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Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling

  • M. Adimy (a1) (a2) and F. Crauste (a1) (a2)


We illustrate the appearance of oscillating solutions in delay differential equations modeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising as consequences of a destabilization of the system, for instance through a Hopf bifurcation. Models of hematopoietic stem cell dynamics are considered for their abilities to describe periodic hematological diseases, such as chronic myelogenous leukemia and cyclical neutropenia. After a review of delay models exhibiting oscillations, we focus on three examples, describing different delays: a discrete delay, a continuous distributed delay, and a state-dependent delay. In each case, we show how the system can have oscillating solutions, and we characterize these solutions in terms of periods and amplitudes.


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[1] Adamson, J.W.. Regulation of red blood cell Production. Am. J. Med., 101 (1996), S4S6.
[2] Adimy, M., Crauste, F.. Global stability of a partial differential equation with distributed delay due to cellular replication. Nonlinear Analysis, 54 (2003), 14691491.
[3] Adimy, M., Crauste, F.. Modelling and asymptotic stability of a growth factor-dependent stem cells dynamics model with distributed delay. Discrete and Continuous Dynamical Systems Series B, 8 (2007), No. 1, 1938.
[4] Adimy, M., Crauste, F.. Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulation. Mathematical and Computer Modelling, 49 (2009), 21282137.
[5] Adimy, M., Crauste, F., El Abdllaoui, A.. Asymptotic Behavior of a Discrete Maturity Structured System of Hematopoietic Stem Cells Dynamics with Several Delays. Mathematical Modelling of Natural Phenomena, Vol 1 (2006), No. 2, 122.
[6] Adimy, M., Crauste, F., El Abdllaoui, A.. Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia. J. Biol. Syst., 16 (3) (2008), 395424.
[7] Adimy, M., Crauste, F., Hbid, M.L., Qesmi, R.. Stability and Hopf bifurcation for a cell population model with state-dependent delay. SIAM J. Appl. Math, 70 (5) (2010), 16111633.
[8] Adimy, M., Crauste, F., Marquet, C.. Asymptotic behavior and stability switch for a mature-immature model of cell differentiation. Nonlinear Analysis: Real World Applications, 11 (2010), 29132929.
[9] Adimy, M., Crauste, F., Ruan, S.. A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia. SIAM J. Appl. Math., 65 (2005), 13281352.
[10] Adimy, M., Crauste, F., Ruan, S.. Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics. Nonlinear Analysis: Real World Applications, 6 (2005), No. 4, 651670.
[11] Adimy, M., Crauste, F., Ruan, S.. Periodic Oscillations in Leukopoiesis Models with Two Delays. J. Theo. Biol., 242 (2006), 288299.
[12] Adimy, M., Crauste, F., Ruan, S.. Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases. Bulletin of Mathematical Biology, 68 (8) (2006), 23212351.
[13] Aiello, W., Freedman, H., Wu, J.. Analysis of a model representing stage-structured population growth with stage-dependent time delay. SIAM Journal of Applied Mathematics 52 (1992), 855869.
[14] an der Heiden, U.. Delays in physiological systems. J. Math. Biol. 8 (1979), 345364.
[15] Alarcon, T., Tindall, M.J.. Modelling Cell Growth and its Modulation of the G1/S Transition. Bull. Math. Biol., 69 (2007), 197214.
[16] Apostu, R., Mackey, M.C.. Understanding cyclical thrombocytopenia: a mathematical modeling approach. J. Theor. Biol., 251 (2008), 297316.
[17] Batzel, J.J., Kappel, F.. Time delay in physiological systems: Analyzing and modeling its impact. Math. Biosciences, 234 (2011), No. 2, 6174.
[18] Bélair, J., Mackey, M.C., Mahaffy, J.M.. Age-structured and two-delay models for erythropoiesis. Math. Biosci., 128 (1995), 317346.
[19] Beretta, E., Kuang, Y.. Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal., 33 (2002), 5, 11441165.No.
[20] Bernard, S., Belair, J., Mackey, M.C.. Sufficient conditions for stability of linear differential equations with distributed delay. Discrete Contin. Dyn. Syst. Ser. B., 1 (2001), 233256.
[21] Bernard, S., Bélair, J., Mackey, M.C.. Oscillations in cyclical neutropenia: new evidence based on mathematical modeling. J. Theor. Biol., 223 (2003), 283298.
