Table Of ContentMonographs in
1
Computational Science
and Engineering
Editors
TimothyJ.Barth
MichaelGriebel
DavidE.Keyes
RistoM.Nieminen
DirkRoose
TamarSchlick
Joakim Sundnes Glenn Terje Lines Xing Cai
Bjørn Fredrik Nielsen Kent-Andre Mardal
Aslak Tveito
Computing the Electrical
Activity in the Heart
With99Figuresand23Tables
ABC
JoakimSundnes
GlennTerjeLines
XingCai
BjørnFredrikNielsen
Kent-AndreMardal
AslakTveito
SimulaResearchLaboratory
P.O.Box134
1325Lysaker,Norway
email:[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
[email protected]
AslakTveitohasreceivedfinancialsupportfromtheNFF–Norskfaglitterærforfatter-ogoversetterforening
LibraryofCongressControlNumber:2006927369
MathematicsSubjectClassification:35Q80,65M55,65M60,92C50,92C0510
ISBN-10 3-540-33432-7SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-33432-3SpringerBerlinHeidelbergNewYork
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Preface
Theheartisafantasticmachine;duringanormallifetimeitbeatsabout2.5billion
times and pumps 200.000 tons of blood through an enormous system of vessels
extending160.000kilometresthroughoutthebody.
Forcenturies,manhastriedtounderstandhowtheheartworks,butthereremain
many unsolved problems, problems that have captured the attention of thousands
of researchers worldwide. There is, for example, a huge amount of research being
devotedtotheanalysisofsingleheartcells.Otherareasofresearchincludetrying
tounderstandhowitworksasacompletemuscle,andhowbloodflowsthroughthe
heart.Theentireprocessisextremelycomplex.
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
thebody[142].HisdogJimmyearnedaplaceinhistorybybeingthefirsttohave
hisheartmeasuredinthisway;seeFigure1.1.In1903WillemEinthoven[34]de-
veloped the first commercial device for recording electrocardiograms (ECGs); see
Figure1.2.
Thisbookisaboutcomputingtheelectricalactivityintheheart.Inordertodo
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
aretransportedthrough theheart anddistributedinthebody. Allthesemodels are
formulatedintermsofdifferentialequations.Basedonthesemodels,wewillderive
discretemodelssuitablefornumericalsimulations.Wewillbaseourmethodsona
finiteelementapproachandwillsolvetheequationsthatariseonparallelcomputers.
Inordertounderstandthisbook,youwillneedtohaveabasiccourseinpartial
differential equations, and it is definitely an advantage to know a bit about finite
elementmethods;butthatisaboutit.Wewillexplainallthebiologyyouwillneed.
Wewillprovideadetailedintroductiontoparallelcomputingandhowtosolvelinear
systemsofequations.
Thisbookisaboutcomputationalscience;infact,itsmainobjectiveistopresent
alargeprojectincomputationalscience.Wewillbeginbygivingyouthenecessary
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?
ThesequestionsarediscussedbrieflyinChapter1.
In Chapter 2 we jump straight to the mathematical models. A wonderful fea-
tureofmathematicsisitsabilitytophraseextremelycomplexphenomenainrather
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
physicsoftheprocess,rangingfrommodelsofwhathappenswithinasinglecellto
acompletemathematicalmodeloftheelectricalactivityintheentireheart.
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,
butmattersbecomecomplicatedduetothecomplexityofthemathematicalmodels.
Chapter3willmotivatethreespecificproblemsthathavetobeconsideredcarefully.
First,thefiniteelementmethodleadstolargesystemsoflinearequations.Itiswell
knownfromotherfieldsthattheseequationsmaybeverychallengingtosolve,and
inChapter4weintroduceandanalyzepreconditioningtechniquestosolvethelinear
systemthatarisesfromthefiniteelementdiscretizations.Second,systemsofordi-
narydifferentialequationsthatmodeltheprocessesgoingoninsinglecellshaveto
besolved.WediscussappropriatemethodsforthisinChapter5.Third,theproblem
thatwearegoingtosolveishuge,andthereforehastobesolvedonparallelcom-
puters.WeintroducesuchcomputersinChapter6anddemonstratetheirusefulness
fortheproblemsathand.
Our primary goal is to contribute to the development of machinery that can
computethelocationandgeometryofamyocardialinfarction.Thisisahugetask
occupying the minds of many researchers around the world, but a technologically
feasiblesolutionstillliesinthefuture.InChapter7weinitiateadiscussionofhowto
utilizeourabilitytosimulatetheelectricalactivityintheheartandthebodyinorder
todetectaninfarction.Thisis,infact,aninverseproblem;wemeasuretheresultof
aprocessandwethencomputeitscause.Inthepresentsetting,themeasurements
areelectrocardiogramsandthecauseistheinfarctedareaoftheheart.Theprocess
isthetransportofelectricalsignalsintheheartandthedistributionoftheseinthe
body.
