Mathematical Models of Therapeutical Actions Related to Tumour and Immune System Competition
24
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe et accède à tout notre catalogue !
Découvre YouScribe et accède à tout notre catalogue !
24
pages
English
Documents
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
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.
Voir
- between cells
- modeling process
- di- vided into
- into
- interaction
- ability can
- can then
- field modeling
Publié par
Langue
English
MathematicalModelsofTherapeuticalActionsRelatedtoTumourandImmuneSystemCompetitionElenaDeAngelis(1)andPierre-EmmanuelJabin(2)(1)DipartimentodiMatematica,PolitecnicodiTorinoCorsoDucadegliAbruzzi24,10129Torino,Italyemail:elena.deangelis@polito.it(2)E´coleNormaleSupe´rieureDe´partementdeMathe´matiquesetApplications,CNRSUMR855345rued’Ulm,75230ParisCedex05,Franceemail:jabin@dma.ens.frAbstractThispaperdealswiththequalitativeanalysisofamodelrelatedtothede-scriptionoftwomedicaltherapieswhichhavebeenintensivelydevelopedinrecentyears.Inparticular,werefertothemodelingoftheactionsappliedbyproteins,toactivatetheimmunedefense,andtothecontrolofangiogenesis,tocontrastthegrowthoftumourcellsbypreventingthefeedingactionsofendothelialcells.Thetherapeuticalactionswhichareobjectofthemodelingprocessdevelopedinthispaperhavetoberegardedasappliedwithintheframeworkofthecompetitionbetweentheimmunesystemandtumourcells.WeprovetheexistenceofsolutionstotheCauchyproblemrelatedtothemodel.Theefficiencyofthetherapiesandtheasymptoticbehaviourintimeofoursolutionsisalsoinvestigated.Keywords:Vlasovkinetictheory,Cauchyproblem,cellpopulation,tumour-immunecompetition1IntroductionMethodsofthemathematicalkinetictheoryhavebeenappliedinthelastfewyearstomodelthecompetitionbetweentumourandimmunecells.Theliteratureinthefieldandacriticalanalysisontheexistingresultsandopenproblemscanbefoundinthereviewpapers[2]and[3].Mathematicalmodelsareexpectedtodescribetheinteractionsandcompetitionbe-tweentumoursandtheimmunesystem.Theevolutionofthesystemmayendupeitherwiththeblow-upofthehost(withinhibitionoftheimmunesystem),orwiththesuppressionofthehostduetotheactionoftheimmunesystem.Themathematicalstructureoftheequationssuitabletodealwiththemodelingoftheabovesystemhave1
beendevelopedandcriticallyanalyzedin[1],whilemotivationsfromscientistsoperat-inginthesciencesofimmunologyinfavorofdevelopmentofmethodsofnonequilibriumstatisticalmechanicscanberecovered,amongothers,in[12]and[13].Themathematicalmethodsarethosetypicalinnonequilibriumstatisticalmechanicsandgeneralizedkinetictheory.Thegeneralidea,asdocumentedin[4],consistsinderivinganevolutionequationforthefirstdistributionfunctionoverthevariablede-scribingthemicroscopicinternalstateoftheindividuals.Generally,thisvariablemayincludepositionandvelocity,butitcanalsorefertosomeadditionalspecificmicro-scopicfeatures.Interactionsbetweenpairshavetobemodeledtakingintoaccountnotonlymechanicalrulesbutalsomodificationsofthenon-mechanicalphysical(internal)state.Specificallyweareinterestedinadevelopmentofthemodelproposedin[4],whosean-alyticalpropertieshavebeenstudiedin[5],towardthedescriptionofmedicaltherapieswhichhavebeenintensivelydevelopedinrecentyears.Morepreciselyamodelofcom-petitionbetweentheimmunesystemandtumourcellswasproposedin[4],thismodeldidnotincludetheeffectofanytherapy.In[5],itwasprovedthatthereexistsglobalsolutionstothismodelandthepossibleasymptoticbehavioursintimeweredetailed.Theaimofthispaperisfirsttopresentageneralframework(inwhichthemodelof[4]fits)whichisthenusedtobuildanewmodel.Thedifferencewiththepreviousmodelof[4]isthattheeffectoftwodifferenttherapiesisnowtakenintoaccount.Thelastpartofthepaperstudiesthepropertiesofthenewmodel;Existenceofglobalsolutionsisaconsequenceoftheproofgivenin[5]andconsequentlyweonlystatetheresult.Howevertheanalysisoftheasymptoticbehaviourintimeismorecomplicatedforthenewmodelthanwhatitwasfortheoneof[4],thusrequiringnewideasandnewproofswhicharepresented.Concerningthetherapies,werefertothemodelingoftheactionsappliedbyproteinstoactivatetheimmunedefense[15]andtothecontrolofangiogenesis[9],[10],tocontrastthegrowthoftumourcellsbypreventingthefeedingactionsofendothelialcells.Allabovetherapeuticalactionswhichareobjectofthemodelingprocessdevelopedinthispaperhavetoberegardedasappliedwithintheframeworkofthecompetitionbetweentheimmunesystemandtumourcells.Referringtotheliteratureontheimmunecom-petition,developedwithinmedicalsciences,theinterestedreaderisaddressedtothesurvey[6].Anotherdevelopmentofthemodelintroducedin[4]ispresentedin[7],wherethecapacityofthebodytorepaircellsdamagedistakenintoaccount.Theideaisthatforalongtimerangethebodyproducesnewcellstoreplacethedeadonesinordertotrytoreachitsnormalhealthystate.Afterthisintroduction,thecontentsofthispaperareorganizedinfivemoreSections:–Section2dealswiththederivationofamathematicalframeworkforthedesignofspecificmodelssuitabletodescribethetherapeuticalactionsindicatedabove.Thismeansderivingaclassofintegro-differentialequationstodescribe,bymeth-odsofthemathematicalkinetictheory,theevolutionoverthebiologicalstateofthecells;2
