Continuum model for flow diverting stents in 3D patient-specific simulation of intracranial aneurysms

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Continuum model for flow diverting stents in 3D patient-specific simulation of intracranial aneurysms

LI, Sha, CHOPARD, Bastien, LATT, Jonas

Abstract

Flow diverters stents (FDS) are a family of stents commonly used by clinicians to treat cerebral aneurysms. The purpose of a FDS is to modify the blood flow within the aneurysmal cavity and induce thrombosis and tissue remodeling. To predict the success of a given FDS in a patient specific situation, or to design more effective FDS, numerical simulations are a promising approach. Unfortunately, a flow simulation in which the details of the FDS are fully resolved requires a high spatial and temporal, and thus large computational time. Coarse grained approaches have been proposed in the literature, assuming that a FDS can be described as a porous media, obeying Darcy's law. However, FDS are not 3D porous media and a solution based on the so-called screen models is more adapted. In this paper we develop a new screen-based model to represent FDS at a coarse scale, with local porosity. It is shown to accurately predicts the flow in patient specific stented aneurysms geometries, while reducing the simulation by one order of magnitude or more. Our approach could be integrated to medical imaging devices to provide clinicians with [...]

LI, Sha, CHOPARD, Bastien, LATT, Jonas. Continuum model for flow diverting stents in 3D patient-specific simulation of intracranial aneurysms. Journal of Computational Science , 2019, vol. 38, p. 101045

DOI : 10.1016/j.jocs.2019.101045

Available at:

http://archive-ouverte.unige.ch/unige:128680

Disclaimer: layout of this document may differ from the published version.

1 / 1

Sha Li, Bastien Chopard

, Jonas Latt

DepartmentofComputerScience,UniversityofGeneva,Carouge1227,Switzerland

a r t i c l e i n f o

Articlehistory:

Received12October2019 Accepted30October2019 Availableonline18November2019

Keywords:

CFDsimulation

LatticeBoltzmannmethod Aneurysm

Stent Screenmodel

a b s t r a c t

Flowdivertersstents(FDS)areafamilyofstentscommonlyusedbyclinicianstotreatcerebralaneurysms.

ThepurposeofaFDSistomodifythebloodflowwithintheaneurysmalcavityandinducethrombosis andtissueremodeling.TopredictthesuccessofagivenFDSinapatientspecificsituation,ortodesign moreeffectiveFDS,numericalsimulationsareapromisingapproach.Unfortunately,aflowsimulation inwhichthedetailsoftheFDSarefullyresolvedrequiresahighspatialandtemporal,andthuslarge computationaltime.Coarsegrainedapproacheshavebeenproposedintheliterature,assumingthata FDScanbedescribedasaporousmedia,obeyingDarcy’slaw.However,FDSarenot3Dporousmediaand asolutionbasedontheso-calledscreenmodelsismoreadapted.Inthispaperwedevelopanewscreen- basedmodeltorepresentFDSatacoarsescale,withlocalporosity.Itisshowntoaccuratelypredictsthe flowinpatientspecificstentedaneurysmsgeometries,whilereducingthesimulationbyoneorderof magnitudeormore.Ourapproachcouldbeintegratedtomedicalimagingdevicestoprovideclinicians witharapidandreliableestimateoftheefficiencyoftheFDSwhichtheyplantoimplementinapatient.

©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Ananeurysmisaweakspotonabloodvesselwallthatcausesa balloon-likebulging.Aneurysmstendtoincreaseinsize,andcanbe atriskofrupturing,leadingtointernalbleeding,withseverecon- sequences[1,2].Inparticular,ruptureofintracranialaneurysmsis aneventwithahighmortalityrate,andaboutonethirdofthesur- vivorssufferfrompermanentneurologicalorcognitivedeficits[3].

Surgicalclippingandcoilembolizationarethemostcommonmeth- odsoftreatment[4],butendovascularapproachesareincreasingly adopted,astheyarelessinvasivecomparedtoopensurgery[3].

Platinumcoilwhichisdeliveredthroughamicrocatheterintothe aneurysmhasbeenwidelyusedforthetreatmentofaneurysms.

Smalleraneurysmsaresuccessfullytreatedbycoils,butlargeand complexaneurysmsaremoredifficulttotreat[5].Flowdiverter stents(FDS)provideanewmethodforcomplexshapedandlarge aneurysms.Theyhavearelativelylowporosityandaretherefore abletoreducetheflowintheaneurysmwithrespecttothatof theartery.Thishastheeffecttoinduceathrombuscapableofsta- bilizingtheaneurysminadefinitivemanner[6].Flowdiverters

∗Correspondingauthor.

E-mailaddress:Bastien.Chopard@unige.ch(B.Chopard).

arebraidedorlaser-cutbyverythinwires(10–50␮m),verysmall pores(∼100␮m)andasingleoramulti-layerstructure[7].

Computationalfluid dynamics(CFD)hasshown tobeareli- abletooltostudytheeffectofagivenflow-divertingstentonthe flowinsideananeurysm.Amongtheabundantliteratureonthis topic,somestudieshavebeenpublishedwhichfocusoncomputer- generatedaneurysmwithanidealizedshape[8,9].Toachievemore reliableresultsusingactualpatientdata,otherauthors[10,3]use 3Drotationalangiographytechniquestoconstructpatient-specific geometriesofbloodvesselswithcerebralaneurysms,whichare then usedasaninputfor CFDsimulations.Appanaboyina etal.

[10]proposeanunstructuredembeddedgridapproachtodealwith thecomplexshapeofbloodvesselsflowdiverters.Janiga[3]intro- ducesanewmethodthatcombines3DCFD-basedoptimization witharealisticdeploymentofavirtualflowdivertingstentfora patient-specificaneurysm.

In mostof theCFDstudiesmentioned above,flow diverting stents arefullyresolved,resultinginlong computationaltimes, duetothelargedifferenceinscalebetweenthestentstrutonone side,andthebloodvesselsandtheaneurysmontheotherside.As aresponsetothisproblem,Augsburger[7]proposesanalterna- tivestrategywhichmodelsaFDSasastatisticallyisotropicporous media.Thismodelishoweverrelativelysimplistic,asitonlycon- sidersonetypeofdesign.Inthisway,itfailstoaccountforthe generalimportant geometrical properties,suchas theporosity,

https://doi.org/10.1016/j.jocs.2019.101045

1877-7503/©2019TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.

