Contents lists available at ScienceDirect
Medical
Engineering
and
Physics
journal homepage: www.elsevier.com/locate/medengphy
A
coupled
mitral
valve—left
ventricle
model
with
fluid–structure
interaction
Hao
Gao
a ,∗,
Liuyang
Feng
a,
Nan
Qi
a,
Colin
Berry
b,
Boyce
E.
Griffith
c,
Xiaoyu
Luo
aaSchoolofMathematicsandStatistics,UniversityofGlasgow,Glasgow,UK bInstituteofCardiovascularandMedicalScience,UniversityofGlasgow,Glasgow,UK
cDepartmentsofMathematicsandBiomedicalEngineeringandMcAllisterHeartInstitute,UniversityofNorthCarolina,ChapelHill,NC,USA
a
r
t
i
c
l
e
i
n
f
o
Articlehistory:
Received16February2017 Revised13June2017 Accepted24June2017
Keywords:
Mitralvalve Leftventricle
Fluid–structureinteraction Immersedboundarymethod Finiteelementmethod Softtissuemechanics
a
b
s
t
r
a
c
t
Understandingtheinteractionbetweenthevalvesandwallsofthe heartisimportantinassessingand subsequentlytreatingheartdysfunction.Thisstudypresentsanintegratedmodelofthemitralvalve(MV) coupledtotheleftventricle(LV),withthegeometryderivedfrominvivoclinicalmagneticresonance im-ages.NumericalsimulationsusingthiscoupledMV–LVmodelaredevelopedusinganimmersed bound-ary/finiteelementmethod.Themodelincorporatesdetailedvalvularfeatures,leftventricularcontraction, nonlinearsofttissuemechanics,and fluid-mediatedinteractionsbetweentheMVand LVwall.Weuse themodeltosimulatecardiacfunctionfromdiastoletosystole.NumericallypredictedLVpumpfunction agreeswellwithinvivodataoftheimagedhealthyvolunteer,includingthepeakaorticflowrate,the systolicejectionduration,andtheLVejectionfraction.InvivoMVdynamicsarequalitativelycaptured. Wefurtherdemonstratethatthediastolicfillingpressureincreasessignificantlywithimpaired myocar-dialactiverelaxationtomaintainanormalcardiacoutput.Thisisconsistentwithclinicalobservations. Thecoupledmodelhasthepotentialtoadvanceourfundamentalknowledgeofmechanismsunderlying MV–LVinteraction,andhelpinriskstratificationandoptimisationoftherapiesforheartdiseases.
© 2017TheAuthor(s).PublishedbyElsevierLtdonbehalfofIPEM. ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)
1. Introduction
The mitral valve (MV) has a complex structure that includes two distinct asymmetric leaflets, a mitral annulus, and chordae tendineaethatconnecttheleafletstopapillarymusclesthatattach tothewalloftheleftventricle(LV).MVdysfunctionremainsa ma-jormedicalproblembecauseofitscloselinktocardiacdysfunction leadingtomorbidityandprematuremortality[1] .
Computational modelling for understanding MV mechanics promisesmoreeffectiveMVrepairsandreplacement[2–5] . Biome-chanical MV models have been developed for several decades, starting from the simplified two-dimensional approximation to three-dimensional models, and to multi-physics/-scale models [6–12] .Mostpreviousstudieswerebasedonstructuraland quasi-staticanalysisapplicabletoaclosedvalve[13] ;however,MV func-tion during the cardiac cycle cannot be fully assessed without modellingtheventricular dynamicsandthe fluid–structure inter-action(FSI)betweentheMV,ventricles,andthebloodflow[13,14] .
∗ Correspondingauthor.
E-mailaddress:[email protected](H.Gao).
