HAL Id: hal-01826726
https://hal.archives-ouvertes.fr/hal-01826726
Submitted on 29 Jun 2018
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Universal ventricular coordinates: A generic framework for describing position within the heart and transferring
data
Jason Bayer, Anton Prassl, Ali Pashaei, Juan F. Gómez, Antonio Frontera, Aurel Neic, Gernot Plank, Edward Vigmond
To cite this version:
Jason Bayer, Anton Prassl, Ali Pashaei, Juan F. Gómez, Antonio Frontera, et al.. Universal ventricular coordinates: A generic framework for describing position within the heart and transferring data.
Medical Image Analysis, Elsevier, 2018, 45, pp.83-93. �10.1016/j.media.2018.01.005�. �hal-01826726�
ContentslistsavailableatScienceDirect
Medical Image Analysis
journalhomepage:www.elsevier.com/locate/media
Universal ventricular coordinates: A generic framework for describing position within the heart and transferring data
Jason Bayera,b, Anton J. Prasslc,Ali Pashaeia,b, Juan F. Gomeza,b,Antonio Fronteraa,d, Aurel Neicc,Gernot Plankc, EdwardJ. Vigmonda,b,∗
aLIRYC Electrophysiology and Heart Modeling Institute, Bordeaux Fondation, avenue du Haut-Lévèque, Pessac 33600, France
bIMB Bordeaux Institute of Mathematics, University of Bordeaux, 351 cours de la Libération, Talence 33405, France
cGottfried Schatz Research Center, Biophysics, Medical University of Graz, Neue Stiftingtalstrasse 6, 8010 Graz, Austria
dDepartment of Electrophysiology, Hôpital Haut Lévèque, 1 avenue Magellan, Pessac 33100 France
a rt i c l e i n f o
Article history:
Received 10 July 2017 Revised 16 January 2018 Accepted 22 January 2018 Available online 2 February 2018 Keywords:
Mapping Coordinates Volumetric meshes Deformation
a b s t r a c t
Beingabletomapaparticularset ofcardiac ventriclestoageneric topologicallyequivalentrepresen- tationhas manyapplications,includingfacilitatingcomparison ofdifferenthearts,as wellas mapping quantitiesandstructuresofinterestbetweenthem.InthispaperwedescribeUniversalVentricularCoor- dinates(UVC),whichcanbeusedtodescribepositionwithinanybiventricularheart.UVCcomprisefour uniquecoordinatesthatwehavechosentobeintuitive,welldefined,andrelevantforphysiologicalde- scriptions.Wedescribehowtodeterminethesecoordinatesforanyvolumetricmeshbyillustratinghow toproperlyassignboundaryconditionsandutilizesolutionstoLaplace’sequation.UsingUVC,wetrans- ferredscalar,vector,andtensordatabetweenfourunstructuredventricularmeshesfromthreedifferent species.Performingthemappingswasveryfast,ontheorderofafewminutes,sincemeshnodeswere searchedinaKDtree.Distanceerrorsinmappingmeshnodesbackandforthbetweenmesheswereless thanthesizeofanelement.Analytically derivedfiberdirectionswerealsomappedacrossmeshes and compared,showing <5°differenceovermostoftheventricles.Theabilitytotransfergradientswasalso demonstrated.Topologicallyvariablestructures,likepapillarymuscles,requiredfurtherdefinitionoutside oftheUVCframework.Inconclusion,UVCcanaidintransferringmanytypesofdatabetweendifferent biventriculargeometries.
© 2018 The Author(s). Published by Elsevier B.V.
ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.
(http://creativecommons.org/licenses/by-nc-nd/4.0/)
1. Introduction
Theventriclesofmammalianheartssharemanycommonchar- acteristics.Theseincludeabiventriculargeometry,Purkinjesystem (PS),andhelicalmyocardialfiberorientation(Streeteretal.,1969).
Theyalsoexpresselectricalheterogeneitywithrespecttotransmu- ral (Lou et al., 2011; Sabir et al., 2007), apicobasal (Janse et al., 2012), andleft-right gradients (Panditet al., 2011; Volders etal., 1999).
It is recognized that describing such scalar, vector, and ten- sor data on a generic heart framework is important, especially
∗ Corresponding author.
E-mail addresses: jason.bayer@ihu-liryc.fr (J. Bayer), anton.prassl@medunigraz.at (A.J. Prassl), ali.pashaei@u-bordeaux.fr (A. Pashaei), juan.gomez@ihu-liryc.fr (J.F.