0/).

thediameterofthestentstruts,ortomodel localdeformations ofthestentcausedbythedeploymentprocedure.Zhangetal.[5]

extendtheporousmediamodelbyaddingacentrifugalforceterm, buttheyusethesamemodelparametersas[7],andaresubject tothesamelimitationsas thelatter.Asa generalizationofthe porous-mediaapproachof[7]and[5],Raschi[11]proposesand evaluatesamethodwhichfullyconsidersthegeometricalparam- etersofthestent,andtesttheproposedmodelwiththehelpof severalpatient-specificarterieswithaneurysms.However,Raschi’s modelisstillstatisticallyisotropicandassumesthattheporosity isauniform,globalparameter.Inapreviouspublication[12],the presentauthorshavedemonstratedwiththehelpof2Dsimula- tionsthattheassumptionofstatisticalisotropydoesnothold,as thetangentialdragforceobeysadifferentlawfromthenormalone.

Thepublication[12]consequentlyproposesanewmodelforthe macroscopicmodelingofflowdivertingstents,basedonso-called screenmodelsapproach,whichwereferhereasscreen-basedflow divertermodel,orSFDM.IntheSFDMapproach,theflowdiverter isexplicitlymodeledasathinporoussurface,anditseffectonthe flowistreateddifferentlyinthenormalandtangentialdirections.

In-depth2DvalidationsoftheSFDMarereportedin[13].

Inthiswork,wetaketheSFDMproposedin[12],whichhasbeen derivedusing2Dsimulationdata,andadjustitto3Dsimulations.It isthenvalidatedusinggeometrydatafromactualpatients.Likein 2D,theSFDMproposesalocalforceterm,whichiscomputedfrom thelocalflowvelocityandappliedtothefluidinthevicinityofthe stentsurface.Additionallytotheequationrelatingflowvariables totheforceterm,thepresent articleprovidesthedetailsofthe technicalprocedurerequiredtodeploythemodel,startingwith agivenartery/aneurysmgeometryandgeometryofthedeployed stent.Thesedetailsincludethecomputationofthe3Dhullofthe stent,thecomputationofthelocalstentporosity,andthealgorithm forthedistributionoftheforcetermtofluidnodesinthevicinity ofthestenthull,andtheyaimtoguaranteethereproducibilityof thepresentedsimulationsbythereader.

2. Stentmodel

Inthissection,wefirstremindthereaderofthe2DSFDMpre- sentedin[12].Extensionsarethenproposedtosolvethe3Dcase, andthenewmodelisvalidatedonsimplesituations.

2.1. Theflowdivertermodelin2D

TheSFDMproposedin[12]proposesseparateformulasforthe normalforce,whichrelatestothepressuredropacrossthestent surface,andforthestresstermtangentialtothestent.Thepressure- droptermissummarizedinEq.(1):

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

P = 1 2kⁿu²n

kⁿ =k0(Reⁿ_d)

k0 =

12.25(1−ˇ)+41.32

1 Re_d

+

1.41(1−ˇ)+1.11

ˇ =Aopen/A_total

(1)

wherek0andkⁿarethedragcoefficientsofthestent.Thecoefficient k0isvalidwhenthestentsurfaceisnormaltotheflowdirection, whilekⁿappliestothecaseofastentwithanarbitraryinclination.

Theparameterˇisthelocalporosityofthestent,definedbythe arearatio.Here,Aopenistheorthogonallyprojectedopenareaofthe screenandAtotalisthetotalcross-sectionalarea.Itshouldbenoted

thatin[12],wealsoproposeamoresophisticatedequationforkⁿ, whichexplicitlydependsontheanglebetweentheincomingflow directionandthestentsurfacenormal,andwhichproducesslightly moreaccurateresultsinstentswithlargeporosities.Thedifference ishoweverminor,andinthepresentarticlewechoosethesimpler equation,asshownabove,forsimplicity.ThepressuredropP, whichrepresentsthenormalforceterm,iscomputedfromthedrag coefficientkⁿ.TheReynoldsnumberinEq.(1)isdefinedasfollows:

⎧ ⎪

⎨

⎪ ⎩

Re_d=Ud Reⁿ_d= und

,

(2)

wheredis thediameterofthestentstruts.TheparameterUis thenormofthelocalflowvelocity,whileun isthenormofthe normalvelocitycomponent.Thesuperscript“n”forkⁿandReⁿ_din Eqs.(1)and(2)bothrefertothenormalcomponent.Thisnotation highlightsthefactthatthenormalforcedependsonlyonthenormal velocitycomponent.

Adifferentrelationisusedforthetangentialstress,aspre- sentedbyEq.(3),whichdependsnotonlyonthenormal,butalso onthetangentialcomponentsofthevelocity.

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

=Bunut/r

B =1+ kⁿ

3.4−

3.4

+0.797

r =1+aReⁿ_d−

(aReⁿ_d) 3 2 +1

2 3 ,

(3)

where

a=7.68ˇ²−8.842ˇ+2.734

ThedeflectioncoefficientBrepresentsthedeviationoftheflow velocity.Itiscomputedfromthedragcoefficientkⁿ,andtherefore onlydependsonthenormalcomponentofthevelocity.Theveloc- ityreductioncoefficientr standsfortheratiobetweenthelocal velocityutandtheupstreamvelocityu^∞_t inthetangentialdirection.

Finally,thetangentialstressdependsnotonlyonBandr,butalso explicitlyontheproductofthenormalandtangentialcomponent ofthelocalvelocities.Onecanthereforeconcludethatdepends ontheflowstructureinamorecomplexmannerthanthenormal force.

TheSFDMintroducedabovehasbeenderivedusing2Dnumer- icalexperiments.Toextenditto3D,wesetupaseriesofsimple validationcases,similartotheonesusedin2Din[12].Inthese cases,thestentsarerepresentedbyplanarmeshes,withastruc- turesimilartothewovenwiremeshoftypicalstents,whichextend infinitelyinbothspacedirections.Thesimulationsrunwithafully resolvedversionoftheseidealizedstents,withoutSFDM,andthe numericalresultsarecomparedwiththepredictionsoftheSFDM.

Fig.1showsthethreeplanarmeshesthatwereinvestigatedinthe simulations.Inpractice,thesimulationsinstantiateonlyanele- mentarypieceofthemesh,asshownonFig.2,whichisextended periodicallyintheplaneofthemesh.

2.2. Numericalexperimentsin3D

Fig.2illustratesthecomputationaldomainforScreenA.The X-axisextendsalongthenormaldirectiontothescreensurface, whiletheY-andZ-axesresideintheplaneofthescreen.Thewhite

Fig.1. Screensstudiedin3D.

