A mathematical model of multistage hematopoietic cell lineages

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January 2014, 19(1): 1-26. doi: 10.3934/dcdsb.2014.19.1

A mathematical model of multistage hematopoietic cell lineages

Mostafa Adimy ^1, , Oscar Angulo ^2, , Catherine Marquet ^3, and Leila Sebaa ^4,

1.	INRIA Rhône-Alpes, Dracula team, Université Lyon 1, Institut Camille Jordan, UMR 5208, 43 Bd. du 11 novembre 1918, F-69200 Villeurbanne Cedex, France
2.	Departamento de Matemática Aplicada, ETSIT, Universidad de Valladolid, Pso. Belén 15, 47011 Valladolid, Spain
3.	Université de Pau, Laboratoire de Mathématiques Appliquées, CNRS UMR 5142, Avenue de l'université, 64000 Pau, France
4.	Laboratoire des systèmes dynamiques, Faculté de Mathématiques, USTHB, BP 32, El-Alia, Bab-Ezzouar, 16111 Alger, Algeria

Received October 2012 Revised July 2013 Published December 2013

Abstract
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We investigate a mathematical model of blood cell production in the bone marrow (hematopoiesis). The model describes both the evolution of primitive hematopoietic stem cells and the maturation of these cells as they differentiate to form the three kinds of progenitors and mature blood cells (red blood cells, white cells and platelets). The three types of progenitors and mature cells are coupled to each other via their common origin in primitive hematopoietic stem cells compartment. The resulting system is composed by eleven age-structured partial differential equations. To analyze this model, we don't take into account cell age-dependence of coefficients, that prevents a usual reduction of the structured system to an unstructured delay differential system. We study the existence of stationary solutions: trivial, axial and positive steady states. Then we give conditions for the local asymptotic stability of the trivial steady state and by using a Lyapunov function, we obtain a sufficient condition for its global asymptotic stability. In some particular cases, we analyze the local asymptotic stability of the positive steady state by using the characteristic equation. Finally, by numerical simulations, we illustrate our results and we show that a change in the duration of cell cycle can cause oscillations.

Keywords: delay differential system, Model of hematopoiesis, feedback control, age-structured partial differential equations, numerical simulations., Lyapunov functional, asymptotic stability.

Mathematics Subject Classification: 34D20, 34D23, 34K06, 92C3.

Citation: Mostafa Adimy, Oscar Angulo, Catherine Marquet, Leila Sebaa. A mathematical model of multistage hematopoietic cell lineages. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 1-26. doi: 10.3934/dcdsb.2014.19.1

References:

