domainwallbase

Therelativeroleofpatientphysiologyanddeviceoptimisationincardiacresynchronisationtherapy:AcomputationalmodellingstudyOnlineSupplementThissupplementprovidesadetailedaccountofthecomputationalmodelusedintheabovearticle,andtheprocessforitspersonalisationtoclinicaldata.
1ComputationalModelWedevelopedaweaklycoupled[1]modelofcardiacelectromechanics,combiningfourmaincomponents:amonodomainmodelofelectrophysiology,usingthetenTusschercellmodel[2]ahyperelasticrepresentationofmechanics,usingtheGuccioneconstitutivelaw[3]aphenomenologicalmodelofactivecontraction,basedonthatbyKerckhosetal.
[4]athreeelementWindkesselmodelofafterloadinejection1.
1ElectrophysiologyModelWeusedaspecialisedelectrophysiologyniteelementsoftware,theCardiacArrhythmiaRe-searchPackage(CARP)[5],forthesimulationofcardiacelectrophysiology.
CARPisamatureandhighlyoptimisedsoftwareforbidomainandmonodomainsimulationofcardiacelectro-physiology,developedattheMedicalUniversityofGraz(Graz,Austria)andtheUniversityofBordeaux(Bordeaux,France).
CardiacelectrophysiologywasmodelledinCARPusingthemonodomainequationχCmVt+Iion=·(σV)(1)1whereχisthecellmembranesurfacetovolumeratio,Cmisthecapacitanceofthecellmem-braneperunitarea,Visthetransmembranepotential,Iionistheioniccurrent,andσisthetissueconductivitytensor.
ThetransmembraneioniccurrentIionwasmodelledwiththetenTusschercardiaccellmodel[2].
Thiswasusedasalthoughitisnotthesimplestavailable,isasucientlydetailedhumanmodelthatiscomputationallytractableonanorganscalewhenusinganecientsimulationpackageandhighperformancecomputing(HPC)resource[6].
Atetrahedralmeshrepresent-ingtheanatomyofthemyocardiumwasrequired,withaspatialresolutionof250m.
Thisresolutionrepresentedacompromisebetweentractablemeshsizeandsolutionaccuracy[7,8].
Tissueconductivitywasdenedinthemodelashomogeneousacrosstheventriclesandtrans-verselyisotropicwithrespecttothemyobreorientations.
1.
1.
1InitialConditionsToaccountforthefrequencydependenceofthecellmodel,wepacedanindividualcelltoalimitcycleat1Hzfor500beats.
Theisolatedcelllimitcyclewasusedasanapproximationforthelimitcycleintissue.
Thelimitcyclereachedbythiscellmodelwasusedastheinitialstateofthecellmodelsinthewholeheartsimulation.
Stimuliweredenedas1mm3cubesinthemyocardium,representingboththeintrinsicstimuliintherightventricle(RV)andseptumforleftbundlebranchblock(LBBB)sinusrhythmandthepacingleadsincardiacresynchronisationtherapy(CRT).
Withinthedenedvolumes,astimuluscurrentof100Acm3isaddedtothemonodomainequations,initiatingdepolarisationofthecellmodel.
1.
1.
2SimulationSetupandExecutionSimulationswereinitiallyrunusingCARPonARCHER(http://www.
archer.
ac.
uk/),theUnitedKingdomnationalHPCresource,using288coresfor2.
54hours.
Activationtimes,asde-terminedbythetimeofmaximumgradientofthethetransmembranepotential,wereexportedbythesimulationsoftware.
1.
2MechanicsModelThemodelofcardiacmechanicsincorporatedahyperelasticrepresentationofpassivetissuebehaviourusingtheGuccioneconstitutivelaw[3],aphenomenologicalmodelofactiveten-2siongenerationbasedonthemodelbyKerckhosetal.
[4]andathreeelementWindkesselmodeltorepresenttheafterloadboundaryconditions.
Underthisquasi-staticapproximation,theequationsofmotiontobesolvedare·σ+f=0(2)whereσistheCauchystresstensor,andfisthebodyforceperunitvolume.
ItisconvenienttosubstituteinthesecondPiola-KirchhostresstensorS,whichisrelatedtoσbyσ=J1FSFT(3)whereF=xXisthedeformationgradienttensorandJ=detFistheJacobeandetermi-nant.
InthedenitionofF,thereferencecongurationXisthecongurationinwhichthebodyisstressfree,oftenreferredtoastheunloadedconguration,andxisthedeformed,orloaded,congurationofthebody.
ThesymbolsXandxhererepresentthestatevectorsofthedescribedcongurations[9].
Smaythenbecalculatedasthesumofapassivecomponent,determinedbyastrainenergyfunctionW(E),andanactivecomponentSabyS=12WE+WETPC1+Sa(4)whereC=FTFistherightCauchy-Greendeformationtensor,E=12(CI)istheGreen-LagrangestraintensorandPisthehydrostaticpressure[9].
SimulationswereimplementedusingContinuumMechanics,Imageanalysis,Signalpro-cessingandSystemIdentication(CMISS)(http://www.
cmiss.
org/),developedattheUniver-sityofAuckland(Auckland,NewZealand).
CMISSisamature,parallelisedcode,andwasalreadycapableofsolvinglargedeformationcardiacmechanicswiththeGuccionemodel.
