Mathematically modelling the electrical activity of the heart : from cell to body surface and back again

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010103876	QP112.5.E46 P84 2005	Open Access Book	Book	Searching... Unknown
Searching... PSZ KL	30000010100201	QP112.5.E46 P84 2005	Open Access Book	Book	Searching... Unknown

This book on modelling the electrical activity of the heart is an attempt to describe continuum based modelling of cardiac electrical activity from the cell level to the body surface (the forward problem), and back again (the inverse problem). Background anatomy and physiology is covered briefly to provide a suitable context for understanding the detailed modelling that is presented herein. The questions of what is mathematical modelling and why one would want to use mathematical modelling are addressed to give some perspective to the philosophy behind our approach. Our view of mathematical modelling is broad -- it is not simply about obtaining a solution to a set of mathematical equations, but includes some material on aspects such as experimental and clinical validation.

Preface	p. vii
Foreword	p. ix
1 Introduction	p. 1
1.1 Background	p. 1
1.2 Aims and Target Audience	p. 4
1.3 Why Model?	p. 5
1.4 Historical Perspectives	p. 9
1.5 Anatomy and Function of the Heart	p. 18
1.5.1 Macroscopic Description	p. 18
1.5.2 Basic Cardiac Cellular Electrophysiology	p. 21
1.5.3 Cardiac Structure and Electrophysiology	p. 27
1.5.4 The Electrocardiogram (ECG)	p. 30
1.5.5 Cardiac Contraction	p. 35
1.6 Role of Mathematical Modelling in the Heart	p. 36
1.7 Notation	p. 40
1.8 Open Questions, Issues and Challenges	p. 41
2 Geometric Modelling	p. 43
2.1 Introduction	p. 43
2.2 Finite Element Basis Functions	p. 45
2.2.1 Local Coordinate System	p. 45
2.2.2 Linear Lagrange Basis Functions	p. 46
2.2.3 Quadratic Lagrange Basis Functions	p. 49
2.2.4 Basis Functions in Higher-Dimensions	p. 51
2.2.5 Cubic Hermite Basis Functions	p. 55
2.3 Fitting Techniques	p. 61
2.3.1 Data Projection	p. 61
2.3.2 Linear Field Fitting	p. 63
2.3.3 Iterative Linear Field Fitting	p. 64
2.3.3.1 Sobolev Smoothing	p. 65
2.4 Geometric Models	p. 68
2.4.1 Heart Models	p. 68
2.4.2 Torso Models	p. 71
2.4.3 Patient-Specific Models	p. 71
2.5 Open Questions, Issues and Challenges	p. 73
3 Cell Modelling	p. 77
3.1 Introduction	p. 77
3.1.1 Units	p. 79
3.2 Biophysically-Based Models	p. 81
3.2.1 Cell Membrane	p. 82
3.2.2 Hodgkin and Huxley (HH)	p. 82
3.2.3 The Noble 1962 Model	p. 88
3.2.4 The Beeler-Reuter Model (BR)	p. 91
3.2.5 The Beeler-Reuter-Drouhard-Roberge Model (BRDR)	p. 97
3.2.6 The DiFrancesco-Noble Model (DFN)	p. 101
3.2.7 The Luo-Rudy I Model (LRI)	p. 107
3.2.8 The Luo-Rudy II Model (LRII)	p. 116
3.2.9 The Noble, Varghese, Kohl and Noble Model (NVKN)	p. 122
3.2.10 Biophysical Cell Model Summary	p. 128
3.3 Simplified Models of Cardiac Myocytes	p. 131
3.3.1 Polynomial Model	p. 131
3.3.2 FitzHugh-Nagumo Model	p. 132
3.3.3 Rogers-Modified FitzHugh-Nagumo Model	p. 133
3.3.4 van Capelle-Durrer Model	p. 136
3.3.5 Fenton-Karma Model (FK)	p. 137
3.4 Solution of Cell Models	p. 142
3.5 CellML	p. 146
3.6 Open Questions, Issues and Challenges	p. 147
4 Tissue Modelling	p. 151
4.1 Introduction	p. 151
4.2 Tissue Structure	p. 152
4.3 Modelling Electrical Activity	p. 154
4.3.1 The Cable Model	p. 154
4.3.2 The Bidomain Model	p. 159
4.4 Numerical Solution Techniques	p. 166
4.5 Finite Element-Derived Finite Difference Method	p. 166
4.5.1 Domain Metrics	p. 167
4.5.2 Describing the Microstructure	p. 172
4.5.3 Expressing the Laplacian Terms	p. 174
4.5.4 Numerical Approximations	p. 176
4.5.5 Evaluation of the Bidomain Laplacian Coefficients	p. 179
4.5.6 Element Branching	p. 183
4.5.7 Bidomain Boundary Conditions	p. 184
4.5.8 Implicit and Explicit Formulations	p. 186
4.5.9 Deformation	p. 188
4.6 Alternative Solution Techniques	p. 188
