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A mathematical model of multistage hematopoietic cell lineages
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 |
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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