Fig.2.Computationaldomainforthescreen.

wiredboxdenotesthefullcomputationaldomain.Thefluidenters fromtheleftataconstantvelocity,andleavestherightboundary, subjecttoaconstantpressure.Theboundariesareperiodicinthe Y-andZ-directions.Thedensityandviscosityofthefluidare= 1000kg/m³and=3.3×10⁻³Pa·s.Thediameterofthestrutsis d=40␮m.Thephysicalsizeofthecomputationaldomainis2.5 mm×0.5 mm×0.5 mm.TheReynoldsnumberRe_d,definedin termsofthestrutdiameter,variesfrom0.4to7.5.uxisdetermined byRe_d.Sincetangentialvelocitiesuyanduzdonotaffectthenormal force,wepostponethediscussionabouttheminSection2.2.2.The fulldomainisresolvedby250×50×50nodes.Simulationsare performedwiththePalabosopensourceLatticeBoltzmannsolver [14].ThesolverisbasedonthelatticeBoltzmannmethod(LBM) (seeforinstanceChenandDoolen[15])whichconvenientlyhandles complexgeometries.

2.2.1. Normalforce

First,wecomparetheequationforthepressuredropPfrom the2DSFDM inEq.(1)withthe3Dnumericalresultsofafully resolved idealized stent. Fig. 3 shows the drag coefficient, as obtainedfromthenumericalsimulationorfromtheSFDM,which isafunctionofReⁿ_d.Itisfoundthatalmostallthesimulationspoints agreewiththe2Dscreenmodelcurves. Itsuggeststhatthe2D SFDMmodelforthedragcoefficientisdirectlyapplicableto3D flowdiverters.

2.2.2. Tangentialforce

In2D,thereexistsonlyonetangentialdirection,andtheSFDM consequentlyproposesasingle,scalarrelationforthetangential stressinEq.(3).For3D,weshowthatthetangentialstresscom- ponents˛dependonlyonthecorrespondingvelocitycomponent u˛.Therefore,bothcomponentsofthestresscanbetreatedbyin anindependentrelation,justlikethesinglestresscomponentin the2DSFDM(Eq.(3)).

Toinvestigatethetangentialstresses,werundirectnumerical simulationsinwhichtheinletvelocityisinclinedwithrespectto

Fig.3. Dragcoefficientforthenormaldirection.

thenormalofthestentplane,byanangle^∞intheY-direction:

u^∞x =Ucos^∞,u^∞y =Usin^∞, u^∞z =0.We runsimulations with valuesof^∞equalto30^◦,45^◦and60^◦respectively.Boththedeflec- tioncoefficientBandthevelocityreductioncoefficientrcomputed fromthe3Dscreenarecomparedwiththe2DSFDMequations.In Fig.4,thesymbolsrepresentthesimulationresultsandthelines representthe2DSFDMpredictions.Themodelclearlyoffersabad fitofthedataforboththedeflectionandthereductioncoefficient.

IncaseofthedeflectioncoefficientB(Fig.4a),thenumericalresults distinctlydependontheporosity,whilethe2DSFDM,whichin[12]

wasfittedwith3Dsimulationdata,isporosityindependent.For thevelocityreductioncoefficientr,Fig.4bshowsthatthe2DSFDM overestimatesthetangentialforce,whichisinverselyproportional r.

Duetothelargediscrepancybetweenthe3Dsimulationresult andthe2DSFDM,itisnecessarytoderiveaproper3DSFDMby carryingoutanewfitforBandron3Dsimulationdata.Eqs.(4) and(5)providethenewparametersforthe3DSFDMafterfitting thedependenceofBagainstkⁿandragainstReⁿ_d.

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

˛ =Bunu˛/r

B =1+kⁿ a −

a

+0.797

r =1+aReⁿ_d−

(bReⁿ_d)^1.7+1

,

(4)

Fig.4. Comparisonbetweenthedeflectioncoefficientandthevelocityreductioncoefficientwiththe2Dequations.

with

⎧

⎪ ⎨

⎪ ⎩

r=ut/u^∞_t .

a=1.456ˇ²−1.534ˇ+0.4594 b=7.886ˇ²−8.555ˇ+2.784

(5)

Asamajorqualitativedifferencetothe2DSFDM,onecanpoint outthatin3D,thedeflectioncoefficientBnowdepends onthe porosity.AsshownfromFig.5,thenewfittedcorrelationagrees wellwiththedatafromthe3Ddirectnumericalsimulation.

3. Fullyresolvedflowdivertersimulation 3.1. Patientdata

A numerical procedure providing fully patient-specific sim- ulationsdepends ona sequence of preparatory technicalsteps, includingtheacquisitionandpost-processingofthepatientdata throughmedicalimagery,andadeformationofthestentgeometry toplaceitintothepatientartery(“virtualstentinsertion”).Allthese stepshavebeentreatedelsewhere,aspartoftheThrombusproject [16],asdescribedin[17]andarenotthetopicofthepresentarticle.

WeacknowledgesupportbytheThrombusprojectforprovidingus withafullanonymizeddatasetwhichforthreedifferentpatients includesthegeometryofanarterywithaneurysm,thegeometry ofadeployedstent,andthesignaldataforthepatientflowrate duringa heartbeat.The 3D vesselandaneurysm surfaceswere acquiredbyabiplaneangiographicsystemwith3Dreconstruc- tion.Theupstreamvelocityprofileswereobtainedusingstandard 2Dphase-contrastmagneticresonanceimagesandahome-made softwareapplication.

Fig.6showsthevesselsandaneurysmsofthepatientsemployed inthisstudy.StentsinFig.7aredeployedinthevesselsofFig.6.A cardiaccirclevelocityprofileofthepatientsarepresentedinFig.8.

TheflowrateinFig.8isthemeanvaluesoverthecrosssection.

Thethree-dimensionalandtimedependentnumericalsimula- tionsareperformedwiththehelpoftheopen-sourcefluidsolver Palabos[14],usingaconfigurationofthesoftwaredescribedinthe frameworkoftheThrombusprojectin[18].TheSingle-Relaxation timeBGKwithsecond-orderpolynomialequilibriummodel[19]

withaD3Q19latticeisusedforthesimulation.Acurvedoff-lattice boundaryconditiondescribed inGuo et al.[20] isused forthe bloodvesselwallsandfortheinletandoutlet.Thecomplexstruc- tureofthestentisfullyresolved.Becausethestentissothin,and hassuchafinestructure,anoff-latticeboundary-conditionisnota goodoption:therearenotenoughcellstoproperlyimplementthe

Table1

Characteristicparametersofthethreepatientarteriessimulatedinthisarticle.

Feature Patient2 Patient3 Patient4

Inletdiameter(mm): 4.5 5.2 5.6

Outletdiameter(mm): 3.0 4.3 2.9

Reynoldsnumber: 254.7 395.2 363.0

interpolationscheme,anditwouldbecomputationallyveryexpen- sivetodoso.Ifwedidthat,thespeedupofourmethodwouldbe evenmorespectacular,thusweusebounce-backboundarycon- ditionwhichiscapableofimposingano-slipconditioninaspace asnarrowasasinglelatticenodeinthesesituations(seeThrom- busVPHproject[16]).WeappliedaDirichletvelocitycondition withatime-oscillatingPoiseuilleprofileattheinlet,andconstant- pressureconditionattheoutlet.

