# American Institute of Mathematical Sciences

November  2015, 20(9): 2765-2791. doi: 10.3934/dcdsb.2015.20.2765

## Age-structured and delay differential-difference model of hematopoietic stem cell dynamics

 1 Inria, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, F-69200 Villeurbanne Cedex, France, France 2 Department of Mathematics, University Aboubekr Belkaid, Tlemcen, Algeria

Received  September 2014 Revised  June 2015 Published  September 2015

In this paper, we investigate a mathematical model of hematopoietic stem cell dynamics. We take two cell populations into account, quiescent and proliferating one, and we note the difference between dividing cells that enter directly to the quiescent phase and dividing cells that return to the proliferating phase to divide again. The resulting mathematical model is a system of two age-structured partial differential equations. By integrating this system over age and using the characteristics method, we reduce it to a delay differential-difference system, and we investigate the existence and stability of the steady states. We give sufficient conditions for boundedness and unboundedness properties for the solutions of this system. By constructing a Lyapunov function, the trivial steady state, describing cell's dying out, is proven to be globally asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state, the most biologically meaningful one, and the existence of a Hopf bifurcation allow the determination of a stability area, which is related to a delay-dependent characteristic equation. Numerical simulations illustrate our results on the asymptotic behavior of the steady states and show very rich dynamics of this model. This study may be helpful in understanding the uncontrolled proliferation of blood cells in some hematological disorders.
Citation: Mostafa Adimy, Abdennasser Chekroun, Tarik-Mohamed Touaoula. Age-structured and delay differential-difference model of hematopoietic stem cell dynamics. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2765-2791. doi: 10.3934/dcdsb.2015.20.2765
##### References:
 [1] M. Adimy, O. Angulo, C. Marquet and L. Sebaa, A mathematical model of multistage hematopoietic cell lineages, Discrete and Continuous Dynamical Systems - Series B, 19 (2014), 1-26. doi: 10.3934/dcdsb.2014.19.1. [2] M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis: Theory, Methods & Applications, 54 (2003), 1469-1491. doi: 10.1016/S0362-546X(03)00197-4. [3] M. Adimy and F. Crauste, Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay, Discrete and Continuous Dynamical Systems - Series B, 8 (2007), 19-38. doi: 10.3934/dcdsb.2007.8.19. [4] M. Adimy, F. Crauste, H. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM Journal on Applied Mathematics, 70 (2010), 1611-1633. doi: 10.1137/080742713. [5] M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM Journal on Applied Mathematics, 65 (2005), 1328-1352. doi: 10.1137/040604698. [6] J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math Biosci, 128 (1995), 317-346, URL http://www.sciencedirect.com/science/article/pii/002555649400078E. [7] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [8] S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, Journal of Theoretical Biology, 223 (2003), 283-298, URL http://www.ams.org/mathscinet-getitem?mr=2079467. doi: 10.1016/S0022-5193(03)00090-0. [9] S. Bernard, J. Bélair and M. C. Mackey, Bifurcations in a white-blood-cell production model, C. R. Biol., 327 (2004), 201-210, URL http://linkinghub.elsevier.com/retrieve/pii/S1631069104000381. doi: 10.1016/j.crvi.2003.05.005. [10] 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. doi: 10.1111/j.1365-2184.1970.tb00340.x. [11] C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia, Journal of Theoretical Biology, 237 (2005), 117-132. doi: 10.1016/j.jtbi.2005.03.033. [12] F. Ficara, M. J. Murphy, M. Lin and M. L. Cleary, Pbx1 regulates self-renewal of long-term hematopoietic stem cells by maintaining their quiescence, Cell Stem Cell, 2 (2008), 484-496, URL http://www.sciencedirect.com/science/article/pii/S1934590908001161. doi: 10.1016/j.stem.2008.03.004. [13] K. Gu and Y. Liu, Lyapunov-Krasovskii functional for uniform stability of coupled differential-functional equations, Automatica, 45 (2009), 798-804. doi: 10.1016/j.automatica.2008.10.024. [14] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer New York, 1993. doi: 10.1007/978-1-4612-4342-7. [15] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, San Diego, 1993. [16] J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, Journal of Theoretical Biology, 270 (2011), 143-153. doi: 10.1016/j.jtbi.2010.11.024. [17] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis., Blood, 51 (1978), 941-956. [18] 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. [19] L. Pujo-Menjouet and M. C. Mackey, Contribution to the study of periodic chronic myelogenous leukemia, C. R. Biol., 327 (2004), 235-244, URL http://linkinghub.elsevier.com/retrieve/pii/S1631069104000253. doi: 10.1016/j.crvi.2003.05.004. [20] H. Takizawa, R. R. Regoes, C. S. Boddupalli, S. Bonhoeffer and M. G. Manz, Dynamic variation in cycling of hematopoietic stem cells in steady state and inflammation, J. Exp. Med., 208 (2011), 273-284, URL http://jem.rupress.org/content/208/2/273.full. [21] P. Vegh, J. Winckler and F. Melchers, Long-term "in vitro'' proliferating mouse hematopoietic progenitor cell lines, Immunology Letters, 130 (2010), 32-35, URL http://www.sciencedirect.com/science/article/pii/S0165247810000544. doi: 10.1016/j.imlet.2010.02.001. [22] A. Wilson, E. Laurenti, G. Oser, R. C. van der Wath, W. Blanco-Bose, M. Jaworski, S. Offner, C. F. Dunant, L. Eshkind, E. Bockamp, P. Lió, H. R. MacDonald and A. Trumpp, Hematopoietic stem cells reversibly switch from dormancy to self-renewal during homeostasis and repair, Cell, 135 (2008), 1118-1129, URL http://www.sciencedirect.com/science/article/pii/S009286740801386X.

