Periodic Oscillations in Leukopoiesis Models with Two Delays

Fabien Craustea

Description

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.

• univ-pau


• blood cells


• has been


• dynamics has


• stem cells


• introduction rate


• blood cell diseases


• stem cell

Sujets

• Blood cell
• Has Been
• Célula madre
• Stem cell

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 26
• Langue: English

Extrait

PeriodicOscillationsinLeukopoiesisModels withTwoDelays ∗ MostafaAdimy a ,FabienCrauste a ,ShiguiRuan b Year2006 a LaboratoiredeMathe´matiquesApplique´es,UMR5142, Universite´dePauetdesPaysdel’Adour, Avenuedel’universite´,64000Pau,France. ANUBISproject,INRIA–Futurs Emails:[e-mail](M.Adimy),[e-mail](F.Crauste) b DepartmentofMathematics,UniversityofMiami, CoralGables,FL33124-4250,USA. Email:[e-mail] Abstract Thetermleukopoiesisdescribesprocessesleadingtotheproductionandregulation ofwhitebloodcells.Itisbasedonstemcellsdifferentiationandmayexhibitabnor- malitiesresultinginseverediseases,suchascyclicalneutropeniaandleukemias.We consideranonlinearsystemoftwoequations,describingtheevolutionofastemcell populationandtheresultingwhitebloodcellpopulation.Twodelaysappearinthis modeltodescribethecellcycledurationofthestemcellpopulationandthetimere- quiredtoproducewhitebloodcells.Weestablishsufficientconditionsfortheasymptotic stabilityoftheuniquenontrivialpositivesteadystateofthemodelbyanalyzingroots ofaseconddegreeexponentialpolynomialcharacteristicequationwithdelay-dependent coefficients.WealsoprovetheexistenceofaHopfbifurcationwhichleadstoperiodic solutions.Numericalsimulationsofthemodelwithparametervaluesreportedinthelit- eraturedemonstratethatperiodicoscillations(withshortandlongperiods)agreewith observationsofcyclicalneutropeniainpatients. Keywords:Leukopoiesis,hematopoieticstemcells,delaydifferentialequations,stability,Hopf bifurcation. 1Introduction Theprocessthatleadstotheproductionandregulationofbloodcellsiscalledhematopoiesis. Itconsistsofmechanismstriggeringdifferentiationandmaturationofhematopoieticstem ∗ ToappearinJournalofTheoreticalBiology 1

M.Adimy,F.Crauste,S.RuanLeukopoiesisModelswithTwoDelays cells.Locatedinthebonemarrow,hematopoieticstemcellsareundifferentiatedcells,un- observabledirectly(eventhoughtheycanbetrackedbymarkers),withuniquecapacities ofdifferentiation(theabilitytoproducecellscommittedtooneofbloodcelltypes)and self-renewal(theabilitytoproduceanidenticalcellwiththesameproperties).Underthe actionofgrowthfactors(moleculesactinglikehormonesplayinganactivator/inhibitorrole), hematopoieticstemcellsproducedifferentiatedcellsthroughoutcelldivisionsuntilbloodcells (whitecells,redbloodcells,andplatelets)areformedandreadytoenterthebloodstream. Weareinterestedhereinleukopoiesis(seeFig.1),theprocessbywhichwhitebloodcells (alsoknownasleukocytes)areproduced,whichisasub-processofhematopoiesis.Likeother bloodcells,whitecellsareoriginatedfromapoolofhematopoieticstemcells.Underthe actionofmainlyG-CSF(GranulocyteColonyStimulatingFactor),agrowthfactoronlyacting ontheleukocyteline,hematopoieticstemcellsdifferentiateinprogenitors(theso-calledCFU, ColonyFormingUnits),whichinturnwillproduceprecursorcellsafteraconsequentnumber ofdivisions.Afterafewdivisionslate,leukocytesareformedandleavethebonemarrowto enterthebloodstream. Duetothenumberofdivisionsandthequantityofcellsinvolvedinleukopoiesis(orin hematopoiesisingeneral),issuesmayariseatdifferentcellularlevelsandsometimesresultin diseasesaffectingwhitecells.Amongawidevarietyofdiseasesaffectingleukocytes,cyclical neutropeniaisofgreatinterest.Itischaracterizedbyaperiodicdecreaseinthecirculating neutrophil(whitecells)numbers,fromnormaltolowvalues,sometimesbarelydetectable [21].Usuallyperiodsobservedvarybetween19and21days,butlongerperiodsupto46 dayshavebeenreportedinsomepatients[22].Oscillationsofallleukocytetypes(otherthan neutrophils)havebeenobservedinpatientswithcyclicalneutropenia,usuallywiththesame periods.Theunderstandingofcyclicalneutropeniahasbeengreatlyaidedbythediscovery thatacaninerace,thegreycollie,getsthiscongenitaldisease(withrathershorterperiods, intheorderof11to16days). Anotherseverediseaseofthewhitecellsischronicmyelogenousleukemia[19],acancerof leukocytes.Insomecases,chronicmyelogenousleukemiaalsoexhibitsperiodicoscillations inallbloodcellcounts,withperiodsfallingintherangeof30to110days,butusuallyabout 70-80days. Thesetwobloodcelldiseasesareknownasperiodichematologicaldiseases.Theyboth originatefromthehematopoieticstemcellcompartment:amutationinonehematopoietic stemcelliscarriedonthroughsuccessivecelldivisionsandthediseaseappearsintheblood- stream. Mathematicalmodellingofhematopoieticstemcellsdynamicshasbeenextensivelystud- iedinthepast30years,withattemptstodeterminecausesleadingtoanumberofperiodic hematologicaldiseases.In1978,Mackey[26]proposedamathematicalmodelofhematopoi- eticstemcellsdynamicsinspiredbytheworksofLajtha[24]andBurnsandTannock[12]. Formedbyasystemoftwononlineardelaydifferentialequations,wherethedelaydescribes theaveragecellcycleduration,thismodelstressedtheinfluenceofsomefactors(suchasthe apoptoticrate,theintroductionrate,thecellcycleduration)playinganimportantroleinthe appearanceofperiodicsolutions.Manyauthorsstudiedpropertiesofthemodelintroduced byMackey[26]andothermodelsrelatedtothisone,inordertounderstandtheroleplayed byhematopoieticstemcellsinthestabilityofhematopoiesis.WerefertoMackeyandRey [28,29,30],MackeyandRudnicki[31,32],Dysonetal.[15,16,17,18],Pujo-Menjouetand Rudnicki[37],AdimyandPujo-Menjouet[7],AdimyandCrauste[1,2]andAdimyetal.[4] forstudiesofstructuredmodelsofhematopoiesis,thestructurebeingeitherage,maturity orage-maturity.Wewanttopointoutthatwedonotconsideranystructureinthemodel analyzedinthispaper,butthesereferencescouldbeusedtoimprovethepresentwork. ThemodelofMackey[26]hasbeenrecentlyanalyzedbyPujo-MenjouetandMackey[36] 2