[22] Bodnar, M., Bartłomiejczyk, A.. Stability of delay induced oscillations in gene expression of Hes1 protein model. Nonlinear Analysis: Real World Applications, 13 (2012), 22272239.
[23] Burns, F.J., Tannock, I.F.. On the existence of a G0 phase in the cell cycle. Cell Tissue Kinet., 19 (1970), 321334.
[24] Cheshier, S.H., Morrison, S. J., Liao, X., Weissman, I.L.. In vivo proliferation and cell cycle kinetics of long-term self-renewing hematopoietic stem cells. Proc. Natl. Acad. Sci. USA, 96 (1999), 31203125.
[25] Ciupe, M.S., Bivort, B.L., Bortz, D.M., Nelson, P.W.. Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models. Math Biosci. 200(1) 2006, 127.
[26] Colijn, C., Foley, C., Mackey, M.C.. G-CSF treatment of canine cyclical neutropenia: A comprehensive mathematical model. Exper. Hematol. (2007), 35, 898907.
[27] Colijn, C., Mackey, M.C.. A mathematical model of hematopoiesis – I. Periodic chronic myelogenous leukemia. J. Theor. Biol., 237 (2005), 117132.
[28] Colijn, C., Mackey, M.C.. A mathematical model of hematopoiesis – II. Cyclical neutropenia. J. Theor. Biol., 237 (2005), 133146.
[29] Cooke, L.. Stability analysis for a vector disease model. Rocky Mountain J. Math., 9 (1979), 3142.
[30] Coutts, A.S., Adams, C.J., La Thangue, N.B.. p53 ubiquitination by Mdm2: a never ending tail ? DNA Repair (Amst). 8 (2009), 48390.
[31] Crauste, F.. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Math. Bio. Eng., 3 (2006), No. 2, 325346.
[32] Crauste, F.. Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch. Mathematical Modeling of Natural Phenomena, 4 (2009), No. 2, 2847.
[33] F. Crauste. Stability and Hopf bifurcation for a first-order linear delay differential equation with distributed delay, in Complex Time Delay Systems (Ed. F. Atay), Springer, 1st edition, 320 p., ISBN: 978-3-642-02328-6 (2010).
[34] Crews, L.A., Jamieson, C.H.. Chronic myeloid leukemia stem cell biology. Curr Hematol Malig Rep., 7 (2012), No. 2, 125132.
[35] J.M. Cushing. Integrodifferential Equations and Delay Models in Population Dynamics. Springer-Verlag, Heidelberg, 1977.
[36] Dale, D.C., Bolyard, A.A., Aprikyan, A.. Cyclic neutropenia. Semin. Hematol., 39 (2002), 8994.
[37] Dale, D.C., Hammond, W.P.. Cyclic neutropenia: A clinical review. Blood Rev., 2 (1998), 178185.
[38] J. Dieudonné. Foundations of Modern Analysis. Academic Press, New-York, 1960.
[39] Foley, C., Bernard, S., Mackey, M.C.. Cost-effective G-CSF therapy strategies for cyclical neutropenia: Mathematical modelling based hypotheses. J. Theor. Biol. (2006), 238, 754763.
[40] Foley, C., Mackey, M.C.. Dynamic hematological disease: a review. J. Math. Biol., 58 (2009), 285322.
[41] Fowler, A.C., McGuinness, M.J.. A delay recruitment model of the cardiovascular control system. J. Math. Biol. 51 (2005), 508526.
[42] Fortin, P., Mackey, M.C.. Periodic chronic myelogenous leukaemia: spectral analysis of blood cell counts and a etiological implications. Br. J. Haematol., 104 (1999), 336345.
[43] Fowler, A., Mackey, M.C.. Relaxation oscillations in a class of delay differential equations. SIAM J. Appl. Math., 63 (2002), 299323.
[44] Fuss, H., Dubitzky, W., Downes, S., Kurth, M.J.. Mathematical models of cell cycle regulation. Brief Bioinform., 6 (2005), 163177.
[45] N. Geva-Zatorsky , N. Rosenfeld, S. Itzkovitz, R. Milo, A. Sigal, E. Dekel, T. Yarnitzky, Y. Liron, P. Polak, G. Lahav, U. Alon. Oscillations and variability in the p53 system. Mol Syst Biol (2006), 2.2006.0033.
[46] K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Population. Dynamics, Kluwer Academic, Dordrecht, 1992.
