Modelling Organs, Tissues, Cells and Devices: Using MATLAB and COMSOL Multiphysics

504 Pages·2017·17.293 MB·English

by  Socrates Dokos (auth.)

Lecture Notes in Bioengineering
Socrates Dokos
Modelling Organs,
Tissues, Cells and
Devices
Using MATLAB and COMSOL
Multiphysics
Lecture Notes in Bioengineering
More information about this series at http://www.springer.com/series/11564
Socrates Dokos
Modelling Organs, Tissues,
Cells and Devices
Using MATLAB and COMSOL Multiphysics
123
SocratesDokos
GraduateSchoolofBiomedicalEngineering
University of NewSouthWales
Sydney
Australia
ISSN 2195-271X ISSN 2195-2728 (electronic)
Lecture Notesin Bioengineering
ISBN978-3-642-54800-0 ISBN978-3-642-54801-7 (eBook)
DOI 10.1007/978-3-642-54801-7
LibraryofCongressControlNumber:2017930790
©Springer-VerlagBerlinHeidelberg2017
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Theregisteredcompanyaddressis:HeidelbergerPlatz3,14197Berlin,Germany
To my wonderful Anna for her selfless
support, inspiration, wisdom, and
encouragement over a lifetime in academia;
and to our children Vasilia, Elias, George,
Timothy, Catherine and Raphael, who make
us proud beyond words.
Preface
Computational modeling plays an increasingly important role in biological and
medical research, as well as in the medical device industry. Like other industries,
successful development of medical devices and implants requires not only exten-
sive testing (bench tests, animal experiments and human trials), but also extensive
computational simulations which allow engineers to cost-effectively investigate
system behavior and iterate the device design. These approaches are inherently
multidisciplinary: biomedical engineers are required to understand not only the
fundamental physical principles of their devices, but also how they interact with
complex physiological systems at the cellular, tissue, and whole-organ levels.
Computational models of such systems are typically multiphysics in nature. In the
not too distant future, bioengineering modeling will also be used routinely in the
clinic to tailor a range of individual therapies and treatment strategies based on
patient-specific models.
It is evident that computational modeling is an important skill for biomedical
engineers, and should form an indispensable component of modern curricula in
biomedicalengineering.Thisbookgrewfrommyowncourse,‘ModellingOrgans,
Tissues and Devices’, taught to late-stage undergraduate and postgraduate engi-
neering students at the Graduate School of Biomedical Engineering, UNSW. It
covered a broad range of modeling topics, from electrical stimulation of tissues
through to diffusion, biomechanics, heat transfer and fluid dynamics. Numerical
softwareusedforteachingthecoursewereMATLABandCOMSOLMultiphysics—
theformerduetoitsprevalentuseinengineeringandthelatterduetoitsmultiphysics
capabilities and appealing user interface. For their major project, students were
required to submit a COMSOL model in any bioengineering field of their choice.
Despite the popularity of the course, it became apparent there was no single book
whichcouldbeusedasrecommendedtextwhichcoveredtherangeoftopicstaught,
includingnumerical/analyticalmethodsforsolvingODE/PDEs,anoverviewofthe
variousmultiphysicsprinciplesinvolved, aswell ashowsuchmodels canbeprac-
ticallyimplementedinMATLABandCOMSOL.Hence,theideaforthisbookwas
born!
vii
viii Preface
Thetextisdividedintotwoparts:PartAcoversbasicmodelingconceptsaswell
as analytical and numerical methods for solving ODE/PDE systems. Part B covers
specific physics applications in bioengineering. Each chapter also includes a set of
problems,over50intotal, with detailed answers providedinthesolutionssection,
along with several worked-examples with code listings throughout the main text
itself. Appendices A and B provide an introduction to MATLAB and COMSOL,
respectively.
All models in the text were solved on a base-model MacBook Air 2013 laptop
with8GRAM.Someofthemodelstooktensofminutestosolve,whilstotherstook
only a few seconds. The MATLAB version employed was R2014b, and the
COMSOLversionwas5.2(thelatestatthetimeofwriting).Mostofthesemodels
could be solved using the standard base packages of MATLAB and COMSOL,
whilstothermodelsrequiredoptionaladd-onssuchasMATLAB’sSymbolicMath
Toolbox or COMSOL’s Nonlinear Structural Materials Module. Such instances
whenever they occur arementioned in thetext.
Finally, I wish to acknowledge my family for their unwavering support
throughout the four long years it took to complete this text, as well the encour-
agement and support from my colleagues at UNSW. A special mention to
Jan-Philip Schmidt, my editor at Springer Verlag, for his patience and encour-
agement, as well as my many students and postdocs, past and present, whose
MATLAB and/or COMSOL modeling works have inspired many examples
throughout this text. They include Amr Al Abed, Siwei Bai, Tianruo Guo,