Becauseofthecomplexinteractions betweentheMV,the sub-mitral apparatus, the heart walls, andthe associated blood flow, very few modelling studies have been carried out that integrate theMV andventriclesinasinglemodel[15–17] .Kunzelman, Ein-stein, andco-workersfirstsimulatednormalandpathological mi-tralfunction[18–20] withFSI usingLS-DYNA (LivermoreSoftware TechnologyCorporation,Livermore,CA,USA)byputtingaMVinto astraight tube.Usinga similarmodellingapproach,Lau etal.[21] comparedMVdynamicswithandwithoutFSI,andtheyfoundthat valvular closureconfigurationisdifferentwhen usingtheFSI MV model.SimilarfindingswerereportedbyTomaetal.[22] .Overthe lastfewyears,therehavealsobeenanumberofFSIvalvular mod-elsusingimmersedboundary(IB)methodtostudytheflowacross theMV[23–25] .Inaseriesofstudies,Tomaetal.[22,26,27] devel-opedanFSIMVmodelbasedoninvitroMVexperimentalsystem to study the function of the chordal structure, and good agree-ment was found between the computational model and in vitro experimentalmeasurements.However,noneoftheaforementioned MVmodelsaccountedfortheMVinteractionwiththeLV dynam-ics. Indeed, Lau etal.[21] found that even witha fixed U-shaped ventricle, the flow pattern was substantially different from that estimated using a tubular geometry. Despite the advancements
http://dx.doi.org/10.1016/j.medengphy.2017.06.042
Fig.1. TheCMR-derivedMV–LVmodel.(a)TheMVleafletsweresegmentedfromastackofMRimagesofavolunteeratearly-diastole,(b)positionsofthepapillarymuscle headsandtheannulusring,(c)reconstructedMVgeometrywithchordae,(d)anMRimageshowingtheLVandlocationoftheoutflowtract(AV)andinflowtract(MV),(e) theLVwalldelineationfromshortandlongaxisMRimages,(f)thereconstructedLVmodel,inwhichtheLVmodelisdividedintofourpart:theLVregionbellowtheLV base,thevalvularregion,andtheinflowandoutflowtracts,(g)thecoupledMV–LVmodel,and(h)therule-basedfibreorientationsintheLVandtheMV.(Forinterpretation ofthereferencestocolourinthetext,thereaderisreferredtothewebversionofthisarticle.)
incomputational modellingofindividual MV[12,13] andLV mod-els[28–30] ,it remains challengingto develop an integratedMV– LV modelthat includes thestrong couplingbetweenthe valvular deformation and theblood flow. Reasons for thisincludelimited data for model construction, difficultchoices of boundary condi-tions,andlargecomputationalresourcesrequiredbythese simula-tions.
Wenketal. [15] reportedapurely structuralMV–LVmodel us-ingLS-DYNAthatincludedtheLV,MV,andchordaetendineae.This model waslater extended to studyMV stress distributions using a saddle-shaped and asymmetric mitral annuloplasty ring[16] . A morecompletewhole-heartmodelwasrecentlydevelopedusinga humancardiac functionsimulatorintheDassaultSystèmesLiving Heartproject [17] ,whichincludedfourventricularchambers, car-diacvalves,electrophysiology,anddetailedmyofibreandcollagen architecture. Using the same simulator, effectsof differentmitral annulusringwerestudiedbyRauschetal.[31] .However,this sim-ulatordoesnotaccountfordetailedFSI.
The earliest valve–heart coupling model that includes FSI is credited to Peskin and McQueen’spioneering work in the 1970s [32–34] using the classical IB approach[35] . Using this same method, Yin etal. [36] investigated fluid vortices associated with the LV motionasa prescribed moving boundary. Recently, Chan-dran and Kim[37] reported a prototype FSI MV dynamics in a simplified LVchamber modelduringdiastolicfilling usingan im-mersedinterface-likeapproach.Oneofthekeylimitationsofthese coupledmodelsisthesimplifiedrepresentationofthe biomechan-ics of the LV wall. To date, there has been no work reported a
coupledMV–LVmodelwhichhasfullFSIandbasedonrealistic ge-ometryandexperimentally-basedmodelsofsofttissuemechanics. ThisstudyreportsanintegratedMV–LVFSImodelderivedfrom in vivo images of a healthy volunteer. Although some simplifi-cations are made, this is the first three-dimensional FSI MV–LV modelthatincludesMVdynamics,LVcontraction,and experimen-tally constrained descriptions of nonlinear soft tissue mechanics. Thisworkis builtonourprevious modelsofthe MV [24,25] and LV[29,38] . The model is implemented using a hybrid immersed boundarymethodwithfiniteelementelasticity(IB/FE) [39] .
