by the species conservation equation ∇ 2 c = Da 2 c,

(4)

where c is the O3 concentration  in the RTLF (made dimensionless with C0 ). The 2 Damkohler number, Da = krDl , represents the ratio of the characteristic time for diffusion to that for chemical reaction, with kr denoting the rate constant for the pseudo-first order reaction,  the thickness of the RTLF layer, and Dl the diffusivity of O3 in the RTLF. The thickness of the RTLF layer is typically much smaller than the airway radius. It decreases with longitudinal position from about 10 μm in the upper airways to about 0.1 μm in the respiratory zone, while the airway radii vary in the range 0.02–0.09 cm depending on the airway generation [9]. As such, a planar (rather than annular) view of the RTLF can be adopted in conjunction with the lubrication approximation to reduce Eq. (4) to ∂ 2 c(y, z) = Da 2 c(y, z), ∂y 2

(5)

where y and z denote the coordinates (made dimensionless with ) normal and tangent to the gas-RTLF interface, respectively. The boundary conditions consist of vanishing O3 concentration at the RTLFtissue interface, and local equilibrium between the O3 concentrations on the two sides of the gas-RTLF interface. The latter can be expressed as c = αC, where α is the equilibrium partition coefficient. The solution of Equation (4) subject to these boundary conditions yields the following O3 concentration profile within the RTLF: αC(z) sinh[Da(1 − y)] (6) c(y, z) = sinh(Da) The resulting expression for the dimensionless O3 flux into the RTLF layer is given by ∂c N = − |y=0 = −αC(z)Da coth(Da). (7) ∂y Requiring that the O3 flux be continuous across the gas-RTLF interface then leads to the effective wall reaction condition    Dl R N = −n · ∇C = KC ; K = αDa coth(Da), (8) Dg  which can be used as a boundary condition on the airway walls for Eq. (3) in simulations based on the pseudo-first order reaction model. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

128 Modelling in Medicine and Biology VIII

Figure 1: Streaklines of flow, color-labeled by local O3 concentration, for an infinitely-fast reaction (Re = 230, P e = 195).

3 Results Using the reported results for the daily air intake of a monkey based on its body weight [10], a range of inspiratory flow rates was estimated for rhesus monkeys. Simulations were performed for a tracheal Reynolds number of 230, corresponding to the upper end of the range of Reynolds numbers for quiet breathing condition. Based on a Schmidt number of Sc = 0.85 for the ozone/air mixture [11], the corresponding value of Peclet number in the simulations was 195. For the pseudofirst order reaction model, a value of Da = 15 was used is the simulations, corresponding to a rate constant on the order of 10, 000 s −1 which represents the lower end of the values reported in the literature. Proper convergence and accuracy of the computations were verified by performing the requisite mesh refinement studies. Streaklines of flow, color-labeled by the local O3 concentration (with red representing high concentration and blue denoting low concentration), are shown in Figure 3 for an infinitely-fast reaction. In this case, the O3 concentration at most of the outflow boundaries is nearly zero, and very little ozone leaves through WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 2: Streaklines of flow, color-labeled by local O3 concentration, for a slow first-order reaction with Da = 15 (Re = 230, P e = 195).

the distal airways. The corresponding results for a slow first-order reaction with Da = 15, but otherwise identical values of the dimensionless parameters, are shown in Figure 3. In contrast to the case of an infinitely fast reaction, a significant amount of ozone remains unreacted in the airways even after thirteen generations along the main path. In this case, most of the O3 is transported to the airways beyond the distal end of the airway structure considered here. In the absence of the larynx, local maxima in O3 wall flux occur just downstream of the carina of each bifurcation (specially in the first three bifurcations of the conducting airways), with the magnitudes of the spikes decreasing with each branching generation [5, 11]. The appearance of the spikes in wall flux just downstream of bifurcations is due to the development of thin concentration boundary layers on the airway walls, starting from the carina. As the flow enters downstream branches, the local Reynolds number of the flow decreases. At lower Reynolds number, the thickness of the concentration boundary layer on the airway wall is larger at any given distance downstream of the carina, thereby leading to a smaller O3 flux at the airway wall. As expected, the magnitudes of the hotspots of O3 flux for the infinitely-fast reaction are much larger than those for the slow firstorder reaction. While the hotspots of O3 wall flux just downstream of the airway bifurcations persist even in the presence of the larynx, their intensities are reduced WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

130 Modelling in Medicine and Biology VIII

Figure 3: The O3 wall flux distribution for a slow first-order reaction with Da = 15 (Re = 230, P e = 195). due to both the substantial uptake of ozone within the larynx, and the effect of the larynx on the downstream flow pattern. The structure of the larynx significantly affects the resulting flow pattern in the trachea. The larynx geometry includes a narrow opening between the vocal cords (called rima glottidis) where a sharp and sudden change in the cross-sectional area of the larynx occurs. For both reaction models, the flow enters the airways with uniform velocity through the inlet of the larynx. The sudden reduction in the cross-sectional area of the larynx just upstream of the rima glottidis leads to the formation of a curved sheet-like laryngeal jet that impinges on one side of the tracheal wall a short distance downstream of the larynx (see the magnified images in Figs. 1 and 2). The bending of the laryngeal jet causes asymmetry in the flow structure extending to the distal trachea. It produces a complex flow pattern with skewed velocity profiles and recirculation regions in the trachea. It should WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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be noted that the complex secondary flow patterns observed in the trachea are produced based on laminar flow simulations. Since the maximum local Reynolds number associated with the Laryngeal jet is about 1, 900 for the inspiratory flow rate considered in this study, the assumption of laminar flow regime in the airways is reasonable. The asymmetry of the three-dimensional airway structure near the first two or three bifurcations leads to significant asymmetry in partitioning of ozone among the airways downstream of those bifurcations. In addition, bending of the airways beyond those bifurcations causes the concentration profiles to be skewed toward one side of the airways. Asymmetry of the velocity and O3 concentration distributions gives rise to an asymmetric O3 wall flux distribution, as shown in Fig. 3 for the case of a slow first-order reaction. The O3 flux distribution on the trachea wall is significantly affected by the complex flow pattern produced by the larynx. In particular, the impingement of the laryngeal jet on the tracheal wall produces a hotspot of ozone flux in the vicinity of the impingement region. Overall, the secondary flow patterns produced in the trachea by the presence of the larynx result in a more uniform O3 wall flux distribution within the trachea, compared to the corresponding simulations in the same airway structure without the larynx.

Acknowledgement This work was supported by the National Institute of Environmental Health Sciences research grant 1P 01 ES 11617.

References [1] Castleman, W.L., Tyler, W.S. & Dungworth, D.L., Lesions in respiratory bronchioles and conducting airways of monkeys exposed to ambient levels of ozone. Exp Mol Pathol, 26(3), pp. 384–400, 1977. [2] Review of national ambient air quality standards for ozone, assessment of scientific and technical information. US Environmental Protection Agency, Office of Air Quality Planning and Standards OAQPS Staff Paper EPA452/R-96-007, June 1996. [3] Cross, C., van der Vliet, A., O’Neill, C.A., Louie, S. & Halliwell, B., Oxidants, antioxidants, and respiratory tract lining fluids. Environ Health Perspect, 102(Suppl. 10), pp. 185–191, 1994. [4] Chang, L., Miller, F.J., Ultman, J.S., Huang, Y., Stockstill, B.L., Grose, E., Graham, J.A., Ospital, J.J. & Crapo., J.D., Alveolar epithelial cell injuries by subchronic exposure to low concentrations of ozone correlate with cumulative dose. Tox Appl Pharm, 109, pp. 219–234, 1991. [5] Keshavarzi, B., Numerical Prediction of the Focal Sites of Ozone-Induced Tissue Injury in the Respiratory Tract. Ph.D. thesis, The Pennsylvania State University, 2009. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

132 Modelling in Medicine and Biology VIII [6] Kimbell, J.S. & Miller, F.J., Regional respiratory-tract absorption of inhaled reactive gases: a modeling approach. In: D E Gardner, J D Crapo, R O Mcclellan, eds Toxicology of the Lung 3rd ed Philadelphia: Taylor & Francis, 102, pp. 557–597, 1999. [7] Overton, J.H., Graham, R.C. & Miller, F.J., A model of the regional uptake of gaseous pollutants in the lung: ii. the sensitivity of ozone uptake in laboratory animal lungs to anatomical and ventilatory parameters. Tox Appl Pharm, 88(3), pp. 418–432, 1987. [8] Miller, F.J., Menzel, D.B. & Coffin, D.L., Similarity between man and laboratory animals in regional pulmonary deposition of ozone. Environ Res, 17(1), pp. 84–101, 1978. [9] Miller, F.J., Overton, J.H., Jaskot, R.H. & Menzel, D.B., A model of the regional uptake of gaseous pollutants in the lung: I. the sensitivity of the uptake of ozone in the human lung to lower respiratory tract secretions and exercise. Toxicol Appl Pharmacol, 79(1), pp. 11–27, 1985. [10] Blackburn, K., Recommendations for and documentation of biological values for use in risk assessment. IEPA/600/6-87/008, pp. 4–16, 1996. [11] Taylor, A.B., Borhan, A. & Ultman, J.S., Three-dimensional simulations of reactive gas uptake in single airway bifurcation. Ann of Biomed Eng, 35(2), pp. 235–249.

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Computational hemodynamics coupled with mechanical behaviour of the surrounded materials, in the specific case of the brachial artery R. Paulus1,2, S. Erpicum1,2, B. J. Dewals1,2,3, S. Cescotto2 & M. Pirotton1,2 1

HACH (Hydrology, Applied Hydrodynamics and Hydraulic Constructions), University of Liège, Belgium 2 ArGEnCo (Architecture, Geology, Environment and Constructions), University of Liège, Belgium 3 F.R.S.-FNRS (Belgian Fund for Scientific Research)

Abstract Blood pressure is an essential measure when it comes to peoples’ health. All around the world a high number of people are suffering from hyper- or low tension, and knowing that these diseases can lead to serious complications it is essential to measure blood pressure with high accuracy. The present methods for the measurement of the arterial pressure, by induction of a pressure through an armband (with a control device called sphygmomanometer) are known to introduce some significant errors. Those are caused partly by the inaccuracy and inadequacy of the armband and its dimensions (either the length or the circumference). The objective of the research presented in this paper is to study and simulate the discharge of the blood in the brachial artery. Based on these modelling results, the response of the fluid to the external pressure of the band can be obtained, and finally appropriate corrective factors between the true pressure and the read one could be assessed. From this perspective, research is carried out with the aim of sharing medical and engineering views on this subject. The artery can be modelled as a particular deformable pipe, when the blood is actually a fluid with specific properties. Thus the two complementary and interconnected domains are covered, so be it the solid mechanics (to obtain analytic relations between the strains and the deformations, using either linear or non-linear theories) and the fluid mechanics (to study the discharge of the blood considering a particular deformable pipe, using finite volume methods). Finally, many factors such as BC, for example, or other specific parameters have to be investigated deeply. In this paper, only the hydraulic results will be presented, discussed and analysed. The mechanical developments, being for now essentially analytical, will be mostly left out. Keywords: blood discharge in vessels, vessels’ behaviour, blood pressure, capillaries network. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090131

134 Modelling in Medicine and Biology VIII

1

Introduction

Blood pressure is by far one of the most essential measures when it comes to human health. Although it is well known that hypertension and low tension diseases are widespread all around the world, one does not always know the impact of a physiologically abnormal measure. For instance, an increase of 5mmHg will double the risks of cardiovascular diseases, it is therefore essential not to underestimate the blood pressure. But it is also important not to overestimate it, cause that would lead to useless treatments, which are costly and sometimes risky if the patient is in good health. The principal method used currently is based on Korotkoff’s discoveries.

(a) Figure 1:

(b)

Operating mode – (a) Surgical appliances – (b) Evolution of the different pressures during the measurement.

It requires the use of a sphygmomanometer (which is a control device of pressure composed of a manometer and a peer allowing the pressure to be induced), an armband and a stethoscope (cf. figure 1(a)). First, a pressure must be induced through the armband, in order to stop the blood flow. Then, by letting this pressure decreasing, the flow resumes and makes some noise called “Korotkoff’s noise”. The doctor then supplies two numbers. The first one corresponds to the pressure when the flow begins (first noise) and the second one to the pressure when the flow stabilizes (no external effect anymore, the noise disappears gradually during the deflation and finally stops). They are respectively called the systolic and diastolic pressure (cf. figure 1(b)). A patient called physiologically normal shall have a pressure around [120, 80] mmHg, what the doctor will call 12/8. The objective at the University of Liège has been to simulate this operating mode. With a numerical access to the phenomenon, one could avoid the problem of surgical appliances’ inaccuracy, and in this way find corrective factors to interpret the falsified measures.

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135

Solution method

The goal intended is to simulate the discharge of the blood in the arteries, more particularly in the brachial artery during blood pressure measurement. Ingeniously speaking, the artery can be modelled as a particular deformable pipe, when the blood is actually a fluid with specific properties. Thus the research coves two complementary and interconnected domains, so be it the solid mechanics and the fluid mechanics, both studied at the MS²F sector. With the solid mechanics, analytic relations between the external strains (not only the armband’s pressure, but also the blood pressure) and the deformations can be obtained, using either linear or non-linear theories. In this manner, the way the artery changes its shape may be determined. Given the mathematically known deformations for the artery, it remains to study the discharge of the blood in this particular deformable pipe, what has been done with specific original VisualBasic routines, but also with the modelling system WOLF from the HACH (Applied Hydrodynamics and Hydraulic Constructions), which is a finite volume flow simulation model developed at the University of Liège.

Figure 2:

Scheme of the solution method.

Behind these theoretical studies, bigger challenges arise. It effectively appears that while the medical understanding of the phenomenon is real, the knowledge of many factors is quite poor. Thus many elements, e.g. the boundary conditions or the mechanical parameters of the living tissues (depending on the chosen material model), have to be investigated using experiments. In this way, the authors have conducted a number of parallel researches about anatomical and physiological data, with the final purpose of completing and calibrating the models, but also to finally validate their accuracy.

3

Equations from solid mechanics analysis

Under geometrical (cf. figure 3) and phenomenological (the contact between the materials are assumed to be perfect) idealization, mathematical analyses have WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

136 Modelling in Medicine and Biology VIII been performed in order to determine the way the artery deflates under specific strains (the internal and external pressures). Thus the artery has been imagined as a particular pipe, and mathematical developments have led to finally get analytic relations between the section and the pressures.

(a) Figure 3:

(b)

Transverse section of the arm – (a) Actual situation – (b) Idealized situation.

To obtain the analytic results, two main hypotheses have to be made, one concerning the materials’ properties and the other the materials’ behaviour under strain; the muscle and the artery are supposed to be transversely isotropic and to follow a plane state of strain. These two assumptions come from observations about the materials’ structure, as seen in figure 4, and are rather acceptable. In fact, the structures of both the artery and the muscle are concentric and present a cylindrical symmetry [1]. Moreover, the stresses are perpendicular to the solids’ generatrix, and one can reasonably accept that the distortions are nil in the longitudinal direction [1]. The materials’ behaviour must also be assumed. In this way it has been chosen to study the two materials from linear and non-linear perspectives. Despite the fact that the known response toward external strains follows nonlinearity (because of the big deformations actually arising) the linear analysis can give a practical and prompt insight of the problem, and has thus been studied as well. The linear analysis gives a first approach of the problem, and the different developments are rather obvious [1]. The results represent the radial displacement for the artery (eq.(1)) ur ( r = ri ) =

1 1  1  2 G1  rc2 − ri2

 3 2 x x 2  (1 − 2ν 1 ) ri pi − rc ri pc + pi − pc ri rc 

(

(

) (

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)

)

(1)

Modelling in Medicine and Biology VIII 1  1  1  1  1 −ν1 ) ri2 pi + 1 − ν 2 ) re2 pe  r 2 − r 2  (  r 2 − r 2  ( G G 1 2 i  c   c  e p cx = 2  1  1  2 1  1  2 2 2    r (1 − 2ν 2 ) + re +  2 2  rc (1 − 2ν 1 ) + ri  G2  r 2 − r 2  c G − r r 1 e c c i     

(

)

(

137

(2)

)

   

where ur [m] is the radial displacement, Gj [MPa] is the shear modulus of the materials, νj [-] is the Poisson coefficient of the materials, rk [m] is the radius of the zones, pk [Pa] is the pressure of the zones, pcx [Pa] is the equilibrium contact pressure, j = (1,2) = (artery,muscle), k = (i,c,e) = (artery,contact,muscle). Similar results can be obtained for the non-linear analysis (see [1]). However, in order to physically interpret the results of the model in this second case, it would also be necessary to thoroughly investigate the non-linear parameters, which are for now almost absolutely unknown.

(b)

(a) Figure 4:

4

Material coaxial structure – (a) Muscle – (b) Artery.

Specificities of blood

The blood is actually a particular fluid. Before going further into the subject, it is of great importance to be precise about what makes it so specific. Its composition, of plasma and formed elements, lead its viscosity to be nonNewtonian. In fact, the presence of a solid part (the plasma) influences the behaviour of the fluid. However, in large vessels, the fluid can be considered as Newtonian and homogeneous [3–4], as the boundary layer is very small compared to the vessel’s WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

138 Modelling in Medicine and Biology VIII radius. In the capillaries, this assumption is no longer acceptable, the flow being determined mainly by viscous stresses. Beside, one can reasonably assume that the blood is incompressible, the Mach number being globally quite small. Finally, the turbulent effects have been neglected. It is well known that some source of instabilities create local turbulence, but analysis of both Reynolds and Womersley numbers show that this last hypothesis is acceptable [5]. For the smaller vessels, however, these assumptions have to be reconsidered, but for now, they have not been updated.

5

Mathematical model

The equations of the fluid mechanics problem are the Navier-Stokes equations (eq.(3)).  ∂Ω  ∂t + ∇ ( ρ ⋅ Ω ) (3) ∂  U + U ⋅∇U = −1 ⋅ ∇p + ν ⋅ ∇ 2U ρ  ∂t They represent, respectively, the conservation of the mass and the momentum. The two variables are the section Ω and the velocity U. 5.1 Integration of the equations on the area In the specific framework of blood discharge, some behavioural observations can be made. First, the flow is mostly one-dimensional. Second, the vertical accelerations are negligible, the pressure being thus hydrostatic. Finally, despite the fact that the streamline curvature can not actually be considered as small, no serious mistakes are made if these are neglected [5]. With these three acceptable assumptions in mind, the Navier-Stokes equations can be area-integrated [6] into the following form:  ∂Ω ∂Q  ∂t + ∂x = ql  (4)  2    ∂Q + ∂  ϕ Q + gI1  = gI 2 + 1 ( Fx + S x ) + Uql   ∂t ∂x  Ω ρ   with

I1 ( h ) =

hf

∫ ( h − ξ ) l ( x,ξ )dξ ;

− hb

I2 ( h) =

hf

∫ (h −ξ )

− hb

∂l ( x, ξ ) ∂x

dξ

(5)

where t [s] is the time, x [m] is the axial position, Ω [m²] is the section, Q [m³/s] is the discharge, ql [m²/s] is the lateral in- or out-flow, φ [-] is the parameter of non-uniform velocity repartition on the section, g [m/s²] is the gravity acceleration, ρ [kg/m³] is the density of blood, Sx [kg/s²] is the integrated effect of Reynolds axial stresses, Fx [kg/s²] is the global effect of roughness and WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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U [m/s] is the axial velocity. One must keep in mind that this set of equations is valuable under the assumptions made about the blood in section 4. 5.2 The Preismann Slot The presented equations are usually used to characterize free-surface flows. The Preismann Slot model [7] introduces an ingenious device to allow the modelling of both free surface and pressurized flows through the same set of equations. It consists in considering a narrow slot on the top of a closed pipe. The width of the slot (Tf [m]) is chosen in order to represent the behaviour of the conceptual free surface flow, with a gravity wavespeed represented by c =

6

gΩ (6). Tf

Numerical resolution

The set of equations introduced above (cf. section 5.1) requires the use of a numerical resolution. As for most of the hydraulic problems, the finite volume method seems to be the most appropriate way to solve the problem. In this section, we will first present the algorithm itself. Second, the boundary conditions required will be discussed. 6.1 Algorithm The set of equations can be written as follow ∂ ∂ U + F (U ) = S (U ) ∂t ∂x

(7)

where the vectors of conservative variables, flux and sources are respectively ql   Q   Ω     ; S (U ) = U =   ; F (U ) = Q ² 1   ϕ + gI1  Q  gI 2 + ρ ( S x + Fx ) + Uql   Ω   

(8)

Finally, the integration of (7) lead to the following conservative formulation:  Fi + 1 − Fi − 1  2 2 U in +1 = U in − ∆t  + S  ∆x   

(9)

The numerical resolution uses a Runge-Kutta time-splitting scheme of the third order, with the three following sub-step evaluation

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140 Modelling in Medicine and Biology VIII Uin ,1

Uin,2

( )

( )

 F 1 Uin − F 1 Uin  i+ i− 2 2 − ∆t  + S Uin  ∆x   n n ,1 ,1  F 1 Ui − F 1 Ui i+ i− 2 2 = Uin − ∆t  + S Uin,1 ∆x 

= Uin

F 1 i+ 2 Uin ,3 = Uin − ∆t  

( )

(

)

(

)

(

)−F (

Uin ,2

Uin ,2

i− 1

∆x

2

)



(

)

(

)

(10)

  + S Uin ,2  

And the final solution is given by U in +1 = α U in ,1 + β U in ,2 + γ U in,3

(11)

with α + β + λ = 1

6.2 Boundary conditions The numerical resolution requires the imposition boundary conditions, actually one per each variable. The habit would be the forcing of the section and the pressure, either at the downstream or the upstream. However, these quantities are not measurable; in fact, having access to such numerical values would require probing or in vivo tests, which is out of our skills. At the MS²F, the fact that the behaviour of both velocities and pressures are known in the capillaries (cf. figure 5) has been used. The boundary conditions are thus implemented as capillaries conditions, which can be done using branches following the architecture of vessels in the arm (cf. figure 6(a) and (b)).

(a) Figure 5:

(b)

Evolution in the vessels – (a) Systolic and diastolic pressures – (b) Section and velocity.

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(b) (a)

Figure 6:

7

141

(c)

Architecture of the vessels – (a) Transition between arteries and veins – (b) Capillaries network – (c) Simplified Capillaries network.

Modelling

On the one hand, the flow through the artery is modelled considering a single rectilinear pipe, with the only boundary condition being the upstream discharge. On the other hand, localized capillary-pumping (as shown in figure 6) is considered such that the flow entering the micro-vascular network behaves as observed on figure 5, so that its oscillation is damped and the pressure decays. For now, the complete modelling (from the entrance of the artery to the exit after the multiple capillaries network) has not been done, but severe tests have been performed in order to characterize the behaviour of both the artery and the capillaries network on their own. For the final integration to be done, it remains to determine the pumping law, so the way the volume of blood is distributed from the artery to the smaller vessels as a function of the arterial pressure. Moreover, the integration of the walls’ behaviour remains to be done. At the moment, the modelling of the blood discharge has been made with walls free of movement.

8

Results

As said before, the only results we have for now concern the mechanical behaviour of the vessels, and on the hydraulic behaviour of both the artery and the capillaries network on their own. Those are successively introduced and examined in this section. 8.1 Behaviour of the materials The established equations give the response of both the artery and the muscle to the external strains. Severe results and their analysis can be found in [1]. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

142 Modelling in Medicine and Biology VIII

Figure 7:

Figure 8:

Evolution of the heat [m] as a function of the axial position [m].

Evolution of the heat [m] as a function of the time [s] for the downstream and the upstream.

8.2 Modelling of the artery For the artery on its own, the numerical resolution needs to assume the downstream pressure, since the capillaries conditions have not been implemented yet. With a hypothetic constant pressure at the downstream and a periodic WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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sinusoidal-type discharge at the upstream, the numerical model gives results similar to the one presented below (see figures 7 and 8). The behaviour is typical for the described flow, whose regime is clearly subcritical. The presented results correspond to the following data: Lmax = 15m (length); Φmean = 0.25m (mean diameter for an empty slot); ∆x = 0.1m; Nc = 0.1 (CFL number); Tf = 0.001m; fRoughness = 0.02; RK1 = 0.15; RK2 = 0.45; RK3 = 0.4; QUpstream,mean = 8m³/s. The fluid characteristics are the water’s one. As can be seen above, the parameters do not fit at all with the physiologic values that can be found in the literature. At this point of the research, it has been chosen to analyse the problem with a qualitative point of view, in order to preserve the model from any scale problem.

Figure 9:

Evolution – (a) of the discharge [m³/s] and – (b) the heat [m] as a function of the axial position [m].

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144 Modelling in Medicine and Biology VIII 8.3 Modelling of the capillaries network For the modelling of the capillaries network, the resolution scheme seen in equations (7)–(11) must be reconsidered. The network is actually simplified as shown in figure 6(c), and the model focus on only one path. There are thus no considerations of lateral outflow, but the junctions are taken into account by considering a revaluation of the incoming flux. The goal of this modelling is to well represent the behaviour noticed in section 6.2. By choosing appropriate values for the roughness coefficient and the Preismann slot (values that are for now dictated only by a qualitative representation will) the dampening of the heat can be well represented. In fact, these two parameters are able to translate in a certain way how the surrounded materials behave. Once again, a hypothetic constant pressure at the downstream and a periodic sinusoidal-type discharge at the upstream have been chosen. The numerical model gives results similar to the one presented below (see figures 9 and 10). The important point here is not the evolution in respect to the axial position, but rather the one with respect to the time (see figure 10). It is effectively clear, by a comparison of these results and the one presented in section 8.2, that the consideration of a wide capillaries subdivision combined with appropriate parameters that represent the dissipative behaviour of the materials lead to the observation of the wanted dampening. The presented results correspond to the following data: Lmax = 15m; Φmean = 0.25m; ∆x = 0.025m; Nc = 0.1; Tf = 0.01m; fRoughness = 0.2; RK1 = 0.15; RK2 = 0.45; RK3 = 0.4; QUpstream,mean = 4m³/s; lcap = 0.5m (length of one intermediate segment). The fluid characteristics are once again the water’s one.

Figure 10:

Evolution of the heat [m] as a function of the time [s] for the downstream and the upstream.

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145

Conclusions

For now, the research is still at its early stage, and there are plenty of perspectives already in our mind. From the precision of the parameters to the final integration of the two presented models, the best is yet to come. However, several interesting results have already been found, and allow us to think about the remaining work with confidence. The boundary conditions deserve to be highlighted, the results presented in section 8.3 showing clearly the expected behaviour. Moreover, as the parameters have not been clarified at this time, it is expected that the quantitative aspect of this specificity will also be pointed as soon as this obstacle is overcome. Finally, the developed mechanical equations are ready to be implemented, while on the other hand the developed model has been elaborated such that the incoming modifications should be easy.

References [1] Paulus, R. Analyse des effets mécaniques et hydrauliques induits dans le bras par un brassard de mesure de la pression artérielle. 2007. Master thesis, University of Liège, Belgium. [2] Paulus, R., B.J. Dewals, S. Erpicum, S. Cescotto, and M. Pirotton. Numerical Analysis of Coupled Mechanical and Hydraulic Effects Induced by a Blood Pressure Meter. 2008. Proceedings of the 4th International Conference on Advanced Computational Methods in Engineering (ACoMEn), Liège, Belgium. [3] Fung, Y.C. Biomechanics, Circulation. 1996. Springer Science, New York, USA. [4] Fung, Y.C. Biomechanics, Mechanical Properties of Living Tissues. 1993. Springer Science, New York, USA. [5] Pochet, T. Ecoulement pulsatoire d’un fluide dans une conduite à parois déformables. 1986. Master thesis, University of Liège, Belgium. [6] Archambeau, P. Contribution à la modélisation de la genèse et de la propagation des crues et innondations. 2006. Thesis, University of Liège, Belgium. [7] Preismann, A. Propagation des intumescences dans les canaux et rivières. First Congress of the French Association for Computation. 1961. Grenoble. France.

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Section 3 Biomechanics

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Biomechanical consideration for dorsal-lumbar and lumbar sagittal spine disorders C. Bignardi1, A. Ramieri2 & G. Costanzo2 1

Department of Mechanics, Politecnico di Torino, Torino, Italy Department of Orthopaedics, University “La Sapienza”Polo Pontino ICOT of Latina and Don Gnocchi Foundation, Rome, Italy

2

Abstract Lower back pain is one of the most important socioeconomic diseases and one of the most important health care issues today. On average, 50-90% of the adult population suffers from lower back pain and lifetime prevalence of lower back pain is 65-80%. The causes of lower back pain often remain unclear and may vary from patient to patient. It is estimated that 75% of such cases are associated with lumbar degenerative intervertebral disc disease. Dysmorphisms or sagittal disorders of the rachis mean back (kyphosis) or front (lordosis) pathological deviations, irreducible in various measures, resulting from structural diskligament and vertebral alterations of various etiologies. Three numerical models of the dorsal-lumbar spine, respectively a physiological model, a ipolordotic model and a kyphotic model, have been realized considering bone, disks and ligaments with their specific mechanical characteristics. For the load and boundary conditions chosen, joint facets and intervertebral disk stresses and disk bulge have been compared for the three spine situations. When it has been possible, the obtained results have been validated with data available in literature regarding both experimental and computational studies. In conclusion it can be assumed that dorsal-lumbar and lumbar sagittal spine disorders can determine premature disks and joints alterations. In particular, it seems that dorsal-lumbar kyphosis, more than lumbar ipolordosis, can expose joint facets and disks to non physiological loads. In addition, if the genetic role of disk degeneration is acknowledged, it is probable that the correction, at a precocious age, of sagittal spine imbalance, can prevent or slacken the disk-joint lumbar-sacral degenerative phenomena. The obtained results agree with the clinical experience. Keywords: spine sagittal disorders, biomechanics, FEM. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090141

150 Modelling in Medicine and Biology VIII

1

Introduction

The concept of sagittal alignment and spine balancing has been introduced in literature in order to define the limit of normality of kyphosis and spinal lordosis. Different studies have been carried out regarding the thoracic and lumbar spine, but few have been carried out concerning the cervical spine, with evaluations of asymptomatic subjects, often volunteers, during growth or as adults. Only quite recent observations have highlighted how spinal-pelvic balancing alterations are responsible for premature lumbar intervertebral disk degenerations, lumbago or spondylolisthesis [3,4]. The basic loading modes acting on the spine, while performing daily activities, are axial compression, flexion/extension, lateral bending and torsion. Load is transferred from one vertebral end-plate to the next by means of nucleus pulposus and annulus fibrosus. Load produces complex stresses within the annular ring. In order to compare joint facets and intervertebral disks stresses and disks bulging for different spine situations, three numerical models of the dorsallumbar spine, respectively a physiological model, an ipolordotic model and a kyphotic model, have been realized considering bone, disks and ligaments with their specific mechanical characteristics. The greatest difficulty in the numerical modelling of the spine is found in the simulation of the intervertebral disk. Compression testing has been the most commonly used method for the study of mechanical behaviour of the intervertebral disk [5–8], but many experiments have been also done subjecting the intervertebral disk to bending and torsional loads and to pure share loading [9–13]. Its viscoelastic nature has also been determined [14]; typically, all viscoelastic structures exhibit hysteresis, intervertebral disks also show this phenomenon in which there is loss of energy after repetitive loading-unloading cycles. Hysteresis has been observed to vary with the applied load, age and disc level [15]. Considering the complex structure of the intervertebral disk and the diverse stresses to which it is subjected under physiological loading conditions, it is clear that experimental techniques alone are not sufficient to fully characterize the overall mechanical behaviour of the motion segment. This is corroborated by the technical complexities that precluded the measurement of the stress state, deformation and disk bulge at different locations throughout the motion segment. This provided the motivation for the development of numerical methods, such as finite element analysis, to expand the experimental data in order to characterize the intervertebral disk parameters, which may be difficult to measure experimentally. Many researchers have simulated the intervertebral disk mechanics using the finite element method. Belytschko et al [16] were the first to use the finite element method for understanding the intervertebral disk mechanics. The disk-body unit was assumed to be an axisymmetric object and annulus as a linear orthotropic material. This model was further expanded to accommodate the nonlinear (orthotropic) properties for the annulus, keeping all other parameters unchanged [17]. Different approaches were followed by Lin et al [18] and by Spilker [19]. The first attempt to make a realistic finite element model of the lumbar intervertebral disk, considering the composite nature of the WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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annulus fibrosus, was made by Shirazi et al [20]. For the first time, this model accounted for both material and geometric nonlinearities along with the representation of the annulus as a composite of collagen fibres embedded in a matrix of ground substance. The nucleus was modelled as an incompressible, inviscid fluid. The model was based on the lumbar L2-L3 functional spinal unit. This model was compared to the experimental observations of load-displacement behaviour, disk bulge, end-plate bulge and intradiscal pressure. Stress and strain distribution in the cortical/cancellous bones, end-plates, annulus fibres and annulus ground substance were reported under compressive load. The same model was expanded to assess the effect of axial torque in combination with compression [21] and sagittal plane moments [22]. There were also significant efforts by other researchers to understand the intervertebral disk mechanics by taking into consideration the experimental results and physiological conditions of other spine components. Crisco and Panjabi [23] compared the lateral stabilizing potential of the lumbar spine muscles as a function of the architecture. Stabilizing effects of muscles on the overall mechanics of a lumbar spine were observed by Goel et al [24]. A viscoelastic model to study the changes in load sharing during the fast and slow loading rate was analyzed by Wang et al [25]. It has been known for a long time that the biphasic nature (solid and fluid phase) of the disk components plays a major role in the loading mechanism of the hydrated intervertebral disk. In the late 1980s, there was an increasing interest in the modelling of the disk as a saturated porous media by using the poroelastic approach [26–28]. Important models are those worked out by Eberlein et al [29], Vena et al [30], Lavaste et al [31] and Cao et al [32]. Many authors have studied pathological conditions of the functional spine unit related to disk degeneration processes, osteoporotic condition, end-plate fractures and different resections of the iatrogenic nature [33–38], but at the moment, as far as we know, the influence on joint facets and intervertebral disk stresses and the disk bulge of spinal-pelvic balancing alterations has not been studied.

2

Materials and methods

Starting from a CAD model of a spine available in the web [39], realized with CT images regarding a healthy 35-year-old male subject, a FEM model of the spine zone T11-S1, that later on we will call “physiological”, has been worked out. The external three-dimensional geometry of vertebral bodies and disks from the starting CAD model have been utilized; spongious bone, nucleus pulposus and annulus fibrosus, constituents of the intervertebral disk, endplates, ligaments and muscles have been drawn afterwards. In order to verify the geometric reliability of the CAD model, it has been scaled and placed upon the RX and MR images that we had at our disposal, regarding a healthy subject, see fig. 1a), b). Afterwards the model was modified, based on RX and MR images that we had at our disposal, making the spinal curve pathologic, respectively ipolordotic and kyphotic; two further FEM models have been so realized, see fig. 2. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

152 Modelling in Medicine and Biology VIII

a) Figure 1:

a) Figure 2:

b)

a) Lumbar zone of the “physiological” FE model realized placed upon an MR image of a healthy subject, b) detail of T11-T12 spinal unit that shows ligament distributions and applied loads.

b)

c)

d)

a) RM image of an ipolordotic spine, b) FE model of an ipolordotic spine, c) RM image of a dorsal-lumbar kyphotic spine, d) FE model of a dorsal-lumbar kyphotic spine that compensates with an ipolordosis.

As regards the modelling of annulus fibrosus, the type of model suggested by Shirazi et al [20] has been adopted; three concentric layers of fibrous tissue in which bundles of connective fibres runs sideways from the nucleus to the outside, alternating their directions in the contiguous layers so as to form respectively 45-70-120º angles with regard to a horizontal plane, have been simulated. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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The modelled materials are listed in table 1; mechanical properties agree with data found in the literature [20]. Ligaments and muscles considered in the modelling, fig. 1b), which give intrinsic stability to the spine, are shown in table 2 with their geometric and mechanical characteristics according with literature [23,24,40–42]. Table 1:

Mechanical properties of materials.

Material Cortical bone Spongious bone Endplate Annulus matrix Annulus fibbers Nucleus Table 2:

Young’s modulus [MPa] 12000 100 23000 4 450 4

Poisson’s ratio 0.3 0.3 0.3 0.4 0.499 0.499

Geometric and mechanical properties of ligaments and muscles.

Material Interspinous ligaments Intertransversal ligaments Anterior longitudinal ligament Posterior longitudinal ligament Sovraspinosus ligament Spinal muscles

Cross section area [mm2] 36.3 2 25.5 9.2 75.7 75.7

Young’s modulus [MPa] 11.6 58.7 20 20 15 0.1

Poisson’s ratio 0.3 0.3 0.3 0.3 0.3 0.3

Solid elements have been used for the discretization of bone, nucleus pulposus, annulus matrix and endplates, while bar elements have been used for annulus fibres, ligaments and muscles. The three models have been bound in correspondence with the sacrum and loaded with a vertical distributed force (350N) in correspondence with T11. For the modelling, the commercial softwares Rhinoceros (Robert McNeel & Associates, Seattle, USA) and Patran/Nastran (MSC Software Corporation, Santa Ana, USA) have been used.

3

Results and discussion

Joint facets and intervertebral disks stresses and disk bulge have been compared for the three spine situations. In figure 3 the results are shown as regards joint facets, in terms of mean von Mises stress values computed in the contact areas.

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154 Modelling in Medicine and Biology VIII Joint facets mean von Mises stresses 60

[MPa]

50 40

Physiological

30

Ipolordotic

20

Kyphotic

10

S1

T1 1 lo T1 w 2 T 1 up 2 lo L1 w L1 u p lo L2 w L2 up lo L3 w L3 up lo w L4 u L4 p lo w L5 u L5 p lo w

0

Location

Figure 3:

Comparison among mean von Mises stresses on joint facets, lower and upper, for the three sagittal spine configurations.