[1]	J. W. Adamson, Regulation of red blood cell production, Am. J. Med., 101 (1996), S4-S6. doi: 10.1016/S0002-9343(96)00160-X.
[2]	M. Adimy, O. Angulo, F. Crauste and J. C. Lopez-Marcos, Numerical integration of a mathematical model of hematopoietic stem cell dynamics, Computers & Mathematics with Applications, 56 (2008), 594-560. doi: 10.1016/j.camwa.2008.01.003.
[3]	M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis, 54 (2003), 1469-1491. doi: 10.1016/S0362-546X(03)00197-4.
[4]	M. Adimy and F. Crauste, 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), 19-38. doi: 10.3934/dcdsb.2007.8.19.
[5]	M. Adimy and F. Crauste, Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulation, Mathematical and Computer Modelling, 49 (2009), 2128-2137. doi: 10.1016/j.mcm.2008.07.014.
[6]	M. Adimy, F. Crauste and A.El Abdllaoui, Asymptotic behavior of a discrete maturity structured system of hematopoietic stem cell dynamics with several delays, Journal of Mathematical Modelling and Natural Phenomena, 1 (2006), 1-19. doi: 10.1051/mmnp:2008001.
[7]	M. Adimy, F. Crauste and A. El Abdllaoui, Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia, Journal of Biological Systems, 16 (2008), 395-424. doi: 10.1142/S0218339008002599.
[8]	M. Adimy, F. Crauste, H. Hbid and R. Qesmi, Stability and hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math, 70 (2010), 1611-1633. doi: 10.1137/080742713.
[9]	M. Adimy, F. Crauste and C. Marquet, Asymptotic behavior and stability switch for a mature-immature model of cell differentiation, Nonlinear Analysis: Real World Applications, 11 (2010), 2913-2929. doi: 10.1016/j.nonrwa.2009.11.001.
[10]	M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM J. Appl. Math., 65 (2005), 1328-1352. doi: 10.1137/040604698.
[11]	M. Adimy, F. Crauste and S. Ruan, Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 651-670. doi: 10.1016/j.nonrwa.2004.12.010.
[12]	M. Adimy, F. Crauste and S. Ruan, Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases, Bulletin of Mathematical Biology, 68 (2006), 2321-2351. doi: 10.1007/s11538-006-9121-9.
[13]	M. Adimy, F. Crauste and S. Ruan, Periodic oscillations in leukopoiesis models with two delays, Journal of Theoretical Biology, 242 (2006), 288-299. doi: 10.1016/j.jtbi.2006.02.020.
[14]	M. Adimy and C. Marquet, On the stability of hematopoietic model with feedback control, Comptes Rendus Mathématique, 350 (2012), 173-176. doi: 10.1016/j.crma.2012.01.014.
[15]	M. Adimy and L. Pujo-Menjouet, Asymptotic behavior of a singular transport equation modelling cell division, Discret. Cont. Dyn. Sys. Ser. B, 3 (2003), 439-456 . doi: 10.3934/dcdsb.2003.3.439.
[16]	R. Apostu and M. C. Mackey, Understanding cyclical thrombocytopenia: A mathematical modeling approach, J. Theor. Biol., 251 (2008), 297-316. doi: 10.1016/j.jtbi.2007.11.029.
[17]	J. J. Batzel and F. Kappel, Time delay in physiological systems: Analyzing and modeling its impact, Math. Biosc., 234 (2011), 61-74. doi: 10.1016/j.mbs.2011.08.006.
[18]	A. Bauer, F. Tronche, O. Wessely, C. Kellendonk, H. M. Reichardt, P. Steinlein, G. Schutz and H. Beug, The glucocorticoid receptor is required for stress erythropoiesis, Genes. Dev., 13 (1999), 2996-3002. doi: 10.1101/gad.13.22.2996.
[19]	J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math. Biosci., 128 (1995), 317-346.
[20]	S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, J. Theor. Biol., 223 (2003), 283-298. doi: 10.1016/S0022-5193(03)00090-0.
[21]	S. Bernard, J. Bélair and M. C. Mackey, Bifurcation in a white-blood-cell production model, C. R. Biologies, 327 (2004), 201-210. doi: 10.1016/j.crvi.2003.05.005.
[22]	F. J. Burns and I. F Tannock, On the existence of a $G_{0}$ phase in the cell cycle, Cell Proliferation, 3 (1970), 321-334.
[23]	C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: 1. Periodic chronic myelogenous leukemia, J. Theor. Biol., 237 (2005), 117-132. doi: 10.1016/j.jtbi.2005.03.033.
[24]	C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: 2. Cyclical neutropenia, J. Theor. Biol., 237 (2005), 133-146. doi: 10.1016/j.jtbi.2005.03.034.
[25]	J. Dyson, R. Villella-Bressan and G. F. Webb, A nonlinear age and maturity structured model of population dynamics, I: Basic theory, J. Math. Anal. Appl., 242 (2000), 93-104. doi: 10.1006/jmaa.1999.6656.
