Dokos Modelling Organs, Tissues, Cells and Devices

Using MATLAB and COMSOL Multiphysics
1. Auflage 2017
ISBN: 978-3-642-54801-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 504 Seiten

Reihe: Lecture Notes in Bioengineering

This book presents a theoretical and practical overview of computational modeling in bioengineering, focusing on a range of applications including electrical stimulation of neural and cardiac tissue, implantable drug delivery, cancer therapy, biomechanics, cardiovascular dynamics, as well as fluid-structure interaction for modelling of organs, tissues, cells and devices. It covers the basic principles of modeling and simulation with ordinary and partial differential equations using MATLAB and COMSOL Multiphysics numerical software. The target audience primarily comprises postgraduate students and researchers, but the book may also be beneficial for practitioners in the medical device industry.

Autoren/Hrsg.

Dokos, Socrates

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;Acronyms;13
4;Part I Bioengineering Modelling Principles, Methods and Theory;14
5;1 Introduction to Modelling in Bioengineering;15
5.1;1.1 Modelling and Simulation in Medicine and Biology;15
5.2;1.2 The Modelling Process;16
5.3;1.3 Mathematical Model Types;17
5.3.1;1.3.1 Linear Versus Non-linear;18
5.3.2;1.3.2 Dynamic Versus Static;19
5.3.3;1.3.3 Deterministic Versus Stochastic;19
5.3.4;1.3.4 Continuous Versus Discrete;21
5.3.5;1.3.5 Rule-Based;24
5.4;1.4 Dimensional Analysis;28
5.4.1;1.4.1 Dimensions and Units;28
5.4.2;1.4.2 Buckingham -Theorem;31
5.5;1.5 Model Scaling;33
5.6;References;39
6;2 Lumped Parameter Modelling with Ordinary Differential Equations;41
6.1;2.1 Overview of Ordinary Differential Equations;41
6.2;2.2 Linear ODEs;43
6.3;2.3 ODE Systems;47
6.3.1;2.3.1 Example Model 1: Cardiac Mechanics;49
6.3.2;2.3.2 Example Model 2: Hodgkin--Huxley Model of Neural Excitation;54
6.4;2.4 Further Reading;58
6.5;References;65
7;3 Numerical Integration of Ordinary Differential Equations;66
7.1;3.1 Taylor's Theorem;66
7.2;3.2 One-Step Methods;71
7.2.1;3.2.1 Backward-Euler Method;74
7.2.2;3.2.2 Trapezoidal Method;76
7.2.3;3.2.3 Runge--Kutta Methods;77
7.2.4;3.2.4 The Generalized-? Method;84
7.3;3.3 Multistep Methods;93
7.3.1;3.3.1 Predictor-Corrector Methods;97
7.3.2;3.3.2 Backward Differentiation Formulas;104
7.3.3;3.3.3 Numerical Differentiation Formulas;107
7.4;3.4 ODE Solver Implementations in Matlab and COMSOL;108
7.5;3.5 Further Reading;111
7.6;References;114
8;4 Distributed Systems Modelling with Partial Differential Equations;116
8.1;4.1 Modelling with PDEs;116
8.1.1;4.1.1 The Gradient;116
8.1.2;4.1.2 The Divergence;119
8.1.3;4.1.3 The Curl;123
8.1.4;4.1.4 The Divergence Theorem;124
8.1.5;4.1.5 Conservation Law Formulation;128
8.1.6;4.1.6 The Laplacian;130
8.1.7;4.1.7 PDE Boundary Conditions;131
8.2;4.2 Basic Analytical and Numerical Solution Techniques;134
8.2.1;4.2.1 Separation of Variables;134
8.2.2;4.2.2 Finite Difference Method;150
8.2.3;4.2.3 Method of Lines;158
8.3;4.3 Further Reading;164
8.4;References;168
9;5 The Finite Element Method;169
9.1;5.1 Finite Elements for 1D Systems;169
9.1.1;5.1.1 Weak Form PDE Equivalent;170
9.1.2;5.1.2 Basis Function Approximation;174
9.1.3;5.1.3 Higher-Order Basis Functions;185
9.2;5.2 Finite Elements for 2D/3D Systems;189
9.2.1;5.2.1 Weak Form Description;190
9.2.2;5.2.2 Basis Function Approximation;193
9.3;5.3 FEM Numerical Implementation;200
9.3.1;5.3.1 Assembly of System Matrices;201
9.3.2;5.3.2 Gaussian Quadrature;202
