A coupled 3D-1D numerical monodomain solver for cardiac electrical activation in the myocardium with detailed Purkinje network

Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

A

coupled

3D–1D

numerical

monodomain

solver

for

cardiac

electrical

activation

in

the

myocardium

with

detailed

Purkinje

network

Christian Vergara

a

,

∗

,

Matthias Lange

b

,

Simone Palamara

a

,

Toni Lassila

b

,

Alejandro

F. Frangi

b

,

Alfio Quarteroni

c

a_{MOX,DipartimentodiMatematica,PolitecnicodiMilano,Italy}

b_{CISTIB,DepartmentofElectronicandElectricalEngineering,TheUniversityofSheffield,UnitedKingdom} c_{ChairofModellingandScientificComputing,ÉcolePolytechniqueFédéraledeLausanne,Switzerland}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received10April2015

Receivedinrevisedform22November2015 Accepted7December2015

Availableonline21December2015 Keywords:

Computationalelectrocardiology Purkinjefibers

Monodomainequation Pullandpusheffect

Wepresentamodelfortheelectrophysiologyinthehearttohandletheelectrical propa-gationthroughthePurkinjesystemandinthemyocardium,withtwo-waycouplingatthe Purkinje–musclejunctions.Inboththesubproblemsthemonodomain modelisconsidered, whereasatthejunctionsaresistorelementisincludedthatinducesanorthodromic prop-agationdelayfromthePurkinjenetworktowardstheheartmuscle.Weproveasufficient conditionforconvergenceofafixed-pointiterativealgorithmtothenumericalsolutionof thecoupledproblem.Numericalcomparisonofactivationpatternsis madewithtwo dif-ferentcombinationsofmodelsforthecoupledPurkinje network/myocardiumsystem,the eikonal/eikonalandthemonodomain/monodomainmodels.Testcasesareinvestigatedfor bothphysiologicalandpathologicalactivationofamodelleftventricle.Finally,weprove thereliabilityofthemonodomain/monodomaincouplingonarealisticscenario.Ourresults underlietheimportanceofusingphysiologicallyrealisticPurkinje-treeswithpropagation solvedusingthemonodomainmodelforsimulatingcardiacactivation.

©2015ElsevierInc.All rights reserved.

1. Introduction

ThePurkinjefibers areadensenetworkofspecializedcellslocatedundertheinnersurfaceoftheheart(theendocardium) and are responsible for the fast conduction of the activation signal from the atrioventricular node to the heart muscle (myocardium). Theinclusion ofthePurkinjefibers incomputationalmodels ofelectrocardiologyhasbeeninrecentyears recognized asfundamental toaccuratelydescribing theelectricalactivationintheleft ventricle

[1–7]

.Thesefibers forma network,whichrepresentstheperipheralpartoftheconductionsystem.

The electrical propagation in the Purkinje fibers has been treated with different mathematical models derived from those commonly usedfor theelectrical propagation inthe myocardium. We citefor examplethe eikonal model [4–6,8], the monodomain model [9,10], and the bidomain model [11,12]. In normal electrical propagation, the electrical signal, originating fromtheatrioventricular (AV) node, travelsalong thisnetwork andenters theventricular muscle through the

*

Correspondenceto:PiazzaLeonardodaVinci32,20133,Milan,Italy.Tel.:+390223994610.

E-mailaddresses:christian.vergara@polimi.it(C. Vergara),m.lange@sheffield.ac.uk(M. Lange),simone.palamara@polimi.it(S. Palamara), t.lassila@sheffield.ac.uk(T. Lassila),a.frangi@sheffield.ac.uk(A.F. Frangi),alfio.quarteroni@epfl.ch(A. Quarteroni).

http://dx.doi.org/10.1016/j.jcp.2015.12.016 0021-9991/©2015ElsevierInc.All rights reserved.

Purkinje–musclejunctions (PMJ).Inpathologicalsituations,suchastheWolff–Parkinson–White (WPW)syndrome,thesignal mayenterthemyocardiumfromdifferentregionssothattwofrontspropagateatthesametime,onefromthenetwork to-wardsthemyocardiumandanotheroneintheoppositedirection.Capturingthecouplednatureofpropagationarisingfrom theinteractionbetweenthePurkinjenetworkandthemyocardiumisafundamentalmodeling issue.Inthisregard,different coupledmodelshavebeenconsideredintheliterature.Wecitethecoupledeikonal/eikonalmodel[13,5,6,8](thefirstmodel referstotheoneusedforthenetwork,whereasthesecondtotheoneusedforthemyocardium),theeikonal/monodomain model[4],the monodomain/bidomain model[9],the monodomain/monodomainmodel [10], andthebidomain/bidomain model[11,12].Whenmonodomainorbidomain modelsare considered,theissueofthecouplingbetweencardiac muscle andPurkinje network shouldbe properly addressed. Indeed,dueto the parabolicnature ofthese models,an explicit al-gorithm based onthe successivesolution ofthe propagationin the networkandin the muscleonly once per time step could notreproduceaccurate resultswhen multiplefrontsare propagating(ashappensforWPW).Theworkscitedabove consideredexplicitcouplingstrategieswhichgiveaccurateresultsonlyforanormalpropagation.

Inthiswork,westartfromthemonodomainmodelproposedforthePurkinjenetworkin

[9]

andconsiderthecoupling withthe monodomain model in themyocardium, obtaining a monodomain/monodomain coupled problem(Sect. 2). We observethatwiththemonodomainformulationwearenotabletostudysomespecificfeatureofpathologicalpropagations, suchastheheartfibrillation.However,thismodelisabletocapturemanycharacteristicfeaturesoftheelectricalpropagation in the heart, in particular it is suited in view of the electro-mechanical coupling. Since it allows to highly reduce the computationaltime withrespecttothebidomain model,inthisworkwechose thissimplifiedmodel.Thisallowedusto perform a well-posednessanalysis ofthe coupledproblem(difficultly applicable to the bidomain context). Moreover, we introduce a semi-implicit time discretization andan iterative algorithm forthe solution of the coupled problem arising ateach time step(Sect.3). We observethatthe proposed algorithmallows ustotreat implicitlythecouplingconditions betweenPurkinjenetworkandcardiacmuscle,thussolvingatrulycoupledproblemwhichisabletodescribealsosituations wheremultiplefrontspropagate.

Finally,wepresentseveralnumericalresultswiththeaimofassessingtheeffectivenessoftheproposedalgorithmand comparing the solutions with the onesobtained withthe eikonal/eikonal model.In particular, we discussthe choice of the conduction velocities in the eikonal andmonodomain models required to obtain comparable results (Sect. 4.1). We alsoperformacomparisonforabenchmarktestbetweentheeikonal/eikonalandthemonodomain/monodomainstrategies, highlightingthe“pullandpush”effect(Sect.4.2)andthenconsiderbothnormalandpathological(WPWsyndrome) prop-agationsinan idealellipsoidalmodeloftheventricle (Sect.4.3).Interestingly,withouralgorithmwe areabletorecover, withoutanyaprioriimposition,someofthemoreinterestingfeaturesoftheelectricalpropagationintheheart,suchasthe pullandpusheffect,andthedelayatthePMJ.Finallywe applythemonodomain/monodomainmethodologytoarealistic case(Sect.4.4).