TheReynoldsnumbersinthethreepatienttest-cases,defined withrespecttotheinletdiameterandtheaverageinletvelocity overtime,arelistedinTable1.Thestrutdiameteris26 ␮mfor allthreestents.Thedensityandviscosityofthebloodare=1000 kg/m³and=3.3×10⁻³Pa·s.ThevelocityinFig.8aretheaver- agevelocityofthePoiseuilleprofileattheinlet.

Thewallshearstress(WSS)hasbeenshowntobeaimportant indicatorofthestentefficiency[21].Here,itismeasuredatfour pointsontheinterioroftheaneurysmwall,asshowninFig.9.

Theacousticwavesaregeneratedbytheinitialcondition.PointA islocatedattheneckoftheaneurysm,andPointDisalwaysthe top-mostpoint.PointBandCaresetrespectivelyattwoopposite locationsoftheaneurysm.Theaccuracyofthesimulationswillbe evaluatedquantitativelyatthesefourmeasurementpoints.

3.2. Gridconvergence

TheWSSovertwocardiaccyclesarepresentedforthethree patientsinFigs.10,11and12,withdifferentresolutionsatthe selectedmeasurementpoints.Inthesesimulations,thecharacter- isticvelocityinlatticeunitsuissetto0.04forPatient2and3,and 0.02forPatient4.Thischaracteristicvelocityisproportionaltothe timestep,anditsoptimalvaluewaschoseninaseparatetime-step convergencestudywhichisnotreportedhere.Inthesimulationof Patient2and3,twocardiaccycleswerecomputed.ForPatient4, thesimulationrequiresalargenumberofmeshnodes,withupto 218.7×10⁶nodesatthesmallestresolution,dx=21.0␮m,caus- ingthesimulation runaboutthree weekson160CPUcoresfor onecardiaccycle.Toavoida wasteofcomputationalresources, andgiventhattheresultsofthesecondpulseforPatient2and3

Fig.5.ThedeflectioncoefficientandthevelocityreductioncoefficientafterfittingwithEqs.(4)and(5)gr5

Fig.6.Thevesselandaneurysmofthepatients.

Fig.7. Theflowdiverterasdeployedineachpatient.

Fig.8. Illustrationoftheflow-ratecurvesforthethreepatients.

appearedtobeidenticaltothefirstone,exceptinashortinitialtime window,wechosetosimulateonlyonecardiaccycleforPatient4.

Fig.10to12comparetheWSSofdifferentresolutionsforthe threepatients.InFig.10,theWSSatdx=28.7␮misclosetothe WSSatdx=21.5␮m.Wethereforedecidedthatgridconvergence

isreachedatdx=28.7␮m,andtookthistobethereferencevalue forfullyresolvedsimulationsofthePatient2testcase.Inthecase ofPatient3,theresultsatdx=27.5magreeperfectlywiththe resolutiondx=18.3minpointAandB,butdeviateslightlyat pointsCandDaroundthepeakoftheWSScurve.Thesamephe-

Fig.9. Measurementpointsinthethreepatientaneurysms:PointA(red),B(blue),C(green),andD(yellow).

Fig.10. TheWSSatdifferentresolutionsforPatient2.

Fig.11. TheWSSatdifferentresolutionsforPatient3.

Fig.12. TheWSSatdifferentresolutionsforPatient4.

nomenonalsoshowedinthePatient4testcase.Weconsequently chosetoadoptthefinestresolutionsforthePatient3and4test cases,namely18.3␮mand21.0␮mrespectively.

4. Implementationofstentmodel

TheSFDMconsiderstheflowdivertingstentasaporous,curved surfaceembeddedintheflow,anditrepresentsitseffectonthe flowthroughabodyforceterm,actingontheflowinthevicinityof thesurface.TheadvantageoftheSFDMisthattheflow-diverting stentdoesnotneedtobefullyresolved,andinsteadisreplacedbya

“poroussurfaceregion”,or“stentregion”,representedbyacurved

surface.Toachievethislevelofrepresentation,itisnecessarytofirst computeastenthullfromtheprovidedstentgeometry.Itshould beunderstoodthat,whiletheoriginalstentgeometryconsistsof multiple,verythintubesforallthestruts,thehullrepresentation disregardsthethicknessofthestruts,andinsteadapproximatesthe shapeofthefullstentbyasinglecurvedtube.Fig.13ashowsboth thesurfaceoftheoriginalstentand,intransparence,thesurfaceof thestenthull.

Inthefollowing,itisassumedthatboththeoriginalstentand thehullarerepresentedbyatriangularsurfacemesh,whichmeans, asetoftriangleswithsharedvertices.Thealgorithmsdescribed below,forthecomputationofthelocalstentporosityandforthe

applicationoftheSFDMforce,requireasinput(1)thecoordinates ofthetrianglevertices,(2)theareaateachvertex,definedasa thirdofthesumofareasofalladjacenttriangles,and(3)thenor- malateachvertex,definedasthenormalizedsumofnormalsof alladjacenttriangles.Theresolutionofthesurfacemesheshasa bigimpactontheaccuracyofthealgorithms,andagoodtrade- offbetweenefficiencyandaccuracymustbefound.Therefore,the meshesaregeneratedatdifferentresolutionsatthetwostagesof theSFDMdeployment:Fortheporositycomputation,boththestent meshandthehullmesharepreparedinsuchawaythatthemaxi- mumlengthofthetriangleedgesissmallerthanhalfthefluid-mesh spacingdxofthefullyresolvedsimulation.FortheSFDMforcecom- putation,onlythehullmeshisneeded,andispreparedinsuchaway thatthemaximumlengthofthetriangleedgesissmallerthanhalf thefluid-meshspacingdxofthecoarse-grainedsimulation.

Fortheporositycomputation, boththestentmeshandthehull mesh are prepared in sucha way that the maximum lengthofthetriangleedgesissmallerthanhalfthefluid- meshspacingdxofthefullyresolvedsimulation.

FortheSFDMforcecomputation, onlythehullmeshisneeded, andispreparedinsuchawaythatthemaximumlength ofthetriangleedgesissmallerthanhalfthefluid-mesh spacingdxofthecoarse-grainedsimulation.

4.1. Generationofthestenthull

Inthisproject,weobtainedthegeometriesofthepatientarter- ies,aswellasthegeometryofthealreadydeployed,fullyresolved stents,asinputparameterstothesimulations.Thedatawaspro- videdtousbytheThrombusproject,andthenumericalprocedure forthedeploymentofthestentinthepatientarteryispublished in[17].Theprovidedgeometryfullydescribedtheshapeofthe individualstrutsofthedeployedstentinformofatriangularsur- facemesh.Usingthisinputdata,thesoftwarePalabosiscapable ofinstantiatingafullyresolvedsimulation,byfillingtheinterior ofthestrutswithsolidnodes,onwhichtheso-calledbounce-back boundaryconditionisapplied.