Gomez), a.frontera@gmail.com (A. Frontera), aurel.neic@medunigraz.at (A. Neic), gernot.plank@medunigraz.at (G. Plank), edward.vigmond@u-bordeaux.fr (E.J. Vig- mond).
in the clinical context. This prompted the American Heart Asso- ciation to define a standard 17 sector map over a decade ago (Cerqueira etal., 2002). However, it canbe difficult toaccurately transferhigh-resolution data andcompare resultsbetweenvastly different heart geometries. This difficulty arises from the highly variable size and relative proportion of hearts within the same speciesorbetweendifferentspecies.
Modern day methods for mapping ventricular data between heartsarenamelybasedonlargedeformationdiffeomorphicmet- ric mapping (LDDMM) applied to anatomical data from mag- netic resonance imaging (MRI) and computed tomography scans (Begetal.,2005).LDDMMhasshownsuccessformappingmyocar- dialfiberorientation(Vadakkumpadanetal.,2012)andanatomical positioning (Miller etal., 2014) betweenhearts. However, several drawbackshavelimitedLDDMM’swidespreadusetoalargearray ofother problems.Forexample,LDDMM worksdirectlywithvox- elizedimagedataandiscomputationallydemanding(hours)when thetemplateandtargetheartsareofsignificantlydifferentresolu-
https://doi.org/10.1016/j.media.2018.01.005
1361-8415/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license.
( http://creativecommons.org/licenses/by-nc-nd/4.0/ )
tions,particularlyifatleastoneofthedatasetshasasubmillime- terresolution.Thismakes transferringlargequantitiesofdatabe- tweenhistology,imaging,andcomputermodels extremelydaunt- ing.Furthermore,unstructuredmeshes,whichareextensivelyused for electromechanical modeling, need to be converted back and forth betweenan image stack format. This significantly increases thestorageandprocessingtimeforLDDMM,aswellasintroduces samplingerrorsatthesurfacesofthemesh.
We propose a novel global positioning system for three- dimensionalbiventricularheartgeometriestoovercomethelimita- tionsofLDDMM.Accordingly,wedefineasetofgenericventricular coordinatesto define positionwithin anybiventricularheart that takesonlyminutes toformulate on <12 central processingunits (CPU),evenformeshresolutionsatthesubmillimeterscalethatre- quiresignificantinput/output.Themethodalsoworksequallywell forregularorirregularmeshes.
To help foster more expansive data sharing, the coordinates areintuitivesothatexperimentalistscaneasilyestimate themfor theirdata,whichalsofacilitatestransferringexperimentaldatadi- rectlytocomputermodelingstudies.Furthermore,thecoordinates varysmoothlyinspacesothatthey canbe usedasargumentsto functionsthat assign myocardialpropertiesbasedon position.Fi- nally,despitegeometrieswhichcanbequitedifferent,subjectivity isminimizedwhenassigningthecoordinatessothatinterobserver variabilityisreduced.
To develop our UVC, we build upon aspects from previous computational cardiac modeling studies. To determine position inthe left ventricular (LV) geometry formechanical simulations, Costaetal.(1996)usedprolatespheroidalcoordinates.Forthepur- poseofassigningactionpotential heterogeneityina biventricular meshforelectrocardiogram genesis,Potseetal.(2006) usedmin- imal distance parameterizations betweenepicardial and endocar- dial surfaces to define the myocardial transmural direction, and Kelleretal.(2012)extendedthisapproachtodefinetheapicobasal direction. Bayeret al.(2012) parameterized the same transmural and apicobasal directions in biventricular geometries for assign- ing rule-based myocardial fiber orientation, but they determined thesedirectionsmoreaccurately fromsolutionstoLaplace’sequa- tioninsteadofminimaldistance.Paunetal.(2017)alsofolloweda Laplaceapproach formappingcomplex endocardialanatomy. De- spitethesedevelopments,noneprovidea completeparameteriza- tionforarbitrarybiventriculargeometries,particularlyforthesep- tumanditsjunctionwiththeLV,rightventricle(RV),andapex.
In this article,we first present the rationaleand methods for determiningUVCforarbitrary volumetricheart meshes.We then demonstrate how to use UVC for transferring scalar, vector, and tensor data between different biventricular meshes. To evaluate thealgorithm’sperformance,weanalyzeerrorsassociatedwiththe transformationprocess.Theresultsfromthisnewapproachshow greatpromiseforitswidespreadusetodetermineuniqueposition- ingintheheartandfacilitatethetransferofdatabetweenhearts.