From the analysis of the diagram it can be observed how, in the physiological model, stress values are highest in correspondence with the thoracic zone and the sacrum and lower in correspondence with the lumbar zone. As regards the kyphotic model, generally intervertebral facets appear more stressed than the other two models, except in correspondence with the sacrum, where we find a reduction of stress state compared with the physiological model. This result could be caused by the fact that in that zone rachis compensates with an ipolordosis. As regards the ipolordotic model, stresses on facets appear in every zone of the spine lower than in the other two models. Results have been compared and validated with results obtained by Goto et al [43] as regards a physiological model of a L4-L5 functional spinal unit. In figure 4 the results are shown as regards intervertebral disks, in terms of mean von Mises stresses; the values shown represent the average stress of the whole intervertebral disk, including nucleus pulposus and annulus fibrosus. Both pathological models show a remarkable increase of stress values in correspondence of all zones of the spine if compared with those obtained with the physiological model. For the kyphotic model in particular, it is evident that there is high stress in correspondence of L5-S1 where, what’s more, the MR images at our disposal showed a disk degeneration. In this case we compared and validated our results with those obtained by Natarajan and Andersson [44] as regards a physiological model of a L3-L4 functional spinal unit. As regards disk bulge, both pathological models show a general increase, both in the sagittal and in the frontal plane, if compared with the disk bulge obtained for the physiological model; in particular, the increase is higher in correspondence of those zones that RX and MR images suggest at the risk of disk degeneration. For example, in table 3 data are shown obtained for the L4-L5 WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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0,6 0,5 0,4 0,3 0,2 0,1 0

Physiological

Ipolordotic

L5_S1

L4-L5

L3_L4

L2_L3

L1_L2

T12_L1

Kyphotic

T11_T12

[MPa]

Intervertebral disks mean von Mises stresses

Location

Figure 4:

Comparison among intervertebral disks; mean von Mises stresses for the three sagittal spine configurations.

Table 3:

Sagittal disk bulge [mm] for the three spine configurations in L4-L5.

Physiological model Front Back 0,99 0,99

Kyfotic model Front Back 1,32 1,32

Ipolordotic model Front Back 1,32 2,47

functional spinal unit. In this case we compared and validated our results with those obtained by Shirazi et al [21] as regards a physiological model of a L2-L3 functional spinal unit.

4

Conclusions

The analysis carried out proves how geometric variations of the rachis curve, both at lumbar and thoracic level, conditions a lot of the stress distribution of the intervertebral disk and of the joint facets. In particular, simulations carried out with the realized models have shown a local increase of disk stresses and bulge in those zones where RX and MR images at our disposal showed degenerated disks. In the ipolordotic model, joint facet stresses are on average lower than in the physiological model, while in the kyphotic model, stresses reach higher values in correspondence of T12-L1. Intervertebral disk stresses are on average higher in the ipolordotic and in the kyphotic models than in the physiological model. Nevertheless, stresses are particularly higher in the kyphotic model in correspondence of T12-L1 and L5-S1. As regards bulge, the highest values are in L4-L5 and L5-S1. In conclusion it can be assumed that dorsal-lumbar and lumbar sagittal spine disorders can determine premature disk and joint alterations. In particular, it seems that the dorsal-lumbar kyphosis, more than the lumbar ipolordosis, can expose joint facets and disks to non physiological loads. In addition, if the genetic role of disk degeneration is acknowledged, it is probable that the correction, at a precocious age, of sagittal spine imbalance, can prevent or slacken the disk-joint lumbar-sacral degenerative phenomena. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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References [1] Heliovaara, M., Makela, M., Knekt, P., Impivaara, O. & Aromaa, A. Determinants of sciatica and low-back pain. Spine, 16(6), pp. 608-614, 1991. [2] Manchikanti, L. Epidemiology of low back pain. Pain Physician, 3(2), pp. 167-192, 2000. [3] Rajnics, P., Templier, A., Skalli, W., Lavaste, F. & Illes, T. The importance of spinopelvic parameters in patients with lumbar disc lesions. International Orthopaedics, 26(2), pp. 104-108, 2002. [4] Rajnics, P., Templier, A., Skalli, W., Lavaste, F. & Illés, T. The association of sagittal spinal and pelvic parameters in asymptomatic persons and patients with isthmic spondylolisthesis. Journal of Spinal Disorders & Techniques, 15(1), pp. 24-30, 2002. [5] Brown, T., Hanson, R. & Yorra, A. Some mechanical tests on the lumbosacral spine with particular reference to the intervertebral discs. The Journal of Bone and Joint Surgery, 39A (5), pp. 1135-1164, 1957. [6] Farfan, H.F. Mechanical Disorders of the Low Back, Lea and Febiger, Philadelphia, 1973. [7] Hirsch, C. The reaction of intervertebral discs to compression. The Journal of Bone and Joint Surgery, 37A (6), pp.1188-1896, 1955. [8] Hirsch, A. & Nachemson, A.L. New observation on the mechanical behaviour of the lumbar discs. Acta orthopaedica Scandinavica, 23, pp. 254-283, 1954. [9] Galante, J.O. Tensile properties of the human annulus fibrosus. Acta orthopaedica Scandinavica, Suppl. 100, pp. 4-91, 1967. [10] Fujita Y., Duncan, N.A. & Lotz, J.C. Radial tensile properties of the lumbar annulus fibrosus are site and degeneration dependent. Journal of Orthopaedic Research, 15(6), pp. 814-819, 1997. [11] Roaf R. A study of the mechanics of spinal injuries. The Journal of Bone and Joint Surgery, 42B, pp. 810-823, 1960. [12] Farfan, H.F., Cossette, J.W., Robertson, G.H., Wells, R.V. & Kraus, H. The effect of torsion on the lumbar intervertebral joints: the role of torsion in the production of disc degeneration. The Journal of Bone and Joint Surgery, 52A, pp. 468-497, 1970. [13] White, A.A. & Panjabi, M.M. Clinical Biomechanics of the Spine. II ed. J.B. Lippincott Company, Philadelphia, 1990. [14] Markolf, K.L. & Morris, J.M. The structural components of the intervertebral disc: a study of their contributions to the ability of the disc to withstand compressive forces. The Journal of Bone and Joint Surgery, 56A, pp. 675-687, 1974. [15] Virgin, W. Experimental investigations into physical properties of intervertebral disc. The Journal of Bone and Joint Surgery, 33B(4), pp. 607-611, 1951.

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[16] Belytschko, T.B., Kulak, R.F., Schultz, A.B. & Galante, J.O. Finite element stress analysis of an intervertebral disc. Journal of biomechanics, 7(3), pp. 277-285, 1974. [17] Kulak, R.F., Belytschko, T.B., Schultz, A.B. & Galante, J.O. Nonlinear behaviour of the human intervertebral disc under axial load. Journal of biomechanics, 9(6), pp. 377-386, 1976. [18] Lin, H.S., Liu, Y.K. & Adams, K.H. Mechanical response of the lumbar intervertebral joint under physiological (complex) loading. The Journal of Bone and Joint Surgery, 60A, pp. 41-55, 1978. [19] Spilker, R.L. A simplified finite element model of the intervertebral disc. In Finite Elements in Biomechanics, ed. Simon, B.R., Tuscon, University of Arizona, pp. 729-747, 1980. [20] Shirazi, A., Shrivastava, S.C. & Ahmed, S.M. Stress analysis of the lumbar disc-body unit in compression. Spine, 9(2), pp. 120-134, 1984. [21] Shirazi, A., Ahmed, A.M. & Shrivastava, S.C. Mechanical response of a lumbar motion segment in axial torque alone and combined with compression. Spine, 11(9), pp. 914-927, 1986. [22] Shirazi, A., Ahmed, A.M. & Shrivastava, S.C. A Finite Element study of a lumbar motion segment subjected to pure sagittal plane moments. Journal of Biomechanics, 19(4), pp. 331-350, 1986. [23] Crisco, J.J. & Panjabi, M.M. The intersegmental and multisegmental muscles of the lumbar spine: a biomechanical model comparing lateral stabilizing potential. Spine, 16(7), pp. 793-799, 1991. [24] Goel, V.K., Kong, W., Han, J.S., Weinstein, J.N. & Gilbertson, L.G. A combined Finite Element and optimization investigation of lumbar spine mechanics with and without muscles. Spine, 18(11), pp. 1531-1541, 1993. [25] Wang, J.L., Parnianpour, M., Shirazi, A., Engin, A.E., Li, S. & Patwardhan, A. Development and validation of a viscoelastic Finite Element model of an L2/L3 motion segment. Theoretical and Applied Fracture Mechanics, 28, pp. 81-93, 1997. [26] Simon, B.R., Carlton, M.W., Wu, J.S.S., Evans, J.H. & Kazarian, L.E. Structural models for human spinal motion segments based on a poroelastic view of the intervertebral disc. Journal of Biomechanical Engineering, 107, pp. 327-335, 1985. [27] Laible, J.P., Pflaster, D.S., Krag, M.H., Simon, B.R., Pope, M.H. & Haugh, L.D. A poroelastic swelling Finite Element model with application to the intervertebral disc. Spine, 18(5), pp. 659-670, 1993. [28] Argoubi, M. & Shirazi, A. Poroelastic creep response analysis of a lumbar motion segment in compression. Journal of Biomechanics, 29(10), pp. 1331-1339, 1996. [29] Eberlein, R., Holzapfel, G.A. & Fröhlich, M. Multi-segment FEA of the human lumbar spine including the heterogeneity of the annulus fibrosus. Computational Mechanics, 34, pp. 147–163, 2004. [30] Vena, P., Franzoso, G., Gastaldi, D., Contro, R. & Dallolio, V. A finite element model of the L4-L5 spinal motion segment: biomechanical

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[31] [32] [33] [34] [35]

[36] [37] [38]

[39] [40] [41] [42] [43]

[44]

compatibility of an interspinous device. Computer methods in biomechanics and biomedical engineering, 8(1), pp. 7-16, 2005. Lavaste, F., Skalli, W., Robin, S., Roy-Camille, R. & Mazel, A. Threedimensional geometrical and mechanical modelling of the lumbar spine. Journal of. Biomechanics, 25(10), pp.1153-1164, 1992. Cao, K.D., Grimm, M.J. & Yang, K. Load sharing within a human lumbar vertebral body using the Finite Element Method. Spine, 26(12), pp. 253260, 2001. Mizrahi, J., Silva, M.J., Keaveny, T.M., Edwards, W.T. & Hayes, W.C. Finite Element stress analysis of the normal and osteoporotic lumbar intervertebral body. Spine, 18(14), pp. 2088-2096, 1993. Natarajan, R.N., Ke, J.H. & Andersson, G.B.J. A model to study the disc degeneration process. Spine, 19(3), pp. 259-265, 1994. Goel, V.K., Monroe, B.T., Gilbertson, L.G. & Brinckmann, P. Interlaminar shear stresses and laminae separation in a disc: Finite Element analysis of the L3-L4 motion segment subjected to axial compressive loads. Spine, 20(6), pp. 689-698, 1995. Lu, Y.M., Hutton, W.C. & Gharpuray, V. Can variations in intervertebral disc height affect the mechanical function of the disc? Spine, 21(19), pp. 2208-2217, 1996. Natarajan, R.N. & Andersson, G.B.J. Modelling the annular incision in a herniated lumbar intervertebral disk to study its effect on disk stability. Computers and Structures, 64(5/6), pp. 1291-1297, 1997. Pitzen, T., Geisler, F., Matthis, D., Muller-Storz, H., Barbier, D., Steudel, W. & Feldges, A. A Finite Element model for predicting the biomechanical behaviour of the human lumbar Spine. Control Engineering Practice, 10, pp. 83-90, 2002. http://homepages.ulb.ac.be/~anatemb/exchange.htm Pintar, F.A., Yoganandan, N., Myers, T., Elhagediab, A. & Sances, A.J. Biomechanical properties of human lumbar spine ligaments. Journal of Biomechanics, 25(11), pp.1351-1356, 1992. Brolin, K. & Halldin, P. Development of a Finite Element Model of the upper cervical spine and parameter study of ligament characteristics. Spine, 29(4), pp. 376-385, 2004. Natarajan, S., Dargahi, J. & Heidari B. Biomechanical effect of posterior elements and ligamentous tissues of lumbar spine on load sharing. Biomedical materials and engineering, 15(3), pp.145-158, 2005. Goto, K., Tajma, N., Chosa, E., Totoribe, K., Kuroki, H., Arizumi, Y. & Arai, T. Mechanical analysis of the lumbar vertebrae in a three-dimensional Finite Element Method model in which intradiscal pressure in the nucleus polposus was used to establish the model, Journal of Orthopaedic Science, 7, pp. 243-246, 2002. Natarajan, R.N. & Andersson, G.B.J. The influence of lumbar disc height and cross-sectional area on the mechanical response of the disc to physiologic loading, Spine, 24(18), pp. 1873-1881, 1999.

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A new brace for the treatment of scoliosis M. G. Antonelli1, P. Beomonte Zobel1, P. Raimondi1, T. Raparelli2 & G. Costanzo3 1

Department of Mechanical Engineering, Energy and Management, University of L’Aquila, Italy 2 Department of Mechanics, Turin Polytechnic, Italy 3 Department of Orthopaedic Surgery, University of Rome “La Sapienza” polo pontino, Italy

Abstract In this paper a new brace for the treatment of scoliosis is proposed. It uses pressurized air, by many air pockets, to apply the corrective thrusts to the rib cage and it can apply the thrusts at many levels of the spine. Moreover, the thrusts can be modified and monitored. In this way the corrective action on the spine can be more effective. The design of the prototype of the new brace is presented together with the design, the prototyping and the validation of the air pocket. Finally the first experimental tests on a simplified prototype of the brace are presented. Keywords: scoliosis brace, spinal orthoses, brace design, pneumatic pad.

1

Introduction

Scoliosis is a complex structural deformity of the spine in which there is an abnormal curvature of the vertebral column with respect to the 3 spatial axes. It appears as a lateral curvature on the frontal plane, a modification of the curves on the sagittal plane and a vertebral rotation on the horizontal plane [1]. Idiopathic scoliosis is the most common type and its origin is unknown. Brace treatment for idiopathic scoliosis has good support in published studies [1,2]. The brace is a tentative to slow the progression of the curve. Several different braces are used in the treatment of scoliosis, and most of them work on the curve via the pressure they exert on the rib cage, usually on 3 points, “to push against” the progressive abnormal curvature of the spine. The choice of the brace depends on the age, on the spine condition, but overall the doctor will select one based on WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090151

160 Modelling in Medicine and Biology VIII his experience with the different orthoses. A few centers treat young children, with severe idiophatic scoliosis, with a body cast that is fixed on the body and can be removed only permanently (it is periodically substituted because of the growth of the child and of the evolution of the scoliosis). Very common for the treatment of scoliosis are the removable braces, and many different types are on the market. The most used are Milwaukee and Boston, designed in the USA, Cheneau and Lyon, designed in France, and La Padula, designed in Italy. In this paper a new brace for the treatment of scoliosis is proposed. It uses pressurized air, by many air pockets, to apply the corrective thrusts to the rib cage and it can apply the thrusts at many levels of the spine. Moreover, the corrective pressure inside the pockets can be modified and monitored. In this way the action on the spine can be more effective. The design of the prototype of the new brace is presented together with the design, the prototyping and the validation of the air pocket. Finally, the first experimental tests on a simplified prototype of the brace are presented.

2

The use of brace in the treatment of scoliosis

Since 400 b.c Ippocrate tried to reduce the scoliosis curvature by a table, to which the patient was tied, and a person that hopped on him pushing the spine. In the following centuries many scientists (Galeno, Parrè, Delpech and others) were engaged in finding a therapeutic method for scoliosis: the application of mechanical pushing to convex parts of the spine, the extension of the spine by a traction force or by the weight of the patient. But only in the 1840s did the activity of orthopaedic medical centers in France specifically devoted to the correction of the scoliosis start. In the following decades the scientists move from the traction fixed devices to different ideas of brace. The brace of ShanzMilwaukee (1945) and the optimised version of Blount and Schmidt (1958) are the most significant ones of that period. The final part of the XX century saw a growing interest in research activities on scoliosis and on the braces. Many different braces were proposed in this period, often “specialised” for a specific curvature, also considering the age of the patient: the Lyon brace, the Boston brace, the Cheneau brace, the Lapadula brace, etc. The use of removable braces is recommended in scoliosis curvatures with the Cobb angle between 20±5° and 40±5° [1], depending on the authors. In this case the therapy is most effective with physical rehabilitation. Therapeutic rehabilitation is addressed for Cobb angle less than 15±5°, while the fixed brace and the surgical treatment is the therapy for the Cobb angle greater than 40±5° [1]. The prognosis of scoliosis is based on some clinical data (hump, Cobb angle, torsional angle, age, etc.) and on its localization and curve pattern. After this step, the clinician specializing in spinal diseases will choose the specific brace, if necessary and useful. The brace works via the thrust it exerts on the rib cage to push against the abnormal curvature of the spine and to reduce the local spasticity. The brace applies thrusts directly onto the rib cage, primary correction effect; consequently, some reaction thrusts arise at the contact area between the brace and thorax. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

a) Figure 1:

b) a) Sketch to show how a Milwaukee brace works [6], b) Milwaukee brace [7].

b)

a) Figure 2:

161

Some curvatures of the b) thoracolumbar, c) lumbar.

c) scoliotic

spine:

a)

thoracic,

Removable brace treatment for idiopathic scoliosis is a fundamental therapy, but the brace has to be light, patient compliant, constructed by an orthotist specializing in the construction of the prescribed brace system. Many brace systems have been proposed, as previously described. Each brace has its own characteristic for a specific correction of the spine. For example the Lyon brace is indicated for lumbar and for thoracolumbar curves. The Cheneau brace has three different versions: for single lumbar curves, for lumbar curves and for thoracic curve with an apex at L4/T10. The Milwaukee brace is the most common one for the treatment of scoliosis. It is used for thoracolumbar curves and for double curves. This brace uses two distinct principles: the 3 points thrust and the extension of the spine. The brace is linked to the pelvic region and has a top part to constrain the chin. The patient has to stay with the chin far from the constraint, and so a traction force is applied to the spine, fig. 1a). To obtain this traction force the Milwaukee brace has a pelvic shell modelled to reduce the lumbar lordosis. To understand how the brace works it is useful to see fig. 1b), WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

162 Modelling in Medicine and Biology VIII where the typical parts of a Milwaukee brace are shown: the pelvic shell, the top part to constrain the chin, the lateral shell to apply the main thrust. In the figure 2, three different curvatures of the scoliotic spine are shown. In this figure are depicted the principal thrust, black arrow, and the reaction thrusts, the gray ones, that the brace applies to the spine. The reaction thrusts born in the pelvic shell and in the top part of the brace. The principal and the reaction thrusts do not succeed to apply the load to all the spine but just on the apex of the curvature and on the pelvic and on the neck regions. To increase the surface of the rib cage where the thrusts are applied, so that the correction pushing is applied to a greater part of the spine, some plates are used.

Figure 3:

Scoliotic spine in a trasversal plane: on the right the main thrust and on the left the 2 reactions thrusts.

Figure 4:

a) Axonometric view of the scoliotic spine model, b) horizontal view of the s.s.m., c) axonometric view of scoliotic spine and thorax.

3

The new brace

It is important to consider that the spine, and obviously the scoliotic one too, is formed by vertebrae and each of them has 6 degrees of freedom (dof) with respect to the adjacent one. These 6 dof have a limited movement range, but there exists the possibility of a collapse of the spine, where it has the maximum value of the instability, when the patient wears a brace with 3 thrusts, fig. 3. The elicoidal shape of the scoliotic spine needs more thrusts relating to the extension of the curvature, figs. 4a), 4b). In fact, the scoliotic curvature extended for 10 vertebrae requires more thrusts than a curvature extended for only 5 vertebrae. From our point of view the spine can be considered as made by WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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24 cylindrical bodies, the vertebrae; the ends of each body articulate by pads of elastic or cartilaginous tissue with those of adjacent ones. This cylinder of 24 segments with a specific configuration, depending on the shape of the vertebrae, is linked to the rear part of the thorax, from the internal side, with a rotational joint, so that the spine can swing relative to the thorax, fig. 4c). The braces use the thrusts on the rib cage to obtain the displacement, i.e. traslation and rotation, of the spine for correcting it. Greater is the surface where the thrust is applied lesser is the possibility that the spine moves in unwanted directions. 3.1 The design of the brace The brace proposed is based on the “global thrust” on the entire thorax. The idea is to have thrusts at many levels of the spine, not only at 3 levels as with a traditional brace. To obtain this behaviour the structural part of the brace is made with plastic shell, and in the internal part of the shell a covering of air pockets is fixed to it. The covering is made by rubber air pockets, in contact with the thorax, that are pressurized individually to obtain the required value of thrust. Moreover the corrective pressure inside the pocket can be modified and monitored. The use of the brace is as follows. At first the value of thrusts to be applied to the rib cage for reduction of the scoliotic curvature has to settle. Then the value of pressure of the air pockets to obtain those thrusts has to be calculated for each point of the scoliotic curvature. Finally each air pocket can be inflated at the specific pressure value. In this way many thrusts are applied to the scoliotic curvature of the spine, one for each vertebra and composed by more radial components, except for the expanding part of the thorax. 3.2 The design and the prototype of the air pocket The knowledge of values of the thrust in a scoliosis brace is necessary to design the air pocket. For this goal some experimental tests have been scheduled with 4 scoliotic patients, that were selected from the clinician, to measure the contact pressure, i.e. the thrust, between the pads and the rib cage. The utility of knowing the thrust value at the pad interface is due to the conception of the new brace. The air pockets are very similar to the pads, with the difference that they can be distributed all around the rib cage. All these patients have the same characteristics: a diagnosis of scoliosis and a therapy with a 3-point Cheneau brace. Other personal information was registered in a database but was not used to select them, because the goal of these tests was only to acquire typical values of thrust applied by the pads of a scoliosis brace. The measurement set-up was made by a sensors’ matrix, a conditioning unit, all manufactured by Medizintechnik Gmbh, a data acquisition board NIDAQ 6036E of National Instruments and a Personal Computer. The sensors’ matrix has a dimension of 80 x 85 mm and a thickness of 2.3 mm. It is made by 8 couples of pressure and temperature sensors that are positioned in two rows and each couple of pressure and temperature sensors are annealed in a gel cell to distribute the pressure on the sensor and to protect against the shock. Each couple of sensors appears as a WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

164 Modelling in Medicine and Biology VIII single sensor and it has a dimension of 11,5 x 27,5 mm. The maximum value of the pressure that each sensor can measure is 2 bars with a maximum error of 5% (i.e. ± 50 mbar). The figure 5 shows a photo of the matrix of sensors and a draw with the disposition of the 8 sensors. A simple protocol was defined to conduct this test. The protocol is as follows: 1. the clinician gives the patient complete information about the experimental test (goal, procedure, risk) and the patient accepts by signing a consensus agreement; 2. the patient wears the brace in the correct way; 3. the sensor is inserted in a disposable sterile small bag; 4. a data acquisition starts to test the sensor and to read the offset, i.e. the zero position of the meter, before inserting the sensor; 5. the brace is removed from tension to insert the sensor between the thrust point and the rib cage. The positioning of the sensor is a delicate step because it is necessary to avoid the presence of creases and the sensor has to remain finely in contact with the thorax and the pad; 6. the brace is tensioned and the first data acquisition starts with the patient in erect position; a check on the acquired data is made, so that if something is wrong the test can be repeated; 7. a second acquisition starts with the patient in seated position; a check on the acquired data is made, so that if something is wrong the test can be repeated; 8. is it the first measurement point? If the answer is yes, go back to the point 5. to go on with the second measurement point. If not, removes the brace from tension to recover the sensor: the test with the patient is completed.

Figure 5:

The 8 sensors’ matrix and the position of these sensors.

A grid of the back was used to locate the position of the measurements points for each patient, fig. 6a). The measurement’s points of the patients, from 1 to 4, were respectively: 2C and 4B, 2C and 3B, 3C and 4A, 1B and 1C. An example of acquired data from the test is reported in fig. 6b)’, 6b)” (sample frequency of 1 kHz and about 10.000 samples acquired), where it is possible to see the acquired pressure raw data for each of the 8 sensors, that measure the pressure between pad and rib-cage, and the total force F of the sensors matrix calculated with this formula: 8

F = A ⋅ ∑ pi ⋅ 1

WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

(1)

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where pi is the pressure measured from the sensor i and A is the area of each sensor (316 mm2 = 11,5 x 27,5). The data are oscillating because of the breathing, i.e. the change of the volume of the rib cage modifies the contact pressure and the thrusts. The maximum value of thrust that was measured is 37 N (patient n. 1 in seated position, point 2C). In all the tests the maximum pressure was measured in seated position. This is due to the kyphotic position of the upper back in many people when seated. Some measurements have shown very low level of pressure, less than 4 N. Probably, in these cases at the interface pad rib cage a contact-no contact situation occurs. Paziente 1 zona 2‐C seduto 0,035

] a 0,03 P [M0,025 er 0,02 su se 0,015 r p 0,01 tc a t 0,005 n o 0 C

b’) Force [N]

a)

b”) Figure 6:

‐0,005

Sensor 1 Sensor 2 Sensor 3 Sensor 4 Sensor 5 Sensor 6 Sensor 7 0

2

4

6

8

10

12

Sensor 8

Time [s]

40 35 30 25 20 15 10 5 0 0

5

10

15

Time [s]

a) Grid of the back to locate the measurement points, b’) example of the output of the 8 sensors’ matrix and, b”) total thrust applied.

After this fundamental preliminary step, the technical issues of the thrust element can be defined as: o maximum level of force, on a area of about 80x80 mm2, not less than 40 N; o possibility to modify the thrust force during the brace therapy; o pneumatic technology to fix the value of pressure, i.e. the thrust, at the desired value; o architecture of many thrust elements, integrated each other, so that a covering of air pockets inside the shell is possible, to apply the thrusts where they need; o soft contact with the thorax, for maximum comfort of the brace. The designed thrust element is an air pocket with a squared shape. The air pocket is made of two layers of elastomeric material. To obtain a chamber that can be pressurised, the two layers are linked each other on the edge and a hole is used to inlet and to outlet the air from the pocket by a leak free valve. This valve is also used as fixing device to the shell. Pressurizing the air inside the pocket the volume grows as a balloon, so that a thrust is obtained “contrasting” the expansion of this volume. The silicone rubber was used as material, for its softness and its compatibility with the skin, and the manufacturing process was defined and tested by a technological set up. Because of the silicone behaviour WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

166 Modelling in Medicine and Biology VIII the output force depends on both the pressure inside the pocket and its volume. In fact the elastomeric materials have a non-linear σ-ε relationship. For this reason it was useful to define a numerical model to obtain a useful design tool. In this way the relationship among the main design parameters of the air pocket with the goal of optimising the driver design can be studied. The numerical model was defined using the Finite Element Technique by the ANSYS code [3]. Other details can be found in [4]. To validate the model a silicone prototype was constructed by the same technology that was used to make the air pockets for the brace, fig. 7a.

a)

b) dH=0mm

] N [ ce r Fo

120

dH=4mm

100

dH=8mm

80

dH=12mm dH=16mm

60

dH=20mm

40

dH=24mm

20

dH=28mm

0 ‐20

c’)

Figure 7:

2 0 , 0

6 0 , 0

0 1, 0

4 1 , 0

8 1, 0

2 2 , 0

6 2 , 0

0 ,3 0

4 3 , 0

8 3 , 0

2 4 , 0

6 4 , 0

0 5 , 0

4 5 , 0

8 5 , 0

Pressure [bar]

F = 19,6 N F = 29,4 N F = 49 N F = 98,1 N

0

dH=32mm

dH=36mm dH=40mm

F = 9,8 N

60

] 50 m40 m [ 30 in ar 20 tS 10

c”)

5 ,0 0

0 ,1 0

5 ,1 0

0 ,2 0

5 ,2 0

0 ,3 0

5 ,3 0

0 ,4 0

5 ,4 0

0 ,5 0

5 ,5 0

0 ,6 0

Pressure [bar]

a) Prototype of the air pocket, b) set up for the isometric tests, c’) results obtained in isometric and c’’) isotonic tests.

The design of the air pocket and the tests on the prototype gave the following results: thickness: 8 mm; square dimension: 100x100 mm; maximum force: 100 N; maximum test pressure: 0.60 bar; burst pressure, at null deformation: 10 bar. Some problem arose in doing the burst test, because of the valve that is fixed to the wall by a nut. When the volume of the driver grows the fixing hole of the valve become larger. For this reason with free deformation of the air pocket the valve is ejected at a pressure of about 0.55 bar. In the application inside the brace the maximum deformation at design level is less than 10 mm. At this value the burst pressure is more than 4 bar, so that it is correct to settle the maximum operative pressure at 0.65 bar. Other experimental tests on the prototypes of air pocket were made in the two characteristic conditions: isometric and isotonic. The design and construction of a simple test bed was necessary to perform these tests. The isometric test consists in constant deformation experiments. The air pocket keeps its deformation by constraints and the force versus pressure inside the chamber is computed. The constraints are made by two simple aluminium plates. The deformation is fixed at different values by a simple screw-nut system that links the two plates. The deformation is measured by a measuring rod and a load cell on the screw is used WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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for the traction force. The pneumatic components are a pressure regulator, connected to the pneumatic line, and a flexible tubing for the connection with the pocket. A manometer measures the air pressure and a portable tension-meter gives the tension value of the load cell that is used to calculate the traction force. The figure 7b shows the experimental set up for the isometric tests of the device. The isotonic test consists in constant force experiments. The device is free to extend and the displacement versus pressure are computed. The test bed is similar to the previous one. The aluminium plates are not linked each other. The constant load is obtained by gravitational mass located on the aluminium plate. The isotonic tests were performed in this manner. On the upper aluminium plate a mass is loaded and during the test the pressure p versus the displacement Dh of this plate is recorded. The test starts with no pressure inside the chamber and opening the pressure regulator in quasi-static manner the pressure reaches a maximum value of 0.6 bar. The isometric tests were performed by regulating the displacement Dh of the two aluminium plates with the screw-nut link from 0 to 40 mm. The pressure inside the chambers is regulated in the same manner of the previous test, and recorded with the output of the load cell. During both the tests, isotonic and isometric, the device shows an hysteretic behaviour, i.e. two distinct lines are obtained increasing the pressure inside the chambers from zero to a certain value and then decreasing to zero. This phenomenon is already described in literature for elastomeric materials [5], and the cause of the hysteresis seems to be the viscoelastic phenomenon. In this application it is neglected and the curves are constructed calculating the medium values. 3.3 The construction of the brace and the first experimental tests The construction of the shell was obtained by a traditional CAD (Computer Aided Design) system normally used for the base frame of other brace, as Cheneau. The shell was modelled on the trunk of an healthy volunteer, inserting an offset, i.e. a backlash, of about 10 mm on the upper part. This offset was necessary because of the thickness of the air pocket (8 mm). A grid of holes were performed on the rear side of the shell, with a correct arrangement to assembly the air pockets by valve and nut from the internal side. Two air pocket prototypes were used for this first experimental tests. Wearing the brace, made by the shell and the 2 air pockets one close to the other, the impression was of too large size, with the drivers in a contact-no contact condition. The measurement made by the sensors matrix confirmed this condition. For this reason a covering was necessary to have a uniform thickness in the internal side of the shell. This problem was due to the use of only 2 air pockets, but it will disappear with the construction of the complete prototype that needs something like 30 air pockets. A layer of material, similar to that used for the pad, with a thickness of about 5 mm was used. It was glued on the internal side of the shell, except where are assembled the 2 air pockets. After this modification the volunteer had good sensation wearing the brace, with a normal contact between thorax and brace, fig. 8a. The measurements confirmed this good behaviour of the brace, giving values of the thrust less than 0.5 N. After this step the first experimental tests on WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

168 Modelling in Medicine and Biology VIII the brace started to measure the thrusts due to a single air pocket. The test was carried out in two different position of the air pocket, approximately 3B and 4B, increasing the pressure value from 0 to 0,65 bar. The measured thrusts, by the sensors matrix, in the two different positions of the pocket versus the air pressure are very similar, with a maximum value of 25 N at 0,65 bar. The graph that shows the thrust of one air pocket versus the air pressure is in the fig. 8b). The result shows that the maximum value of thrust does not meet the design value of 40 N, but it is important that the shape of the graph is approximately linear. The deficit in the maximum value of the thrust that the driver can apply means that a rectangle of 4 air pockets, 200x200 mm2, can apply a maximum thrust of 100 N, instead of 160 N. That’s mean that this new brace could be really interesting considering that the complete brace has a covering of air pockets in the internal side.

b)

a) Figure 8:

4

a) Prototype of the brace, b) total thrust applied by one air pocket.

Conclusion

In this paper a new brace for scoliosis is proposed. The basic idea of the new brace born to overcome the limitation of the 3 points thrust, typical of the most common removable braces used for scoliosis therapy. To obtain a better efficacy of the therapy, and to avoid the collapse of the spine, the thrusts should be applied at many levels of the spine. In this way the possibility that the spine moves in unwanted directions is avoided. Moreover the knowledge of the values of the thrusts is very useful to have quantitative data for the success of the brace therapy. The structural part of the new brace is made with plastic shell and a covering of air pockets is fixed to the internal part. The pressure inside each air pocket can be regulated at the desired value. The paper shows the design, the prototyping and the experimental validation of the air pocket. Finally the first experimental tests on a healthy person with a simplified prototype of the brace with only 2 air pockets are shown. The results show that the idea of this new brace can be really interesting and the feasibility study here presented has given a positive answer. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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The work is in progress and the next step is a complete prototype of the new brace and, finally, a clinical validation.

Acknowledgements The authors are thankful to Mr. Carlo Fricano and Mr. Alessandro Cipriani, as well as to Ortopedia Picena di Guiducci Quirino, Ascoli Piceno, Italy, for their kind support.

References [1] Negrini S, Aulisa L, Ferraro C, Fraschini P, Masiero S, Simonazzi P, Tedeschi C, Venturin A., Italian guidelines on rehabilitation treatment of adolescents with scoliosis or other spinal deformities, Eura Medicophys 2005, 41. [2] Labelle H, Dansereau J, Bellefleur C, Poitras B. Three-dimensional effect of the Boston brace on the thoracic spine and rib cage, Spine. 1996 Jan 1;21(1). [3] ANSYS Theoretical manual, Swanson Analysis system Inc., Houston, USA, 1989 [4] Raparelli, T., Beomonte Zobel, P., Durante, F., Costanzo, G., Development of a pneumatic lumbar unloading device, IMG04 - International Conference on Intelligent Manipulation and Grasping, July 1-2, 2004, Genoa, Italy, pagg. 197-201 [5] Ferry J. D., Viscoelastic properties of polymers, John Wiley & Sons, Inc., New York, 1980 [6] Viladot, R., Cohi, O., Clavel, S., Ortesi e protesi dell’apparato locomotore, (in italian) Ed.Verduci, 1988 [7] Simposi Clinici CIBA, (in italian), 1973

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Parameters of kinaesthesis during gaits derived from an ultrasound-based measuring system R. M. Kiss Department of Structures, Budapest University of Technology and Economics, Hungary

Abstract In order to analyze kinaesthesis of motion, the author used a ZEBRIS ultrasoundbased measuring system with single active markers. In essence, the measurement involves the determination of the spatial positions of different anatomical points on the human body while a person is stepping on firm and non-firm plates, which also have active markers attached. This type of measurement can model the correction ability of the person being examined. The provocation test, when the plate is suddenly swivelled out with a provocation unit, can model the equilibration ability of the person being investigated. In this study the kinematic and oscillation (swing) characteristics of 50 healthy persons were identified during stepping and provocation. The kinematic characteristics of stepping were identified by measuring the range of knee motion in all three directions of space and the range of motion of the shoulder girdle and pelvis in a local coordinate system of segments (shoulder girdle and pelvis). The oscillation (swing) characteristics of provocation were identified by average logarithmic decrement, Lehr’s damping ratio, frequency and own frequency. Kinaesthesis can be described by analyzing the results of the measurements performed. Keywords: ultrasound-based measuring system, kinaesthesis, proprioception, swing parameters.

1

Introduction

Kinaesthesis (dynamic perception) is the perception of the relationships of the moving parts of the body to each other. When examining kinaesthesis, we analyze the reiterative accuracy of well-known movements. This method is suitable for examining patients with infantile cerebral paresis and degenerative WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090161

172 Modelling in Medicine and Biology VIII clinical patterns. As regards the investigation of kinaesthesis, gait analysis is very widely used. The standard deviations of different kinematic parameters (temporal, spatial, angular parameters) can model the reiterative accuracy of motion [1], which is equivalent to modelling kinaesthesis. However, the gait analysis cannot be used a short time after the injury or after the reconstruction. Kinematic parameters are not suitable for modelling the equilibration ability of the person, which means modelling the oscillation of motion after a sudden change in motion. The ZEBRIS ultrasound-based motion analyzer system (ZEBRIS Medizintechnik GmbH, Isny, Germany), using individual markers, a measurement control program and a biomechanical model, is suitable for determining the positions of individual markers attached to different anatomical points on the human body while the person is stepping on firm (stable) ground as well as on non-firm (unstable) ground. By using a non-firm plate, the oscillation of motion can be determined after a provocation, when the plate is suddenly swivelled out with a provocation unit. The accuracy of the measuring method enables us to calculate the different kinematic as well as the oscillation (swing) parameters. The measuring method is suitable for analyzing correction and equilibration abilities, and it may be used shortly after reconstruction surgery. This research was aimed at jointly producing a test method and corresponding biomechanical parameters suitable for modelling the kinaesthesis of motion. A further objective was to generate a databank of parameters from healthy young subjects as a basis for further investigations.

2

Materials and methods

2.1 Subjects The positions of the anatomical points were determined for 50 healthy subjects using the method described below. Only people without any clinical history of lower extremity or shoulder joint disease or injury were involved in the study. There were 25 males (average age 22.1 5.1 years, average height 171.9 4.9 cm, average weight 72.1 3.4 kg) and 25 females (average age 24.6 6.45 years, average height 168.9 7.3 cm, average weight 63.1 7.5 kg). The tests were authorized by the Science and Research Ethics Committee of Semmelweis University. Each voluntary subject provided informed written consent to perform the tests in advance. 2.2 Measurement method, biomechanical model The spatial coordinates of certain anatomical points during motion were measured by a ZEBRIS CMS10 computer-controlled, ultrasound-based motion analysis system located in the Biomechanical Laboratory of the Department of Orthopaedics of Semmelweis University. The active, individual markers attached to the anatomical points are transmitters and emit ultrasound signals at specific intervals, which are recorded WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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by three markers (microphones) built in the measuring head (the measurement frequency being 100Hz). Knowing the speed of the ultrasound waves, the distances between each active marker and the microphones on the measuring head can be calculated from the time delays of the transmission. From the distances between the three microphones of the measuring head and each of the active markers and knowing the spatial coordinates of the three microphones on the measuring head, the spatial coordinates of the active markers at each moment of time during the measurement can be calculated using the method of triangulation [2] (Figure 1). The main criteria of the measurement procedure are standardization and reproducibility. Therefore, the locations on the body of the active sensors that emit the ultrasound signals must be selected so that the possibility of displacement during measurement is excluded; the anatomical points must be properly determinable and palpable through the skin. The active markers are fixed in place, using two-sided plaster tape, to the tuberositas tibiae in order to test the motion of the lower limb and to the spina iliaca anterior superior and acromion scapulae in the shoulder girdle to test the motion of the upper body. These anatomical points are particularly useful because there is relatively little motion of the skin over osseous anatomical points during gait and other types of motion. An important element of the examination of kinaesthesis (dynamic perception) is modelling the correction ability on non-firm (unstable) ground. The most suitable device for this is the ZEBRIS Posturomed® plate, secured by eight springs with the same spring strength in each direction (Figure 1). The Posturomed® plate can be used as a firm plate (so that the springs do not allow motion of the plate in any direction) or as a non-firm plate (so that the springs allow motion of the plate in both horizontal directions). Using the Posturomed® plate the oscillation of motion can be modelled with different oscillation parameters (frequency, damping parameters, etc.) after a provocation. During the provocation the plate is suddenly swivelled out with a provocation unit (Figure 1), and the tension on the springs is released when they are in an extreme position. The oscillation (swing) parameters can be calculated from the positions of the markers attached to the plate as a function of time. 2.3 Test and assessment parameters for characterizing the kinaesthesis The tests are performed with males stripped to the waist and with females in bras, so that the anatomical points around the shoulder and on the lower limb are easy to access. Each of the points involved in the investigation are anatomical and anthropometrical points for the person performing the examination. Three types of tests have been set up for kinaesthesis. 2.3.1 Test for the lower limbs: knees The position of tuberositas tibiae are measured during 15-20 cycles of stepping on a firm (stable) and a non-firm (unstable) plate. The ultrasound-based measuring head is located in front of the subject at 90 degrees (Figure 1).