[26]	J. Dyson, R. Villella-Bressan and G. F. Webb, A nonlinear age and maturity structured model of population dynamics, II: Chaos, J. Math. Anal. Appl., 242 (2000), 255-270. doi: 10.1006/jmaa.1999.6657.
[27]	C. Foley and M. C. Mackey, Dynamic hematological disease: A review, J. Math. Biol., 58 (2009), 285-322. doi: 10.1007/s00285-008-0165-3.
[28]	P. Fortin and M. C. Mackey, Periodic chronic myelogenous leukaemia: Spectral analysis of blood cell counts and a etiological implications, Br. J. Haematol., 104 (1999), 336-345.
[29]	A. Fowler and M. C. Mackey, Relaxation oscillations in a class of delay differential equations, SIAM J. Appl. Math., 63 (2002), 299-323. doi: 10.1137/S0036139901393512.
[30]	M. E. Gurtin and R. C. MacCamy, Nonlinear age-dependent population dynamics, Arch. Rat. Mech. Anal., 54 (1974), 281-300.
[31]	J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
[32]	N. D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Soc., 25 (1950), 226-232.
[33]	C. Haurie, D. C. Dale and M. C. Mackey, Cyclical neutropenia and other periodic hematological disorders: A review of mechanisms and mathematical models, Blood, 92 (1998), 2629-2640.
[34]	C. Haurie, R. Person, D. C. Dale and M. C. Mackey, Hematopoietic dynamics in grey collies, Exp. Hematol., 27 (1999), 1139-1148. doi: 10.1016/S0301-472X(99)00051-X.
[35]	C. Haurie, D. C. Dale and M. C. Mackey, 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), 401-409. doi: 10.1016/S0301-472X(98)00061-7.
[36]	K. Kaushansky, The molecular mechanisms that control thrombopoiesis, The Journal of Clinical Investigation, 115 (2005), 3339-3347. doi: 10.1172/JCI26674.
[37]	D. S. Krause, Regulation of hematopoietic stem cell fate, Oncogene, 21 (2002), 3262-3269. doi: 10.1038/sj.onc.1205316.
[38]	L. G. Lajtha, On DNA labeling in the study of the dynamics of bone marrow cell population, (Ed. F. Jr. Stohlman), The kinetics of Cellular Proliferation, Grune and Stratton, New York, 1959, 173-182.
[39]	J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, J. Theor. Biol., 270 (2011). doi: 10.1016/j.jtbi.2010.11.024.
[40]	M. C. Mackey, Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956.
[41]	M. C. Mackey, Periodic auto- immune hemolytic anemia: An induced dynamical disease, Bull. Math. Biol., 41 (1979), 829-834. doi: 10.1016/S0092-8240(79)80019-1.
[42]	M. C. Mackey and A. Rey, Transitions and kinematics of reaction-convection fronts in a cell population model, Physica D,, 80 (1995), 120-139.
[43]	M. C. Mackey and A. Rey, Propagation of population pulses and fronts in a cell replication problem: Non-locality and dependence on the initial function, Physica D, 86 (1995), 373-395.
[44]	J. M. Mahaffy, J. Bélair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependant delay, J. Theor. Biol., 190 (1998), 135-146. doi: 10.1006/jtbi.1997.0537.
[45]	J. G. Milton and M. C. Mackey, Periodic haematological diseases: mystical entities of dynamical disorders? J. R. Coll. Phys., 23 (1989), 236-241.
[46]	L. Pujo-Menjouet, S. Bernard and M. C. Mackey, Long period oscillations in a $G_{0}$ model of hematopoietic stem cells, SIAM J. Appl. Dyn. Syst., 4 (2005), 312-332. doi: 10.1137/030600473.
[47]	L. Pujo-Menjouet and M. C. Mackey, Contribution to the study of periodic chronic myelogenous leukemia, Comptes Rendus Biologies, 327 (2004), 235-244. doi: 10.1016/j.crvi.2003.05.004.
[48]	M. Z. Ratajczak, J. Ratajczak, W. Marlicz et al., Recombinant human thrombopoietin (TPO) stimulates erythropoiesis by inhibiting erythroid progenitor cell apoptosis, Br J. Haematol., 98 (1997), 8-17. doi: 10.1046/j.1365-2141.1997.1802997.x.
[49]	S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
[50]	M. Santillan, J. Bélair, J. M. Mahaffy and M. C. Mackey, Regulation of platelet production: The normal response to perturbation and cyclical platelet disease, J. Theor. Biol., 206 (2000), 585-603. doi: 10.1006/jtbi.2000.2149.
[51]	S. Tanimukai, T. Kimura, H. Sakabe 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), 1025-1033.
[52]	G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and textbook in Pure Appl. Math., 89, Marcel Dekker, New York, 1985.

