A Mathematical Study of the Hematopoiesis Process with Applications to Chronic

-

English
26 pages
Obtenez un accès à la bibliothèque pour le consulter en ligne
En savoir plus

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

Informations

Publié par
Nombre de lectures 103
Langue English
Signaler un problème
AMathematicalStudyoftheHematopoiesisProcesswithApplicationstoChronicMyelogenousLeukemiaMostafaAdimy,FabienCrausteandShiguiRuanYear2004LaboratoiredeMathe´matiquesApplique´es,FRE2570,Universite´dePauetdesPaysdel’Adour,Avenuedel’universite´,64000Pau,France.E-mail:mostafa.adimy@univ-pau.fr,fabien.crauste@univ-pau.frDepartmentofMathematics,UniversityofMiami,P.O.Box249085,CoralGables,FL33124-4250,USA.E-mail:ruan@math.miami.eduAbstractThispaperisdevotedtotheanalysisofamathematicalmodelofbloodcellsproduc-tioninthebonemarrow(hematopoiesis).Themodelisasystemoftwoage-structuredpartialdifferentialequations.Integratingtheseequationsovertheage,weobtainasys-temoftwononlineardifferentialequationswithdistributedtimedelaycorrespondingtothecellcycleduration.Thissystemdescribestheevolutionofthetotalcellpopula-tions.ByconstructingaLyapunovfunctional,itisshownthatthetrivialequilibriumisgloballyasymptoticallystableifitistheonlyequilibrium.Itisalsoshownthatthenontrivialequilibrium,themostbiologicallymeaningfulone,canbecomeunstableviaaHopfbifurcation.Numericalsimulationsarecarriedouttoillustratetheanalyticalre-sults.Thestudymaybehelpfulinunderstandingtheconnectionbetweentherelativelyshortcellcycledurationsandtherelativelylongperiodsofperipheralcelloscillationsinsomeperiodichematologicaldiseases.Keywords:Bloodcells,hematopoiesis,differentialequations,distributeddelay,asymptoticstability,Lyapunovfunctional,Hopfbifurcation.1IntroductionCellularpopulationmodelshavebeeninvestigatedintensivelysincethe1960’s(see,forex-ample,Trucco[33,34],Nooney[25],Rubinow[28]andRubinowandLebowitz[29])andstillResearchwaspartiallysupportedbytheNSERCofCanadaandtheCollegeofArtsandSciencesattheUniversityofMiami.1
M.Adimy,F.CrausteandS.RuanHematopoiesisModelinterestalotofresearchers.Thisinterestisgreatlymotivated,ononehand,bythemedi-calapplicationsand,ontheotherhand,bythebiologicalphenomena(suchasoscillations,bifurcations,travelingwavesorchaos)observedinthesemodelsand,generallyspeaking,inthelivingworld(MackeyandGlass[19],MackeyandMilton[20]).Hematopoiesisistheprocessbywhichprimitivestemcellsproliferateanddifferentiatetoproducematurebloodcells.Itisdrivenbyhighlycoordinatedpatternsofgeneexpressionundertheinfluenceofgrowthfactorsandhormones.Theregulationofhematopoiesisisabouttheformationofbloodcellelementsinthebody.Whiteandredbloodcellsandplateletsareproducedinthebonemarrowfromwheretheyenterthebloodstream.Theprincipalfactorstimulatingredbloodcellproductionisthehormoneproducedinthekidney,callederythropoietin.About90%oftheerythropoietinissecretedbyrenaltubularepithelialcellswhenbloodisunabletodeliversufficientoxygen.Whenthelevelofoxygenintheblooddecreasesthisleadstoareleaseofasubstance,whichinturncausesanincreaseinthereleaseofthebloodelementsfromthemarrow.Thereisafeedbackfromthebloodtothebonemarrow.Abnormalitiesinthefeedbackareconsideredasmajorsuspectsincausingperiodichematologicaldiseases,suchasauto-immunehemolyticanemia(Be´lairetal.[4]andMahaffyetal.[23]),cyclicalneutropenia(Haurieetal.[14]),chronicmyelogenousleukemia(FowlerandMackey[12]andPujo-Menjouetetal.[26]),etc.Cellbiologistsclassifiedstemcellsasproliferatingcellsandrestingcells(alsocalledG0-cells)(seeMackey[16,17]).Proliferatingcellsarecommittedtoundergomitosisacertaintimeaftertheirentranceintotheproliferatingphase.Mackeysupposedthatthistimeofcytokinesisisconstant,thatis,itisthesameforallcells.Mostofcommittedstemcellsareintheproliferatingphase.TheG0-phase,whoseexistenceisknownduetotheworksofBurnsandTannock[8],isaquiescentstageinthecellulardevelopment.However,itisusuallybelievedthat95%ofpluripotentstemcellsareintherestingphase.Restingcellscanexitrandomlytoeitherentryintotheproliferatingphaseorbeirremediablylost.Proliferatingcellscanalsobelostbyapoptosis(programmedcelldeath).ThemodelofMackey[16]hasbeennumericallystudiedbyMackeyandRey[21]andCrabbetal.[9].Computersimulationsshowedthatthereexiststrangebehaviorsofthestemcellpopulation,suchasoscillationsandbifurcations.Recently,Pujo-MenjouetandMackey[27]provedtheexistenceofaHopfbifurcationwhichcausesperiodicchronicmyelogenousleukemiaandshowedthegreatdependenceofthemodelontheparameters.Inthispaper,basedonthemodelofMackey[16],weproposeamoregeneralmodelofhematopoiesis.Wetakeintoaccountthefactthatacellcyclehastwophases,thatis,stemcellsinprocessareeitherinarestingphaseoractivelyproliferating.However,wedonotsupposethatallcellsdivideatthesameage,becausethishypothesisisnotbiologicallyreasonable.Forexample,itisbelievedthatpluripotentstemcellsdividefasterthancommittedstemcells,whicharemorematurecells.Therearestrongevidences(seeBradfordetal.[7])thatindicatethattheageofcytokinesisτisdistributedonaninterval[τ,τ]withτ0.Hence,weshallassumethatτisdistributedwithadensityfsupportedonaninterval[τ,τ],with0τ<τ<+.Theresultingmodelisasystemoftwodifferentialequationswithdistributeddelay.Asimplermodel,dealingwiththepluripotentstemcellpopulationbehavior,hasbeenstudiedbyAdimyetal.[1].Someresultsaboutstabilityofdifferentialequationswithdistributeddelaycanbemen-tioned.In[6],Boesestudiedthestabilityofadifferentialequationwithgamma-distributeddelay.Gammadistributionshavethepropertytosimplifythenatureofthedelayandthissituationisclosetotheonewithdiscretedelay.Anderson[2,3]showedstabilityresultslinkedtothedifferentmoments(especiallytheexpectationandthevariance)ofthedistri-bution.Kuang[15]alsoobtainedgeneralstabilityresultsforsystemsofdelaydifferentialequations.Morerecently,sufficientconditionsforthestabilityofdelaydifferentialequations2