Surface flattening of the human left atrium and proof of concept clinical applications

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ComputerizedMedicalImagingandGraphics38(2014)251–266

ContentslistsavailableatScienceDirect

Computerized

Medical

Imaging

and

Graphics

jo u r n al ho me p ag e: w w w . e l s e v i e r . c o m / l o c a t e / c o m p m e d i m a g

Surface

ﬂattening

of

the

human

left

atrium

and

proof-of-concept

clinical

applications

Rashed

Karim

a,∗

,

YingLiang

Ma

a

,

Munjung

Jang

a

,

R.

James

Housden

a

,

Steven

E.

Williams

a,b

,

Zhong

Chen

a,b

,

Asghar

Ataollahi

c

,

Kaspar

Althoefer

c

,

C.

Aldo

Rinaldi

a,b

,

Reza

Razavi

a,b

,

Mark

D.

O’Neill

a,b

,

Tobias

Schaeftter

a

,

Kawal

S.

Rhode

a

a_Division_of_Imaging_Sciences_and_Biomedical_Engineering,_King’s_College_London,_SE1_7EH,_UK b_Department_of_Cardiology,_Guy’s_and_St._Thomas’_NHS_Foundation_Trust,_London_SE1_7EH,_UK c_Centre_for_Robotics_Research,_Department_of_Informatics,_King’s_College_London,_SE1_7EH,_UK

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received3May2013 Receivedinrevisedform 20December2013 Accepted23January2014

Keywords:

Leftatrialsurfaceparameterization Lesionvisualization

Surfaceﬂattening

Catheterablationplanningandnavigation Atrialﬁbrillation

a

b

s

t

r

a

c

t

Surfaceflatteninginmedicalimaginghasseenwidespreaduseinneurologyandmorerecentlyin cardiol-ogytodescribetheleftventricleusingthebull’s-eyeplot.Themethodisparticularlyusefultostandardize thedisplayoffunctionalinformationderivedfrommedicalimagingandcatheter-basedmeasurements. Wehypothesizedthatasimilarapproachcouldbepossibleforthemorecomplexshapeoftheleftatrium (LA)andthatthesurfaceflatteningcouldbeusefulforthemanagementofpatientswithatrialfibrillation (AF).Weimplementedanexistingsurfacemeshparameterizationapproachtoflattenandunfold3DLA models.Mappingerrorsgoingfrom2Dto3Dandtheinversewereinvestigatedbothqualitativelyand quantitativelyusingsyntheticdataofregularshapesandcomputertomographyscansofan anthropomor-phicphantom.Testingoftheapproachwascarriedoutusingdatafrom14patientsundergoingablation treatmentforAF.3DLAmesheswereobtainedfrommagneticresonanceimagingandelectroanatomical mappingsystems.Thesewereunfoldedusingthedevelopedapproachandusedtodemonstrate proof-of-conceptapplications,suchasthedisplayofscarinformation,electricalinformationandcatheterposition. Theworkcarriedoutshowsthattheunfoldingofcomplexcardiacstructures,suchastheLA,isfeasible andhasseveralpotentialclinicalusesforthemanagementofpatientswithAF.

1. Introduction

1.1. Background

Planar mapping or surface parameterization of

three-dimensional(3D)meshesisawellstudiedproblemincomputer graphics[1].Ithasawiderangeofapplicationsinvariousﬁelds ofscienceand engineeringwiththemostimportantapplication beingmappingtexturestopolygonalmeshmodelsforenhancing their visual quality. Parameterizations are known to introduce distortions in either angles or areas and it is desirable that a mappingminimizesthesedistortions.Anisometricmappingisone thatpreservesdistancesandisalmostimpossibletoachievefor

∗_{Corresponding}_author_at:_Division_of_Imaging_Sciences_and_Biomedical Engineer-ing,TheRayneInstitute,King’sCollegeLondon,LambethWing,St.Thomas’Hospital, LondonSE17EH,UK.Tel.:+442071883052.

E-mailaddresses:[email protected],[email protected](R.Karim).

complex surfaces. Others such as conformal (angle-preserving) orauthalic(area-preserving)arepossible.Floater[2–4]produced several works in surface parameterization by extending a pre-viously known geometrical theorem due to Tutte [5]. Floater’s

methods imposed ﬁxed boundaries in the resulting

parame-terization.Free boundary parameterizationmethods have been proposedbySchefferand deSturler[6]andlater byLevyetal.

[7]. Desbrun et al. [8] employed conformal energy functions with linear solutions capable of both free and ﬁxed boundary parameterization.

Surfaceparameterizationhasbeenappliedtoproblemsarising inmedicalimageprocessing.Planarmappingofthecerebralcortex hasimportantapplicationsandhavebeenproposedinHurdaletal.

[9],Hakeretal.[10]andGuetal.[11]forcomputingconformal maps.TheuseofRicciﬂowinﬂatteningthecortexhasalsobeen investigated[12].Morerecently,planarmapshavealsobeen com-monlyusedforregisteringimagesofthebrain,e.g.[13].Acommon techniqueistomapthecortextoasphericalsurface.Thesphere providesa standardplatformforcomparisonbeforeregistration canbeaccomplished.ThisisthemethodimplementedinFreeSurfer

[14],aleadingandwidelyusedtoolforbrainsurfacereconstruction. However,FreeSurfersuffersfromaseverecomputationalburden

Open access under CC BY license.

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252 R.Karimetal./ComputerizedMedicalImagingandGraphics38(2014)251–266

causingittobeveryslow.Mostoftheexistingmethodsrelyon someformofdimensionalityreductionandthisusuallyinvolves solvinganeigenvalueproblem.Thesizeofthemeshcanoftenbea limitingfactorandthereisusuallyatrade-offbetweenspeedand meshsize.Oftenthismeansdecimatingameshandthuslowering theresolutionoftheunderlyingdata.