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174 Modelling in Medicine and Biology VIII The range of motion of the knees is the difference between the minimum and maximum values of the spatial coordinates of the anatomical point of tuberositas tibiae to be determined during one period, in the spatial directions x, y and z, which means in the lateral, anterior-posterior and up-down directions, respectively. xa1,ya1,za1

active marker, transmitter, xt,yt,zt

xa2,ya2,za2

d1 d2

xa3,ya3,za3

Figure 1:

d3

measuring head with three microphones

Calculated position of the active marker (transmitter) (xt, yt,zt) from the positions of the three microphones of the measuring head (xai, yai, zai), and from the distances between the three microphones and an active marker (di) by the triangulation method. The picture shows the measurement arrangement during the test for the lower limbs.

2.3.2 Test for the upper body: shoulder girdle and pelvis The position of acromions and ASIS are measured during 15-20 cycles of stepping on a firm (stable) and a non-firm (unstable) plate. The ultrasound-based measuring head is located in front of the subject at 90 degrees (Figure 2a). Before starting the measurement, in the calibration phase, the spatial position of the left and right acromion must be recorded with the subject in a natural posture. Using the positions of the acromions the local coordinate system of the shoulder girdle can be determined, as the axis  of the local coordinate system of the shoulder girdle is the line connecting the left and right acromions. Axis of the local coordinate system is a line perpendicular to axis  at the midpoint of the line connecting the left and right acromions. Axis  of the local coordinate system is a line perpendicular to the plane defined by axes  and  at the intersection of the two axes (Figure 3a). Flexion-extension of the shoulder girdle (), which occurs upon rotation of the line connecting the left and right acromions around the axis  of the local coordinate system, that is, in the  plane, corresponds to the angle enclosed by the line connecting the left and right acromions and the axis  (Figure 3a) during motion. The tilt of the shoulder WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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girdle (), which changes with the rotation of the line connecting the left and right acromions around the axis  of the local coordinate system, that is, in the  plane, corresponds to the angle enclosed by the line connecting the left and right acromions and the axis (Figure 3a) during motion. The rotation angle of the shoulder girdle (), which changes upon rotation of the line connecting the left and right acromions around the axis  of the local coordinate system, that is, in the  plane, corresponds to the angle enclosed by the line connecting the left and right acromions and the axis  (Figure 3a) during motion. The ranges of the different motions of the shoulder girdle correspond to the differences between the minimum and maximum values of the angles (flexion-extension, tilt, rotation) of the shoulder girdle that are determined during one period. The local coordinate system of the pelvis can be determined – similarly to the local coordinate system of the shoulder girdle – before starting the measurement, from the spatial positions of the anatomical points of the left and right spina iliaca anterior superior (Figure 3b). The range of flexion-extension of the pelvis (), the tilt of the pelvis (), the rotation angle of the pelvis () is the difference between the minimum and maximum values of the angles that are determined during one period.

a) Figure 2:

b)

Measurement arrangement a) during the test for the upper body (test for shoulder girdle and pelvis) and b) during the provocation.

2.3.3 Testing for provocation Test for provocation includes three separate measurements, when the subject stands on both limbs or on the right or the left foot. The springs are released, and the plate suddenly moves in a horizontal direction, so that the subject has to balance and re-equilibrate as the plate moves. The spatial coordinates of the two active sensors placed on the moving plate are recorded. The ultrasound-based measuring head is located to the side of the subject at 30 degrees (Figure 2b).The lateral displacement of the measuring plate as a function of time can be calculated from the spatial coordinates of the markers fastened to the measuring plate as a function of time, where the coordinate x (in the lateral direction, which is parallel to the direction of provocation) as a function of time constitutes a damping curve (Figure 4).

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176 Modelling in Medicine and Biology VIII 







  

  a) Figure 3:





b)

Definition of the local coordinate system a) of the shoulder girdle and motions of the shoulder girdle and b) of the pelvis and motions of the pelvis (in the positive direction of the coordinate axis a clockwise angle will be positive).

As regards the analysis of kinaesthesis, the value of logarithmic decrement is only nearly constant as the system is not closed and there may be a number of disturbing factors. The amplitude measured at the moment t = 0 of the test may be particularly disturbed due to the start-up of the system. It is expedient to commence the test at the moment t1 and to determine the average value of the dynamic characteristics. The average logarithmic decrement is the average of the natural-based logarithm of the ratio of the amplitudes in an identical direction of two periods remote in time: 1 K i  ln 1 , i  1 Ki where K1 is the amplitude at the moment t = t1 (Figure 4) and Ki-1 is the amplitude at the moment t = ti-1 (Figure 4). The frequency [1/s] is the number of repetitions in a unit of time, or the rapidity of swing: i 1 f  , ti  t1 where T is the swing time in seconds. Lehr’s damping ratio (damping ratio, ratio D, damping factor) is the ratio of actual and critical damping:  , D 2   4 2 where is the logarithmic decrement and π is 3.14. Own frequency is 1 f0  , T 1  D2 where T is the swing time in seconds and D is Lehr’s damping number. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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2.4 Statistical analysis

Data processing and statistical analyses were performed using MS Excel based software that we developed. In the case of each subject examined, we calculated the average and the standard deviation of the biomechanical characteristics from the measurement results of the motion cycles recorded, and these data were further processed. The uniformity of the standard deviations was checked by an F-test; significance levels of the differences between the average values of identical parameters were determined by a one-sample t-test, applying a symmetrical critical range on the dominant and opposite sides.

3

Results

The results of the test are summarized in Tables 1-3.

4

Discussion

The standard deviations of different kinematic determined during gait can model kinaesthesis. However, the correction ability cannot be analyzed during gait on a fixed, stable walkway or a treadmill. Our measuring method is able to model the correction ability of the subject investigated because the motion of the subject can analyzed while a stepping on a non-firm (unstable) plate. The kinematic parameters are not suitable for modelling the equilibration ability after a sudden change in motion. During the provocation test, when the plate is suddenly swivelled out with a provocation unit in the horizontal direction, the subsequent motion of the plate is determined. The value of the coordinate x (parallel with the provocation direction) as a function of time is a damping curve; from it the different oscillation (swing) parameters could be calculated. It is known from the literature [3] that the ranges of motion of the knee during a gait cycle are 2-3 cm in the lateral direction and 3-5 cm in elevation, and these values are similar to our results for the lower extremities test when subjects were stepping on a firm plate (Table 1). By analyzing the results of the lower extremities test, it can also be established that the correction for non-firm ground can be solved with smaller anterior-posterior and up-down motions and wider right-left motions (Table 1). The standard deviation of the motion of the knee is bigger when stepping on a non-firm plate than it is when stepping on a firm plate (Table 1). From both results it can be established that the reiterative accuracy of motion on a non-firm plate is failing. The trend of these results is similar to the trend of gait parameters measured after injury [4, 5], in elderly people [6, 7] or in cases of arthritis [8, 9]. The tilt and rotation of the shoulder and pelvis (Table 2), established while stepping on a firm plate, are similar to the results found in the literature [3, 6, 7]. The flexion-extension of the pelvis during stepping (Table 2) is smaller than the flexion-extension during gait [3, 6, 7] because the flexion-extension of the pelvis plays a less important role in the forward motion of the gait. The lack of reiterative accuracy of motion on a non-firm plate establishes the greater motion of the pelvis and shoulder on a non-firm plate. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

178 Modelling in Medicine and Biology VIII

Figure 4:

Horizontal (x) coordinate of the active sensors attached on the plate during tests of provocation. Coordinate x as a function of time yields a damping curve. Factors required for calculating the swing parameters (the amplitude Ki at the moment t = ti, T is the swing time).

Table 1:

Range of knee motion [mm] while stepping on firm and non-firm plates (average ± standard deviation).

Direction

†

x (rightleft) y (antpost) z (updown)

Stepping on firm plate Dominant side Non-dominant side 27.63±1.82 25.4±2.36

Stepping on non-firm plate Dominant side Non-dominant side 42.36±4.47† 38.62±4.42†

191.65±9.16

183.12±10.45

149.98±15.51†

142.24±19.40†

46.23±3.13

44.86±8.34

24.56±6.94†

19.24±7.76†

Significant difference between the average values of parameters compared to parameters while stepping on a firm plate. The oscillation (swing parameters), as the average logarithmic decrement, Lehr’s damping ratio, the frequency and the own frequency enable modelling of the equilibration ability. The subjects could damp the motion of the plate when Lehr’s damping ratio was less than 1 in all three examinations (Table 3). The results show that the dominant side plays an important role in damping of the plate motion because it is significantly higher than the Lehr’s damping ratio of

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Table 2:

Range of motion of pelvis and shoulder girdle [degree] while stepping on firm and non-firm plates (average ± standard deviation).

Type of motion Flexionextension Tilt Rotation

Table 3:

179

Stepping on firm plate Shoulder Pelvis 0.91±0.23 1.42±0.34

Stepping on non-firm plate Shoulder Pelvis 1.16±0.41 1.67±0.53

3.32±0.57 3.86±0.67

4.21±0.79† 6.96±1.07†

2.66±0.46 5.58±0.58

7.38±0.73† 7.46±0.88†

Oscillation (swing) characteristics during the provocation test (average ± standard deviation).

Standing on both limbs dominant limb non-dominant limb

 1.321 ±0.044 1.278 ±0.095

D 0.2058 ±0.0164 0.1992 ±0.0248

f [1/s] 0.003068 ±0.000344 0.003116 ±0.000343

f0 [1/s] 0.003132 ±0.00358 0.003173 ±0.000348

0.483 ±0.212 †, ‡

0.0766 ±0.0419†,‡

0.003181 ±0.000375

0.003187 ±0.000355

- average logarithmic decrement, D - Lehr’s damping ratio, f – frequency, f0 – own frequency. †Significant difference between the average values of parameters compared to parameters measured while standing on both limbs. ‡Significant difference between the average values of parameters compared to parameters measured while standing on the dominant limb. standing on a non-dominant limb (Table 3). From these results it can be established that the damping effect of the subject depends on the ability of the dominant side because the damping ratio when standing on both limbs is similar to that when standing on the dominant side (Table 3). On the basis of our test, the average logarithmic decrement and Lehr’s damping ratio prove the dependence on lateral dominance. The frequency and the own frequency are independent of lateral dominance because there was no significant difference between standing on the dominant or on the non-dominant side (Table 3). These parameters depend on the weight of the subject and their ability to balance their whole body because there was no significant difference between the frequency and the own frequency, or between the parameters determined while standing on both limbs or on a single limb (Table 3). The measurement method developed and the corresponding kinematic and oscillation (swing) parameters enable determination of the kinaesthesis of the motion. The measuring method can be performed shortly after injuries, including after orthopaedic injuries. Stepping on a non-firm plate can model the correction ability of investigated persons. The provocation test and the swing parameters calculated from it can model the equilibration ability. Lehr’s damping ratio and WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

180 Modelling in Medicine and Biology VIII the own frequency are characteristic of the subject examined; they are substantially affected by various orthopaedic injuries, surgeries, age and sporting activities.

Acknowledgement The research described in this paper was supported by a grant from the National Scientific Research Found (OTKA), Hungary, project No: T049471.

References [1] Kirtley, C., Clinical gait analysis. Theory and practice. Elsevier Churchill Livingstone: Edinburgh, 2006. [2] Kiss, R.M., Kocsis, L. & Knoll, Zs. Joint kinematics and spatial temporal parameters of gait measured by an ultrasound based system. Medical Engineering &Physics 26, pp.611-620, 2004. [3] Winter, D.A., Biomechanics and motor control of human movement. WileyInterscience Publication: New York, 1990. [4] Theoret, D. & Lamontage, M., Study on three-dimensional kinematics and electromyography of ACL deficient knee participants wearing a functional knee brace during running. Knee Surgery Sports Traumatology and Arthroscopy 14, pp.555-563, 2006. [5] Torry, M.R., Decker, M.J. & Ellis, H.B., Mechanisms of compensating for anterior cruciate ligament deficiency during gait. Medicine and Science in Sports and Exercises 36, pp.1403-1412, 2004. [6] Bejek, Z. Paróczai, R., Illyés, Á. & Kiss, R.M., The influence of walking speed on gait parameters in healthy people and in patients with osteoarthritis. Knee Surgery Sports Traumatology Arthroscopy 14, pp. 612622, 2006. [7] Riley, P.O., DellaCroce, U. & Kerrigan, D.C., Effect of age on lower extremity joint moment contributions to gait speed. Gait and Posture 14, pp. 264-270, 2001. [8] Al-Zahrani, K.S. & Bakheit, A.M. A study of gait characteristic of patients with chronic osteoarthritis of the knee. Disabil Rehabil 24, pp. 275-280, 2002. [9] Börjesson, M., Weidenhielm, L., Atsson, E. & Olsson, E., Gait and clinical measurements in patients with knee osteoarthritis after surgery: prospective 5-year follow-up study. The Knee 12, pp.121-127, 2005.

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Evaluating elbow joint kinematics with the Stewart Platform Mechanism M. Alrashidi1, İ. Yıldız2, K. Alrashdan1 & İ. Esat1 1

School of Engineering and Design, Brunel University, UK Mechanical Engineering Department, Yıldız Technical University, Turkey

2

Abstract Joint measurement is necessary for studying joint laxity. Joint laxity in elbows is a problem which normally comes with age. However, it can increase to critical levels with rupture or damage to the ligaments of the elbow and affects the stability and capabilities of the joint, interfering even with daily activities. A new Stewart platform based elbow joint measurement was developed. The study focused on flexion, extension, valgus and varus motions and all experimental data were taken from a real experimental setup. The centre of rotation and motion of the angles of the forearm were found by the kinematics of the Stewart Platform Mechanism with a simmechanics based program algorithm. The joint motions of two male subjects were measured and the results of these measurements were compared with graphs. Comparisons demonstrate that the Stewart platform based measurement device sufficiently measures all motions of the elbow with six axes. Keywords: joint laxity, Stewart platform, elbow kinematics.

1

Introduction

Kinematics of the elbow joint occupy a considerable place in orthopedic surgery. Many devices have been constructed with this aim. Hand goniometers were formerly employed for measuring elbow kinematics [1]. Morrey and Chao studied the motions of the elbow joint [2]. They measured elbow flexion and forearm rotation by using an electronic goniometer. Another study was published by Morrey and Chao for calculating elbow joint motion

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182 Modelling in Medicine and Biology VIII with the help of biplanar roentgenograms [3]. They obtained three-dimensional kinematics of the joint in their research. Tanaka et al. used electromagnetic motion tracking data and described the first three-dimensional elbow kinematic [4]. Lateral roentgenograms used a kinematic analysis of elbow kinematics by London [5]. In this research, London used a special Reuleaux technique for analysis. The Reuleaux technique [11] was first used by Fisher to obtain he location of the axis of elbow flexion [6]. Bottlang et al. [7] used direct electromagnetic motion tracking to trace the passive and dynamic motion of the natural elbow joint. With improving technology and silicone technology, inertial and magnetic sensors have also been employed for measuring human joints [8]. In this study, a Stewart platform based device was developed for measuring elbow kinematics. The Stewart platform mechanism was first proposed as a flight simulator in 1965 by Stewart [9].

2

Materials and methods

2.1 Stewart Platform Mechanism The Stewart Platform Mechanism (SPM) is a six-axis parallel mechanism. This type of parallel mechanism is often applied in robotics and also in the medical field. A Stewart platform type robot has one moving plate, six parallel actuators and one fixed plate. Actuators are usually mounted between the moving plate and the fixed plate by spherical or universal joints. This configuration allows the mechanism to have three translational and three rotational motions. In this study a Stewart platform type measurement device was used. This device also has one moving plate and one fixed plate, but instead of actuators it has six linear potentiometers (Fig. 1). (a)

(b) (c)

Figure 1:

Stewart platform based elbow joint measurement device (a), arm fixation apparatus (b,c).

2.2 Measurement method The SPM is a kind of kinematic chain. In this study, potentiometers were used as the legs of the SPM. Potentiometer data were used for calculating the elbow kinematics. An illustrative block diagram to illustrate the method of measurement method is found in Fig. 2. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 2:

183

Measurement steps.

As shown in Fig. 2, the first step is to fix the subject’s forearm to the SPM. The next step involves gathering the potentiometer data from the data acquisition device while the forearm is moving. Calculating the position of the centre of gravity of the SPM using the model of SPM derived from Matlab Simmechanics constitutes the third step in measuring. The Matlab Simmechanics model includes the forward dynamics and kinematics of the SPM. The forward kinematics method is one of the critical parts of the measuring. In parallel mechanisms such as the SPM, it is extremely difficult to derive the positions of the centre of gravity from the leg lengths. Many methods have been developed for solving this problem. The most important one is the Newton-Rhapson method, which entails an iterative solution [10]. The block diagram of a virtual Simmechanics model of SPM is shown in Fig. 3. The final step allows us to obtain the elbow kinematics from the position of the SPM.

Figure 3:

Matlab-Simmechanics model of the SPM.

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184 Modelling in Medicine and Biology VIII 2.3 Elbow kinematics The SPM has three translational ( x, y, z ) and three rotational (  ,  ,  ) motions with respect to its base frame, as shown in Fig. 5a. Rotational motions  ,  , represent the rotations of the mobile platform along the x, y, z axes, respectively. Four main motions of the forearm (flexion, extension, varus and valgus) were observed and measured on the subjects. The relationship between these motions and the SPM motions are shown in Fig. 4.

( )

Figure 4: zm

 ym

Angle definitions of forearm.

xm



 Yc

u

rc

Xc

Figure 5:

rc

z





Zc

(a) Base and mobile frames; (b) side view of setup; and (c) top view of mobile platform.

Obtaining the coordinates of the rotation centre is significant when treating diseases of the forearm. Almost all elbow implants have one rotation centre WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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because they are fully mechanical structures and not biological structures. The normal elbow is a flexible joint thanks to its ligaments and this means that it has no exact rotational centre. The laxity of the elbow joint can be expressed in the space of the centre of rotation. The advantage of the Stewart measurement device is that it makes all motions of arm measurable not only evident motions such as flexion and extension, but also negligible motions, including all six of its axes. In both the side and the top view of SPM, it is clearly shown (see Fig. 5b-c) that the forearm draws an imaginary circle which has the radius ( rc ). According to the cosine theorem,

u 2  rc2  rc2  2rc rc cos 

(1)

u indicates the distance between the initial point and current point of the centre of gravity of the mobile platform and  is the angle of flexion. The rotation radius can be obtained by extracting rc in Equation 1.

u2 2  2 cos 

rc 

(2)

Finally, the coordinates of centre of rotation of the elbow joint can be expressed as:

X c  rc . cos 

Yc  rc .sin 

(3)

Z c  Z  rc .sin  x

y

-9

0

50

100 150 sample xa

200

-20

250

35

0

50

100 150 sample ya

200

30

250

0

50

100 150 200 sample Center Of Rotation(C.O.R.)X c

15

250

C.O.R. y (mm)

-140 -160

0

50

Figure 6:

100 150 sample

200

250

0

50

100 150 200 sample Center Of Rotation(C.O.R.)Y c

-4

250

50

100 150 200 sample Center Of Rotation(C.O.R.)Z c

250

0

50

250

10

10 5 0 -5 -10

250

0

15

-120

200

-2

C.O.R. z (mm)

-100

0

100 150 sample za

2

a

a

17 16

-4

50

4 Z (deg)

Y (deg)

-2

0

6

18

0

a

X (deg)

40

19

2

C.O.R. x (mm)

0 -10

4

-180

10

z (mm)

y (mm)

x (mm)

-12

-6

45

20

-11

-13

z

30

-10

0

50

100 150 sample

200

250

0

-10

-20

100 150 sample

200

Motions of SPM and centre of rotation during the vargus and valus motions.

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3

Results

The forearms of two subjects were tested by the SPM based measurement mechanism. The motions of the SPM and centre of rotation can be seen in Fig. 6. In Fig. 6, the x, y , z graphs demonstrate the translational motion of the SPM.

X a , Ya , Z a depict the angular motions of the SPM, which are  ,  ,  . The last three graphs illustrate the centre of rotation of the forearm. All the graphs in Fig. 6 were created from the data taken from one subject during the vargus and valus motions and were not created with the aim of comparison. They show that the SPM-based measuring mechanism allows all six axes to be measured. Flexion and extension motions are shown in Fig. 7. The flexion angles are limited to about 80 degrees because the lengths of potentiometers are themselves limited.

Figure 7:

Flexion and extension angles.

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The varus and valgus motions are shown in Fig. 8. In the first graph, the subject starts with varus motion and follows it with valgus motion. Both graphs illustrate that the varus and valgus motion capacities are limited to a minimum of 5 degrees.

Figure 8:

Valgus and varus angles.

Fig. 9 illustrates the centre of rotation of the forearm on the YZ axis during valgus and valus motions, exposing joint laxity. Joint laxity is directly related to the bounds of the links so this is the main reason for selecting both valgus and varus motion. Values are a little bigger than expected, because fixing both forearm and shoulder to the set up is insufficiently exact.

4

Conclusion

A Steward platform based elbow joint measurement device is tested with basic motions of the forearm. The mechanism was tested with two subjects and WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

188 Modelling in Medicine and Biology VIII

Figure 9:

Joint laxity: valgus and varus.

succeeded in measuring all the motions of the forearm and ascertaining the centre of rotation. Tests were executed with the help of another person to ensure that the subjects made the specified motions. The accuracy of the device was found insufficient because of the way in which the arm was fixed. Using special clamps should make the device more accurate. In further works, a new SPM based joint measurement device with forced feedback should more accurate and should gather more comparable data with the help of forced feedback and linear actuators.

References [1] Boone D.C. and Azen S.P.: Normal Range of Motion of Joints in Male Subjects. J. Bone and Joint Surg., 61-A:756-759, July 1979 WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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[2] Morrey B.F., Askew L.J. and Chao E.Y.: A Biomechanical Study of Normal Functional Elbow Motion. J. Bone and Joint Surg., 63-A:872-877, July 1981 [3] Morrey B.F. and Chao E.Y.: Passive Motion of the Elbow Joint. J. Bone and Joint Surg., 58:501-508, 1976 [4] Tanaka S., An K.N., Morrey B.F.: Kinematics and Laxity of Ulnohumeral Joint under Varus-Valgus Stress. J. Musculoskel Res 2:45-54, 1998 [5] London J.T.: Kinematics of the Elbow. J Bone and Joint Surg., 63:529535,1981 [6] Fisher G.: cited in R. Fick: Handbuch der Anatomie und Mechanik der Gelenke unter Berücksichtigung der Bewegenden Muskeln, Vol.2. p. 299, 1911 [7] Bottlang M., Madey S.M., Steyers C.M., Marsh J.L., Brown T.D.: Assessment of Elbow Joint Kinematics in Passive Motion by Electromagnetic Motion Tracking. J. Orthop Res, Vol.18:195-202,2000 [8] Cutti A.G., Giovanardi A., Rocchi L., Davalli A., Sacchetti R.: Ambulatory Measurement of Shoulder and Elbow Kinematics through Inertial and Magnetic Sensors. Med Bio Eng Comput. 46:169-178,2008 [9] Stewart, D., A Platform with Six Degrees-of-freedom, Proceedings of Mechanical Engineering Part I, Vol.180, pp. 371-386, 1965-1966. [10] Harib K. and Srinivasan K.: Kinematic and Dynamic Analysis of Stewart Platform Based Machine Tool Structures. Robotica Vol:21,pp.541554.2003 [11] Reuleaux, Franz: Kinematics of Machinery: Outlines of Theory of Machines. pp. 56-80.Translated and edited by A.B. Kennedy. New York, Dover.1963

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Dynamics of the human stomach R. Miftahof1 & N. Akhmadeev2 1 2

College of Medicine and Medical Sciences, AGU, Bahrain Kazan Medical University, Russia

Abstract A mathematical model of the stomach as a soft electromyogenic biological shell is developed, which is based on detailed biological data of the structure and function of the organ. The dynamics of the spread of the wave of depolarization and concomitant stress-strain changes in the smooth muscle syncytia of the stomach were analyzed. Numerical results revealed that the fundus, the body and the antrum of the organ always experience biaxial stress-strain states, while the cardia and the pylorus undergo uniaxial stretching with crease development. The circular smooth muscle layer generated greater total forces throughout the dynamic process in comparison to the longitudinal layer. The body of the organ along the lesser curvature and the cardia-fundal area was constantly overstressed. Although the theoretical results qualitatively resemble patterns of electrical and mechanical activity observed in vivo and in vitro, there is currently no affirmative experimental evidence to conduct a detailed quantitative evaluation of the results. Keywords: human stomach, soft thin shell, biomechanics.

1

Introduction

Most hollow organs of the human body, including the eyeball, the esophagus, the stomach, the gallbladder, the uterus, the ureter, and the bladder, can be viewed as thin shells. Their high endurance and enormous functionality depend on biomechanical properties of the tissues they are made of and specific arrangements of internal constituents. The distinctive anatomical appearance of organs is strongly influenced by the force systems to which they are subjected and is correlated with their structural advantages: i) containment of optimal space within and outside, ii) efficiency of physiological (biomechanical) WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090181

192 Modelling in Medicine and Biology VIII function, iii) a high degree of reserved strength and structural integrity, and iv) a high strength-to-weight ratio. All of the above mentioned hollow organs share these properties. Consider, for example, that the stomach is the organ of the gastrointestinal tract that is located in the left upper quadrant of the abdomen immediately below the diaphragm. Its prime role is to accommodate and digest ingested food. Even with a small wall thickness, which in normal subjects varies from 3 to 5 mm, with the characteristic radius of the curvature of the middle surface ranging within10 < Ri < 15 (cm), it is capable of holding 2-5 liters of mixed gastric content without increasing intraluminal pressure. Large distensions and extensive mechanical activity of the organ do not affect function of the adjacent structures such as the pancreas, liver, and spleen. With the latest advancement in mathematical modeling of complex biological systems, it has become possible to develop complex models of the abdominal viscera and to gain insight into the hidden physiological mechanisms of their function [1-4]. Such models will have enormous implications for our understanding of the pathophysiology of various diseases, e.g., gastroparesis, gastric dysrhythmias, gastroesophageal reflux disease, dumping syndrome, etc, and their pharmacological treatment. The aims of this study are: i) to model the stomach as a thin soft biological shell, and ii) to study the dynamics of the propagation of electromechanical waves within the organ under normal physiological conditions.

2

Physiology of the stomach

The shape of the stomach is greatly modified by changes within itself and in the surrounding viscera that no one form can be described as typical. The chief configurations are determined by: i) the amount of the stomach contents, ii) the stage of the digestive process, iii) the degree of development of the gastric musculature, and iv) the condition of the adjacent loops of the small and large intestines. The stomach is more or less concave on its right side, convex on its left. The concave border is called the lesser curvature; the convex border, the greater curvature. The region that connects the lower esophagus with the upper part of the stomach is called the cardia. The uppermost adjacent part to it is the fundus. The fundus adapts to the varying volume of ingested food and it frequently contains a gas bubble, especially after a meal. The largest part of the stomach is known simply as the body. It functions as a reservoir for ingested food and liquids. The antrum, the lowermost part of the stomach, is usually funnel-shaped, with its narrow end connecting with the pyloric region. It empties into the duodenum - the upper division of the small intestine. The pyloric portion of the stomach tends to curve to the right and slightly upward and backward and thus gives the stomach its J-shaped appearance. The effectiveness and diversity of physiological responses of the stomach to internal and external stimuli depend on the inherent activity of smooth muscle cells, neurons, interstitial cells of Cajal, and their topographical organization in gastric tissue. Smooth muscle cells are embedded into a network of collagenous and elastin fibers and are coupled via gap junctions into three distinct syncytia WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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(muscle layers). The external longitudinal muscle layer continues from the esophagus into the duodenum. The middle uniform circular layer is the strongest and completely covers the stomach. The circular fibers are best developed in the antrum and pylorus. At the pyloric end, the circular muscle layer greatly thickens to form the pyloric sphincter. The innermost oblique muscular layer is limited chiefly to the cardia-fundal regions and progressively weakens as it approaches the pylorus. The longitudinal, oblique and circular smooth muscle syncytia react with a variety of rhythmical movements which are a result of an electrochemical coupling phenomenon. The role of pacemakers belongs to interstitial cells of Cajal, which are located within the myenteric nervous plexus and smooth muscle syncytia [5]. Contractions in the smooth muscle are initiated by myosin light chain phosphorylation via the activation of calcium calmodulin-dependent myosin light chain-kinase. The key player in the process is free cytoplasmic calcium [6]. Elastin and collagen fibers are high-molecular-weight structural proteins built in a three dimensional loosely woven network. Elastin is soft and may be stretched to 250% of the unloaded configuration. Collagen is relatively inextensible stiff reinforcing structural component, and the main load carrying element. Collagen fibers are usually undulated and they become stiff when straightened under the action of applied loads. The strength of tissues is strongly correlated with the collagen content. The uttermost tunica serosa coats the entire organ and provides its final shape. The submucous coat and mucous membrane of the stomach consist of epithelial and glandular cells. Their role in the biomechanics of the stomach and load distribution, in particular, is negligible. Motor propulsive activity in the stomach originates in the upper part of the body of the organ. Three types of mechanical waves are observed: i) small isolated contraction waves, and ii) peristaltic waves that slowly move from the point of origin down toward the pyloric sphincter. These types of contractions produce slight or deep indentations in the wall and serve as mixing, crushing and pumping mechanisms for the gastric contents. The third type of wave is nonpropagating in nature and is a result of the tonic simultaneous contraction of all muscle layers that are normally superimposed on small and peristaltic contractions [7].

3

Material characteristics

The mechanical properties of biocomposites are highly specific and depend on the organ, the topographical site, the respective function, and species and greatly influenced by environmental risk factors and age. The soft tissue that makes the wall of the stomach is characterized as transversely anisotropic non-homogenous viscoelastic biocomposite that undergoes finite strains. This has been convincingly demonstrated in quasistatic uniaxial tests on specimens collected from different regions of the organ [8]. The force-ratio of elongation (T-) data indicate that the tissue is initially compliant, 1.0 < < 1.3-1.4, and stiffens at WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

194 Modelling in Medicine and Biology VIII higher loads. For 1.4 < < 2.0 the tissue behaves nearly elastically until it breaks at Tmax = 1.5 N/cm. Stretching of specimens excised in the directions of the longitudinal, circumferential and oblique layers has shown properties of transverse anisotropy. The wall has lower extensibility along the orientation of longitudinal smooth muscle fibers and the highest along of the circumferential layer. Experiments under biaxial tension have shown that the shear force applied to the tissue is significantly less, 0.01Tmax, compared with the tangential force. The nonlinear behavior of the stomach has also been studied under complex loading, i.e., a combined inflation with subsequent external local compression of the anterior surface of the organ [9]. Instantaneous intraluminal pressure, volume and stress-strain recordings indicate that in the low pressure domain 0.1< P< 3.0 (kPa) the stomach experiences biaxial stress-strain states and smooth continuous deformations in all regions. In the higher pressure range 3.0
4

Model assumptions

The following assumptions are made in the formulation of a mechanical model of the stomach based on the anatomical, physiological and mechanical data of its appearance and function: 1) the stomach is a closed thin soft shell of complex geometry; it satisfies the criterion of thin shells: max h/Ri<1/20, where h is the thickness and Ri are the radii of the curvature of the middle surface of the shell, and the conditions of softness: the wall of stomach does not resist compression and bending, the shear stresses are negligibly small, and creases may be formed during deformation; 2) the wall of the stomach is formed of two distinct longitudinal and circular smooth muscle layers embedded into a network of elastin and collagen fibers; the layers display an orthogonal type of weaving in the undeformed configuration; 3) the biocomposite is a nonlinear viscoelastic transversely anisotropic continuum undergoing finite strains; its “passive” component depends on the mechanics of inactive smooth muscle syncytia, collagen and elastin fibers; the “active” component – the forces of contraction and relaxation, is the result of electromechanical conjugation at the level of intracellular contractile proteins; the key biochemical factor in the process of conjugation is free calcium ions; 4) muscle layers are electrically excitable biological media; the longitudinal smooth muscle layer possesses anisotropic cable electrical characteristics, while the circular layer is electrically isotropic; 5) the initial conditions assume that the stomach is in the resting unexcited state; a single pacemaker or multiple spatially distributed pacemakers are associated with interstitial cells of Cajal; their discharge rates and intensity are assumed to be known a priori; they provide excitatory signals to the system;

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6) clamped edge boundary conditions are realized at the cardiac and pyloric ends of the shell throughout simulations. Mathematical formulation of the model is given elsewhere [10].

5

Results

5.1 Inflated stomach The strain distribution in the wall of the bioshell inflated by intraluminal pressure p = 20 kPa in the state of dynamic equilibrium is shown in Fig. 1. The maximal elongation of the longitudinal muscle fibers, l = 1.35, is observed in the cardiofundal region along the greater curvature of the organ. A part of the fundus bordering the body experiences the maximal biaxial extension with l = 1.35, c = 1.04 in that area. The cardia and the body of the stomach along the lesser curvature, both undergo biaxial deformation with l = 1.01, c = 1.24, while the antrum-pyloric region along the greater curvature is under uniaxial strain, l = 0.5, c = 1.16. Negative circumferential deformations are registered in the cardia and the pylorus. Thus, the fundus and the body of the stomach undergo biaxial distension and the cardia and antrum-pyloric areas are subjects to uniaxial elongation. Analysis of the total force distribution in the bioshell demonstrates that maxTl = 9.3 mN/cm, Tc = 14.5 mN/cm are recorded in the fundus and the body. The maximum Tc = 34.7 mN/cm is registered in a small area of the body along the lesser curvature of the stomach. In the antrum and the lower cardia, total forces of average intensity Tl = 4.1 mN/cm, Tc = 14.5 mN/cm are observed. The proximal part of the cardia remains unstressed with Tl = Tc = 0. 5.2 Electromechanical wave activity Let two identical pacemaker cells be located to the longitudinal and circular smooth muscle layers in the upper body along the greater curvature of the stomach. The cells discharge simultaneously multiple impulses (n = 5) of amplitude φ0 = 100 mV and duration td = 0.1 s. The activation of ion channels on the membrane of smooth muscle causes the generation of the waves of depolarization of amplitude φ(l,c) = 65–70 mV. The velocity of the propagation of excitation varies between the syncytia and regions of the organ. Thus the wave φl quickly spreads within the longitudinal muscle fibers to encase a narrow zone within the anterior surface of the organ (t = 0.8 s). The wave φl sustains short wave-length and a constant amplitude 18 mV throughout 2.0
196 Modelling in Medicine and Biology VIII

Figure 1:

Static stress-strain distribution in the stomach.

The cardiac and pyloric regions experience extensive depolarization with a maxφc = 63.3 mV. At t = 5.6 s the longitudinal and circular smooth muscle syncytia of the stomach show a similar pattern in the distribution of depolarization with the average amplitude φl ≈ 3.7mV, φc ≈ 7.5mV. There is a smooth distribution of active forces of contraction in the fundus and the body of the stomach where Tal = 6.1 mN/cm and Tac = 7.4 mN/cm are generated during 0
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t = 0.8 s Figure 2:

197

t = 5.6 s

Dynamics of the force distribution in the stomach at times as indicated.

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198 Modelling in Medicine and Biology VIII The pattern of total force distribution in the shell is similar to that observed at the state of dynamic equilibrium. There is a rise in the intensity of forces, which is consistent with the process of the generation of active forces of contraction by the muscle syncytia. Thus, maxTl = 15.1 mN/cm, Tc = 22.4 mN/cm are recorded in the body, and Tl = 2.7 mN/cm, Tc = 5.3 mN/cm in the cardia and the pylorus of the stomach. 5.3 Time lag in firing of pacemakers We studied the effect of time delay (ti) in discharges of pacemaker cells on the dynamics of the force – elongation development in the bioshell. The number of impulses, their amplitude and duration correspond to the experimental conditions as described above. The pacemaker located in the longitudinal muscle syncytium fires first, followed by a discharge of the pacemaker on the circular muscle syncytium at ti = 0.75 s. There are no significant differences in the intensity of force development. However, the delay in activation of the circular smooth muscle layer results in early wrinkling (t = 0.8 s) in the body along the lesser curvature which persists throughout. The delayed generation of contractions in the circular smooth layer results in a reciprocal contraction-relaxation relation between the two bisyncytia. The pattern of electromechanical activity resembles peristalsis, i.e., a propagating wave of contraction – relaxation that satisfies the condition of reciprocity (Fig. 3).

Active force (mN/cm)

16.0

13.0

10.0

Circumferential layer

Longitudinal layer

7.0

4.0 0.0

2.5

5.0

7.5

10.0

12.5

15.0

Time (s)

Figure 3:

6

Changes in the active force dynamics in the longitudinal and circumferential smooth muscle layers at a “control” point in the body of the anterior surface of the human stomach.

