Mathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition

Elena De Angelis

Description

Mathematical Models of Therapeutical Actions Related to Tumour and 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 de- scription of two medical therapies which have been intensively developed in recent years. In particular, we refer to the modeling of the actions applied by proteins, to activate the immune defense, and to the control of angiogenesis, to contrast the growth of tumour cells by preventing the feeding actions of endothelial cells. The therapeutical actions which are object of the modeling process developed in this paper have to be regarded as applied within the framework of the competition between the immune system and tumour cells. We prove the existence of solutions to the Cauchy problem related to the model. The efficiency of the therapies and the asymptotic behaviour in time of our solutions is also investigated. Keywords: Vlasov kinetic theory, Cauchy problem, cell population, tumour-immune competition 1 Introduction Methods of the mathematical kinetic theory have been applied in the last few years to model the competition between tumour and immune cells.

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Sujets

• De Angelis
• Interaction

Informations

• Publié par: pefav
• Nombre de lectures: 9
• Langue: English

Extrait

MathematicalModelsofTherapeuticalActions RelatedtoTumourandImmuneSystemCompetition ElenaDeAngelis (1) andPierre-EmmanuelJabin (2) (1) DipartimentodiMatematica,PolitecnicodiTorino CorsoDucadegliAbruzzi24,10129Torino,Italy email:[e-mail] (2) E ´ coleNormaleSupe´rieure De´partementdeMathe´matiquesetApplications,CNRSUMR8553 45rued’Ulm,75230ParisCedex05,France email:[e-mail] Abstract Thispaperdealswiththequalitativeanalysisofamodelrelatedtothede- scriptionoftwomedicaltherapieswhichhavebeenintensivelydevelopedinrecent years.Inparticular,werefertothemodelingoftheactionsappliedbyproteins, toactivatetheimmunedefense,andtothecontrolofangiogenesis,tocontrast thegrowthoftumourcellsbypreventingthefeedingactionsofendothelialcells. Thetherapeuticalactionswhichareobjectofthemodelingprocessdevelopedin thispaperhavetoberegardedasappliedwithintheframeworkofthecompetition betweentheimmunesystemandtumourcells.Weprovetheexistenceofsolutions totheCauchyproblemrelatedtothemodel.Theefficiencyofthetherapiesand theasymptoticbehaviourintimeofoursolutionsisalsoinvestigated. Keywords:Vlasovkinetictheory,Cauchyproblem,cellpopulation,tumour-immunecompetition 1Introduction Methodsofthemathematicalkinetictheoryhavebeenappliedinthelastfewyearsto modelthecompetitionbetweentumourandimmunecells.Theliteratureinthefield andacriticalanalysisontheexistingresultsandopenproblemscanbefoundinthe reviewpapers[2]and[3]. Mathematicalmodelsareexpectedtodescribetheinteractionsandcompetitionbe- tweentumoursandtheimmunesystem.Theevolutionofthesystemmayendup eitherwiththeblow-upofthehost(withinhibitionoftheimmunesystem),orwith thesuppressionofthehostduetotheactionoftheimmunesystem.Themathematical structureoftheequationssuitabletodealwiththemodelingoftheabovesystemhave 1

beendevelopedandcriticallyanalyzedin[1],whilemotivationsfromscientistsoperat- inginthesciencesofimmunologyinfavorofdevelopmentofmethodsofnonequilibrium statisticalmechanicscanberecovered,amongothers,in[12]and[13]. Themathematicalmethodsarethosetypicalinnonequilibriumstatisticalmechanics andgeneralizedkinetictheory.Thegeneralidea,asdocumentedin[4],consistsin derivinganevolutionequationforthefirstdistributionfunctionoverthevariablede- scribingthemicroscopicinternalstateoftheindividuals.Generally,thisvariablemay includepositionandvelocity,butitcanalsorefertosomeadditionalspecificmicro- scopicfeatures.Interactionsbetweenpairshavetobemodeledtakingintoaccountnot onlymechanicalrulesbutalsomodificationsofthenon-mechanicalphysical(internal) state. Specificallyweareinterestedinadevelopmentofthemodelproposedin[4],whosean- alyticalpropertieshavebeenstudiedin[5],towardthedescriptionofmedicaltherapies whichhavebeenintensivelydevelopedinrecentyears.Morepreciselyamodelofcom- petitionbetweentheimmunesystemandtumourcellswasproposedin[4],thismodel didnotincludetheeffectofanytherapy.In[5],itwasprovedthatthereexistsglobal solutionstothismodelandthepossibleasymptoticbehavioursintimeweredetailed. Theaimofthispaperisfirsttopresentageneralframework(inwhichthemodelof[4] fits)whichisthenusedtobuildanewmodel.Thedifferencewiththepreviousmodel of[4]isthattheeffectoftwodifferenttherapiesisnowtakenintoaccount.Thelast partofthepaperstudiesthepropertiesofthenewmodel;Existenceofglobalsolutions isaconsequenceoftheproofgivenin[5]andconsequentlyweonlystatetheresult. Howevertheanalysisoftheasymptoticbehaviourintimeismorecomplicatedforthe newmodelthanwhatitwasfortheoneof[4],thusrequiringnewideasandnewproofs whicharepresented. Concerningthetherapies,werefertothemodelingoftheactionsappliedbyproteinsto activatetheimmunedefense[15]andtothecontrolofangiogenesis[9],[10],tocontrast thegrowthoftumourcellsbypreventingthefeedingactionsofendothelialcells.All abovetherapeuticalactionswhichareobjectofthemodelingprocessdevelopedinthis paperhavetoberegardedasappliedwithintheframeworkofthecompetitionbetween theimmunesystemandtumourcells.Referringtotheliteratureontheimmunecom- petition,developedwithinmedicalsciences,theinterestedreaderisaddressedtothe survey[6]. Anotherdevelopmentofthemodelintroducedin[4]ispresentedin[7],wherethe capacityofthebodytorepaircellsdamagedistakenintoaccount.Theideaisthatfor alongtimerangethebodyproducesnewcellstoreplacethedeadonesinordertotry toreachitsnormalhealthystate. Afterthisintroduction,thecontentsofthispaperareorganizedinfivemoreSections: –Section2dealswiththederivationofamathematicalframeworkforthedesign ofspecificmodelssuitabletodescribethetherapeuticalactionsindicatedabove. Thismeansderivingaclassofintegro-differentialequationstodescribe,bymeth- odsofthemathematicalkinetictheory,theevolutionoverthebiologicalstateof thecells; 2