2. Mathematicalmodelsfortheelectricalactivation

Inthissectionweprovidethemathematicalmodelsconsideredinthisworkforthedescriptionoftheelectricalactivation inthemyocardiumandinthePurkinjenetwork,andthecorrespondingcoupledproblem.Wewillusethesubscriptsm and p tocharacterizethequantitiesrelatedtothemyocardiumandtothePurkinjenetwork,respectively.

2.1. Activationinthemyocardium

2.1.1. Monodomainmodelinthemyocardium

Thebidomain model, whichaccountsforthepropagationoftheextra- andintra-cellular potentials(see,e.g.,

[14–16]

), isthe mostcommonlyusedmodel todescribe theelectrical activation inthe myocardium.To reducethe high computa-tionalcostsassociatedtousingthebidomainmodel,thesimplermonodomain model,whichdescribestheevolutionofthe transmembranepotential Vm inthemyocardiumdomain



m,isoftenused.Itreadsasfollows:

GivenVm,0andwm,0,findVm

: 

m

× (

0,T

]

→ R

andwm

: 

m

× (

0,T

]

→ R

dm,suchthat

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

χ

m



C

m

∂

Vm

∂

t

+

I m ion

(

Vm

,

wm

)



− ∇ · (∇

Vm

)

=

I in



m

× (

0

,

T

),

dwm dt

=

fm

(

Vm

,

wm

)

in



m

× (

0

,

T

),

(

∇

Vm

)

n

=

0 on

∂

m

× (

0

,

T

),

Vm

(

x

,

0

)

=

Vm,0

(

x

),

wm

(

x

,

0

)

=

wm,0

(

x

)

in



m

,

(1)

where



istheconductivitytensorgivenby

(

x

)

=

σ

tI

+



σ

f

−

σ

t

af

(

x

)

af

(

x

)

T

,

σ

t and

σ

f aretheconductivitiesintheorthogonalandlongitudinaldirectionswithrespecttothefibers,andaf istheunit

vectoralignedwiththefibers.

χ

misthesurface-to-volumeratioofthecellmembrane,

C

misthemembranecapacitance,Iionm

representstheioniccurrents(more precisely,currentdensitiespersurface unit), wm istheunknownvectorthat includes

thegatingandionconcentrationvariablesoftheODEsystemrepresentingasuitablecellmodel,andthevectorialfunction f_misanon-lineartermwhichdeterminestheevolutionofwm.Wehaveconsideredtheno-fluxboundarycondition

(1)

3on the ventricle

[1]

.The forcingterm I,representingan externalcurrent(moreprecisely,a currentdensitypervolumeunit), willbespecifiedoncewecouplethissystemwiththe1Dmonodomainone,seeSection2.3.1.

Forthesakeofexposition,inwhatfollowswewillcompactlywriteproblem

(1)

asfollows

Pm

(

Vm

,

wm

,

I

)

=

0

.

The monodomain model is based on the assumption of equal anisotropy ratio between the intra- and extra-cellular domains. If thereis no injectionof currentintothe extracellular domain,this modelis indeeda good approximation of themorecomplexbidomainone

[17,18]

.Noticethat

(1)

isacoupledproblem,sincethetransmembranepotentialandthe gating/ionconcentrationvariablesappearinboththedifferentialproblemsthroughthecouplingtermsIion

m and fm.

2.1.2. Eikonalmodelinthemyocardium

If one is interested only in the ventricular activation times, defined as the time at which the potential reaches the intermediate value between the maximum and the resting potential [19,20], then a further simplified model could be considered, namelytheeikonalmodel, thatprovides ateachpoint theactivation time.Thismodeldiscards allthe cellular kineticsanddescribesonlythemacroscopicspreadingoftheexcitationwavefronts.Assuch,itdoesnotrequireafinespatial resolution,makingitpossibletosimulatetheactivationoflargevolumesofcardiactissueatlowcomputationalcosts.Itis indeedagoodapproximationofthebidomainmodel

[21]

forthecomputationofactivationtimes,whereas itisunsuitable todescribere-entrantphenomenasuchasarrhythmias.

Inthisworkweconsidertheanisotropic eikonalequation,whichreads:

Givenum,0,findtheactivationtimesum

: 

m

→ R

suchthat

Cf



(

∇

um

)

T D

∇

um

=

1 x

∈ 

m

,

um

(

x

)

=

um,0

(

x

)

x

∈ 

m

,

(2) where



m isthe set of boundary points generatingthe front, D

(

x

)

models the anisotropic tensor that accountsfor the

presenceofthemuscularfibers,andCf

(

x

)

representsthevelocityofthedepolarizationwavealongthefiberdirection.We

usethefollowingexpression

[20]

D

(

x

)

=

k2I

+ (

1

−

k2

)

af

(

x

)

af

(

x

)

T

,

(3)

where k isthe ratiobetweenthe conductionvelocities inthe orthogonalandlongitudinal directionswithrespect to the fibers.

Note thatsince we didnot consideranydiffusivetermin theeikonal problem,our modeldoesnottake into account theeffectsofwavefrontcurvatureortheinteractionbetweenawavefrontwitheitherthedomainboundariesorwithother fronts. This is justified by observing that in our case the myocardial activation is regulated by the Purkinje fibers, and becauseoftheirhighdensity,thediffusiontermgivesasmallcontributionwithrespecttotheadvectionone.

Problem

(2)

canbesolvedveryefficientlybythefastmarchingmethod

[22]

andhasbeensuccessfullyusedforclinical applications,see

[13,5,6]

.

2.2. ActivationinthePurkinjenetwork 2.2.1. Monodomainmodelinthenetwork

Both thecardiomyocytes in themyocardium and theones inthe Purkinje networkare electrically connected by gap-junctions, intercellular channelsproviding a low resistancepathwayfor thespreading ofthe actionpotential

[23]

.Unlike what is usually donein themyocardium where theeffectof thegap-junctions,asa consequenceof thehomogenization process, is hidden in the conductivitytensor D, in [9] the authors proposed to explicitlymodel a gap-junctions in the Purkinje networkasaresistor placedbetweentwoPurkinje cells.Thisallows usto easilywrite theKirchhofflaws atthe bifurcation points ofthe network,sincethepotential andthecurrentinthePurkinjecell/gap junctionunitare treatedas independentvariables.Moreover,ashighlightedin

[9]

,this“discrete”approach,inantithesistoahomogenizedone,allows onetodescribethesawtootheffect,see

[9]

formoredetails.

Fig. 1. Schematic representation of a gap-junction linking two Purkinje cells.

Inthisworkweusethediscretemodelproposedin

[9]

.Asaconsequence,thegap-junctionresistanceneedstobe com-patiblewiththehomogenized conductiontensorofasingle Purkinjecell/gapjunctionunit.Tothisaim,let

σ

p∗ denotethe

conductivityinthecells,andRg theresistanceoverthegap-junction.Bothtogether determinetheequivalent conductivity

σ

p

= (

σ

p∗l

)/(

l

+

σ

p∗Rg

πρ

2

)

ofasinglecell/gap-junctionunit,wherel isthelengthofthecelland

ρ

itsradius.Itisimportant

tonotethat,incontrastto

[9]

,thischoiceoftheequivalentconductivitydependsonthephysicalpropertiesofthecelland notonthenumericalparameters(spacediscretizationstep).