Discussion

A first biomechanical model of the organ as a soft multilayer biocomposite shell was constructed by Miftakhov in 1983 [10]. Under general assumptions of curvilinear orthotropy and physical and geometrical nonlinearity, a mathematical formulation and numerical investigations of the dynamics of stress-strain distribution in the organ under simple and complex loadings were performed. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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The dynamics of the development of uniaxial stress-strained states in the cardia and pylorus as a function of intraluminal pressure was demonstrated computationally and supported experimentally. Results offered a valuable insight into the mechanism of blunt abdominal trauma with rupture of the anterior wall of the stomach and gave a biomechanical explanation for the Mallory-Weiss syndrome. It was thought previously that atrophic changes in the gastric mucosa and submucous layer were responsible for longitudinal tears in the cardia-fundal region and life threatening intragastric bleeding. The model study proved that the anatomical structure and configuration of the stomach per se makes these regions more susceptible, than the others, to linear submucous breaks and ruptures. The current model is an extension of the previous model and incorporates cable electrical properties of the longitudinal and circular smooth muscle syncytia along with mechanical nonlinearity. We continue to adopt the phenomenological approach to describe stomach behavior and, thus, we exclude from considerations a detailed modeling of intricate intracellular signaling mechanisms involved in complex physiological responses of the organ. Special emphasis in this study was given to the questions of electromechanical coupling in smooth muscle syncytia and cable electromechanical wave activity. In the current study we restricted our attention to the cable, rather than to more general oscillatory, properties of smooth muscle syncytia. The network of interstitial cells of Cajal – pacemakers of gastric motility, was substituted by a single cell or a set of spatially distributed cells. They discharged a priori known electrical impulses. Even with these constructive simplifications, the model reproduced and predicted: 1) patterns of the propagation of the wave of excitation within the electrically anisotropic longitudinal and electrically isotropic circular smooth muscle syncytia; 2) small isolated contraction waves in the two smooth muscle syncytia and non-propagating tonic simultaneous contractions of both muscle layers superimposed on small contractions; 3) the development of wrinkles in the cardia, the body along the lesser curvature and the pylorus, and 4) high levels of tension in the body of the stomach along the lesser curvature. Care should be taken though in transferring the results of simulations to explain real biomechanics of the human stomach. One has to bear in mind that the biological plausibility of the model is constrained by the model assumptions, despite the fact that the theoretical results resemble qualitatively patterns of electrical and mechanical activity that are observed in mainly animal studies in vivo and in vitro. At the moment there is no direct affirmative experimental evidence obtained on human subjects to run a detailed quantitative evaluation and comparison of the computational data.

References [1] Miftahof, R., Nam, H.G. & Wingate D. L. Mathematical modeling and simulation in enteric neurobiology, World Sci., 2008 (in press) [2] Pullan, A., Cheng, L., Yassi, R. & Buist, M. Modeling gastrointestinal bioelectric activity. Prog. Biophys. & Mol. Biol. 85, pp. 523-550, 2004 WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

200 Modelling in Medicine and Biology VIII [3] Cheng, L., Komuro, R., Austin, T.M., Buist, M.L. & Pullan A. J. Anatomically realistic multiscale models of normal and abnormal gastrointestinal electrical activity. World J. Gastroenterol. 7, pp. 13781383, 2007 [4] Corrias, A. & Buist, M.L. A quantitative model of gastric smooth muscle cellular activation. Annals Biomed. Eng. 35, pp. 1595–1607, 2007 [5] Hirst, D. G. S. & Suzuki, H. Involvement of interstitial cells of Cajal in the control of smooth muscle excitability. J. Physiol. 576, pp. 651-652, 2006 [6] Bárány, M. Biochemistry of Smooth Muscle Contraction. Academic Press, p. 418, 1996 [7] Alvarez, W.C. & Zimmermann, A. Movements of the stomach. Am. J. Physiol. 84, pp. 261-270, 1928 [8] Miftakhov, R.N. 1981a Age changes of the 'quasi-equilibrium' module of elasticity of the human stomach. In: Shell interactions with fluids, Acad. Sci. USSR. pp. 197-204 (in Russian) [9] Miftakhov, R.N. 1983b Experimental investigations of the stomach under complex loading. In: Hydroelasticity of shells, Acad. Sci. USSR, pp.172181 (in Russian) [10] Miftakhov, R.N. Experimental and numerical investigations of soft shells. Ph.D. thesis. Kazan State Univ., 1983, (in Russian).

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Section 4 Physiological processes

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The human body exposed to a magnetotherapy device magnetic field D. Poljak1, S. Sesnic1, D. Cavka1, M. Titlic2 & M. Mihalj2 1 2

University of Split, Split, Croatia Clinical Hospital Centre, Split, Croatia

Abstract This paper deals with human exposure to extremely low frequency (ELF) magnetic fields generated by a magnetotherapy device. The problem is twofold, i.e. it implies the assessment of an external ELF magnetic field and other internal parameters related to the human body response to this field (total current, power density, total power). Of particular importance is the current density induced inside the human body as the basic restriction proposed by the relevant international bodies for non-ionizing radiation, such as ICNIRP. ELF magnetic field levels generated from a magnetotherapy device are obtained from measurements. Knowing the external magnetic field generated by a magnet, the circular current density induced in the human body is obtained by the use of the analytical formula arising from the disk model of the human torso. On computing the circular current density, it is possible to assess other parameters of interest related to body response. The obtained values of both the magnetic field and the internal current density are compared to exposure limits in terms of reference levels and basic restrictions proposed by ICNIRP. Keywords: magnetotherapy, human exposure to magnetic fields, disk model of the body, induced current density.

1

Introduction

Magnetotherapy based on extremely low frequency (ELF) magnetic fields is considered to be a non-invasive technique, widely used in the treatment of muscle pain [1]. The sale of magnetic field based devices such as bracelets, insoles, pillows, etc. makes more than 1 billion dollars worldwide per year [1].

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204 Modelling in Medicine and Biology VIII Human exposure to ELF magnetic fields generated by magnetotherapy devices has been analyzed in a number of studies in last decade [1-5]. Of particular interest was a possible effect of such exposures to the cardiovascular system [1]. Contradictory findings have been reported in many previous papers. Some studies claim that ELF magnetic fields cause slowing of the heart rate, while other studies do not confirm those effects. For example, the results published in [4] indicate that 150-200μT ELF magnetic field at the subject’s heart position causes slowing of the heart rate. Some authors claim that the slowing of the heart rate due to the exposure to an ELF magnetic field is related to the interaction with the control mechanisms for thermoregulation and blood pressure [2]. Findings in [2] imply the magnetic field 28μT/50Hz affects the normal heart rate [3]. It is important to emphasize that this value is below the relevant guidelines proposed by ICNIRP for public exposure. There are also experimental studies reporting no significant effects of ELF magnetic fields to which humans are normally exposed daily on heart rate [3]. As the displacement currents at extremely low frequencies are negligible, the electric and magnetic fields can be analyzed separately and can be determined via the computations or/and measurements in [6-10]. In the case of magnetic field exposure, the internal currents form close loops, contrary to electric field exposure where the currents induced in the body have the axial character [7]. This paper deals with an analysis of human exposure to ELF magnetic fields generated by magnetotherapy devices using the analytical approach based on the disk model of the human body [11]. First, the spatial distribution of the magnetic field is determined from measurements and then the circular current density induced in the human body, as a main parameter for the estimation of low frequency exposure effects proposed by ICNIRP basic restrictions [12], is computed. This current density is evaluated by using analytical formula featuring the disk model of the human body. The obtained values for both external magnetic induction and internal current density are compared to exposure limits proposed by ICNIRP. The knowledge of the internal current density provides further analytical evaluation of the induced total current, power density and the total power dissipated in the human body exposed to ELF magnetic field. It is worth noting that the main feature of the model is efficiency and rather rapid estimation of the phenomena.

2

The disk model of the human body

If human being is exposed to ELF magnetic field the circular current density is induced inside the body due to component of the magnetic flux density normal to the body. This internal current density can be assessed by using the simplified disk model of the human body [11], Fig 1. The disk is assumed to be homogeneous with radius a and conductivity σ. The analytical relation for the current density can be directly derived from Maxwell equation:

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 B

 J ρ

a

Figure 1:



 a

The disk representation of the human body.

  B xE   t



(1)

where B and E denotes the external magnetic and internal electric field, respectively. Using the constitutive equation:

  J  E

(2)

 where J denotes the internal current density, and integrating over an arbitrary surface it follows:

   B  S xJdS  S  t dS

(3)

Applying the Stokes theorem, for time harmonic fields assuming the constant value of magnetic induction, equation (3) becomes:

    J d s   j  B   dS c

(4)

S

i.e., it follows:  2

2

 J  d   jB   dd z

0

(5)

0 0

where ω=2πf is the angular operating frequency. Taking into account the rotational symmetry of the geometry from Fig 1 the straightforward integration yields: WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

206 Modelling in Medicine and Biology VIII

J  2   jBz  2

(6)

and the value of the induced current inside the disk is simply given by:

J   f  B z

(7)

Once knowing the current density, taking the straightforward integration it is possible to calculate the total current flowing through the disk, as shown in Fig 2: a a   aa a3 I   JdS     fBz  d  dz   fBz    d dz   fBz 2 S 0 0 0 0

(8)

It is worth noting that the thickness of the disk is not specified by the ICNIRP guidelines, as the total current is not related to basic restriction. In this work, the thickness is assumed to be equal a (the disk radius).

 B

 J

ρ

Figure 2:

a

dρ

Integration over the disk cross-section.

The power density dissipated inside the body, for time-harmonic excitations, is defined, as follows:

2 1  J Pd  EJ  2 2

(9)

Combining (7) and (9), for ρ=a, yields:

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(10)

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Furthermore, the total power dissipated in the disk body model is given by:

2 J 1  P   Pd dV   EJdV   dV  2 2 V V V

(11)

Combining (7) and (11) after integration, one obtains:

1 P 2

2 a a

  

1 4

 2 a 2 Bz2  d  dzd   3 f 2 Bz2 a5

2

0 0 0

(12)

Finally, a rate of magnetic energy increase in the human body can be computed from the expression:

Wmag t



 1   H  B dV t V 2





(13)

As the human body represents a linear magnetic material, the magnetic field strength

  H is proportional to flux density B so equation (13) simplifies into: Wmag  B 2   dV (14) t t V 2

For the time-harmonic variation of external ELF magnetic induction:

B (t )  B0 cos t

(15)

Wm  1 a 3 2 B (t ) dV   B02 sin 2t   2 t t V 2

(16)

It follows:

Note that the energy change oscillates twice faster than the external ELF magnetic flux density.

3

The results

The measurements of the ELF magnetic field generated by the magnetotherapy device have been carried out via the magnetic field meter AARONIA AG SPECTRAN NF 5020 at the frequencies commonly used for physical therapy, i.e. at 50Hz, 75Hz and 100Hz. The duration of the physical therapy is usually between 15 and 20 minutes. The magnetic flux density measurement has been performed in the vicinity of the device. The obtained values of the magnetic induction are mostly from 100μT to 500μT which directly correspond to ICNIRP reference levels for general and professional population, respectively. Once the magnetic flux density is determined, the internal current density can be calculated using the disk model of the human body. The disk radius is a=0.14m while the conductivity is σ=0.5S/m. Furthermore, knowing the internal current density provides the assessment of additional parameters describing the human body response in terms of total current, power density and total power dissipated inside the human body. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

208 Modelling in Medicine and Biology VIII The computed values for various parameters representing the human body response to ELF magnetic field are presented in Tables 1 to 3 for different frequencies. Finally, Figs 3 to 5 show the rate of change of magnetic energy in the disk body model. Table 1: B0 [μT] 100 200 300 400 500 Table 2: B0 [μT] 100 200 300 400 500 Table 3:

Various parameters of human body response to ELF magnetic field exposure at f=50 Hz. ICNIRP ICNIRP J [mA/m2] [μT] [mA/m2] 1,10 500 (workers) 2,20 10 (workers) 3,30 100 2 (general public) (general 4,40 public) 5,50

I Pd [mA] P [nW] [μW/m3] 0,01 5,21 1,21 0,02 20,8 4,84 0,03 46,9 10,9 0,04 83,4 19,3 0,05 130, 07 30,2

Various parameters of human body response to ELF magnetic field exposure at f=75 Hz. ICNIRP ICNIRP J [mA/m2] [μT] [mA/m2] 1,65 333 (workers) 3,30 10 (workers) 4,95 67 2 (general public) (general 6,60 public) 8,25

I Pd [mA] P [nW] [μW/m3] 0,02 11,7 2,72 0,03 46,9 10,9E 0,05 106 24,5 0,06 188 43,5 0,08 293 68

Various parameters of human body response to ELF magnetic field exposure at f=100 Hz.

B0 ICNIRP J ICNIRP [μT] [μT] [mA/m2] [mA/m2] 100 1,10 250 200 (workers) 4,40 10 (workers) 50 300 6,60 2 (general public.) 400 (general 8,80 500 public) 11,00

I Pd [mA] P [nW] [μ W/m3] 0,01 5,21 1,21 0,04 83,4 19,3 0,06 188 43,5 0,09 334 77,4 0,11 521 121

From the obtained results it is visible that for some scenarios the values of both magnetic flux density and internal current density exceed the exposure limits, for both general population and workers. However, this statement should be considered rather carefully, namely:  According to ICNIRP guidelines the reference levels are assumed to be spatially averaged over the whole body, provided that the basic restrictions on localized exposure are not exceeded. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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

On the other hand, ICNIRP guidelines define public exposure as all kind of exposures to which general population is exposed exceeding occupational exposure and exposure during medical treatment.  Therefore, an attention should be focused to exposure of workers, i.e. medical personnel. Scenario of interest is twofold: - exceeded reference levels for the case of entire body exposure - exceeded basic restrictions for the case of localized exposure Consequently, careful future studies on professional exposure to ELF magnetic fields would be of interest.

B [μT]

(dWm/dt)x10e16

200 150 100 50 0 -50 -100 -150 -200 0

5

10

15

20

25

30

35

40

t [ms]

Figure 3:

Magnetic induction and magnetic energy change at f=50Hz.

B [μT]

(dWm/dt)x10e16

300 200 100 0 -100 -200 -300 0

5

10

15

20

25

t [ms]

Figure 4:

Magnetic induction and magnetic energy change at f=75Hz.

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210 Modelling in Medicine and Biology VIII

4

Concluding remarks

In this work human exposure to ELF magnetic fields generated by magnetotherapy devices is analyzed using the simplified disk model of the human body. The problem is twofold involving the assessment of both magnetic flux density and the internal parameters related to the human body. These parameters are internal current density, power density and total power dissipated in the body. The magnetic field generated by the magnetotherapy device is determined from measurement. The current density induced inside the human body is computed featuring the analytical formula arising from the disk body model. The assessment of the internal current density provides the calculation of further parameters of the human body exposed to ELF magnetic fields in terms of total internal current, power density and total power dissipated inside the body. The results obtained from measurement and calculations for the magnetic flux density and the internal current density, respectively, are below the reference levels and basic restrictions, respectively, proposed by the ICNIRP guidelines in most cases. B [μT]

(dWm/dt)x10e16

400 300 200 100 0 -100 -200 -300 -400 0

5

10

15

20

t [ms]

Figure 5:

Magnetic induction and magnetic energy change at f=100Hz.

Some scenarios may lead to values of external magnetic flux density and internal current density exceeding the exposure limits. Thus, of particular interest would be further investigation on professional exposures to ELF magnetic fields used in physical therapy treatments.

References [1] M. Fernandez, P.J. Watson, D.J. Rowbotham, Effect of Pulsed Magnetic Field Therapy on Pain Reported by Human Volunteers in a Laboratory Model of Acute Pain, British Journal of Anaesthesia 99 (2), pp. 266-269, 2007. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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[2] M. L. Sait, A.W. Wood, H.A. Sadafi, A study of Heart Rate Variability in Human Subjects Exposed to Occupational Levels of 50Hz Circularly Polarised Magnetic Fields, Medical Engineering and Physics 21, pp. 361369, 1999. [3] Y. Kurokawa, H. Nitta, H. Imai, M. Kabuto, Can Extremely Low Frequency Alternating Magnetic Fields Modulate Heart Rate or its Variability in Humans?, Autonomic Neuroscience: Basic and Clinical, pp. 53-61, 2003. [4] Z. Tabor, J. Michalski, E. Rokita, Influence of 50Hz Magnetic Field on Human Rate Variability: Linear and Nonlinear Analysis, Bioelectromagnetics 25, pp 474-480, 2004. [5] L. Finegold, B.L. Flamm, Editorial: Magnet Therapy, Extraordinary claims, but no proved Benefits, BMJ, Vol, 332, Jan 2006. [6] O.P. Gandhi, J.Y. Chen, Numerical Dosimetry at Power Line Frequencies Using Anatomically Based Models, Bioelectromagnetics Suppl., Vol. 1, pp. 43-60, 1992. [7] D. Poljak, Human Exposure to Electromagnetic Fields, SouthamptonBoston: WIT Press, 2003. [8] R.W.P. King, Fields and Currents in the Organs of the Human Body When Exposed to Power Lines and VLF Transmitters, IEEE Trans. Biomedical Eng., Vol. 45, No 4, pp. 520-530, April 1998. [9] Poljak, D., Rashed, Y., The Boundary Element Modelling of the Human Body exposed to the ELF Electromagnetic Fields, Engineering Analysis with Boundary Elements, 26, pp 871-875, 2002. [10] D. Poljak, A. Peratta, C.A. Brebbia, A 3D BEM Modelling of Human Exposure to ELF Electric Fields, BEM XVII, Incorporating Electrical Engineering and Electromagnetics, pp. 441-451, Orlando, USA, March 2005. [11] IEC 62226-2-1: Exposure to electric or magnetic fields in the low and intermediate frequency range – Methods for calculating the current density and internal electric field induced in the human body, 1st edition, November 2004 [12] ICNIRP Guidelines for Limiting Exposure to Time-Varying, Electric, Magnetic and Electromagnetic Fields (up to 300GHz), Health Phys., Vol. 74, 4 (1998), 494-522

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Electronic active model for saccadic eye movements O. Terán & E. Suaste Bioelectronics Section, Department of Electrical Engineering, CINVESTAV-IPN, Mexico

Abstract The design, development and construction of an electronic active model (EAM) were proposed. The model was based on the reciprocal innervation mechanical model for horizontal eye movements and was intended to emulate the biodynamic rotatory properties of a human eye plant caused by neurologic activation signals on the lateral and medial extraocular muscles. Biodynamic properties of the mechanical eye plant, such as viscosity, inertia and elasticity, were included in the EAM in order to obtain a similar response. Apart from these features, voltage controlled resistors (VCRs) based on MOSFET to keep the nonlinearity of the plant were also included. These active elements, together with passive components, constitute the structure of the extraocular muscles, eyeball and tissues around the eyeball. The activation of agonist-antagonist signals were generated by two voltage controlled sources to reproduce the neurological activity. In this way, applying the appropriate signals to generate saccadic movements was obtained by a voltage signal proportional to the velocity of the eye torque. This signal allows one to obtain information relating to acceleration and position in order to validate the EAM. Due to this capability to emulate saccadic movements, the EAM makes it possible to reproduce involuntary eye movements caused by pathologies such as Horizontal Nystagmus. Finally, some of the EAM’s advantages lie in its response velocity and the ease of obtaining continuous records for the dynamic eyeball response in a signal that features ease of recording and application in biomedical areas, and is particularly relevant to specialists such as Neuro-ophthalmologists, ophthalmologists and optometrist, and even for medical education. Keywords: eye plant, electronic model, saccadic eye movements.

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214 Modelling in Medicine and Biology VIII

1

Introduction

Nowadays, models and instruments that offer valuable information regarding the visual system and make it possible to reproduce phenomenological events (generally in a particular section) are becoming more frequently found in the biomedical sciences, such as ophthalmology, neurology and physiology, which are directly linked with human vision,. One of these sections has been the oculomotor system, which is the final stage in the human control of eye movements. Descartes was the first to propose the idea of the conjugate muscles action to produce rotation of the eyeball. This principle of reciprocal innervations opens the way to new ideas that simulate the prediction of the dynamic response of eye movements [1]. A mechanical model that obtains this result was presented by Clark and Stark [2] for saccadic horizontal eye movements. Based on Descartes’ principle and Hill’s model to muscles [3], a mechanical model is capable of predicting magnitude, velocity and acceleration of saccadic eye movements from the primary position to a new desired position of the eye. This model has probed the efficacy of simulating the eye plant due to the considerable nonlinear properties of the extraocular eye muscles [4]. In this manner, the importance of the eye plant models lie in obtaining a new form to analyze disorders in people with some class of congenital or pathological disorder in the oculomotor system that most of the time causes a diminution in visual acuity, as occurred in the case of Nystagmus, which can only be treated to reduce it effects. For this reason, the presented design focuses on offering a tool that can be directly applied in diagnostic or research related to the human visual system. In addition, the electronic active model (EAM) is a useful tool in medicine or ophthalmologic schools for didactical purposes

2

Method

The sixth order model developed by Clark and Stark [4] has been studied by other authors [5–7] and the results have demonstrated that it can reproduce realistic eye position, velocity and acceleration trajectories of human saccades. For this reason, our design is based on this nonlinear mechanical model for horizontal eye movements (Fig. 1). The extraocular muscles are based on the Hill muscle model. Both contain nonlinear dashpots BAG and BAT, the viscosity of which change as a function of force and velocity in the muscles. Inside each muscle, there is an active-state tension generator (ATG) responsible for producing the necessary force to the extraocular muscle contraction [8,9]. This force is the ideal physiological force generated into the muscle and cannot be measured directly. The force produced by an active state tension generator defines the magnitude and duration of saccadic eye movements; therefore it is important to control these properties. In order to simplify this control, the force produced by the ATGs is obtained from filtering a pulse-step signal with a lowpass filter, which changes its time constant in the falling edge of the signal. This is related with the activation and deactivation of cells inside the muscle [4].

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The model element JG represents the inertia due to mass of the eye globe; KG and BG are the elasticity and viscosity caused by the tissue that surrounds the eye globe, including the lateral rectus muscles. KSE is related with the instantaneous change of length of the muscle in response to an instantaneous change of force. KP(AG) and KP(AT) are the passive elasticity of the muscle. The ATGs FAT and FAG depend directly on the pulse step signals NAG and NAT. In Fig. 2 the input and output signals of the agonist and antagonist extraocular rectus muscle and the moment when the time constant change occurs in the filter are represented. θ1

JG Extraocular rectus muscle KSE NAG

θ2 Filter

KP(AT)

KP(AG) KG

FAG BAG

Extraocular rectus muscle

Eye globe

BG

KSE

θ3

FAT BAT

NAT Filter

Figure 1:

Sixth order nonlinear eye plant model for horizontal saccadic eye movements [3], BAG and BAT represent the nonlinear force-velocity relationship.

Figure 2:

Controller signals transformed into active tensions by the first order filter with the two activation and deactivation constants.

The equations that govern the entire system are: 

FAG  K SE ( 2  1 )  B AG  2 WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

(1)

216 Modelling in Medicine and Biology VIII 

FAT  K SE (1   3 )  BAT  3

(2) 



K SE ( 2  1 )  K SE (1   3 )  K P1  BP  1  J G  1

(3)

where:    1.2 FAG /(900   2 )  2  0 B AG    3FAG / 900 2 0   3  0 3FAT / 900 BAT     1.2 FAT /(900   3 )  3  0

(4)

(5)

In eqn. (3) the term KP represents the elements KP(AG), KP(AT) and KG that were simplified in a spring connected to the eye globe. Considering the equation of the system and the characteristics of the mechanical model, the analog circuit was obtained directly (Fig 3). According to the terminology of the mechanical model, components were named in respect to their corresponding mechanical analogies.

Figure 3:

Simplified electronic model where R1 and R2 are nonlinear resistors.

The resultant circuit has three networks, which senses current on each one to represent the direction of the eye movements. As a consequence of the electrical equivalence, the magnitude of position θ1 is represented in the circuit by the electrical charge q. Therefore I1=dq/dt represents the velocity of the eye movements. To obtain magnitude and acceleration it is necessary to integrate and derive this current signal. The voltage sources VAG and VAT generate the equivalent tension produce by the ATGs (Fig. 2). The pulse-step signal and filter are included inside the voltage sources. The eye plant model contains two nonlinear dashpots to satisfy the nonlinear force-velocity relationship in the muscle. In the circuit (Fig. 3) these mechanical elements are represented by nonlinear resistors (R1, R2). Their values are given by the voltage and current in each network, as shown in the equations of the circuit. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

C SE

1

where:

 ( I

V AG  C SE

1

V AT  C SE

1

 (I  (I

2

1

217

 I 1 ) dt  R1 ( I 2 )

(6)

 I 3 )dt  R2 ( I 3 )

(7)



2  I1 ) dt   ( I1  I 3 )dt  C EQ

1.25 V AG  900  I 2 R1    3  V AG  900 1.25  V AT  900  I 3 R2    3  V AT  900

1



 I1dt  RG I1  LG I 1

(8)

I2  0 (9)

I2  0 I3  0 (10)

I3  0

R1 and R2 are voltage controlled resistors (VCRs) that are circuits capable of emulating an electric resistor, which have the advantage of changing resistance with a control voltage. For this reason, a VCR solves the problem of a timevariant resistor present in our EAM (Fig. 4). The VCR designed is based on the principle of resistive mirror [10]. This technique linearizes the trans-impedance response of common BJT, FET or MOSFET transistors as a function of the voltage control. To generate these voltages (Vf1=R1 and Vf2=R2) for each VCR requires independent circuits of control (f1 and f2) designed to response according eqns. (9) and (10). Other important blocks of the circuit are the filters, which change their time constant commuting resistors (τAC, τDE) at the same time

Figure 4:

Block diagram of the EAM where I1 represents the eye velocity for saccadic movements.

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218 Modelling in Medicine and Biology VIII

Figure 5:

Position curves obtained by the EAM on CP for saccades with amplitudes of 1, 5, 10, 20, 30, 40, 50 and 60 deg.

Time (s)

Figure 6:

Velocity curves obtained by the EAM on RG for saccades with amplitudes of 1, 5, 10, 20, 30, 40, 50 and 60 deg.

(Fig. 2). Thus, a microcontroller coordinates the electronic switches inside the filter with the pulse-step signal (NAT, NAG), modifying the time constants correctly. This requires that the microcontroller also controls the amplitude and duration of NAT and NAG. Both parameters are easy to control digitally using a digital-analog converter for the amplitude of the signal and a timer inside the microprocessor for duration; this also guarantees the repeatability of the input stimulation voltage signal to the EAM.

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219

Results

The final circuit of the EAM was probed in a range of movement of 1 to 60 degrees from an initial position of 0 degrees. The final values of the electronic circuit are: CP = 1160uF, RG = 15Ω, LG = 47mH, CSE = 560uF, RAC= 1.8kΩ, RDE=3.6kΩ, CAG = 2.2uF and CANT = 2.2uF. All of these values were obtained from the original mechanical model multiplied by a factor of 1000 to simplify the design. The electronic ATGs are related to the mechanical force in a scale of 1 gf = 0.01 V. Position (Fig. 5), velocity (Fig. 6) and acceleration (Fig. 7) curves were obtained through voltage on CP, RG and LG, respectively, taking advantage of the capability of these components to derivate (LG), integrate (CP) or leave the wave form of the current without change (RP). However, it is necessary for LG to have a high quality factor in order to minimize the error caused by electric resistance of the wire.

Figure 7:

Acceleration curves obtained by the EAM on LG for saccades with amplitudes of 1, 5, 10, 20, 30, 40, 50 and 60 deg.

To obtain the appropriate magnitude scale, position voltages need to be multiplied by a factor of CP, RG for velocities and 1/LG for acceleration voltage curves and each one multiplied by 100000 to compensate for the reduction in the voltage of the ATGs and increments in the values of the components in the circuit. In order to evaluate the goodness of the EAM results, the voltage curves were analyzed according with the main-sequence diagrams for human eye movements [1] and compared (Fig. 8) with saccade eye movement signals obtained by high velocity video-oculography [11], the curve positions and the velocity obtained at the maximum deviation of ±10 ms for the duration of the saccades, ±3 deg from the desired position, +100 deg/s. Many of these EAM curves do not match with the experimental curves due to biological variations that modify the dynamical responses of the human eye plant. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

220 Modelling in Medicine and Biology VIII

a)

b) Figure 8:

4

a) The solid line is the postion estimation generated by the EAM, the dotted line is the position trajectory obtained by high velocity video-oculography, both curves for a 10 deg saccade. b) The solid line is the velocity estimation directly generated by the EAM, the dotted line is a curve velocity obtained from the derived position curve in a), both curves were generated from a 10 deg saccade.

Discussion

The designed EAM is capable of emulating the saccadic eye movements as suggested by the comparison with experimental results. According with experimental researches by several authors [1, 5, 6, 12–14], 10º saccades generated by EAM comply with 50º ms of duration, peak velocity of 500 deg/s and acceleration of 40000 deg/s2. The EAM has been based on nonlinear models complying with the nonlinear force-velocity relationship present in extra ocular eye muscles. A VCR was designed with the experimental response in the WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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required resistive range and with the greater linearity in the saccadic signals bandwidth [15]. The resistive range of the VCR was determined by the experimental probes in an arrangement of saccade amplitude of 1 to 60 degrees. Due to the amplitude of the saccadic eye movement signals obtained by the EAM, these do not need to be conditioned to be measured by an instrument, such as an oscilloscope, except where the application has a great demand for current. The EAM may function completely without the necessity to use a computer because the activation state signals are easily generated and manipulated with the aid of a microcontroller or another source can be used to generate the desired stimulation signals.

References [1] Bahill, A. T., Bioengineering: Biomedical, Medical and Clinical Engineering, Prentice-Hall, Inc, Englewood Cliffs, New Jersey, pp. 112200, 1981, 1981. [2] Clark, M. R. & Stark, L., Control of Human Eye Movements: III. Dynamic Characteristics of the Eye Tracking Mechanism. Mathematical Biosciences, 20, pp. 239-265, 1974 [3] Clark, M. R. & Stark, L., Control of Human Eye Movements: I. Modeling of Extraocular Muscle. Mathematical Biosciences, 20, pp. 191-211, 1974. [4] Clark, M. R. & Stark, L., Control of Human Eye Movements: II. A Model for the Extraocular Plant Mechanism. Mathematical Biosciences, 20, pp. 213-238, 1974. [5] Enderle, J. D., Wolfe, J. W., Time Optimal Control of Saccadic Eye Movements. IEEE Transactions on Biomedical Engineering, 34(1), pp. 4355, 1987. [6] Bahil, A. T., Latimer, J. R., Troost, B. T., Linear Homeomorphic Model for Human Movement. IEEE Transactions on Biomedical Engineering, 27(11), pp. 631-639, 1980. [7] Martin, C. F., Schovanec, L., Muscle Mechanics and Dynamics of Ocular Motion. Journal of Mathematical Systems, Estimation and Control, 8(2), pp. 1-15, 1998. [8] Robinson, D. A., The Mechanics of human Saccadic Eye Movement. J. Physiol., 174, pp. 245-264, 1964. [9] Enderle, J. D., Bronzino, J. D., Blanchard, S. M., Physiological Modeling (Chapter 12). Introduction to Biomedical Engineering, Academic Press, pp. 693-797, 2005. [10] Tadic, N., Resistive Mirror-Based Voltage controlled Resistor with Generalized Active Devices. IEEE Transactions on Instrumentation and Measurement, 42(2), pp. 587-591, 1998. [11] Villamar L A and Suaste E (2008) High Velocity Videoculography to Determination of the Pupil Dynamics, American Institute of Physics Conf. Proc. Vol. 1032, Tenth Symposium on Med. Phys., Melville, New York, pp. 276-279, 2008.

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222 Modelling in Medicine and Biology VIII [12] Enderle, J. D., Wolfe J. W. & Yates J. T., The Linear Homeomorphic Saccadic Eye Movement Model-A Modification. IEEE Transactions on Biomedical Engineering, 31(11), pp. 717-720, 1984. [13] Clark, M. R., Stark, L., Time Optimal of Human Saccadic Eye Movements. IEEE Transactions on Automation Control, 20(3), pp. 345-348, 1975. [14] Pfann, K. D., Keller, E. L. & Miller, J. M., New Models of the Oculomotor Mechanics Based on Data Obtained with Chronic Muscle Force Transducers. Annals of Biomedical Engineering, 23, pp. 346-358, 1995 [15] Zuber, B. L., Semmlow, J. L. & Stark, L., Frequency characteristics of the Saccadic Eye Movement. Biophysical Journal, 8(11), pp. 1288-1298, 1986.

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Vestibular apparatus: dynamic model of the semicircular canals L. Gastaldi, S. Pastorelli & M. Sorli Dipartimento di Meccanica – Politecnico di Torino, Italy

Abstract The vestibular apparatus plays a fundamental role in equilibrioception. It is located in the inner ear and it substantially comprises otoliths, which indicate linear acceleration, and semicircular canals, which detect rotational acceleration. In this paper models of different complexity of the semicircular canals are reported and discussed: a monocanal hydro-elastic model and a three-canals model, in which interactions between canals are considered. Different trends are obtained by implementing the two models in a Matlab environment. The numerical results of the monocanal model are compared with the classical torsion-pendulum model and with literature data; while, for the three-canals model, responses to angular velocity directed parallel to the physiological axes are reported. Keywords: semicircular canals, hydroelastic model, three-canals model, mechano-transduction.

1

Introduction

The vestibular system is the apparatus that detects information about spatial position and movement of the head and of the body. This information is a fundamental input to control the posture, the upright position and to coordinate eye and head movement. The vestibular is quite a complex system and not all of the mechanical-nervous transduction mechanisms are completely known; this is also due to the impossibility of directly measuring human nervous signals. Mauro et al. developed a model of the vestibular apparatus to be used for the motion cueing algorithm of a movement simulator [1], but it can be also effectively employed to better understand the functioning of the system itself or with diagnostic aims to evaluate pathologies of the balance system. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090211

224 Modelling in Medicine and Biology VIII In this paper a model of the semicircular canals, which detect only rotational movements, is presented. Since its discovery at the end of the XIX century, the transduction biophysics of the vestibular apparatus has become an object of interest and in literature there are some models that describe the mechanics of the semicircular canals with different approximation levels: mass-spring-damping lumped parameters models [2, 3] that describe the macro-mechanics of the phenomenon and monocanal [4] or three-canals [5] fluidodynamic models. The aim of the development of a new model is to obtain a rigorous mathematical description that can be used to validate simpler or linearised formulations or as a component in more complete models that account for the vestibular-ocular reflex or the mechanical-electric transactions peculiar to the semicircular canals. Regarding the geometrical and physiological data of the semicircular canals, due to the intrinsic difficulty of a direct measurement, the authors referred to literature, considering and comparing, whenever possible, different sources and average values were taken.

2

Physiology of the semicircular canals

Peripheral apparatus of the vestibular system comprises different organs that perform distinct sensorial functions and perceive static and/or dynamic movements according to the six d.o.f. that characterise a body that moves in a three-dimensional space. To register head movements, or more precisely head movement change, the vestibular apparatus comprises two different sensorial structures, both placed in the labyrinth of the inner ear:  otoliths to detect linear acceleration;  semicircular canals to detect angular acceleration. Both sensorial organs contains the same receptors, the hair cells, but they are characterised by two different peculiar anatomical structures that determine the specificity in detecting linear or angular movements. The semicircular canals (fig. 1) are three three-quarter circular ducts that intercommunicate in the utricle and they are filled with a fluid called endolymph. According to their position, canals are designated as horizontal, posterior and anterior. They are aligned approximately orthogonally to one another and the horizontal canal forms a 30° angle with the physiological transverse plane, while the superior and posterior canals are aligned roughly at a 45° angle to the sagittal plane. Each canal has an expanded end, the ampulla, which opens into the utricle and the superior and posterior canals join together at one extremity. In the internal epithelium of the ampulla, next to the utricle, a gelatinous structure named the cupula is present; it introduces a discontinuity in the canal and the hair cells’ cilia are imbedded inside. The cilia cells operate the transduction of a mechanical deformation due to acceleration into nervous impulses that will be elaborated by the central nervous system.

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horizontal canal anterior canal

posterior canal

ampulla utricle cupula

Figure 1:

Physiology of the semicircular canals.

The physical principle is the following: when twisting the head, the bone structure of the semicircular canals that are integral to it accelerate with the same law, while the endolymph, due to inertial effects, moves with a lag, causing a relative displacement between the duct and the endolymphatic fluid. This flow causes a cupula deformation and a variation of the rest position of the hair cells and hence a change of the nerve impulse discharges rate carried by the vestibular nerve fibres to the brain stem.

3

Torsion-pendulum model

The first mathematical description of the physiology of angular motion detection is represented by the torsion-pendulum model first introduced by Steinhausen [6]. The single semicircular canal is described using a damped mass-spring model subject to an inertial force proportional to the mass. Through the years several integrations have been made to this formulation, as measurements on vestibular models pointed out that sensations detected during a rotational movement are more complex than those predicted by the torsion-pendulum model. Young and Oman [7] introduced an adaptation operator in cascade to resolve the difference between the perceived responses experimentally measured and the ones predicted by the torsion-pendulum model. Zacharias [8] suggested the introduction of a further term to also consider neural transduction dynamics. Let us consider the canal schematisation of fig. 2. k

c

e

m

Figure 2:

c

Torsion-pendulum model of a semicircular canal.