This manuscript concerns application of planar mapping to the heart. The use of this has been so far restricted to the relatively simplyshaped left ventricle (LV).The LV myocardial wallis oftentransformedintoa planarbull’s-eye plot[15].This is a polar plot of different sections of the wall that allows simpliﬁcation and standardization. It can be particularly use-ful for visualization and reporting of functional information about the LV, such as wall motion, perfusion and distribu-tion of scarring. The bull’s-eye plot has been adopted by the

American Heart Association as the standard way to

repre-sent the LV. In this manuscript we implement, validate and test planar mapping of the more complex left atrium (LA) to assist in the management of patients with atrial ﬁbrillation (AF).

1.2. Clinicalmotivation

AFisthemostcommonheartrhythmdisturbanceandoriginates intheLA.Itaffectsapproximately2.3millionpeopleintheUSA[16]. Sinceitwasshownthatectopicbeatsfromwithinthepulmonary veins(PV)commonlyinitiateAF[17],catheterablationstrategies haveevolvedtoelectricallyisolatethePVsfromtheLAbody.By controlleddestructionofcardiactissueinacircumferential pat-ternaroundtheleft andrightPV pairs,the PVsareelectrically disconnectedfromtheLA and therefore cannotinﬂuence atrial electricalbehaviour.Althoughthistreatmentcanprovideacurefor manypatients,arrhythmiarecurrencesarecommon,particularlyin patientswithpersistentAF[18–20],leadingtoarepeatprocedure inupto50%ofthischallengingpatientgrouptoachievethebest clinicaloutcomes.

Recurrences of AF following a circumferential PV isolation procedureare invariably associated with recovery of PV to LA conduction which is dependentlargely on a failure toachieve continuous and permanent lesions. Multiple factors inﬂuence theoperator’sabilitytoachievesuchalesionincluding,butnot limitedto,catheterstability,power,tissuethickness,contactand temperatureachieved,forceanddurationofradiofrequency(RF) application,cardiacmotionandlocalheatsinkeffect.Manyofthe factorsimpedingeffectivelesiondeliverycanbemitigatedinpart byoperatorexperiencebut theycertainly cannotbeeliminated

[21] asevidenced byhigh volumecentrescontinuing toreport levelsofarrhythmiarecurrenceinexcessof10–15%atﬁveyears afterablation[19,20].Clearly, there remains a needtoidentify waysin whichimprovementsin ablationtechniquecanleadto betterclinicaloutcomes.

CatheterablationproceduresforAFareperformedbycardiac electrophysiologists.Thevastmajorityofcardiac electrophysiol-ogistsutilizeanelectroanatomicalmappingsystemtoguidethe procedure[21].BoththeCARTO(BiosenseWebster)andVelocity (St.JudeMedical)systemspermitreconstructionofan anatomi-calrepresentationoftheLA,electrophysiologicalcharacterization ofthechamber(voltage,activationtime)anddisplayinreal-time ofthecatheterpositionwithinthechamber.3Delectroanatomical modelsofthetargetcardiacchambersobtainedbylivecatheter tracking, pre-procedural imaging, such as magnetic resonance imaging(MRI)orcomputedtomography(CT),oracombinationof bothareusedtoguidetheinterventions.Inthesesystems,3D mod-elsdisplayedonatwo-dimensional(2D)screenrequirefrequent interactionfromtheend-userfordisplayingthecurrentregionof interest.WehypothesizedthattheuseofplanarmappingoftheLA

couldassistinthedisplayoftheanatomicalandfunctional informa-tionthatiscommonlyusedinthemanagementofpatientswithAF.

1.3. Proposedwork

Inearlierwork[22]thatfocusedonplanarmappingoftheLV, webrieflydemonstratedthataB-splinebasedLAflatteningmaybe possible.Inthisworkweinvestigateasurfaceflatteningtechnique developedprimarilyfortexturemappingincomputergraphics[8]

andapplyittoﬂatten3DLAmeshes.Themathematicalbackground isdescribedinSection2.1.Wequalitativelyandquantitatively val-idatedthetechniquebyapplicationtosyntheticdataofregular shapes(Section2.2.1)andtoananthropomorphicphantomimaged withCT(Section2.2.2).Finallywetestourmethodusing14data setsfrom patientsthatunderwent RFablationfor treatmentof atrialﬁbrillationandusethesedatatodemonstrateseveral proof-of-conceptapplicationsoftheplanarmapping(Section2.2.3).

2. Methods

2.1. Mathematicalframeworkforsurfaceunfolding

WedeﬁneheretheproblemofunfoldingandﬂatteningtheLA mesh.Essentiallythisisaparameterizationofapiece-wisemesh. GivenapiecewiselinearmeshMoftheLA,theproblemofmesh parameterizationistoobtainalinearmappingϕbetweenMand aplanartriangulationU∈R2_._The_mapping_is_isomorphic_meaning

UmustinheritthesameintrinsicandstructuralpropertiesasM. MeshMisnon-closedduetoacutmadeatthemitralvalveannulus andcontainsholesateachostiumsincethePVsmustbetruncated beforetheunfold.The3Dpositionoftheithvertexinthemeshis denotedbyx_i=(xi,yi,zi)tandinthe2DmeshUasui=(ui,

v

i)t.The

mappingϕcreatesaone-to-onecorrespondencebetweenthe3D meshandits2Dunfold,essentiallyﬂatteningthemeshtoaplanar map.