Toinstantiate coarsegrainedsimulations withourmodel or withtheRaschiapproach,itisnecessarytoapproximatethestent byasimplersurface,asinglecurvedtubethatenclosesallstent struts,whichinthefollowingwerefertoasthestenthull.Wegen- eratethestenthullusingthedataofthedeployedstentaccording tothefollowingprocedure.Firstasetofsamplingpointsresiding insidethestrutsisgenerated,fromwhichahullenclosingthese pointsisgeneratedinformofanorientedsurfacemesh.Finally, thegeometryissmootheduntiltheunderlyingstrutmeshisno longervisible,andtheendsarecuttorevealtheinletandoutlet.

AppendixAoutlinesthestepsrequiredtoimplementthisproce- durewiththeopen-sourcesoftwareMeshLab.Theproducedstent hullisalsodescribedintheformofatriangularsurfacemesh.

Itcanbementionedthat,fortheimplementationoftheWSS, onlypartofthestenthull,invicinityoftheostiumtheaneurysm, isreallyrequired.Tosavecomputationaltime,theremainingparts ofthestenthullcanberemoved.ForthePatient2geometry,we usedthecompletestenthull.Fortheothertwopatients,onlythe ostiumregionwasused.

Itshouldfinallybepointedoutthatthemethodforgenerat- ingastenthulliscertainlynotunique.Theguidelinesprovidedin thissectionsimplysummarizethetechnicalchoicesmadebythe authors,inthehopetheymayhelpinclinedreadersinapplyingthe SFDMfortheirownneeds.

4.2. Calculationofthelocalporosity

TheforcetermpredictedbytheSFDMdependscriticallyonthe localstentporosity. For a stentdeployedina blood vessel,the porosityisspacedependentandneedstobecomputedlocallyfor eachvertexofthestenthull.Tocomputetheporosityatagivenhull vertexv,theproposedalgorithmgathersallhullverticesandstrut verticesinsideaboxcenteredatvandofside-length20dx,where dxisthefluidmeshspacingofthefullyresolvedsimulations.Then, thelocalareasAhullandAstrutaredefinedasthesumofareasof thegatheredhullandstrutverticesrespectively.Thelocalporosity dependsontheratiobetweentheprojectionareaofthestrutson thehulltotheareaofthehull:

ˇlocal=1−Astrut/ Ahull

(6)

Here,theareaAstrutisdividedbytocomputetheprojection areaofthestrutundertheassumptionthatthecross-sectionofthe strutiscircular.

Theporosityprofileonthestenthullcomputedbythisalgorithm isshowninFig.13b.Itcanbeseenthattheporosityishigherin theperipheralpartsandlowerintheinnerstronglycurvedpart.

Thepicturealsoshowsthatthecomputedporosityisnotsmooth, asdetailsofthestrutpatternremainvisible.Thisstructuregets howeversmoothedoutastheporosityvaluesaretransferredto thehullsurfacemeshwithcoarserresolution,whichisusedinthe courseofanSFDM-basedsimulation.Thistransferoftheporosity fromthefinetothecoarsesurfacemeshesrealizedbytwosteps:

1Generatetheverticesforbothafineandcoarserepresentationof thesurfacemeshofthestenthull.

2Foreachvertexnonthecoarsegrid,gatherallfine-gridvertices containedinacubeofside-length20dxandcenteredatv,where dxistheresolutionofthefinefluid-mesh.Computetheaverage porosityofallgatheredvertices.

Fig.13cshowsthataftertransferringtheporosityvaluestothe coarsemesh,thedistributionofporosityappearsmuchsmoother.

In thisexample,thePatient2 datasetisused,witha fine-grid

resolutionofdx=28.72␮m,andascalefactorof3,leadingtoa coarse-gridresolutionofdx=86.16␮m.

4.3. Implementationoflocalforces

TheequationsinSection2provideallSFDMforcetermsinunits ofpressure,andtheyareconvertedtoaforce,actingonavertexof thestenthull,throughamultiplicationbythevertexarea.When theforceistransmittedtothefluid,itisappliedinaregionoffinite volume,athinshellwithanaveragethicknessofdx.Theforceterm isthereforeadditionallydividedbydx,toconvertittoadensity- of-forceindirectionofthehullnormal.Everyhullvertexapplies theforcetoasinglefluidcellwhichislocatedmostclosely.Afluid cellcanthereforereceivecontributionsfromvarioushullvertices, andtheforceactingonafluidcellisexpressedasasumoverthese contributions:

⎧ ⎪

⎨

⎪ ⎩

fn= avP dx ft= av dx

(7)

wherefnandftarethenormalandtangentialbodyforcesandavis thevertexarea.

4.4. ImplementationoftheRaschimodel

Inorder tobetterevaluatethequality ofthe3DSFDM, it is comparedagainstthemodelbyRaschietal.[11].Thismodelis summarizedbythefollowingequations:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎩

P = 1 2Ku²n

K = 22

Re_h+1.3(1−ˇ)+(1 ˇ−1)

2

Re_h = udh

(8)

whereRe_histheReynoldsnumberbasedonthehydraulicdiameter dh,whichisdefinedasfourtimestheratiooftheconnectedvoid volumetothewettedsurfaceareaAw.

dh=4Vvoid

Aw

(9)

Raschimodelisisotropic inalldirections. Forthetangential stressesyandz,thevelocityuxjustneedstobereplacedbyuy

anduz.

Forthismodel,wecomputethelocalporosityinthesameway asfortheSFDM,whereasin theRaschi’s article[11],auniform porosityforthewholestentisused.Themodelalsodependson thehydraulicdiameterdh,whichisdifferentfromthestrutdiame- terusedintheSFDM.Fig.14illustratestheprocedureweadopted tocomputeanapproximatevalueofthehydraulicdiameter,by computingaprojectionofthestent.

Inthissketch,disthediameterofstrut,A0andA1aretheareas enclosedbetweentheredlinesandbluelines,whichrepresentthe centerlinesofthestruts.AccordingtoEq.(9),Vvoidand Aw are requiredforthecomputationofdH.V_voidisapproximatedbythe productbetweenthevoidprojectionareaA0andd.TocomputeAw, weconsiderthe3Dsolidpartsenclosedinthebluerhombicareas

Fig.14.TheillustrationforthecomputationofhydraulicdiameterforRaschi’s model.

Table2

Theresolutionforallsimulations.