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226 Modelling in Medicine and Biology VIII The transfer function of the mechanics of the semicircular canals can be derived from the balance of the forces acting on the system, which, using the Laplace transform, may be written as: e  c s (1)  c k  s2  s  m m where:  angular velocity of the canal c angular displacement of the semicircular canal duct, considered integral with the head and referred to the inertial system; angular displacement of the endolymph, referred to the inertial system; e k elastic constant c coefficient of kinematic viscosity m mass of the endolymph; Due to the viscous characteristics of the endolymph, this is an over-damped system and eqn. (1) becomes:

e  c   1 2 s    1s  1 2 s  1

(2)

introducing: 1  c / k long time constant 2  m / c short time constant The short time constant is dominant for high frequencies and is defined by the ratio of the mass and the viscous damping. On the contrary the long time constant, defined as the ratio of the viscous damping term and the stiffness term, influences the system behaviour for the low frequency range. The numerical value of the short time constant can be derived by the semicircular canal physiology and there is a substantial agreement in assuming for man 2=0.005 s [9], while for the value of the long time constant data reported in bibliography are discordant. Van Egmond et al. [10], according to verbal response of humans subjected to various motion inputs, assumed a long time constant of 10 s; Mayne [11] used the audiogyral illusion and estimated a value between 8 and 11 s, while Groen [12], using nystagmus records, obtained 16 s. The estimation of the long time constant based on subjective responses is quite difficult, as it has to be purged of the effect of neural processes. Assuming that no neural process takes place for vestibular-ocular nystagmus, Schmidt et al. [13] calculated 1=18 s; as this result is consistent with the consideration of other authors [2, 12, 14] we assumed this value to estimate the frequency response of the torsion-pendulum model. Even if the parameters of this lumped parameter model are not easily identifiable, this mathematical description remains the most popular tool in analysing the behaviour of the semicircular canals. However, the torsionpendulum model is unable to describe the displacement of the cupula and its interaction with the endolymph. To evaluate the real displacement that describes the hair cells models, fluid mechanics are needed. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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4

227

Hydroelastic-fluidodynamic monocanal model

A detailed analysis of the fluid mechanics in a toroidal duct subjected to a pressure gradient was presented by Van Buskirk and Grant [4]. Another detailed model was obtained by Rabbitt and Damiano [5], using a perfect toroid and approximating the cupula as a linear elastic plate. In the present study, in order to develop the monocanal model, the authors started the treatment from a straight duct with a circular section subjected to an axial motion law. Assuming that the fluid motion in the duct is laminar and that the endolymph is a newtonian and incompressible fluid, the Navier-Stokes equation for the fluid axial component is: u 1  P  u  2u       2   v (3) t   z r r r  where: v duct acceleration u duct-endolymph relative velocity  endolymph density  endolymph dynamic viscosity P fluid pressure z axial coordinate r radial coordinate t time The solution of the differential equation, expressed in the Laplace dominium is:   J r    1 P (4)  1  v   0 u r , s      J 0 rt    s z where rt is the duct radium, J0 is zero order Bessel’s function and  2    s /  . The semicircular canals can be considered, at least on first approximation, as a plane toroid; the canal and the utricle are represented with two different sections, subtended respectively by the angles  and , and with internal radius rc and ru, as represented in fig. 3. R is the radius of the curvature of the toroid, considered constant, while z is the curvilinear coordinate along the duct axis with the origin in the cupula. As the fluid velocities in the canal are relatively low and supposing a laminar flow, forces and accelerations of the fluid that are not axially directed may be neglected. Hence we can assimilate the toroidal lengths to rectilinear ducts with a velocity equal to the peripheral velocity of the duct itself. When considering an angular velocity  of the semicircular canal around the axis perpendicular to the toroid and passing in the curvature centre, the velocity of the fluid relative to the canal is:   J r   1 Pc  (5)  1 u c r , s      R   0   J 0 rc    s z where

Pc is the axial pressure gradient in the canal. z

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228 Modelling in Medicine and Biology VIII canal



R

 utricle

z cupula

Figure 3:

Discretization of the semicircular canal for the monocanal model.

In eqn. (5) the duct radius is neglected, being R»rc, and the curvilinear coordinate varies in the range 0  z  R . Similarly, for the utricle length, where the duct radius ru cannot be neglected anymore if compared to the toroid radius R, eqn. (4) becomes:    J r   1 Pu  1 uu r , s      R  r   0   J 0 ru    s z

(6)

Pu is the axial pressure gradient in the utricle and the curvilinear z coordinate varies in the range R  z     R . Observing that the semicircular canal is a closed circuit, the following relation between the variation of pressure Pc in the canal length, the variation of pressure Pu in the utricle and the pressure difference Pm on the cupula membrane can be written:

where

Pu  Pc  Pm

(7)

For the utricle, the pressure variation Pu can be derived from eqn. (6) and, considering the conservation of flow, it can be expressed as a function of the fluid velocity relative to the canal velocity, being Ac/Au the ratio of the canal and utricle cross sections:   J r  A s uc c  s R  r  0  1 Au  J 0 ru    R (8) Pu   J 0 r    1    J 0 ru   To evaluate the pressure drop on the cupula, this had been modelled as an elastic circular membrane clamped at the ampullary wall along its entire perimeter and subjected to a distributed load Pm in the axial direction. Assuming that cupula and endolymph have the same density [4, 5], the membrane deflection can be expressed in the Laplace domain as uc / s , representing with h the membrane thickness, with rm the external radius of the membrane, with Em and m Young’s and Poisson’s modulus, the difference of pressure Pm is: WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

Pm 



uc

s rm 2  r 2

16 E m h 3 31   m 



2

229 (9)

Substituting functions (8) and (9) in eqn. (7) an expression of Pc can be derived. By introducing it in eqn. (5), the transfer function of u c /  can be pointed out and hence the transfer function between the angular displacement  corresponding to the membrane deflection and the canal angular velocity is: u  r , s  Gs  c   sR Ms 2  Cs  K where coefficient G, M, C and K are respectively:



G  16 R r 2  rm2

 R  R  r    r1 2



2 c



1  1 1  2 2  2 r  ru r

(10)

   

2   2  r A  r2 M  2 R r 2  rm2    2  1   c  2  1     Au  rc     ru





2  1  1 2 r A  r2 C  8R r 2  rm2    2  1 2   c  2  1 2     r  Au  rc  rc  u    ru



K 



256 Em h 3   1 1  1 1   2  2  2  2    3 1   m    rc r  ru r 

Comparing expressions (1) and (10), it can be noticed that the transfer function of the presented hydroelastic fluidodynamic model has a structure analogous to the one derived with the more simple torsion-pendulum model, but with the advantage that the characteristic time constants  1  C K ,  2  M C and the gain G K can be expressed as functions of the system physiological parameters. In fig. 4 the graphs of the frequency response  /  for the torsion-pendulum model described by eqn. (2) and for the hydroelastic fluidodynamic model are shown. For the latter, the mean angular displacement of the cupula is reported. The values of the physiological parameters used for the numeric simulation and listed in tab. 1 have been extracted from literature [4, 5, 15]. For a further comparison in fig. 4 the frequency response of the Van Buskirk and Grant [4] model is also reported. The frequency responses are consistent, also considering the uncertainty of the physiological parameters. The perception band of the semicircular canal is in the range of 0.1 Hz and 100 Hz, with an attenuation at lower and higher frequencies. In the perception band the cupula deflection and the velocity are in counter phase, while for lower and higher frequencies the system presents respectively a lag of - 90° and - 270°.

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230 Modelling in Medicine and Biology VIII

Figure 4:

Frequency response of the mean deflection and angular velocity.

Table 1:

Anatomical and physiological data used in the numerical models. canal curvature radius internal radius of the canal internal radius of the utricle angle subtended by the canal angle subtended by the utricle membrane radius membrane thickness membrane Young’s modulus membrane Poisson’s modulus dynamic viscosity of the endolymph density of the endolymph

5

R rc ru

 

rm h Em

m  

3.175 mm 0.15 mm 1.19 mm 1.4 rad 0.42 rad 0.9 mm 0.57 mm 0.04 N/m2 0.5 10-3 Pas 1000 kg/m3

Hydroelastic-fluidodynamic three-canals model

The orientation of the three semicircular canals determines the vestibular apparatus sensibility to angular acceleration directed in any direction [16]. In general, given a stimulus, each canal answers in a different manner. To evaluate the responses it is necessary to individuate the acceleration components directed at the canals’ toroids axes and then to study the system fluid-dynamic behaviour. As depicted in fig. 5, the vertical canals, anterior and posterior, join in a common segment, with a cross section that is about twice the section of the single canal, before converging in the utricle and they came out of it at opposite ends. We want to consider, for calculation reasons, a simplified geometry that is still close to the anatomical one, analogously to what had been done for the monocanal model; the canals had been described as toroids with a circular planar section, with each being orthogonal. These planes are oriented in respect of the WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

231

Modelling in Medicine and Biology VIII

physiological axes, as shown in fig 5. The utricle is modelled as a segment of the horizontal canal with an enlarged section; the anterior and posterior canals share the vertical segment with the double section and each shares half a utricle with the horizontal canal. ant. canal z

pos. plane

vertical segment

y x ant. plane hor. plane

utricle pos. canal

Figure 5:

hor. canal

Discretization of the vestibular anatomical geometry.

Based on the schematisation reported in fig. 5, it is possible to write the relations between pressure in the different segments of the semicircular canals: horizontal canal posterior canal

Pm / h  Pu  Pc / h

Pm / p   Pu 2  Pv  Pc / p

(11) (12)

anterior canal

Pm / a  Pu 2  Pv  Pc / a

(13)

where Pu is the pressure variation at the utricle ends, Pm the pressure drop on the cupula, Pv the pressure variation in the vertical segment shared by the anterior and posterior canals and Pc the pressure variation along the canal. Subscripts /h /p and /a indicate magnitudes relative respectively to horizontal, posterior and anterior canals. Analogously to the monocanalar model, the endolymph is assumed to be a newtonian incompressible fluid governed by the Navier-Stokes equations. Using continuity of pressure and conservation of flow, after some algebra, the relation (14) between the membrane displacement vector of the three canals [h p a] and the angular velocity vector projected in the directions perpendicular to the canals planes [h p a] becomes:  h   A     a  p  1  a  a 2

b1 B b2

c1  h  c2   p  C  a 

(14)

where the matrix coefficients, calculated for the cupula maximum deflexion, which is in correspondence to the canal axis, are:

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232 Modelling in Medicine and Biology VIII

A

 A    c A u 

 16 Em h 3   A     s 2  4  2  c 2  s  Au ru    rc 31   m rm4 Rh       s

 A  A  32 Em h 3    A  1 Ac   2   c  c   s 2  8  2  c s   Au 2 Av    rc Au 2 ru2 Av rv2  31   m rm4 R p   B  2    2 s  A  A  32 Em h 3    A  1 Ac    2   c  c   s 2  8  2  c s 2 2 Au 2 Av    rc Au 2 ru Av rv  31   m rm4 Ra   C  2     2 s

a1 

  4   s 2  r    u

b1 

 Ac R p    4    s  Au Rh      ru2  

c1 

 Ac Ra    4   s 2   Au Rh      ru  

a2 

b2 

Ac Rh  Au 2    2  R p

Ac Rh  Au 2     2  Ra

 4    2s   r 2   u 

 A  1  R p  Ac   1   2s   c   2  2 s   2  A r  2    2  Ra  Au 2  ru v  v  

c2 

1

Ra 2    2  R p

1

A   1   A  1  c  2  2 s   c   2  2 s   Au 2  ru   Av  rv

The nomenclature used in the coefficients expressions has the same significance it had in the monocanal model, with the addition of subscripts h p a and v to indicate respectively the horizontal, posterior and anterior canal and the vertical shared segment; the angle  is the arc subtended by the common vertical segment. The frequency responses  h /  ,  p /  and  a /  of the cupulae mean deflection for three cases of head rotation along the physiological axes: roll (axis x), yaw (axis y) and pitch (axis z) are shown in fig. 6. Anthropometrical and physiological data are assumed equal for the three canals and are the same as listed in tab. 1, with the addition of  = 0.21 rad. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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a

233 b

c

Figure 6:

Frequency response of the three canals average cupula deflection and angular velocity for: a) roll; b) pitch and c) yaw motion.

The pass band is still the same as the monocanal model, but gain values change according to the direction of the motion. In particular the horizontal canal gain is prevalent for the yaw motion, while for the roll motion it is lower than –100 dB. Vertical canals present a behaviour very similar in all three motion laws tested.

6

Conclusions

In this paper the fluidodynamic analysis of the semicircular canals based on a simplified geometry is presented. Both a monocanal model and a three-canals model had been described. For the first model results of the frequency response between the cupula deflection and the motion law are consistent with the trends of the simpler torsional-pendulum model and with other fluidodynamic models presented in literature. For the three-canal model in literature experimental results are not present, so it would be necessary to collect some data and this is a challenging goal. In fact, direct experimental measures are not possible and for the indirect measures, exploiting the nystagmus evaluation, it is necessary to also take into account the neural answer dynamic. The collection of experimental data is still an open question in research about the vestibular system. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

234 Modelling in Medicine and Biology VIII

References [1] Mauro S., Mattiazzo G., Pastorelli S., Sorli M., Development of a flight simulator: motion cueing algorithms, XVII Congresso AIMETA, Firenze, Italy, pp. 11-15, 2005. [2] Ormsby C., Model of Human Dynamic Orientation, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1974 [3] Young, L.R., Dynamic control models of the semicircular canals, Dynamic Response of Biomechanical Systems, pp.133-145, 1970. [4] Van Buskirk W. C., Grant J. W., Vestibular Mechanics, The Biomedical Engineering Handbook, Second Edition, pp. 36.1-36.14, 2000 [5] Rabbitt R. D., Damiano E. R., A Hydroelastic Model of Macromechanics in the Endolymphatic Vestibular Canal, Journal of Fluid Mechanics, 238, pp. 337-369, 1992 [6] Steinhausen W. Über die beobachtungen der cupula in der bognegangsampullen des labyrinthes des libenden hecths. Pfluegers Arch 232, pp.500–512, 1933. [7] L. R. Young, C. M. Oman, Model for Vestibular Adaptation to Horizontal Rotation, Aerospace Medicine, 40, pp. 1076-1080, 1969 [8] Zacharias, G.L., Motion Sensation Dependence on Visual and Vestibular Cues, 1977, Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA. [9] Steer, R.W., The influence of angular and linear acceleration and thermal stimulation on the human semicircular canal, Thesis, M.I.T. [10] Van Egmond, A.A., Groen, J. J., and Jongkees, L. B. W., The Mechanics of the Semicircular Canal. Journal of Physiology. 110, pp. 1-17, 1949 [11] Mayne, R.A., A Systems Concept of the Vestibular Organs, Handbook of Sensory Physiology, Vestibular System, Springer-Verlag: New York, pp. 493-560, 1974. [12] Groen J.J., The mechanics of the semicircular canals, J. Physiol. Lond., 110, pp.1-17, 1957 [13] Schmid, R.M., Stefanelli, M., Mira, E., Mathematical modelling. Acta Otolaryngol. 72, pp. 292-302, 1971. [14] Goldberg, J.M., and Fernandez, C., Physiology of Peripheral Neurons Innervating Semicircular Canals of the Squirrel Monkey. II. Response to Sinusoidal Stimulation and Dynamics of Peripheral Vestibular System. Journal of Neurophysiology, 34(4): p. 661-675, 1971. [15] Sato H., Sando I., Takahashi H., Computer-aided three-dimensional measurement of the human vestibular apparatus, Otolaryngology - Head and Neck Surgery, 107(3), pp. 405-409, 1992. [16] Rabbitt R.D., Directional Coding of Three-Dimensional Movements by the Vestibular Semicircular Canals, Biological Cybernetics, 80, pp. 417-431, 1999.

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A mathematical model to predict the performance of advanced therapies in wound healing J. Ko1, S. Dickman2 & V. W. Li3,4 1 Department of

Mathematics, Brown University, USA University, Providence, USA 3 Institute of Advanced Studies, The Angiogenesis Foundation, USA 4 Angiogenesis and Wound Healing Center, Department of Dermatology, Brigham and Women’s Hospital, Harvard Medical School, USA 2 Brown

Abstract Wound healing is a complex, dynamic process. The ability to simulate this process using mathematical models that incorporate quantitative data on growth factors, tissue repair cells and matrix components would be a powerful tool to predict, analyze, and optimize new therapies. We present such a mathematical framework based on a system of ordinary differential equations and wound healing parameter values from the established literature. In contrast to conventional therapy, advanced modalities can augment certain components of the healing process in a measurable fashion. The performance of specific wound therapies can be simulated and compared to other therapies. We have enhanced the model by incorporating parameters of clinical practice used in the real world setting. This approach has application to predictive performance analysis and optimization of new advanced modalities and determination of best clinical practice. Keywords: wound healing, differential equations, bifurcation.

1 Introduction Wound healing is a complex process involving the interaction of many cell types and signaling molecules. In the U.S. alone, 6 million patients have chronic wounds. In many conditions, such as diabetes, the wound healing process is complicated by biochemical imbalances that often lead to wounds that do not heal. Each year, WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090221

236 Modelling in Medicine and Biology VIII 5% of diabetics develop a leg ulcer, of which 53% do not heal even after 20 weeks. Several mathematical models have been proposed to predict healing behavior for normal vs. diabetic wounds. These models have been proposed to be able to estimate the effectiveness of various clinical treatments such as skin grafts and growth factor therapy. Little work has been done, however, to compare these models with clinical evidence and to ensure that the models and the parameter values accurately represent the current best clinical practices. In addition, recent clinical and laboratory research findings call for revision to previous models.

2 Model description We describe here a general framework for a wound healing system based on the models of Waugh and Sherratt. Each of the numerous variables in the wound healing process – cell types, matrix components, growth factors, and other signaling molecules – are modeled as concentrations over time. Adapting the classic predator-prey population model, the rate of change of a cell population is modeled by the rate of cell migration, cell mitosis and cell death. Similarly, the rate of change of a growth factor, matrix component or chemical mediator population is modeled by the rate of production minus the rate of decay. The basic mathematical model in [1] is given by the following system of equations: ⎧ ˙ ⎪ ⎨φI = αK(T ) + k1 k2 φI (1 − k3 (φI + φR )) − d1 φI φ˙R = (1 − α)K(T ) + k1 k2 φR (1 − k3 (φI + φR )) − d1 φR ⎪ ⎩˙ T = k 4 φ I − d2 T

(1)

where the variables φI , φR and T are the densities of inflammatory macrophages, repair macrophages and the growth factor TGF-β, respectively; ki , d1 are growth and decay rates of the variables; K(T ) is the effect of monocyte migration due to TGF-β as seen in controlled-interaction studies; α ∈ [0, 1] is the fraction of monocytes becoming inflammatory macrophages and will also be seen to be a bifurcation parameter for this system. This system attempts to capture an essential part of the wound healing process, as illustrated in Figure 1. Significantly, the system uses a surrogate endpoint of healing– the “zero equilibrium point”– at which the inflammatory macrophage population has stabilized to its pre-wound level. This zero equilibrium point generally corresponds to the cessation of wound inflammation and proliferative activity that accompanies closure of the wound cavity and surface. The contributions of epidermal keratinocytes, platelets, and blood vessel endothelial cells – ignored in previous models – can be captured their impact on TGF-β, fibroblasts and PDGF. A bifurcation analysis of this model shows that there exists an α∗ such that for α ∈ [0, α ∗ ), there is one stable equilibrium and for α ∈ [α ∗ , 1], there are three equilibria, with a sequence of stable-unstable-stable. This analysis underlies the initial choice of α in the model: an initial choice of α ∈ [0, α ∗ ) corresponds to WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

Modelling in Medicine and Biology VIII

Keratinocytes

MMPs

237

TIMP

Monocytes TGF-ß1 MØInflamm ( )

Fibroblasts

HA

PDGF

MØRepair

Collagen I, III, V Proteoglycans Fibronectin Angiogenesis Epithelialization

Platelets

Figure 1: Modeled interactions in the wound healing process. Adapted from [2].

wound healing in normal skin and α ∈ [α∗ , 1] corresponds to healing in diabetic skin. A subsequent 7-variable revision in [3] builds upon this basic model, with the additional variables of the densities of fibroblasts (F ), hyaluronan (H A), collagen (C), and PDGF (P ). An equilibrium analysis shows a similar bifurcation structure as in the simplified model (1). Fixing a value of αnormal = 0.5 and αdiabet ic = 0.8, and substituting all parameter values, we can solve for the equilibrium points (φI , φR , T , P , F, C, H ). In both cases, we found more than 100 equilibrium points. Most equilibrium points were disregarded because they contained complex coordinates and were thus nonphysical. Moreover, equilibrium points with negative coordinates could also be disregarded, as these too represent a nonphysical representation. Restricting to real-valued equilibrium points, a linear analysis readily shows that the situation is similar to that of (1): for α = αnormal, the one real-valued equilibrium is stable, and for α = αdiabetic, three real-valued equilibria exist, with the sequence being stable-unstable-stable. A persistent feature of this revised model is the relative magnitude of the quantities φI , φR , T at each of the equilibrium points in the diabetic case relative to magnitude at the normal state. The first diabetic equilibrium point has coordinate values similar to those of the stable equilibrium point in the normal case. This first equilibrium point shows that diabetic wounds are capable of reaching a healing state. The second equilibrium point corresponds to the chronic inflammation state. The last equilibrium point, which has elevated levels of macrophages φI , φR WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

238 Modelling in Medicine and Biology VIII (where φI  φR ), T , and P relative to the levels at equilibrium in the normal case, corresponds to a chronic non-healing wound. In this paper, we describe a model which reflects ongoing clinical research findings. Our model is a system which incorporates the effect of suppressed levels of PDGF in diabetic wounds and the presence of epidermal keratinocytes in one of the skin graft therapies (Apligraf) by modeling its effect on TGF-β. Importantly, we show how data from controlled experiments can be incorporated into this model in lieu of modeling the evolution of a variable for which only limited information is known. Our model is given by the following system of equations: ⎧ φ˙I = αK(T ) + k1 k2 φI (1 − k3 (φI + φR ) − k5 F − k6 C) − d1 φI ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ φ˙R = (1 − α)K(T ) + k1 k2 φR (1 − k3 (φI + φR ) − k5 F − k6 C) − d1 φR ⎪ ⎪ ⎪ ⎪T˙ = k4 φI + k7 KC (t)F − d2 T ⎪ ⎨ P˙ = k8 (α)(φI + φR ) + k9 F − d3 P (2) ⎪ ⎪ ⎪ ˙ ⎪ F = M(P ) + k10F (1 − k3 (φI + φR ) − k5 F − k6 C) − d4 F ⎪ ⎪ ⎪ ⎪ ⎪C˙ = k11F + Ff (T )g(C) − d5 F C ⎪ ⎪ ⎩˙ H = k12F − d6 H where ki , di refer to growth and decay rates, M(P ), f (T ), g(C) are measured migration effects as seen in controlled-interaction studies, and KC , which is nonzero only for the treatment Apligraf, is the measured effect of keratinocytes on TGF-β in the presence of fibroblasts. Exact parameter values used in the simulations are provided in the Appendix.

3 Testing the model against clinical data As an initial test for our model, we compared data from wound healing clinical trials with predictions made by our wound healing model. 3.1 Topical growth factor therapy for diabetic ulcers One clinical trial was designed to examine the effectiveness of rhPDGF-BB 0.01% gel (becaplermin, Systagenix, London), a topical growth factor therapy which dramatically increases local PDGF levels at the wound site, in treating diabetic ulcers. The ulcers had been unhealed for at least 60 days before treatment began. One group of patients was treated with the medication daily for 20 weeks, while a control group was given placebo control. Applying these conditions, our model predicts that wounds in the control group remains at an unhealed steady state, while the wounds receiving the topical growth factor reaches the stable equilibrium in 23 weeks (Figure 2). This is in good agreement to the results of the clinical trial, in which patients receiving medication healed significantly (P = 0.01) faster than the control group. Roughly 50% of patients receiving medication were healed within WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Equilibrium at zero reached at 23 weeks

Daily PDGF x 20 wks

pg/mm3

239

Unhealed for 60 d

Days

Days

pg/mm3

Figure 2: Topical PDGF therapy for chronic diabetic ulcers. Left: dosing; right: time to surrogate endpoint for healing.

Daily PDGF x 3 weeks

Equilibrium reached 20% earlier

Days

Days

Figure 3: Topical PDGF therapy for acute wounds.

20 weeks, compared to 25% of those in the control group, demonstrating a 43% increase in the incidence of healing (P = 0.007) [4]. 3.2 Topical growth factor therapy for acute wounds Another clinical trial was undertaken to examine the effectiveness of PDGF growth factor therapy (becaplermin gel) on acute, non-diabetic wounds, compared to the topical antibiotic bacitracin. Participants in the experimental group received daily applications of PDGF for three weeks immediately following the wound event. Figure 3 shows the prediction of our model: the group receiving growth factor therapy achieves equilibrium 20% earlier than the control group. Clinical data, by comparison, showed that growth factor therapy helped achieve complete wound closure 32% faster than bacitracin [5]. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

240 Modelling in Medicine and Biology VIII

zero equilibrium reached at 14 wks

Days

Figure 4: Bioengineered cryopreserved dermal fibroblast therapy for chronic diabetic ulcers and time to surrogate endpoint for healing.

3.3 Dermagraft therapy for diabetic ulcers A clinical trial was designed to test the effectiveness of Dermagraft (Advanced BioHealing, Westport, CT), a bioengineered cryopreserved dermal fibroblast skin construct used to treat chronic diabetic wounds. In this trial, patients with diabetic foot ulcers which had remained unhealed for at least six weeks were randomized into a control group and a treatment group receiving Dermagraft. Figure 4 shows the prediction of our model: patients receiving Dermagraft treatment reach the surrogate healing endpoint at week 14 of treatment. In comparison, the clinical trial data showed that after 12 weeks of treatment, 30% of patients receiving Dermagraft had completely healed, compared to 18.3% of patients in the control group [6]. 3.4 Apligraf therapy for diabetic ulcers Apligraf (Graftskin, Organogenesis, Canton, MA) is a bioengineered, cultured, bilayered living skin construct consisting of both epidermal keratinocytes and dermal fibroblasts. A clinical trial was conducted to examine the effectiveness of Apligraft treatment on diabetic foot ulcers. In our model, Apligraf was applied monthly over three months, following current best clinical practice, to ulcers which had been unhealed for at least six weeks. In this case, it is noteworthy to compare our results with those using the Waugh-Sherratt model in [3], which applied Apligraf weekly, not monthly and did not take into consideration the contribuWIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Previous model: no zero equilibrium

Apligraf applied monthly x 3 Fibroblasts + Keratinocytes effects Persistence of Apligraf cells 1 mo

New model w/ keratinocyte effects: zero equilibrium achieved

Unhealed for 6 wks

Days

Days

Figure 5: Bioengineered cultured bilayered skin therapy for chronic diabetic ulcers. Left: dosing of Apligraf; right: comparison of results predicted by previous model [3] and our model.

tion of epidermal keratinocytes in Apligraf. These differences may account for the previous model’s inability to predict wound closure using Apligraf, instead achieving an unhealed steady state. Our model, taking into account the effect of keratinocytes on TGF-β, predicts that the zero equilibrium point is reached after 140 days (Figure 5). These results are consistent with actual clinical data. The pivotal trial showed that with three monthly applications, 56% of patients receiving Apligraft treatment had achieved complete wound closure at 12 weeks (84 days), compared to 39% in the control group, a P = 0.0026 difference [7]. A separate study found that Apligraft treatment led to 70% of patients achieving complete wound healing with an average of two applications [8].

4 Discussion and future work Our wound healing model uses a population-based system of equations to predict the level of certain cell types and chemicals as functions of time. By comparing predictions from the model with clinical data, we have shown that the model captures certain key features of the wound healing process. By looking at current clinical research findings, we were able to improve how the model predicts the behavior of specific treatments on healing. This type of calibration is critical and should be an ongoing process in light of new research findings. In addition, a thorough review of all equation coefficients and parameters was undertaken to increase the accuracy of the model’s predictions. The system does not model the spatial distribution of tissue cells within the healing wound. Thus, the effect of cells such as epidermal keratinocytes are manifest in its influence on TGF-β levels. Multiple factors may impair the wound healing process. For instance, proteolysis can be stimulated by bacterial infection. In turn, excessive enzymatic proteolyWIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

242 Modelling in Medicine and Biology VIII sis will disrupt the healing process. Other factors which slow the healing process include cellular senescence, sustained inflammation, moisture imbalance, physical pressure, compromised patient status, impaired perfusion, insufficient angiogenesis, and deficient growth factors [9]. The deficiencies of growth factors like PDFG and TGF-β can also impair wound healing. All mathematical models which attempt to approximate a complex biological process like wound healing have limitations. Not all the variables included in this model have known accurate values. In addition, the heterogeneity between individual patients is a difficult factor to incorporate into a mathematical model. Future planned work are: (i) continually improve the accuracy of the parameter values as new data emerges from scientific and clinical studies; (ii) to introduce more variables into the model that can better represent the complexity of wound healing (eg. MMP, other growth factors); (iii) address directly the endpoint of healing via the spatio-temporal dynamics. In summary, this model can provide important insights into the fundamental drivers that affect wound healing. This model can also be used to understand how perturbations in certain parameters can optimize healing. A practical use for this model might eventually be to guide the rational design of drug and device prior to significant investment in commercial research and development.

Appendix A A.1 Variables The variables in this model are the same as those used in [3], and are described below: φI : Inflammatory macrophages (cells/mm3 ) φR : Repair macrophages (cells/mm3) α: Proportion of migrating monocytes differentiating into φI (cells/mm3 ) k1 : % macrophages undergoing mitosis k2 : macrophages growth rate k3 : inverse max macrophage density (/cells/mm3 ) k4 : macrophage TGF-β production rate K(T ): monocyte migration due to TGF-β P : PDGF (pg/mm3) d3 : PDGF decay rate(/day) T : TGF-β (pg/mm3) k7 : TGF-β production rate by fibroblasts (pg/cells/day) KC (t): scaled effect of keratinocytes on TGF-β in the presence of fibroblasts k4 : TGF-β production rate by macrophages (pg/cells/day) d2 : TGF-β decay rate (/day) F : fibroblasts (cells/mm3 ) k10 : fibroblast growth rate (/day) d4 : fibroblast death rate (/day) k5 : inverse max fibroblast density (/cells/mm3 ) WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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kf(t) 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0

5

10

15

20

25

30

Days

Figure 6: kf (t), the measured effect of keratinocytes on TGF-β in the presence of fibroblasts.

k9 : fibroblast PDGF production rate (pg/cells/day) k8 : macrophage PDGF production rate (pg/cells/day) M(P ): fibroblast migration due to PDGF C: collagen (μg/mm3 ) k11 : fibroblast collagen rate (μg/cells/day) f (T ): fibroblast synthesis of collagen due to TGF-β g(C): collagen synthesis due to collagen density k6 : inverse max collagen density (μg/mm3 ) d5 : collagen remodeling rate (/day) H : hyaluronan k12 : hyaluronan synthesis by fibroblasts (μg/cells/day) d6 : hyaluronan decay rate (/day) A.2 Determination and refinement in parameter values We describe here any changes to parameters that were also used in [3] as well as the determination of new parameter values in our model. – Start treatment time points are defined at a point where the wound has become chronic (e.g. one month = 28 days), which is the inclusion criteria for most clinical trials in wound healing. – k8 = k8 (α): Recent research in [10] shows that diabetic limb tissues are deficient in the growth factor PDGF by up to 40%. – The number of fibroblasts and collagen in Apligraf and Dermagraft were determined based on the fibroblast density and thickness of the product, according to personal communication with Katie Faria, Organognesis and Gary Gentzkow (Advanced Tissue Sciences) WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

244 Modelling in Medicine and Biology VIII Table 1: Parameter values used in simulations. Variable

Initial

φI

200

φR

200

Normal

Diabetic

α (initial)

0.5

0.8

k1

0.05

0.05

k2

0.693

0.693

k3

0.002

0.002

k4

0.07

0.07

d1

0.2

0.2

P

2

d3 T

4.0 6 0.004

0.004

k4

0.07

0.07

d2

9.1

9.1

k10

0.924

0.924

d4

1.0

2.5

k5

0.0025

0.0025

k9

0.0015

0.0015

k8

0.015

0.003

2.5

0.015

C

6

10

k11

20

5

k6

0.0004

0.0004

d5

1.50e-05

1.50e-05

d6

0.7

0.7

k12

0.008

0.001

H

Dermagraft

Regranex

1

1

650

4

0.4

5100

14000

9.1

8.5

7.45

80

4 .2

k7

F

Apligraf

0.01

– The effect of PDGF on fibroblast migration was determined by calculating cells per field (400x magnification) in studies [11]. – The concentration of PDGF was determined through personal communication with Liza Ovington, Johnson & Johnson Wound Management. – KC (t): the presence of keratinocytes in one product (Apligraf) is taken into consideration through its effect on suppressed TGF-β production by fibrobWIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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lasts in studies of co-cultures [12]. To find KC (t), we take the curve kf = kf (t) derived from the curve that measures the effect of keratinocytes and fibroblasts on TGF-β normalized by the base curve that measures the effect of fibroblasts alone over the course of 30 days. The curve for kf used in our simulations is shown in Figure 6. KC (t) can be taken to be: KC (t) = ((kf )(t) +

1 (kf ) (t))ek7 ∗t k7

In our simulations, we take the variable functions α(H ), K(T ), M(P ), f (T ) and g(C) to be the same as in [3].

Acknowledgement Many thanks to Jonathan Sherratt for his interest, his expertise, and his generosity.

References [1] Waugh, H.V. & Sherratt, J.A., Macrophage dynamics in diabetic wound healing. Bull Math Biol, 68(1), pp. 197–207, 2006. [2] Pierce, G.F., Mustoe, T.A. & Lingelbach, J.et al.., Platelet-derived growth factor and transforming growth factor-beta enhance tissue repair activities by unique mechanisms. J Cell Biol, 109, pp. 429–440, 1989. [3] Waugh, H.V. & Sherratt, J.A., Modelling the effects of treating diabetic wounds with engineered skin substitutes. Wound Rep Reg, 15(1), pp. 556– 565, 2007. [4] Wieman, T., Smiell, J. & Su, Y., Efficacy and safety of a topical gel formulation of recombinant human platelet-derived growth factor-BB (becaplermin) in patients with chronic neuropathic diabetic ulcers. a phase iii randomized placebo-controlled double-blind study. Diabetes Care, 21(5), pp. 822–827, 1998. [5] Li, V.W., Ma, J. & Ko, J.et al.., Dynamics of acute wound healing following topical rhPDGF therapy. J Am Acad Dermatol, 52(3), p. P213, 2005. [6] Marston, W.A., Hanft, J., Norwood, P. & Pollak, R., The efficacy and safety of Dermagraft in improving the healing of chronic diabetic foot ulcers: results of a prospective randomized trial. Diabetes Care, 26, pp. 1701–1705, 2003. [7] Veves, A., Falanga, V., Armstrong, D.G. & et al., Graftskin, a human skin equivalent, is effective in the management of noninfected neuropathic diabetic foot ulcers: a prospective randomized multicenter clinical trial. Diabetes Care, 24, pp. 290–295, 2001. [8] Novartis US08 Non-Controlled Study. Unpublished. [9] CMS Medical Coverage Advisory Committee Meeting on Usual Care of Chronic wounds. Baltimore, MD, 2005. [10] Tanii, M., Yonemitsu, Y. & Fujii, T.et al.., Diabetic microangiopathy in ischemic limb is a disease of disturbance of the platelet-derived growth WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

246 Modelling in Medicine and Biology VIII factor-BB/protein kinase C axis but not of impaired expression of angiogenic factors. Circ Res, 98, pp. 55–62, 2006. [11] Facchiano, A., De Marchis, F. & Turchetti, E.et al.., The chemotactic and mitogenic effects of platelet-derived growth factor-BB on rat aorta smooth muscle cells are inhibited by basic fibroblast growth factor. J Cell Sci, 113 (Pt 16), pp. 2855–2863, 2000. [12] Le Poole, I.C. & Boyce, S.T., Keratinocytes suppress transforming growth factor-beta1 expression by fibroblasts in cultured skin substitutes. Br J Dermatol, 140, pp. 409–416, 1999.

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Modeling of capacitance relaxation phenomena in a malignant membrane T. K. Basak1, K. Bhattacharya1, S. Halder2, S. Murugappan3, V. Cyril Raj4, T. Ravi4, G. Gunasekaran4 & P. Shaw1 1

Department of Electrical Engineering, Jadavpur University, Kolkata, India 2 Department of Electrical & Electronics Engineering, ICFAI Tech, ICFAI University, Tripura, India 3 Department of Computer Science and Engineering, Annamalai University, Chennai, India 4 Dr. M.G.R. Educational and Research Institute, (Dr. M.G.R. Deemed University) Chennai, India

Abstract Electrical Impedance scanning (EIS) is a new technique in which moderate variations in capacitance values are reflected by the cells of various type of normal membrane. The malignant membrane, in contrast, in EIS demonstrates significant capacitance relaxation phenomena concomitant with α dispersion in the low audio frequency range. The author has endeavored to establish the above stated phenomena. In his experimental setup, the lipoprotein constituent of the membrane was dissolved in acetone solvent and the interaction of the dissolved lipoprotein in the presence of electrical stimuli in the lower audio frequency range at room temperature of about 25ºC was studied. The capacitance relaxation phenomena concomitant with α dispersion in the malignant membrane for subjects below and above 50 years has been simulated in MATLAB 6.5. The model incorporates a modified homeostat comprising of a homeostat and transduction phase in the feedback path and it represents the interaction of the lipoprotein in a malignant membrane associated with α dispersion mediated through capacitance relaxation phenomena. Results from the output of the model are in close conformity with the capacitance relaxation phenomena, which suggest an adjunctive detection modality in the differentiation of membrane malignancy that is equivocal on Ultrasonography. Keywords: capacitance relaxation, malignant membrane, α dispersion, modified homeostat. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090231

248 Modelling in Medicine and Biology VIII

1

Introduction

This paper describes a comprehensive model [1] of capacitance relaxation phenomena concomitant with α dispersion. This α dispersion is linked with the interaction of lipoprotein mediated through α receptor proliferation (mR-α) in the malignant cell [1,12]. It adopts a new approach to modeling. Using artificial neural networks (ANN) it is not easy to obtain a dynamic model that reflects dependence on membrane dynamics. Therefore, the authors have endeavored to design a new approach in this respect based on criteria of electrical control system. The model has been simulated in MATLAB 6.5 R13. In the model (Fig 1) the homeostat is a controller that performs interactions with in a cell for maintaining a static or dynamic cooperation in the environment of the cell concerned [1]. The modified homeostat incorporates cellular transduction phase in a feedback path. Since the capacitance relaxation phenomenon [10] is dependent on age, the transduction of the malignant cell requires a modified homeostat. The block diagram of the modified homeostat is shown in the Fig1. The measurement of capacitance of bioelectric membrane has been widely used in morphological research [2,3]. As a means of assessing changes in lipoprotein organization, it is generally assumed that the dielectric properties of the normal membrane consisting of lipoprotein are constant in the lower audio frequency range, so that the changes in the membrane area reflect changes in the lipoprotein interactions [4,5]. From the classical papers of Debye, Cole and Cole and others, it transpires that the behavior of the dipoles associated with the integral lipoproteins of membrane is affected by the electric field with consequent on settings of α dispersions in the low audio frequency range which can be characterized by relative capacitive increment against frequency [5,8,9]. Taking into account of the finding described above and considering capacitive increment associated with α dispersion in the low audio frequency range, it can be stated the complex capacitance of the membrane has two parts namely a frequency dependent capacitance and a static d.c. capacitance associated with α dispersion. The capacitance measured in the lower audio frequency range can vary because of the change in membrane area concomitant with hydrophobic interaction characterizing the lipoprotein organization [6,7].