2. Methodology
2.1. MV–LVmodelconstruction
The MV geometrywasreconstructed from LVOT MV cine im-agesatearly-diastole,justaftertheMVopens.Theleaflet bound-aries were manually delineated from MR images, as shown in Fig. 1 (a), inwhichthe heads ofpapillarymuscle andthe annulus ringwereidentifiedasshownin Fig. 1 (b).TheMVgeometryandits sub-valvularapparatuswerereconstructedusingSolidWorks (Das-saultSystèmes SolidWorks Corporation, Waltham, MA, USA). Be-causeit is difficult to seethe chordal structuralin the CMR,we modelled the chordae structure using sixteen evenly distributed chordae tendineae running through the leaflet free edges to the annulusring,asshownin Fig. 1 (c), followingprior studies[24,25] . Ina similar approachto theMV reconstruction, theLV geometry wasreconstructedfromthesamevolunteeratearly-diastoleby us-ingboththeshort-axis andlong-axiscineimages[29,40] . Fig. 1 (d) showstheinflowandoutflowtractsinoneMRimage.TheLVwall wasassembled fromthe short-/long-axisMRimages(Fig. 1 (e))to formthethreedimensionalreconstruction(Fig. 1 (f)).TheLVmodel wasdividedintofourregions:theLVwall,thevalvularregion,and theinflowandtheoutflowtracts,asshownin Fig. 1 (g).
The MV model was mounted into the inflow tract of the LV modelaccordingtotherelativepositionsderivedfromtheMR im-agesin Fig. 1 (g). The left atrium wasnot reconstructed but mod-elledasatubular structure,andthegapbetweentheMVannulus ringandtheLVmodelwasfilledusingahousingdiscstructure.A three-elementWindkesselmodelwasattachedtotheoutflowtract oftheLV model toprovide physiological pressureboundary con-ditionswhenthe LVisinsystolic ejection[40] .The chordaewere not directly attached to the LV wall since the papillary muscles werenotmodelled,asinourpreviousstudy[25] .Themyocardium has a highly layered myofibre architecture, which is usually de-scribedusing a fibre–sheet–normal(f, s, n) system. A rule-based methodwasusedtoconstructthemyofibreorientationwithinthe LVwall. Themyofibre anglewasassumedtorotatefrom−60°to 60° fromendocardium to epicardium, representedby the red ar-rowsin Fig. 1 (h).In a similar way, the collagenfibres in the MV leaflets were assumed to be circumferentially distributed, paral-lel along the annulus ring, represented by the yellow arrows in Fig. 1 (h).
2.2.IB/FEframework
ThecoupledMV–LVmodelemploys anEuleriandescriptionfor the blood, which is modelled as a viscous incompressible fluid, along with a Lagrangian description for the structure. The fixed physical coordinates are x=
(
x1,x2,x3)
∈,
and the Lagrangian reference coordinate system is X=(
X1,X2,X3)
∈U. The exterior unitnormalalong∂
U is N (X ). Letχ
(X ,t) denotethephysical po-sitionofanymaterialpoint X attimet,sothatχ
(
U,t)
=s
(
t)
is thephysicalregionoccupiedbytheimmersedstructure.TheIB/FE formulationoftheFSIsystemreadsρ
∂
u∂
t(
x,t)
+u(
x,t)
·∇
u(
x,t)
=−
∇
p(
x,t)
+μ
∇
2u(
x,t)
+fs(
x,t)
, (1)∇
·u(
x,t)
=0, (2)fs
(
x,t)
=
U
∇
·Ps
(
X,t)
δ
(
x−χ
(
X,t))
dX−
∂U
Ps
(
X,t)
N(
X)
δ
(
x−χ
(
X,t))
dA(
X)
, (3)∂
χ
∂
t(
X,t)
=u
(
x,t)
δ
(
x−χ
(
X,t))
dx, (4)where
ρ
isthefluid density,μ
isthefluid viscosity, u isthe Eu-lerian velocity, p is the Eulerian pressure, and f s is the Eulerian elasticforce density,whichisdetermined fromthe deformed im-mersed structureand its activecontraction. Notethe IB formula-tion employs a common momentum equation for both the fluid andthesolid,inwhichtheadditionalsolid stressesareaccounted forby f sin Eq. (1) .InteractionsbetweentheLagrangianand Eule-rianfields are achievedby integral transformswitha Diracdelta function kernelδ
