Linear Algebraic Transformations of the Bidomain E

CS-1992-11LinearAlgebraicTransformationsoftheBidomainEquationsandTheirImplicationsonNumericalMethodsN.Hooke,C.Henriquez,P.LanzkronandD.RoseDepartmentofComputerScienceDukeUniversityDurham,NorthCarolina27708-0129September30,1992LinearAlgebraicTransformationsoftheBidomainEquationsandTheirImplicationsonNumericalMethodsN.Hooke,C.Henriquez,P.LanzkronandD.RoseSeptember30,1992AbstractInthispaperwegiveamathematicalframeworkforthetreatmentofthebidomainequationsusedtomodelpropagationincardiactissue.Weshowhowpreviousworktsintothisframeworkanddiscusstheimplicationsofvariouslineartransformationsoftheequationsonthenumericalmethodsofsolution.1IntroductionThevastmajorityofmodelstostudypropagationincardiactissuearebasedontheassumptionthatthetissuecanberepresentedasanintracellularregionseparatedfromagroundedinterstitialorextracellularregionbyamembrane.Thismonodomainrepresentationeectivelyignoresanyeectofcurrentowoutsidethecellsontheresultingtransmembranepotential.Analternatemodelforcardiactissueattemptstodescribetheaveragecurrentowinsideandoutsideofcellsbyconsideringeachregionasseparateandlinkedonlythroughthetransmembranecurrentdensity.Thebido-mainrepresentationhassomedistinctadvantagesoverthetraditionalmonodomainmodel.First,theextracellulareldsinsideandoutsidethetissuearisingfromthepropagatedactivitycanbecomputeddirectly.Second,theeectofexogenousex-tracellularstimulioreldsontheresultingbehaviorcanbeinvestigated.Finally,separateboundaryconditionscanbeappliedtotheintracellularandinterstitialre-gions,accountingforpresenceofanadjoiningbathorastructuralanomaly.Whilethebidomainmodelhasagreaterrangeofapplicationsthanthemon-odomainmodel,itisconsiderablymorediculttoevaluate.Anumberofapproaches1havebeenused,eachpredicatedbysomelineartransformationofthebidomainequa-tionstoobtainanumericallytractableform.Thegoalofthispaperistoformalizethesevariouslineartransformationsandtodeterminewhichformleadstothemostecientnumericalscheme.2BidomainequationsThebidomainmodelrepresentsthetissueasanintracellularspaceandaninterstitialspacewhoseelectricpotentialsarebothdenedonthesamespatialdomain.Bothspacesareconsideredtobecontinuouseverywherebutseparatedfromeachotherateverypointbyamembraneofunspeciedtopology.Eachdomainisapassivevolumeconductorwhoseonlysourcesofelectriccurrentarelocatedinthemembrane,thedomainboundary,andanysourcesplacedinthedomainforexperimentalpurposes.Consequentlytheelectricpotentialintheinteriorofeachdomainsatisestheconservationequationsr(iri)=ImIsi(1)r(ere)=ImIse(2)whereiandearetheelectricpotentials(V)oftheintracellularandinterstitialdomainsrespectively,iandearethecorrespondingconductivitytensors(Sm1),Imisthetransmembranecurrentdensity(Am2),isthesurface/volumeratioofthemembrane(m1),andIsiandIsearetheexternallyimposedcurrentsources(Am3).Theparametersi,oandrepresentthediscretestructureofthetissueaveragedoveralengthscaleofmanycell-lengths.Thetransmembranepotential,m,isdenedasm=ie:(3)Ingeneralwemakenoassumptionsaboutthenatureofiore.Forthisdiscussionweconsiderslabsofparallelberswhereiandeareindependentofpositionandsharethesameprincipalaxes.Underthisassumption,thex,yandzaxesmaybechosensuchthatiandearediagonalandr(iri)=gix@2i@x2+giy@2i@y2+giz@2i@z2;r(ere)=gex@2e@x2+gey@2e@y2+gez@2e@z2;fornonnegativeconstantsgix,giy,giz,gex,geyandgez.Inthatcasethefollowinganisotropyratiosarefrequentlyused[1]x=gixgex;y=giygey;z=gizgez:2Thecasex=y=ziscommonlyreferredtoasequalanisotropy,i.e.,i=ke,kaconstant.Itisageneralizationofisotropy[2,3]andallowsthebidomaintobereducedtoanequivalentmonodomain.Thebidomainequationssimplifytothewidely-usedmonodomainequationr(rm)=ImIsundertheassumptionthateiseectivelyinnite(eisuniformlyzero)orundercertainassumptionsregardingtheanisotropiesofiande,describedbelow.Forthepurposesofinvestigatingactionpotentialbehaviour,thetransmembranecurrentmaybedescribedbyacapacitivecomponentandanioniccurrentgovernedbyaHodgkin-Huxley-likemodel[4,5,6,7,8,9].Suchmodelsdeneanioniccurrentwithnonlineardependenceonmandthemembranestate,whichincludesensembleaveragesofproteinscontrollingionicpermeabilityandmayincludecompartmentalmodelsofionicconcentrationsadjacenttothemembrane.Foramembranemodelwithmsuchstatevariables,itsdynamicsmaybedescribedbythesystemofdierentialequationsd~qdt+~M(~q;m)=0(4)where~q2Rmisdenedthroughoutthedomainandwhere~M:RmR7!Rmisanonlinearfunction.ThetotaltransmembraneioniccurrentImisthendenedasIm=Cmdmdt+Iion(~q;m)(5)whereIion:RmR7!Risanonlinearfunctionrepresentingthesumofalltheindividualtransmembraneioniccurrents.Thecompletesetofbidomainequationsforsuchmembrane-basedmodelsisthenr(iri)=Cm@m@t+Iion(~q;m)Isi;(6)r(ere)=Cm@m@tIion(~q;m)Ise;(7)d~qdt+~M(~q;m)=0:(8)BothDirichletandNeumanboundaryconditionshavebeenappliedtoequa-tions(6)and(7).LikePoisson’sequation,thebidomainequationshavenouniquesolutionifpureNeumanboundaryconditionsareapplied.If~q(x;y;z;t),i(x;y;z;t)ande(x;y;z;t)representasolutionsatisfyinggiveninitialandNeumanboundarycondi