CardiacmechanicsweresolvedusingtricubicHermitebasisfunctionstodescribethege-ometryanddisplacementsofthemyocardium.
CubicHermiteelementsprovideasuccinct,C1continuousdescriptionofthecardiacanatomy.
Inaddition,whentheincompressibilityconstraintisaddedtothemodel,anadditionalhydrostaticpressureeldmustbeintroducedtotheconstitutiveequations.
Theinterpolationschememustbeofalowerorderthanthatofthegeometry,soacubicbasispermitsuseofalinearbasisforthehydrostaticpressurevariable,andthuscontinuousstressesacrosselementboundaries[10].
31.
2.
1PassiveTissueModelTomodelthepassivebehaviourofthemyocardium,weusedamodiedGuccioneconstitu-tivelaw[11],whichconsidersmuscletissuetobehyperelasticandanisotropicwithprincipalcomponentsalignedwiththemyobrestructure.
ThestrainenergyWisgivenbyW=C1eQ1(5a)whereQ=C2E2+C3E2ss+E2nn+2E2ns+2C4E2fs+E2nf(5b)E,EssandEnnarethecomponentsoftheGreen-LagrangestraintensorEinthebre,sheetandsheetnormaldirections,respectively,andEns,EfsandEnfarethecorrespondingshearstrains.
Ci(i∈{1,2,3,4})arethemodelparameters.
1.
2.
2ActiveTensionWeusedaphenomenologicalmodelofactivetension,originallydevelopedbyKerckhosetal.
[4],andmodiedtoreducethenumberofparameterswhilemaintainingrelevancetoten-siongenerationandsystolicfunction[12].
TheactivetensioninthedirectionofthemyocardialbresTawasgivenbyTa=T0φtanh2tctrtanh2tmaxtctdif0(6a)wheretcisthetimeaftertheonsetofcontraction,φisthenonlinearlengthdependentfunctionφ=tanh(a6(λa7))(6b)inwhichλisthestretchratiointhebredirection,andtr,whichregulatestherisetimeofthetensiontransient,isgivenbytr=tr0+a4(1φ)(6c)Thereare7parametersinthismodel;thepeakisometrictensionT0,thedurationoftensiongenerationtmax,thebaselineupstroketimeconstanttr0,thedownstroketimeconstanttd,thelengthdependenceoftheupstroketimeconstanta4,thedegreeoflengthdependencea6,andtherelativesarcomerelengthwherenoactivetensionisgenerateda7.
4Thetimeofmechanicalcontractionwasdeterminedbymappinglocalactivationtimesfromthemodelofcardiacelectrophysiologytothemechanicsmesh,plusaxedelectromechanicaldelay,whichisthedelayfromthedepolarisationofthemyocytecellmembranetothestartoftensiongenerationinthemyobres.
Thelocaltimerelativetotheonsetofmechanicalcontractiontcwasthereforecalculatedbytc=t(ta+tdelay)(7)wheretaisthelocalactivationtime,tdelayistheelectromechanicaldelay,andtisthecurrenttimeinthesimulationrelativetotheonsetofventriculardepolarisation.
LocalactivationtimestaweremappedfromthenodesofthetetrahedralmeshusedforelectrophysiologytotheGaussianquadraturepointsofthecubicHermitemeshusinganearestneighbourapproach.
ActivetensionwasincludedinthelargedeformationmechanicsequationsbyaddingitintothesecondPiola-KirchhostresstensorasshownaboveinEq.
4.
1.
2.
3BoundaryConditionsIsovolumetricconstraintswereimposedontheventricularcavitiesduringtheisovolumetriccontraction(IVC)andisovolumetricrelaxation(IVR)phases.
Inejection,thethreeelementWindkesselmodelwasusedtoregulateoutowinaphysiologicalway.
Thismodelisde-scribedbytheordinarydierentialequation(ODE)dUdt=PZRC+1ZdPdt1ZC+1RCU(8)whereUistheowrateofbloodoutoftheventricle,Pistheventricularpressure,Zistheaorticresistance,andRandCaretheresistanceandcomplianceoftheperipheralarterialcirculationrespectively.
Aswewereprimarilyinterestedinsystolicfunctioninthisstudy,weadoptedasimpliedandfastmodelofdiastoletoallowustocompletethefullheartcycle.
Theventricularpressureissetwithaprescribedfunctionalform.
Inadditiontotheaboveventricularcavityboundaryconditions,werestrictthemotionofthenodesatthebaseofthecomputationalmesh.
Withthemeshalignedsuchthatthebaseliesinthexyplane,motionofthebaseinthezdirectionispreventedbyconstrainingboththenodaldisplacementsandoutofplanederivativesofthedisplacementeld.
Thisrepresentstheeectofthestivalveplaneoncardiacmechanicsandsignicantlyimprovesthenumericalstabilityofthemodel.
Twonodesarealsorestrictedintheirmotionwithintheplane,toprevent5arbitraryrigidbodymotionandrotation,withoutimposingarticialconstraintsonthesolvedstate.
1.
2.
4SolutionProcedureSimulationsofcardiacmechanicswithCMISSwereexecutedontheHPCresourceattheDepartmentofBiomedicalEngineeringatKing'sCollegeLondon,using4coresfor1420hours.
2PersonalisationWorkowThepersonalisationofourmodelofcardiacelectromechanicsinvolvedmanysteps,butcangenerallybebrokendownintothreemajorsections;generationofapersonalisedcardiacgeometry,personalisationofelectrophysiologymodelinputsandparameters,andttingofmechanicsmodelinputsandparameters.
