Periodic Oscillations in Leukopoiesis Models with Two Delays

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Periodic Oscillations in Leukopoiesis Models with Two Delays? Mostafa Adimya, Fabien Craustea, Shigui Ruanb Year 2006 a Laboratoire de Mathematiques Appliquees, UMR 5142, Universite de Pau et des Pays de l'Adour, Avenue de l'universite, 64000 Pau, France. ANUBIS project, INRIA–Futurs Emails: (M. Adimy), (F. Crauste) b Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA. Email: Abstract The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnor- malities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time re- quired to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions.

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PeriodicOscillationsinLeukopoiesisModelswithTwoDelays∗MostafaAdimya,FabienCraustea,ShiguiRuanbYear2006aLaboratoiredeMathe´matiquesApplique´es,UMR5142,Universite´dePauetdesPaysdel’Adour,Avenuedel’universite´,64000Pau,France.ANUBISproject,INRIA–FutursEmails:mostafa.adimy@univ-pau.fr(M.Adimy),fabien.crauste@univ-pau.fr(F.Crauste)bDepartmentofMathematics,UniversityofMiami,CoralGables,FL33124-4250,USA.Email:ruan@math.miami.eduAbstractThetermleukopoiesisdescribesprocessesleadingtotheproductionandregulationofwhitebloodcells.Itisbasedonstemcellsdiﬀerentiationandmayexhibitabnor-malitiesresultinginseverediseases,suchascyclicalneutropeniaandleukemias.Weconsideranonlinearsystemoftwoequations,describingtheevolutionofastemcellpopulationandtheresultingwhitebloodcellpopulation.Twodelaysappearinthismodeltodescribethecellcycledurationofthestemcellpopulationandthetimere-quiredtoproducewhitebloodcells.Weestablishsuﬃcientconditionsfortheasymptoticstabilityoftheuniquenontrivialpositivesteadystateofthemodelbyanalyzingrootsofaseconddegreeexponentialpolynomialcharacteristicequationwithdelay-dependentcoeﬃcients.WealsoprovetheexistenceofaHopfbifurcationwhichleadstoperiodicsolutions.Numericalsimulationsofthemodelwithparametervaluesreportedinthelit-eraturedemonstratethatperiodicoscillations(withshortandlongperiods)agreewithobservationsofcyclicalneutropeniainpatients.Keywords:Leukopoiesis,hematopoieticstemcells,delaydiﬀerentialequations,stability,Hopfbifurcation.1IntroductionTheprocessthatleadstotheproductionandregulationofbloodcellsiscalledhematopoiesis.Itconsistsofmechanismstriggeringdiﬀerentiationandmaturationofhematopoieticstem∗ToappearinJournalofTheoreticalBiology1

M.Adimy,F.Crauste,S.RuanLeukopoiesisModelswithTwoDelayscells.Locatedinthebonemarrow,hematopoieticstemcellsareundiﬀerentiatedcells,un-observabledirectly(eventhoughtheycanbetrackedbymarkers),withuniquecapacitiesofdiﬀerentiation(theabilitytoproducecellscommittedtooneofbloodcelltypes)andself-renewal(theabilitytoproduceanidenticalcellwiththesameproperties).Undertheactionofgrowthfactors(moleculesactinglikehormonesplayinganactivator/inhibitorrole),hematopoieticstemcellsproducediﬀerentiatedcellsthroughoutcelldivisionsuntilbloodcells(whitecells,redbloodcells,andplatelets)areformedandreadytoenterthebloodstream.Weareinterestedhereinleukopoiesis(seeFig.1),theprocessbywhichwhitebloodcells(alsoknownasleukocytes)areproduced,whichisasub-processofhematopoiesis.Likeotherbloodcells,whitecellsareoriginatedfromapoolofhematopoieticstemcells.UndertheactionofmainlyG-CSF(GranulocyteColonyStimulatingFactor),agrowthfactoronlyactingontheleukocyteline,hematopoieticstemcellsdiﬀerentiateinprogenitors(theso-calledCFU,ColonyFormingUnits),whichinturnwillproduceprecursorcellsafteraconsequentnumberofdivisions.Afterafewdivisionslate,leukocytesareformedandleavethebonemarrowtoenterthebloodstream.Duetothenumberofdivisionsandthequantityofcellsinvolvedinleukopoiesis(orinhematopoiesisingeneral),issuesmayariseatdiﬀerentcellularlevelsandsometimesresultindiseasesaﬀectingwhitecells.Amongawidevarietyofdiseasesaﬀectingleukocytes,cyclicalneutropeniaisofgreatinterest.Itischaracterizedbyaperiodicdecreaseinthecirculatingneutrophil(whitecells)numbers,fromnormaltolowvalues,sometimesbarelydetectable[21].Usuallyperiodsobservedvarybetween19and21days,butlongerperiodsupto46dayshavebeenreportedinsomepatients[22].Oscillationsofallleukocytetypes(otherthanneutrophils)havebeenobservedinpatientswithcyclicalneutropenia,usuallywiththesameperiods.Theunderstandingofcyclicalneutropeniahasbeengreatlyaidedbythediscoverythatacaninerace,thegreycollie,getsthiscongenitaldisease(withrathershorterperiods,intheorderof11to16days).Anotherseverediseaseofthewhitecellsischronicmyelogenousleukemia[19],acancerofleukocytes.Insomecases,chronicmyelogenousleukemiaalsoexhibitsperiodicoscillationsinallbloodcellcounts,withperiodsfallingintherangeof30to110days,butusuallyabout70-80days.Thesetwobloodcelldiseasesareknownasperiodichematologicaldiseases.Theybothoriginatefromthehematopoieticstemcellcompartment:amutationinonehematopoieticstemcelliscarriedonthroughsuccessivecelldivisionsandthediseaseappearsintheblood-stream.Mathematicalmodellingofhematopoieticstemcellsdynamicshasbeenextensivelystud-iedinthepast30years,withattemptstodeterminecausesleadingtoanumberofperiodichematologicaldiseases.In1978,Mackey[26]proposedamathematicalmodelofhematopoi-eticstemcellsdynamicsinspiredbytheworksofLajtha[24]andBurnsandTannock[12].Formedbyasystemoftwononlineardelaydiﬀerentialequations,wherethedelaydescribestheaveragecellcycleduration,thismodelstressedtheinﬂuenceofsomefactors(suchastheapoptoticrate,theintroductionrate,thecellcycleduration)playinganimportantroleintheappearanceofperiodicsolutions.ManyauthorsstudiedpropertiesofthemodelintroducedbyMackey[26]andothermodelsrelatedtothisone,inordertounderstandtheroleplayedbyhematopoieticstemcellsinthestabilityofhematopoiesis.WerefertoMackeyandRey[28,29,30],MackeyandRudnicki[31,32],Dysonetal.[15,16,17,18],Pujo-MenjouetandRudnicki[37],AdimyandPujo-Menjouet[7],AdimyandCrauste[1,2]andAdimyetal.[4]forstudiesofstructuredmodelsofhematopoiesis,thestructurebeingeitherage,maturityorage-maturity.Wewanttopointoutthatwedonotconsideranystructureinthemodelanalyzedinthispaper,butthesereferencescouldbeusedtoimprovethepresentwork.ThemodelofMackey[26]hasbeenrecentlyanalyzedbyPujo-MenjouetandMackey[36]2

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