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

Abstract

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. Preface; Contents; Acronyms; Part I Bioengineering Modelling Principles, Methods and Theory; 1 Introduction to Modelling in Bioengineering; 1.1 Modelling and Simulation in Medicine and Biology; 1.2 The Modelling Process; 1.3 Mathematical Model Types; 1.3.1 Linear Versus Non-linear; 1.3.2 Dynamic Versus Static; 1.3.3 Deterministic Versus Stochastic; 1.3.4 Continuous Versus Discrete; 1.3.5 Rule-Based; 1.4 Dimensional Analysis; 1.4.1 Dimensions and Units; 1.4.2 Buckingham -Theorem; 1.5 Model Scaling; References; 2 Lumped Parameter Modelling with Ordinary Differential Equations. 3.4 ODE Solver Implementations in Matlab and COMSOL3.5 Further Reading; References; 4 Distributed Systems Modelling with Partial Differential Equations; 4.1 Modelling with PDEs; 4.1.1 The Gradient; 4.1.2 The Divergence; 4.1.3 The Curl; 4.1.4 The Divergence Theorem; 4.1.5 Conservation Law Formulation; 4.1.6 The Laplacian; 4.1.7 PDE Boundary Conditions; 4.2 Basic Analytical and Numerical Solution Techniques; 4.2.1 Separation of Variables; 4.2.2 Finite Difference Method; 4.2.3 Method of Lines; 4.3 Further Reading; References; 5 The Finite Element Method; 5.1 Finite Elements for 1D Systems. 5.1.1 Weak Form PDE Equivalent5.1.2 Basis Function Approximation; 5.1.3 Higher-Order Basis Functions; 5.2 Finite Elements for 2D/3D Systems; 5.2.1 Weak Form Description; 5.2.2 Basis Function Approximation; 5.3 FEM Numerical Implementation; 5.3.1 Assembly of System Matrices; 5.3.2 Gaussian Quadrature; 5.3.3 Non-Linear Systems; 5.4 Further Reading; References; Part II Bioengineering Applications; 6 Modelling Electrical Stimulation of Tissue; 6.1 Electrical Stimulation; 6.1.1 Maxwell's Equations; 6.1.2 Electrostatic Formulations; 6.1.3 Volume Conductor Theory. 6.1.4 Example: Cell Culture Electric Field Stimulator6.1.5 Example: Access Resistance of Electrode Disc; 6.2 Modelling Electrical Activity of Tissues; 6.2.1 Continuum Models of Excitable Tissues; 6.2.2 Example: Modelling Spiral-Wave Reentry in Cardiac Tissue; 6.2.3 Modelling PDEs/ODEs on Boundaries, Edges and Points; 6.2.4 Example: Axonal Stimulation Using Nerve Cuff Electrodes; 6.3 Further Reading; References; 7 Models of Diffusion and Heat Transfer; 7.1 Diffusion; 7.1.1 Fick's Laws of Diffusion; 7.1.2 Example: Diffusion and Uptake into a Spherical Cell; 7.1.3 Convective Transport. 2.1 Overview of Ordinary Differential Equations2.2 Linear ODEs; 2.3 ODE Systems; 2.3.1 Example Model 1: Cardiac Mechanics; 2.3.2 Example Model 2: Hodgkin--Huxley Model of Neural Excitation; 2.4 Further Reading; References; 3 Numerical Integration of Ordinary Differential Equations; 3.1 Taylor's Theorem; 3.2 One-Step Methods; 3.2.1 Backward-Euler Method; 3.2.2 Trapezoidal Method; 3.2.3 Runge--Kutta Methods; 3.2.4 The Generalized-α Method; 3.3 Multistep Methods; 3.3.1 Predictor-Corrector Methods; 3.3.2 Backward Differentiation Formulas; 3.3.3 Numerical Differentiation Formulas. Preface; Contents; Acronyms; Part I Bioengineering Modelling Principles, Methods and Theory; 1 Introduction to Modelling in Bioengineering; 1.1 Modelling and Simulation in Medicine and Biology; 1.2 The Modelling Process; 1.3 Mathematical Model Types; 1.3.1 Linear Versus Non-linear; 1.3.2 Dynamic Versus Static; 1.3.3 Deterministic Versus Stochastic; 1.3.4 Continuous Versus Discrete; 1.3.5 Rule-Based; 1.4 Dimensional Analysis; 1.4.1 Dimensions and Units; 1.4.2 Buckingham -Theorem; 1.5 Model Scaling; References; 2 Lumped Parameter Modelling with Ordinary Differential Equations5058 2.1 Overview of Ordinary Differential Equations2.2 Linear ODEs; 2.3 ODE Systems; 2.3.1 Example Model 1: Cardiac Mechanics; 2.3.2 Example Model 2: Hodgkin--Huxley Model of Neural Excitation; 2.4 Further Reading; References; 3 Numerical Integration of Ordinary Differential Equations; 3.1 Taylor's Theorem; 3.2 One-Step Methods; 3.2.1 Backward-Euler Method; 3.2.2 Trapezoidal Method; 3.2.3 Runge--Kutta Methods; 3.2.4 The Generalized-α Method; 3.3 Multistep Methods; 3.3.1 Predictor-Corrector Methods; 3.3.2 Backward Differentiation Formulas; 3.3.3 Numerical Differentiation Formulas.