Patient2 Patient3 Patient4 Fineresolutiondx(␮m) 28.718 18.338 20.975 Timestep(F)dt(s) 6.2157×10⁻⁶ 2.9522×10⁻⁶ 1.9628×10⁻⁶ Coarseresolutiondx(␮m) 86.153 55.015 73.414 Timestep(C)dt(s) 1.8647×10⁻⁵ 8.8565×10⁻⁶ 6.8698×10⁻⁶

Resolutionratio(C/F) 3.0 3.0 3.5

ashalfofastraightcylinderwithdiameterdandlengthL.Then fromtheinformationabove,weconclude:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

A0/A1=ˇ Vvoid=A0d A1−A0=d

2L Aw=d

2L

(10)

Fromthis,thehydraulicdiameteriscomputesas:

dH=4Vvoid

Aw = 4dˇ

(1−ˇ) (11)

5. Results

Inthissection,theresultsofsimulationsofafullyresolvedstent arecomparedwithcoarse-grainedsimulationsusingourSFDMand theporous-mediamodelbyRaschietal.Thefullyresolvedsimu- lationusedthefineresolution,whichwasdeterminedthrougha gridconvergencestudyinSection3.Theothertwomodelswere executedatacoarserresolution,withascalefactorof3inevery spacedirectionforPatient2andPatient3,andascalefactorof 3.5forPatient4.Thesescalefactorswereexperimentallyfoundto beoptimal,becauseathigherscalefactors,significantdifferences betweenthefullyresolvedandcoarse-grainedsimulationsoccur.

AllresolutionsaresummarizedinTable2.

Allcomparisonsbetweenfullyresolvedandcoarse-grainedsim- ulationsarecarriedoutatthepointduringthecardiaccyclewhen theflowrateishighest,asindicatedbyaredpointforeachpatient caseinFig.8.Itshouldbementionedthatfortechnicalreasons,

Fig.15.Isovelocitysurfacewithbackview(top)andsideview(bottom).Fromlefttoright,theresultsofthefullyresolvedstent,theSFDM,andtheRaschimodelareshown.

Fig.16.Positionoftheslices.Right:Slice1.Left:Slice2.

theexactmaximumwasslightlymissedinthecaseofPatient4 (seeFig.8),becauseourerroranalysiswasperformedinapost- processing step,using voluminoussimulation dataproduced at specifictimeintervalsonly.

Inthefollowing,theresultsobtainedinthepatientcases2,3, and4arebepresentedandanalyzedindividually,butthefigures areintegratedtogetherforthereasonofspace.

5.1. Patient2

Fig.15aprovidesabackandasideviewoftheiso-velocitysur- faceatV=0.05 m/sforPatient2testcase.Theimagesfocuson theaneurysm,whichfortheinvestigatedmedicalproblemsisthe regionofinterest.Clearly,theSFDMreproducestheiso-velocity surfaceofthefullyresolvedsimulationquiteprecisely,whilein theRaschimodel,theiso-surfaceextendstoofarindirectionofthe aneurysmwall.

Fig.17displays,atthesamemomentinthecardiaccycle,iso- contoursofthevelocitynormontwoslicesthroughtheaneurysm (thepositionoftheslicesisshowninFig.16a.InSlice1,theiso- velocitylinesoftheSFDMaregenerallysimilartothoseofthefully resolvedsimulation.Slice2showsthattheSFDMdoesnotexhibit substantialdifferencesintheregionclosetotherightwall,whileon theleftside,thevelocitycontourofthevortexat84mm/sissmaller thantheoneofthefullyresolvedsimulation(withamaximumof 112mm/s).ButtheSFDMstillpresentsvelocitypatternveryclose totheoneofthefullyresolvedstent.TheRaschimodelontheother handexhibitsmuchdenseriso-velocitylinesnearthewall,andthe velocitiesinsidetheaneurysmaresubstantiallytoohigh.

Fig.18aand Table3 compare thesimulationsquantitatively over thefirst twocardiaccycles. Exceptfor ashortinitialtran- sientbehavior,thedatainbothcardiaccyclesisexactlyidentical,an observationthatwillbeexploitedtorestrictthecomputationally moreintensivecaseofPatient4toasinglecardiaccycle.Fig.18a displaystheWSSinthefourmeasurementpoints(definedinFig.9).

TheSFDMshowssomediscrepancyinpointAandBfromthefully resolvedstentsimulation,whilePointCandDcorrespondverywell.

ThemostimportantdiscrepancyisobservedinpointA.Thispoint ishoweverlocatedontheaneurysmneck,veryclosetothestent, anditisnotunexpectedthatinthisareadiscretepropertiesofthe fullyresolvedstenthaveanoticeableeffectontheflow.Theresults oftheRaschimodelontheotherhanddeviatesubstantiallyfrom thefullyresolvedsimulation.

Table 3 shows the WSS values of the fully resolved stent (FRS)andthedifferencesoftheSFDM(D_SFDM)andRaschi’smodel (D_Raschi),fortheaverage,themaximum,andtheminimumvalues oftheWSSduringthesecondcardiaccycle.Thedifferenceisthe absolutedifferencebetweentherelevantmodelandtheFRSsimu- lation,showninEq.(12).Asexpected,thelargesterrorsareinPoint A,whichissituatedontheaneurysmneck.Inallotherpoints,most valuesoftheSFDMareveryclosetothoseofthefullyresolvedstent simulationwithdifferencessmallerthan0.05Pa.Raschi’smodelon theotherhandshowsmuchlargerdifferences,especiallyinPoint C,withvaluesthatareuptothreetimeslargerthanthoseofthe FRS.

Dmodel=|WSS_model−WSS_FRS| (12)

5.2. Patient3

Fig.15bshowstheiso-contourofthevelocitynormat0.15m/s forthetestcasePatient3.Thefullyresolvedsimulationpredicts theformationofadome,theshapeofwhichisunderestimatedby theSFDM,whiletheRaschimodeloverreachesitsextent.Itcanbe concludedthattheSFDMoverestimatesthedragforcesgenerated bythestent,whiletheRaschimodelunderestimatesthem.Fig.17b showstheiso-contoursofthevelocityontwoslicesthroughthe aneurysm.OnSlice1,whichhasalargercross-sectionalarea,the SFDMproducesresultssimilartothefullyresolvedsimulation,but onsliceswithasmallercross-sectionalarealikeSlice2,thehigh- velocitycontours(140mm/s)arenotreproducedinbulkareasof theaneurysmandareonlyfoundintheregionclosetotheneck.This furtherconfirmsthattheSFDMslightlyunderestimatestheveloc-

Fig.17.Velocitynormontwoslicesinsidetheaneurysm.

Table3

WSScomparisonforPatient2(FRSrepresentsfullyresolvedstent).

