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A dynamic finite element surface model for segmentation and tracking in multidimensional medical images with application to cardiac 4D image analysis

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A dynamic finite element surface model for segmentation and tracking in multidimensional medical images with application to cardiac 4D image analysis
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  ToappearintheJournalofComputerizedMedicalImagingandGraphics,1994.  ADynamicFiniteElementSurfaceModel forSegmentationandTrackingin MultidimensionalMedicalImageswith ApplicationtoCardiac4DImageAnalysis  TimMcInerneyandDemetriTerzopoulos  DepartmentofComputerScience,UniversityofToronto,Toronto,ON,CanadaM5S1A4  Abstract  Thispaperpresentsaphysics-basedapproachtoanatomicalsurfacesegmentation, reconstruction,andtrackinginmultidimensionalmedicalimages.Theapproachmakes useofadynamic\balloon"model|asphericalthin-plateundertensionsurfacespline whichdeformselasticallytottheimagedata.Thettingprocessismediatedby internalforcesstemmingfromtheelasticpropertiesofthesplineandexternalforces whichareproducedfromthedata.TheforcesinteractinaccordancewithLagrangian equationsofmotionthatadjustthemodel'sdeformationaldegreesoffreedomtot thedata.Weemploytheniteelementmethodtorepresentthecontinuoussurface intheformofweightedsumsoflocalpolynomialbasisfunctions.Weuseaquintic triangularniteelementwhosenodalvariablesincludepositionsaswellastherst andsecondpartialderivativesofthesurface.Wedescribeasystem,implementedon ahighperformancegraphicsworkstation,whichappliesthemodelttingtechniqueto thesegmentationofthecardiacLVsurfaceinvolume(3D)CTimagesandLVtracking indynamicvolume(4D)CTimagestoestimateitsnonrigidmotionoverthecardiac cycle.Thesystemfeaturesagraphicaluserinterfacewhichminimizeserrorbyaord- ingspecialistusersinteractivecontroloverthedynamicmodelttingprocess.  Keywords:  3D/4DMedicalImageAnalysis,DeformableModels,FiniteElements, Dynamics,CardiacLVSegmentation,NonrigidMotionTracking,Visualization,Inter- action.  1Introduction  CT,MRI,PETandothernoninvasivemedicalimagingtechnologiescanprovideexceptional viewsofinternalanatomicalstructures,butthecomputeraidedvisualization,manipula- tion,andquantitativeanalysisofthemultidimensionalimagedatatheyproduceisstill 1   limited.State-of-the-artmedicalimagersgeneratemassivedatabasesofstaticvolume(3D) anddynamicvolume(4D)images.Thesedatasets,whichusuallysuerfromsampling artifacts,spatialaliasing,andnoise,areessentially\blocksofgranite"withmeaningfulem- beddedstructures.Animportantproblemistoextractthesurfaceelementsbelongingtoan anatomicalstructure(thesegmentationstep)andtointegratethesesurfaceelementsinto agloballycoherentsurfacemodelofthestructure(thereconstructionstep).Certaindiag- nosticproceduresalsorequirethetrackinganddeformationanalysisofnonrigidlymoving anatomicalsurfacese.g.,thestretchingoftheleftventricle(LV)duringthecardiaccycle isdirectlyrelatedtoheartcondition.Theeaseandaccuracyofsuchprocedurescanbe criticallydependentuponthemodelused.Dynamicmodelsareneededwhicharerobust againstnoise-corrupteddataandwhicharecapableofaccuratelyrepresentingthecomplex geometriesofanatomicalsurfaceswhilepermittingthequantitativemeasurementofhighly nonrigidtissuekinematics. Thispaperdescribesaphysics-basedmodelingapproachthataddressesthesurfaceseg- mentationandreconstructionproblems,aswellasthegeometricanalysisandnonrigidmo- tionestimationproblems.Wedevelopadynamic,elasticallydeformablesurfacemodelwhose