2

Modeling and methods

The lipoprotein constituent of the membrane was extracted from the subject and preserved in formaldehyde. After taking out from the formaldehyde the membrane was cleaned and then dried. After that a small weighted amount of the membrane (10 mg) was dissolved in measured quantity (10 ml) of acetone solvent in a container fitted three electrode system made of silver–silver chloride. The capacitance measurement was done by a LCR Meter (HP 4284A) at temperature 25ºC corresponding to an excitation voltage of 100 mV [10, 11]. The relative capacitance of a membrane is the ratio of the measured value of capacitance of the solvent to that of lipoprotein extract in solvent. It has been established that in a malignant membrane, there is distinct relaxation jump, WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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which is age dependent. It was observed during the experiment that α dispersion occurs in the malignant membrane in the lower audio frequency (around 200 Hz) at which the relative capacitance abruptly jumps to a high value. It is interesting to observe α dispersion concomitant with capacitance relaxation concerning the status of receptor proliferation [1] in malignant cell. The model (Fig 1) comprises of homeostat and a transduction phase in the feedback path through which the cellular transduction in membranes is reflected through α dispersion. The transduction phase [1] in the feedback path of the homeostat represents physiological status for subjects below and above 50 years in order to reflect α dispersion concomitant with respective capacitance relaxations. The input to the homeostat is an electrical signal, which is analog to the measured value of relative capacitance shown in the Fig 1.

Figure 1:

3

Block diagram of the modified homeostat.

Discussion and conclusion

In Fig 2, the input to the model is capacitance relaxation of the malignant cells for the subject below 50 years and corresponding α dispersion is the output of the model in Fig 3. In Fig 4 and Fig 5 the similar input and output data are represented for subject above 50 years. In the present experiment for subjects 6 out of 8 with malignancy were correctly detected and verified with Ultrasonography. It has been established by the author that in a malignant membrane there is distinct relaxation jump, which is age dependent. This relaxation jump is concomitant with α dispersion in the lower audio frequency range (around 200 Hz), which is responsible for strong interaction of lipoprotein in malignant cells and this effect is reflected on the membrane transport and behavior associated WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

250 Modelling in Medicine and Biology VIII

Figure 2:

Capacitance relaxation phenomena for malignant subject below 50 years.

Figure 3:

α dispersion concomitant with capacitance relaxation phenomena for malignant subject below 50 years.

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Figure 4:

Capacitance relaxation phenomena for malignant subject above 50 years.

Figure 5:

α dispersion concomitant with capacitance relaxation phenomena for malignant subject above 50 years.

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252 Modelling in Medicine and Biology VIII hydrophobic interaction. It is interesting to note that the α dispersion with the corresponding capacitance relaxation are different for subjects with malignancy below 50 years and above 50 years. The p.u. (per unit) scale values signify normalization of the curve to correlate a particular physiological phenomenon. Results of the model concomitant with capacitance relaxation suggest the potential usefulness of EIS with the corresponding α dispersion as an adjunctive imaging modality in the differentiation of lymphadenopathy that is equivocal on Ultrasound. From capacitance relaxation associated with α dispersion it is possible to determine the signaling pathway in tumor growth and angiogenesis. The vascular endothelial growth factor is dependent on α dispersion. It is well established that it is one of the key regulator of the tumor growth and meta- static dissemination for which molecular basis of tumor angiogenesis has been of keen interest in the field of cancer research [12]. It is experimentally observed that for normal membrane there is no capacitance relaxation phenomena with the significant rise of capacitance values present in the malignant membrane and as such α dispersion will almost remain constant [10].

4 Authors' contributions Professor T. K. Basak received a third world scientist award from ICTP, Trieste, Italy and worked with Professor A. Glilozzi in the Dept of Biophysics, University of Genoa, Italy in 1985. He furnished the innovative idea in the present paper. Mr. Suman Halder, S. Murugappan, V. Cyril Raj, T. Ravi, and G. Gunasekaran are the registered Ph.D. candidates in Jadavpur University under Prof. T. K. Basak. Prof. K. Bhattacharya is also the co guide of Mr. Suman Halder. Dr. P. Shaw has completed PhD. Under Prof. T. K. Basak and now he is associated with research activities with Prof. T. K. Basak.

Acknowledgements The authors are grateful to the authorities of Jadavpur University. Prof. T. K. Basak is particularly indebted for inspiration received from his late wife, Mala Basak who is in the heavenly abode of Shree Shree Ramakrishna Paramhansa.

References [1] Basak Tapas K, Halder Suman, Kumar Madona, Sharma Renu and Midya Bijoylaxmi. A topological model of biofeedback based on catecholamine interactions. Theor Biol Med Model., March 21,2005. doi: 10.1186/17424682-2-11 [2] Gliozzi A, Bruno S, Basak TK, Rosa MD, Gambacorta A. Organization and Dynamics of Bipolar Lipids from Sulfobus Solfataricus in Bulk Phases and in Monolayer Membranes. System Appl Microbiol. 1986;7:266–27. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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[3] Awayda M.S., Van Driessche W., Helman S.I. Frequency – Dependent capacitance of the apical membrane of frog skin: Dielectric Relaxation processes. Biophysics J, January 1999, P 219-232, Vol. 76, No. 1 [4] Coster, H.G.L., and Smith J.R., The molecular organization of bimolecular lipid membranes. A study of the low frequency Maxwell-Wagner impedance dispersion. Biochim. Biophys. Acta. 1974, 373: 151–154, [Medline] [5] Daniel V.V., Dielectric Relaxation, 1967, Academic Press, London. [6] Pething R., Dielectric and electronic properties of biological materials. 1979, John Willey & Sons, New York. [7] Schwan H.P. Electrical properties of tissues and cell suspensions. In Advances in Biological and Medical Physics, 1957, p.147–209 Academic Press, New York, [8] Helman S.I., Thompson S.M, Interpretation and use of electrical equivalent circuits in studies of epithelial tissues, Am. J. Physiol. 1982, 243:F519F531, [Medline] [9] Kell D.B., and Harris C.M., On the dielectrically observable consequences of the diffusional motions of lipids and membranes. I. Theory and overview. Eur. Biophysics J. 1985, 12: 181–197 (Medline) [10] Shaw P., Basak T.K., Ghosh N.C. Capacitance Relaxation Phenomena in Cartilaginous Membrane Everyman’s Science, Vol. XL No. 6,February’06– March’06 [11] Shaw P., Basak T.K. & Ghosh N.C., Novel Method for Detecting Malignancies in Membrane, Science & Culture, No.5-6, May – June 2006 [12] Dinel J. Hicklin Lee M. Ellis Roll of the Vascular Endothelial Growth Factor Pathway in Tumar Growth and Angiogenesis, Journal of Clinical Oncology, vol 23, No 5 (February 10), 2005: pp. 1011–1027.

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Section 5 Data acquisition and analysis

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A novel 3D torso image reconstruction procedure using a pair of digital stereo back images A. Kumar & N. Durdle Department of Electrical & Computer Engineering, University of Alberta, Canada

Abstract This paper presents a novel procedure for creating a 3D torso image from a pair of stereo digital 2D back images. The aim of this procedure is to obtain 3D images that can be used for assessment of external spinal deformities in scoliosis. Scoliosis is a condition characterized by lateral deviation of the spine coupled with rotation of individual vertebra resulting in visible torso asymmetries. The procedure provides clinicians with a cost effective and mobile setup of acquiring 3D images. To improve the registration process, a novel approach combining tree weighted colour based image segmentation and differential geometry was developed. Image reconstruction involved pre-processing, triangulation and texture application to obtain a 3D image. Analysis was performed using human subjects and objects of known dimension. Evaluation of system performance was done against existing stereovision procedures and range scanning systems. The final 3D image was compared to that obtained from the Konica Minolta Vivid 700 laser scanner. Each image was divided into 360 cross sections for evaluation against size and shape. The 3D image reconstructed from this novel procedure was 75–100% accurate when compared against the 3D image from the laser scanner. The results demonstrate that the procedure is a cost effective clinical tool for assessing torso shape and symmetry. Keywords: scoliosis, registration, image reconstruction, belief propagation, differential geometry, disparity map.

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1

Introduction

Scoliosis is a condition characterized by lateral deviation of the spine coupled with rotation of individual vertebra resulting in visible torso asymmetries [1]. The assessment of severity of scoliosis is traditionally done using radiographs of the spine. However, radiographs do not describe the visible torso deformity associated with scoliosis [2]. Many three dimensional data acquisition techniques have been investigated for developing a system to assist clinicians in the evaluation of external scoliosis deformities. This is because most scoliosis patients and their families are more concerned with the shape of the torso than the internal alignment of the spine. Traditional procedures for assessment of torso shape are based on landmarks. Since the back surface is smooth and featureless, it becomes very difficult to locate these landmarks in real time. Other techniques such as difference mapping, Moiré topography, ISIS scanning, Quantec system scanning and laser scanning have been developed over the years for assessment of torso shape [2]. Disadvantages in these methods range from poor resolution images to expensive processes. In light of these problems and due to advancements in stereo computer vision, there is a need to develop a cost effective, accurate technique that can be clinically used for assessing scoliosis. Progress in computer vision and availability of faster computer processors has lead to development of stereo vision algorithms for simultaneous stereo camera capture, calibration and reconstruction. Stereo capture systems consist of digital or TV cameras positioned with known geometry. Significant advances have been recently made in the area of computer vision, as a result of publically available performance testing such as the Middlebury data set [3], which has allowed researchers to compare their algorithms against all state-of-the-art algorithms [4]. Stereo correspondence or registration of stereo images is one of the most active research areas in computer vision [3]. In the area of stereo vision research, stereo images refer to images captured at different viewpoints using cameras with known geometry [5]. In order to register the stereo images, we need to determine the closest (least error) or best point-to-point correspondence between the two images. Stereo registration algorithms can be used on stereo images captured using calibrated digital or TV cameras to obtain three dimensional (3D) point set data. A triangulation algorithm is primarily applied on point set data to obtain a 3D surface. 3D surface reconstructions of scenes or localized objects in a scene using stereo vision has modern applications in 3D modelling, computer graphics, facial expression recognition, surgical planning, architectural structural design etc. [3,6]. The 3D scene geometry established from this process is used to reconstruct 3D torso images using stereo reconstruction to study scoliosis. However a problem associated with the existing stereo registration algorithms such as that developed by Klauss et al. [7] is that it assumes frontal parallel plane geometry. This means that it assumes depth is constant (with respect to the rectified stereo pair) over a region under consideration [8]. Reconstruction of smooth and curved surfaces where depth is constantly changing violates this WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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assumption. Li and Zucker [6] have tried to solve this problem using differential geometry but they have not applied their algorithm to the one developed by Klauss et al. [7]. Belief Propagation and Graph Cuts are the most commonly used methods for refinement of the registration process [3]. Tree Re-weighted Message Passing, which might become a serious rival to Belief Propagation and Graph Cuts [9], has not been used with differential geometry. Tree Reweighted Message Passing’s improvement over Belief Propagation and Graph Cuts becomes significant for more difficult functions [9]. Stereo reconstruction algorithm is applied to registered images. Since, the registration process leaves stray points due to errors, this need to be removed. Triangulation to connect the 3D points obtained through registration into polygons and texture application is the last step to obtain a fully reconstructed 3D object.

2

Objective

The objective of this paper is to present a procedure to reconstruct a 3D image from 2D stereo images of the torso using a stereo camera setup. The 3D image can facilitate the assessment of scoliosis clinically. This requires the process to require minimal user input in order to prevent errors; to be error correcting in order to prevent stray data points; to be cheaper and more accurate than existing methods. The aim of this paper is two-fold. Firstly, to create a novel procedure by investigating and improving on existing methods in computer vision for stereo image registration (correspondence matching in a pair of stereo digital images). Finally, the registered image is preprocessed, leading to a reconstructed 3D image.

3

Materials

A stereo digital camera setup is required to obtain 2D images of the torso. The cameras used for the setup are two 3 Megapixel Nikon digital cameras. The acquired images are processed using an Intel® Core 2 Duo 2.4 GHz, 4GB RAM PC to obtain the reconstructed 3D torso image. The stereo digital camera setup is shown in fig. 1. The vertical camera setup is used opposed to the horizontal camera setup primarily because of the shape and size of the torso. Since, the torso is longer (in length) than it is wider, the vertical camera setup allows for images to be captured in portrait orientation. This increases the total number of pixels in the digital images that represent the torso in the vertical setup as opposed to the horizontal setup.

4

Methods

The 3D torso image reconstruction procedure can be divided into 3 stages as shown in fig. 2

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Figure 1:

Vertical stereo digital camera setup.

4.1 2D Stereo back image acquisition Stereo image acquisition is performed using the vertical stereo camera setup. Image calibration and rectification are not required in this process. This saves the total time required for 3D image reconstruction procedure. Image calibration and rectification defines the relationship between pixels on a particular image to 3D coordinate in world space. In this reconstruction procedure, the relative position of each of the 3D points in the final reconstructed torso image is what defines the size and shape of the torso. Therefore, absolute positions of the 3D point in world space are not relevant. Fig. 3 shows images taken using the vertical camera setup.

Figure 2:

Stages of torso image reconstruction.

4.2 Stereo back image registration Stereo registration methods are assessed using univalued disparity function of one image with respect to the other (referred to as reference image). When first introduced in human vision literature, disparity was used to describe the difference in location of corresponding features seen by the left and right eye [3]. The procedure for obtaining disparity is divided into three stages: mean shift colour segmentation, adaptive local pixel matching, and differential geometry in a tree reweighted belief propagation procedure. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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The x, y spatial coordinates of the disparity space are taken to be coincident with the pixel coordinates of a reference image. The correspondence between a pixel in a reference image and pixel in the matching image is linearly related to each other by the disparity function. The disparity function obtained over the 2D images is known as the disparity space image (DSI). DSI gives a 3D point set data used in the final stage for 3D Torso Point generation. It represents the confidence or log likelihood of a match implied by the disparity function. The values of DSI are converted into z spatial coordinate values using the focal length of the cameras. We use 128 levels of disparity (which implies the same number of distinct z values). We find using these many disparity levels provides accuracy and speed in the reconstruction procedure.

Figure 3:

Images acquired from the top and bottom cameras respectively of a vertical digital camera setup.

The process of colour segmentation is to decompose the reference image into regions of colour or greyscale [7]. The mean-shift analysis approach is essentially defined as a gradient ascent search for maxima in a density function defined over a high dimensional feature space [7]. Comaniciu and Meer’s [9] mean shift segmentation is insensitive to differences in camera gain. Fig. 4 shows the mean shift colour segmented image obtained using Comaniciu and Meer’s method. The next step involving local pixel matching is an essential step for defining a disparity plane. The aim of this step is to provide an initial estimate of the disparity space image (DSI). The disparity plane is based on 3D x-y-d space supporting slanted and curved surfaces where x, y are spatial coordinates and d is the inverse depth or disparity [10]. The disparity planes are calculated using local pixel matching. Local pixel matching requires calculation of a matching score and an aggregation window [7]. Matching score is obtained using a self-adapting dissimilarity measure that combines the sum of absolute pixel intensity differences (SAD) and a gradient-based measure as implemented by Klauss et al. [7]. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 4:

Mean shift segmented images of top and bottom cameras respectively.

Finally, the calculation of the final disparities is performed. The algorithms that perform well in this stage are based on an energy minimization framework. This means that we need to choose at each pixel the disparity associated with the minimum cost value [3]. E(d) = Edata(d) + λEsmooth(d).

(1)

It involves minimizing two separate energy functions that are summed together to calculate the final energy minimization term as given by the eqn (1) where d represents disparity. The symbol Edata(d) in eqn (1) measures how well the disparity function agrees with the input image pair and is given by the summation of matching score over the spatial coordinates. The formulation of Edata(d) follows in eqn (2) where C is the matching score. Edata(d) = Σ C(x, y, d(x, y)).

(2)

The symbol Esmooth(d) in eqn (2) encodes smoothness in the image by measuring the differences between the neighbouring pixels’ disparities [3]. Esmooth(d) can be described by eqn (3) where ρ is some monotonically increasing function of disparity difference. Esmooth(d) = Σ ρ(d(x, y) − d(x+1, y)) + ρ(d(x, y) − d(x, y+1)).

(3)

The Esmooth(d) operates on the frontal parallel assumption that is altered in this procedure as suggested by Li and Zucker using “floating” disparities [11]. Li and Zucker apply “Floating disparities” on the Max-Product Belief Propagation framework. Tree-Reweighted Message Passing as defined by Kolmogorov [12] is used in this procedure as the energy minimization framework. The key subroutine of the Tree-Reweighted Message Passing algorithm is Max-Product Belief Propagation [12]. The “floating” disparities [11] are therefore added to the Max-Product Belief Propagation component of Tree-Reweighted Belief Propagation. The Tree-Reweighted Belief Propagation is advantageous because WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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messages are passed in a sequential order rather than a parallel order requiring half the space. Convergence is reached in two passes rather than having a convergence condition as in the case of Max-Product Belief Propagation. Therefore applying differential geometry to Tree-Reweighted Belief Propagation produces better results. The result of the image registration is the DSI shown in fig. 5 where the disparity levels on the top image are shown using the bottom image as a reference image. We can see that using the above technique to acquire DSI leads to a smooth variation in disparity across the back image.

Figure 5:

DSI of top camera image using bottom camera image as reference.

4.3 3D torso point generation and triangulation This is the last step of the image reconstruction process and involves using eqn (4) to obtain the z spatial coordinate using the above-calculated DSI at a known x, y spatial coordinate.

z

b f . d x, y 

(4)

In eqn (4), b (baseline) represents the distance from the optical centre of the top camera to that of the bottom camera, f is the focal length of the cameras and d(x,y) is the disparity at that x, y location on the image. The x, y, z values are plotted using Visualization Toolkit (VTK) software [14]. The stray points are removed since they either comprise of errors in calculation of the DSI, errors in calculation of z spatial coordinate or belong to regions of the image that do not represent the torso. There are also holes and occlusions in the image due to missing z data points caused to errors noted above. Pre-processing the 3D point set needs is done to remove stray data points and fill in occlusions. Triangulation is also done to join the 3D points into polygons that represent the 3D surface of the torso.

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264 Modelling in Medicine and Biology VIII Firstly, using an edge detection algorithm in VTK on the DSI we determine the region that represents the torso. All the 3D points outside this region are discarded. This helps us to eliminate most of the stray points. Now all the 3D points are joined together by connecting lines between points of nearest Euclidean distance. This gives us a 3D image of the torso, which has a few stray points and occlusions. The image is of the same format and characteristics as that obtained from range scanning systems. The task of further processing and triangulating the 3D image is implemented in VTK using the technique described by Kumar et al. [14]. The resultant 3D torso image is shown in fig. 6.

5

Results

The output of image registration stage was compared to 2 existing registration procedures, segment-based adaptive belief propagation (adaptive BP) and colour-weighted hierarchical belief propagation (hierarchical BP). It outperformed existing methods, particularly for high curvature regions and significantly large cross sections. Its accuracy of reconstruction ranged from 85– 100% compared to 75-100% for existing methods.

Figure 6:

Reconstructed 3D torso image.

The final 3D image was compared to that obtained from the Konica Minolta Vivid 700 laser scanner. The image of a human subject was divided into 360 lateral cross sections for evaluation against size and shape. The 3D image reconstructed from this novel procedure was 75–100% accurate when compared against the 3D image from the Konica Minolta® Vivid 700 laser scanner. The results demonstrate that the procedure is a cost effective clinical tool for assessing torso shape and symmetry. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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265

Discussion

The 3D torso reconstruction procedure provides a cost effective alternative for assessing torso shape and symmetry. Future work will focus on testing with more human subjects and with optical imaging methods other than the laser scanner. It will also consist of more comprehensive testing of the vertical camera setup under varying light conditions.

References [1] Ajemba, P.O, Durdle, N.G, Hill, D.L & Raso, V.J., Re-positioning Effects of a Full Torso Imaging System for the Assessment of Scoliosis. Canadian Conf on Electrical and Computer Engg. 2004, Vol: 3, pp. 1483- 1486. [2] Ajemba, P.O, Durdle, N.G, Hill, D.L & Raso, V.J., A Torso Imaging System for Quantifying the Deformity Associated with Scoliosis. Instrumentation and Measurement, IEEE Transactions on, Vol. 56, Issue 5, Oct. 2007, pp. 520–1526. [3] Scharstein, D. & Szeliski, R., A Taxonomy and Evaluation of Dense TwoFrame Stereo Algorithms. International Journal of Computer Vision, 47(1/2/3), pp. 7-42, April-June 2002. [4] Yang, Q., Wang, L., Yang, R., Stewenius, H. & Nister, D., Stereo Matching with Colour-Weighted Correlation, Hierarchical Belief Propagation and Occlusion Handling. Computer Vision and Pattern Recognition, 2006 IEEE Computer Society Conference on, Vol. 2, pp. 2347- 2354 [5] Sun, J., Zheng, N. & Shum, H., Stereo Matching Using Belief Propagation. Pattern Analysis and Machine Intelligence, IEEE Transactions on, Vol 25, Issue 7, July 2003, pp. 787–800 [6] Li, G. & Zucker, S.W., Stereo for Slanted Surfaces: First Order Disparities and Normal Consistency. Proc. of EMMCVPR, LNCS, 2005 - Springer [7] Klaus, A., Sormann, M. & Karner, K., Segment-Based Stereo Matching Using Belief Propagation and a Self-Adapting Dissimilarity Measure. Pattern Recognition, ICPR 2006, Vol. 3, pp. 15-18 [8] Li, G. and Zucker, S.W., Differential Geometric Consistency Extends Stereo to Curved Surfaces. Proc. of 9-th European Conference on Computer Vision, ECCV (3) 2006, pp. 44-57 [9] Comaniciu, D. & Meer, P., Mean Shift: A Robust Approach toward Feature Space Analysis. IEEE Transactions on Pattern Analysis and Machine Intelligence, May 2002 (Vol. 24, No. 5), pp. 603-619. [10] Bleyer, M. & Gelautz, M., Graph-based surface reconstruction from stereo pairs using image segmentation. Proc. of the SPIE, Volume 5665, pp. 288299, 2004 [11] Li, G. & Zucker, S.W., Surface Geometric Constraints for Stereo in Belief Propagation. Computer Vision and Pattern Recognition, 2006 IEEE Computer Society Conference on, Vol 2, 2006, pp. 2355–2362

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266 Modelling in Medicine and Biology VIII [12] Kolmogorov, V., Convergent Tree-Reweighted Message Passing for Energy Minimization. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 28, No. 10, October 2006. [13] Visualization Toolkit, www.vtk.org [14] Kumar A., Ajemba P., Durdle N. & Raso J., Pre-processing Range Data for the Analysis of Torso Shape and Symmetry of Scoliosis Patients. Studies in health technology and informatics, 123(), pp. 483-487, 2006.

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Computer simulations and modeling in oncology: methods and applications C. Guiot1 , P. Paolo Delsanto2 & A. S. Gliozzi2 1 2

Department of Neuroscience, Università di Torino, Torino, Italy Department of Physics, Politecnico di Torino, Torino, Italy

Abstract Computational models and simulations can be powerful tools for gaining an insight into the extremely complex mechanisms governing tumoral growth. In order to be relied upon, however, they must be validated by comparison with sufficiently long strings of experimental or observational data. For obvious ethical reasons it is virtually impossible to obtain such data “in vivo”. It may be, therefore, expedient to study the growth of tumoral lines “in vitro” or “ex vivo”, i.e. by transplanting them into lab animals (e.g., mice). In fact, experiments with as many as 900 successive transplants into new healthy mice have been performed. Using a recently proposed technique for the analysis of experimental datasets (the Phenomenological Universalities Approach), we have succeeded to reproduce, to an excellent level of reliability, the results of such “multipassage” growth and to explain quantitatively why the growth curves become progressively steeper at each new transplant. We believe that our method could also be applied to study metastatic diffusion and suggest new experiments to further validate our approach and results. Keywords: tumor models, growth models, data analysis.

1

Introduction

Computational models can be very useful in many subfields of biomedicine and, in particular, in oncology, due to the extreme complexity of the mechanisms governing tumoral growth, such as angiogenesis, invasion of the surrounding tissues, metastatic diffusion, etc. In fact, such models allow theoretical understanding of the processes involved, by varying the details of the proposed model or their parameters, or by adding new ingredients and/or eliminating ineffectual ones. If a satisfactory agreement is found with the experimental or observed phenomenology, the models may help to reach a good comprehension WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090251

268 Modelling in Medicine and Biology VIII of the neoplastic development. As a corollary, it becomes possible to perform virtual experiments of selected therapies and to predict or optimize the outcome of suggested therapeutic protocols. In some cases they can also reduce the need of much more expensive and objectionable experiments on lab animals. The current relevance of mathematical and computational modeling is due to a combination of related factors [1]. Among them: the advent of systemsbiology-driven concepts in biomedicine that draw from an ever increasing volume of molecular data [2–5], the introduction of novel and cancer-focused interdisciplinary funding programs at the NIH (such as the Integrative Cancer Biology Program [6]) and the decreasing cost of the computational power necessary to run large and clinically relevant simulations. Physical models, based on a comparison with well known phenomena, which present formal analogies with some aspects of the tumoral development, may also be extremely useful, since they may suggest new mechanisms to be tested and analysed. This “crossfertilization” can be efficiently achieved by means of the Phenomenological Universalities (PUN) approach, recently proposed by Delsanto et al. [7–9]. As an example of physical models, we wish to mention here the study of tumor invasiveness, based on the analogy with two well known physical mechanisms, i.e. the mechanical insertion of a solid inclusion in an elastic material specimen or the impinging of a water drop on a solid surface [10]. Finally, biological models are essential to validate the results obtained by means of the theoretical models, both for what concerns the understanding of the phenomenology and the applications for diagnostic and/or therapeutic purposes. They include the implementation of selected tumor lines in “in vitro” or “exvivo” experiments on lab animals, such as mice. For a broad list of recent articles and other information on this topic, we refer to the repository of mathematical models and corresponding computational codes assembled within the framework of the Center for the Development of a Virtual Tumor (CViT) Project (http://www.cvit.org), belonging to the US NIH-NCI ICBP (integrative Cancer Biology Program) [6]. There exists a large number of computational models and simulation techniques, such as cellular automata, finite difference methods, LISA (Local Interaction Simulation Approach), etc. [11–16]. They generally consist of ‘‘mesoscopic’’ formulations that help us to connect the macroscopic and microscopic points of view, i.e. what is mainly of clinical interest from what can be learned from the bio-chemo-physics of the cells, e.g. by means of “ab initio” calculations [17]. Such an understanding is necessary not only to predict the emergence of macroscopic phenomena out of microscopic laws, but also to correlate microscopic and macroscopic parameters [18,19].

2 The PUN approach The PUN approach consists of the search of best fitting functions based only on the experimental datasets available and without any reference to the field of application. The most important PUN classes studied to date are the aboveWIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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mentioned classes UN, which at the first level (N=0) correspond to unrestricted exponential growth. At the level N=1, they yield the Gompertz law. Finally, at the level N=2 (i.e. U2), they successfully predict the fractal properties of the solution of the growth equation at larger times. From a purely applicative point of view, PUNs can be described as a tool for solving the following problem: let us assume that we have an experimental dataset: yi = y (ti ) , where t can be the time (or any other independent variable) and y any observable depending on it. The usual procedure is to perform a fitting of the data, but the choice of the fitting function is generally arbitrary. As a result, the analysis is, in general, only of qualitative value, and often based on the visual inspection of the plots. By contrast, we wish to proceed here in a way that is justified in the framework of a “universal” approach, i.e., totally independent of the field of application. If the nature of the problem suggests that it can be reduced to a first order ODE, we aim to analyse it starting from the nonlinear growth equation:

y = a( y, t ) y (t ) where y =

(1)

dy and a represent the growth rate. dt

Equation (1) is, however, not limited to the modelling of growth problems, since there is no restriction on the nature of the variables y and t. Equation (1) in its complete generality cannot take us too far. In order to use it for a quantitative analysis, it is necessary to restrict its generality by means of some “constraint”, which, although arbitrary, at least are independent of the particular field of application. Let us then assume that a is a function solely of z = ln( y ) and that its derivative with respect to z may be expanded as a set of powers of a, i.e.

b = a = b(a ) =

da da ∞ = ∑αnan z = a dz dz n =1

(2)

If a satisfactory fit of the experimental data is obtained by truncating the set at the N-th term (or power of a), then we state that the underlying phenomenology belongs to the Universality Class UN. It can be easily shown that the Universality Class U1, i.e. with N=1 and b = α1a , represents the well-known ‘Gompertz’ law, which has been used for more than a century to study all kinds of growth phenomena. The class U2 includes, besides Gompertz as a special case, all the growth models proposed to date in all fields of research, i.e., besides the already mentioned model of West et al. [20,7], also the exponential, logistic, thetalogistic, potential, von Bertalanffy, etc. (see, for a review, Ref. [21]). By solving the differential equations z = a and a = b , with b written, for brevity, in the case N=2

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b = α a + β a2

(3)

we find the U2 solution

 β  y = 1 + (1 − eα t )   α 

−

1

β

(4)

It is interesting to observe that Eq.(4) can be written as

u = c1 + c2τ

(5)

which shows that the scaling invariance, which was lost due to the nonlinearity of a ( z ) , may be recovered if the fractal-dimensioned variable u = y

τ = exp(α t )

−β

and

are considered. In fact is, in general, non-integer. In Eq. (5) c1 β

and c2 are constants: c2 = − α , c1 = 1 − c2 . It may also be useful to note that y is the solution of the ODE:

y = γ 1 y p − γ 2 y, where p = 1 + β .

γ1

and

γ2

are two constants:

(6)

γ2 =α / β

and

γ1 = 1− γ 2 .

Their sum is equal to 1, due to the chosen normalization y (0) = 1 . Equation (6) coincides with West's universal growth equation [20], except that here p may be totally general, while West and collaborators adopt Kleiber's prescription (p=3/4) [22], which seems to be well supported by animal growth data. For other systems different choices of p may be preferable: in particular C. Guiot et al. suggest a dynamical evolution of p in the transition from an avascular phase to an angiogenetic stage in tumors [23]. Equation (6) has a very simple energy balance interpretation, with representing the input energy (through a fractal branched network),

γ1 y p

γ2y

the

metabolism and y the asymptotically vanishing growth. In fact all UN’s (at least up to N=3) correspond to energy conservation (or, equivalently, to the first Principle of Thermodynamics). However, in U1 there is no fractal dimensionality and both the input energy and metabolism are proportional to y. In U2, as we have seen, there is fractal dimensionality in the energy input term. In U3 there is also a fractal dimensionality term (with the same exponent p ) in the growth.

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Another very important property of complex systems is cyclicity, which seems to be an almost unavoidable consequence of the feedback system from the surrounding environment, particularly in biomedical and socio-economical sciences. In order to include it in the treatment, we must also consider the case of dependence of a on both z and t [24]. We assume that

a( z, t ) = a ( z ) + a (t ) in which a is assumed to be the sum of two contributions to the growth rate, one ( a ( z ) ) which depends only on z (or y ) and the other ( a (t ) ) solely timedependent. Then, by writing,

y = y (t ) y (t ),

(6)

it follows

y = ( a + a ) yy , which shows that Eq.(1) can be split into a system of two uncoupled equations

y = a ( z ) y

(7)

and

y = a (t ) y

(8)

Eq.(7) can be solved as before (for the case a( z ) ), giving rise to the classes UN. For the solution of Eq.(8) we can assume ∞

a = z = ∑ An En

(9)

n =1

where En = exp[i ( nωt + Ψ n )] . Then, if the sum in Eq.(3) can be truncated to the M-th term, we will state that the corresponding phenomenology belongs to the class UN/TM.

3

PUNs and the multicellular tumor spheroids (MTS) growth

A convenient experimental tool that captures some of the most relevant features of tumor growth kinetics while allowing for a manageable description are the multicellular tumor spheroids (MTS) [17,19]. MTS are spherical aggregations of tumor cells that may be grown under strictly controlled conditions. Their simple

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272 Modelling in Medicine and Biology VIII geometries and the ability to produce them in large quantities has led to interesting new insights into cancer research. In order to understand the “basics” of tumor evolution, we recall that it is generally assumed that tumors originate from a “seed” and grow by cell duplication, therefore following in a first phase an exponential growth law. As long as no mechanical or nutritional restrictions apply, they go on replicating with a constant duplication time. After a while, however, host and other constraints force the development of a necrotic core, and growth slows down towards some asymptotic level of saturation. This behaviour is well described by the well-known Gompertz law, which has been heuristically used for more than a century in biology and other disciplines. Most aggressive “in vivo" or "ex-vivo" tumors overcome nutrients deprivation by means of angiogenesis, and the neovascular network partly supports growth, as discussed by Delsanto et al. [25], following the model of West et al. [20] and West and Brown 22]. This third phase is complemented by the processes of tumor invasion and metastasis. In Fig.1 three regions may be well identified. In the first one, corresponding to the PUN class U0, there is an almost perfect exponential growth without necrotic core formation. In the second one, corresponding to U1, a bending of the growth curve towards some asymptotic level of saturation can be clearly observed: it is due to the decreasing availability of nutrients for the growing MTS. In the third phase (U2), a further decrease in the growth rate occurs, since the MTS approaches the borders of the culture medium in a non-uniform fashion, giving rise to some fractal structures at its surface and/or core interface. Simulation data Experimental data U0 fit U1 fit U2 fit

MTS

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Log of the total number of cells in the MTS

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10 160 10

Time [Hours]

The three phases of growth of MTS. Temporal evolution of a MTS made of EMT6/Ro mouse mammary carcinoma cells grown in a confined culture medium. The experimental data (triangles) are taken from [26]. The “squares” and “circles” correspond to the total numbers of MTS cells and necrotic cells, respectively. They have been obtained from a mesoscopic simulation, based on the model of Delsanto et al. [18,19].

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273

PUNs and the multipassaged tumors in mice.

By applying the PUN approach to the data of Steel [27] and McCredie and Sunderland [28], we have found that the class U2 describes extremely well both their datasets, although with accelerated time scales. If we transplant a tumor seed in a mouse flank and, after it has grown for a few days, we collect a small fraction m0 of the tumor and reimplanted it to another mouse etc. each time only for a short time T (e.g. ten days) tumor is allowed to grow; then we can assume at each ‘passage’ n an experimental growth law with approximately the same rate a (since free growth, in a healthy and nutrient-rich tissue is always occurring, and can write at time t=nT+∆t . In fact, experiments with as many as 900 successive transplants into new healthy mice have been performed. It was observed that tumors grow rate apparently increases at successive transplant (see Fig 2(a)). Actually, Eq.(10) shows that at each transplant the exponential trend is corrected by a term which accounts for the real age of the tumor which, by increasing at each transplant, thus accelerates the growth. In other words, the growth rate a remains the same, provided the time is properly renormalized (see Fig 2(b)). m(t) = m0 (1+exp (naT) (exp(a∆t)-1))

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Results from Steel [27]. Up to 10 transplants of cells from the tumoral line rat fibroadenoma have been performed, but the curves corresponding to only three of them have been reported [9] in logarithmic scale. As discussed in the text, the averaged growth curves become increasingly steeper with successive passages, due to the aging of the newly transplanted tumor cells. By rescaling the time, we obtain a plot (Fig.2b), in which all the curves are collapsed into a unique one.

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References [1] Deisboeck TS et al. (2009) In silico cancer modeling:is it ready for prime time. Nature Clinical Practice 6: 34 [2] Liu ET et al. (2006) In the pursuit of complexity: systems medicine in cancer biology. Cancer Cell 9: 245–247 [3] Kitano H (2002) Computational systems biology. Nature 420: 206–210 [4] Hornberg JJ et al. (2006) Cancer: a systems biology disease. Biosystems 83: 81–90 [5] Coffey DS (1998) Self-organization, complexity and chaos: the new biology for medicine. Nat Med 4: 882–885 [6] Deisboeck TS et al. (2007) Advancing cancer systems biology: introducing the Center for the Development of a Virtual Tumor, CViT. Cancer Informatics: 1–8 [7] Delsanto PP, ed., Universality of Nonclassical Nonlinearity with applications to NDE and Ultrasonics (Springer, 2007). [8] Castorina P, Delsanto PP, and Guiot C. Classification Scheme for Phenomenological Universalities in Growth Problems in Physics and Other Sciences, Phys. Rev. Lett. 2006, 96:188701. [9] Delsanto PP, Guiot C and Gliozzi AS. Scaling, growth and cyclicity in biology: a new computational approach, Theor. Biol. Med. Modell. 2008, 5:5. [10] Guiot C, Pugno N, Delsanto PP and Desiboeck TS. Physical aspect of cancer invasion, Phys. Biol 2007,4:1-6. [11] Capogrosso Sansone B, Delsanto P, Magnano M, Scalerandi M. (2001). Effects of anatomical constraints on tumor growth. Phys. Rev. E, vol. 64; p. 021903, ISSN: 1063-651X [12] Scalerandi M, Pescarmona GP, Delsanto P, Capogrosso Sansone B. (2001). A LISA Model of the Response of the Vascular System to Metabolic Changes of the Cells Behavior. Phys. Rev. E, vol. 63; p. 11901, ISSN: 1063-651X [13] Delsanto P, Romano A, Scalerandi M, Pescarmona GP. (2000). Analysis of a Phase Transition Between Tumor Growth and Latency. Phys. Rev. E, vol. 62; p. 2547-2554, ISSN: 1063-651X [14] Scalerandi M, Romano A, Pescarmona GP, Delsanto P, Condat CA. (1999). Nutrient Competition as a Determinant for Cancer Growth. Phys. Rev. E, vol. 59; p. 2206-17, ISSN: 1063-651X [15] Delsanto P, Gliozzi AS, Guiot C. (2008). Scaling, growth and ciclicity in biology: a new computational approach. Theor. Biol. and Med. Mod., vol. 5; p. 5, ISSN: 1742-4682 [16] Delsanto P, Gliozzi AS, Bruno CLE, Pugno N, Carpinteri A. (2008). Scaling laws and fractality in the framework of a phenomenological approach. Chaos, solitons & fractals, ISSN: 0960-0779 [17] Chignola R, Del Fabbro A, Dalla Pellegrina C and Milotti E, Ab initio phenomenological simulation of the growth of large tumor cell populations, Phys. Biol. 2007, 4:114-33. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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[18] Delsanto P, Griffa M, Condat CA, Delsanto S, Morra L. (2005). Bridging the gap between mesoscopic and macroscopic models: the case of multicellular tumor spheroids. Phys. Rev. Lett., vol. 94; p. 148105, ISSN: 0031-9007. [19] Delsanto P, Condat CA, Pugno N, Gliozzi AS, Griffa M (2008). A Multilevel Approach to Cancer Growth Modeling. J. Theor. Biol., vol. 250; p. 16-24, ISSN: 0022-5193. [20] West GB, Brown JH and Enquist BJ. A general model for ontogenetic growth. Nature 2001, 413:628-631 [21] De Vladar HP, Density-Dependence as a Size-Independent Regulatory Mechanism, J. Theor. Biol., 238, 245-256 (2006). [22] West GB and Brown JH. Life’s universal scaling laws. Physics Today 2004, 57(9): 36–43; Savage VM, Deeds EJ, Fontana W. Sizing Up Allometric Scaling Theory. PLoS Comput Biol 2008, 4(9): e1000171. [23] Guiot C, Delsanto PP, Carpinteri A, Pugno N, Mansury Y, Deisboeck TS, The dynamic evolution of the power exponent in a universal growth model of tumors, J Theor Biol (2006), 240:459-463. [24] Delsanto PP, Gliozzi AS, and Guiot C, Scaling, Growth and Cyclicity in Biology: a New Computational Approach, Theoretical Biology and Medical Modelling (2008), 5:5. [25] Delsanto PP, Guiot C, Degiorgis PG, Condat AC, Mansury Y and Desiboeck TS. Growth model for multicellular tumor spheroids. Appl. Phys. Lett. 2004, 85: 4225–4227. [26] Freyer JP and Sutherland RM. Regulation of growth saturation and development of necrosis in EMT6/Ro multicellular spheroids by the glucose and oxygen supply, Cancer Res. 1986, 46:3504. [27] Steel GG. Growth Kinetics of Tumors, Clarendon Press, Oxford, 1977. [28] McCredie JA, Sutherland RM. Differences in growth and morphology between the spontaneous C3H mammary carcinoma in the mouse and its syngeneic transplants. Cancer 1971, 27:635-642.