2. Methods 2.1.UVCrationale
Position isdescribedwithinarbitraryventriclesby acombina- tion of four parameters called Universal Ventricular Coordinates (UVC).As mentionedintheintroduction,therationale forUVCis basedonhowclinicalandexperimentalstudiesdefinespatialdata within the ventricles.Accordingly, the first coordinate ofUVC, z, represents the distance traveled along the long axis of the ven- triclesfromthe apexto thebase. Thesecond coordinate ofUVC,
ρ,isthedistancefromtheendocardium toepicardium,i.e.trans- murality. The third coordinate of UVC, φ, is the circumferential distance around the long axis of the LV and RV. The φ is nec-
essary to distinguish between the posterior and anterior regions oftheventriclesinzandρ.Thefinal coordinate,ν,distinguishes between the LV andRV for the other three coordinates of UVC.
Inthe followingsections,necessaryuserinputsandUVCare dis- cussedindetailutilizingthebiventricularhumanmeshdescribed inMorenoetal.(2011),Bayeretal.(2016).
2.2. UserinputsforUVC
TocomputeUVC, theusermustfirst providethefollowingin- puts:(i)asingleepicardialapexsurfacepoint;(ii)asingleLVen- docardialsurfacepoint; (iii)asingleRV septalsurfacepoint;and (iv)thesurfacepointsrepresentingtheventricularbaseattheatri- oventricularjunction.Therestofthealgorithmisfullyautomatic, withonlyone tolerance(TSEPT) toadjust foroptimalresults.TSEPT definestheRVseptalsurfaceandisdescribedlaterinmoredetail.
Withtheuserinputsabove,thefollowingsurfacesareobtained tocomputeUVCinabiventricularmesh.
1. Base 2. Epicardium 3. LVendocardium 4. RVendocardium 5. RVseptum
Theprocessofobtainingeachsurfaceisasfollows.Foravolu- metricmesh,theentiresurfaceisknownbyidentifyingallelement faces not shared by another tissue element. On this surface, the userdefinesthebase,whichthenleavesthreedistinctisolatedre- gions:theepicardium;LVendocardium;andRVendocardium.The epicardialsurfaceisidentifiedbyfindingthesurfacecontainingthe epicardial apexpoint (input by theuser). The LVendocardium is identifiedbyfindingtheendocardialsurfacecontainingtheLVen- docardialsurfacepoint (inputbytheuser)withthethirdremain- ingsurfacelabeledasRVendocardium.TheRVendocardiumisfur- ther subdivided into its septal and free wall endocardialregions using thechange in thesurface normalatthe interface between the RVseptum andendocardium. Morespecifically, startingfrom the user input septal node on the RV surface, the algorithm de- fineseachconnectedelementasendocardiumandthengrowsthe regionbylookingatsurfacenormals.Ifthenormalofasurfaceel- ementunderconsiderationdoesnotdifferbymorethanthetoler- anceTSEPTfromitsneighboringelement,itisaddedtotheregion.
ATSEPTof0.05forthedotproductofneighboringnormalsisused forallmeshesinthisstudy.Inthecasetheresolutionofthemesh istoolow todetect prominentchanges insurfacenormalsatthe junctionof theRV septum withthefree wall, manual segmenta- tion wouldbe straightforward giventhe oversimplified geometry atsuchresolutions.Manualinterventionmayalsobe usedforthe baseinputifitisdifficulttodefineautomaticallyusingsimplenu- mericalapproaches.
2.3. SeparatingtheventriclesattheLV-RVjunction
To facilitate the computation of UVC, the LV, RV, and LV-RV junctiondomainsare identifiedinthebiventriculargeometry.Ac- cordingly,the boundaryconditionsshowninFig. 1A areassigned to thesurfacesof thebiventricularmesh,where φ =0onthe LV endocardium,φ =0.5ontheRVseptum,andφ =1ontheRVen- docardium.SolvingLaplace’sequationusingtheseDirichletbound- aryconditionsgivesthesolutionshowninFig.1B.Usingthisscalar field,theLVisdefinedbyallscalarvalues ≤0.5(Fig.1C)andthe RVbyallscalarvalues >0.5(Fig.1D).