deformationisgovernedbybasiclawsofnonrigidmotion.Theformulationofthemotion equationsincludesastrainenergy,simulatedforces,andotherphysicalquantities.Thesur- facestrainenergystemsfromathin-plateundertensionvariationalspline.Deformation resultsfromtheactionofinternalsplineforceswhichimposesurfacecontinuityconstraints andexternalforceswhichtthesurfacetotheimagedata.Theinherentlydynamicformu- lationofthemodelmakesitsuitablebothforstaticanatomicalsurfacereconstructionand forproblemsinvolvingthereconstructionandtrackingofnonrigidlymovingorgans. Todealwithclosedanatomicalsurfaces,weformulateadeformable\balloon"modelthat istopologicallyisomorphictoasphere.Weemploytheniteelementmethodtospatially discretizetheballoon,uniformlytessellatingitintoasetofconnectedtriangularelement domains.Theniteelementmethodprovidesananalytic,piecewisepolynomialsurface representationthatis(  C  1  )continuousacrosstriangles.Weuseaquinticniteelementwhose nodalvariablesincludenotonlythenodalpositions,butalsotherstandsecondparametric partialderivativesofthesurface.Theelementisnaturallysuitedtothesurfaceenergy functionalbecausethesesamepartialderivativesoccurinthethin-plateundertensionenergy expression.Theexistenceofparametricderivativenodalvariablesfacilitatesthecomputation ofthedierentialpropertiesofthemodeledsurface.Inparticular,thenodalvariablesand theirtimederivativescanbeusefulforcomputingthesurfacecurvature,enclosedvolume, andmotionpropertiesofanatomicalsurfaces. Wehaveimplementedasystemonahighperformancegraphicsworkstationwhichapplies thedynamicmodelttingtechniquetothesegmentationoftheLVsurfaceincardiacvolume (3D)CTimagesandLVtrackingindynamicvolume(4D)CTimagesinordertoestimate nonrigidLVmotionoverthecardiaccycle.Thesystemincludesagraphicaluserinterface whichprovidesinteractivevisualizationandaordscontroloverthemodelttingprocess. Theinterfaceallowsausertoselecttheinitialsizeandlocationofthemodelandtoexert interactiveforcesonthemodelasitdeformstotthedata.Thistypeofinteractivecontrolis desirableinmedicalimageanalysisapplicationswherethereislowtoleranceforinaccuracy, becauseitallowsspecialistuserstoexploittheirknowledgetocorrectmodelttingerrors. 2   2Background  Theliteratureonsegmentationandsurfacereconstructionin3Dmedicalimagesincludes bothmanualandautomatictechniques.Thedominantmanualmethodisslice-editing.In manualslice-editingaskilledoperator,usingacomputermouse,pen,ortrackballtracesthe regionofinterestoneachsliceofthevolume.Thislaborintensivemethodsuersfrommany drawbacks,suchasdicultiesinachievingreproduciblesegmentationresults,dicultiesin comparingmeasurementsfromdierentoperators,anddicultiesdeducing3Dstructure from2Dslices.Thetechniquecanbespeededupandmademorereproducible,however, throughtheuseofcontourextractionmethodssuchasinteractivesnakes1,2]. Thetraditionalautomaticsegmentationmethods,suchasdensitythresholdingandthe applicationof(2Dor3D)edgeoperators,havemanywell-knownproblems.Edgedetection andthemorerecentmarchingcube3]techniquereducevolumedataintosomethingthat ismorereadilydisplayedthrough3Dgraphics,suchassurfaceelements.However,they employonlythelocalpropertiesoftheimagedatahence,theyraisethedicultproblemof establishingtheconnectivityofsurfacetraceelementsinordertoassemblesensibleglobal