Patient2 AverageWSS(Pa) MinimumWSS(Pa) MaximumWSS(Pa)

FRS DSFDM DRaschi FRS DSFDM DRaschi FRS DSFDM DRaschi

PointA 1.58 1.15 0.99 1.02 0.74 0.62 3.08 2.37 2.19

PointB 0.25 0.04 0.13 0.12 0.00 0.07 0.67 0.25 0.16

PointC 0.13 0.02 0.15 0.06 0.01 0.22 0.26 0.02 0.28

PointD 0.42 0.01 0.13 0.24 0.00 0.10 0.80 0.05 0.30

Fig.18.WSSinthemeasurementpoints.Finthelegendmeansfineresolution,Ccoarseresolution.

ityinsomeregionsoftheaneurysm,buttheoverallresultsremain relativelyclosetothepredictionsofthefullyresolvedsimulation.

Thevelocities generated in theaneurysmby theRaschimodel, ontheotherhand,aredistinctivelytoolarge,andevenproduce aqualitativelydifferentpattern.

Fig. 18b and Table 4 present a quantitative comparison for Patient3.ItcanbepointedoutthatinPointA,veryclosetothe stent,thetwoforcemodels,runningincoarsesimulations,pro- ducelargeoscillationintheWSS.ThedifferencesfortheSFDMand Raschi’smodelinTable4areevaluatedoverthesecondcardiac cycle,asforPatient2.FortheSFDM,thedifferencesareconsistent withthoseofFig.18b.ThedifferencesinPointBarequitelarge,as theobtainedvaluesoftheWSScorrespondtoapproximatelyhalf ofthoseobtainedintheFRS.ThedifferencesinPointC,whichare thesmallestamongthefourpoints,arenegligible.Themaximum valuesarealwayslargerthanthevaluesoftheothertwocolumn, buttherelativeerroriscomparabletotheonefortheaverageand minimumvalues.Raschi’smodelexhibitsexceedinglylargediffer- ences,exceedingtheexpectedWSSvaluesbyasmuchasafactor three.

5.3. Patient4

Fig.15cshowstheisosurfaceofthevelocitynormat0.10 m/s.

WhiletheSFDMoverestimatestheshapeoftheisosurfaceonly slightly,theRaschimodeloverreachestheshapesubstantially.The isocontoursonslicesfortheSFDMinFig.17cshowasimilarphe- nomenonasforPatient3.Forlargecross-sectionslikeSlice1,the SFDMproducesaverysimilarvelocitypatternasthefullyresolved simulation.ButforSlice2,thehighvelocitycontoursfor210mm/s and175mm/sarenotreproducedbytheSFDM,andthe140mm/s

contourissmaller,indicatinganoveralllowervelocitythaninthe fullyresolvedsimulation.We concludefromtheseobservations thattheSFDMoverestimatesthedragexertedbythestentonthe flow.WiththeRaschimodel,thevelocityisagainsubstantiallytoo high,asit canbeclearlyseenbythelargecircleformedbythe 210mm/sisocontour.

TheWSSsatPatient4measurementpointsareshowninFig.18c.

ThematchbetweenSFDMand fullyresolvedsimulation isvery good,althoughsomeminordiscrepanciesarevisibleinPointsB andD.Table5showsthedifferencesoftheaverage,maximum,and minimumWSS,which,giventhatweonlysimulatedonecardiac cycleforPatient4,werecomputedinthetimeintervalfrom0.5s to1.5stoexcludetheinitialtransients.Asbefore,thebiggestdiffer- encesareshowninPointA,whichisontheneckoftheaneurysm.

Generallyspeaking,thedifferencesofSFDMareinsignificant,as indicatedinFig.18c.PointCstillhasthesmallesterrors,whichare 0.01,0.00and0.07Pacorrespondingtotheaverage,minimumand maximumWSSs.TheaverageWSSdifferencesofPointAandDare around0.05,andtheminimumonesarenegligible.Themaximum onesarearound0.2,whicharenotsignificantcomparedwiththe relevantFRSWSSs.Raschi’smodelperformsbetterinthiscasethan forPatient3,butstillproducesdiscrepanciesthatareseveraltimes abovethoseoftheSFDM.

5.4. Thecomputationaltime

ThemainpurposeoftheSFDMistoreducethecomputational time of a blood flow simulation in anartery/aneurysm with a deployedstent.Table6showsthecomputationaltime thatwas requiredinallofoursimulationsofthisstudy.Allofthemwere executedon160CPUsofthesameHPCcluster.Sincetheexecution

Table4

WSScomparisonforPatient3(FRSrepresentsfullyresolvedstent).

Patient3 AverageWSS(Pa) MinimumWSS(Pa) MaximumWSS(Pa)

FRS DSFDM DRaschi FRS DSFDM DRaschi FRS DSFDM DRaschi

PointA 0.96 0.24 0.13 0.39 0.26 0.24 2.08 0.96 0.91

PointB 0.54 0.24 0.42 0.14 0.09 0.14 1.49 0.60 1.86

PointC 0.19 0.01 0.74 0.07 0.02 0.78 0.67 0.11 1.50

PointD 0.88 0.20 1.86 0.07 0.03 0.79 3.09 1.21 3.52

Table5

WSScomparisonforPatient4(FRSrepresentsfullyresolvedstent).

Patient4 AverageWSS(Pa) MinimumWSS(Pa) MaximumWSS(Pa)

FRS DSFDM DRaschi FRS DSFDM DRaschi FRS DSFDM DRaschi

PointA 0.32 0.16 0.29 0.02 0.12 0.38 1.57 0.13 0.07

PointB 0.33 0.06 0.46 0.08 0.02 0.35 1.03 0.23 0.43

PointC 0.11 0.01 0.15 0.03 0.00 0.06 0.29 0.07 0.17

PointD 0.32 0.04 0.45 0.07 0.00 0.21 0.88 0.17 0.60

Table6

Characteristicparameterforthepatients.

Patient2 Patient3 Patient4

Resolutionratio(C/F): 3.0 3.0 3.5

Fullyresolvedstentsimulation(onecycle) 15.2hrs(0.63d) 312.1hrs(13.0d) 530.7hrs(22.1d)

SFDMsimulation(onecycle) 0.7hrs 4.1hrs 5.4hrs

SpeedupusingSFDM(TF/TC) 21.7 76.1 98.3

timewiththeRaschimodelisalmostthesameastheonewiththe SFDM,weonlylistthedatafortheSFDM.Theresolutionratiois theratioofthediscretespacingdxonthecoarselatticeoverthe spacingdxonthefinelattice.Thespeedupindicatesbyhowmuch thesimulationwasacceleratedusingtheSFDM.