We hope this book can serve as an introduction to this field for applied math-
ematiciansandcomputationalscientistsandalsoforresearchersinbioengineering
whoareinterestedinusingthetoolsofcomputationalscienceintheirresearch.The
book presents the authors’ view of the field, with a strong emphasis on the math-
ematical models and how to solve them numerically. There are few new scientific
resultsinthebook.Mostofthematerialpresentedhereisbasedonalreadypublished
material.
Fornebu,Norway JoakimSundnes
March2006 GlennTerjeLines
XingCai
BjørnFredrikNielsen
Kent-AndreMardalandAslakTveito
Table of Contents
1 PhysiologicalBackground................................. 1
1.1 TheElectrocardiogram ........................................ 2
1.1.1 PhysicsandPhysiology ................................ 5
1.1.2 CellularElectricalActivity.............................. 9
1.1.3 SignalConduction..................................... 11
1.1.4 DiagnosisandtheInverseProblem ....................... 12
1.2 ComputerSimulations ........................................ 14
1.2.1 WhySimulate? ....................................... 14
1.2.2 StateoftheArt........................................ 15
2 MathematicalModels ..................................... 21
2.1 ModellingtheBodyasaVolumeConductor ...................... 21
2.1.1 AVolumeConductorModel ............................ 21
2.2 AModelfortheHeartTissue................................... 24
2.2.1 ExcitableTissue....................................... 24
2.2.2 TheBidomainModel .................................. 25
2.2.3 TheMonodomainModel ............................... 30
2.3 CouplingtheHeartandtheBody ............................... 32
2.4 ModelsfortheIonicCurrent ................................... 33
2.4.1 TheFitzHugh-NagumoModel........................... 34
2.4.2 TheCellMembrane ................................... 36
2.4.3 TheNernstEquilibriumPotential ........................ 38
2.4.4 ModelsforIonicFlux.................................. 40
2.4.5 ChannelGating ....................................... 42
2.4.6 TheHodgkin-HuxleyModel ............................ 44
2.4.7 AModelforCardiacCells .............................. 46
2.4.8 ModelsforVentricularCells ............................ 48
2.4.9 SecondGenerationModels ............................. 51
2.5 SummaryoftheMathematicalModel............................ 53
3 ComputationalModels .................................... 57
3.1 TheFiniteElementMethodfortheTorso......................... 57
3.1.1 ASimplifiedModelProblem............................ 57
3.1.2 ADipoleModeloftheHeart ............................ 63
3.1.3 KnownPotentialontheHeartSurface .................... 67
3.2 TheHeartEquations .......................................... 70
3.2.1 OperatorSplitting ..................................... 71
3.2.2 OperatorSplittingfortheMonodomainModel ............. 75
3.2.3 OperatorSplittingfortheBidomainModel ................ 78
3.3 CouplingtheHeartandtheTorso ............................... 82
VIII TableofContents
3.4 NumericalExperiments ....................................... 85
3.4.1 ConvergenceTests..................................... 85
3.4.2 Simulationona2DSlice ............................... 92
3.4.3 NormalPropagation ................................... 93
3.4.4 Ischemia............................................. 94
4 SolvingLinearSystems ................................... 99
4.1 Overview ................................................... 99
4.2 IterativeMethods.............................................100
4.2.1 TheRichardsonIteration ...............................101
4.2.2 TheFDMDiscretizationPoissonEquationin1D ...........103
4.2.3 TheRichardsonIterationRevisited.......................106
4.2.4 Preconditioning .......................................109
4.2.5 TheJacobiMethod ....................................110
4.2.6 TheRelaxedJacobiMethod.............................112
4.2.7 TheExactandtheInexactBlockJacobiMethods ...........113
4.2.8 TheGauss–SeidelMethod ..............................114
4.2.9 TheRelaxedGauss–SeidelMethod.......................115
4.2.10 TheSymmetricGauss–SeidelMethod ....................115
4.2.11 TheExactandtheInexactBlockGauss–SeidelMethod......115
4.3 TheConjugateGradientMethod ................................116
4.3.1 TheCGAlgorithm ....................................116
4.3.2 ConvergenceTheory...................................120
4.3.3 NumericalExperiments ................................121
4.4 Multigrid ...................................................123
4.4.1 Idea.................................................123
4.4.2 TheoreticalFramework.................................125
4.4.3 ConvergenceTheory...................................127
4.4.4 NumericalExperiments ................................128
4.5 DomainDecomposition .......................................128
4.6 PreconditioningRevisited......................................133
4.6.1 Idea.................................................133
4.6.2 SpectralEquivalence...................................133
4.6.3 TheRichardsonIterationRe-Revisited....................134
4.6.4 PreconditionedConjugateGradientMethod ...............135
4.6.5 ConvergenceAnalysisRevisited .........................136
4.6.6 VariableCoefficients...................................137
4.6.7 NumericalExperiments ................................138