AbreakdownofthewholeworkowcanbeseenvisuallyinFig.
1.
2.
1GeometryPersonalisationOurmodelofcardiacelectromechanics,asintroducedinSection1,usedtheniteelementmethodtosolvetheequationsdescribingcardiacelectrophysiologyandmechanicalcontrac-tion.
Therstpartofourpersonalisationworkowwasthereforethegenerationofacomputa-tionalmeshwhichisanatomicallyaccurateandpatientspecic.
Asdescribedinthissection,thiswasderivedfromthepatient'sanatomicalmagneticresonanceimaging(MRI),whichprovidesadetaileddescriptionoftheventricularanatomy.
2.
1.
1MRISegmentationTheanatomicalMRIwassegmented,producingabinaryimagestackoftheventricles.
Seg-mentationwasperformedeitherusingitk-SNAP[13],anopensourceapplicationformanualorsemi-automaticsegmentationof3Dmedicalimages,orlaterintheprojectwasperformedusingafullyautomaticsegmentationsoftwaredevelopedbyourcollaboratorsatUniversityCollegeLondon[14].
AscanbeseeninFig.
2,theautomatedtoolsegmentedoutmanyotherregionsoftheheart,butasimplepostprocessingstepallowedabinaryimageofthemyocardialvolumetobegenerated.
6Figure1:Aowchartofthepersonalisationworkowusedinthisproject.
Ascanbeseenhere,thevariousclinicaldatawereprocessedbeforeintegrationintothemodel.
SeeSections2.
1,2.
2and2.
3fordetailofprocessingstepsingeometry,electrophysiologyandmechanicspersonalisationrespectively.
7Figure2:AutomatedsegmentationofcardiacMRI[14].
Predenedanatomicalregionswereassigneddierenttags(a),resultinginadetailedstructuralmodeloftheheart(b).
2.
1.
2MechanicsMeshGenerationThenextstepinthegeometrypersonalisationprocesswasthegenerationofaniteelementmesh.
Inourworkow,weusedacubicHermitemeshforthesimulationofcardiacmechanics.
MeshesweregeneratedusingtheautomatedcubicHermitettingapplicationdevelopedbyLamataetal.
[15,16].
Briey,atemplatemeshwitharegularellipsoidalshapewasalignedwiththesegmentationofthemyocardium.
Thistemplatewasthenwarpedtomatchthesegmentation,asillustratedinFig.
3.
Thiswasachievedbycreatingabinaryimagestackofthedomainofthemesh,thencalcu-latingawarpingeldbetweentherasterisedmeshvolumeandthebinarysegmentationusingtheSheeldImageRegistrationToolkit(ShIRT)[17].
Thiswarpingeldwasthenassim-ilatedbackontothetemplatemeshusingavariationaltechniquedescribedin[15].
Thisprocesswasrepeatedseveraltimesinaniterativeapproach,successivelyimprovingthematchbetweenmeshandsegmentation.
Oncethispersonalisationprocesswascomplete,analpost-processingstepcroppedthemeshtoensureaatbaseplane,facilitatingthelaterimpositionofboundaryconditionsinsimulations.
2.
1.
3ElectrophysiologyMeshGenerationRelativelycoarse,highordermeshescreatedbythemethoddescribedinSection2.
1.
2aresuitableforthesimulationoflargedeformationmechanics,wherestressandstraineldsare8Figure3:Customisationofatemplatemeshtothebinarysegmentation.
Anellipsoidaltem-platemesh(a)wasalignedwiththesegmentation(b),thenwarpedtomatchitsgeometryusingthetechniquedescribedin[15,16](c).
Thettedmeshisshowninpanel(d).
generallysmoothlyvarying,butsimulationofcardiacelectrophysiologyrequiresameshwithamuchnerspatialresolution.
Closetothewavefrontofelectricalactivation,therearesharpspatiotemporalgradientsinthetransmembranepotentialwhichmustberesolved.
Previousstudieshaveshownthatsub-millimetrespatialresolutionisrequiredtoachieveconvergenceinthesimulationresultwhenusingthemonodomainequations[8,18].
Forthisreason,wegeneratedaseparate,highresolutionmeshtailoredforsimulationofcardiacelectrophysiol-ogy.
MeshesweregeneratedbyrstrenderingthettedcubicHermitemeshasabinaryimagevolumewitharesolutionof200m.
ThemeshingpackageTarantula(http://www.
meshing.
at/)usedthisbinaryimagevolumetocreateahighresolution(mean250medgelength)tetra-hedralmeshoftheventricles.
AsTarantulaalsomeshedtheventriclecavitiesandtheregionsurroundingtheheart,wenallypostprocessedtheoutputmeshtoremovenon-myocardium9elements,leavingahighqualitytetrahedralmeshoftheventricles.
2.
1.
4FibreMappingImagingofmyobreorientationsinvivowasnotavailableinourpatientcohort.
Toapproxi-matetheheterogeneousbredistributionthathasbeenobservedacrossthemyocardium,weintroducedrulebasedbreeldorientationsbasedonhuman[19]andcanine[20–22]mea-surements.
Settingupbredirectionsinthemodelwasthereforenotinthetruesenseapersonalisationstep,butthebreorientationwasdependentonthepatientgeometryandsoapersonalisedrulebasedbreeldhadtobegeneratedforeachpatient.