Since the LA is an anatomical structure with some distinct anatomicalfeatures,suchastheappendage,itisimportantto pre-serve,whenunfolding,asmuchofthenaturalorintrinsicproperties oftheLAmeshaspossible.ThedistortionmeasureEbetweenM andUisafunctionalthattakesinasinputtwotriangulationsand returnsarealvalue:

E:T×T→R (1)

This distortion measure evaluates how much the structural propertiesofthemeshesaredistortedduetotheunfolding.In[8],a distortionmeasurewasdevelopedbyensuringafewbasic proper-tiesarefulfilledbyapredefinedfunctional.Thetwomostimportant propertiesrelevanttoLAmeshesarelistedhere:(1)invariancedue torigidtransformationsofthemesh,and(2)preservescontinuity duetofinertriangulations.Theformerisimportantastwomeshes exactlyalikebuthavingundergonerotationsandtranslationsmust yieldazerodistortion,therebyallowingtheparameterizationtobe independentofanyrigidtransformations.Thelatterensuresthat thisdiscreteversionofdistortion(duetothediscretenatureofLA meshes)convergestoitscontinuousformaswegetfinerandfiner triangulationsofthemesh.We restrictourdiscussionof distor-tionmeasuresto1-ringneighbourhoods.A1-ringisasubsetofthe 3DLAmesh,consistingofavertexandallitsadjacenttriangles.See

Fig.1foranillustrationofa3D1-ringneighbourhoodwithits corre-spondingunfoldplanar2D1-ring.Thedistortionmeasureandthus theparameterization,discussedfurtherinthistext,isonewhich preservesanglesandareasof1-ringneighbourhoods.

2.1.1. Angle-preservation

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R.Karimetal./ComputerizedMedicalImagingandGraphics38(2014)251–266 253

Fig.1. LAmeshpatchor1-ringneighbourhoodsMin3Dwithitsassociatedﬂattenedplanar1-ringUin2D.

function.InDifferentialGeometry,duetotheminimalsurface prob-lem,itiswellknownthattheminimizationofaso-calledDirichlet energyleadstoa conformalmapping[23].TheDirichletenergy canbedeﬁned ona mesh patchand capturesangledistortions betweencorrespondingtrianglesofthepatchin3Dand2D.Itcan beshowntofulﬁlthedesirablepropertiesoutlinedaboveforthe unfold[23].ThetotalDirichletenergyforthepatchisgivenby:

EA=

ringedges(i,j)

(cot_˛_ij₊cot_ˇ_ij)_|u_i₋u_j_|2 ₍₂₎

wherethelength|u_i−u_j|isinthe2Dparameterdomainandangle ˛ijistheoppositeangleto(xi,xj)in3D(seeFig.1).Thederivation

isomittedherebutisastraightforwarddifferentiationofalinear mapandcanbefoundin[23].Therearetwoadvantagesofusing thisfunction:(1)itdependsonlyontheangleofthe3Dmesh(i.e. cotangentterm),and(2)theﬁrst-orderderivativeoftheenergy termisconvenientlylinear:

∂EA

∂u_i =

ringedges(i,j)

(cot˛ij+cotˇij)(ui−uj)=0 (3)

Theoptimalplacementofvertexuicanthusbesolvedbysolving thelinearsysteminEq.(3).TheDirichletenergyinEq.(2)thus givesameasureofdistortionin-termsoftheanglesbetweenthe meshpatchin3Danditsunfoldin2D.IftheDirichletenergyis minimized,aconformal(i.e.anglepreserving)transformation is possible,however,sinceweaimtoﬁxtheborderofthe2Dunfold toasquareborder,achievingafullyconformaltransformationmay beimpossible.ThisDirichletenergytermbecomespartoftheﬁnal distortionmeasure.

2.1.2. Area-preservation

Asidefrompreservingangles,thereisalsoanarea-preserving component.Considerthetipanglearoundxiofthe3DmeshM inFig.1;thegradientofthisanglecanbeeasilyderivedfromits individualtrianglesshowninFig.2,thegradientoftheangleεw.r.t. placementofvertexAisgivenby:

∇

ε=cot˛ |AB|2AB+

cotˇ

|AC|2AC (4)

NotethatbothtermsofεinEq.(4)arefunctionsofthetriangle’s area.ForinstanceifprojectionofContoABisCthentheﬁrstterm’s co-efﬁcientcot˛/|AB|2_is_equal_to_the_area_of_CBC_/₂_T2_where_T

istheareaofABC.Bysumminganglesaroundthe3DmeshinM, itcanbeshownthat:

∇

=

j∈N(i)

cotij+cotıij

|x_i−x_j|2 (xj−xi) (5)

whereisgradientw.r.t.vertexxiandisthetotalanglearoundxi andnotnecessarily2duetoitspositioningin3D.Notethatsimilar toEq.(4),theco-efﬁcientsinEq.(5)arefunctionsofthearea.

FollowingfromEq.(5)Desbrunetal.[8]obtainanoveldistortion measure:

Ex=

j∈N(i)

cotij+cotıij |x_i−x_j|2 (ui−uj)

2

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whereanglesijandıijaredeﬁnedinFig.1.Thiscombinesareaof themeshManditsunfoldingU.Notetheonlyunknownisvertex

uiandthusitsplacementintheparameterspace.Onceagain,the optimalplacementofuicanbedeterminedbysolving∂EX/∂ui=0

andthisconvenientlyresultsinalinearproblem.

2.1.3. Finalparameterization

Thetwodistortionmeasuresdescribedabovearecombinedinto asinglemeasure.Inadditiontothisaﬁxedboundaryconditionis imposedrequiringtheborderoftheunfoldingtobeasquare.The combineddistortionmeasurecanthusbewrittenas:

E= EA+(1− )EX (7)

where isa constantweighingtheinfluenceofeachdistortion measure.Toobtainthefinalparameterization,positionsofinternal verticesu_i inthe2Dunfoldaresolvedfor.Theboundary condi-tionsarestatedbeforehand,i.e.thesquareborderoftheparameter spaceismatchedwiththeborderofthe3DLAmesh.Thestepstocut opentheLAmeshandthusdefineitsbordersaredescribedbelow. Theoptimalpositionsofallu_icanbeobtainedbyminimizingthe combineddistortionmeasureandthussolvingasparselinear sys-temfollowingfrom∂EX/∂ui=0and∂EA/∂ui=0:

MU=C (8)