(x)[35] , in Eqs. (3) and (4 ). Different from the classical IBapproach [35] , heretheelastic force density f s is de-terminedfromthefirstPiola–Kirchoff stresstensor P s.Thisallows thesolid deformationsto be describedusinga nonlinear hypere-lastic constitutive law;see Section 2.3 below. Formore details of the hybridIB/FEframework, readers arereferred to thepaper by GriffithandLuo[39] .2.3. Softtissuemechanics
ThetotalCauchystress(
σ
)inthecoupledMV–LVsystemisσ
(
x,t)
=σ
f(
x,t)
+σ
s(
x,t)
forx∈s,
σ
f(
x,t)
otherwise, (5)where
σ
fisthefluid-likestresstensor,definedasσ
f(
x,t)
=−pI+μ
[∇
u+(
∇
u)
T]. (6)σ
s isthesolid stress tensorobtainedfromthenonlinear soft tis-sueconstitutive laws.The firstPiola–Kirchhoff stresstensor Ps in Eq. (3) isrelatedtoσ
sthroughPs=J
σ
sF−T, (7)inwhichF=
∂
χ
/∂XisthedeformationgradientandJ=det(
F)
. In the MV–LV model, we assume the structure below the LV baseiscontractile(Fig. 1 (g)),theregionsabovetheLVbasalplane, includingtheMVanditsapparatuses,arepassive.Namely,Ps=
Pp+Pa belowthebasalplane,
Pp abovethebasalplane, (8)
wherePa andPp aretheactive andpassivePiola–Kirchhoff stress tensors, respectively. The MV leaflets are modelled asan incom-pressiblefibre-reinforcedmaterial withthestrain energyfunction
WMV=C1
(
I1−3)
+av
2bv
(
exp[bv
(
max(
Ifc,1)
−1)
2]−1)
, (9)in which I1=trace
(
C)
is the first invariant of the right Cauchy– Green deformation tensor C = F TF , Icf =f c
0·
(
C fc0)
is the squared stretchalong thecollagen fibredirection, andfc0 denotes the col-lagenfibre orientationin thereferenceconfiguration.Themax(•) functionensurestheembeddedcollagennetworkonlybearsloads whenstretched,butnotincompression.C1,av,andbvarematerial parameters adopted froma prior study[25] and listed in Table 1 . ThepassivestresstensorPpintheMVleafletsisthen
Pp=
∂
WMV∂
F −C1F−T+β
slog(
I3)
F−T, (10) where I3=det(
C)
, andβ
s is the bulk modulus for ensuring the incompressibility of immersed solid, and the pressure-like term C1F−T ensurestheelasticstressresponseiszerowhenF=I.Wemodelthechordaetendineaeasaneo-Hookeanmaterial,
Wchordae=C
(
I1−3)
, (11)Table1
MaterialparametervaluesforMVleaflets,chordaeandthemyocardium. MVleaflets C1(kPa) av(kPa) bv(kPa)
Anterior 17.4 31.3 55.93 Posterior 10.2 50.0 63.48 Chordae C(kPa)
Systole 9000 Diastole 540 Myocardium
Passive a(kPa) b af(kPa) bf as(kPa) bs a8fs(kPa) b8fs
0.24 5.08 1.46 4.15 0.87 1.6 0.3 1.3 Active Treq= 225kPa
Thepassive responseoftheLVmyocardium isdescribedusing theHolzapfel–Ogdenmodel[41] ,
Wmyo=
a
2bexp[b
(
I1−3)
]+i=f,s
ai 2bi
{
exp[bi
(
max(
I4i,1)
−1)
2]−1}
+ afs
2bfs
{
exp[bfs
(
I8fs)
2]−1}
,(12)
inwhicha,b,af,bf,as,bs,afs,bfsarethematerialparameters,I4f,I4s and I8fs arethe strain invariants relatedto the myofibre orienta-tions.Denoting themyofibredirectioninthereferencestateby f 0 andthesheetdirectionby s 0,wehave
I4f=f0·
(
Cf0)
,I4s=s0·(
Cs0)
,and I8fs=f0·(
Cs0)
. (13)Themyocardialactivestressisdefinedas
Pa=JTFf
0f0, (14)
whereTistheactivetensiondescribedbythemyofilamentmodel of Niederer etal.[42] , using a set of ordinary differential equa-tions involvingthe intracellularcalcium transient(Ca2+), sarcom-ere length,andtheactive tensionatthe restingsarcomerelength (Treq).
The constitutive parameters in Eqs. (9) , (11 ) and (12 ), sum-marised in Table 1 , are obtained from our previous studies [25,29] which showed that the simulated MV and LV dynamics matched the in vivo measurements well. In the active contrac-tion model (Eq. (14) ), we adopted the same parameters asused by Niedereretal. [42] ,exceptthatTreqissetto225kPa tomatch thecontractionsobservedintheimagedvolunteer.