Inthemechanicsmesh,breorientationsweredenedbyanglesrelativetothelocalξcoordinatesoftheniteelementmesh,correctedtoensureorthonormality.
TheanglesusedaregivenbelowinTab.
1.
Theseangleswereinterpolatedacrosstheelementsofthemeshwithlinearbasisfunctions.
InterpolatinganglesrelativetolocalξcoordinatesratherthanusingCartesianvectorshastheadvantagethattheirorientationcanalsobeeasilyevaluatedindeformedcongurations.
Table1:Genericbreanglesfromhumanandcaninedata[19–22]usedinthemodel.
Thebredirectionwasdenedatthefollowinganglestotheanticlockwisecircumfer-entialdirection(ξ1),whenviewedfromthebasaldirection.
Positiveanglesindicatebredirectionstowardsthebase.
RegionAngle(degrees)EndoMidEpiLVfreewallbase60060LVfreewallapex832435LVseptumbase60--LVseptumapex83--RVfreewallbase60-60RVfreewallapex60-35RVseptumbase60--RVseptumapex60--Ontheelectrophysiologymesh,breorientationsweremappedfromthemechanicsmesh.
Todeterminethebredirectionforeachelement,itscentroidwasevaluatedandthecorre-spondingelementandlocalξcoordinateinthemechanicsmeshcalculated.
ThebreanglesweretheninterpolatedatthatpointandthecorrespondingCartesianvectorevaluated.
102.
2ElectrophysiologyPersonalisationWepersonalisedandsolvedourmodelofcardiacelectrophysiologyindependentlyofmechan-ics.
Thisthereforeformsthesecondmajorpartofourpersonalisationworkow.
Beforetheinputsandparametersofthemodelwerepersonalised,theclinicaldatatobeusedforthisprocesswasprocessedintoamoredirectlyusefulformat.
2.
2.
1X-Ray–MRImageFusionWeusedX-rayimagesfromangiographytodeterminethepositionsofpacingandnon-contactmapping(NCM)studycatheters.
ThespecialisedX-ray–MRI(XMR)setupusedintheseclinicalcasesallowedcatheterlocationstobetransformedintotheMRIscannercoordinatespace,andthuswithourcomputationalmeshes.
ThisregistrationworkwasdonepreviouslybyourcolleaguesintheBiomedicalEngineer-ingdepartmentatKCL[23–25].
ProvidedwithcatheterpositionsfromtheX-rayimages,andtheappropriatetransformationmatricestoMRIscannercoordinates,thepacingcathetersandNCMpotentialmapscanbevisualisedalongwithourcomputationalmesh,asillustratedinFig.
4.
2.
2.
2Non-ContactMappingProcessingTheEnSiteNCMsystem(St.
JudeMedical,St.
Paul,MN,USA)providedthefunctionalityofexportingvirtualendocardialpotentialtracesandendocardialgeometry,butnotmapsofdepolarisationtime.
However,duetopreviouslycompletedworkonthisdataset[26],wewereabletocalculatetheseactivationtimemapsfromthepotentialtraces.
Thevirtualpotentialtraceswereprocessedusingoneoftwoalgorithms.
Therstcalculatedtheactivationtimeforeachvirtualpotentialtraceinisolationbyndingthetimeofthemax-imumpositivegradientofthesignal.
Thiswasknownastheunipolarmethod.
Thesecondalgorithm,knownastheLaplacianmethod,constructedadiscreteLaplacianoperatorusingtheNCMgeometryandappliedittothevirtualpotentialsignals.
ThetimeofthemaximumLaplacianofthesignalwastakenastheactivationtime.
Fig.
5illustratestheoutcomeofthisprocessing.
WheretheLaplacianalgorithmproducedreasonableresults,withanactivationpatterncon-sistentwithavisualinspectionofthepotentialmapsovertime,theywereusedintheremainderoftheworkow.
Otherwise,theunipolaralgorithmwasused,whichgenerallyproducedless11Figure4:RegistrationofcatheterlocationsandvirtualendocardialsurfacefromNCMwithapersonalisedniteelementmesh(red,anteriorwallremoved).
Sucharegistrationallowedtheintegrationofrealpacingsitesandendocardialpotentialinformationintothemodel.
ColoursshownontheNCMgeometryrepresentthevirtualunipolarpotential,tunedtohighlightthepropagatingdepolarisationwavefrontfromlefttoright.
spatiallysmoothresultsbutwasmorerobustinmatchingtheobservedactivationpattern.
2.
2.
3StimuliElectricalstimuliwereaddedtothemodelforbothintrinsicandpacedactivationsites,andweredenedassmall(1mm3)cubesembeddedintheappropriatesideoftheventriclewall.
Thelocationandrelativetimingofthesestimuliweredeterminedasdetailedbelow.
Stimulustimesweredeterminedrelativetosinoatrial(SA)activationasareferencepoint,althoughaswedidnotmodeltheatriaonlytherelativetimingofthestimuliwasincludedinthesimulation.
IntrinsicActivationForintrinsicactivation,wedidnotexplicitlymodelthePurkinjenetwork,butinsteaddenedactivationsitesatlocationsintheRVfreewallandseptumbelievedtocaptureitseect.