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Fig.3. (a)InitializationoftheunfoldingprocessbeginswithamanualcutaroundthemitralvalveannulusoftheLAandfourveinsandappendageasindicatedbydashed lines,(b)LAafterthecut,(c)theborderofthemitralvalvecutasindicatedusingredarrowsismatchedwiththe2Dunfold’sboundary.(d)Sameas(c)butseenfroma differentviewplane.(e)Boundaryofmitralvalvecutin3Dismappedtoboundaryofunfoldin2D.Thiscompletestheinitializationprocess.Abbreviations:RIV,rightinferior vein;RSV,rightsuperiorvein;LIV,leftinferiorvein;LSV,leftsuperiorvein;APP,appendage.

where Mis theco-efﬁcientmatrix carrying co-efﬁcients of the respectivedistortionmeasures,i.e.∂EA/∂uiand∂EX/∂ui.Cisthe

vectorofboundaryconditions:[0 Cboundary]Tandthepositionsof internalverticesaresolvedforinU.Thesystemcanbeefﬁciently solvedusingConjugateGradientandSSOR(SymmetricSuccessive Over-relaxation)preconditioning[24].Forthis work,the imple-mentationofthesurfaceparameterizationof[8]istheonefound intheComputational GeometryAlgorithmsLibrary(CGAL) [25]

withwrapperswritteninC++andusingVTKforvisualizationand mappingprocedures.

2.1.4. Methodinitialization

Prior to computing the parameterization for the unfolding, boundaryconditionsmustbeset.Theboundaryofthe2Dunfold mustbemappedinitiallytoa boundaryinthe3DLAmesh.An idealboundaryistheLAmitralvalveannuluswheretheLAjoins theleftventricle.TheLAisthusmanuallycutinthisregionand anillustrationisprovidedinFig.3(a)showingthelocationofsuch

acut.Theboundaryofthecutisgenerallydiscshapedowingto theLA’ssphericallyshapedchamber(seeFig.3(b)–(d)).Thedisc ismappedtothe2Dunfoldboundary(Fig.3(e))andthis initial-izestheunfoldingprocess.Furthermore,additionalcutsaremade ateachPVandtheappendage.However,thisisonlytomapvein andappendagelocationsintheunfoldwithholes.Themitralvalve cutaffectstheﬁnalparameterizationresultandthusthequalityof unfold.AsablationlesionsareonlymadeinandaroundPVs,the cutismadeappropriatelytocovervitalregionsoftheLA.Thecut wasperformedusingmesheditingsoftware(MeshLab,ISTI–CNR, Pisa,Italy).

2.2. Experimentalprotocol

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Fig.4. Commongeometricalshapeswithregulartexturepatternsthatwereusedforevaluatingtheunfoldtechnique.Theshapeswereahemisphere(toprow),paraboloid (middlerow)andatruncatedellipsoid(bottomrow).

2.2.1. Regularshapes

Toevaluatetheextentofdeformationswiththeunfold tech-nique,someregularshapeswithsimpletextureswereunfolded andthedeformationsundergonebythetextureswerecomputed. A total of three shapes were constructed using a commercial meshprocessingandeditingsoftware(Rhinoceros,McNeelNorth America,WA,USA).Thesewereahemisphere,aparaboloidanda truncatedellipsoid(Fig.4).Theshapeswerechosenbecauseoftheir resemblancetocardiacstructuresofinterestsuchastheatriumor ventricle.Allshapesweregivenatextureconsistingofstripestoaid theinterpretationoftheeffectoftheunfold.Thetexturedirection waseitherinthesamedirectionastheshape’saxis(hemisphere)or orthogonaltoit(paraboloidandellipsoid).Itisnotalwayspossible todeﬁneperfectlystraightlinesinthetexture(seetruncated ellip-soidinFig.4)duetotheirintersectionwiththearrangementofthe mesh’svertices.Eachshapewasunfoldedbymappingitsborderto thatoftheunfolded2Dsquare.Asallthreeshapesareopen, bor-derswerealreadypresentandthusacutasshowninFig.3wasnot necessary.Distortionindistanceduetotheunfoldwasmapped oneachshape’sunfolded2Dsquare.Tocomputethisdistortion map,foreveryvertexontheunfold,theratio2D/3Dofthemean distancetoadjacentverticeswasplotted.Thisillustratesthe dis-tortionundergonebyeverypointneighbourhoodontheunfoldand thusdemonstratewhichregionsaredistortedmorethanothers.

2.2.2. Glass-heartphantom

Itisimportantthatdistancesbetweenpointsonthe2Dunfold are comparable toactual distances in physical space and thus validated against ground truth. To validate this, a glass heart phantom(Farlow’sScientiﬁcGlassblowing,GrassValley,CA,USA) wasimaged(Fig.5).Thephantomconsistsof allchambersand great vesselsof theheart. It wasscannedby a 512×512×384 voxelCTscanwith0.68mm_×0.68mm_×1.00mmvoxelresolution (BrillianceiCT,PhilipsHealthcare,TheNetherlands).Amanual seg-mentationoftheLAwasobtainedanda3Dsurfaceextractedusing

themarchingcubesalgorithm.The3Dsurfacewascutatthemitral valveannulus(Fig.3)andunfoldedontoa2Dsquarebymapping theborderofthecuttothatofthe2Dsquare.Thirteenseparate

measurementswere madeonthephantom bythree observers.

Themeasurementsweredistancesbetweenlocationsofanatomical interest.Thesemeasurementswerecarefullyselectedby consid-eringtheirrepeatabilityandare shownin Fig.6.Theobservers measuredeachlocationthreetimesandthemeanvaluewas cal-culated.Thesewerecomparedtotheircorrespondingdistanceson the2Dunfold.Similartothedistortionmapextractedforregular shapes,adistortionmapforthephantomwascomputedto illus-trateregionsundergoingdistortionsduetotheunfoldingprocess.

2.2.3. Clinicalcases

2.2.3.1. Patientdata. 14patients(10malesand4females,mean age=60±8years)withsymptomatic,drugrefractoryparoxysmal AF undergoingPV isolation withcircumferential catheter abla-tionprocedurewereincludedinthestudy.Sixmonthsfollowing catheter intervention,acardiac MRIscanwasperformedwhich includedDE-MRIscanstoquantifytheextentofpermanent post-ablationatrialinjury.7ofthesepatientssubsequentlyunderwent redo ablationprocedures due to recurrenceof arrhythmia.The datafortheredointerventionfromtheelectroanatomicalmapping systemwereavailable,includingthereconstructed3DLAsurface meshesandvoltagemaps.