Wethereforehaveasequenceofelementaryunitscomposedby twoPurkinjecellsconnectedbyagapjunction,which are characterized by the same spatial coordinates (see Fig. 1). Each of these units is characterized by the extra-cellular andintra-cellularpotentialsandby thecurrentsrelatedto thecellsattheleft andattheright(identifiedwiththeindex

−

and

+

, respectively) and to the gap junction (identified with the index g). We assume here that the extra-cellular potential

φ

e is constant for each unit, so that we can consider the transmembrane potential as the effective potential

unknown.Thus,foreachunittheunknownsoftheproblemarethetransmembranepotentialsVg

,

V+p

,

V−p andthecurrents

Ig

,

I+p

,

I−p.

We assume that thebifurcation andintersection points ofthe networkare located incorrespondenceof some ofthe gap-junctions.Kirchhofflawsatthegenericbifurcationorintersectionyield

p



j=1

Ig,j

=

0

,

Vg,1

= . . . =

Vg,p

,

(4)

where p is thenumberof branchesissuingfromthebifurcation orintersection andwe adopttheusual convention that enteringcurrentsarepositiveandexitingcurrentsarenegative.

FromOhm’slawatthegap-junctions,wealsohave

Ig

= ±

Vg

−

V±p

Rg

/

2

.

(5)

TheintracellularcurrentI±_p thatflowsinthePurkinjecellcanbewrittenas

I±_p

= −

πρ

2

σ

p

∂

V_p±

∂

l

,

where

ρ

istheradiusofthePurkinjecell,and

σ

p istheequivalentintracellularconductivity

[9]

.Thankstotheconservation

ofcurrentsatthegap-junction, Ig

=

I+

=

I−,wehave

Ig

= −

πρ

2

σ

p

∂

V+_p

∂

l

=

πρ

2

_σ

p

∂

V−_p

∂

l

.

(6)

Tosummarize,themonodomainmodelwithgap-junctionsinthePurkinjenetworkisgivenbythemonodomainequation writtenin eachsegment ofthenetwork,together withtherelations atthe gap-junctions

(5)

–(6) andwiththe continuity relationsatthebifurcation/intersectionpoints

(4)

:

GivenVp,0

,

wp,0andhA V,findV±p,i

:

Si

× (

0,T

]

→ R,

Vg,i

:

Si

× (

0,T

]

→ R

andwp,i

:

Si

× (

0,T

]

→ R

dp

,

i

=

1,

. . . ,

P ,such

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

χ

p

C

p

∂

V±_p,i

∂

t

+

I ion p

(

Vp±,i

,

w±p,i

)



−

∂

∂

l

σ

p

∂

V_p±_,i

∂

l



=

0 in Si

× (

0

,

T

],

i

=

1

, . . . ,

P

,

∂

w±_p,i

∂

t

+

fp

(

V ± p,i

,

w±p,i

)

=

0 in Si

× (

0

,

T

],

i

=

1

, . . . ,

P

,

Vg,i

=

V+_p,i

+

Ig,₂iRg

=

V_p−_,i

−

Ig,₂iRg in Si

× (

0

,

T

],

i

=

1

, . . . ,

P

,

Ig,i

= −

πρ

2

σ

p ∂V+_p,i ∂l

=

πρ

2

_σ

p ∂V−_p,i ∂l in Si

× (

0

,

T

],

i

=

1

, . . . ,

P

,



ik pk i=ik₁Ig,i

=

0 at bk

,

k

=

1

, . . . ,

P

,

t

∈ (

0

,

T

],

V_g,ik 1

= . . . =

Vg,ikpk at bk

,

k

=

1

, . . . ,

P

,

t

∈ (

0

,

T

],

−

π σ

p

ρ

2

∂

V_p±

∂

l

(

s0

)

=

hA V t

∈ (

0

,

T

],

−

π σ

p

ρ

2

∂

V_p±

∂

l

(

sj

)

=

Nj j

=

1

, . . . ,

N

,

t

∈ (

0

,

T

],

V±_p

=

Vp,0

(

x

)

in



p

,

w±_p

=

wp,0

(

x

)

in



p

,

(7)

where Siarethesegmentsofthenetworksuchthat



iP=1Si

= 

p,



p beingthePurkinjenetworkdomain,

χ

p the

surface-to-volumeratioofthecellmembrane,Iion

p theioniccurrents(orcurrentdensitiespersurfaceunit),

C

p

(

x

)

isthemembrane

capacitance,l isthecurvilinearcoordinatealongthenetwork,s0thecoordinateoftheatrioventricularnode,sj

,

j

=

1,

. . . ,

N,

thecoordinatesofthePMJ,bk thecoordinatesofthebifurcationandintersectionpoints,andik1

,

. . . ,

ikpk arethe pk indices

relatedtothe potentialsandcurrentsinvolvedatthebifurcation/intersection pointbk.Equations

(7)

8 representNeumann boundary conditionsatthePMJ, which areeitherinlets oroutletsforthe system.We leave forthemomentthe data Nj

unspecified:theywillbeprovidedbythecouplingwiththemyocardialactivation,seeSect.2.3.1. Forthesakeofexposition,inwhatfollowswewillcompactlywriteproblem

(7)

asfollows

Pp

(

V+p

,

Vp−

,

Vg

,

Ig

,

w+p

,

w−p

,

N

)

=

0

,

wheretheunknownsaredefinedgloballyinallthenetworkstartingfromtheirvalueoneachsegment Si.

Acomputational convergenceanalysisofthenumericalsolutiontowards theexactoneforproblem

(7)

inthePurkinje network hasbeen performed by us in [24]. This is the first attempt to validate the fullydiscrete representation ofthe network given by gap-junction/Purkinje cell units. The results showed convergence of the solution both for steady and pulsatiletestcases.

2.2.2. Eikonalmodelinthenetwork

Inthecaseofanetworkofone-dimensionallinesegmentsrepresentingthePurkinjefibers,we canconsideragainthe eikonalmodelwithoutdiffusion:

Givenup,0,findtheactivationtimesup

: 

p

→ R

suchthat

⎧

⎨

⎩

Cp





∂

up

∂

l



 =

1 x

∈ 

p

,

up

(

x

)

=

up,0

(

x

)

x

∈ 

p

,

(8) where



p isthesetofpointsgeneratingthefrontinthenetwork(forexample,inanormalpropagation,theAVnode)and

Cp theconductionvelocity(5–10timesgreaterthanthemuscularone

[25]

).Again,weneglectthediffusiontermsincethe

highadvectiontermVp dominatesanydiffusionprocess.

2.3. Coupledproblems

ThePurkinjefibersformasubendocardialnetworkcharacterizedbyahighconductionvelocityandareisolatedfromthe muscle,exceptattheirendpoints,thePMJ,whicharelocatedontheendocardium.ThroughthePMJ,thesignalcouldeither entertheventriclefromthenetwork,asinanormalpropagation(orthodromicpropagation), orenterthenetworkfromthe myocardium, as happensforsome pathologicalconditions (antidromicpropagation), see,e.g.

[6]

. Inboth cases a delay at thePMJisobserved,inparticularanorthodromicdelaydo ofabout5–15 msandanantidromicdelayda ofabout2–3 ms [26,27].Thus,wehaveacoupledproblembetweentheelectricalpropagationinthe1Dnetworkandinthe3Dmyocardium wherethecouplingpointsarethePMJ.

In what follows, we describe two possible coupled strategies, namely the monodomain/monodomain (MM) and the eikonal/eikonal(EE)ones.