12Figure5:ExampleNCMreconstructionsurfacewith(a)samplepotentialmapsatsuccessivetimepoints(redindicatingdepolarisation)and(b)thecorrespondingprocessedactivationtimemap,usingtheLaplacianalgorithm(colourspectrumwithorangeindicatingearlyacti-vationandbluelateactivation).
PatientspecicinformationdescribingthelocationoftheearliestsiteofactivationintheRVwasnotavailable,sotheintrinsicRVfreewallactivationsitewasestimatedfromanelectrophysiologicalstudyofisolatedhumanheartsintheliterature[27].
InourmodelthetimeofstimulationofthispointwaschosentocorrespondtothebeginningoftheQRScomplexonelectrocardiogram(ECG),whichindicatesventriculardepolarisation.
Intheseptum,weaddedastimulusattheearliestlocationandtimeofdepolarisation,asseenonNCMatsinusrhythm.
PacingLeadsThelocationsofpacingleadsfromXMRimagefusionwereusedtochooseappropriatestim-ulussitesinthemodel.
Duringstandardbiventricularpacing,thecoronarysinus(CS)andrightventricleapex(RVA)leadswerepaced100msafterSAnodepacingbythehighrightatrium(HRA)lead.
TherelativetimingoftheventricularpacingleadswiththeHRAleadallowedsynchronisationofthepacedandintrinsicstimuli.
13AnexampleofpersonalisedintrinsicandpacedstimuluslocationsisshowninFig.
6.
Figure6:Stimulusvolumesweredenedassmall(1mm3)cubes(blue)embeddedinthemyocardium(red).
TheseptumandRVintrinsicactivationsiteswerederivedfromNCMandliteraturerespectively,whiletheLVandRVpacinglocationswereplacedbasedonX-rayderivedcatheterlocations(yellow).
2.
2.
4ConductionBlockNoneofthepatientsinthismodellingstudyexhibitedscaronMRI,sothisdidnotneedtobeincorporatedintothemodel.
However,insomecasesNCMrevealedalong,narrowregionontheanteriorwalloftheLVthroughwhichtheactivationwavedidnotpropagate.
Instead,theactivationwavehadtotaketheslowerpatharoundthisblockregion.
Theseregionsappearedtobeconsistentbetweenthevariousactivationmodesforwhichdatawasavailable.
Inordertocharacterisethisbehaviourinourmodel,wedenedthintransmuralregionsofverylowtissueconductivity.
AnexampleofsuchablockregionisshowninFig.
7.
2.
2.
5TissueConductivityInordertopersonalisethemodelofcardiacelectrophysiologytoeachpatient,wettedthetissueconductivitytotheavailableelectrophysiologicaldata.
Activationmapsdidnotshowcontinuousprogressionoftheactivationwave,howevernoiseinthesignalattributableto14Figure7:AnexampleofalowconductionregiontoreplicateelectricalblockobservedonNCM.
Allelementsinsidetheblueslabwereassignedaverylowconductivity.
cathetermotionartefactsanddilatedLVsizemeantthatwedidnothavesucientcondenceinthemeasurementstotaregionalconductionvalue.
Tomitigatetheimpactofsignalnoiseasingleconductivityvaluewasttedacrossthemyocardium.
Wemodelledthetissueconductivityastransverselyisotopicwithrespecttothemyobreorientations.
Weassumedaxedratioofbretocrossbreconductivitybasedonpreviousmeasurementsin[28],asttingbothconductivitiesledtoanunderconstrainedoptimisationproblem.
TheQRSduration(QRSd)wasmeasuredmanuallyfromtheECGrecordedbytheNCMsystem,usingthecalliperfeatureinthesoftwareinterface.
ThestartofthecalliperwasplacedatthelocalminimumimmediatelybeforetheQRScomplex,orimmediatelybeforethebe-ginningoftheprincipalupstrokeifnominimumwasidentied.
TheendofthecalliperwasplacedbythesamecriteriaimmediatelyfollowingtheQRScomplex,andthetimedurationcalculatedbytheNCMsystemwasrecorded.
QRSdurationsweremeasuredforthreerepre-sentativebeatsofeachactivationmode,andtheaveragewasusedfortting.
Thexedratioconductivitieswerethenttedbyasimplealgorithmbasedonsuccessivelinearinterpolation:1.
Twoinitialsimulationswererunwithlongitudinalconductivitiesof0.
3and0.
5Sm1,encompassingareasonablephysiologicalrange.
2.
Thetotalactivationtimeoftheventricleswascalculatedfromthesesimulationswithanautomaticpostprocessingstep,andcomparedwiththetargetvalue(theQRSduration).
153.
Anewtrialvalueoftheconductivitywasestimatedbyinterpolationbetweenthetwosimulationsboundingthetargettotalactivationtime,orextrapolationfromtheclosesttwoifthetargetwasnotbounded.
4.
Thistrialvaluewasthensimulatedandthecorrespondingtotalactivationtimecalcu-lated.
5.
Steps3and4wererepeateduntiltheestimatedtrialvaluewasequaltothepreviousonetothreesignicantgures.
2.
2.
6ValidationSimulatedactivationtimemapsfortheLVendocardiumwerecomparedqualitativelywiththeactivationtimemapsfromNCM.
WesimulatedthefourpacingmodalitiesmentionedinSection2.
2.
3,usingtheconductivitiesttedatsinusrhythm.
WealsocheckedforagreementbetweenthesimulatedandmeasuredtotalandtotalLVendocardialactivationtimes.