2.4. Boundaryconditionsandmodelimplementation
Because only the myocardium below the LV basal plane con-tracts, we fix the LV basal plane along the circumferential and longitudinal displacements, but allow the radial expansion. The myocardium below the LV basal plane is left free to move. The valvular regionisassumedto bemuch softerthanthe LVregion. In diastole, a maximum displacement of 6mm is allowed in the valvularregionusingatetheringforce.Insystole,thevalveregion is gradually pulled back to the original position. The inflow and outflowtractsare fixed.BecausetheMV annulusringis attached to a housing structure which is fixed, no additional boundary conditions are applied to the MV annulus ring. Fluid boundary conditionsareappliedtothetopplanesoftheinflowandoutflow tracts.Thefunction oftheaorticvalveissimplymodelledas:the aortic valve is eitherfully openedor fullyclosed, determined by thepressuredifferencebetweenthevaluesinsidetheLVchamber and the aorta. After end-diastole, the LV region will contract simultaneously triggeredbya spatiallyhomogeneously prescribed intracellular Ca2+ transient [29] , as shown in Fig. 3 . The flow boundaryconditionsinacardiaccyclearesummarisedbelow.
• Diastolic filling: A linearly ramped pressure from 0 to a population-based end-diastolicpressure (EDP=8mmHg)is ap-pliedtotheinflowtractover0.8s,whichisslightlylongerthan
the actual diastolic duration of the imaged volunteer (0.6s). In diastole about80% of diastolicfilling volume is dueto the “sucking” effect oftheleft ventricleinearly-diastole [43] .This negative pressurefield insidethe LV cavityis dueto the my-ocardialrelaxation.Wemodelthis“sucking” effectusingan ad-ditionalpressureloadingappliedtotheendocardialsurface, de-noted as Pendo, which is linearly rampedfrom 0 to 12mmHg over 0.4s, andthenlinearlydecreased tozeroatend-diastole. The value of Pendo ischosen by matching the simulated end-diastolicvolumetothemeasureddatafromCMRimages.Blood flow is not allowed tomove out ofthe LV cavitythrough the inflow tractindiastole.Zeroflowboundaryconditionsare ap-pliedtothetopplaneoftheoutflowtract.
• Iso-volumetriccontraction:Alongthetopplane oftheinflow tract,theEDPloadingismaintained,andtheflowiscontrolled by the MV leaflet dynamics. Note during the iso-volumetric contraction,regurgitationmayoccurduetotheMVclosure ac-tionandthedysfunctionoftheMVapparatus.However,asmall backflow before the MV is fully closed may be deemed nor-mal[44] . Zero flow boundary conditions are retained for the outflowtract.Thedurationoftheiso-volumetriccontractionis determined by themyocardialcontraction andendswhen the aorticvalveopens.TheaorticvalveopenswhentheLVpressure ishigherthanthepressureintheaorta,whichisinitiallysetto be the cuff-measured diastolic pressurein thebrachial artery, 85mmHg.
• Systolicejection:Whentheaorticvalveopens,athree-element Windkessl model is coupled to the top plane of the outflow tracttoprovideafterload.Thevolumetricflowratesacrossthe top plane of the outflow tract is calculated from the three-dimensionalMV–LVmodel,andfedintotheWindkesselmodel [45] ,whichreturnsanupdatedpressurefortheoutflowtractin the nexttime step.Thesystolicejection phaseendswhen the left ventriclecannot pumpanyflowthroughtheoutflowtract, andtheWindkesselmodelisdetached.
• Iso-volumetric relaxation:Zeroflow boundary conditionsare appliedtoboththetopplanesoftheoutflowandinflowtracts untilthetotalcycleendsat1.2s.
ThecoupledMV–LVmodelisimmersedina 17cm× 16cm× 16cmfluidbox.Abasictimestepsize
t0=1.22×10−4sisused in the diastolic and relaxation phases, a reduced time step size (0.25
t0)isusedintheearlysystolewithadurationof0.1s,and an even smaller time step of0.125
t0 is used in theremainder ofthesystolicphase.Becauseexplicittimesteppingisusedinthe numericalsimulations [39] ,weneedtouseatimestepsizesmall enoughtoavoidnumericalinstabilities,particularlyduringthe sys-tolicphasetoresolvethehighlydynamicLVdeformation.