[47] Glass, L., Beuter, A., Larocque, D.. Time delays, oscillations, and chaos in physiological control systems. Mathematical Biosciences, 90 (1988), 111125.
[48] Guerry, D., Dale, D., Omine, D.C., Perry, S., Wolff, S.M.. Periodic hematopoiesis in human cyclic neutropenia. J Clin Invest. 52 (1973), 32203230.
[49] J. Hale, S.M. Verduyn Lunel. Introduction to functional differential equations. Applied Mathematical Sciences 99. Springer-Verlag, New York, 1993.
[50] Haupt, Y., Maya, R., Kazaz, A., Oren, M.. Mdm2 promotes the rapid degradation of p53. Nature 387 (1997), 296299.
[51] Haurie, C., Dale, D.C., Mackey, M.C.. Cyclical neutropenia and other periodic hematological disorders: A review of mechanisms and mathematical models. Blood, 92 (1998), 26292640.
[52] Haurie, C., Dale, D.C., Mackey, M.C.. Occurrence of periodic oscillations in the differential blood counts of congenital, idiopathic, and cyclical neutropenic patient before and during treatment with G-CSF. Exp. Hematol., 27 (1999), 401409.
[53] Haurie, C., Dale, D.C., Rudnicki, R., Mackey, M.C.. Modeling complex neutrophil dynamics in the grey collie. J Theor Biol. 204 (2000), 505519.
[54] Haurie, C., Person, R., Dale, D.C., Mackey, M.C.. Hematopoietic dynamics in grey collies. Exp. Hematol., 27 (1999), 11391148.
[55] Hayes, N.D.. Roots of the transcendental equation associated with a certain difference-differential equation. J. London Math. Soc., 25 (1950), 226232.
[56] Hearn, T., Haurie, C., Mackey, M.C.. Cyclical neutropenia and the peripheral control of white blood cell production. J. Theor. Biol. 192 (1998), 167181.
[57] Hirata, H., Yoshiura, S., Ohtsuka, T., Bessho, Y., Harada, T., Yoshikawa, K., Kageyama, R.. Oscillatory Expression of the bHLH Factor Hes1 Regulated by a Negative Feedback Loop. Science 298 (2002), 840843.
[58] Y. Kuang. Delay Differential Equations with Applications in Population Dynamics. Academic Press, INC., San Diego, CA (1993).
[59] L.G. Lajtha. On DNA labeling in the study of the dynamics of bone marrow cell populations, in: Stohlman, Jr., F. (Ed), The Kinetics of Cellular Proliferation, Grune and Stratton, New York (1959), 173–182.
[60] Lei, J., Mackey, M.C.. Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia. J. Theor. Biol., 270 (2011), 143153.
[61] Li, J., Kuang, Y., Mason, C.. Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two time delays. J. Theoret. Biol., 242 (2006), 722735.
[62] Longobardo, G.S., Cherniack, N.S., Fishman, A.P.. Cheyne–Stokes breathing produced by a model of the human respiratory system. J. Appl. Physiol. 21 (1966), 18391846.
[63] N. MacDonald. Time Lags in Biological Models. Springer-Verlag, Heidelberg, 1978.
[64] Mackey, M.C.. Unified hypothesis of the origin of aplastic anaemia and periodic hematopoiesis. Blood, 51 (1978), 941956.
[65] Mackey, M.C.. Periodic auto- immune hemolytic anemia: an induced dynamical disease. Bull. Math. Biol., 41 (1979), 829834.
[66] Mackey, M.C.. Cell kinetic status of haematopoietic stem cells. Cell Prolif., 34 (2001), 7183.
[67] Mahaffy, J.M., Bélair, J., Mackey, M.C.. Hematopoietic model with moving boundary condition and state dependant delay. J. Theor. Biol., 190 (1998), 135146.
[68] Mallet-Paret, J., Nussbaum, R.D., Paraskevopoulos, P.. Periodic solutions for functional differential equations with multiple state-dependent time lags. Topol. Methods Nonlinear Anal., 3 (1994), 101162.
[69] Milton, J.G., Mackey, M.C.. Periodic haematological diseases: mystical entities of dynamical disorders ? J.R. Coll. Phys., 23 (1989), 236241.
[70] Monk, N.A.M.. Oscillatory expression of Hes1, p53, and NF-k B driven by transcriptional time delays. Curr. Biol. 13 (2003), 14091413.