Fig. 2. Schematicrepresentationofagenericmyocardialdomainmandofagenericnetworkp.Thenodes0representstheAVnode,whereasthenode

s1ands2arethePMJ,whichactassourcetermsforthemyocardiumthroughthespheresofradiusr centeredinthePMJ.

2.3.1. Monodomain/monodomaincoupling

The MM strategy hasbeen introduced in

[9]

,and isbased on using(1) forthe myocardium and(7)for thePurkinje network. However, in that work the authors considered an explicit coupling between the two subproblems which was basedontheirsequentialsolution(networkfirstandthenmyocardium)onlyoncepertimestep.Thisdidnot allowedthe authorstotreatcaseswhere,besidesthefrontpropagatingfromtheAVnode,otherfrontsoriginatefromthemuscleasin pathologicalconditions.Oneofthemajornoveltiesofthepresentworkistoconsideranimplicitcouplingbetweenthetwo subproblems,asdetailedinwhatfollows.

Towritethecoupledsystem,we needtointroduceamodeldescribingthepropagationoftheelectrical signalthrough thePMJ.Fromhistologicalinspection,PMJsappeartobecomposedbytransitionalcellsconnectingtogether thedistalpart ofthe Purkinje fibers andthesurrounding myocardialcells [28]. Adetailedmodel ofthe PMJis presentedin [29], with the aimofstudyingthe conductiondelay atthe PMJ. However, inthiswork we considera simplermodel,based onthe introduction ofa PMJ resistance

[11,9]

,which provides a good approximation ofthe real behavior ofthe PMJ asshown in[30].Theinfluenceofthe PMJonthetwo subdomains(the myocardiumandthePurkinje network)hasbeen modeled intermsofexchange ofcurrents.Ononehand,thePMJs actassources forthemyocardiumthrough regions ofinfluence modeled asspheresofradiusr centeredinthePMJforasuitable r (see

Fig. 2

)

[11]

.Ontheotherhand,thePMJsprovide thecurrenttothenetworkthroughtheprescriptionofNeumannboundaryconditionsforproblem

(7)

(rememberrelation

(7)4).

Asdiscussed,thePMJhasbeenmodeledasaresistanceelement,sothatthecurrent

γ

j atthe j-thPMJcanbewritten

thankstotheOhm’slawasfollows

γ

j

=

V+p(sj)+V−p(sj) 2

−

1 Ar



Br(sj) Vmdx RPMJ j

=

1

, . . . ,

N

,

t

∈ (

0

,

T

],

(9)

where

B

r

(

sj

)

istheballofradiusr centeredatthepointsj, Ar thevolumeofthisballandRPMJtheresistanceofthePMJ

(supposedto bethe sameforallthe PMJ).The potentialappearing atthenumeratorofthe previousequation isnothing butthe jump betweenthe Purkinjenetwork potential and themyocardial potential atthe PMJ. Notice that the value of the potential fromthePurkinje network side havebeen chosen astheaverage ofthe two potentials V+_p and V_p− atthe terminalnodeofthenetwork(thePMJ),sincealsoherewehaveusedagap-junctionmodel.Instead,thepotentialfromthe myocardiumsidehasbeencomputedastheaverageofthemyocardialpotential Vm overtheballinvolvedintheexchange

ofthecurrent.

Summarizing,byusingthenotationintroducedintheprevioussubsections,thecoupledMMproblemreadsasfollows: Findforeacht,V+_p

,

V−_p

,

Vg

,

Vm

,

Ig

,

wp+

,

w−p

,

wm+wm− and

γ

j

,

j

=

1,

. . . ,

N,suchthat

⎧

⎪

⎪

⎨

⎪

⎪

⎩

Pm



Vm

,

wm

,



N j=1A1r

I

Br(sj)

γ

j

+

I ext



₌

₀

_,

Pp



V+_p

,

V−_p

,

Vg

,

Ig

,

w+p

,

w−p

,

γ

=

0

,

PPMJ



V+_p

,

V−_p

,

Vm

,

γ

=

0

,

(10)

where PPMJ

=

0 representsrelations

(9)

,

I

Y isthecharacteristicfunctionrelatedtotheregionY

⊂ 

m,andIextanexternal

current.

2.3.2. Eikonal/eikonalcoupling

Adifferentstrategyconsistsincouplingtheeikonalproblems

(2)

and

(8)

(EEstrategy).Againthecouplingisprovidedat thePMJ,sothattheset



m in

(2)

and



p in

(8)

couldcontainalsosomeofthePMJ.

UnliketheMMstrategy, inthiscaseitwasnecessarytoidentifytheorthodromicPMJs,that istheonesthat bringthe signalfromthenetworktothemyocardium,andtheantidromicPMJsthatbringthesignalfromthemyocardiumtothe net-work.Indeed,thesolutionsoftheeikonalproblemsrepresentfrontspropagatingfromtheirsourcepoints.Then,inourcase wehadingeneraltwofronts,onecomingfromtheAVnodeandanotheronegeneratedinthemyocardiumdueto patholog-icalconditions(suchastheWPWsyndromeortheleftbundlebranchblock).Wereferthereaderto

[6,8]

forfurtherdetails.

3. Numericalsolutionofthemonodomain/monodomaincoupledproblem

In this section we propose an algorithm for the numerical solution of the MM coupled problem (10). In particular, in Section 3.1we firstintroduce thetime discretization followedby afixed point algorithm,whose convergenceanalysis is carried out inSection 3.2. Finally,in Section 3.3we provide details aboutthe numericalsolution ofthe monodomain subproblemsarisingateachiterationofthefixedpointalgorithm.

ForthenumericalsolutionoftheEEcoupledproblemweadoptherethestrategyproposedin

[8]

foranormal propaga-tion,andextendedtotreatalsopathologicalconditionsin

[6]

.Wereferthereadertotheseworksforfurtherdetails. 3.1. Numericalalgorithm

Forthe3Dproblem

(1)

1weproposeasemi-implicittimediscretization,withthediffusivetermtreatedimplicitlythrough thebackwardEulermethod,andthecouplingtermIm_iontreatedexplicitly.Theequation

(1)

2isdiscretizedwiththeforward Eulermethod:

⎧

⎨

⎩

χ

m

C

m



t Vm

− ∇ · (∇

Vm

)

=

χ

m

C

m



t V n m

−

χ

mImion

(

Vmn

,

wnm

)

+

I in



m

,

wm

=

wnm

− 

t fm

(

Vmn

,

wnm

)

in



m

,

(11) wherewehavedroppedthecurrentindexn

+

1 intheunknownsonthelefthandsideforthesakeofsimplicity.

Thesameapproachwasconsideredforthetimediscretizationofthe1Dproblems

(7)

1 and

(7)

2:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

χ

p

C

p



t V ± p,i

−

∂

∂

l

σ

p

∂

V±_p,i

∂

l



=

χ

p

C

p



t



V_p±_,i



n

−

χ

pIionp



V±_p,i



n

,



w±_p

n



in Si

,

i

=

1

, . . . ,

P

,

w±_p,i

=



w±_p,i



n

− 

t f_p



V±_p,i



n

,



w±_p,i



n



in Si

,

i

=

1

, . . . ,

P

.