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PKAIN: an artificial immune network for parameter optimization in pharmacokinetics L. Liu1, C.-H. Lai2, S.-d. Zhou3, F. Xie3 & H.-w. Lu3 1

School of Information Technology, Jiangnan University, Wuxi, Jiangsu, P. R. China 2 School of Computing and Mathematical Science, University of Greenwich, London, UK 3 Suzhou University Affiliated Forth People’s Hospital, The Forth People’s Hospital of Wuxi, Wuxi, Jiangsu, P. R. China

Abstract The PKAIN algorithm is an artificial immune network, which has been designed to optimize parameters of linear pharmacokinetic models in our previous work. In this paper, the algorithm is modified to optimise parameters of nonlinear pharmacokinetic models. To evaluate parameters, the numerical inverse Laplace method is adopted to calculate drug concentrations of the dynamic system. The initial solutions of pharmacokinetic parameters are generated randomly by the PKAIN algorithm in a given solution space. Memory cells to be used in the search of global optimal parameters are generated. The optimal mechanism of the algorithm is based on artificial immune network principles and simplex mutation. In addition, a distributed version of the PKAIN algorithm is proposed to improve its efficiency. Keywords: pharmacokinetic model, distributed computing, artificial immune network, numerical inverse Laplace, simplex.

1

Introduction

The artificial immune system [1] is a novel soft computing paradigm, which simulates powerful abilities of biological immune systems in mutual action to defend against pathogenic organisms. Studies on the comparison of optimization performances between the artificial immune system and other heuristic algorithms [2], such as simulated annealing algorithm, genetic algorithm, and WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090261

278 Modelling in Medicine and Biology VIII evolution programming, have demonstrated that the artificial immune system has outstanding characteristics of population diversity and global convergence. Furthermore, the artificial immune network has been successfully used in multimodal function optimization [3], and dynamic environment optimization [4]. Various applications of the artificial immune network are currently being studied by various research groups. In order to understand absorption, distribution, metabolization and elimination processes of a drug, a pharmacokinetic model is required to be built with suitable model parameters. Optimised parameters are usually determined through the use of observed drug concentrations after a period of time of applying the drug. Therefore the optimisation process is the most fundamental and important task in establishing a robust pharmacokinetic model. In our previous work, parameters of a linear compartment model are optimized [5]. First, the Laplace transform is applied to the linear compartment model. Analytical inverse Laplace transform is used to deduce the drug concentration function. The artificial immune network PKAIN was developed to optimise parameters of the concentration function. In contrast to Gauss-Newton and simplex methods, the PKAIN is capable of obtaining the global optimal solution and the process is insensitive to initial solutions. In this paper, the PKAIN method is extended to optimise parameters of nonlinear pharmacokinetic models. Unlike linear compartment models, the concentration function of nonlinear pharmacokinetic model tends to be unavailable in its analytic form. Therefore the numerical inverse Laplace method is used to obtain discrete concentrations in a given temporal period for the given nonlinear model. In order to optimize parameter, the PKAIN is modified as follows. First, fitness calculation integrated with numerical inverse Laplace method is developed. Second, a distributed version of the PKAIN algorithm is proposed to improve its efficiency.

2

The artificial immune network for parameter optimization of nonlinear pharmacokinetics

In the PKAIN algorithm, a set of parameters describing a given pharmacokinetic model is encoded into a memory cell of the artificial immune network. The procedure of the PKAIN artificial immune network for parameter optimization of nonlinear pharmacokinetics is described as follows. Initially, the memory cells of the artificial immune network are randomly generated in the solution space. For each cell, clone selection and mutation are used to generate new cells. The fitness of each cell is calculated by evaluating the ‘goodness’ of the set of parameters and are encoded into them. In the network suppression process, similar cells are deleted to maintain a relatively smaller network scale. Certain proportional cells with lower fitness are updated dynamically and a new generation of the network is generated. The network is allowed to evolve until certain stopping criterion has been achieved. Finally, the memory cell with the highest fitness is decoded and the optimal set of parameters of the pharmacokinetic model is obtained. In the following subsections, the fitness WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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calculation, clone selection and simplex mutation processes of the PKAIN are described. 2.1 Fitness calculation Let c = {θ1 ,θ 2 ," , θ l } denotes a set of l parameters describing a pharmacokinetic model. A candidate solution of the parameters is encoded into a memory cell. The fitness of a network cell is related to an objective function which is used to evaluate the goodness of the cell. In the context of pharmacokinetic model optimization, it is a measure of how well the drug concentration computed by using the set of parameters of the model would fit with the observed drug concentration. The fitness is measured as a numerical value ranging from 0 to 1 and the closer the fitness value is to 1 means that the better the set of parameters encoded in a memory cell is to fit the observation. Let x i , i = 1,2,..., n be the drug concentration observed at time t i and X i the calculated drug concentration using

c for the nonlinear pharmacokinetic model, the fitness of a memory cell η (c) ,

may be defined as follow: n

n

i =1

i =1

η (c) = 1 − ∑ wi ( xi − X i )2 / ∑ ( xi − x ) ,

(1)

where x is the average observed drug concentration. wi is set to the value 1, otherwise 1 / xi , or 1 / xi 2 when x i exhibits a wide range of values. Consider the nonlinear pharmacokinetic model of which the drug concentration Xi = X (ti ) is required： dX = A( X ) + H ( X ) + f (t ) ， (2) dt where A(X ) is a linear function of X , H(X ) is a nonlinear function of X , and f (t )

is a known function of t . To obtain numerical solutions of the nonlinear pharmacokinetic model, the iterative coefficient-inverse Laplace method (ICIL), which was successful used in our previous work [6] for nonlinear Black-Scholes equations in financial computing, is adopted. At each time step, a linearization of the nonlinear term H (X ) is obtained by ~ computing H (X ) using the approximation X~ , which is updated within an iterative update process in order for X~ to converge to X~ . Each step of the iterative update process involves a numerical solution to the equation, dX ~ = A( X ) + H ( X ) + f (t ) , dt

(3)

where Laplace transformation may be applied easily. Let X p and X p +1 be the numerical solutions of eq. (3) at t = t p and t = t p +1 respectively. Also let U (λ ) and F (λ ) be the Laplace transforms of X and f (t ) respectively, where λ is the parameter introduced after taking the Laplace transform. Applying the Laplace transform to eq. (3), being defined in the time interval t ∈ [ t p , t p +1 ] , in its differential form leads to ~ U (λ ) = [ H ( X ) / λ + F (λ ) + X p ] /(λ − A), (λ − A ≠ 0)

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280 Modelling in Medicine and Biology VIII The iterative update process to obtain the numerical solution X p + 1 , using X p as the initial approximation to X and assuming m parametric functions of U (λ j ) , j = 1, 2,..., m , is described in the ICIL algorithm as below. FUNCTION to find X

p +1

Initial approximation: X

(0) p +1

= X

p

//Iterative Inverse Laplace Method in t ∈ [t p , t p +1 ] k=0; ITERATE X = X (pk+)1 ; Compute U (λ j ) = [ H ( X ) / λ j + F (λ j ) + X p ] /(λ j − A) , j = 1, 2,..., m ln 2

where λ j = j t − t ; p+1 p (k ) Find X p +1 ≈

ln 2 t p +1 − t p

m

∑ w U (λ ) j =1

j

j

ln 2

where λ j = j t − t ; p+1 p w j = (−1)

k m / 2 (2k )! ; k = (1+ j ) / 2 ( m / 2 − k )! k! ( k − 1)! ( j − k )! ( 2 k − j )!

m + j min( j , m / 2 ) 2

∑

k = k +1 ; (k ) ( k −1) UNTIL X p +1 − X p +1 < ε X p +1 = X p( k+)1 ; END-FUNCTION 2.2 Clonal selection and simplex mutation The clonal selection process of an artificial immune network is a computational implementation of the clonal selection principle in solving optimization problems, emphasizing multimodal and combinatorial optimization [7]. The process of clonal selection is composed of clone, mutation and selection steps. In the clone step, identical off springs of memory cells are generated. Then new cells are created through the mutation step. The PKAIN algorithm executes clone and fitness-based mutation steps as below. c new = c + αN (0,1) (4)

( )

α = 1 β exp(−η * (c))

(5)

where cnew is a new memory cell mutated from the cell c , N (0,1) is a Gaussian random variable of zero mean with deviation σ = 1 . The mutation factor α is inversely proportional to the clone constant β . η * (c) is the normalized fitness of WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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the cell c . In this case, cells with low fitness mutate heavily so that they have many chances to be better than their parents to improve the diversity of the population. On the contrary, cells with higher fitness mutate relatively little to reserve their priority. In order to improve local optimal capability, the simplex mutation is designed. In the classical simplex method, a simplex is constructed by its L + 1 vertices in the L dimensional Euclidean space. Then, through a sequence of elementary geometric transformations (reflection, contraction, expansion and shrinking), the initial simplex moves, expands and contracts. In such a way that it adapts itself to the function landscape and finally surrounds the optimum [8]. Since the simplex method is an excellent method for local optimization, it has been used with other searching techniques in a hybrid fashion. Yen et al. [9] developed a variant of the concurrent simplex method which begins with L + Ω points, where Ω > 1 , instead of L + 1 points as in the classical simplex method. In this paper, a partition-based concurrent simplex mutation is examined. The new cells cnew generated by the clone and mutation steps of the clonal selection process for a given memory cell c are considered as a natural partition group. The number of simplex mutated cells is denoted as N c , N c > L + 1 . After executing the concurrent simplex method to obtain cnew , there are N c − L number of new cells that have been updated. The partition-based concurrent simplex mutation is described as the following steps. (1) Order Order the cells of cnew to satisfy η (c1 ) ≥ ... ≥ η (cL ) ≥ ... ≥ η (cL +i ) ≥ ... ≥ η (cN c ), i = 1, 2,..., N c − L . Calculate the centroid of L number of cells with higher fitness values, c = (c1 + c2 + ... + c L ) / L . (2) Reflection For each c L +i cell, compute the reflection point c r by using

c r = c + ρ (c − c L +i )

(6)

Calculate the fitness η r = η (c r ) . If η (cL ) < η r < η (c1 ) , accept

c L +i = c r and

terminate the operation. (3) Expansion e

r If η r ≥ η (c1 ) , expand the point c to c by using

c e = c + ρχ (c − cL+i )

(7)

Calculate the fitness η e = η (ce ) . If η e > η r , accept c L +i = c and terminate the e

r operation; otherwise, accept cL +i = c and terminate the operation.

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282 Modelling in Medicine and Biology VIII (4) Outside Contraction o If η (cL +i ) < η r < η (cL ) , compute the outside contraction point c as

c o = c + ργ (c − cL +i )

(8)

Calculate the fitness η = η (c ) . If η > η , accept cL +i = c and terminate the operation. o

o

o

r

o

(5) Inside Contraction i If η r < η (cL ) and η (cL +i ) ≥ η r , compute an inside contraction point c as

c i = c − γ (c − c L + i )

(9)

i i Calculate the fitness η i = η (ci ) . If η > η (cL +i ) , accept c L+i = c and terminate the operation. There are coefficients of reflection ( ρ ), expansion ( χ ), and contraction ( γ ). The usual choices of these coefficients are ρ = 1, χ = 2, γ = 0.5 . The shrinking operator of the classical simplex method is replaced with the mutation step of the clonal selection process.

3 A distributed PKAIN method In order to obtain numerical solution of eq. (3) accurately by means of a temporal integration method, temporal intervals [t p , t p +1 ] should be small. Unfortunately, concurrent computation of all time steps in a temporal integration method is impossible. It seems that to achieve a distributed algorithm to yield a de-coupling of the original problem is almost impossible. However in the ICIL algorithm, the numerical solutions X p , p = 1, ..., N may be computed concurrently and, thus, the total computational time of becomes significantly reduced. In this paper, we examine a two-level temporal decomposition method which stems from our previous work of concurrency in time domain computation [10, 11]. Assuming the N temporal steps of serial calculation are equally divided into N coarse parts, each represents a coarse temporal interval ∆T . In the secondary temporal decomposition, each ∆T is divided into N fine parts of finer temporal intervals ∆t , ∆ T = N fine .∆ t . First, a number of numerical solutions of the nonlinear equation, using the concept of the Laplace transform and its iterative inverse, are obtained on a coarser temporal mesh, t ∈ [ T p − 1 , T p ] , p ∈{1,…, N coarse} . Second, each coarse temporal mesh t ∈ [T p −1 , T p ] can be into several finer temporal meshes t ∈ [T p −1 , T p −1 + t i ] ， i ∈ {1, … , N fine − 1} . The numerical solutions on the coarser temporal mesh obtained in the former step are used as initial conditions on the finer temporal mesh. Solutions defined on a finer temporal mesh for each of the coarse temporal mesh are now being obtained concurrently using a temporal integration method. This linearization leads to a distributed version of the fitness calculation process decomposed

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for the PKAIN. Suppose the fitness calculation can be distributed into p processors, the pseudocode of the fitness calculation is described as below. FUNCTION Distributed fitness calculation Decode the network cell c into parameters. FOR p = 1 TO N coarse , do coarser temporal mesh on t ∈[T p−1 , T p ] Find X p from X p −1 ; END-FOR Distribute p = 0 TO N coarse − 1 ， X p to processors， For i = 1 To N fine − 1 , do finer temporal mesh on t ∈ [Tp , Tp + ti ] Find X p + i from X p + (i −1) ; END-Distribute and receive X p + i . n

n

i =1

i =1

η = 1 − ∑ wi ( xi − X i ) 2 / ∑ ( xi − x )

END-FUNCTION

4

Experiments and discussion

A typical nonlinear pharmacokinetic model for drug concentration is often described by Michaelis-Menten equation. In this section, the PKAIN algorithm is used to optimize Michaelis-Menten pharmacokinetic parameters of the bolus intravenous examples 1 and 2 as described in [12]. Optimal parameters are compared to the solutions given by the accurate linear regression (ALR), improved Hanes-Woolf method (HW), and combined Runge-Kutta method (RKPS) in the literature. The equation of drug concentration described by Michaelis-Menten equation consists of two parameters is given as follow. V X dX =− m (10) dt Km + X Table 1: Method PKAIN HW ALR RK-PS

Comparison of optimal parameters for example 1. Km

Vm

R

10.907 10.5135 9.3936 9.9804

4.016 3.9615 3.9573 3.9994

0.042690 2.260105 0.293910 0.111829

In other words, candidate solutions of c = {Vm , Km} are encoded into memory cells. Optimal parameters for example 1 obtained by PKAIN, ALR, HW, and RK-PS methods are shown in Table 1. Optimal parameters for example 2 are shown in Table 2. In order to compare these parameters, the relative residual WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

284 Modelling in Medicine and Biology VIII R = ∑ (( xi − X i ) 2 / xi ) is

calculated. The smaller the weighted residual is, the better the parameters are. The results demonstrated that the PKAIN method outperforms HW, ALR, and PK-PS algorithms in parameter optimization of nonlinear pharmacokinetic models.

5

Table 2:

Comparison of optimal parameters for example 2.

Method PKAIN HW ALR RK-PS

Km

Vm

R

33.2223 28.0806 27.5851 29.5126

13.4414 11.2978 11.2828 13.0842

1.5718282 9.292684 8.509200 1.817844

Conclusions

In this paper, the artificial immune network PKAIN is designed to optimize nonlinear pharmacokinetic parameters. The method to obtain numerical solutions of nonlinear system is integrated into its fitness calculation process. Experimental results obtained by the PKAIN algorithm are better than those obtained by HW, ALR, and PK-PS methods. Together with our previous work, the PKAIN algorithm is capable of obtaining optimal parameters for both linear and nonlinear pharmacokinetics. In addition, the two-level temporal decomposition method is used to parallelize the nonlinear pharmacokinetic model using an iterative inverse Laplace transformation. The efficiency of the PKAIN algorithm can be greatly improved when it is implemented in a distributed environment due to the fact that there is no data dependence of the fine temporal mesh computation.

Acknowledgement The authors acknowledge support from innovation team project JNIRT0702 of Jiangnan University.

References [1] de Castro, L.N., Timmis, J., Artificial immune systems as a novel soft computing paradigm. Soft Computing, 7(8), pp.526-544, 2003. [2] Xie, K.G., Zeng, X.H., Li, C.Y., et al., Comparative analysis between immune algorithm and other random searching algorithms. Journal of Chongqing University, 26(11), pp.43-47, 2003, (in Chinese). [3] de Castro, L.N, Timmis, J., An artificial immune network for multimodal function optimization. Proc. of IEEE Congress on Evolutionary Computation, IEEEE Service Center, Honolulu, USA, pp.674-699, 2004. [4] de Franca, F.O., Von Zuben, F.J., de Castro, L.N., An artificial immune network for multimodal function optimization on dynamic environments. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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[5] [6] [7] [8] [9]

[10] [11]

[12]

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Proc. of the GECCO conf, ACM Press, Washington DC, pp. 289-296, 2005. Liu, L., Zhou, S.D., Lu, H.W., et al., Parameter optimization of pharmacokinetics based on artificial immune network. Applied Mathematics And Mechanics-English Edition, 29(4), pp.549-559, 2008. Lai, C.H., Parrott, A.K., Rout, S., A Distributed Algorithm for European Options with Nonlinear Volatility. Computers and Mathematics with Applications, 49, pp.885-894, 2005. de Castro, L.N., Von Zuben, F.J., Learning and optimization using the clonal selection principle. IEEE Trans on Evol Comp, 6(3), pp.239-251, 2002. Jeffrey, C.L., James, A.R., Margaret, H.W., et al., Convergence properties of the Nelder-Mead simplex method in low dimensions. Society for Industrial and Applied Mathematics, 9(1), pp.112-147, 1998. Yen, J., Liao, J., Randolph, D., Lee, B., A Hybrid approach to modelling metabolic systems using a genetic algorithms and the simplex method. IEEE Transactions on Systems; Man; and Cybernetics; 28(2), pp.173-191, 1998. Lai, C.H., On transformation methods and concurrency in time domain computation. Proc. of the DCABES 2007, pp. 5-6, 2007. Lai, C.H., Numerical solutions of certain nonlinear models in European options on a distributed computing environment. Nonlinear Models in Mathematical Finance: New Research Trends in Option Pricing, ed. Matthias Ehrhardt, Nova Science Publisher, pp.283-298, 2008. Su, Y.F., Optimizing method of Michaelis-Menten pharmacokinetic parameters of bolus intravenous administration. Chin J Clin Pharmacol Ther 10(10), pp.1198-1200, 2005, (in Chinese).

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Readability of reassigned scalograms and extraction of spectra features for signal analysis S. Mekaoui1, A. Houacine1 & T. Gharbi2 ¹Department of Telecommunications, L.C.P.T.S., USTHB, Algiers, Algeria ²Institut des Microtechniques, Laboratoire d’ Optique P.M. Duffieux, UFR Besançon, France

Abstract In this paper, a wavelet scalogram is used. The wavelet scalogram presents some disadvantages. This is particularly true for time-frequency analysis and representation, which present inconvenient cross-terms. Mechano-myogram (MMG) signals are acquired via a home probe highly sensitive optical sensor. The data obtained from two healthy subjects and two patients tested under drastic conditions are analyzed to characterize the dynamic properties of the MMG and also to determine their frequency contents. We developed the reassignment form of the scalogram, which improves its resolution and readability. A plot of the scalogram contours is also presented to test the direct readability of the scalogram representations. Spectra features are extracted and relevant parameters are assessed, such as the power spectral density, the mean frequency, the average frequency and the well known ratio HF% that characterizes the dynamic characteristics of the tested muscles. For that purpose, the number of subjects had been increased to 24 healthy subjects and up to 18 patients affected by different specific muscular diseases. Keywords: mechano-myogram (MMG) signals, reassigned wavelet scalogram, power spectral density, mean and average frequencies, MMG rms value, MMG average value, HF% ratio value.

1

Introduction

Muscular sounds actually known as mechano-myogram (MMG) signals are acquired with the help of a highly sensitive optical sensor. These signals are nonWIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090271

288 Modelling in Medicine and Biology VIII stationary and of random form with a very low amplitude. The study is focused on the characterization of their frequency contents and the extraction of spectra features that represent their dynamic properties. For that purpose, we tested the forearm muscles (e.g. flexors) of many healthy subjects and patients. Timefrequency analysis was implemented to overcome the shortcomings of the FFT analysis. Among several TFA methods was the most popular, called the Hitherto method [1–3], which initiated the time-frequency plane. The Hitherto method is specifically devoted to the analysis of non-stationary signals [2, 3]. The TFA methods revealed their limitations on finding a good trade-off between time and frequency resolutions. The limitation imposed by the Heisenberg-Gabor inequality [1–6, 10], which made the trade-off unavoidable, compelled the authors to find a solution. Thus, a compromise was to be found between time and frequency resolutions for whatever non-stationary signal. To overcome these drawbacks, other authors [2–6, 10, 14] proposed other time varying signal analysis tools on a concept of scale rather than frequency, such as the Wavelet scalogram [2, 6]. Other tools, such as the affine smoothed version of the PseudoWigner-Ville distribution [2, 10], were also implemented. However, bilinear time-frequency distributions such as the Wigner-Ville distribution have good concentration in the time-frequency plane, but present the disadvantage of interference terms (cross-terms) that can blur the readability in the time frequency plane of auto-terms (this is significant). Many attempts had been tried by the authors to overcome these inconvenient drawbacks. Unfortunately, those attempts were all tending toward a loss in time-frequency concentration [9–12, 14–16]. The wavelet scalogram is limited by the Heisenberg-Gabor inequality and presents the same weaknesses in the time scale-plane. To remove these shortcomings, the authors implemented a modified form of the wavelet scalogram called the reassignment method of the wavelet scalogram [1–4, 6, 12]. This method preserved energy properties and made both time and scale resolutions rightly enhanced.

2

Wavelet scalogram background

2.1 Continuous wavelet scalogram The concept of the continuous wavelet scalogram is to subdivide a signal x(t) into a set or a family of zero mean functions called the “wavelets”, derived from an elementary function Ψ (the “mother wavelet”), by translation in time and dilation in scale of the later. Then the following relation is tenable, [1, 2, 5, 6, 12, 14–16]: +∞

CWTx (t , a,ψ ) = ∫ x(t )ψ t*,a (s )ds ,

with:

 s −t   . (1)  a 

ψ tx,a (x ) = (a −1/ 2 )ψ 

−∞

The parameter a corresponds to a scale factor. Time and frequency resolutions are limited by the Heisenberg-Gabor inequality [1–5].

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2.2 Wavelet scalogram The wavelet scalogram is then defined by:

SC ( x, T , a ) = CWTx (t , a,ψ ) , 2

(2)

where ψ is the wavelet function. The scalogram is interpreted as the smoothed version of the Pseudo-WignerVille distribution. One should refer to [2, 6, 10, 12, 14–16] to learn more. 2.3 Reassignment method of the wavelet scalogram The concept of reassignment was based on the previous assumptions. As depicted, one has to find a compromise between time and frequency resolutions. It appears that it was necessary to enhance the readability of the scalogram and make the concentration of significant terms greatly localized in the scalogram and get more improved readability reducing the maximum cross-terms. So, we chose the reassignment form to attain these goals [2, 6].

3

Methods

3.1 Subjects Four adults, two healthy subjects and two patients, were tested in this study. The MMG signals were acquired via a home probe. Many other patients affected with muscle diseases, such as current dystrophies and atrophies, were tested. Some of them were affected by Steinert and one by Marie-Charcot-Tooth disease. 3.2 Data analysis 3.2.1 Power spectral density Power spectral density is to be extracted from the MMG’s spectra. This parameter is estimated using the Welsh method and then noted as:

S( f ) =

1 L ∑ Sl ( f ) , L l

(3)

where l represents the index of the interval in respect to the frequency limits of the MMG’s range (e.g. 0–45 Hz). 3.2.2 The average frequency The average frequency in a determined range of frequencies is defined by: f1

Fav = where S ( f

∑ f .S ( f )

f = f0 f1

∑ S( f )

,

f = f0

) is the power spectral density.

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(4)

290 Modelling in Medicine and Biology VIII 3.2.3 Mean frequency The mean frequency is given by: Fmean

∑ S( f ) =

f = f0

f1

∑ S( f ) ,

(5)

f = Fmean

where Fmean is the mean frequency and S ( f ) is the power spectral density. 3.2.4 Mean value of the MMG signal The mean value of the MMG corresponds to the mean time average value of the rectified MMG signal (in micrometers) and is given by: rectified Vmean = YMMG ,

(6)

where YMMG is the amplitude of the MMG signal. 3.2.5 Root-mean-square value of the MMG The root-mean-square value of the MMG signal is calculated from the previous equation of the power spectral density in the limits of the MMG’s determined frequency range 0–45 Hz and is determined by:

Vrms =

f1 =45

∑ S ( f ) ; where: f 0 =0

f 0 = 0 Hz and f1 = 45 Hz.

(7)

3.2.6 The ratio HF% This ratio was defined as the ratio of the power spectral density in the range of 0–45 Hz to the power spectral density in the range of 6–45 Hz for whatever type of muscle, thus: 45

∑S( f )

HF%= f45=15 . ( ) S f ∑

(8)

f =6

This ratio reflects the contribution of the fast fibers. 3.2.7 Statistics Twenty-four healthy subjects and up to 18 patients were tested in a clinical environment. Then, for each extracted feature, a statistical analysis is produced with the help of the well-known Origin 6.1 software.

4

Results and discussion

4.1 Reassigned wavelet scalogram results The results presented in this section concern two healthy subjects and two patients respectively. The first sub-section of results is organized so that for each MMG signal there exists four sub-windows. The first sub-window gives the WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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display of the acquired MMG signal, whereas the second shows the power spectral density of the analyzed signal. The third sub-window displays the reassigned wavelet scalogram and finally the fourth illustrates the contour plot of the scalogram. The healthy subjects are named Heal.1 and Heal.2 whereas the patients are named Pat.1 and Pat.2. Considering Figure 1, which gives the results for the healthy subjects, it can be noticed that there are good concentrations of energy peaks around 10 Hz and 20 Hz and a poor concentrations around 12 Hz and 15 Hz (Heal.1). The reassigned wavelet contour plot clearly shows this assertion and reveals several frequencies at the same instant. The power spectral density displayed shows a concentration of frequencies in the lower range of the power spectrum whose frequency axis is normalized to the highest value. These observed differences in frequencies at different instants are mainly due to the fact that force increases along with the increasing number of fibers that are recruited since contraction. This process seems to evolve steadily until the subject is in a state of total exhaustion. The readability of reassigned wavelet scalograms for Heal.2 indicates that energy peaks are distributed around 5 Hz and 12 Hz with a high level of brightness, and are relatively poor around 10 Hz, 15Hz and 20 Hz (Heal.2). Nevertheless, we should notice the appearance of the blurring 7 Hz (Heal.2), well known by the clinicians to correspond to only muscle tremors. The examination of the power spectral density yields the same observation as in the case of Heal.1. In the case of the patients, Figure 2 gives the patients results, which are organized as in the case of the healthy subjects. It can be seen from the displayed reassigned scalograms that Pat.1 had developed a very poor effort probably due to the nature of the muscular disease and hence an awful grasping of the strain gauge as the scalogram shows few energy peaks at around 5 Hz and 10 Hz and a very small number with very poor energy at around 15 Hz and 20 Hz. Only tremors and clearly noticeable large transients are observed in the acquired MMG signal. The contours plot neatly reveals this fact. Real exhaustion at the beginning of the measurement protocol is obvious. Similar observations are noticed in the second case (Pat.2) who did his best to firmly grasp the strain gauge, but was unsuccessful and was unable to fulfill the fixed force consigns of the experimental protocol. The generated frequencies revealed by both the scalogram and the contours plot are confusedly dispersed around 5 Hz and 20 Hz with lower intensities. Obviously, only fast fibers responded to the excitation as the frequencies are localized in the high frequency range of the most significant of the MMG’s frequency range. Patients were not able to stand more than 15 seconds of experimentation. 4.2 Statistical results and analysis Figures 3–6 provide graphic representations of the extracted spectra features of the MMG signals namely: average frequency versus average force, mean frequency versus average force, MMG amplitude rms value versus maximum force, and HF% ratio versus maximum force, for either healthy subjects or patients. The subject group consists of 24 healthy subjects and 18 patients. The group of patients was organized in accordance with the nature of the WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

292 Modelling in Medicine and Biology VIII

Figure 1:

Results of two healthy subjects. For each of them, the first subwindow gives the acquired MMG signal, the second sub-window the power spectral density of the MMG signal. The third and fourth illustrate respectively the reassigned wavelet scalogram of the MMG signal and its contours plot.

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Figure 2:

293

Results of two unhealthy subjects. For each of them the first subwindow gives the acquired MMG signal, the second sub-window the power spectral density of the MMG signal. The third and fourth illustrate respectively the reassigned wavelet scalogram of the MMG signal and its contours plot.

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294 Modelling in Medicine and Biology VIII

Average

Frequency

in

Hz

(a) 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Y = 9.48743 + (0.00451)*X (linear regression) (standard deviation) with : σ = 1.43973 R = 0.14666 (regression coefficient)

0 20 40 60 80 100 120 140 160 180 200 220 240 Average force in Newton

Average

Frequency in Hz

(b)

Figure 3:

10,0 9,5 9,0 8,5 8,0 7,5 7,0 6,5 6,0 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0

Y = 5.45159 + (0.04152)*X (linear regression) (standard deviation) with: σ = 1.60381 R= 0.49068 (regression coefficient)

0

10

20 30 Average

40 50 60 70 80 force in Newton

90

Average frequency in Hz versus average force in Newton, (a) in the case of healthy subjects, (b) in the case of patients.

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Mean frequency in

Hz

Modelling in Medicine and Biology VIII

13 12 11 10 9 8 7 6 5 4 3 2 1 0

(a)

Y = 8.73749 + (0.00457)*X (linear regression) with : σ = 1.91959 (standard deviation) R = 0.11204 (regression coefficient)

0 20 40 60 80 100 120 140 160 180 200 220 240

Mean

frequency

in

Hz

Average

10,0 9,5 9,0 8,5 8,0 7,5 7,0 6,5 6,0 5,5 5,0 4,5 4,0 3,5 3,0 2,5 2,0 1,5 1,0 0,5 0,0

force

in

Newton

(b)

Y= 3.82434 + (0.05038)*X (linear regression) with : σ = 1.79811 (standard deviation) R = 0.52035 ( regression coefficient )

0

10

20

30

Average

Figure 4:

295

40

50

force in

60

70

80

90

Newton

Mean frequency versus average force in (a) healthy subjects, (b) patients.

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296 Modelling in Medicine and Biology VIII

MMG rms value ( in µm )

(a)

1,2 1,1 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0

-5

-7

2

Y = 0.82228+(-7.65784*10 )*X + (4.62216*10 )*X (quadratic) with: σ = 0.13662 ( standard deviation ) 2 R = 0.05233 (quad. reg. coefficient )

0 50 100 150 200 250 300 350 400 450 500 550 600

MMG rms value ( in µm )

Maximum force in Newton

(b)

1,8 1,7 1,6 1,5 1,4 1,3 1,2 1,1 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0

-6

2

Y = 1.01514+(-0.00162)*X+(5.78208*10 )*X (quadratic) with : σ = 0.44966 ( standard deviation ) 2 R = 0.06897 ( quad. reg. coefficient )

0

50 100 150 200 250 300 350 400 450 500

Maximum force in Newton

Figure 5:

MMG rms value versus maximum force in (a) healthy subjects, (b) patients.

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HF% ratio ( in percentages )

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(a)

32,5 30,0 27,5 25,0 22,5 20,0 17,5 15,0 12,5 10,0 7,5 5,0 2,5 0,0 0

50 100 150 200 250 300 350 400 450 500 550 600

Maximum force in Newton

(b)

HF% ratio ( in percentages )

30 25 20 15 10 5 0

0

50

100 150 200 250 300 350 400 450 500

Maximum

Figure 6:

force in Newton

HF% ratio versus maximum force in (a) healthy subjects, (b) patients.

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298 Modelling in Medicine and Biology VIII muscular diseases and named Pat.1 through to Pat.18 and distributed as follows: Steinert (Pat.1, 2, 13, 14, 15 and 16); Belt dystrophies (Pat.3, 4, 6, 7, 8, 10, 11, 12, 17 and 18); Charcot-Marie-Tooth (Pat.5 only). Figure 3 gives the statistical variation of the average frequency versus average force in both groups, which made the comparison easier. This feature characterizes the evolution of the MMG signal amplitude in the period of stability of the muscular contraction. The linear regression estimated yields a positive slope. It can be observed that in the case of the healthy subjects few values are dispersed whereas in the case of the patients, we observed dispersed values with a greater standard deviation. In both cases, the average frequency varies linearly with the force. In order to check on this tendency we have chosen to study the mean frequency of the MMG signal versus average force. Figure 4 gives the results of this second spectrum feature and its assessment. So, we noticed in the case of the healthy subjects that the mean frequency takes smaller values and a greater standard deviation when the regression is still linear with a positive slope inferring to a linear function of the average force. In the case of the patients, the mean frequency seems to behave similarly. Globally the values of the mean frequency are well correlated with a higher regression coefficient. Figure 5 illustrates the MMG rms value in terms of maximum force for either the healthy subjects or the patients. The only relevancy is the poor and dispersed values in the case of the patients and the quadratic form of this feature. This compelled us to test and estimate another interesting feature that the MMG acquired and which in fact best characterizes the activity of fast muscle fibers. This parameter is called the HF% ratio. Figure 6 gives the variations of this important feature in terms of maximum force. Examining figure 6, we observed that this ratio varies from 5% to 35%. In the case of patients affected with different muscular diseases it appears that the maximum of this ratio is 30% and the minimum is around 5%. In this case we did not notice significant differences with the healthy subjects except that for special diseases like Steinert and Charcot-Marie-Tooth the values of HF% are smaller than in the case of healthy subjects.

5

Conclusion

The first aim of this work was to show the readability of the reassigned wavelet scalogram of the MMG signals acquired from the flexors forearm muscles of many healthy subjects and patients affected with some well-known muscular diseases. The most relevant fact is the improvement of the readability of the reassigned wavelet scalograms and hence, a better concentration of the most significant frequencies is obtained in both cases. The contours plot emphasizes these observations and the estimated power spectral densities confirmed the frequency range of the MMG signals. Moreover, tremors were read on the reassigned wavelet scalograms, particularly in the case of the patients. We found that these tremors were revealed by concentration of frequencies in the vicinity of 7 Hz and were due to awful adaptation with the grasping of the gauge. Also, as had been observed for the patients, the power spectra were shifted to the lower WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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frequencies during fatigue. Moreover, reassigned scalograms showed a better localization in time of the frequency contents. The second part of this work dealt with the statistical assessment of some spectra features that can best characterize the muscle dynamic properties such as average frequency, mean frequency, average value and the rms value of the MMG signal, and finally the important HF% ratio. The average and rms value of the MMG amplitude are known as features that represent the evolution of the MMG amplitude for whatever healthy or patient subject. The analysis of these parameters clearly illustrated that the frequency and amplitude of MMG signals are in linear relationship with force for both cases of subjects. We also found that the disparity of values of the rms MMG and its average are in the same order whereas for the average frequency it is smaller than the mean frequency for both groups. In addition, these two parameters are in linear relationship with force whereas those previously cited are in quadratic relationship with force. Then, we implemented the HF% ratio, which can serve as a good tool to assess the contribution of fast fibers as a peculiar indicator for affected muscles.

