Qualitative Analysis of a Mean Field Model of Tumor Immune System Competition
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Qualitative Analysis of a Mean Field Model of Tumor-Immune System Competition Elena De Angelis(1) and Pierre-Emmanuel Jabin(2) (1)Dipartimento di Matematica, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy email: (2)Ecole Normale Superieure Departement de Mathematiques et Applications, CNRS UMR 8553 45 rue d'Ulm, 75230 Paris Cedex 05, France email: Abstract This paper deals with the qualitative analysis of a model related to the immune response to the evolution of the progression of endothe- lial cells which have lost their differentiation and start their evolution toward methastatic states. We prove the existence of solutions to the Cauchy problem related to the model. The asymptotic behavior in time of our solutions is also investigated. Keywords: Vlasov kinetic theory, Cauchy problem, cell population, tumor-immune com- petition 1 Introduction This paper deals with the qualitative analysis of the initial value problem for a mathematical model designed to describe the interaction and the com- petition between immune and cancer cells mediated by environmental cells which provide the feeding for the growth related to the mitosis process. The mathematical model was proposed in [1] derived on the basis of math- ematical methods typical for nonequilibrium statistical mechanics and gen- eralized kinetic theory.
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QualitativeAnalysisofaMeanFieldModelofTumor-ImmuneSystemCompetitionElenaDeAngelis(1)andPierre-EmmanuelJabin(2)(1)DipartimentodiMatematica,PolitecnicodiTorinoCorsoDucadegliAbruzzi24,10129Torino,Italyemail:deangeli@calvino.polito.it(2)´EcoleNormaleSupe´rieureDe´partementdeMathe´matiquesetApplications,CNRSUMR855345rued’Ulm,75230ParisCedex05,Franceemail:jabin@dma.ens.frAbstractThispaperdealswiththequalitativeanalysisofamodelrelatedtotheimmuneresponsetotheevolutionoftheprogressionofendothe-lialcellswhichhavelosttheirdifferentiationandstarttheirevolutiontowardmethastaticstates.WeprovetheexistenceofsolutionstotheCauchyproblemrelatedtothemodel.Theasymptoticbehaviorintimeofoursolutionsisalsoinvestigated.Keywords:Vlasovkinetictheory,Cauchyproblem,cellpopulation,tumor-immunecom-petition1IntroductionThispaperdealswiththequalitativeanalysisoftheinitialvalueproblemforamathematicalmodeldesignedtodescribetheinteractionandthecom-petitionbetweenimmuneandcancercellsmediatedbyenvironmentalcellswhichprovidethefeedingforthegrowthrelatedtothemitosisprocess.Themathematicalmodelwasproposedin[1]derivedonthebasisofmath-ematicalmethodstypicalfornonequilibriumstatisticalmechanicsandgen-eralizedkinetictheory.Thegeneralidea,asdocumentedin[2],consistsinderivinganevolutionequationforthefirstdistributionfunctionoverthe1
variabledescribingthemicroscopicinternalstateoftheindividuals.Gen-erally,thisvariablemayincludepositionandvelocity,butitcanalsorefertosomeadditionalspecificmicroscopicstates.Interactionsbetweenpairshavetobemodelledtakingintoaccountnotonlymechanicalrulesbutalsomodificationsofthenon-mechanicalphysical(internal)state.Thismodelreferstotheearlystageoftumordevelopmentwhentumorcellsarenotyetaggregatedintoasolidform.Thissituationalsooccurwhencellsareresiduallydispersedintheenvironmentafterremovalofsolidform.Developmentsofmodelingmethodsforlargedispersedsystemsbymethodsofthemathematicalkinetictheoryisdocumentedinthecollectionofsur-veyseditedin[3]and[4].SpecificallythemathematicalmodeldealtwithinthispaperisbasedonameanfielddescriptioncorrespondingtotheVlasovequation.Thisgeneralizationisappliedtovariousfieldsofappliedsciences,anexampleisthemodelingofdispersedbubblesinafluid,see[5]and[6].Cellularphenomena,relatedtoinnerpropertiesofthecell,playarelevantroleintheevolutionofthephysicalsystemwearedealingwith,seee.g.Greller,TobinandPoste[8].Thecellularscalereferstothemain(interac-tive)activitiesofthecells:activationandproliferationoftumorcellsandcompetitionwithimmunecells.Inparticular,proliferationoftumorcells,whichareoftendegeneratedendothelialcells,happenswhenanenviron-mentalcellloosesitsdeathprogramand/orstartstoundergoingmitosiswithoutcontrol.Moreover,competitionwiththeimmunesystemstartsiftumorcellsarerecognizedbyimmunecells.Activationandinhibitionoftheimmunecellsintheircompetitionwithtumorisregulatedbycytokinesignals.Tumorcellscanbeadditionallyactivatedtowardsproliferationduetonutrientsupplyfromenvironmentalcells.Specificallythepresentpaperaimstoanalyzethemodelproposedin[1]throughaqualitativeanalysiswhosemainresultreferstoexistenceanduniquenessofthesolutiontotheinitialvalueproblemundersuitableas-sumptionsontheinitialdata.Anasymptoticanalysisisalsodevelopedtobetterunderstandtherelevanceofsomephenomenadescribedbythemodel.Particularattentionisgiventothequalitativeanalysisoftheasymptoticbe-haviorofthesolutionswhichmayeithershowtheblowupoftumorcells,ortheprogressivedestructionoftumorcellsduetotheactionofimmunecells.Indeed,thequalitativeanalysisdevelopedinthispaperpointsouttheabovebehaviorwithreferencetotheparameterswhichcharacterizethemodel.Asitwillbeshowninwhatfollowsaninterestingbiologicalinterpretationcanbegivenwithreferencetotheaboveasymptoticanalysis.Thecontentsofthispaperareproposedinfivesections:2
