Compartmental Model for Cancer Evolution Chemother

CompartmentalModelforCancerEvolution:ChemotherapyandDrugResistanceJ.J.Westman,B.R.Fabijonas,D.L.Kern,andF.B.HansonDepartmentofMathematics,UniversityofCalifornia,Box951555,LosAngeles,CA90095-1555,USADepartmentofMathematics,SouthernMethodistUniversity,P.O.Box750156,Dallas,TX75275-0156,USAInstituteforMathematicsandItsApplications,UniversityofMinnesota,207ChurchStreetSE,Minneapolis,MN55455,USALaboratoryforAdvancedComputing,UniversityofIllinoisatChicago,851MorganSt.;M/C249,Chicago,IL60607-7045,USADecember20,2001AbstractInthispaper,amodelispresentedthatexplorestheroleofdrugresistanceintheevolutionofcancersubjecttotreatmentwithasinglecytotoxicagent.Animportantfeatureofthismodelisthatitisdesignedtostartfromasinglecellandgeneratetheevolutionofthecancer.Fromthisperspective,therolesofintrinsicornaturaloccurringresistanceaswellasthatofacquiredresistancefromchemotherapycanbeseenatanystageofthedevelopmentofthecancerortreatmenthistory.Usingthismodel,anempiricalfitcanbemadetothatofagivenpatientbasedonestimatedparameterswhichcanbeupdatedasmoreobservationsbecomeavailable.Afocusisplacedontheheterogeneousnatureofthecancerandtheeffectsoftreatments,whichsuggeststhatthismodelshouldprovideareasonabletooltoexploretreatmentoptionsforcancerpatients.Keywords:Cancer;Chemotherapy;Drugresistance;Compartmentalmodel1IntroductionOneoftheleadingcausesofdeathintheworldiscancer.Eventhoughthereareanumberoftreatmentoptionsforcancerpatientssuchassurgery,chemotherapy,immunotherapy,andradiotherapy,thelifeexpectancyforthecancerpatientwillbediminishedduetothediseaseandquitepossiblythetreatmentsaswell.Thesetreatmentmodalitiescannotingeneralprovideacureforcancerbutmaybringaboutremissionthatcanlaterrelapse.TheeffectsoftheseCorrespondingauthor:JohnJ.Westman,DepartmentofMathematics,UniversityofCalifornia,Box951555,LosAngeles,CA90095-1555,USA,Phone:[phone],Fax:[phone],e-mail:[e-mail]theNationalScienceFoundationGrantDMS-99-73231.1treatmentscanvaryfromcancertocancerandindividualtoindividual,whichfurthercomplicatesthesituationforeffectiveeradicationofcancerinanygivenpatient.Currently,thereexistsaneedforfundamentalunderstandingofthepropertiesandmechanismsinvolvedinthegenesisandevolutionofcancers.Furthermore,therolesbywhichvarioustreatmentmodalitieseffectthegrowthandspreadofcancerneedstobeunderstoodingreaterdetail.Thisisanexceptionallydifficulttasksincecancersarecomposedofavarietyofcelltypeswhichmaybeindifferentphasesofthecellcycle.Themacroscopicpropertiesforagiventypeofcancercanchangedependingonthenumberofcancerouscellspresent,withvariationamonghowandwherethesepropertiesaredetermined.Thisleadstoconfusionintermsofwhichvaluestouseforparameterssincethereisalackofconsistentdataaboutthepropertiesforagivencancer.Forexample,Panetta[33]andreferencesthereinestimatethedoublingtimeforbreastcanceras14.65daysfromonegroupofstudiesand150daysfromanother.In[2]and[3],theintrinsicgrowthratesusedforaGompertziandynamicforleukemiahaveerrorboundsthatareanorderofmagnitudegreaterthantheestimateused,whichisassembledfromseveralsources.Clearly,thistypeofdiscrepancywouldleadtoradicallydifferenttreatmentmethodssincetheunderlyingcharacteristicsofthehowcancerwouldevolveisradicallydifferent.Twoofthemostcrucialconcernswiththeuseofchemotherapiesandradiotherapiesaretoxicityandresistance.Toxicitylimitsthedoseandfrequencybywhichthesetreatmentsmaybeadministered.Resistance,whetherintrinsicoracquired,limitstheeffectivenessofthetreatments.Therefore,iftherolesoftoxicityandresistancecanbeunderstoodleadingtoamodelwhichwouldrelatetheireffectstotheevolutionofthecancer,thenthisinformationcanbeusedtoselectappropriatetreatmentssoastominimizethespreadofcancerandresistancewhileadheringtotoxicitylimitsforagivenpatientwithagivencancer.Thispaperexaminestheevolutionofcancersubjecttotreatmentwithasinglecytotoxicagentanddrugresistance–toxicitywillnotbeexploredhere.InSection2,biologicalbackgroundandmotivationaregiventosupporttheformofthemodelspresentedinSection3.AnumericalstudyispresentedinSection4andadiscussionofthemodelandthenumericalresultsisinSection5.2Basicbiologicalcelldynamics,cancerevolution,andchemotherapyThecelllifecycleisshowninFigure1.Abriefdescriptionisgivenhere,andamoredetaileddiscussioncanbefoundin[11,16,21,37]andreferencestherein.ThecelllifecyclebeginsinthegapphaseinwhichproteinandRNAsynthesesareactive.Oncetheseprocessesarecomplete,thecellthenentersthesynthetic()phasewherenewDNAisgenerated.AfterthenewDNAisformed,theduplicatedchromosomescondenseinthegapphasewhilecyclin-mediatedactivitiescreateorganellesandmoleculesnecessaryforthenextandfinalphase,mitosis().Mitosisconsistsofasetofstepswhichleadstothedevelopmentoftwodaughtercells.InallofthesephasesthereareanumberofcheckpointsthatthecellmustpassthroughtoinsuretheintegrityoftheDNA.Shouldthecellfailtomeetthenecessaryrequirementsatthevariouscheckpoints,thecelleitherrepairsordestroysitself,aprocessreferredtoasapoptosisorprogrammedcelldeath.Shouldoneofthesecheckpointmechanismsfail,theresultcanbeam