Forthefullyresolvedsimulations,thedomainsizeofPatient3is muchlargerthanthatofPatient2duetothelongerartery.Patient 4requiresaveryhighresolutiontoachieve convergency,which maycausedbythelargesizeofaneurysm.Therefore,therunning timeofthefullyresolvedsimulationofPatient3andPatient4are ismuchlongerthanthatofPatient2.

It shouldbepointed out that in thecoarse simulations,the timestepisincreasedbyafactorproportionaltothespacingdx, because we apply convective scaling(the inlet velocity in lat- ticeunitsremains constant).Therefore, thetheoretical,optimal speedupthatcouldbeobtainedwithSFDMsimulationsisof3⁴=81 forthePatient2andPatient3cases,and3.5⁴=150forthePatient4 case.Theactualspeedupswewereabletoachievewithourimple- mentationinthePalabos framework are distinctlybelowthese optimalvalues.Thisisinpartexplainedbythefactthatthepar- allelspeedupcurveofthecodeisbelowlinear,andthatthecoarse simulationthereforerunswithlowerparallelefficiencythanthe highresolutionone.Butthebiggestlimitingfactorappearstoorigi- natefromthecomputationaloverheadofimplementingtheSFDM.

Thisobservationmotivatedourdecision tocutthestentforthe SFDMimplementation,totherelevantareainfrontoftheaneurysm neck,forthePatient3andPatient4testcases,astrategywhich allowedthespeedupoftheSFDMimplementationtobeimproved byapproximatelyafactorfour.Weexpectfurtherspeedimprove- mentsinthefuture,asourfirstimplementationoftheSFDMhas notyetbeenoptimized.

Notwithstandingthediscrepancybetweentheoreticalspeedup andcurrentlyachievedvalues,itisclearthatevenwiththecur- rentimplementationthegoaloftheprojectisreached:whilefully resolvedsimulationstakeseveraldays,whichistoolongforaprac- tical,patient-specificdecision-makingprocess,theSFDMreduces

thecomputationstoamuchmoreacceptabletimeofjustafew hours.

6. Discussion

Thisstudyisacontinuationoftheworkpresentedin[12],in whicha2DSFDMisproposed,whichreplacesafullyresolvedstent byaforceterm,appliedonacoarse-grainedlattice.Inthepresent work,theSFDMisextendedto3Dthrough,andthemodelparam- etersarenewlyfittedforthispurpose.Patient-specificsimulations demonstratethatthe3Dmodeliscapableofreproducingtheblood flowintheaneurysmofastentedvesselquiteaccuratelyatcoarse resolutions,asshownbythevelocityprofilesandbyWSSvalueson theaneurysmwall.

TheequationsforthenormalforceoftheSFDMareidenticalin the2Dandthe3Dmodel,whilethoseofthetangentialforcepresent significantdifferences.Firstofall,thedeflectioncoefficientBiscon- sideredindependentoftheporosityin2D,whileanexplicitporosity dependencemustbeaccountedforin3D.Furthermore,thecoeffi- cientsforthevelocityreductioncoefficientrmustberecomputed throughfitswithnumericalexperimentsin3D,giventhatthe2D modelunderestimatesthevaluesforthe3Dreductioncoefficient.

Thisdiscrepancycanbeexplainedbythefactthat2Dstentshave onlyonetangentialaxisandcanthereforenotfullyrepresentthe physicsofastentembeddedina3Dfluid.

Severaltechnicalchallengeshadtobeovercometorunpatient- specificsimulationsusingacoarse-grainedSFDM.Theseinclude generationofthestenthull,computationofthelocalporosity,and applicationof thebody-force termto thefluid. Inthe studyof Raschietal.([11]),thestenthull,referredtoastheporousmedium layer,isobtainedbyextractingthecenterlineoftheparentvessel andgeneratingacylindricalhostsurface.Ourapproach,inwhich thestenthullisgenerateddirectlyfromthedeployedstent,relies moreheavilyonproperpre-processingof thepatientdataper- formedbyanindependenttoolchain,astheoneoftheThrombus projectusedinourcase,andnotonanin-housestentfittingpro-

thedifferenceisquiteinsignificant,whiletheRaschimodelsignif- icantlyunderestimatesthevelocityreductioncausedbythestent.

For quantitative comparison, we measured the wall-shear- stressWSSinfourdifferentpointsontheaneurysmwalloverafull heartcycle.Asexpected,thepredictionsoftheSFDMarepooronthe aneurysmneck,closetothestent.ThedataproducedbytheWSS ishoweverofexcellentqualityintheotherareasoftheaneurysm, especiallyinpointsthatareopposedtothestentoropposedtothe zonewheretheflowenterstheaneurysm(wherethrombusfor- mationusuallystarts).Aswecomparedtheminima,maxima,and averagevaluesoftheWSSalongthecurve,itwasfoundthatmost differencesoftheSFDMsimulationsforPatient2and4aresmaller than0.05Pa,whilePatient3’sdifferencesarealittlebigger.Com- paredwiththeaverageandminimumWSSs,themaximumvalues havethebiggestabsolutedifferences,butmostofthemarenomore than30%ofthecorrespondingWSSsofthefullyresolvedstentsim- ulation.ThedifferencesofRaschi’smodelarealwayslargerthan theWSSsofthefullyresolvedstent,whichmeansthatthepercent- ageerrorsaremorethan100%.Fromthequantitativecomparison, wecansaythatSFDMreducestheerrorofthesimulationtoan acceptablelevel.

Bymeasuringthetimeofcomputationofourimplementationof theSFDM,weconcludedthatthesimulationprocesscanbedramat- icallyshortenedbytheprocedure.Inoneofthetestcases,Patient 4,thetimewasreducedfrom22daystoonly5.4hours.Wefur- therforesee additionalreductionsofthecomputationaltime in thefuture,asthequalityof theimplementationoftheSFDMis improved.Allinall,theSFDMholdsuptoitspromisetoproposea toolforpatient-specificmedicaldecisionmakingduringaneurysm treatment.

Acknowledgement

We acknowledge partial funding and access to high per- formance computing resources from the CADMOS center (http://www.cadmos.org) and partial funding from the Euro- peanUnionHorizon2020researchandinnovationprogrammefor the CompBioMed project (http://www.compbiomed.eu/) under grantagreement675451.WefinallythanktheThrombusproject for providing patient-specific data for the test cases, and Guy Courbebaisseforprovidingguidanceontheusageofthedata.

AppendixA. Thegenerationofthestenthull

Thisappendixprovidesapracticalguideonthegenerationof thestenthull,asdescribedinSection4.1,usingtheopen-source softwareMeshLab.Thegenerationofthestenthullconsistsofthe followingsteps:

1Importthetriangular surfacemeshes for allstentstruts, and mergethemintoasinglefileifthisisnotalreadythecase.

smoothingsteps.

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