surfacestructures4].Thesedicultieshavepromptedsomeresearcherstosettleformerely visualizingthevolumedatainitsoriginalformusingmorphology5]orvolumerendering techniques6].Unlikeglobalsurfacemodels,however,thesevoxel-displayrepresentationsdo notattempttocapturethegeometricstructureofanatomicalstructureshence,theydonot treatthedatainamannerconsistentwiththephysicalpropertiesoftheimagedobjects. Deformablesurfacemodelsareapromisingapproachtoextractinganatomicallymean- ingfulstructuresfromvolumedata.Thedynamicformofthedeformablemodeltting techniquedescribedinthispaperwasrstintroducedbyTerzopoulos,Witkin,andKass7]. Theyproposedadynamicdeformablecylindermodelconstructedfromgeneralizedsplines, alongwithforceeldtechniquestotthemodeltoimagedata.Thisdynamicapproachis beingpursuedbyseveralresearchersincomputationalvision8,9,10,11,12,13,14].The useofniteelementrepresentationsforvariationalproblemsinvisionwererstexplored in15].Ourformulationappliestheniteelementmethodtothethin-plateundertension splineproposedin16]inordertoderivediscretenonrigiddynamicsequations.Thenite elementrepresentationyieldspiecewisecontinuousdeformablesurfacemodelsthatgenerally requirefewervariablesforsimilaraccuracycomparedtonitedierenceapproaches. OurworkisrelatedtothatofYoung17]18]andCohenandCohen19,20]whoalso develop3Ddeformablesurfacemodelswhicharebasedonthethin-plateundertensionspline. YoungtsanopenbicubicHermitianniteelementbasedsurfacetothe3Dlocationsofthe coronaryarteriesatdiastasis.Theparametersofthetime{varyingdisplacementeldwere thenttedtothetrackeddisplacementsofthebifurcationpointsofthecoronaryarteries. CohenandCohentacylindrical,bicubicHermitianniteelementbasedsurfacetoMRI imagesoftheLV.Anotherrelevantdeformablemodelisthediscretemodeldevelopedby Miller  etal.  21],whichissubdividedandttedtoCTvolumeimagesbyarelaxation processthatminimizesasetofconstraintssuchasthedistancetothedataorthelocal curvatureofthemodel. Inourworkwedevelopaclosed3Dsurfacemodelbasedonaquintictriangularnite elementwithpositionandderivativenodalvariables.Themodelbeginsasauniformlytessel- latedicosahedronwhichmaysubdividerepeatedlytoattainthedesiredgeometricresolution. Ourmodelisdynamicinthesensethatitundergoesdeformationsthataregovernedbynon- 3   (a)(b)(c) Figure1:Balloonmodelswithvaryingelasticityandpulledbyaspringpointforce. rigidLagrangianmechanics.Note,however,thatalthoughthesedynamicsequationsserve wellinmodelttingandtrackingusingmultidimensionaldatasets,wemakenoattemptto modeltheactualbiomechanicalpropertiesoftheanatomicalstructureunderconsideration (suchasthecardiacLVsee,e.g.,22]).  3DynamicDeformableBalloonModel  Theballoonmodelthatwedevelopinthispaperisconstructedofthesimulatedthin-plate materialundertension.Thedeformationenergyofthematerialservesasaconstraintwhich compelsthemodeltovarysmoothlyalmosteverywhere.Theballoonisrepresentedasa vector-valuedparametricfunction  x  (  uv  )=  x  (  uv  )  y  (  uv  )  z  (  uv  )] >  wherevector  x  repre- sentsthepositionsofmaterialpoints(  uv  )relativetoareferenceframeinEuclidean3-space. Thedeformationenergyofthethin-plateundertensionmaterialisgivenbytheenergy functional  E  p  (  x  )=  ZZ    10   @  x  @u   2  +    01   @  x  @v   2  +    20   @  2  x  @u  2   2  +    11   @  2  x  @u@v   2  +    02   @  2  x  @v  2   2  dudv:  (1)  E  p  isacontrolled-continuitysplinedenedin16].Thenonnegativeweightingfunctions    ij  (  uv  )and    ij  (  uv  )controltheelasticityofthematerial.The    