Mathematically Modelling the Electrical Activity of the Heart

The following sections are included:

• Preface

• Foreword

• Contents

The following sections are included:

• Background

• Aims and Target Audience

• Why Model?

• Historical Perspectives

• Anatomy and Function of the Heart
  • Macroscopic Description
  • Basic Cardiac Cellular Electrophysiology
  • Cardiac Structure and Electrophysiology
  • The Electrocardiogram (ECG)
  • Cardiac Contraction

• Role of Mathematical Modelling in the Heart

• Notation

• Open Questions, Issues and Challenges

The following sections are included:

• Introduction

• Finite Element Basis Functions
  • Local Coordinate System
  • Linear Lagrange Basis Functions
  • Quadratic Lagrange Basis Functions
  • Basis Functions in Higher-Dimensions
  • Cubic Hermite Basis Functions

• Fitting Techniques
  • Data Projection
  • Linear Field Fitting
  • Iterative Linear Field Fitting
    • Sobolev Smoothing

• Geometric Models
  • Heart Models
  • Torso Models
  • Patient-Specific Models

• Open Questions, Issues and Challenges

The following sections are included:

• Introduction
  • Units

• Biophysically-Based Models
  • Cell Membrane
  • Hodgkin and Huxley (HH)
  • The Noble 1962 Model
  • The Beeler-Reuter Model (BR)
  • The Beeler-Reuter-Drouhard-Roberge Model (BRDR)
  • The Difiancesco-Noble Model (DFN)
  • The Luo-Rudy I Model (LRI)
  • The Luo-Rudy II Model (LRII)
  • The Noble, Varghese, Kohl and Noble Model (NVKN)
  • Biophysical Cell Model Summary

• Simplified Models of Cardiac Myocytes
  • Polynomial Model
  • FitzHugh-Nagumo Model
  • Rogers-Modified FitzHugh-Nagumo Model
  • van Capelle-Durrer Model
  • Fenton-Karma Model (FK)

• Solution of Cell Models

• CellML

• Open Questions, Issues and Challenges

The following sections are included:

• Introduction

• Tissue Structure

• Modelling Electrical Activity
  • The Cable Model
  • The Bidomain Model

• Numerical Solution Techniques

• Finite Element-Derived Finite Difference Method
  • Domain Metrics
  • Describing the Microstructure
  • Expressing the Laplacian Terms
  • Numerical Approximations
  • Evaluation of the Bidomain Laplacian Coefficients
  • Element Branching
  • Bidomain Boundary Conditions
  • Implicit and Explicit Formulations
  • Deformation

• Alternative Solution Techniques
  • Finite Difference Solution of the Bidomain Equations
  • Finite Element Solution of the Bidomain Equations
  • Finite Volume Solution of the Bidomain Equations

• Testing the Solution Method
  • The Laplacian Operator
  • One-Dimensional Propagation
  • Deforming Fibre
  • Three-Dimensional Isotropic Propagation
  • Anisotropic Three-Dimensional Propagation

• Examples of Tissue Excitation
  • Excitation in Two Dimensions
  • Excitation in Three Dimensions

• General Comments

• Open Questions, Issues and Challenges

The following sections are included:

• Introduction

• Equivalent Source Models

• Empirical Models

• Bidomain Models

• Tissue Types

• Illustrative Examples
  • Ventricular Excitation
  • Atrial Excitation

• Open Questions, Issues and Challenges

The following sections are included:

• Introduction

• The Electrocardiogram

• Electrical Activity in the Torso
  • Torso Boundary Conditions
  • Summary of the Integrated Model

• Geometric Torso Model

• Torso Solution – The Finite Element Method (FEM)
  • Gaussian Quadrature
  • Analytic Test Problem

• Torso Solution – The Boundary Element Method (BEM)
  • Numerical Solution Procedures for the Boundary Integral Equation
  • Numerical Evaluation of Coefficient Integrals
  • Conductivity Tensor
  • The Derivative BEM
    • Fundamental Solution Derivatives
    • Derivative Boundary Element Identities
    • Singularity on the Domain Boundary
    • Discretisation
    • Conductivity Tensor
  • Source Terms
  • Accuracy and Computational Efficiency

• From Cell to Body Surface
  • Common Approaches
  • Dipole Source Calculation
  • Coupled and Uncoupled Solutions
  • Coupling Approaches
    • Boundary Iteration Method
    • Direct Assembly
  • Two-Dimensional Fully-Coupled Forward Simulations

• Three-Dimensional Torso Simulations
  • Dipole from Experimental Recordings
  • Dipoles from Cellular Current Density
  • Solution Visualisation

• Open Questions, Issues and Challenges

The following sections are included:

• Introduction

• The Inverse Problem

• Transfer Matrices

• Singular Value Decomposition (SVD)

• Potential-based Inverse Algorithms
  • Tikhonov Regularisation
  • Truncated SVD (TSVD)
  • Greensite Potential-Based Inverse

• Activation-based Inverse Algorithm

• Determining the Regularisation Parameters
  • Optimal Criterion
  • L-Curve Criterion
  • CRESO Criterion
  • Zero-Crossing Criterion

• Validation Approaches
  • Simulation Studies
  • Experimental Validation

• Modelling Aspects

• The Outlook

• Open Questions, Issues and Challenges

The following sections are included:

• Ventricular Mechanics

• Electro-mechanics

• Ventricular Blood Flow

• Coronary Blood Flow

• Ischaemia

• Re-entry and Cardiac Arrhythmias

• Re-entry and Mechanics

• The Future of Cardiac Modelling

• Open Questions, Issues and Challenges

The following sections are included:

• Appendix A Finite Element Example

• Bibliography

• Index