2.
3MechanicsPersonalisationCardiacmechanics,includingtissuepassivestiness,Windkesselmodelboundaryconditionsandactivecontractionmodelparameters,werettedtoavailableclinicaldata.
However,someofthisdatawasnotdirectlyusableinthemodelttingprocesssorsthadtobeprocessedintoamoredirectlyusableformat.
2.
3.
1VentricularVolumeVolumetransientswerederivedfromasegmentationofthepreclinical3Dechocardiogram(ECHO).
ThiswasdonebyexperiencedcliniciansusingtheTomTecanalysissoftware(TomTecImagingSystems,Unterschleissheim,Germany),whichallowedforrapidsemi-automatictrackingoftheLVendocardiumacrossthewholeheartcycle.
TheTomTecsoftwarepro-videdusefulclinicalmetricssuchasejectionfraction,butimportantlyalsoprovidedatraceoftheventriclevolumeovertime.
2.
3.
2VentricularPressureVentricularpressurewasrecordedduringsinusrhythmandeachpacingprotocol,withsev-eralrecordingsmadeforeachmodeofactivation.
ThePhysioMonsoftware(RadiMedical16Systems,Uppsala,Sweden)savesthisdatainaneasilyparsableformat.
Eachpressurerecordingwasrstsplitupintoseparatepressurebeats,usingmarkersde-terminedbyPhysioMonandexportedalongwiththepressuredata.
Anyectopicbeats,char-acterisedbyearly/delayedonsetofsystoleorbyabnormallyhighorlowpeakpressure,werediscarded,alongwiththetheprecedingandfollowingbeats.
Theremainingbeatswerere-sampledandaveragedforeachrecording,providingasmoothed,representativebeatforthatactivationmode.
2.
3.
3Pressure–VolumeSynchronisationPressuredatawasrstsynchronisedwithECGbyuseofdatarecordedintheNCMstudy.
Fromamanualinspectionofthe3DECHOdatafromwhichthevolumetraceswerecalcu-lated,anosetwiththepeakoftheRwaveonECGwasdetermined.
OncepressureandvolumeweresynchronisedwitheachotherthroughtheECG,apressure-volume(PV)loopwasplotted.
AnexampleofthisisshowninFig.
8.
Figure8:Resultofpressure–volumedatasynchronisationforonepatient.
Oncethepressureandvolumetraceshavebeensynchronisedintime(a),aPVloopcanbeplotted(b).
2.
3.
4WindkesselModelTheWindkesselmodelwastdirectlytoPVdata.
Usingthepressuretransientduringtheejectionphaseasaninput,thecorrespondingventricularvolumewascalculatedbyintegrating17thethethree-elementWindkesselmodelODE(Eq.
8)andtherelationofoutowUtoven-tricularvolumeU=dVdt.
TheseequationsweresolvednumericallyusingtheODEintegratorfromtheSciPy(http://www.
scipy.
org/)scienticPythonlibrary.
ThethreeparametersoftheWindkesselmodelwerettedbycalculatingaresidualbe-tweenthesimulationandclinicalvolumetrace.
Thisresidualwasthel2normoftheerrorsbetweenthesimulationandclinicalvolumetransients,augmentedbyadditionalconstraintsontheejectionfractionanddurationofejection.
Importantly,aconstraintforcingretrogradeowattheendofejectionwasintroduced.
Thisensuredphysiologicalbehaviourinthefullcoupledmodelofelectromechanicsattheendofejection,whensuchaowreversalisdetectedandthesimulationmovestotheIVRphase.
AnexampleofamodelttedusingtheaugmentedconstraintcanbeseeninFig.
9.
Figure9:ExampleofaWindkesselmodelsimulationwithparametersttedusingtheim-provedcostfunctionenforcingthepredictionofretrogradeowattheendofejec-tion.
Ascanbeseenonthesimulatedvolumetrace(a)andPVloop(b),themodelpredictstheowofbloodbackintotheventriclefromtheaorta,asindicatedbythedashedline.
Thisreverseowisnotseeninarealheartcycle,astheaorticvalveclosestopreventretrogradeow,andtheheartenterstheisovolumetricrelaxationphase.
TheWindkesselmodelparametersandthetimeofthestartofejectionwerettedtotheclinicalvolumetracewithacombinationofparametersweepsandlocalrenementusingtheimplementationofthedownhillsimplexalgorithmintheSciPyscienticPythonlibrary.
Weranalargesetofoptimisationsfromdierentinitialguesses,whichwerechosenusingafull18factorialexperimentaldesign.
Usingthisapproach,WindkesselmodelparameterswerettedfortheLV,takingaround15minutespercaseusingasinglecoreonaworkstationcomputer.
RVparameterswerenotpersonalisedduetothelackofavailablepressuredata,soweredenedrelativetotheLVparametersusingratiossourcedfromcanineandporcineexperiments[29–31].
Theseratioswere0.
35forZ,0.
125forRand4.
5forC.
2.
3.
5TissueStiffnessAsintroducedinSection1.
2.
1,themodiedGuccioneconstitutivelawbyOmensetal.
[11]has4parameters,C1toC4,whichgovernthematerial'spassivedeformation.
Wedidnothaveenoughdatatouniquelyconstraintheconstitutiveparameters,soinsteadassumedxedanisotropyratiosbasedonpreviousexperimentalmeasurementsinordertoimprovetheiden-tiabilityoftheparameterestimation[12].