9.3.3;5.3.3 Non-Linear Systems;204
9.4;5.4 Further Reading;205
9.5;References;207
10;Part II Bioengineering Applications;208
11;6 Modelling Electrical Stimulation of Tissue;209
11.1;6.1 Electrical Stimulation;209
11.1.1;6.1.1 Maxwell's Equations;209
11.1.2;6.1.2 Electrostatic Formulations;211
11.1.3;6.1.3 Volume Conductor Theory;212
11.1.4;6.1.4 Example: Cell Culture Electric Field Stimulator;215
11.1.5;6.1.5 Example: Access Resistance of Electrode Disc;218
11.2;6.2 Modelling Electrical Activity of Tissues;223
11.2.1;6.2.1 Continuum Models of Excitable Tissues;223
11.2.2;6.2.2 Example: Modelling Spiral-Wave Reentry in Cardiac Tissue;225
11.2.3;6.2.3 Modelling PDEs/ODEs on Boundaries, Edges and Points;233
11.2.4;6.2.4 Example: Axonal Stimulation Using Nerve Cuff Electrodes;234
11.3;6.3 Further Reading;242
11.4;References;243
12;7 Models of Diffusion and Heat Transfer;244
12.1;7.1 Diffusion;244
12.1.1;7.1.1 Fick's Laws of Diffusion;244
12.1.2;7.1.2 Example: Diffusion and Uptake into a Spherical Cell;245
12.1.3;7.1.3 Convective Transport;248
12.1.4;7.1.4 Example: Drug Delivery in a Coronary Stent;249
12.2;7.2 Heat Transfer;254
12.2.1;7.2.1 Heat Conduction and Convection;255
12.2.2;7.2.2 The Bioheat Equation;257
12.2.3;7.2.3 Example: RF Atrial Ablation;258
12.3;7.3 Further Reading;265
12.4;References;267
13;8 Solid Mechanics;269
13.1;8.1 Biomechanics;269
13.2;8.2 Tensor Fundamentals;269
13.2.1;8.2.1 Tensor Definition;269
13.2.2;8.2.2 Indicial Notation;271
13.2.3;8.2.3 Tensor Transformation Law;272
13.2.4;8.2.4 Tensor Invariants;274
13.3;8.3 Mechanics Principles;277
13.3.1;8.3.1 Stress;277
13.3.2;8.3.2 Strain;281
13.4;8.4 Linear Elasticity;287
13.4.1;8.4.1 Example: Detecting Tension in a Respirator Strap;290
13.5;8.5 Linear Viscoelasticity;296
13.6;8.6 Hyperelastic Materials;297
13.6.1;8.6.1 Example: Myocardial Shear;300
13.7;8.7 Further Reading;305
13.8;References;308
14;9 Fluid Mechanics;310
14.1;9.1 Fluid Motion;310
14.1.1;9.1.1 Example: Laminar Flow Through a Circular Tube;311
14.2;9.2 Navier-Stokes Equations;316
14.2.1;9.2.1 Example: Drug Delivery in a Coronary Stent Revisited;319
14.3;9.3 Non-laminar Flow;326
14.4;9.4 Modelling Blood Flow;329
14.4.1;9.4.1 Electric Circuit Analogues for Blood Flow;329
14.4.2;9.4.2 Example: Aortic Blood Flow;330
14.4.3;9.4.3 Blood as a Non-newtonian Fluid;334
14.4.4;9.4.4 Example: Axial Streaming of a Blood Cell;335
14.5;9.5 Further Reading;344
14.6;References;346
15;Appendix A Matlab Fundamentals;347
15.1;A.1 Matlab Overview;347
15.2;A.1.1 User Interface;347
15.3;A.1.2 Working with Variables and Arrays;348
15.4;A.1.3 Matlab Programming;350
15.5;A.1.3.1 Scripting;351
15.6;A.1.3.2 Conditional Branching and Loops;352
15.7;A.1.3.3 Code Debugging;353
15.8;A.1.4 Solving Linear Systems of Equations;354
15.9;A.1.5 User-Defined Functions;355
15.10;A.1.6 Solving Systems of ODEs in Matlab;356
16;Appendix B Overview of COMSOL Multiphysics;358
16.1;B.1 COMSOL Basics;358
16.2;B.1.1 User Interface;358
16.3;B.1.2 Specifying Models;360
16.4;B.1.2.1 The Model Wizard;360
16.5;B.1.2.2 Creating a Geometry;361
16.6;B.1.2.3 User-Defined Parameters, Functions and Variables;362
16.7;B.1.2.4 Assigning Materials;365
16.8;B.1.2.5 Physics and User-Defined Equation Settings;365
16.9;B.1.2.6 Component Couplings;367
16.10;B.1.3 Solving and Visualisation;369
16.11;B.1.3.1 Mesh Settings;369
16.12;B.1.3.2 Solver Settings;370
16.13;B.1.3.3 Visualisation of Results;370
16.14;B.2 Example Model: Cardiac Defibrillation;372
17;Solutions;383
18;Index;497