A Mathematical Study of the Hematopoiesis Process with Applications to Chronic

Fabien Crauste

Description

Niveau: Supérieur, Doctorat, Bac+8
A Mathematical Study of the Hematopoiesis Process with Applications to Chronic Myelogenous Leukemia Mostafa Adimy†, Fabien Crauste† and Shigui Ruan? Year 2004 †Laboratoire de Mathematiques Appliquees, FRE 2570, Universite de Pau et des Pays de l'Adour, Avenue de l'universite, 64000 Pau, France. E-mail: , ?Department of Mathematics, University of Miami, P. O. Box 249085, Coral Gables, FL 33124-4250, USA. E-mail: Abstract This paper is devoted to the analysis of a mathematical model of blood cells produc- tion in the bone marrow (hematopoiesis). The model is a system of two age-structured partial differential equations. Integrating these equations over the age, we obtain a sys- tem of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration. This system describes the evolution of the total cell popula- tions. By constructing a Lyapunov functional, it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium. It is also shown that the nontrivial equilibrium, the most biologically meaningful one, can become unstable via a Hopf bifurcation. Numerical simulations are carried out to illustrate the analytical re- sults. The study maybe helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases.

• general stability results


• blood cells


• maximal age ?


• can also


• cellular gain


• proliferating cells


• age


• rate ?


• stem cell

Sujets

• Blood cell
• Stem cell

Informations

• Publié par: mijec
• Nombre de lectures: 70
• Langue: English

Extrait

AMathematicalStudyoftheHematopoiesis ProcesswithApplicationstoChronic MyelogenousLeukemia MostafaAdimy † ,FabienCrauste † andShiguiRuan ∗ Year2004 † LaboratoiredeMathe´matiquesApplique´es,FRE2570, Universite´dePauetdesPaysdel’Adour, Avenuedel’universite´,64000Pau,France. E-mail:[e-mail],[e-mail] ∗ DepartmentofMathematics,UniversityofMiami, P.O.Box249085,CoralGables,FL33124-4250,USA. E-mail:[e-mail] Abstract Thispaperisdevotedtotheanalysisofamathematicalmodelofbloodcellsproduc- tioninthebonemarrow(hematopoiesis).Themodelisasystemoftwoage-structured partialdifferentialequations.Integratingtheseequationsovertheage,weobtainasys- temoftwononlineardifferentialequationswithdistributedtimedelaycorresponding tothecellcycleduration.Thissystemdescribestheevolutionofthetotalcellpopula- tions.ByconstructingaLyapunovfunctional,itisshownthatthetrivialequilibrium isgloballyasymptoticallystableifitistheonlyequilibrium.Itisalsoshownthatthe nontrivialequilibrium,themostbiologicallymeaningfulone,canbecomeunstableviaa Hopfbifurcation.Numericalsimulationsarecarriedouttoillustratetheanalyticalre- sults.Thestudymaybehelpfulinunderstandingtheconnectionbetweentherelatively shortcellcycledurationsandtherelativelylongperiodsofperipheralcelloscillations insomeperiodichematologicaldiseases. Keywords:Bloodcells,hematopoiesis,differentialequations,distributeddelay,asymptotic stability,Lyapunovfunctional,Hopfbifurcation. 1Introduction Cellularpopulationmodelshavebeeninvestigatedintensivelysincethe1960’s(see,forex- ample,Trucco[33,34],Nooney[25],Rubinow[28]andRubinowandLebowitz[29])andstill ∗ ResearchwaspartiallysupportedbytheNSERCofCanadaandtheCollegeofArtsandSciences attheUniversityofMiami. 1