Fig.2. FlowratesacrosstheAVandMVfromdiastoletosystole.Diastolicphase: 0sto0.8s;systolicphase:0.8sandonwards.Positiveflowrateindicatestheblood flowsoutoftheLVchamber.
(https://libmesh.github.io ), PETSc (http://www.mcs.anl.gov/petsc ), andhypre (http://www.llnl.gov/CASC/hypre ). All simulations were carriedout on a Linux workstation atthe University of Glasgow, with10Intel(R)Xeon(R)CPUcores(2.65GHz)and32GBRAM.The simulationtimeforonecardiaccycleisaround240h(10days).
3. Results
3.1.Pumpfunction
Fig. 2 showsthe computedvolumetric flow rates through the MV andthe AV frombeginning of diastole to end-systole. In di-astole,thevolumetricflow ratethroughtheMVlinearlyincreases withincreasedpressureloadingintheendocardialsurface,witha maximumvalueof210mL/sat0.4s.Diastolicfillingismaintained bythe increasedpressurein theinflowtract, butwithdecreased flow rates until end of diastole at 0.8s. The negative flow rate in Fig. 2 indicates the flow is entering the LV chamber. After end-diastole,themyocardiumstartstocontract,andthecentralLV pressureincreasesuntilitexceedstheaorticpressure(initiallyset tobe85mmHg)at0.857s. Duringiso-volumetric contraction,the MVcloseswithatotalclosureregurgitationflowof7.2mL,around 10% ofthetotal fillingvolume, whichis comparableto thevalue reportedby Laniado etal.[44] . There is only minor regurgitation across the MV during systolic ejection after the iso-volumetric contraction phase. Blood is then ejected out of the ventricle throughthe AV,andthe flow ratethroughthe AV duringsystole reachesapeakvalueof468mL/s(Fig. 2 )(theCMRmeasuredvalue is498mL/s).The total ejectionduration is 243ms (the measured duration is 300ms) with a stroke volume of 63.2mL. The total blood ejected out of the LV chamber, including the regurgita-tionthrough the MV, is 72.1mL. This corresponds to an ejection fractionof51%,andisclosetotheCMRmeasuredvalueof57%.
Fig. 3 shows the profiles ofthe normalised intracellular Ca2+, LVcavityvolume,centralLVpressure,andtheaveragemyocardial active tension from diastole to systole. Until mid-diastole (0s to 0.56s),thecentral LVpressure isnegative,andthe associated di-astolic filling volume is around 65mL, which is 90% ofthe total diastolicfilling volume. In late-diastole,the LV pressure becomes positive. There is a delaybetween the myocardial active tension andtheintracellular Ca2+ profile,butthecentral LVpressure fol-lowstheactive tensionclosely throughoutthecycleasshown in
Fig.3. NormalisedintracellularCa2+ ,LVcavityvolume,centralLVpressure,and
averagemyocardialactivetension.Allcurvesarenormalisedtotheirownmaximum values,whichare:1μMolfor Ca2+ ,145mLfor LVcavityvolume,162mmHgfor
centralLVpressure,and96.3kPaforaveragemyocardialactivetension.
Fig. 3 .The peak systolicLV pressureis162mmHg, comparableto thecuff-measuredpeakbloodpressure150mmHg.
3.2. Intracardiacflowpattern
Fig. 4 (a–d)showsthestreamlinesatearly-diastolic filling, late-diastolic filling, when the MV is closing (iso-volumetric contrac-tion), and mid-systolic ejection when the left ventricle is eject-ing.During thediastolicfilling (Fig. 4 (a)), thebloodflows directly through the MV into the LV chamber towards the LV apex, in late-diastolein Fig. 4 (b),theflowpatternbecomeshighlycomplex. Wheniso-volumetriccontractionends,theMVispushed back to-wardstheleft atrium.Inmid-systole, thebloodispumpedoutof theLVchamberthroughtheaorticvalveintothesystemic circula-tion,formingastrongjetasshownin Fig. 4 (d).
3.3. MVandmyocardialdynamics
Fig. 5 shows the deformed MV leaflets along with the corre-sponding CMRcine imagesatearly-diastole (the referencestate), end-diastole, and mid-systole. In general, the in vivoMV and LV dynamics fromdiastole to systole are qualitatively capturedwell bythecoupledMV–LVmodel.However,adiscrepancyisobserved duringthediastolicfilling,whentheMVorificeinthemodelisnot openedaswidelyasintheCMRcineimage(Fig. 5 (b)).Inaddition, themodelledMVleafletshavesmallgapsnearthecommissure ar-easevenin thefullyclosurestate.This ispartiallycausedby the finitesizeoftheregulariseddeltafunctionattheinterfaceand un-certaintiesinMVgeometryreconstructionusingCMRimages.