TheseratioswerexedatC3=12C2andC4=14C2.
Thisallowedustorecastthestrainenergyfunction(Eq.
5)asW=C(eαQ1)(9a)whereC=C1andQ=E2+12E2ss+E2nn+E2fs+E2fn+E2ns(9b)WethenonlyneededtotthetworemainingparametersCandα.
ThiswasachievedbyttingtheLVpressure-volumerelationshiptoclinicaldatainlatediastole,whenthemyocar-diumisassumedtobequiescentandwecanneglecttheeectsofactivetension.
PersonalisedReferenceGeometryInSection2.
1wegeneratedapatientspecicmodelgeometrybasedontheenddiastolicstate.
However,whenmodellingcardiacdeformationswerequireanunloaded,orstressfree,geometry.
Toaccountforthenon-zeroenddiastolicpressureload,weperformedadeationsteptoestimatethestressfreeconguration.
Areformulationoftheniteelementequationsasdescribedin[32]wasusedtocalculatethereferencestatefromadeformedstate.
Inthisformulation,theresidualfunctionwasposedintermsofthereferencestateratherthanthedeformedstate,whichwasthensolvednumerically.
ImposingthebaseplaneboundaryconditionsintroducedinSection1.
2.
3,thereferencestatewascalculatedusingtheabovemethod.
PressuredatawasonlyavailablefortheLVsoRV19pressurewasapproximatedas50%oftheLVpressure.
Thiswasbasedontheratioofdiastolicpressuresinthetwoventriclesrecordedincanineexperiments[33,34]andintheclinic[35].
Thesolvedreferencecongurationdependedonthetissuestinessparameters,sothisstepmustberepeatedforeachstinessweexaminewhiletting.
PassiveInationCalculationofthepassivepressure-volumerelationshipfortheventricleswasdonebyapas-siveinationsimulation.
Startingfromthereferencestate,weincreasedthecavitypressurein0.
2kPaincrementsuptotheenddiastolicpressure.
Ateachpressurelevel,thevolumeoftheventriclecavitieswascalculated,andtheresultingpressure-volumerelationshipwasrecordedforlateranalysis.
AllreferencestateandsubsequentpassiveinationsimulationswererunusingCMISSonaworkstationcomputer,using4cores.
ParameterFittingTottheconstitutivelawparametersinEq.
9,weranparametersweepsonCandα.
Foreachsampledpairofparameters,wecalculatedtheassociatedreferenceconguration,andranapassiveinationsimulationuptotheenddiastolicstate.
AcostfunctionwasevaluatedwhichsampledthePVrelationshipuniformlybetween95%and100%oftheenddiastolicvolume,andcomputedthel2normofthedierencebetweentheclinicalandsimulatedpres-sures.
WealsoincludedaconstrainttoenforceaminimumLVcavityvolumeinthereferencecongurationofV50%,thevolumeat50%ejection(12VIVC+12VIVR).
A5*5fullfactorialparametersweeponCandαwasrun,withvaluesevenlyspacedontheranges10kPato20kPaand10to20respectively.
Ifasuitablematchwasnotfoundontherstsweep,extensionsorrenementsofthesweepwererun.
2.
3.
6ActiveTensionActivetensionmodelparameterswerettedtotheclinicalpressureandvolumedata.
UsingthesetupdescribedinSection1.
2.
4,simulationsofcardiacelectromechanicswererunforgivensetsofactivetensionparametervalues.
Thesimulationseachgeneratedalewithventriclepressuresandvolumesovertime,facilitatingacomparisonwithclinicaldatasuchasthatseeninFig.
10.
Notethatthedierenceinclinicalandsimulatedvolumesintherst150msinFig.
10bistobeexpected,asduringthisperiodtheLVisstillindiastole.
Sincewesimulatedonlya20Figure10:Comparisonofclinicalandsimulated(a)pressuretransients,(b)volumetransientsand(c)correspondingPVloops.
singlebeat,theLVwasplacedinitsenddiastolicstateatt=0,andremainedquiescentforsometimeduetothedyssynchronouscontractionofLBBB.
ThesimulationwasterminatedattheendofIVRasdiastolicpassivellingdoesnotaectthecalculatedcostfunctionforactivetensionmodeltting.
CostFunctionDesignAcostfunctionwasdevelopedbasedonthecalculationofmetricsdescribinggeometricalfeaturesofthepressureandvolumetransients.
Thecostfunctiontargetedglobalfeaturesofthepressureandvolumetransientsasana¨vel2normofthedierencebetweensimulatedandclinicaltransientswasfoundtobeoverlysensitivetospecicfeatures,forexampletheearlyupstrokeofthepressuretransient.
Thecalculatedmetrics,asseeninFig.
11,weredenedinasucientlyrobustwaysothatthesamealgorithmcouldbeappliedtobothclinicalandsimulationdata.
Our'geometric'costfunctioncombinedthesemetrics{pi}togetherusingthefractionaldierenceofthesimulationvaluewiththeclinicalvaluetomitigatetheeectofverydierent21Figure11:Calculatedmetricsusedinthe'geometric'costfunction.
Fromthepressuretran-sient(a),peakpressureandpeakdPdtontheupstrokewerecalculated,inadditiontoupstrokeanddownstroketimes,startingandnishingfromthetimeatwhichpressurewas5%betweenthestartingpressureandpeakpressure.