(12)

Withthisinmind,wecanintroducesuitableoperators



Pmand



Pp andcompactlywritethediscretized-in-timeproblems (11)and

(12)

as



Pm

(

Vm

,

I

)

=

0 and



Pp



V+_p

,

V−_p

,

Vg

,

N

=

0,respectively(N isagaintheNeumanndataprescribedatthe PMJ).Notice thatwedidnotexplicitlyindicatethedependenceoftheprevious operatorson wm

,

w+p

,

w−p sincethey are

not involved directlyin thecoupling, andthat thedependence of



Pp and



Pm on thequantities atprevious time step is

understood.Thisallowsustowritethediscretized-in-timeversionoftheMMproblem

(10)

asfollows:

Findforeachn,V_p+

,

V−_p

,

Vg

,

Vm

,

Igand

γ

j

,

j

=

1,

. . . ,

N,suchthat

⎧

⎪

⎪

⎨

⎪

⎪

⎩



Pm



Vm

,



N j=1A1r

I

Br(sj)

γ

j

+

I ext



₌

₀

_,



Pp



V_p+

,

V−_p

,

Vg

,

γ

=

0

,

PPMJ



V+_p

,

V−_p

,

Vm

,

γ

=

0

.

(13)

Forthesolutionofthediscretized-in-timeMMcoupledproblem

(13)

we proposeafixedpoint strategy,whereateach iteration thecurrents

γ

j computedatthepreviousiteration areusedto solvethe3D andthe1D problems,andthen the

valuesofthepotentialsareusedtoupdatethePMJcurrents.Thisideaissummarizedin

Algorithm 1

reportedbelow. 3.2. Analysis

In this section we provide a convergence analysisof Algorithm 1in the particular caseof



p composed by a single

branch,thuswithnobifurcations/intersections.Thankstothetimediscretizationused,ateachtimestepweobtainalinear problem, hencewe can restrict ourselves to analyzethe convergence towards the nullsolution in the caseof vanishing forcing terms. We can therefore set Iext

=

0,hA V

=

0, and nullinitial conditions, and set to zero the quantities at the

previous time step. Moreover, we notice that since we do not have anybifurcation/intersection points, there are no Vg

and Ig inthis caseandwehave only Vp to describe thecell potential, insteadof V+p and Vp−. Finally,we cantranslate

the solution Vp and Vm into Vp

=

0 and Vm

=

0 correspondingtotherestingpotential conditions.Moreover, weassume

thattheioniccurrentsarezerowhenthetransmembranepotentialequalstherestingpotential.Inview oftheanalysis,we introducetheweakformulationsofthemonodomainproblems.Thus,ourfixedpointstrategycanberewrittenasreported in

Algorithm 2

.

Algorithm1Solutionofthediscretized-in-timeMMcoupledproblem.

Let k be the iteration index within each time step. Set k=0 and γj(0) =γ0,j :=

 V+p n (sj)+  V−p n (sj) 2 − 1 Ar  Br(sj) Vn mdx RPMJ , j=1,. . . ,N, with  V+p n ,V−p n ,Vn

mtheconvergedsolutionattheprevioustimestep,andchooseatoleranceε>0;thenatiterationk+1

whileγ(k)_−γ(k−1)_{ >ε}

1. Solvethediscretized-in-timemonodomainproblem(1)inthemyocardiumwithappliedcurrentsgivenbyγ(k)_,thatis

Pm ⎛ ⎝Vm(k+1), N  j=1 1 ArIBr(sj) γ(_jk)+Iext ⎞ ⎠ =0;

2. Solvethediscretized-in-timemonodomainproblem(7)inthePurkinjenetworkwithNeumannboundaryconditionsatthePMJgivenbyγ(k)_,thatis

Pp  (V+_p)(k+1), (V−_p)(k+1),V(gk+1),γ(k)  =0; 3. Compute γ(k+1) j =  V+p (k+1)_(s j)+  V−p (k+1)_(s j) 2 − 1 Ar  Br(sj) Vm(k+1)dx RPMJ , j₌1, . . . ,N_; (14) 4. Setk=k+1. end

Algorithm2SolutionofthereducedMMdiscretized-in-timecoupledproblem. Letk betheiterationindexwithineachtimestep.Setk=0 andγ(0)_=γ

0,andchooseatoleranceε>0;thenatiterationk+1 whileγ(k)_−γ(k−1)_{ >ε}

1. Solvethefollowingdiscretized-in-timemonodomainprobleminthemyocardiumwithappliedcurrentsgivenbyγ(k)_:

FindVm∈H1(m)suchthat

 m χmCm Vm(k+1) t Wmdx+  m D∇Vm(k+1)· ∇Wmdx= N  j=1 1 Ar  Br(sj) γ(k) j Wmdx, (15) forallWm∈H1(m);

2. Solvethefollowingdiscretized-in-timemonodomainprobleminthePurkinjenetworkwithNeumannboundaryconditionsatthePMJgivenbyγ(k)_:

FindVp∈H1(p)suchthat

 p χpCp V(pk+1) t Wpdl+  p σp ∂V(pk+1) ∂l ∂Wp ∂l dl= − 1 πρ2 N  j=1 γ_j(k)Wp(sj), (16) forallWp∈H1(p);

3. Computethevalueofγ(jk+1)with(14);

4. Setk=k+1. end

Thecoupledproblemin

Algorithm 2

canberewrittenasfollows

⎧

⎪

⎪

⎨

⎪

⎪

⎩

Vm(k+1)

=

Fm

(

γ

(k)

)

in



m

,

V(_pk+1)

=

Fp

(

γ

(k)

)

in



p

,

γ

(k+1)

₌

_F PMJ



V(_pk+1)

,

V_m(k+1)

,

where Fm

: R

N

→

H1

(

m

),

Fp

: R

N

→

H1

(

p

),

and FPMJ

:

H1

(

m

)

×

H1

(

p

)

→ R

N providetheexplicitexpressions ofthe

unknowns obtainedfrom (15), (16) and(14). Algorithm 2 can be written in compact formas the following fixed point iteration

where

F

: R

N

_{→ R}

N_s.t.

_F

₍

_γ

₎

₌

_F

PMJ

(

Fp

(

γ

),

Fm

(

γ

)).

Toprovetheconvergenceofthepreviousiterations,weneedtoshowthatthereexistsaconstantC

∈ [

0,1)suchthat



F

(

γ

(k)

)

 ≤

C



γ

(k)

 ∀

k

,

(17)

foreach

γ

(0)_,where

_

_{· }

_{istheusualEuclideannorm.Thisiswhatisprovedinthefollowingresult.}

Proposition1.Underthefollowingassumptions: – Thereexisttwoconstants0

<

b

<

B suchthat

b

ξ ≤ ξ

tD

(

x

)ξ

≤

B

ξ

2

,

∀ξ ∈ R

2

,

(18)

fora.e.x

∈ 

m;

– Theparameters

σ

pandb satisfy

σ

p

≥

4N3/2_C2 T

πρ

2_R2 PMJ

,

b

≥

4N 3/2 A3r/2R2PMJ

,

(19)

whereCTisthetraceconstantfortheSobolevspaceH1

(

p

)

;

– Thetimestep



t

>

0 ischosensuchthat



t

≤

min



χ

p

C

p

σ

p

;

χ

m

C

m b



;

(20)

then,thereexistsaconstantC

∈ [

0,1)suchthat

(17)

issatisfied.