show all references

References:

[1]	J. W. Adamson, Regulation of red blood cell production, Am. J. Med., 101 (1996), S4-S6. doi: 10.1016/S0002-9343(96)00160-X.
[2]	M. Adimy, O. Angulo, F. Crauste and J. C. Lopez-Marcos, Numerical integration of a mathematical model of hematopoietic stem cell dynamics, Computers & Mathematics with Applications, 56 (2008), 594-560. doi: 10.1016/j.camwa.2008.01.003.
[3]	M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis, 54 (2003), 1469-1491. doi: 10.1016/S0362-546X(03)00197-4.
[4]	M. Adimy and F. Crauste, 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), 19-38. doi: 10.3934/dcdsb.2007.8.19.
[5]	M. Adimy and F. Crauste, Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulation, Mathematical and Computer Modelling, 49 (2009), 2128-2137. doi: 10.1016/j.mcm.2008.07.014.
[6]	M. Adimy, F. Crauste and A.El Abdllaoui, Asymptotic behavior of a discrete maturity structured system of hematopoietic stem cell dynamics with several delays, Journal of Mathematical Modelling and Natural Phenomena, 1 (2006), 1-19. doi: 10.1051/mmnp:2008001.
[7]	M. Adimy, F. Crauste and A. El Abdllaoui, Discrete maturity-structured model of cell differentiation with applications to acute myelogenous leukemia, Journal of Biological Systems, 16 (2008), 395-424. doi: 10.1142/S0218339008002599.
[8]	M. Adimy, F. Crauste, H. Hbid and R. Qesmi, Stability and hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math, 70 (2010), 1611-1633. doi: 10.1137/080742713.
[9]	M. Adimy, F. Crauste and C. Marquet, Asymptotic behavior and stability switch for a mature-immature model of cell differentiation, Nonlinear Analysis: Real World Applications, 11 (2010), 2913-2929. doi: 10.1016/j.nonrwa.2009.11.001.
[10]	M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM J. Appl. Math., 65 (2005), 1328-1352. doi: 10.1137/040604698.
[11]	M. Adimy, F. Crauste and S. Ruan, Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 651-670. doi: 10.1016/j.nonrwa.2004.12.010.
[12]	M. Adimy, F. Crauste and S. Ruan, Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases, Bulletin of Mathematical Biology, 68 (2006), 2321-2351. doi: 10.1007/s11538-006-9121-9.
[13]	M. Adimy, F. Crauste and S. Ruan, Periodic oscillations in leukopoiesis models with two delays, Journal of Theoretical Biology, 242 (2006), 288-299. doi: 10.1016/j.jtbi.2006.02.020.
[14]	M. Adimy and C. Marquet, On the stability of hematopoietic model with feedback control, Comptes Rendus Mathématique, 350 (2012), 173-176. doi: 10.1016/j.crma.2012.01.014.
[15]	M. Adimy and L. Pujo-Menjouet, Asymptotic behavior of a singular transport equation modelling cell division, Discret. Cont. Dyn. Sys. Ser. B, 3 (2003), 439-456 . doi: 10.3934/dcdsb.2003.3.439.
[16]	R. Apostu and M. C. Mackey, Understanding cyclical thrombocytopenia: A mathematical modeling approach, J. Theor. Biol., 251 (2008), 297-316. doi: 10.1016/j.jtbi.2007.11.029.
[17]	J. J. Batzel and F. Kappel, Time delay in physiological systems: Analyzing and modeling its impact, Math. Biosc., 234 (2011), 61-74. doi: 10.1016/j.mbs.2011.08.006.
[18]	A. Bauer, F. Tronche, O. Wessely, C. Kellendonk, H. M. Reichardt, P. Steinlein, G. Schutz and H. Beug, The glucocorticoid receptor is required for stress erythropoiesis, Genes. Dev., 13 (1999), 2996-3002. doi: 10.1101/gad.13.22.2996.
[19]	J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math. Biosci., 128 (1995), 317-346.
[20]	S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, J. Theor. Biol., 223 (2003), 283-298. doi: 10.1016/S0022-5193(03)00090-0.
[21]	S. Bernard, J. Bélair and M. C. Mackey, Bifurcation in a white-blood-cell production model, C. R. Biologies, 327 (2004), 201-210. doi: 10.1016/j.crvi.2003.05.005.
[22]	F. J. Burns and I. F Tannock, On the existence of a $G_{0}$ phase in the cell cycle, Cell Proliferation, 3 (1970), 321-334.
[23]	C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: 1. Periodic chronic myelogenous leukemia, J. Theor. Biol., 237 (2005), 117-132. doi: 10.1016/j.jtbi.2005.03.033.
[24]	C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: 2. Cyclical neutropenia, J. Theor. Biol., 237 (2005), 133-146. doi: 10.1016/j.jtbi.2005.03.034.
[25]	J. Dyson, R. Villella-Bressan and G. F. Webb, A nonlinear age and maturity structured model of population dynamics, I: Basic theory, J. Math. Anal. Appl., 242 (2000), 93-104. doi: 10.1006/jmaa.1999.6656.