2.2.3.2. MR image acquisition. All scanning was performed on

a 1.5T Achieva scanner (Philips Healthcare, The Netherlands).

A 3D whole heart image was acquired by using a 3D

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Fig.5.Imagesoftheglass-heartphantomusedtoobtaingroundtruthmeasurementsondistances.Clockwisefromleft:(aandb)X-rayprojectionimages.(c)Amid-section slicethroughaCTscanand(d)aphotographicimage.

ﬁeld echo(TFE) withrespiratory-navigation, cardiac-gating and withwholeheartcoverage(0.6mm×0.6mm×4.0mmacquired, 0.6mm×0.6mm×2.0mmreconstructed,3minduration).

2.2.3.3. 3DLA meshextractionfromMRI. TheLAwassegmented

fromtheanatomicalbSSFPscanforeachofthe14patientsusing anautomaticapproachbasedonastatisticalshapemodel[26].The modelisa3DmeshbuiltfromtrainingdataandadaptedtotheLA anatomicalMRimage.ThemodelincludesPVswhichare approx-imately14mminlengthandwithanaveragediameterof12mm. ItalsoincludestheLAappendageanditslengthisapproximately 15mmfromLAbody.Themodelﬁttingerrorreportedin[23]is 0.72mmover42datasets.Forourwork,theautomatic segmenta-tionwasveriﬁedbyahumanraterandmanualcorrectionswere madewhenevernecessary.

2.2.3.4. Unfolding. Eachofthe14MRI-derivedLAsurfacemeshes wasunfoldedusingthemethodsdescribedinSection2.1.Toassess theamountofdistortionduringtheunfoldprocessaquantitative andqualitativeanalysisofdistortionwascarriedout.Quantitative analysiscompareddistortionindistancesandareasbetween3D and2DontheMRI-derived3Dmeshes.Fordistances,thegeodesic distancebetweenpoints on3D totheirmappeddistancein 2D wascompared.Forareas,areasofregionsin3Dwerecompared totheirmappedregionsin2D.Asthedeformationsarenon-linear witheveryregionoftheatriumundergoingdifferentamountsof deformation,theatriumwassub-dividedintoﬁvedistinctregions. These regions were the left and right sides, the anterior and

posteriorendsandtheroof.Fig.7illustratestheseregionswith thepointsselectedtoperform theanalyses.Distances between eachpairsofpointswereconsideredandareasenclosedbyevery possibletriangleusingthesepointswereconsidered.Subdividing theLAintoregionsallowedasegmentalcomparisonofdistortions in lengths and area. For a qualitative analysis of distortion in clinicalcases,distortionmapsweregeneratedineachcase. Fur-thermore,distortionwasalsovisualizedusingtexturemapping. Regular textured patterns such as a checker-board or striped patternsweregenerated ineachLA 3Dmeshandthenmapped to2D.Deformationsinthetextureswerevisualizedandassessed atanterior,posteriorandroofsectionsofLA.The2Dunfoldwas furtherinvestigatedand evaluatedtoseeifit couldbeusedto select ablation lesions. Lesions were marked circumferentially aroundeachpulmonaryvein.Theirmappingto3Dwasvalidated bycomparingeachselected point’sdistancetotheveinostium in2Dand3D.Thisinvestigatedwhethertheunfoldmapoverlaid withsofttissueinformationfromDE-MRIcanbeaccuratelyused tomapablationlesionpoints.

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Fig.6. 3DLAsurfaceextractedfromglassheartphantomCTscan.Forgroundtruthtesting,variousmeasurementsphysicallyandontheunfold:(a)diametersofeach veinandappendage,(b)measurementsmadeacrosstheroof.Thesearedistancesbetweenthemid-anteriortomid-posteriorandmid-lefttomid-right,(c)basalview.(d) Distortionmapofglassheartphantomillustratingdistortionratiosofdistancesin2D/3D.LandmarksareplottedonLAshellandunfoldandcorrespondencesareindicated usingcolouredarrows.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthearticle.)

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Fig.8.Regulargeometricalshapesunfoldedfrom(a)hemisphere,(b)paraboloidand(c)truncatedellipsoid.FortheseshapesrefertoalsoFig.4.(d)Distortionmapfor hemisphere,(e)distortionmapforparaboloid,(f)distortionmapfortruncatedellipsoid.

3. Results

3.1. Regularshapes

Mappingregulartexturepatternsfromthe3Dto2Dforregular shapesgivesanunderstandingofthenatureofdeformationdue totheunfoldexpectedonacardiacchambersuchastheLA.For theopenhemisphere,thecircularborderwasmappedtoasquare borderonthe2Dunfold.Thisdeformationofacircletoasquareis evidentinthedeformationundergonebythetextureinFig.8(a), withthecirclesstretchedtoﬁllthesquare’scorners.Deformations appeartobeclosetolinearthroughoutexceptatthecornersofthe

2Dunfold.Intheopenparaboloid,textureswereoriented orthog-onallytothecircularborder.Thesetexturedeformationsillustrate thenon-linearityoftheunderlyingtransformation(seeFig.8(b)) –thecornersgenerallyexperiencegreaterstretching.Thewavy deformationinthetextureexhibitsthisphenomenon.Itis impor-tanttonotewhythestripesrunalongthediagonalofthesquare asopposedtobeinghorizontalorverticallyaligned.Thisissimply becauseofhowthesectionsofthecircularborderin3Dmatchwith thesectionsofthe2Dsquareonunfold.Deformationintextures onthetruncatedellipsoidtransformation(seeFig.8(c))illustrate thatshapeswhichenclosemorespaceundergogreaterdeformation simplybecausealargersurfaceareanowneedstobeﬁttedwithina

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constrainedspaceofthe2Dsquare.Distortionmapswerealso gen-eratedforeachshapeandgiveninFig.8(d)–(f).Thesemaps illus-trateincreasedstretchinggoingfromthehemispheretoellipsoid.