Proof. Fromthedefinitionof

F

wecanwrite



F

(

γ

(k)

₎

_

2

_{= }

_F PMJ

(

Vm(k+1)

,

V(pk+1)

)



2

=

N



j=1







V(pk+1)

(

sj

)

−

_A1_r



Br(sj)V (k+1) m dx RPMJ







2

≤

2 R2_PMJ N



j=1

⎛

⎜

⎝



V(_pk+1)

(

sj

)



2

+

⎛

⎜

⎝

_A1 r



Br(sj) V_m(k+1)dx

⎞

⎟

⎠

2

⎞

⎟

⎠ ,

(21)

whereweusedtheinequality

(

a

+

b

)

2

≤

2a2

₊

_2b2 _a

_,

_b

_{∈ R.}

Regarding thefirst termatthe righthandsideof

(21)

,we canapply thetrace theorem(see [31]). Wenoticethat inour casetheboundaryoftheprobleminthenetworkisgivenbythePMJsj andbytheAVnode s0,sowehave

N



j=1



Vp(k+1)

(

sj

)



2

≤

N



j=1



Vp(k+1)

(

sj

)



2

+



V(pk+1)

(

s0

)



2

≤

CT



V(pk+1)



2_H1_( p)

.

(22)

Regardingthesecondtermattherighthandsideof

(21)

,weusethefollowinginequalityholdingforeveryboundeddomain



and0

≤

p

≤

q

≤ ∞

:



z



Lp_()

≤ ||

1

p−1q



_z



Lq_()

,

providedthatz

∈

Lq

()

andwhere

||

isthesizeofthedomain.Inourcaseweset



=

B

r

(

sj

),

p

=

1,q

=

2,soweobtain



Vm(k+1)



L1_(Br(s j))

=



Br(sj)

|

Vm(k+1)

|

dx

≤



Ar



Vm(k+1)



L2_(Br(s j))

.

(23)

Therefore,wehavethefollowingestimateforthesecondtermattherighthandsideof

(21)

N



j=1

⎛

⎜

⎝

_A1 r



Br(sj) Vm(k+1)dx

⎞

⎟

⎠

2

=

1 Ar2 N



j=1



Vm(k+1)



2_L1_(B r(sj))

≤

Ar A2 r N



j=1



Vm(k+1)



2_L2_(Br(s j))

≤

1 Ar N



j=1



Vm(k+1)



2_L2_( m)

=

N Ar



V_m(k+1)



2_L2_( m)

≤

C1



Vm(k+1)



2_H1_(m)

,

(24)

withC1

=

ANr.Then,owingto

(22)

and

(24)

,

(21)

reads



F

(

γ

(k)

)



2

≤

2 R2_PMJ



CT



V(pk+1)



2H1_( p)

+

C1



V (k+1) m



2H1_( m)



.

(25)

Now,wehavetofindsuitableestimatesfortherighthandsideof

(25)

intermsof



γ

(k)

_

_{.Tothisaim,wetake W} p

=

V(pk+1)asatestfunctionin

(16)

obtaining

χ

p

C

p



t





V_p(k+1)





2 L2_( p)

+

σ

p







∂

V(pk+1)

∂

l







2 L2_(p)

= −

1

πρ

2 N



j=1

γ

_j(k)V_p(k+1)

(

sj

).

(26) Thus,wehave C2





V(pk+1)





2 H1_( p)

≤

NCT

πρ

2





V(_pk+1)





H1_( p) N



j=1

|

γ

_j(k)

|,

withC2

=

min

{

χpC_tp

;

σ

p

}

,andthen





V(pk+1)





H1_( p)

≤

C3



γ

(k)

,

(27) withC3

=

N 3/2_C T C2πρ2.

We proceed now by considering the equation inthe myocardium

(15)

, andwe take Wm

=

Vm(k+1) asa test function,

obtainingfrom

(18)

theestimate

χ

m

C

m



t



V (k+1) m



2_L2_(m)

+

b

∇

V (k+1) m



2_L2_(m)

≤

N



j=1 1 Ar

γ

_j(k)



Br(sj) Vm(k+1)dx

.

(28)

Then,owingto

(23)

,wehave

N



j=1 1 Ar

γ

_j(k)



Br(sj) Vm(k+1)dx

=

1 Ar N



j=1

|

γ

_j(k)

| 

Vm(k+1)



L1_(B r(sj))

≤

√

Ar Ar N



j=1

|

γ

(_jk)

| 

Vm(k+1)



L2_(Br(s j))

≤

1 Ar N



j=1

|

γ

_j(k)

| 

Vm(k+1)



L2_( m)

=

N Ar



γ

(k)

_{ }

_V(k+1) m



H1_( m)

.

Thepreviousinequalitytogetherwith

(28)

gives





Vm(k+1)





H1_( m)

≤

C4



γ

(k)

,

(29) withC4

=

!

N Ar 1 min{χm Cm t ;b} .

Thus,puttingtogether

(25)

,

(27)

and

(29)

,weobtain

(17)

with

C

=

2 R2 PMJ

(

CTC3

+

C1C4

)

=

2 R2 PMJ

N3/2C2_T min

{

χpCp t

;

σ

p

}

πρ

2

+



N Ar



3/2 ₁ min

{

χmCm t

;

b

}



.

Dueto

(20)

,weobtain C

=

2 R2_PMJ

N3/2_C2 T

σ

p

πρ

2

+

1 b



N Ar



3/2



,

whichislessthanonebecauseof

(19)

.

2

Remark1. Wenotice that theassumptions on the parameters

σ

p andb given by

(19)

depend on thevalue of thetrace

constant CT,which isnotcomputableforgeneraldomains.Thereforewe cannotdetermineexplicitlythevalue of

σ

p and

b that guarantee that

F

is a contraction. Nevertheless, in all the numerical experiments reported in what follows, we experienced thattheproposed algorithmnot onlyconverges,butitdoesso(withinmachineaccuracy) inafinitenumber ofiterations.

Remark2.Therestrictionon



t givenby

(19)

shouldbematchedwiththeonerequiredforstability oftheforwardEuler methodsfortheODEsystems

(1)

2 and

(7)

2.Thus,theeffective



t isthesmallerofthesetwo.

3.3. Solutionofthestand-alonesubproblems

In thissection we detail thenumerical strategiesused to solvethe 3D andthe 1D monodomainsubproblems arising at each iteration of

Algorithm 1

.For the solutionof the3D subproblem,we consider Lagrangian finite elements andan implicit/explicit method,see [1]. Forthe solutionof the1D subproblem we followthe methodology presentedin [9]. In particular,we assumetohaveasystemofgap-junction/Purkinjecellsforeachnode ofthemesh.Foreachsegmentofthe network Si,weknowthevaluesofVn_g,i andIn_g,iattheprevioustimesteptn.Then,thenumericalschemetocompute Vg,i

andIg,i foreachsegment Siattimetn+1 canbedividedintofoursteps:

1. Recoveringthetransmembranepotential



V±_p,i



n.By considering (7)3, we can recover thevalue ofthe transmembrane potentialasfollows:



V±_p,i



n

=

Vn_g,i

∓

I n g,iRg 2

;

2. Operatorsplitting–firstpart.Wecomputetheintermediatepotentials



V_p±_,i



n+1/2 asfollows:

C

p



V±_p,i



n+1/2

−



V_p±_,i



n



t

= −

I ion p



V±_p,i



n

,



w±_p,i



n



;

(30)

3. Updateof Vg and Ig.We computetheintermediatevaluesV_gn+1_,i /2 andIn_g,+1_i /2 withthefollowingexpressionsobtained

bymanipulatingthetwoequationsin

(7)

3:

In_g+_,i1/2

=



V+_p,i



n+1/2

−



V−_p,i



n+1/2 Rg

,

Vn_g,+_i1/2

=



V+_p,i



n+1/2

+



V_p−_,i



n+1/2 2

;

(31)

4. Operatorsplitting–secondpart.Thesecondpartoftheoperatorsplittingshouldbegivenby

χ

p

C

p V_p±_,i

−



V±_p,i



n+1/2



t

−

∂

∂

l

σ

p

∂

V±_p,i

∂

l



=

0

.