4.6.1 Finite Difference Solution of the Bidomain Equations	p. 189
4.6.2 Finite Element Solution of the Bidomain Equations	p. 191
4.6.3 Finite Volume Solution of the Bidomain Equations	p. 195
4.7 Testing the Solution Method	p. 196
4.7.1 The Laplacian Operator	p. 196
4.7.2 One-Dimensional Propagation	p. 199
4.7.3 Deforming Fibre	p. 202
4.7.4 Three-Dimensional Isotropic Propagation	p. 203
4.7.5 Anisotropic Three-Dimensional Propagation	p. 205
4.8 Examples of Tissue Excitation	p. 206
4.8.1 Excitation in Two Dimensions	p. 207
4.8.2 Excitation in Three Dimensions	p. 213
4.9 General Comments	p. 219
4.10 Open Questions, Issues and Challenges	p. 221
5 Whole-Heart Modelling	p. 223
5.1 Introduction	p. 223
5.2 Equivalent Source Models	p. 224
5.3 Empirical Models	p. 225
5.4 Bidomain Models	p. 228
5.5 Tissue Types	p. 231
5.6 Illustrative Examples	p. 231
5.6.1 Ventricular Excitation	p. 232
5.6.2 Atrial Excitation	p. 234
5.7 Open Questions, Issues and Challenges	p. 236
6 Organ in the Body - The Forward Problem of Electrocardiology	p. 239
6.1 Introduction	p. 239
6.2 The Electrocardiogram	p. 240
6.3 Electrical Activity in the Torso	p. 245
6.3.1 Torso Boundary Conditions	p. 247
6.3.2 Summary of the Integrated Model	p. 248
6.4 Geometric Torso Model	p. 249
6.5 Torso Solution - The Finite Element Method (FEM)	p. 250
6.5.1 Gaussian Quadrature	p. 257
6.5.2 Analytic Test Problem	p. 259
6.6 Torso Solution - The Boundary Element Method (BEM)	p. 260
6.6.1 Numerical Solution Procedures for the Boundary Integral Equation	p. 268
6.6.2 Numerical Evaluation of Coefficient Integrals	p. 270
6.6.3 Conductivity Tensor	p. 274
6.6.4 The Derivative BEM	p. 275
6.6.4.1 Fundamental Solution Derivatives	p. 276
6.6.4.2 Derivative Boundary Element Identities	p. 276
6.6.4.3 Singularity on the Domain Boundary	p. 277
6.6.4.4 Discretisation	p. 283
6.6.4.5 Conductivity Tensor	p. 287
6.6.5 Source Terms	p. 288
6.6.6 Accuracy and Computational Efficiency	p. 292
6.7 From Cell to Body Surface	p. 296
6.7.1 Common Approaches	p. 297
6.7.2 Dipole Source Calculation	p. 298
6.7.3 Coupled and Uncoupled Solutions	p. 299
6.7.4 Coupling Approaches	p. 300
6.7.4.1 Boundary Iteration Method	p. 301
6.7.4.2 Direct Assembly	p. 304
6.7.5 Two-Dimensional Fully-Coupled Forward Simulations	p. 308
6.8 Three-Dimensional Torso Simulations	p. 316
6.8.1 Dipole from Experimental Recordings	p. 317
6.8.2 Dipoles from Cellular Current Density	p. 320
6.8.3 Solution Visualisation	p. 321
6.9 Open Questions, Issues and Challenges	p. 326
7 The Inverse Problem of Electrocardiology	p. 329
7.1 Introduction	p. 329
7.2 The Inverse Problem	p. 332
7.3 Transfer Matrices	p. 333
7.4 Singular Value Decomposition (SVD)	p. 339
7.5 Potential-based Inverse Algorithms	p. 340
7.5.1 Tikhonov Regularisation	p. 342
7.5.2 Truncated SVD (TSVD)	p. 343
7.5.3 Greensite Potential-Based Inverse	p. 345
7.6 Activation-based Inverse Algorithm	p. 347
7.7 Determining the Regularisation Parameters	p. 354
7.7.1 Optimal Criterion	p. 354
7.7.2 L-Curve Criterion	p. 355
7.7.3 CRESO Criterion	p. 356
7.7.4 Zero-Crossing Criterion	p. 357
7.8 Validation Approaches	p. 357
7.8.1 Simulation Studies	p. 357
7.8.2 Experimental Validation	p. 364
7.9 Modelling Aspects	p. 369
7.10 The Outlook	p. 371
7.11 Open Questions, Issues and Challenges	p. 372
8 Modelling Other Cardiac Processes	p. 375
8.1 Ventricular Mechanics	p. 375
8.2 Electro-mechanics	p. 378
8.3 Ventricular Blood Flow	p. 378
8.4 Coronary Blood Flow	p. 381
8.5 Ischaemia	p. 383
8.6 Re-entry and Cardiac Arrhythmias	p. 383
8.7 Re-entry and Mechanics	p. 388
8.8 The Future of Cardiac Modelling	p. 388
8.9 Open Questions, Issues and Challenges	p. 390
Appendix A Finite Element Example	p. 393
A.1 Element Stiffness Matrix	p. 394
A.2 Global Stiffness Matrix	p. 395
A.3 Boundary Conditions	p. 396
A.4 Irregular Geometries	p. 397
Bibliography	p. 399
Index	p. 417

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