[71] Morley, A.. Periodic diseases, physiological rhythms and feedback control-a hypothesis. Aust. Ann. Med. 3 (1970), 244249.
[72] Morley, A., Baikie, A.G., Galton, D.A.G.. Cyclic leukocytosis as evidence for retention of normal homeostatic control in chronic granulocytic leukaemia. Lancet, 2 (1967), 13201322.
[73] A. Morley, E.A. King-Smith, F. Stohlman. The oscillatory nature of hemopoiesis. In: Stohlman, F. (Ed.), Hemopoietic Cellular Proliferation. Grune & Stratton, New York, (1969), 3–14.
[74] Nelson, P.W., Murray, J.D., Perelson, A.S.. A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci., 163 (2000), 201215.
[75] Pørksen, N., Hollingdal, M., Juhl, C., Butler, P., Veldhuis, J. D., Schmitz, O.. Pulsatile insulin secretion: Detection, regulation, and role in diabetes. Diabetes, 51 (2002), S245S254.
[76] Pujo-Menjouet, L., Bernard, S., Mackey, M.C.. Long period oscillations in a G0 model of hematopoietic stem cells. SIAM J. Appl. Dyn. Systems, 4 (2005), No. 2, 312332.
[77] Pujo-Menjouet, L., Mackey, M.C.. Contribution to the study of periodic chronic myelogenous leukemia. Comptes Rendus Biologies, 327 (2004), 235244.
[78] Ratajczak, M.Z., Ratajczak, J., Marlicz, W., et al. Recombinant human thrombopoietin (TPO) stimulates erythropoiesis by inhibiting erythroid progenitor cell apoptosis. Br J. Haematol., 98 (1997), 817.
[79] Santillan, M., Bélair, J., Mahaffy, J.M., Mackey, M.C.. Regulation of platelet production: The normal response to perturbation and cyclical platelet disease. J. Theor. Biol., 206 (2000), 585603.
[80] Smith, B.R.. Regulation of hematopoiesis. Yale J Biol Med., 63 (1990), No. 5, 371380.
[81] Smith, H.L.. Reduction of structured population models to threshold-type delay equations and functional differential equations: a case study. Math. Biosc., 113 (1993), 123.
[82] Sturis, J., Polonsky, K. S., Mosekilde, E., Van Cauter, E.. Computer model for mechanisms underlying ultradian oscillations of insulin and glucose. Am. J. Physiol., 260 (1991), E801E809.
[83] Sturrock, M., Terry, A.J., Xirodimas, D.P., Thompson, A.M., Chaplain, M.A.J.. Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways. J. Theor. Biol., 273 (2011), 1531.
[84] Tanimukai, S., Kimura, T., Sakabe, H. et al. Recombinant human c-Mpl ligand (thrombopoietin) not only acts on megakaryocyte progenitors, but also on erythroid and multipotential progenitors in vitro. Experimental Hematology, 25 (1997), 10251033.
[85] E. Terry, J. Marvel, C. Arpin, O. Gandrillon, F. Crauste. Mathematical Model of the primary CD8 T Cell Immune Response: Stability Analysis of a Nonlinear Age-Structured System. J. Math. Biol. (to appear).
[86] Tolic, I.M., Mosekilde, E., Sturis, J.. Modeling the insulin-glucose feedback system: The significance of pulsatile insulin secretion. J. Theoret. Biol., 207 (2000), 361375.
[87] J.J. Tyson, B. Novak. Regulation of the Eukaryotic Cell Cycle: Molecular Antagonism, Hysteresis, and Irreversible Transitions. J. theor. Biol., 210 (2001), pp. 249–263.
[88] Vainchenker, W.. Hématopoïèse et facteurs de croissance. Encycl. Med. Chir., Hematologie, 13000 (1991), M85.
[89] Walther, H.O.. The solution manifold and C1-smoothness of solution operators for differential equations with state dependent delay. J. Differential Eqs., 195 (2003), 4665.
[90] G.F. Webb. Theory of Nonlinear Age-Dependent Population Dynamics. Monographs and textbook in Pure Appl. Math., 89, Marcel Dekker, New York (1985).
[91] Weissman, I.L.. Stem cells: units of development, units of regeneration, and units in evolution. Cell, 100 (2002), 157168.


Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling

  • M. Adimy (a1) (a2) and F. Crauste (a1) (a2)


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