Fig.4. StreamlinesintheMV–LVmodelatearly-diastolicfilling(a),late-diastolicfilling(b),whenisovolumetriccontractionends(c),andatthemid-systole.Streamlineare colouredbyvelocitymagnitude,theLVwallandMVarecolouredbythedisplacementmagnitude.Red:high;blue:low.(Forinterpretationofthereferencestocolourinthis figurelegend,thereaderisreferredtothewebversionofthisarticle.)
3.4. EffectsofPendo onpumpfunction
From Fig. 3 ,itisclearthatthattheappliedendocardialpressure (Pendo)createsanegativepressureinsidetheLVchamber,similarto theeffectsofthemyocardialactiverelaxation. Wefurther investi-gate how Pendo affects the MV–LV dynamics by varying its value from8mmHgto16mmHg,andtheeffectswithoutPendobutwith anincreasedEDPfrom8mmHgto20mmHg.Weobservethatwith an increasedPendo, thepeak flow rateacross the MV duringthe filling phase becomeshigher with more ejected volume through the aorticvalve. Wealso havea longerejection duration,shorter iso-volumetric contraction time,and higherejection fractionasa result of increasing Pendo. On the other hand, if we do not ap-plyPendo,amuchgreaterandnon-physiologicalEDPisneededfor therequiredejectionfraction.Forexample,withEDP=8mmHg,the ejectionfractionisonly29%.OnlywhenEDP=20mmHg,thepump functioniscomparabletothecasewithEDP=8mmHgandPendo= 16mmHg.Theseresultsaresummarisedin Table 2 .
4. Discussion
This study demonstrates the feasibility of integrating a MV modelwithaLVmodelfromahealthyvolunteerbasedoninvivo CMR images.Thisisthe first physiologicallybased MV–LVmodel withfluid–structureinteractionthatincludesnonlinear hyperelas-tic constitutive modelling of the soft tissue. The coupled MV–LV modelisusedtosimulateMVdynamics,LVwalldeformation, my-ocardial active contraction, as well as intraventricular flow. The
modellingresultsareinreasonablequantitativeagreementwithin vivomeasurementsandclinicalobservations,suchasthepeak aor-ticflowrate(468mL/svs.498mL/s),theejectionduration(243ms vs. 300ms), peak cuff-measured systolic pressure (162mmHgvs. 150mmHg),andLV ejection fraction(51% vs. 57%).Note that the above comparisonsare madewithcomputational vs. imaged val-ues.
Fig.5. ComparisonsbetweentheMVandLVstructuresat(a)referenceconfiguration,(b)end-diastole,and(c)end-systole,andthecorrespondingCMRcineimages(left). Colouredbythedisplacementmagnitude.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)
Table2
EffectsofEDPandtheendocardialpressureloading(Pendo)onMVandLVdynamics.Thecaseforsections
3.1–3.3ishighlightedwithboldfonts.
Cases(mmHg) tiso−con(ms) tejection(ms) Vejection LV (mL) V
filling MV (mL) F
peak
MV (mL/s) LVEF(%)
EDP= 8,Pendo= 8 60 227 52 60.6 412.93 47
EDP= 8,Pendo= 10 58 237 57.6 65.9 442.60 49 EDP=8,Pendo= 12 57 243 63.2 72.1 468.41 51
EDP= 8,Pendo= 14 55 251 67.8 76.8 486.84 53
EDP= 8,Pendo= 16 54 256 72.3 81.3 503.76 54
EDP= 8,Pendo= 0 75 174 20.8 29.5 209.54 29
EDP= 12,Pendo= 0 64 213 41.0 50.9 343.47 42
EDP= 14,Pendo= 0 61 226 50.6 59.8 406.81 47
EDP= 16,Pendo= 0 58 243 61.8 71.9 459.64 51
EDP= 18,Pendo= 0 56 251 68.5 79.3 486.58 54
EDP= 20,Pendo= 0 55 262 75.7 86.2 511.16 55
becausepumpfailure willoccur ifnoothercompensation mecha-nismexists.
