Monographs in Computational Science and Engineering 1

Monographs in Computational Science and Engineering 1 Editors Timothy J. Barth Michael Griebel David E. Keyes Risto M. Nieminen Dirk Roose Tamar Schlick

Joakim Sundnes Glenn Terje Lines Xing Cai Bjørn Fredrik Nielsen Kent-Andre Mardal Aslak Tveito Computing the Electrical Activity in the Heart With 99 Figures and 23 Tables ABC

Joakim Sundnes Glenn Terje Lines Xing Cai Bjørn Fredrik Nielsen Kent-Andre Mardal Aslak Tveito Simula Research Laboratory P.O. Box 134 1325 Lysaker, Norway email: sundnes@simula.no glennli@simula.no xingca@simula.no bjornn@simula.no kent-and@simula.no aslak@simula.no Aslak Tveito has received financial support from the NFF Norsk faglitterær forfatter- og oversetterforening Library of Congress Control Number: 2006927369 Mathematics Subject Classification: 35Q80, 65M55, 65M60, 92C50, 92C0510 ISBN-10 3-540-33432-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33432-3 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors and techbooks using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 11737407 46/techbooks 543210

Preface The heart is a fantastic machine; during a normal lifetime it beats about 2.5 billion times and pumps 200.000 tons of blood through an enormous system of vessels extending 160.000 kilometres throughout the body. For centuries, man has tried to understand how the heart works, but there remain many unsolved problems, problems that have captured the attention of thousands of researchers worldwide. There is, for example, a huge amount of research being devoted to the analysis of single heart cells. Other areas of research include trying to understand how it works as a complete muscle, and how blood flows through the heart. The entire process is extremely complex. The history of bioelectricity can be traced back to the late eighteenth century and the experiments of Luigi Galvani. A century later, in 1887, Augustus Wallers managed to measure the electrical signal generated by the heart at the surface of the body [142]. His dog Jimmy earned a place in history by being the first to have his heart measured in this way; see Figure 1.1. In 1903 Willem Einthoven [34] developed the first commercial device for recording electrocardiograms (ECGs); see Figure 1.2. This book is about computing the electrical activity in the heart. In order to do so, we will need mathematical models of how the electrical signals are generated in the heart. Furthermore, we will need mathematical models of how the signals are transported through the heart and distributed in the body. All these models are formulated in terms of differential equations. Based on these models, we will derive discrete models suitable for numerical simulations. We will base our methods on a finite element approach and will solve the equations that arise on parallel computers. In order to understand this book, you will need to have a basic course in partial differential equations, and it is definitely an advantage to know a bit about finite element methods; but that is about it. We will explain all the biology you will need. We will provide a detailed introduction to parallel computing and how to solve linear systems of equations. This book is about computational science; in fact, its main objective is to present a large project in computational science. We will begin by giving you the necessary physiological background. What is going on in the heart and how can we use a recording on the body surface to say anything about the condition of the heart? These questions are discussed briefly in Chapter 1. In Chapter 2 we jump straight to the mathematical models. A wonderful feature of mathematics is its ability to phrase extremely complex phenomena in rather simple equations. Yet despite this ability, the equations we describe in Chapter 2

VI Preface are rather complicated. These models are absolutely necessary in order to be able to compute the electrical activity on a computer. We guide you through the basic physics of the process, ranging from models of what happens within a single cell to a complete mathematical model of the electrical activity in the entire heart. In Chapter 3 we discuss how to discretize the equations derived in Chapter 2. We will do this using the finite element method. The approach is straightforward, but matters become complicated due to the complexity of the mathematical models. Chapter 3 will motivate three specific problems that have to be considered carefully. First, the finite element method leads to large systems of linear equations. It is well known from other fields that these equations may be very challenging to solve, and in Chapter 4 we introduce and analyze preconditioning techniques to solve the linear system that arises from the finite element discretizations. Second, systems of ordinary differential equations that model the processes going on in single cells have to be solved. We discuss appropriate methods for this in Chapter 5. Third, the problem that we are going to solve is huge, and therefore has to be solved on parallel computers. We introduce such computers in Chapter 6 and demonstrate their usefulness for the problems at hand. Our primary goal is to contribute to the development of machinery that can compute the location and geometry of a myocardial infarction. This is a huge task occupying the minds of many researchers around the world, but a technologically feasible solution still lies in the future. In Chapter 7 we initiate a discussion of how to utilize our ability to simulate the electrical activity in the heart and the body in order to detect an infarction. This is, in fact, an inverse problem; we measure the result of a process and we then compute its cause. In the present setting, the measurements are electrocardiograms and the cause is the infarcted area of the heart. The process is the transport of electrical signals in the heart and the distribution of these in the body. We hope this book can serve as an introduction to this field for applied mathematicians and computational scientists and also for researchers in bioengineering who are interested in using the tools of computational science in their research. The book presents the authors view of the field, with a strong emphasis on the mathematical models and how to solve them numerically. There are few new scientific results in the book. Most of the material presented here is based on already published material. Fornebu, Norway March 2006 Joakim Sundnes Glenn Terje Lines Xing Cai Bjørn Fredrik Nielsen Kent-Andre Mardal and Aslak Tveito