4.7 TheMonodomainModel ......................................140
4.7.1 Multigrid ............................................141
4.7.2 NumericalExperiments ................................141
4.7.3 DomainDecomposition ................................142
4.8 TheBidomainModel .........................................142
TableofContents IX
5 SolvingSystemsofODEs.................................. 149
5.1 SimpleODESolvers..........................................149
5.1.1 TheEulerMethods ....................................149
5.1.2 StabilityAnalysisfortheEulerMethods ..................151
5.2 Higher-OrderMethods ........................................154
5.2.1 MultistepMethods ....................................154
5.2.2 Runge-KuttaMethods..................................156
5.3 SolvingNonlinearEquations ...................................160
5.3.1 Newton’sMethod .....................................161
5.3.2 Newton’sMethodforHigher-OrderSolvers ...............162
5.4 AutomaticTimeStepControl ..................................164
5.5 TheCellModelEquations .....................................168
5.5.1 ExplicitversusImplicitMethods.........................168
5.5.2 SimulationResults ....................................171
6 Large-ScaleElectrocardiacSimulations ................... 175
6.1 TheElectrocardiacSimulator...................................175
6.1.1 TheNumericalStrategy ................................176
6.1.2 SoftwareComponentsoftheElectrocardiacSimulator.......176
6.2 RequirementsforLarge-ScaleSimulations .......................177
6.2.1 TheMemoryRequirement ..............................178
6.2.2 RealisticEstimatesforMemoryandTimeUsage ...........179
6.3 IntroductiontoParallelComputing..............................181
6.3.1 HardwareandProgrammingModels......................181
6.3.2 DivisionofWorkandtheResultingOverhead..............182
6.3.3 SpeedupandParallelEfficiency..........................184
6.4 TwoSimpleExamples ........................................186
6.4.1 AddingTwoVectors ...................................186
6.4.2 InnerProduct .........................................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 DiscretizationbyFiniteDifferences ......................197
6.6.2 ASequentialProgram..................................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 TableofContents
6.8 DomainDecompositionasParallelPreconditioners ................207
6.8.1 AdditiveSchwarzPreconditioner ........................208
6.8.2 ParallelDDfortheMonodomainEquation ................210
6.8.3 ParallelDDfortheBidomainEquations ..................210
6.8.4 ExtensiontotheTorso .................................211
6.9 ParallelizingElectrocardiacSimulations .........................212
6.9.1 TheOverallSimulationProcess..........................212
6.9.2 PartitioningtheDomains ...............................213
6.9.3 StraightforwardParallelizationTasks .....................214
6.9.4 SolvingtheBlockLinearSysteminParallel ...............215
6.10 SimulationonaRealisticGeometry .............................215
6.11 Summary ...................................................216
7 InverseProblems.......................................... 219
7.1 ASimpleExample ...........................................221
7.1.1 ProblemFormulation ..................................222
7.1.2 FourierAnalysis ......................................224
7.1.3 Ill-Posedness .........................................225
7.1.4 DiscretizationandanOutputLeastSquaresFormulation
oftheProblem ........................................229
7.1.5 RegularizationTechniques ..............................231
7.2 TheClassicalInverseProblemofElectrocardiography..............238
7.2.1 MathematicalFormulation ..............................239
7.2.2 ALinearProblem .....................................241
7.2.3 Discretization.........................................243
7.2.4 TheTime-DependentProblem...........................246
7.2.5 NumericalExperiments ................................249
7.3 ComputingtheLocationandOrientationofaDipole ...............253
7.3.1 Preliminaries .........................................255
7.3.2 TheInverseProblem...................................256
7.3.3 ParameterizationsLeadingtoLinearProblems .............257
7.3.4 ANumericalExample .................................261
7.3.5 ParameterizationsLeadingtoNonlinearProblems ..........265
7.3.6 ANumericalExperiment ...............................270
7.4 ComputingtheSizeandLocationofaMyocardialInfarction ........271
7.4.1 ModellingInfarctions,theDirect(Forward)Problem........272
7.4.2 ModellingInfarctions,theInverseProblem ................274
7.4.3 DifferentiationoftheObjectiveFunction..................276
7.4.4 NumericalExperiments ................................281
TableofContents XI
A Color Figures ............................................. 287
B Rate FunctionsandIonicCurrents........................ 297
B.1 TheHodgkin-HuxleyModel ...................................297
B.2 TheNobleModelforPurkinjeCells.............................297
B.3 TheBeeler-ReuterModel......................................298
C CoefficientsfortheImplicitRKSolvers ................... 300
Bibliography ................................................. 301
Index......................................................... 308