Onthevolumetransient(b),ejectionfractionandtimewerecalculated,similarlystartingfromthetimeof5%ejection.
magnitudes.
ItwascomputedasRg=r(10a)whereri=ωipclinicalipsimipclinicali(10b)and{ωi}areasetofweightsthatweremanuallyadjustedtoprioritisefeaturesdeemedmoreimportant.
TheweightsusedaregiveninTab.
2.
Table2:Componentweights{ωi}ofthe'geometric'costfunction(Eq.
10).
Residualcompo-nentsarethoseshowninFig.
11.
ComponentWeightωPeakPressure5.
0UpstrokeTime1.
0DownstrokeTime1.
0PeakdPdt1.
0EjectionFraction5.
0EjectionTime5.
0Itwasfoundnecessarytoaddanadditionalconstrainttothecostfunction,includingpres-suredatafrompacedactivationmodes.
Modelsttedwithoutthisdataexhibitedagoodt22atsinusrhythm,butfailedtoshowanysignicantchangeonpacing.
Tobringourmodelstoparametervaluesreplicatingtheresponseofthehearttopacing,simulationsofcontrac-tionunderstandardbiventricular(BiVSIM)pacingwererunandtheacutehaemodynamicresponse(AHR)calculated.
AHRiscomputedasAHR=maxdPdtpacedmaxdPdtbaselinemaxdPdtbaseline(11)wheremaxdPdtisthepeakrateofchangeofpressureontheupstroke,asseeninFig.
11.
Weposedaresponsecostwhichwasagainthefractionaldierencebetweensimulationandclini-caldataRr=AHRclinicalAHRsimAHRclinical(12)andcombineditwiththegeometriccostRgtoreachthefullcostfunctionR=αRg+(1α)Rr(13)withthenewparameterαprovidingcontrolovertherelativeweightsofthegeometricandresponsecosts.
ParameterSweepsOfthe7parametersintheactivetensionmodel(seeSection1.
2.
2fordetails),6werettedtoclinicaldata(specicallyT0,tr0,td,tmax,a4anda6),whilea7wasxedtoanexperimentallyvalidatedvalueof0.
7[36].
WesampledtheparameterspaceusingLatinhypercubesampling(LHS)[37].
LHSdesignshaveanadvantageovermoreconventionalfullfactorialdesignsinthattheyprovideagoodcoverageofallparameters'ranges,andalsohaveanadvantageoversimplerandomsamplingastheyensureamorehomogeneoussamplingdensity.
LHSalsooersthepracticaladvantagethatthenumberofsamplesisindependentofthedimensionalityoftheparameterspace.
Fittingoftheactivetensionmodelwasdoneusinganiterativeapproach.
Initialsweepswith150sampleswererunusingtheparameterrangesgiveninTab.
3,withlaternarrowersweepsreningthesearch.
23Table3:InitialactivetensionparametersweeprangesusingtheLHSdesign.
ParameterRangeUnitsT080200kPatr010100mstd80150mstmax450600msa42001000msa63712.
3.
7ValidationOncethettingprocessesoutlinedinthissectionwerecompleted,validationwasperformedbycheckingforagreementbetweenthesimulateddeformationoftheheartatsinusrhythmandthepre-implantationcineMRI.
ShortaxiscineMRIstackswereregisteredwiththemodelgeometryusingtheembeddedscannerorientationinformation.
Animageslicehalfwaybetweentheapexandbasewasse-lectedforcomparisonwithsimulationresults,andatissueboundarycontourwasgeneratedfromthemodelinthesameplaneandoverlayedontheimage.
Frameswerethengeneratedat100msintervals,andavisualcomparisondonetovalidatethatthemodelaccuratelyrepro-ducedventriculardeformationsatsinusrhythm.
2.
4EnsuringUniquenessofFitAsourcomputationalmodelofcardiacelectromechanicshasalargenumberofparametersandinputs,wemustbecarefultoensureauniquettomodeldatawhenperformingthemodelpersonalisation.
Thenecessarycomplexityofthemodelresultsinalargeandnonlinearparameterspacewithinwhichitisnotfeasibletoguaranteeuniqueness.
Ourpersonalisationapproachthereforeexploitstheabilitytoseparateseveralpartsofthemodelandtthemtodierentdataorphasesoftheheartcycle.
Asexplainedinthissupplement,tissueconductivityisttedusingECGdata(Section2.
2.
5),theWindkesselmodelisttedusingpressureandvolumetransientsduringejection(Section2.
3.
4),passivetissuestinessisttedusingthelatediastolicpressure-volumerelation(Section2.
3.
5),andthemodelofactivecontractionisttedtosystolicpressureandvolumetransients(Section2.
3.
6).
Thistargeteduseofdatawiththecomponentsforwhichtheyhavegreatestrelevanceincreasedourcapacitytoachieveaconstrainedparameterset.
24Furthermore,ourapproachfocusesonttingthoseparametersthatarepertinenttoourapplication.
Wheredatawasinsucienttopersonaliseallmodelparameters,valuesweredeterminedusingliteraturebasedmeasurementsorratiostootherparametersinordertoensureauniquet.
Thiswasthecasefortheanisotropyratioofthetissueconductivityσx/σf(Section2.
2.
5)andthesarcomerelengthratioatwhichnoactivetensionisgenerateda7(Section2.
3.
6).
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