2.4. Apicobasalcoordinate-z
The apicobasal directionfromthe apex to base ofthe ventri- cles is representedwith the coordinate zfollowing the approach
Fig. 1. Separation of the LV and RV. A: Three Dirichlet boundary conditions applied to the LV endocardium (0.0), RV septum (0.5), and RV endocardium (1.0) of the biventricular human mesh. B: The solution to Laplace’s equation with the boundary conditions in A. The LV-RV interface is defined by the isoline shown at 0.51. C: The separated LV. D: The separated RV.
by Bayeretal.(2012).Accordingly, zisdefinedasthesolutionto Laplace’sequationwithDirichletboundaryconditionsof0applied to theapex and1to thebase (Fig. 2A).All meshesshould share thesamelocation fortheventricularapexto avoiderrorinz.For example,theRVapexandLVapexforallheartsarenotalwaysin thesameplanethatisperpendiculartothelongaxisoftheheart.
Thus,anapicalboundaryconditionisplacedattheLV-RVjunction toensureaconsistentzforallLVandRVapicalorientationswithin biventricular meshes. To obtain this point atthe LV-RV junction, the minimal distance between the LV apex point (input by the user)andtheepicardialsurfaceoftheRVapexinFig.1Disdeter- mined. Thentheclosest pointoneach theLVandRVendocardial surfacestothisnewLV-RVapicalpointontheepicardiumisfound.
Lastly,twovectorsaredefinedby connectingtheepicardialLV-RV apical point to each of the LV andRV endocardial points. Apical boundarynodesareidentifiedbyfindingallmeshnodesthathave adistanceperpendiculartoeitherofthesetwovectorswithamag- nitudelessthan1.5timestheaverageedgelengthforallelements inthe biventricularmesh.The 1.5-foldthresholdforlabelingapi- cal nodesguarantees a smooth apical boundary formeshes with irregularelements.TheapicalboundaryisshowninFig.2Aforthe biventricularhumanmesh.
To ensure the solutionfor z variessmoothly and evenly from theapextothebase,thegeodesicapproachbyPaunetal.(2017)is usedtonormalizez.Withthisapproach,theshortestgeodesicdis- tancefromtheapextothebaseisdetermined(Fig.2B).Theinitial Laplacesolution inFig.2B isthensampledalongthisgeodesic as afunctionofnormalizeddistancetogenerateamappingfromthe LaplacesolutiontothenormalizeddistanceasshowninFig.2C.
Fig. 3. Transmural coordinate - ρ. Two Dirichlet boundary conditions applied to the endocardium (0.0) and epicardium (1.0) of the LV ( A ) and RV ( C ) of the biventricular human mesh. The solution to Laplace’s equation with the boundary conditions in A and C to obtain ρin the LV ( B ) and RV ( D ), respectively.
2.5.Transmuralcoordinate-ρ
Thecoordinateρrepresentsthedistancefromtheendocardium toepicardium.Sincetheinterventricularseptumcanbemoretyp- ical of LV than RV electrophysiology (Morita et al., 2007), ρ is
solvedseparately intheLV andRVwiththeseptum asanexten- sion to theLV free wall. As a result, ρ is 0on the LV and non-
septalRVendocardialsurfaces,andis1ontheepicardialandsep- talRVendocardialsurfaces.TheDirichletboundaryconditionsfor thesesurfacesontheLVandRVare showninFig.3AandC. The solutionstoLaplace’s equationwiththeseboundaryconditionsto obtainρ fortheLVandRVareshowninFig.3BandD.
2.6.Rotationalcoordinate-φ
Thecoordinateφrepresentsthecircumferentialrotationaround thelong axesofthe LVandRV.In ouralgorithm,φ iscomputed
separately in the LV and RV by solving Laplace’s equation with the Dirichletboundary conditions shown in Fig. 4Afor φ in the
LV(Fig.4B), andtheboundary conditionsshowninFig.4C forφ
in the RV (Fig. 4D). In the LV of Fig. 4A, φ rotates from +π in
theLVwall(magenta)to+π/2attheouter (orange)and+π/2.5 attheinner (yellow)surfaces oftheanterior LV-RVjunction,and then to 0 at the middle of the septum (green). The inverse oc- cursfor theposterior regionof theventricles from0at thesep- tumto−πintheLVfreewall.IntheRV,theφsolutioninFig.4D
rotates from+π/2 at the anterior surface of the LV-RV junction (orange) to −π/2 atthe posterior surfaceof the LV-RV junction (blue)inFig.4C.Withthisrepresentationforφ,ventricularpoints inthe anterior regionof theventricles aredefined atφ >0and
Fig. 2. Apicobasal coordinate - z . A: Dirichlet boundary conditions applied to the apex (0.0) and base (1.0) of the biventricular human mesh. B: The initial solution to Laplace’s equation with the boundary conditions in A to obtain z . The dark line is the geodesic for the minimal distance traveled from the ventricular apex to base. C: The normalized z using the distance traveled along the geodesic.