References [1] Z. Peng, F. Chu and Y. He, Vibration Signal Analysis and Feature Extraction based on Reassigned Wavelet Scalograms. Journal of Sound and Vibration, 253 (5), pp. 1087–1100, 2002. [2] F. Auger and P. Flandrin, Improving the Readability of Time-Frequency and Time-Scale Representations by the Reassignment Method. IEEE Transactions on Signal Processing, Vol. 43, N°5, pp. 1068–1089, 1995. [3] B. Gramatikov, J. Brinker, S. Y. Chun, N. V. Thakor, Wavelet analysis and time-frequency distributions of the body surface ECG before and after angioplasty. Elsevier Method and Programs in Biomedicine, 62, pp. 87–98, 2000. [4] I. Djurovic and L. Stankovic, Time-frequency representation based on the reassigned S-Method. J. of Signal Processing, Vol. 77, Issue 1, pp. 115– 120, 1999. [5] O. Rioul and P. Flandrin, Time-Scale Energy Distributions: a general class extending Wavelet Transforms. IEEE Trans. On Signal Processing, SP40(7), pp. 1746–1757, 1992. [6] P. Flandrin, E. Chassande-Mottin, P. Abry, Reassigned Scalograms and their fast Algorithms. Proceedings of the SPIE-95, 2569, pp.152-158, San Diego, USA, 1995. [7] J. Lin, Feature Extraction of Machine Sound using Wavelet and its Application in Fault Diagnosis. Elsevier NDT & E International 34, pp. 25– 30. [8] D. Barchiesi and T. Gharbi, Local spectral information in the near field with wavelet analysis and entropy. J. of Applied Optics, Vol. 38, N° 31, pp. 6587- 6596, 1999. [9] C. Li, C. Zheng and C. Tai, Detection of ECG characteristic points using Wavelet Transforms. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

300 Modelling in Medicine and Biology VIII [10] Leon Cohen, Time-frequency Distributions – A Review, Proceedings of the IEEE, Vol. 77, N°7, [11] T. Gharbi et al, Optical near field data analysis through time-frequency distributions application to characterization and separation of the image content by reassignment. J. of Optical Society of America, Vol.17, N 12, 2000. [12] C. Torrence and G.P. Compo, A Practical Guide to Wavelet Analysis. Bulletin of the American Meteorological Society, 61-78, Vol.79, 5, 1998. [13] J.C. Wood and D.T. Barry, Time-Frequency Analysis of Skeletal Muscle and Cardiac Vibrations. Proceedings of the IEEE Vol. 84, N 9, 1986. [14] Marie Farge, Wavelet Transforms and their Applications to Turbulence. Annual Review Fluid Mechanics, 24, pp.395–457, 1992. [15] G. Kaïser, A Friendly Guide to Wavelets (Book, Sixth printing), (1999) Library of Congress and Cataloguing, Printed by Quin Woodbine, Woodbine N.J. USA. [16] R. Polikar, Fundamental Concepts and an Overview of the Wavelet Theory, Wavelet tutorial (2002), Second edition, Engineering, Rowan, 2002, USA. [17] S.R. Perry et al, Mean Power Frequency and amplitude of the Mechanomyographic and Electromyographic Signals during incremental cycle Ergometry. J. of Electromyography and Kinesiology, 11 pp. 299–305, 2001. [18] P. Madeleine, P. Bajaj, K. Sognard and L.A. Nielsen, Mechanomyography and Electromyography force Relationships during Concentric and Eccentric Contractions. J. of Electromyography and Kinesiology 11, pp. 113–121, 2001. [19] W.A. Mackay, D.J. Gramond, H.C. Kwan and J.J. Murphay, Measurements of Human Forearm Viscoelasticity. Journal of Biomechanics, Vol 19, N°3, pp. 231–238, 1986, UK. [20] M. Ouamer, M. Boiteux, M. Petitjean, L. Travens, A. Salès, Acoustic myography during voluntary isometric contraction reveals non-propagative lateral vibration. Journal of Biomechanics 32, 1279–1285, 1999.

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Section 6 Virtual reality in medicine

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An internal examination training system supporting abnormal labor conditions A. Doi1, K. Noguchi1, K. Katamachi1, T. Ishii2, H. Uno3, S. Mega4 & K. Matsui4 1

Iwate Prefectural University, Japan The Japanese Red Cross Hokkaido College of Nursing, Japan 3 KOKEN Co., Ltd, Japan 4 JFP, Inc., Japan 2

Abstract For obstetricians and midwives, “internal examination” refers to an important diagnostic technique in which the progress of labor is examined using the index and middle fingers inserted into the vagina or rectum. Training of this internal examination technique has been commonly performed using a model of the human body (manikin). However, with this method, it was impossible to determine visually where and how the examining fingers are touching, making it difficult for trainers to teach advanced examination skills efficiently and evaluate training achievements. Against this background, we have developed a training system for internal examination that enables simulation of normal and abnormal conditions of labor by detecting the position and direction of the examining fingers in real-time via tactile and visual perceptions using anatomical and virtual models. This system allows trainees to experience both normal and abnormal fetal descent into the pelvis. Keywords: virtual models, manikin, magnetic sensor, internal examination, a training system.

1

Introduction

Previously developed training systems for internal examination include our own system [1–3], ePelvis [4–6] developed by a group from Stanford University, and the peripartum diagnosis/delivery assistance training system [7]. The ePelvis attaches several sensors inside the mother’s body and is not suitable for close WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090281

304 Modelling in Medicine and Biology VIII monitoring of fingers and the evaluation of examination techniques. The peripartum diagnosis/delivery assistance training system is basically a visual learning system using video images, and is suitable for teaching and explanation but not for training of the internal examination itself. Our approach in our training systems described in references [1–3] is different in comparison with the two systems, ePelvis and the peripartum diagnosis/delivery assistance training system. We utilize magnetic sensors that are attached with two fingers, and monitor the motions of the two fingers during an internal examination. Model Sensor

Location

MFC Wearing of the location sensors

Figure 1:

System configuration.

Our previous system was used to simulate the normal labor condition and thus was not suitable for simulating various abnormalities that can occur during labor. We thus developed a system based on fetal models that could reproduce various abnormal labor conditions (frank breech presentation, complete breech presentation, placenta previa, and face presentation). Our newly developed internal examination training system consists of models of maternal body parts, including the vagina and the uterine ostium, and fetal body parts (anatomical models), a personal computer (PC), and a magnetic sensor (Fig. 1). The magnetic sensor consists of a transmitter (XMTR), a location sensor, and a controller (miniBIRD, Ascension Technology Corp.). Magnetic information transmitted from the transmitter is detected by the location sensor, and the location information is received by the PC. The location sensor is attached and WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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fixed to a fingerstall made of silicon rubber, which is worn on the index and middle fingers, with which the internal examination is performed.

2

System overview

Figure 2 shows the external appearance of the training system. The system supports normal labor conditions of one-finger dilatation, two-finger dilatation, full dilatation and type C (another type of full dilatation) and abnormal labor conditions of frank breech presentation, complete breech presentation, placenta previa and face presentation. Other conditions can be added if necessary. The monitor displays geometric models (i.e. virtual models of the human body, examining fingers, etc.) of the same size as the anatomical models. An advantage of the system is that it can display on the monitor the position and the direction of the examining fingers during an internal examination via the magnetic sensors worn on the examining fingers. The system thus allows trainers to evaluate visually the skill level and accuracy of examination techniques on the monitor. Inside the human body model shown in Figure 2 is a guiding structure to install models of the pelvis and the fetus. An appropriate anatomical model of the fetus should be placed in the human body model before starting training. The skin in the virtual model was created by measuring the skin with a non-contact 3D digitizer, and converting this to a polygonal shape.

Figure 2:

Our internal examination training system.

The entire fetal models are created by a computer aided design (CAD) software. All generated polygonal models are combined, and both color and optical information are added. The anatomical models of the pelvis and the body were created as follows: a 3D image of the pelvis was taken with a 3D-computed tomography (CT) system, the 3D image was appropriately smoothed and isosurfaced, and the surface shape of the pelvis and the body was created. The anatomical model of the pelvis and the fetus was made by a rapid proto-typing device (3D printer). The skin of the mother’s body enclosing the pelvis was created by plastic material and vinyl cloth.

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3

Our training method

Eight anatomical models of the fetus, including four normal and four abnormal models, were prepared. Figure 3 shows four normal cases (left top: type C (near full dilation), right top: full dilation, left bottom: two-finger dilation, right bottom: one-finger dilation). Figure 4 shows four abnormal cases (left top: incomplete foot presentation, right top: complete breech presentation, left bottom: placenta previa, right bottom: face presentation). These models create only the parts that are touched by the examining fingers, and are displayed together with the whole image of the fetus. After a fetal model was placed in the system, the virtual model of the selected fetal model was selected on the dialogue on the display, and the anatomical model and the virtual model were adjusted. The position of both anatomical and virtual models is adjusted by using the location sensor. Figure 5 shows how a fetal model is displayed when it has been changed (before and after change). Figure 6 shows the dialogue for the selection of an anatomical model of a fetus (normal one-finger dilatation has been selected) and the control displays of each model (i.e. models of the mother’s skin, pelvis, vagina, and fingers). The display control allows the selection of the constitutive models to display, the mode of display (i.e. wireframe (display with lines) or shading), transparency, and the display color. The system can also display internal views with four cross-sectional images (left, right, front, and back views) (Fig. 7). Normal and abnormal conditions of different fetuses can be incorporated into the system by performing the following actions: 1) creating an anatomical model of the fetus; 2) measuring or modeling the anatomical model to create a virtual model; and 3) registering the new anatomical and virtual models into the system. The positions of the anatomical and virtual models can be adjusted automatically simply by indicating the zero point of a scale called “station” with an index finger pointer on the screen at the first application start-up.

Figure 3:

Anatomical models of fetus for normal cases.

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Figure 4:

Anatomical models of fetus for abnormal cases.

Fetal

Figure 5: Transparency ratio

model

Change of a fetal model. Display color Display type (Display mode)

Model type (Skin, Bone, Baby, Fingers)

Baby status

Display on/off

Figure 6:

Fetal model selection dialogue and display function.

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Skin of the mother displayed in wireframe model

(a) Figure 7:

4

(b)

(c)

(d)

Display function of the internal examination training system. (a) Cross-section: left view. (b) Cross-section: right view. (c) Crosssection: anterior view. (d) Cross-section: posterior view.

Fetal presentation

Fetal presentation refers to the orientation of the fetus in the uterus and is classified as “longitudinal presentation”, “transverse presentation,” or “oblique presentation”. Longitudinal presentation is further divided based on the position of the fetal head into the “head presentation” (i.e. the fetal head is facing down) and “pelvic presentation” (i.e. the fetal pelvis is facing down; also known as the breech position). All fetal presentations other than head position are considered abnormal. It is known that less than 5% of fetuses are delivered in breech position. Delivery of fetuses in breech position carries a higher risk of experiencing difficulty in pushing out the head and causing compression of the umbilical cord than does delivery of fetuses with their head facing down (i.e. head presentation). Breech position is roughly divided into frank breech presentation, full breech presentation, knee presentation, and foot presentation. The fetal presentation in which fetal feet are facing up and the hips facing down is referred to as the frank breech presentation, that in which both feet are facing down is referred to as complete breech presentation, and that in which only one foot is facing up is referred to as incomplete breech presentation. “Knee presentation” is the condition in which the fetal knee is flexed and facing down during delivery (i.e. the fetus descends with its knee presenting first), and is further divided into “complete knee presentation” (i.e. both knees facing down) and “incomplete knee presentation” (i.e. only one knee facing down). “Foot presentation” is the condition in which the fetus descends with its extended legs presenting first during delivery. Some breech babies can be delivered naturally through the vaginal canal if they are in frank breech presentation or full breech presentation. Breech babies in knee or foot

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presentation have less chance to be delivered naturally and are usually delivered by Caesarian section (cutting) [8].

5

System evaluation

We conduct a survey of eight students at our university, and make inquiries for our system, and evaluate it for understanding where the position of ischiadic thorn, baby head, and uterine ostium. We also check that it is easy to understand the rotation of baby head and the open of the uterine ostium. In the lecture of an internal examination, first we teach an internal examination in the classical method using a manikin only. Next, we teach them by using our system and make inquiries for our system with unregistered style. We check understanding of the students for several questions before and after looking images in our system. The result of several questions shows that the understanding of all students is improved by using our system [9].

Figure 8:

Fetal models of normal conditions (left: one-finger dilatation; right: two-finger dilatation).

Figures 8 and 9 show fetal models of normal conditions supported by the internal examination training system (i.e. one-finger dilatation, two-finger dilatation, type C (almost full dilatation), and full dilatation). Figures 10 and 11 show fetal models of abnormal conditions (incomplete foot presentation (frank breech presentation), complete breech presentation, placenta previa, and face presentation). We have been evaluating the usefulness of the present technique by disseminating its use as a learning tool and presenting it at international meetings and exhibitions. At the exhibition of the 21st Japan Academy of Midwifery Scientific Meeting (Oita prefecture in Japan, March 10 and 11, 2007), we presented and demonstrated our internal examination training system and a number of people experienced the training procedure and answered a questionnaire about the usefulness of the system. Over 100 people attended our presentation, approximately 60 of whom experienced the training procedure and provided feedback. Most of those who experienced the training system at the exhibition of the 21st Japan Academy of Midwifery Scientific Meeting provided positive WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

310 Modelling in Medicine and Biology VIII comments, although some requested improvements or the addition of functions. These requests included the addition of various abnormal labor conditions, more measurement functions (e.g. distance between two fingers and distance to the head of fetus), an automatic calibration function, and an examination rating function.

Figure 9:

Fetal models of normal conditions (left: type C; right: full dilatation).

Figure 10:

Fetal models of abnormal conditions (left: incomplete foot presentation; right: complete breech presentation).

We have satisfied one of the requests by developing 4 abnormal labor conditions (frank breech presentation, complete breech presentation, placenta previa and face presentation) and have included them in the latest training system. We have also added measurement functions using one or two fingers; the location sensors mounted on the index and middle fingers enable the acquisition of position information in space (i.e. spatial position and rotation). Using this measurement function, we can precisely measure spatial distances estimated on the basis of the distance between the two fingers or the measurer’s experience. In future studies, we aim to perform more detailed examinations of abnormal labor conditions. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 11:

6

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Fetal models of abnormal conditions (left: placenta previa; right: face presentation).

Conclusion

We have developed an internal examination training system that supports the learning of abnormal labor conditions. The system provides a unique learning experience that cannot be gained in daily practice. The use of the system in teaching practice will help to nurture competent obstetricians and midwives in a short period. On the basis of specialist feedback, we included 4 abnormal fetal models in the training system. The availability of additional abnormal fetal models will make the system more relevant to actual clinical situations. Although the present system was developed specifically for internal examination of the fetus, it can also be applied for such purposes as medical training and preoperative planning for other parts of the body (e.g. thorax and abdomen) by using different anatomical and virtual models.

References [1] Ishii, T., Doi, A., Katamachi, K., Noguchi, K., and Uno, H., Medical Training Device, Japan Patent Application No. 2005-032614, Reference No. 4-1130, Receipt No. 50500225450, Patent Applicant: KOKEN Co., Ltd. (August 2004) [2] Ishii, T., Doi, A., Katamachi, K., Noguchi, K., and Uno, H., Medical Training Device, International patent application (European Patent Office), Reference No. GP05-1033PCT, Receipt No. 50600235406, Application No. Notification: PCT/JP2006/30219, Patent Applicant: KOKEN Co., Ltd. [3] Doi, A., Matsui, K., Katamachi, K., Noguchi, K., Ishii, T., Uno, H., A computer assisted medical training system for checking status of delivery by using virtual reality technique and physical models, CARS2007, p. 156, 2007. [4] Carla M. Pugh, et al., The E-Pelvis: A Pelvic Examination Simulator, SUMMIT (Stanford University Medical Media & Information

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[5] [6] [7] [8] [9]

Technologies) Project Overview, http://summit.stanford.edu/publications/ tear_index.html SUMMIT (Stanford University Medical Media & Information Technologies, e-Pelvis, http://summit.stanford.edu/research/epelvis.html Shreve, J., ePelvis for simulation of gynecologic internal examination, http://hotwired.goo.ne.jp/news/technology/story/20020320307.html MC Medica Shuppan, 3D CG Peripartum Diagnosis/Delivery Assistance Training System, http://www.medica.co.jp/3d-bunben/#top Yajima, S., Nakano, H., and Taketani, Y., NEW Gynecologic Sciences, Nankodo; 2nd Edition (July 2004). Noguchi, K., and Ishii, T., Computer-assisted Educational Evaluation, Presentation at Japan Academy of Nursing Education, August 2, 2006.

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Development of a training system for interventional radiology M. Ide1 , Y. Fujii2 , B. Fujioka2, T. Komeda2 , H. Koyama2 , S. Yamamoto2 , M. Mohri3 & P. Beomonte Zobel4 1 Shibaura

Institute of Technology, Functional Control Systems, Japan Shibaura Institute of Technology, Systems Engineering, Japan 3 Mohri Hospital, Japan 4 University of L’Aquila,facolta di ingegneria, Laboratorio di Automazione a Fluido, Italy 2

Abstract The objective of the study reported here was to develop a master slave system for catheter-guided vascular surgery conducted by interventional radiology. By using a master slave system, the surgeon is not exposed to x-rays during the operation because the master tool managed by an operator is located away from the slave tool, which is near the patient. The system must provide vivid realism to the surgeon, particularly with regard to force information, because this surgery is performed in three dimensions while the surgeon watches a two-dimensional monitor. In this study, we developed a training system for a catheter guide in order to upgrade the surgeon’s skills because it is difficult to upgrade a master slave system without training. The system consists of a human interface device as the master tool, a control box, and a simulator. This training simulator is for the master slave system, which we developed. The master tool has a force display function using an electrorheological fluid. Two advantages of the fluid actuator are that it can be used without force feedback control and there is mechanical safety, as the surgeon does not experience any accidental force. An open loop control is used to achieve a simple mechanism and algorithm. Our results of preliminary experiments indicated that the output force achieved correlated with that sent from the PC. Three surgeons evaluated this training system under a variety of conditions. The operation of the master tool is simple. The thrust and rotation movements of the catheter can be handled instinctively and without complicated instructions. In addition, accurate force display, response, and stability were achieved with the electrorheological fluid. In the future, the training will need for a realistic depiction of interventional radiology, and the system provides accurate readings for aspiration and blood flow. Keywords: interventional radiology, training system, force feedback. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line) doi:10.2495/BIO090291

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1 Introduction Interventional Radiology (IR) is a minimally invasive surgery. Its use has increased in the last 20 years. IR requires a catheter approach through a blood vessel to the diseased area where the surgery is to be performed with the use of small tools. With this procedure, Digital Subtraction Angiography (DSA) is used to determine the position of the catheter within the body with a vascular contrast medium. DSA transmits a continuous x-ray that is used to determine the position of the catheter, and, as a result, medical personnel are exposed to x-rays even when protection is worn. The motivation to conduct this study was the need to develop a Master Slave System (MSS) for catheter-guided IR surgery through the blood vasculature. The MSS is required to prevent the exposure of medical personnel to x-rays. The operator manages a master tool at a remote location from the slave tool, which is placed near the patient and under the DSA device. The force display between the master and slave is important to warn the surgeons when a surgery error occurs. However, the force display is not only for patient safety. A surgeon can view the vasculature from within with the use of a monitor during the IR procedure; however, the picture is two-dimensional information, making it important to display the force in order to accurately estimate the image depth. In addition, the force display is used for error recovery as well as an assessment of the heartbeat, aspiration, and affected area. These innovations are used to achieve a higher degree of realism. We have developed an MSS for interventional radiology [1]. However, generally, an MSS is difficult to adopt unless training is provided. Specialized training is necessary before this system can be adopted. Training to use the system is a problem due to its cost. A training system has been developed with virtual reality for cost reduction [2]. Research for the training system has indicated that training for catheterization effectively enhances the skill of the surgeon [3]. Therefore, for the purposes of this study, we developed a catheter simulator to use for teaching an advanced technique for catheterization. The operation of catheter is possible to conduct the catheter movement in the twist and insertion direction [4]. During an IR procedure, the operator estimates the depth of the blood vessel while watching a monitor with two degrees of freedom. In this case, the force reflection is important to confirm the contact between the catheter and a blood vessel [5]. The force reflection is an advantage to the surgeon [6]. Therefore, the display of force assists the individual who places the catheter in the field of the blood vessel in three dimensions while viewing the monitor in two dimensions. As a result, patient safety is improved. There are some previous studies involving catheter training systems [7,8]. Most of these studies focus on catheter function and movement. However, none of the existing devices displays the force reflection or heartbeat [9], and there are no specific interfaces to be used with force feedback [10] or to demonstrate differences in the actual IR image [11].

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Recently, Mentice, Inc. (www.mentice.com) developed a Procedicus Vascular Intervention System Trainer (VISTTM ). This system, as a treatment tool, displays force in three dimensions and accurately depicts a patient’s ailment. The simulator, however, does not provide an accurate display of the heartbeat, blood flow, and aspiration. For this study, the training system was developed for an existing MSS [1]. Furthermore, although the device requires specific instruments, such as a computer board, it is generally assumed that it can be used in conjunction with a common PC connected with a universal serial bus (USB). In addition, with the use of an IR, it should be possible to assess the heartbeat, blood flow, and aspiration.

2 Training simulator A system overview is shown in Fig. 1. The configuration diagram of the haptic device, console box, and simulator is shown in Fig. 2. The haptic device is used to accurately recreate an IR procedure with two degrees of freedom for insertion and rotation. The catheter with a two-dimensional image is guided toward the target by the operator. The operators were able to sense heartbeat during the training sessions.

Figure 1: Overview of the training simulator. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 2: Diagram of the configuration of the haptic device, console box, and simulator image.

2.1 Haptic device [1] This haptic device, shown in Fig. 3, has a ball screw; it rotates in synchronicity with all components and identifies the position of the thrust movement with the use of an encoder. The electrorheological (ER) fluid is placed in a case with a disk that is connected to a ball screw. The disk rotates with the ball screw, and the operator senses the force because of the resistance from the rotation caused by the voltage given to the ER fluid. The ER fluid is functional. It changes the shear viscosity or dynamic viscous elasticity according to the supplied electric field. This is the principle generating the force feedback. The ball screw is supported by bearings, and there are two seals used to retain the ER fluid within the case. When using it, the operator grasps the outer frame and moves the nut of the ball screw forward and backward as the thrust movement. At this moment, a disk rotates within the case. If the ER fluid changes viscosity, the disk reacts to the friction created. The fluid prevents smooth movement, and the operator senses force and torque from the ball screw. In comparison to an electric motor, the manner of force display is simpler in this system. The operator receives force feedback when the thrust shafts rotate with the disk. However, the contact points on the ball screw are the ball bearings and the seals so that the friction is at a minimum for the performance of the force feedback. The sensor for position is equipped with two encoders to detect the thrust (360[ppr]) and rotation (360[ppr]) direction. The operator holds the nut of the ball screw with one hand and the outer frame with the other. The operator may conduct two movements. The axis of ball screw is directly connected to the encoder axis and to the disk for the force display. The WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 3: The haptic device has a ball screw that can rotate in synchronicity with all components and detect the position of the thrust movement using an encoder. The ER fluid is in a case with a disk that is connected to a ball screw. The disk rotates with the ball screw, and the operator can feel the force because of the resistance from the rotation caused by the voltage given to the ER fluid.

Figure 4: Volt-force curve of the haptic device.

weight of this device is about 200 [g], the length is 250 [mm], and the diameter is 48 [mm]. The volt-force curve is shown in Fig. 4. 2.2 Console box The console box distributes the commands from the haptic device and the simulator. The catheter on the simulator moves on the basis of the information received from the console box. Moreover, the force on the catheter in the simulator is defined in the program and is then sent to the console box so that the collision WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

318 Modelling in Medicine and Biology VIII force between catheter and the wall of the vessel and interference can be displayed to the operator. Generally, controls or measurements from a PC, such as those obtained from ER fluid, an encoder, or haptic devices, require specific instruments, such as a pulse counter or DA boards. However, the simulator considered here was intended for a general case because it is difficult to use a common PC due to its lack of the required instruments. This system can be connected to a PC with a USB port.

3 Modeling of IR 3.1 Modeling of catheter and wall of vessel A dynamic model for simulation is shown in Fig. 5. The blood vessel assumes a rigid body. When the catheter collides, the wall becomes deformed in the direction of insertion. The tip of the catheter is bent in order to determine the direction for the branch connection of the vessel. Furthermore, the catheter can also be deformed when it collides with a vessel. For realizing this condition, the catheter constructed discrete model that is connected contact point of rigid stick. 3.2 Collision force and simulator image The collision force generally consists of elasticity and viscosity. However, this haptic device uses ER fluid for force reflection so that only viscosity was expressed. Only if viscosity reflection causes, the operator can sense the collision so that the patient’s security can preserve because it is a passive force reflection. The image used is an angiographic picture [12]. The image was displayed on the simulation monitor with a texture mapping method of OpenGL.

Figure 5: Dynamic simulation model. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 6: Principle of collision detection. 3.3 Collision detection The coordinates of the parts of the wall of the vessel face on the image were extracted for collision detection. One of the links of catheter was assumed to be a segment of a line. The wall of the vessel was assumed to be a flat surface. The image of collision detection is shown in Fig. 6. The collision is detected when one joint point put front side and when other joint point put behind side as condition of collision. The joint is indicated by a point, P0 , on the vessel wall; the normal vector of the vessel wall is n; and the points at both ends are P1 , P2 . The collision detection is circulated from the inner product of P0 P1 , P0 P2 , and vector n. The condition equation is shown as Eq. (1). When a collision occurs, the angle of P0 P1 and vector n will be blunt. The value of the inner product is negative. On the other hand, the angle of P0 P2 and vector n will be sharp so that the value of the inner product is positive. When this value is negative, a collision is detected; when it is positive, no collision is detected. −−−→ → −−−→ − → n · P0 P1 × − n · P0 P2 ≤ 0

(1)

3.4 Catheter inflection Although a catheter is generally made of a flexible material, such as nylon or polyurethane, in this study, the catheter model is a polyarticular link mechanism capable of inflection for simplification. As shown in Fig. 7, the operator inserts the catheter in the direction of insertion, and the simulator detects the collision. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 7: Catheter inflection.

Figure 8: View of the calculation of the catheter inflection. The program then calculates the inflection angle of the catheter using collision force. Moreover, the reaction force is calculated by acceleration of the catheter. The acceleration is calculated by a second-order differential of displacement of the catheter. The catheter mass is mc ; the displacement when there is a collision of the catheter is Xc; the reaction force Fcw is shown in Eq. (2). The mass of the catheter is defined as 10 [g]. As shown in Fig. 8, the inflection angle of the catheter is K[deg]; the reaction force by collision is Fcw; the catheter length is lc; the longitudinal elastic modulus of the catheter is Ec; and the geometrical moment of inertia is Ic. The angle of the inclination slope of the cantilever equation is shown in Eq. (3). WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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Figure 9: Vascular bifurcation.

Fcw




d2 X c

=
mc 2

dt

(2)

FcW l2 cos ∅ 2Ec Ic

(3)

K=

Link ab of the tip of the catheter revolves A [deg] around joint b. Link bc revolves B [deg] around joint c. Link cd revolves C [deg] around joint d. After revolution, the detection repeats to the next collision. The guide on the vascular bifurcation is possible, as shown in Fig. 9.

4 Surgeon’s evaluation Three surgeons evaluated the training simulator under a variety of conditions. The haptic device was easy to use. The thrust and the rotation of the catheter could be handled instinctively by the surgeon without complicated instructions. The surgeons could use a suitable thrust velocity of the catheter because it could be controlled with the gain. When the catheter came in contact with the vascular wall, the surgeons could evaluate the pressure with the haptic device.

5 Conclusions For this study, a training system was developed for an existing MSS. Moreover, a console box and a simulation program with a connected haptic device were developed. This system can be connected to a PC with a USB port. Its usefulness was assessed by surgeons. In the future, a simulator capable of more detail would enhance the practicality of the system.

Acknowledgements The author would like to thank ERtec and the Brain Science and Life Technology Research Foundation. WIT Transactions on Biomedicine and Health, Vol 13, © 2009 WIT Press www.witpress.com, ISSN 1743-3525 (on-line)

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References [1] Masaru Ide et al.: “Development of a master slave system for interventional radiology”, International Journal of Computer Assisted Radiology and Surgery, Volume 3, Supplement 1, pp. 343, 2008 [2] Kostas Vlachos, Evangelos Papadopoulos, Senior Member,Design and Implementation of a Haptic Device for Training in Urological Operations, IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 19, NO. 5, pp. 801–809 [3] Scott A. Engum, et al.:Intravenous catheter training system: Computerbased education versus traditional learning method The American Journal of surgery 186, pp. 67–74, 2003 [4] F. Arai, R. Fujimura, T. Fukuda, and M. Negoro: “New Catheter Driving Method Using Linear Stepping Mechanism for Intravascular Neurosurgery,” Proc. of the IEEE Int. Conf. on Robotics and Automation, pp. 2944–2949, 2002 [5] M.Tanimoto, F.Arai, T. Fukuda: “Force Display Method for Intravascular Neurosurgery,” Proc. of the IEEE SMC ’99 Conf., Vol. 4, pp. 1032–1037, 1999 [6] Christopher R. Wagner, Robert D.Howe: “Force Feedback Benefit Depends on Experience in Multiple Dgree of Freedom Robotic Surgery Task,” IEEE Trans on Robotics, Vol. 23, No. 6, pp. 1235–1240, 2007 [7] Julien Lenoir, Stephane Cotin, at al., Interactive physically-based simulation of catheter and guidewire, Computers & Graphics 30, pp. 416–422, 2006 [8] Jan Egger, et al., A Fast Vessel Centerline Extraction Algorithm for Catheter Simulation, Twentieth IEEE Int. Symp. on Computer- Based Medical Systems, pp. 177–182 [9] W. Lawton, et al. Tubes in Tubes: Catheter Navigation in Blood Vessels and its Applications, International Journal of Solids and Structures, Vol.37, Issue 22, pp. 3031–3054, 2000 [10] Suraj Bhat, et al., A physically-based model for guidewire simulation on patient-specfic data, International Congress Series, Vol. 1281, pp. 479–484, 2005 [11] Y.Y.Cai, et al., Tactile VR for hand-eye coordination in simulated PTCA, Computers in Biology and Medicine, Vol. 36, pp. 167–180, 2006 [12] DEAGOSTINI FInside Human BodyCUNIT75

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Author Index Accardo A.................................. 39 Ahmed S. ................................... 83 Akhmadeev N. ......................... 191 Alrashdan K............................. 181 Alrashidi M.............................. 181 Antonelli M. G......................... 159 Averbach M. .............................. 17 Bărbat T. .................................... 93 Basak T. K. .............................. 247 Beomonte Zobel P. .......... 159, 313 Bernad E. ................................... 93 Bernad S. I. ................................ 93 Bhattacharya K. ....................... 247 Bignardi C................................ 149 Borhan A.................................. 123 Bui A.......................................... 83 Cavka D. .................................. 203 Cescotto S. ............................... 133 Chutakositkanon C..................... 17 Corbi G. ..................................... 39 Costanzo G. ..................... 149, 159 Cyril Raj V. ................................247 D’Addio G. ................................ 39 Dewals B. J. ............................. 133 Dickman S. .............................. 235 Doi A. ...................................... 303 Durdle N. ................................. 257 Eitel G...................................... 103 Enomoto Y................................. 71 Erpicum S. ............................... 133 Esat İ. ....................................... 181 Ferrara N.................................... 39 Fujii Y...................................... 313 Fujioka B. ................................ 313 Gastaldi L. ............................... 223 Gharbi T................................... 287 Gliozzi A. S. ............................ 267

Guiot C. ................................... 267 Gunasekaran G. ......................... 247 Halder S. .................................. 247 Houacine A.............................. 287 Hyre M. R.................................. 27 Ide M. ...................................... 313 Imai Y........................................ 49 Ishii T....................................... 303 Ishikawa T. ................................ 49 Katamachi K. ........................... 303 Keshavarzi B. .......................... 123 Kiss R. M................................. 171 Ko J.......................................... 235 Komeda T. ............................... 313 Kondo H. ................................... 49 Koyama H................................ 313 Kudo T....................................... 71 Kumar A. ................................. 257 Lai C.-H................................... 277 Li V. W.................................... 235 Liffman K. ................................. 83 Liu L. ....................................... 277 Lu H.-w.................................... 277 Macpherson A. K....................... 17 Macpherson P. A. ...................... 17 Manasseh R. .............................. 83 Matsui K. ................................. 303 McGilvray K. C.......................... 57 Mega S..................................... 303 Meinke M. ............................... 103 Mekaoui S................................ 287 Miftahof R. .............................. 191 Mihalj M.................................. 203 Mohri M................................... 313 Murugappan S. ........................ 247 Neti S. ........................................ 17 Noguchi K. .............................. 303

324 Modelling in Medicine and Biology VIII Paolo Delsanto P...................... 267 Parameswaran S....................... 115 Pastorelli S............................... 223 Paulus R................................... 133 Pirotton M................................ 133 Poljak D. .................................. 203 Pulliam R. M.............................. 27 Puttlitz C. M. ............................. 57 Raimondi P. ............................. 159 Raj R. ....................................... 115 Ramieri A. ............................... 149 Raparelli T. .............................. 159 Ravi T. ........................................247 Rengo F...................................... 39 Sarkar R. .................................... 57 Schröder W. ............................. 103 Sesnic S.................................... 203 Shaw P. .......................................247 Shimano K. ................................ 71 Shoucri R. M................................ 3

Smirnov S. ............................... 115 Sorli M..................................... 223 Squire J. C. ................................ 27 Suaste E. .................................. 213 Susan-Resiga R.......................... 93 Šutalo I. D.................................. 83 Tan J. ....................................... 115 Terán O.................................... 213 Titlic M.................................... 203 Ultman J................................... 123 Uno H. ..................................... 303 Valleru N. ................................ 115 Xie F. ....................................... 277 Yamaguchi T. ............................ 49 Yamamoto S. ........................... 313 Yıldız İ..................................... 181 Zhou S.-d. ................................ 277

...for scientists by scientists

Human Respiration Anatomy and Physiology, Mathematical Modeling, Numerical Simulation and Applications Edited by: V. KULISH, Nanyang Technological University, Singapore

Books on human respiration are usually written either by physicians or engineers. The product of close collaboration between both, this volume presents the latest developments and major challenges in the area of biomedical engineering concerned with studies of the human respiratory system. The contributors cover the anatomy and physiology of human respiration, some of the newest macro and microscopic models of the respiratory system, numerical simulation and computer visualisation of gas transport phenomena, and applications of these models to medical diagnostics, treatment and safety. Series: Advances in Bioengineering, Vol 3 ISBN: 1-85312-944-5 2006 240pp £95.00/US$165.00/€139.00 eISBN: 978-1-84564-228-0

Modelling in Medicine and Biology VII Incorporating Environmental Electromagnetic Fields Edited by: C.A. BREBBIA, Wessex Institute of Technology, UK

Projections for advances in medical and biological technology will transform medical care and treatment. This is in great part due to the results of interaction and collaborations between the medical sciences and

engineering. These advances will result in substantial progressions in health care and in the quality of life of the population. Computer models in particular have been increasingly successful in simulating biological phenomena. These are lending support to many applications, including amongst others cardiovascular systems, the study of orthopaedics and biomechanics and electrical simulation. Another important contribution, due to the wide availability of computational facilities and the development of better numerical algorithms, is the ability to acquire, analyse, manage and visualise massive amounts of data. Featuring contributions from the Seventh International Conference on Modelling in Medicine and Biology, this book features a broad range of topics which will be of particular interest to medical and physical scientists and engineers interested in the latest developments in simulations in medicine. Topics include: Cardiovascular Systems; Biomechanics; Computational Fluid Dynamics; Intracranial Pressure Dynamics; Exposure to Electromagnetic Fields; Skin and Membranes; Data Acquisition and Analysis and Computer Simulation. WIT Transactions on Biomedicine and Health, Vol 12 ISBN: 978-1-84564-089-7 2007 352pp £116.00/US$232.00/€174.00 eISBN: 978-1-84564-295-2

WITPress Ashurst Lodge, Ashurst, Southampton, SO40 7AA, UK. Tel: 44 (0) 238 029 3223 Fax: 44 (0) 238 029 2853 E-Mail: [email protected]

...for scientists by scientists

Computational Modeling of Tissue Surgery

Wall–Fluid Interactions in Physiological Flows

Edited by: M.E. ZEMAN, Katholieke Universiteit Leuven, Belgium and M. CERROLAZA, Universidad Central de Venezuela, Venezuela

Edited by: M.W. COLLINS, London South Bank University, UK, G. PONTRELLI, C.N.R., Istituto per le Applicazioni del Calcolo, Italy and M.A. ATHERTON, London South Bank University, UK

In recent decades, the evolution of computational modeling has been primarily determined by the increasing power and speed of data transfer, visualization and other technological tools. This positive impact has inspired a new approach to study the human body and its structures. Written by respected researchers from a range of disciplines, this book provides valuable information on different methods of modeling, simulation and analysis of hard and soft tissues. These techniques aim to develop tools that can offer a meaningful input to the medical practice. The applications of computational modeling in biomechanics are vast, with many different trends having been developed worldwide. This book reviews the latest research on a selection of key issues. Series: Advances in Bioengineering, Vol 1 ISBN: 1-85312-749-3 2005 288pp £107.00/US$189.00/€159.00

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All fluid flow problems in the human body involve interaction with the vessel wall. This volume presents a number of studies where primarily mathematical modelling has been applied to a variety of medical wall–fluid interaction problems. The medical applications discussed are highly varied, while some key clinical areas are also addressed. Unusually, a number of important medical challenges involving fluid flow are considered in combination with the relevant solid mechanics. For the researcher this book offers new scope for developing and demonstrating a mastery of the scientific principles involved. Partial Contents: Numerical Simulation of Arterial Pulse Propagation Using OneDimensional Models; Modelling the Reopening of Liquid-Lined Lung Airways; A Finite-Volume Model of the Guldner ‘Frogger’ – A Training Device for Skeletal Muscle in Cardiac Assist Use Both in Training Mode and Coupled to a Ventricular Assist Device; Geometric Constraints in the Feto-Placental Circulation – Umbilical Cord Coiling and Ductus Venosus Dilation; Numerical Modelling of Blood Flow in a Stented Artery. Series: Advances in Computational Bioengineering, Vol 6 ISBN: 1-85312-899-6 2004 204pp £85.00/US$129.00/€125.00

...for scientists by scientists

Modelling in Medicine and Biology VI

Medical Applications of Computer Modelling

Edited by: M. URSINO, University of Bologna, Italy, C.A. BREBBIA, Wessex Institute of Technology, UK, G. PONTRELLI, Istituto per le Applicazioni del Calcolo – C.N.R., Italy and E. MAGOSSO, University of Bologna, Italy

Volume 1 – Cardiovascular and Ocular Systems

Mathematical models and computer simulation techniques have been playing an increasingly important role in biological and medical research during the last decades and their impact is certainly going to increase further in future years. Featuring contributions from the Sixth International Conference on Modelling in Medicine and Biology, this book covers a broad spectrum of topics including: Simulation of Physiological Processes; Cardiovascular System; Neural Systems; Biomechanics; Computational Fluid Mechanics in Biomedicine; Orthopaedics and Bone Mechanics; Simulations in Surgery; Advanced Technology in Dentistry; Data Acquisition and Analysis; and Image Processing. WIT Transactions on Biomedicine and Health, Vol 8 ISBN: 1-84564-024-1 2005 624pp £239.00/US$425.00/€355.50

Volume 2 – The Respiratory System Edited by: T.B. MARTONEN, National Health & Environmental Effects Research Laboratory, and University of North Carolina, USA “...clearly demonstrates how the rigorous application of computational mathematics can illuminate problems in medicine and biology, and contains excellent examples.” PHYSIOLOGICAL MEASUREMENT

reviewing Volume 1 “…covers most of the important topics, and describes how the current use of computer modeling advances our knowledge of aerosolized drug delivery.” JOURNAL OF AEROSOL MEDICINE

reviewing Volume 2

Designed for use in both academic and research environments, these books address applications of computer modelling and fluid dynamics to biological systems. Series: Advances in Computational Bioengineering, Vols 3 & 4 SET ISBN: 1-85312-809-0 2001 £230.00/US$339.00/€345.00

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