Table 1

Parametersusedinthenumericalexperiments,suitablereferences,andphysiologicalranges.

E–E M–M Ref. Range

χm(cm−1) – 1400 [35] – χp(cm−1) – 1467 [36] – Rg(kOhm) – 500 [9] – r(cm) – 0.06 [11,37] [0.01–0.1] ρ(cm) – 0.0017 [36] – RPMJ(kOhm) – 11000 [9,38] [1000–25 000] ε – 10−5 _{– –} do(ms) 5.0 – [26,27] [5–15] da(ms) 2.0 – [26,27] [2–3] σp(kOhm−1cm−1) – 35.0 [25] – σf (kOhm−1cm−1) – 1.334 [35] – σt(kOhm−1cm−1) – 0.176 [35] –

Now,byaddingthesetwoequationsandbydividingby2,weobtainthanksto

(31)

χ

p

C

p Vg,i

−

Vng+,i1/2



t

−

∂

∂

l



σ

p

∂

Vg,i

∂

l



=

0

.

(32)

As finiteelement basisto solvethe previous problemwe usetheone-dimensional cubicHermitebasis,so thatwe can directlyrecoveralsothederivative ofthepotential,whichisrelatedtothecurrent(recalling

(7)

4).Hermitefiniteelements aresuitableforsuchapurposeastheyarebaseduponsolvingthepotentialanditsderivativeateachnode.

Finally,oncewehavedetailedhowtocomputethe valuesof Vg,i andIg,i foreachsingle segmentofthenetwork,we

needtoenforcetheKirchhofflaws

(7)

5–6tocomputetheglobalVgandIg.Tothisaim,wemodifytheglobalfiniteelement

matrixassociatedtothecollectionof

(32)

bysubstituting1’sor0’sintherowsrelatedtobifurcationorintersectionpoints accordinglyto

(7)

5–6.

4. Numericalexperiments

In thissection we presentseveralnumerical results withthe aim ofassessing the reliability of Algorithm 1to solve the MM coupled problem andcomparing the results withthose obtainedwith the EE coupled problem. First of all, in Section4.1wediscusshowtoestimateaconstantconductionvelocityfromthecoupledmonodomainproblemstobeused in the eikonal ones in view of the forthcoming comparison. After this preliminary step, in Section 4.2 we consider an academictestcasewithsimplifiedgeometriestocomparetheresultsobtainedwiththetwodifferentstrategies,whereas, inSection4.3weapplythesestrategiestosimulatebothanormalandapathologicalpropagationinanellipsoidalidealized leftventricle.Finally,inSection4.4,weapply

Algorithm 1

toarealisticgeometry.

AllthenumericalresultsrelatedtotheMMproblemhavebeenobtainedwiththeparallel FiniteElementlibraryLifeV, developed atMOX – Politecnico di Milano, REO/ESTIME – INRIA,CMCS – EPFL, andE(CM)2 – Emory University.For the 3D monodomainproblemweconsidered

P

1 Lagrangianfiniteelements, whereas forthe1D problemcubic Hermitefinite elements.Forboththemonodomainproblems,wechoseatimestep



t

=

0.01 ms.Theionicmodelsusedinournumerical experimentsweretheDiFrancesco–Noble model

[32]

forthePurkinjecells,andtheLuo–Rudy-I model

[33]

forthemyocardial cells.ThenumericalschemesforsolvingthecoupledEEproblemhavebeenimplementedinastand-aloneandserialcode based on the VTK 5.8.0 library [6,8]. For the solution of the single eikonal problems, we considered the fast marching method (FMM) proposed in[22] for the1D problem andthe modified version ofthe FMM proposed in [34] forthe 3D problem.

Ifnototherwise specified,inallthe numericalexperimentsweused thedatacollected inTable 1,wherewe reported alsosuitablereferencesand,forsome,therangesofthevaluesreportedtherein.

Notice thatwe didnot needtoprescribe explicitlythedelay atthePMJinthe MMmodel,since inthiscasethePMJ resistancemodelitselfwasabletointroducesuitabledelays.

4.1. Assessingtheconductivitiesinviewofthecomparisons

Intheset-upoftheforthcomingnumericaltests,wefaced twocriticalpoints:(i) thechoiceofproperquantitiestobe comparedinviewofadiscussionoftheresults,and(ii)theuseofcomparableconductionvelocitiesinboththeMMand EEcoupledproblems.

The first issue is crucial because the output of the monodomain problem is the transmembrane potential, whereas the one of the eikonal problem is the local activation time. Then, in view of the comparisons, we computed from the transmembranepotentialstheactivationtimesprovidedbythemonodomainproblems,definedagainasthetimeatwhich thepotentialreachesthemeanvaluebetweentherestingpotentialandtheplateaupotential. Thisallowed ustocompare thesevalueswiththeonesprovidedbytheEEproblem.Tothisaim,wedenotedwithuM

Fig. 3. Myocardial domain. Test for the estimation of the conduction velocity.

Fig. 4. Conductionvelocitiesinthemyocardiumasafunctionofthelocalcoordinatel estimatedfromthesolutionofamonodomainproblem:Cx t on A B

(left),CtyonAC (center),andCf onA D (right).Testfortheestimationoftheconductionvelocity.

inthenetworkandinthemyocardium,respectively,obtainedbysolvingtheMM problem,andwithuE_p

(

x

)

anduE_m

(

x

)

the activationtimesinthenetworkandinthemyocardium,respectively,providedbytheEEstrategy.

For what concerns point (ii) above, we neededto use comparable parameters in order to obtain meaningful results. In particular, we remark that the propagationvelocities have a differentnature in the monodomain model than in the eikonal one.Indeed,inthefirstcasetheconductionvelocityoftheelectricalsignalisnotconstant intime andspaceand dependsonthesolution.Forexample,thepropagationvelocitychangeswhentwowavefrontscollideorwhenthewavefront interactswiththeboundaryofthedomain.Onthecontrary,intheeikonalproblems,theconductionvelocityisaprescribed parameterofthemodel,andthereforeitdoesnotdependonthesolutionoftheproblem.However,weobservethatinthe caseofa single wavefront,in themonodomaincasesthe conductionvelocityisalmost constant farfromtheboundaries. Thissuggestsastrategytoestimateareferenceconstantconductionvelocityfromthemonodomainmodel,tobethenused intheeikonalmodel.ThiswasdoneforboththePurkinjenetworkandthemyocardium.