M.Adimy,F.CrausteandS.RuanHematopoiesisModel interestalotofresearchers.Thisinterestisgreatlymotivated,ononehand,bythemedi- calapplicationsand,ontheotherhand,bythebiologicalphenomena(suchasoscillations, bifurcations,travelingwavesorchaos)observedinthesemodelsand,generallyspeaking,in thelivingworld(MackeyandGlass[19],MackeyandMilton[20]). Hematopoiesisistheprocessbywhichprimitivestemcellsproliferateanddifferentiateto producematurebloodcells.Itisdrivenbyhighlycoordinatedpatternsofgeneexpression undertheinfluenceofgrowthfactorsandhormones.Theregulationofhematopoiesisisabout theformationofbloodcellelementsinthebody.Whiteandredbloodcellsandplatelets areproducedinthebonemarrowfromwheretheyenterthebloodstream.Theprincipal factorstimulatingredbloodcellproductionisthehormoneproducedinthekidney,called erythropoietin.About90%oftheerythropoietinissecretedbyrenaltubularepithelialcells whenbloodisunabletodeliversufficientoxygen.Whenthelevelofoxygenintheblood decreasesthisleadstoareleaseofasubstance,whichinturncausesanincreaseinthe releaseofthebloodelementsfromthemarrow.Thereisafeedbackfromthebloodtothe bonemarrow.Abnormalitiesinthefeedbackareconsideredasmajorsuspectsincausing periodichematologicaldiseases,suchasauto-immunehemolyticanemia(Be´lairetal.[4]and Mahaffyetal.[23]),cyclicalneutropenia(Haurieetal.[14]),chronicmyelogenousleukemia (FowlerandMackey[12]andPujo-Menjouetetal.[26]),etc. Cellbiologistsclassifiedstemcellsasproliferatingcellsandrestingcells(alsocalledG 0 - cells)(seeMackey[16,17]).Proliferatingcellsarecommittedtoundergomitosisacertain timeaftertheirentranceintotheproliferatingphase.Mackeysupposedthatthistimeof cytokinesisisconstant,thatis,itisthesameforallcells.Mostofcommittedstemcells areintheproliferatingphase.TheG 0 -phase,whoseexistenceisknownduetotheworksof BurnsandTannock[8],isaquiescentstageinthecellulardevelopment.However,itisusually believedthat95%ofpluripotentstemcellsareintherestingphase.Restingcellscanexit randomlytoeitherentryintotheproliferatingphaseorbeirremediablylost.Proliferating cellscanalsobelostbyapoptosis(programmedcelldeath). ThemodelofMackey[16]hasbeennumericallystudiedbyMackeyandRey[21]and Crabbetal.[9].Computersimulationsshowedthatthereexiststrangebehaviorsofthestem cellpopulation,suchasoscillationsandbifurcations.Recently,Pujo-MenjouetandMackey [27]provedtheexistenceofaHopfbifurcationwhichcausesperiodicchronicmyelogenous leukemiaandshowedthegreatdependenceofthemodelontheparameters. Inthispaper,basedonthemodelofMackey[16],weproposeamoregeneralmodel ofhematopoiesis.Wetakeintoaccountthefactthatacellcyclehastwophases,that is,stemcellsinprocessareeitherinarestingphaseoractivelyproliferating.However, wedonotsupposethatallcellsdivideatthesameage,becausethishypothesisisnot biologicallyreasonable.Forexample,itisbelievedthatpluripotentstemcellsdividefaster thancommittedstemcells,whicharemorematurecells.Therearestrongevidences(see Bradfordetal.[7])thatindicatethattheageofcytokinesisτisdistributedonaninterval [τ,τ]withτ≥0.Hence,weshallassumethatτisdistributedwithadensityfsupportedon aninterval[τ,τ],with0≤τ<τ<+∞.Theresultingmodelisasystemoftwodifferential equationswithdistributeddelay.Asimplermodel,dealingwiththepluripotentstemcell populationbehavior,hasbeenstudiedbyAdimyetal.[1]. Someresultsaboutstabilityofdifferentialequationswithdistributeddelaycanbemen- tioned.In[6],Boesestudiedthestabilityofadifferentialequationwithgamma-distributed delay.Gammadistributionshavethepropertytosimplifythenatureofthedelayandthis situationisclosetotheonewithdiscretedelay.Anderson[2,3]showedstabilityresults linkedtothedifferentmoments(especiallytheexpectationandthevariance)ofthedistri- bution.Kuang[15]alsoobtainedgeneralstabilityresultsforsystemsofdelaydifferential equations.Morerecently,sufficientconditionsforthestabilityofdelaydifferentialequations 2