During diastole, the MV–LV model seems to yield a smaller orifice compared to the corresponding CMR images. In our pre-vious study [25] , the MV was mounted in a rigid straight tube, the peak diastolic filling pressure is around 10mmHg, and the peakflowrateacrosstheMViscomparabletothemeasuredvalue (600mL/s).WhileinthiscoupledMV–LVmodel,eventhoughwith additional Pendo, the peakflow rate(200mL/s) ismuch lessthan the measured value. One reason is because of the extra resis-tance from the LV wall, which is absent in the MV-tube model [25] .Thediastolicphasecanbedividedintothreephases[43] :the rapidfilling,slowfilling,andatrialcontraction.Duringrapidfilling, the transvalvularflow isresultedfrommyocardial relaxation(the “sucking” effect),whichcontributesto80%ofthetotal transvalvu-lar flow volume. During slow filling and atrial contraction, the left atrium needs to generate a higher pressureto provide addi-tionalfilling.InthecoupledMV–LVmodel,therampedpressurein thetop plane oftheinflowtractduringlate-diastoleisrelatedto the atrialcontraction,andduringthistime, only 10%of thetotal transvalvularflowoccurs.However,thepeakflowrateinrapid fill-ing phaseismuchlower comparedtothemeasured value,which suggeststhatthemyocardialrelaxationwouldbemuchstronger.
Inaseriesofstudiesbasedoninvitro
μ
CTexperiments,Toma etal. [22,26,27] suggestedthatMVmodelswithsimplifiedchordal structure would not compare well with experimental data, and thata subject-specific3D chordalstructureisnecessary.Thismay explain some of the discrepancies we observe here. A simplified chordal structure is usedin thisstudybecause we are unable to reconstruct the chordal structure from the CMR data. CT imag-ing mayallow the chordae reconstruction butit comes with ra-diationrisk.Patient-specificchordal structureinthecoupledMV– LV modelwouldrequirefurtherimprovements ofinvivo imaging techniques.Several other limitations in the model may also contribute to the discrepancies. These include the uncertainty of patient-specific parameteridentification,the uncertaintyinMVgeometry reconstructionfromCMRimages,thepassiveresponseassumption around theannulusringandthevalvularregionoftheLV model, andthe lackof pre-straineffects.Studies addressing theseissues are already under way. We expect that further improvement in personalisedmodellingandmoreefficienthighperformance com-puting would make the modelling more physiologically detailed yet fast enough forapplications in risk stratification and optimi-sationoftherapiesinheartdiseases.
5. Conclusion
Interaction betweenthe mitral valve andthe ventricular wall plays an essential role in cardiac pump function. In this study,
we have developed a first fully coupled MV–LV model that in-cludes fluid–structure interaction as well as soft tissue mechan-icswith model parameters determined fromin vivo image data. The modelgeometry isderived frominvivo magnetic resonance imagesofahealthyvolunteer,andincorporatesthree-dimensional finiteelementrepresentationsoftheMVleaflets,sub-valvular ap-paratus,andtheLVgeometry.Fibre-reinforcedhyperelastic consti-tutivelaws are used to describe thepassive response ofthe soft tissues, and a myofilament model is used to model the myocar-dial activecontraction.Ourresults showthat thedeveloped MV– LVmodel cansimulateMV–LV interactionwithgoodagreements, including the peak aortic flow rate, the peak systemic pressure, thesystolicejectionduration andtheLVejectionfraction,within vivomeasurements despiteseveralmodellinglimitations.We fur-therfindthat withimpairedmyocardial active relaxation, the di-astolic filling pressure needs to increase significantly in order to maintaina normal cardiac output, whichis consistent with clin-icalobservations inpatientswithimpairedmyocardialrelaxation. Thismodelthereby representsasteptowards awhole-heart mul-tiphysicsmodellingwithatargetforclinicalapplications.
Conflictsofinterest
Theauthorshavenoconflictsofinterest.
Acknowledgements
We are grateful for the funding from the UK EPSRC (EP/N014642/1 , and EP/I029990/1 ) andthe British Heart Founda- tion (PG/14/64/31043 ), andthe National Natural Science Founda- tion of China (no. 11471261 ).In addition,FengreceivedtheChina ScholarshipCouncilStudentshipandtheFeeWaiverProgrammeat the University of Glasgow, Luo is funded by a Leverhulme Trust Fellowship(RF-2015-510 ),andGriffithissupportedbythe National Science Foundation (NSFaward ACI 1450327 )andthe National In- stitutes of Health (NIHaward HL117063 ).
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