Table of Contents 1 Physiological Background... 1 1.1 The Electrocardiogram... 2 1.1.1 PhysicsandPhysiology... 5 1.1.2 CellularElectricalActivity... 9 1.1.3 Signal Conduction....... 11 1.1.4 Diagnosis and the Inverse Problem.... 12 1.2 ComputerSimulations... 14 1.2.1 WhySimulate?... 14 1.2.2 StateoftheArt... 15 2 Mathematical Models... 21 2.1 Modelling the Body as a Volume Conductor... 21 2.1.1 A Volume Conductor Model..... 21 2.2 A Model for the Heart Tissue..... 24 2.2.1 ExcitableTissue... 24 2.2.2 The Bidomain Model.... 25 2.2.3 The Monodomain Model........ 30 2.3 CouplingtheHeartandtheBody... 32 2.4 Models for the Ionic Current..... 33 2.4.1 The FitzHugh-Nagumo Model... 34 2.4.2 TheCellMembrane... 36 2.4.3 The Nernst Equilibrium Potential..... 38 2.4.4 Models for Ionic Flux.... 40 2.4.5 Channel Gating......... 42 2.4.6 The Hodgkin-Huxley Model..... 44 2.4.7 A Model for Cardiac Cells....... 46 2.4.8 Models for Ventricular Cells..... 48 2.4.9 Second Generation Models...... 51 2.5 Summary of the Mathematical Model..... 53 3 Computational Models... 57 3.1 TheFiniteElementMethodfortheTorso... 57 3.1.1 A Simplified Model Problem..... 57 3.1.2 A Dipole Model of the Heart..... 63 3.1.3 KnownPotentialontheHeartSurface... 67 3.2 TheHeartEquations... 70 3.2.1 Operator Splitting....... 71 3.2.2 Operator Splitting for the Monodomain Model..... 75 3.2.3 Operator Splitting for the Bidomain Model.... 78 3.3 CouplingtheHeartandtheTorso... 82

VIII Table of Contents 3.4 Numerical Experiments......... 85 3.4.1 Convergence Tests....... 85 3.4.2 Simulationona2DSlice... 92 3.4.3 Normal Propagation..... 93 3.4.4 Ischemia.... 94 4 Solving Linear Systems... 99 4.1 Overview... 99 4.2 Iterative Methods...100 4.2.1 The Richardson Iteration........ 101 4.2.2 TheFDMDiscretizationPoissonEquationin1D...103 4.2.3 The Richardson Iteration Revisited....106 4.2.4 Preconditioning.........109 4.2.5 The Jacobi Method...... 110 4.2.6 The Relaxed Jacobi Method...... 112 4.2.7 The Exact and the Inexact Block Jacobi Methods...113 4.2.8 The Gauss Seidel Method....... 114 4.2.9 The Relaxed Gauss Seidel Method....115 4.2.10 The Symmetric Gauss Seidel Method........ 115 4.2.11 The Exact and the Inexact Block Gauss Seidel Method.....115 4.3 TheConjugateGradientMethod...116 4.3.1 TheCGAlgorithm...116 4.3.2 Convergence Theory..... 120 4.3.3 Numerical Experiments......... 121 4.4 Multigrid...123 4.4.1 Idea...123 4.4.2 Theoretical Framework..........125 4.4.3 Convergence Theory..... 127 4.4.4 Numerical Experiments......... 128 4.5 Domain Decomposition.........128 4.6 Preconditioning Revisited........ 133 4.6.1 Idea...133 4.6.2 Spectral Equivalence..... 133 4.6.3 The Richardson Iteration Re-Revisited........ 134 4.6.4 Preconditioned Conjugate Gradient Method...135 4.6.5 Convergence Analysis Revisited...... 136 4.6.6 VariableCoefficients...137 4.6.7 Numerical Experiments......... 138 4.7 The Monodomain Model........140 4.7.1 Multigrid...141 4.7.2 Numerical Experiments......... 141 4.7.3 Domain Decomposition.........142 4.8 The Bidomain Model....142

