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 Ind