Modelling Organs, Tissues, Cells and Devices (eBook)

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