3.2. Glass-heartphantom

Physicaldistancesontheglass-heartphantomwerecompared todistancesmeasuredonthe2Dunfold.Therearetwoimportant considerations–whetherthephysicalmeasurementsthemselves

are repeatableand whetherthereis intra-observer variation. It wasthusimportanttoselectanatomicalregionsthatcanbe dis-tinctlylocatedbothonthephantomandunfold.Themeasurements obtainedfromobserversshowedthattherewasgoodagreement betweenthem (see Fig. 9).Furthermore, measurementson the unfoldwereonaveragewithin10%ofthemeanofallobservers’ measurements.Adistortionmapforthephantomwascomputed byplottingtheratioofdistancein2Dto3D.Themapisshownin

Fig.6(d).Overall,themapillustratesstretchingatthebordersand

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Fig.10.(Continued)

cornerswithsomenegativestretchorshrinkingneartheatrialroof. RegionsneartheostiumofPVsaremostlypreservedwithratios closeto1.

3.3. Clinicalcases

Toanalyzedistortionscausedbythemappingprocess,geodesic distancesonthe3Dsurface andtheirmappingon2Dare com-paredquantitativelyinFig.10(seeleftcolumn)andalsoillustrated usingdistortionmapsinFig.11.Distancesaregenerallywell pre-servedintheroof,posteriorandanteriorsectionsoftheLA(R>0.9 andlowratiosindistortionmaps).Therightandleftsides expe-riencegreaterstretchingbecausealargesectionofthecutisin thevicinity,whereasthiswasnotthecasewithpointsontheroof andanterior/posteriorsegments.Recallthattheboundaryofthe cutisstretchedtoﬁttheboundaryofthe2Dsquare.Forthearea comparisoninFig.10(seerightcolumn),regionsofdifferentsizes rangingfrombigtosmallwereusedfortheanalysis.Againthe LAwassubdividedintoseparatesegmentsandeachsegment ana-lyzedseparately.Smallerregionsacrosseverysegmentshowgood preservationoftheirareasfrom3Dto2D,butforlargerregions, there is distortion.This is expectedas small distortionswithin smallsectionsofalargerregioncumulatetogivearesultinglarger distortion.

Theunfoldingprocessonclinicalcasesisalsovalidatedusing texturemapping.Texturesdrawnonthe3Dmeshweremapped

to their respective 2D unfold. As textures on the 3D are gen-erated by utilizing the underlying mesh vertex grid, perfectly regular patterns can be difﬁcult to achieve due to irregularity in themesh structure, i.e. somevertices have 3,5 or 7 neigh-bours depending onthe 3D mesh reconstruction process. This producessmallartefactsinthetexturesin3D(seegapsin spot-tedtexturein Fig.12 and brokenlines onthestriped texture). Nevertheless,thenatureofthedeformation isdemonstrated in theseimagesandit isclearthat theunfolding processof map-pingacircular-likeborderofthecutintheannulus(seeFig.3)toa 2Dsquareresultsinlargerdistortionsattheedgeoftheunfold. However,critical areassuch asthe roof and PV ostia are least affectedandthisis adesirableresultin thecontext ofLA abla-tion.

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Fig.11. Distortionmapsillustrateaccuracyoftheunfoldingprocessinthreeclinicalcases.Distortionisrepresentedasratioofdistancein2Dto3Dandismappedusinga colourscalerangingfrom0(−vestretch)to5(+vestretch).Abbreviationsforveins–byconventionLSV,RSV,LIV,RIV.(Forinterpretationofthereferencestocolourinthis ﬁgurelegend,thereaderisreferredtothewebversionofthearticle.)

4. Clinicalapplications

4.1. MappingDE-MRIontheunfoldforsoft-tissuevisualization

DE-MRIforimagingRFablationlesionsisarapidlyexpanding ﬁeldofresearchwhichseekstoallowamorein-depth characteri-zationoflesionsandrecentstudieshaveinvestigatedthis[27,28]. Currently,itispossibletovisualizetheshapeandsizeof circum-ferentialandlinearlesionsetsonthe3DLAsurface.TheDE-MRI imageintensitiesduetolesionscanbeprojectedontothe3DLA surfaceusingthetechniquedescribedin[29]whichisessentiallya maximumintensityprojection(MIP)method.Itisatwo-step pro-cess.TheﬁrststepinvolvesregisteringthesurfacewiththeDE-MRI imageallowingboththesurfaceandDE-MRIimagetobeinthe sameco-ordinatesystem.Oncefused,DE-MRIimageintensityis mappedtothe3DLAsurfaceusingMIP.Onceonthe3Dsurface, DE-MRIimageintensitycanbemappedtothe2Dunfold.Three examplesfromclinicalcasesareshowninFig.14.DE-MRIimage intensityin Fig.14 ismapped ona yellowtoredlinear colour scalewithredindicatingscar.Circumferentialscarringaroundthe PVsisexpected.GapsinscarringcanleadtoPVreconnectionand recurrenceofarrhythmia.Theentiredistributionofthescarring isrevealedintheunfold,asshowninFig.14,andgapsareeasily evident.Theelectrophysiologistwillseektoclosethesegapsby re-ablatingtheseareas.InFig.14pointshavebeenhypothetically selectedtoclosethesegaps(seepurplepoints).

4.2. Endocardialvoltagemapping

The majority of cardiac electrophysiologists utilize an elec-troanatomicalmappingsystemtoguidetheablationprocedure.