[26]	J. Dyson, R. Villella-Bressan and G. F. Webb, A nonlinear age and maturity structured model of population dynamics, II: Chaos, J. Math. Anal. Appl., 242 (2000), 255-270. doi: 10.1006/jmaa.1999.6657.
[27]	C. Foley and M. C. Mackey, Dynamic hematological disease: A review, J. Math. Biol., 58 (2009), 285-322. doi: 10.1007/s00285-008-0165-3.
[28]	P. Fortin and M. C. Mackey, Periodic chronic myelogenous leukaemia: Spectral analysis of blood cell counts and a etiological implications, Br. J. Haematol., 104 (1999), 336-345.
[29]	A. Fowler and M. C. Mackey, Relaxation oscillations in a class of delay differential equations, SIAM J. Appl. Math., 63 (2002), 299-323. doi: 10.1137/S0036139901393512.
[30]	M. E. Gurtin and R. C. MacCamy, Nonlinear age-dependent population dynamics, Arch. Rat. Mech. Anal., 54 (1974), 281-300.
[31]	J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
[32]	N. D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Soc., 25 (1950), 226-232.
[33]	C. Haurie, D. C. Dale and M. C. Mackey, Cyclical neutropenia and other periodic hematological disorders: A review of mechanisms and mathematical models, Blood, 92 (1998), 2629-2640.
[34]	C. Haurie, R. Person, D. C. Dale and M. C. Mackey, Hematopoietic dynamics in grey collies, Exp. Hematol., 27 (1999), 1139-1148. doi: 10.1016/S0301-472X(99)00051-X.
[35]	C. Haurie, D. C. Dale and M. C. Mackey, 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), 401-409. doi: 10.1016/S0301-472X(98)00061-7.
[36]	K. Kaushansky, The molecular mechanisms that control thrombopoiesis, The Journal of Clinical Investigation, 115 (2005), 3339-3347. doi: 10.1172/JCI26674.
[37]	D. S. Krause, Regulation of hematopoietic stem cell fate, Oncogene, 21 (2002), 3262-3269. doi: 10.1038/sj.onc.1205316.
[38]	L. G. Lajtha, On DNA labeling in the study of the dynamics of bone marrow cell population, (Ed. F. Jr. Stohlman), The kinetics of Cellular Proliferation, Grune and Stratton, New York, 1959, 173-182.
[39]	J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, J. Theor. Biol., 270 (2011). doi: 10.1016/j.jtbi.2010.11.024.
[40]	M. C. Mackey, Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956.
[41]	M. C. Mackey, Periodic auto- immune hemolytic anemia: An induced dynamical disease, Bull. Math. Biol., 41 (1979), 829-834. doi: 10.1016/S0092-8240(79)80019-1.
[42]	M. C. Mackey and A. Rey, Transitions and kinematics of reaction-convection fronts in a cell population model, Physica D,, 80 (1995), 120-139.
[43]	M. C. Mackey and A. Rey, Propagation of population pulses and fronts in a cell replication problem: Non-locality and dependence on the initial function, Physica D, 86 (1995), 373-395.
[44]	J. M. Mahaffy, J. Bélair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependant delay, J. Theor. Biol., 190 (1998), 135-146. doi: 10.1006/jtbi.1997.0537.
[45]	J. G. Milton and M. C. Mackey, Periodic haematological diseases: mystical entities of dynamical disorders? J. R. Coll. Phys., 23 (1989), 236-241.
[46]	L. Pujo-Menjouet, S. Bernard and M. C. Mackey, Long period oscillations in a $G_{0}$ model of hematopoietic stem cells, SIAM J. Appl. Dyn. Syst., 4 (2005), 312-332. doi: 10.1137/030600473.
[47]	L. Pujo-Menjouet and M. C. Mackey, Contribution to the study of periodic chronic myelogenous leukemia, Comptes Rendus Biologies, 327 (2004), 235-244. doi: 10.1016/j.crvi.2003.05.004.
[48]	M. Z. Ratajczak, J. Ratajczak, W. Marlicz et al., Recombinant human thrombopoietin (TPO) stimulates erythropoiesis by inhibiting erythroid progenitor cell apoptosis, Br J. Haematol., 98 (1997), 8-17. doi: 10.1046/j.1365-2141.1997.1802997.x.
[49]	S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
[50]	M. Santillan, J. Bélair, J. M. Mahaffy and M. C. Mackey, Regulation of platelet production: The normal response to perturbation and cyclical platelet disease, J. Theor. Biol., 206 (2000), 585-603. doi: 10.1006/jtbi.2000.2149.
[51]	S. Tanimukai, T. Kimura, H. Sakabe 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), 1025-1033.
[52]	G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and textbook in Pure Appl. Math., 89, Marcel Dekker, New York, 1985.

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