A Mathematical Model of Immune Competition Related to Cancer Dynamics
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A Mathematical Model of Immune Competition Related to Cancer Dynamics Ilaria Brazzoli(1), 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)Universite de Nice-Sophia Antipolis Departement de Mathematiques Parc Valrose 06108 Nice Cedex 2, France email: Abstract This paper deals with the qualitative analysis of a model describing the compe- tition among cell populations each of them expressing a peculiar cooperating and organizing behaviour. The mathematical framework in which the model has been developed is the kinetic theory for active particles. The main result of this paper is concerned with the analysis of the asymptotic behavior of the solutions. We prove that, if we are in the case when the only equilibrium solution if the trivial one, the system evolves in such a way that the immune system, after being activated, goes back towards a physiological situation while the tumor cells evolve as a sort of progressing travelling waves, characterizing a typical equilibrium/latent situation. Keywords: kinetic theory for active particles, Cauchy problem, travelling waves, complex systems, biological systems, cell population, tumour-immune competition, tumor dormancy.
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- tition among cell
- tumour-suppressive mechanism
- th population
- cells evolve
- cell death
- cancer dormancy
- shaping cancer
- overall state
AMathematicalModelofImmuneCompetitionRelatedtoCancerDynamicsIlariaBrazzoli(1),ElenaDeAngelis(1)andPierre-EmmanuelJabin(2)(1)DipartimentodiMatematica,PolitecnicodiTorinoCorsoDucadegliAbruzzi24,10129Torino,Italyemail:ilaria.brazzoli@polito.it,elena.deangelis@polito.it(2)Universite´deNice-SophiaAntipolisDe´partementdeMathe´matiquesParcValrose06108NiceCedex2,Franceemail:Pierre-Emmanuel.JABIN@unice.frAbstractThispaperdealswiththequalitativeanalysisofamodeldescribingthecompe-titionamongcellpopulationseachofthemexpressingapeculiarcooperatingandorganizingbehaviour.Themathematicalframeworkinwhichthemodelhasbeendevelopedisthekinetictheoryforactiveparticles.Themainresultofthispaperisconcernedwiththeanalysisoftheasymptoticbehaviorofthesolutions.Weprovethat,ifweareinthecasewhentheonlyequilibriumsolutionifthetrivialone,thesystemevolvesinsuchawaythattheimmunesystem,afterbeingactivated,goesbacktowardsaphysiologicalsituationwhilethetumorcellsevolveasasortofprogressingtravellingwaves,characterizingatypicalequilibrium/latentsituation.Keywords:kinetictheoryforactiveparticles,Cauchyproblem,travellingwaves,complexsystems,biologicalsystems,cellpopulation,tumour-immunecompetition,tumordormancy.1IntroductionThispaperdealswiththequalitativeanalysisofamodeldescribingthecompetitionamongcellpopulationseachofthemexpressingapeculiarcooperatingandorganizingbehaviour.Indetail,theobjectofourstudyaresomeaspectsofthecompetitionbetweentumorandimmunecells,inpresenceofenvironmental(i.e.endothelial)cells.Suchcompetitionisapartofthecomplexevolutionaryprocessthatisthecancerprogressionandrepresentsatumor-defenceandtumour-suppressivemechanism,atleastattheearlystageoftheprocess.Thetermprogressionrelatedtocancercells,asreportedinthepioneerpaper[10],describesthebiologicalstateofthecellsandconciselyrepresentsawaytodescribehowfarfromthehealthy(freefromdisease)statethecellsare.Wealsoreferto[14]1
forareviewofthedevelopingknowledgeandtheunderstandingofthemanydifferentaspectsofthephenomenonoftumorprogression.Asreportedin[3],tumoursurvivalandmetastasisnecessitateovercomingimmunesurveillanceandregulation,extracellularmatrixbarriersandlimitingnutrients.Thecrucialroleoftheimmunesysteminshapingcancerdevelopmentandcontrollingtumourgrowthhasbeenrecognizedfordecades,seeforexample[1]and[12].Anactiveimmunesystemcandetectearlylesionsandcankeeptheproliferatingtumorcellsatlownumbers.Itcanalsopreventtheexpansionofproliferatingtumorcellsandcaneliminatethem.Atumourmass,althoughangiogenesiscompetent,mightbepreventedfromexpandingandkeptdormant(i.e.inanon-dividingstateandwithquiescentphysiologicalfunctions,astateofequilibrium)byanactiveimmuneresponse,[1].Thisisthemechanismoftheimmunosurveillance.Ontheotherhand,evasionoftheimmunesystemcausestumourmassexpansion:tumorcellswithincreasedcapacitytoattenuatetheimmuneresponsecanescapeequilibriumandmayprogress.Thisprocessasawholeistermedcancerimmunoediting,see[12].Anotherimportantaspectoftumorprogressionconsistsintheevolutionoftumourcellpopulationsthroughsequentialgeneticandepigeneticchanges.Inclinicaltrials,oneofthepossiblestrategytocontrastsuchevolutionconsistsinmobilizingtheim-munesystemagainstacrucialselfantigenthatisupregulatedinthetumour(activeimmunotherapy),[11].Atthemoment,aninterdisciplinaryandamultiscaleapproachtothestudyofcancerprogressionandevolutionseemsarealitymorethenachallenge,asmentionedin[4],[5],[2],[15].Moregenerally,aninterdisciplinaryapproachtothestudyofcomplexsystems,ascancerdynamicscanbeconsidered,isneededinordertocapturethemainessenceofthesuchkindofproblems.Amongdifferentdisciplines,themathematicalapproachingeneralandthemathematicalmodellingapproachinparticular,isaddressedtothephenomenologicaldescriptionofsomeaspectsofthebiologyofthesystemwearedealingwith.Theaimofthispaperifjusttopresentandtocarryoutaqualitativeanalysisofamodeldescribingsomeparticularfeaturesoftheimmunecompetitionrelatedtocancerdynamics.Moreindetail,weproposeandanalyzeamodeldescribingtherelevantphaseofthecancerdormancythatistheimmunosurvelliance.Cancerdormancyisacomplexstageofthediseaseprogressionandtherapeuticre-sponse,butitisstillpoorlyunderstood.Theimmunosurvellianceisoneimportantphaseofthisstage.Otherphasesareangiogenicdormancyandcellulardormancy.Presently,thedetailsoftheentiremechanismscontrollingeachsinglephaseoftumordormancyarestillunknownaswellastheintersectionsamongthedifferentphases.Fromaclinicalpointofview,itiscruciallyimportanttounderstandthemechanismsofthecancerdormancy,becausemetastasesandlocalrecurrencesderivedfromthesedormantcells.Additionally,dormantcellsareresistanttochemotherapysothattheyescapethedrugtreatmentstoinducecelldeath,[1].Themathematicalframeworkinwhichthemodelhasbeendevelopedisthekinetictheoryforactiveparticles:wereferto[4]foracompleteoverviewofthemathematicalframeworkandacomprehensivelistofpapersdevotedtomodelsdescribingtheimmune2