Table of Contents IX 5 Solving Systems of ODEs... 149 5.1 SimpleODESolvers...149 5.1.1 The Euler Methods...... 149 5.1.2 Stability Analysis for the Euler Methods......151 5.2 Higher-Order Methods...154 5.2.1 Multistep Methods...... 154 5.2.2 Runge-Kutta Methods.... 156 5.3 Solving Nonlinear Equations..... 160 5.3.1 Newton smethod...161 5.3.2 Newton s Method for Higher-Order Solvers...162 5.4 AutomaticTimeStepControl...164 5.5 The Cell Model Equations....... 168 5.5.1 Explicit versus Implicit Methods...... 168 5.5.2 SimulationResults...171 6 Large-Scale Electrocardiac Simulations... 175 6.1 The Electrocardiac Simulator..... 175 6.1.1 TheNumericalStrategy...176 6.1.2 Software Components of the Electrocardiac Simulator......176 6.2 Requirements for Large-Scale Simulations....177 6.2.1 The Memory Requirement....... 178 6.2.2 RealisticEstimatesforMemoryandTimeUsage...179 6.3 Introduction to Parallel Computing....... 181 6.3.1 Hardware and Programming Models..........181 6.3.2 Division of Work and the Resulting Overhead......182 6.3.3 Speedup and Parallel Efficiency....... 184 6.4 TwoSimpleExamples...186 6.4.1 AddingTwoVectors...186 6.4.2 Inner Product.... 188 6.5 Domain-BasedParallelization...191 6.5.1 DivisionofDataforFDM...192 6.5.2 DivisionofDataforFEM...194 6.5.3 SummaryofPrinciples...196 6.6 ExplicitFDMinParallel...197 6.6.1 Discretization by Finite Differences...197 6.6.2 A Sequential Program....198 6.6.3 Parallelization...200 6.7 ParallelConjugateGradientIterations...201 6.7.1 ConjugateGradientRevisited...202 6.7.2 ParallelCGandFDM...203 6.7.3 ParallelCGandFEM...203

X Table of Contents 6.8 Domain Decomposition as Parallel Preconditioners....207 6.8.1 Additive Schwarz Preconditioner.....208 6.8.2 Parallel DD for the Monodomain Equation....210 6.8.3 ParallelDDfortheBidomainEquations...210 6.8.4 ExtensiontotheTorso...211 6.9 Parallelizing Electrocardiac Simulations...... 212 6.9.1 The Overall Simulation Process...212 6.9.2 Partitioning the Domains........ 213 6.9.3 StraightforwardParallelizationTasks...214 6.9.4 Solving the Block Linear System in Parallel...215 6.10 SimulationonaRealisticGeometry...215 6.11 Summary...216 7 Inverse Problems... 219 7.1 ASimpleExample...221 7.1.1 ProblemFormulation...222 7.1.2 FourierAnalysis...224 7.1.3 Ill-Posedness.... 225 7.1.4 Discretization and an Output Least Squares Formulation oftheproblem...229 7.1.5 Regularization Techniques....... 231 7.2 The Classical Inverse Problem of Electrocardiography......238 7.2.1 MathematicalFormulation...239 7.2.2 A Linear Problem.......241 7.2.3 Discretization...243 7.2.4 The Time-Dependent Problem....246 7.2.5 Numerical Experiments......... 249 7.3 ComputingtheLocationandOrientationofaDipole...253 7.3.1 Preliminaries...255 7.3.2 TheInverseProblem...256 7.3.3 Parameterizations Leading to Linear Problems.....257 7.3.4 ANumericalExample...261 7.3.5 Parameterizations Leading to Nonlinear Problems......... 265 7.3.6 A Numerical Experiment........270 7.4 Computing the Size and Location of a Myocardial Infarction.......271 7.4.1 Modelling Infarctions, the Direct (Forward) Problem.......272 7.4.2 Modelling Infarctions, the Inverse Problem....274 7.4.3 Differentiation of the Objective Function......276 7.4.4 Numerical Experiments......... 281

Table of Contents XI A Color Figures... 287 B Rate Functions and Ionic Currents... 297 B.1 The Hodgkin-Huxley Model..... 297 B.2 The Noble Model for Purkinje Cells......297 B.3 The Beeler-Reuter Model........ 298 C Coefficients for the Implicit RK Solvers... 300 Bibliography... 301 Index... 308