To this aim, we considered two reference scenarios, one for the myocardium given by the cuboid with dimensions 0.3

×

0.7

×

2.0 cm,see

Fig. 3

,andoneforthenetworkgivenbyasingle Purkinjefiber.Forthemyocardium,weestimated twoconductionvelocities,oneinthedirectionofthefibers(Cf)whichisparallelto A D,andtheotheroneinthedirection

transversetothefiber(Ct).Todothis,wesolvedthemonodomainprobleminthecuboidwithasourcecurrentappliedin

theinternalcornerofsize0.15 cm withoneofthecornerscoincidingwith A andsidesparalleltotheonesofthecuboid, see

Fig. 3

.ThisallowedtoobtaintheactivationtimeuM

m

(

x

)

andtodefinethefollowingvelocities



Cf

(

x

)

:=

1





∂umM ∂z





,



Ci_t

(

x

)

:=



1



∂uM m ∂i





i

=

x

,

y

.

Then, weevaluatedthesequantitiesalongthethreesegments A B, AC and A D,see

Fig. 4

.Thus,we tookasanestimation oftheconductionvelocitiesCf,mandCt,mprovidedbythemonodomainproblemthemeanvalueofthesequantities,

Cf,m

=

1 Nr



xi∈A D



Cf

(

xi

),

and

Fig. 5. ConductionvelocityCp inthePurkinjefibersasafunctionofthelocalcoordinatel estimatedfromthesolutionofthemonodomainproblem.Test

fortheestimationoftheconductionvelocity.

Ct,m

=

1 2

⎛

⎝

1 Nx



xi∈AC



C_tx

(

xi

)

+

1 Ny



xi∈A B



C_ty

(

xi

)

⎞

⎠ ,

whereNr,NxandNy arethenumbersofpointsinA D,AC and A B,respectively.Theconductionvelocitiesfoundwiththese

estimateswerethenusedintheeikonalequation

(2)

–(3).Inparticular,weusedCf

=

Cf,mandk

=

Ct,m

/

Cf,m.Referringto

thedatareportedin

Table 1

,wefoundCf

=

0.067 cm/ms andk

=

0.43.

ForthePurkinjenetworkweproceededinasimilarway.Inparticular,weconsideredthepropagationofawavefrontina singlePurkinjefiberoflength5 cmbysolvingtheone-dimensionalmonodomainproblem.Weappliedattheleftboundary acurrentstrongenoughtotriggertheexcitationofaPurkinjecell,whereasontherightboundaryahomogeneousNeumann condition.ThisallowedtoobtaintheactivationtimeuM

p

(

x

).

Sinceinthiscasewehadonlyonedirectionofpropagation,we

estimatedtheconductionvelocityinthesinglePurkinjefiberasfollows

Cp,m

=

1 Np Np



i=1



Cp

(

xi

),

whereNp isthenumberofnodesofthemeshdiscretizingthePurkinjefiberand



Cp isgivenby



Cp

(

x

)

=

1





∂uMp ∂s





.

(33)

In

Fig. 5

(right)we depicttheevolutionof



Cp inthePurkinjefiber.Inparticular,we usedavalue

σ

p (see

Table 1

)which

allowed ustoobtain aphysiological valueof theconductionvelocity [25]. Weobservethat, farfromthe boundaries,the conductionvelocitywas almostconstantandequalto0.3 cm/ms,whereas,neartotheboundaries, thewavefront interac-tionsresultedinanon-constant conductionvelocity.Wethususedtheestimatedvalue Cp,m asconductionvelocity Cp in

the1Deikonalproblem

(8)

.

For the computations we have used as discretization steps hm

=

0.001 cm for the cuboid, leading to 200 000 nodes

and1.1milliontetrahedralelements, andhp

=

0.0165 cm forthePurkinje fiber.Thevalue ofhp waschosen equaltothe

characteristiclengthofaPurkinjecell

[36]

.

4.2. Numericaltestinacuboid

InthissectionwereporttheresultsofatestinacuboidforacomparisonoftheEEandMMcouplingstrategiesinthe caseoforthodromic propagation. The myocardialgeometrywas thesame considered intheprevious section (see Fig. 3), whereasforthePurkinjefibersweconsideredasimplenetworkcharacterizedbythreesegmentsandonebifurcationpoint. Thisnetworkliedononesideofthecuboiddomain,similartophysiologicalsituationwherethePurkinjefibersarelocated justbeneaththe endocardium,see

Fig. 6

. Thesignal enters fromtheAV node,represented by s0 in

Fig. 6

, left,andthen reachesthePMJs1ands2.ThroughthesetwoPMJs,thesignalentersthemyocardium.

Forthecomparison, we firstcomputedtheactivation maps,whicharerepresented inFig. 6,right. We noticethatthe MMandEEstrategiesdescribeasimilaractivationpatterninboththePurkinjenetworkandinthemyocardium.Toexamine furtherindetailtheactivationpattern, wealsocomputedthecumulativepercentageofactivatedtissue,whichisdepicted in

Fig. 7

.Notethattheslopeofthisquantitygivesususefulinformationaboutthepropagationvelocityofthewavefrontin bothdomains.WebeginbyanalyzingthePurkinjenetwork.In

Fig. 7

(left)thepercentageofactivatedtissueinS1andthen S2isrepresented.Inparticular,inthePurkinjenetworktheEEmodelresultsinaconstantconductionvelocitythroughthe

Fig. 6. RepresentationofthemyocardiumandPurkinjenetworkdomains(left),andactivationmapsinthecaseofMM(top,right)andEE(bottom,right) strategies.Testinthecuboid.

Fig. 7. ComparisonofthepercentageofactivatedtissueinthePurkinjenetwork(left)andinthemyocardium(right)forthetwodifferentcouplingstrategies. Testinthecuboid.

junction,whereastheMMmodelfeaturesadistinctdiscontinuity.Tobetterinvestigatethisphenomenon,wecomputethe followingquantity 1





∂uMp ∂l

(

x

)





,

(34)

whichisanestimateoftheconductionvelocityinthePurkinjenetworkprovidedbythemonodomainproblem.Noticethat thisquantityisdifferentingeneralfromthevalue

(33)

,sincethelatterhasbeencomputedinthecaseofasinglewavefront propagatinginasinglefiber.

Werepresentin

Fig. 8

(left)theevolutionofthequantity

(34)

insegments S1andS2.Inparticular,weobserveaninitial accelerationofthesignalats0,reachingthevalueof0.3 cm/ms,followedbyadecelerationwhenthesignalapproachesthe bifurcation point.Afterthebifurcationthesignalacceleratesagainandtheconductionvelocityassumeslargervalues.This behavioroftheconductionvelocityisknownas“pullandpush” effect

[39,40]

,whichisduetothefactthatthecurrentjust beforethebifurcationpointneedstoincreaseitsvalueinordertobeabletostimulatetheincreasednumberofcellsafter thebifurcation.Duetoenergyarguments,thisproducesadecrementoftheconductionvelocityjustbeforethebifurcation point (“pull”effect).Onthecontrary,theexcited branchesallow theincrease ofthevalueoftheconductionvelocityafter the bifurcation point (“push” effect). Furthermore,we notice a further incrementforl

>

0.95 ofthe conductionvelocity whenthesignalapproachedthePMJ,sincetheresistancetothepropagationdecreaseswhenthewavefrontapproachesthe boundary. Tobetterdescribethe“pullandpush” effect,we alsoran asimulationofanetworkformedonly bytwolevels of bifurcations. In

Fig. 8

(right)we report theconduction velocity asafunction ofthe localcoordinate.We observethat thesignal afterthe“push”effect,returnstothereferencevaluebeforethenext“pull”effect.Wenoticethatthe“pulland push”effectcanonlybecapturedbytheMMmodel,sinceintheEEmodeltheconductionvelocityisprescribedasamodel parameter.