10  and    01  functionscontrol thetensionsinthe  u  and  v  directions,respectively,whilethe    02  and    20  functionscontrol thecorrespondingbendingrigidities,andthe    11  functioncontrolsthetwistingrigidity. Increasingthe    ij  hasatendencytodecreasethesurfaceareaofthematerial,whileincreasing the    ij  tendstomakeitlessexible.Ingeneral,theweightingfunctionsmaybeusedto introducedepthandorientationdiscontinuitiesinthematerial.Inthispaper,however,we donotmakeuseofthiscapabilityandsetthefunctionstoconstantvalues    ij  (  uv  )=    ij  and    ij  (  uv  )=    ij  .Figure1showsthethinplateundertensionballoonpulledradiallyby aspringpointforce(in(a)    ij  =0  :  8and    ij  =0,in(b)    ij  =    ij  =0  :  5,andin(c)    ij  =0 and    ij  =0  :  8): Ageneralandelegantapproachtottingdeformablesurfacemodelstodata,especially whenthedataaretime-varying,istomakethemodels  dynamic  .Adynamicformulation 4   imposesanaturaltemporalcontinuityonthemodel,therebypermittingasmoothlyanimated displayofthedatattingprocess.Italsoallowsausertointeractwiththemodelbyapplying constraintforcestopullitoutoflocalminimatowardsthecorrectsolution. InaLagrangiandynamicsformulation,thepositionsofmaterialpointsbecomesatime- dependentfunction  x  (  uvt  )andweimbuethesimulatedmaterialwithmassanddamping densities.Thedeformationenergyyieldsinternalelasticforces,and  E  p  (  x  )isminimized whentheseforcesequilibrateagainstexternallyappliedforcesandthemodelstabilizes:  @  x  =@t  =  @  2  x  =@t  2  =  0  . Thedynamicbehavioroftheballoonmodelduringthettingprocessisgovernedbythe second-orderpartialdierentialequations   @  2  x  @t  2  +   @  x  @t  +    x  E  p  =  f    (2) wherethersttermrepresentstheinertialforcesduetothemassdensity    (  uv  ),thesecond termrepresentsthedampingforcesduetothedampingdensity    (  uv  ),thethirdterm representstheelasticforcewhichresistdeformation,andnally  f  (  uvt  )representsthe externalforcesderivedfromtheimagedata.The(generallynonlinear)dataforcesmaybe formalizedasstemmingfromadatafunctional  E  d  (  x  )=  ;  ZZ  x  >  f  dudv:  (3)  4FiniteElementRepresentation  Thenitedierencemethodortheniteelementmethodareapplicabletocomputingnu- mericalsolutionstothefunction  x  (  uvt  ).Finitedierencesolutionsapproximatethecon- tinuousfunction  x  asasetofdiscretenodesinspace.Adisadvantageofthenitedierence approachisthatthecontinuityofthesolutionbetweennodesisnotmadeexplicit.The niteelementmethod,ontheotherhand,providescontinuoussurfaceapproximationsthat is,themethodapproximatestheunknownfunction  x  intermsofcombinationsoflocalbasis functions23]. Toapplytheniteelementmethodtoourmodels,wetessellatethecontinuousmaterial domain(  uv  )intoameshof  M  elementsubdomains  E  j  .Weapproximate  x  asaweighted sumofpiecewisepolynomialbasisfunctions  N  i  :  x  (  uvt  )    ^  x  (  uvt  )=  n  X  i  =1  N  i  (  uv  )  q  i  (  t  )    (4) where  q  i  isavectorofnodalvariablesassociatedwithmeshnode  i  . Substituting(4)into(2)yieldsthediscreteequationsofmotion  M    q  +  C  _  q  +  Kq  =  f  q    (5) with  q  =  q  >  1  :::  q  >  i  :::  q  >  n  ] >  ,wherethemassmatrix  M  ,dampingmatrix  C  ,andstiness matrix  K  aresparse,symmetricmatricesandvector  f  q  arenodaldataforces.These  global  matricesmaybeassembledfromtheirassociated  local  elementmatricesbyexpandingeach 5 
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