The mapping system, such as CARTO, is able to acquire

unipolarorbipolarreadingsofvoltagesontheendocardiumusing thecatheterspresentinsidetheheart.Priortovoltagemapping,the 3DLAgeometryofthepatientiscollectedusingatrackedcatheter. Thegeometryisreconstructedfromapointcloudrecordedbythe catheter’stip.Endocardialvoltageismapped,displayedand visual-izedonthe3DLAgeometry.Inarecentstudy[28]itwasshownthat voltageandscarhaveanimportantrelationshipandthusitisuseful tocorrelateandvisualizethetwoespeciallyforre-doprocedures. However,thereisanimportantcomplexitywhencorrelating volt-ageandscar.The3Delectroanatomicalgeometryisfarlessdetailed thanthe3DLAscarmeshobtainedfromMRI,owingtohigh res-olutioninMRIimages.Usingthe2Dunfold,thecomplexityofthe twodissimilargeometriesisreducedandcorrespondencescanbe made.InFig.15,severalre-doclinicalcasesareillustrated.Ineach case,theendocardialvoltagemapsareunfoldedandcomparedwith pre-ablationDE-MRIscarimaging.Notelowvoltageiscomparable tohighintensityDE-MRIandthusthecolourscaleinvoltagemaps areinverted(redtogreen).The2Dunfoldreducesthecomplex differencebetweenthetwogeometriesandpresentssimilar look-ingmapsineachcasewherebyvoltageandscarcanbecompared. Lesionformationcircumferentiallyoneachsideiscomparedwith lowvoltageareas.Cases1–4demonstratelowvoltageareasoneach sideandcorrespondwellwithsigniﬁcantDE-MRIenhancementin thoseareas.

4.3. Cathetervisualizationandnavigation

During ablation procedures, electroanatomical mapping

sys-tems display live catheter positions on a 3D LA mesh. In

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Fig.12.Thenatureofdeformationduetotheunfoldprocessfrom3Dto2Dis ana-lyzedhereusingregulartexturepatterns.Deformationsareseenontwoseparate LAmodelswithaspottedtexture(topandmiddlerow)andstripedtexture(bottom row).TheLAineachcaseisshowninthreedifferentviews(anterior,posteriorand rightside).Abbreviationsforveins–byconventionLSV,RSV,LIV,RIV.

differentangles.The unfoldcan displaycatheter tippositionin a variety of ways depending on whether the catheter is near orin contact withthewall. Withoutany ambiguityor interac-tion,usingtheunfoldtheelectrophysiologist isabletoconﬁrm which side of the atriumthe catheter is located at. In Fig. 16, twoexamplesareillustratedtodemonstratethattheunfoldcan

be used simultaneously with the 3D mesh to provide useful

positional information of a catheter during an ablation proce-dure.

Fig.13. Evaluatingtheamountofstretchcausedduetotheunfoldmappingtolesion pointsmarkedcircumferentiallyontheleftandrightPVs.Theratioofdistancein3D toitsmappeddistancein2Displottedonalogscale.Apositive(+ve)stretchmeans thatthe3Ddistanceisgreaterthanin2Dandanegative(−ve)stretchotherwise.

5. Discussion

Themethodof LA3Dsurfaceparameterizationimplemented in this workcomputes optimallocations for 3D mesh vertices inapre-deﬁned2Dparameterspace.Theoptimalityconditionis achievedbyminimizingalinearcombinationoffunctionsthat pre-serveanglesandareasof eachmeshpatchsubjecttoboundary conditions.Theprimaryaimwastousethemethodtounfoldthe leftatriumofthehearttosupportthemanagementof patients withatrialﬁbrillation.Themethodwasvalidatedusingsynthetic dataandphysical phantomdata.Itwasthen appliedtopatient dataderivedfromMRIandelectroanatomicalmappingsystemsto demonstrateclinicalutility.

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Fig.14. ClinicalapplicationI:mappingsofttissueinformationfromDE-MRItothe 2Dunfoldinthreeseparateclinicalcases.TheLAineachcaseisshowninthree dif-ferentviews(posterior,sideandanterior).Areasinredrepresentsitesofprevious ablation.Correspondingregionsarelabelledwithdottedandﬁlledarrows repre-sentingtwodifferentregions.Atypicallesionsetwouldaimtocompletelyencircle thepulmonaryveinsasindicatedbythestylizedpurplepoints.Abbreviationsfor veins–byconventionLSV,RSV,LIV,RIV.(Forinterpretationofthereferencesto colourinthisﬁgurelegend,thereaderisreferredtothewebversionofthearticle.)

Distortionsduetotheunfoldwerefoundtobeunavoidableand this issimplydue totherestrictions imposedontheunfolding processthroughboundaryconditions(i.e.2Dsquare).By perform-ingsegmentalcomparisonsofdistanceandarea,itwasfoundthat regionsclosertothelocationofthecutinthemitralvalveannulus andontheleftandrightsidesoftheveinsexperiencedthe great-estdistortionandstretching.However,theamountofstretching involved doesnotdistortthelesionpatterntoanextentthatit isunsuitableforvisualization.Thisisclearlydemonstratedinthe lesionmapsofFig.14wherelesionformationsaroundtheveins arepreserved.ThemostcriticalareaslieinandaroundPVsand roofregionandthisworkdemonstratedthatdistortion inthese regionswasnothigh.

Tounderstandhow distortions ofthenature seenin theLA unfoldmightaffectanend-user,itisimportanttoconsidera sce-nariowheretheuserusestheunfoldtoselectlesions.Insuchcases, endocardialvoltageorDE-MRIenhancementinformationonthe unfoldis usedtodecidelesionpositions.Pointsselectedonthe unfoldaremaderelativetofeaturesvisibleintheunfolditselfsuch asdistortedlesionorvoltagepatterns.Asthesefeaturesare them-selvesdistortedtoacertainextent,theend-userwouldhavelittle difﬁcultyidentifyinglocationsofinterestoncethenatureof distor-tionwasunderstoodandappreciated.