MiganooshAbramian,KhalidAlonazi,SinišaSovilj,YousefAlharbi,AzamBakir,
Chin Neng Leong, Bee Ting Chan, Abdulrahman Alqahtani, Mitra Mohd Addi,
Shuhaida Yahud, Shijie Yin, Ben Hui, David Chan and Einly Lim. I have thor-
oughly treasured and learnt much from our interactions.
Sydney, Australia Socrates Dokos
May 2016
Contents
Part I Bioengineering Modelling Principles, Methods and Theory
1 Introduction to Modelling in Bioengineering... .... .... ..... .... 3
1.1 Modelling and Simulation in Medicine and Biology .. ..... .... 3
1.2 The Modelling Process..... .... .... .... .... .... ..... .... 4
1.3 Mathematical Model Types . .... .... .... .... .... ..... .... 5
1.3.1 Linear Versus Non-linear . .... .... .... .... ..... .... 6
1.3.2 Dynamic Versus Static ... .... .... .... .... ..... .... 7
1.3.3 Deterministic Versus Stochastic .... .... .... ..... .... 7
1.3.4 Continuous Versus Discrete ... .... .... .... ..... .... 9
1.3.5 Rule-Based ... ..... .... .... .... .... .... ..... .... 12
1.4 Dimensional Analysis. ..... .... .... .... .... .... ..... .... 16
1.4.1 Dimensions and Units.... .... .... .... .... ..... .... 16
1.4.2 Buckingham p-Theorem .. .... .... .... .... ..... .... 19
1.5 Model Scaling .. .... ..... .... .... .... .... .... ..... .... 21
Problems... .... .... .... ..... .... .... .... .... .... ..... .... 23
References.. .... .... .... ..... .... .... .... .... .... ..... .... 27
2 Lumped Parameter Modelling with Ordinary Differential
Equations.. .... .... .... ..... .... .... .... .... .... ..... .... 29
2.1 Overview of Ordinary Differential Equations .... .... ..... .... 29
2.2 Linear ODEs.... .... ..... .... .... .... .... .... ..... .... 31
2.3 ODE Systems... .... ..... .... .... .... .... .... ..... .... 35
2.3.1 Example Model 1: Cardiac Mechanics ... .... ..... .... 37
2.3.2 Example Model 2: Hodgkin–Huxley Model of Neural
Excitation .... ..... .... .... .... .... .... ..... .... 42
2.4 Further Reading . .... ..... .... .... .... .... .... ..... .... 46
Problems... .... .... .... ..... .... .... .... .... .... ..... .... 46
References.. .... .... .... ..... .... .... .... .... .... ..... .... 53
ix
x Contents
3 Numerical Integration of Ordinary Differential Equations..... .... 55
3.1 Taylor’s Theorem.... ..... .... .... .... .... .... ..... .... 55
3.2 One-Step Methods ... ..... .... .... .... .... .... ..... .... 60
3.2.1 Backward-Euler Method.. .... .... .... .... ..... .... 63
3.2.2 Trapezoidal Method . .... .... .... .... .... ..... .... 65
3.2.3 Runge–Kutta Methods.... .... .... .... .... ..... .... 66
3.2.4 The Generalized-a Method .... .... .... .... ..... .... 73
3.3 Multistep Methods ... ..... .... .... .... .... .... ..... .... 82
3.3.1 Predictor-Corrector Methods... .... .... .... ..... .... 86
3.3.2 Backward Differentiation Formulas.. .... .... ..... .... 93
3.3.3 Numerical Differentiation Formulas.. .... .... ..... .... 96
3.4 ODE Solver Implementations in Matlab and COMSOL..... .... 97
3.5 Further Reading . .... ..... .... .... .... .... .... ..... .... 100
Problems... .... .... .... ..... .... .... .... .... .... ..... .... 100
References.. .... .... .... ..... .... .... .... .... .... ..... .... 103
4 Distributed Systems Modelling
with Partial Differential Equations... .... .... .... .... ..... .... 105
4.1 Modelling with PDEs. ..... .... .... .... .... .... ..... .... 105
4.1.1 The Gradient.. ..... .... .... .... .... .... ..... .... 105
4.1.2 The Divergence..... .... .... .... .... .... ..... .... 108
4.1.3 The Curl. .... ..... .... .... .... .... .... ..... .... 112
4.1.4 The Divergence Theorem . .... .... .... .... ..... .... 113
4.1.5 Conservation Law Formulation. .... .... .... ..... .... 117
4.1.6 The Laplacian. ..... .... .... .... .... .... ..... .... 119
4.1.7 PDE Boundary Conditions .... .... .... .... ..... .... 120
4.2 Basic Analytical and Numerical Solution Techniques.. ..... .... 123
4.2.1 Separation of Variables... .... .... .... .... ..... .... 123
4.2.2 Finite Difference Method . .... .... .... .... ..... .... 139
4.2.3 Method of Lines.... .... .... .... .... .... ..... .... 147
4.3 Further Reading . .... ..... .... .... .... .... .... ..... .... 153
Problems... .... .... .... ..... .... .... .... .... .... ..... .... 153
References.. .... .... .... ..... .... .... .... .... .... ..... .... 157
5 The Finite Element Method .... .... .... .... .... .... ..... .... 159
5.1 Finite Elements for 1D Systems.. .... .... .... .... ..... .... 159
5.1.1 Weak Form PDE Equivalent... .... .... .... ..... .... 160
5.1.2 Basis Function Approximation . .... .... .... ..... .... 164
5.1.3 Higher-Order Basis Functions.. .... .... .... ..... .... 175
5.2 Finite Elements for 2D/3D Systems ... .... .... .... ..... .... 179
5.2.1 Weak Form Description .. .... .... .... .... ..... .... 180
5.2.2 Basis Function Approximation . .... .... .... ..... .... 183