Several proof-of-concept clinical applications were demon-strated.VisualizingscartissuefromDE-MRIonthe2Dunfoldreveal gapsinablationlesionsandprovidesusefulinformationforre-do procedures. Unfoldingof electroanatomicalmaps wasalso pre-sented,inparticularfordisplayingendocardialvoltageinformation. Finally,the2Dunfoldwasdemonstratedasausefulutilityduring catheter-tipvisualizationon3Dtransparentmodels.Ambiguous positionsin3Drequireinteractionwiththemodel–anexample ofpointslyingontheposteriorandanteriorwallisdemonstrated. Onthe2Dunfold,suchambiguityisremovedastheposteriorand anteriorwallscanbesimultaneouslyvisualized.

5.1. Limitations

Theproposedmethodhasanumberoflimitations.Thecutmade atthemitralvalveannulusofthe3DLAmeshduringmethod ini-tializationimposesalimitationontheunfold.Sinceitisdesirable forthePVostiatoliecentrallyandapproximatelysymmetrically onthe2Dunfold, thecutontheleftand rightsidesshouldbe suchthatitsgeodesicdistancefromleftand rightPVsarealike. Thesameappliestotheposteriorandanteriorsidesandtheir dis-tancefromLAroof.However,obtainingtheoptimalcutmanually canbedifﬁcultgiventheaboveconstraints,andfurther,thefolding oftheLAsurfaceattheappendagemakesitchallenging. Conse-quently,insomeinstances,afewsub-optimalcutconﬁgurations wereexploredbeforegeneratingthe2Dunfold.Operatorbiasand thusinter-observervariationwasnotcompletelyeliminated.

Therewerelimitationsinthemethodsusedforvalidatingthe 2Dunfold.A2Dunfoldoftheglass-heartphantomwasgenerated anddistancesonthephantomwerecompared,i.e.physicalversus 2D.Pointcorrespondencebetweenphysicalandvirtual3D/2Dwas establishedinthisworkusingsimplevisualassessment,i.e.fora pointselectedontheglass-heart,itspositiononthe3DLAmesh wasmanuallyselected.Therearenovelwaysofdeterminingwhere exactly a point liesin physical space and this is byemploying magnetic trackersor someformofimaging. Establishing corre-spondencemanuallyintroduceserrorsbutthiswasminimizedby selectingpointsinregionsthatwereanatomicallydistinct.

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Fig.15.ClinicalapplicationII:unfoldon3DelectroanatomicalLAmeshobtainedfrommappingsystems.Fiveclinicalre-doablationcasesareshownherewith electroanatom-icalmap(column1)anditsunfold(column2)andcomparedwitheachcase’spre-ablationMRIreconstructed3DmeshwithDE-MRIscarenhancement(column4)andits unfold(column3).Notelowvoltageinmilli-volts(mV)iscomparabletohighintensityinDE-MRIrepresentedasstandarddeviations(SD)fromhealthytissueintensity. Abbreviationsforveins–byconventionLIV,RSV,APP–appendage.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversion ofthearticle.)

begreaterwhencomputedover largerdistancesand areas,this canbealimitationofthesemaps.However,distortionoverlarger regionswasinvestigatedinthesegmentalevaluation ofclinical caseswithplotsinFig.10.Datapointshaveawiderspreadand deviatefromthelinearrelationshipforlargerdistancesandareas indicatinggreaterdistortion.

All 2D unfold explored in this work was on a square. The squarewas chosenas it canbemore convenienttoworkwith on a ﬂat visual display unit. Other shapes are not explored

and one good candidate for mapping would be on a circle.

Mapping on the square imposes signiﬁcant distortions at cor-ners;acircleofferssmoothergradientsatitsedgesandis thus

expected to have lesser distortions. However, representing it ona ﬂatvisualdisplayunit canpresent somedifﬁculties espe-cially when zooming in whereby a section of circle can get clipped.

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Fig.16. ClinicalapplicationIII:unfoldincathetervisualizationandnavigation:example1showsaninstancewherethecatheterismanoeuvredfromanteriortoposterior alongtheroof(directionindicatedbydashedarrow).Inthe3DposteriorLAview,itisunclearwhetherthecatheterisnearertoposteriororanteriorwalluntilthe3Dmodel isre-oriented(roofview).Theunfoldshowsthecathetertipasitmovesalongtheroof(pinkstylizedpoints)andthereisnoambiguityaboutitslocation.Example2showsa similarsituationwhereitisunclearwhetherthecatheterisontheposteriororanteriorwallinthe3Dviewandtheelectrophysiologistisabletoconﬁrmitslocationusing the2Dunfoldview.(Forinterpretationofthereferencestocolourinthisﬁgurelegend,thereaderisreferredtothewebversionofthearticle.)

6. Conclusions

Proceduralguidanceusinga3Dmeshcanbecumbersomeand demandsconstantinteractionwiththegeometry.Theuseof pla-narmapswithpoint-by-pointcathetertechnologies(manualor robotic) coulddecrease the complexity of navigation and even improvenavigationbylessexperiencedoperators.Theproposed workinvestigatesanapproachtoﬂattenandvisualize3DLA geom-etryalongwithscarand/orendocardialvoltageinformationona2D planarmap.The2Dunfoldisbothquantitativelyandqualitatively validatedandseveralclinicalproof-of-conceptclinicalapplications aredemonstrated. Futureworkwilllookintoemployingplanar mapsinlivecatheterproceduresandtoevaluateclinicalutility.

Acknowledgements

TheauthorswouldliketothankmembersoftheDivisionof BiomedicalEngineeringandImagingSciences,King’sCollege Lon-donwhoassistedwiththisstudy.Inparticular,theywouldliketo thankSamanthaObomforassistancewiththeexperiments.

TheauthorsacknowledgesupportfromtheKing’sCollege Lon-donCentreof Excellencein MedicalEngineeringfunded bythe WellcomeTrustandEPSRC(WT088641/Z/09/Z).Thisresearchwas alsosupportedbytheNationalInstituteforHealthResearch(NIHR) BiomedicalResearchCentreatGuy’sandStThomas’NHS Founda-tionTrustandKing’sCollegeLondon.Theviewsexpressedarethose oftheauthor(s)andnotnecessarilythoseoftheNHS,theNIHRor theDepartmentofHealth.

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