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##### References:
 [1] M. Adimy, O. Angulo, C. Marquet and L. Sebaa, A mathematical model of multistage hematopoietic cell lineages, Discrete and Continuous Dynamical Systems - Series B, 19 (2014), 1-26. doi: 10.3934/dcdsb.2014.19.1. [2] M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis: Theory, Methods & Applications, 54 (2003), 1469-1491. doi: 10.1016/S0362-546X(03)00197-4. [3] M. Adimy and F. Crauste, Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay, Discrete and Continuous Dynamical Systems - Series B, 8 (2007), 19-38. doi: 10.3934/dcdsb.2007.8.19. [4] M. Adimy, F. Crauste, H. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM Journal on Applied Mathematics, 70 (2010), 1611-1633. doi: 10.1137/080742713. [5] M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM Journal on Applied Mathematics, 65 (2005), 1328-1352. doi: 10.1137/040604698. [6] J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math Biosci, 128 (1995), 317-346, URL http://www.sciencedirect.com/science/article/pii/002555649400078E. [7] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. doi: 10.1137/S0036141000376086. [8] S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, Journal of Theoretical Biology, 223 (2003), 283-298, URL http://www.ams.org/mathscinet-getitem?mr=2079467. doi: 10.1016/S0022-5193(03)00090-0. [9] S. Bernard, J. Bélair and M. C. Mackey, Bifurcations in a white-blood-cell production model, C. R. Biol., 327 (2004), 201-210, URL http://linkinghub.elsevier.com/retrieve/pii/S1631069104000381. doi: 10.1016/j.crvi.2003.05.005. [10] 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. doi: 10.1111/j.1365-2184.1970.tb00340.x. [11] C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia, Journal of Theoretical Biology, 237 (2005), 117-132. doi: 10.1016/j.jtbi.2005.03.033. [12] F. Ficara, M. J. Murphy, M. Lin and M. L. Cleary, Pbx1 regulates self-renewal of long-term hematopoietic stem cells by maintaining their quiescence, Cell Stem Cell, 2 (2008), 484-496, URL http://www.sciencedirect.com/science/article/pii/S1934590908001161. doi: 10.1016/j.stem.2008.03.004. [13] K. Gu and Y. Liu, Lyapunov-Krasovskii functional for uniform stability of coupled differential-functional equations, Automatica, 45 (2009), 798-804. doi: 10.1016/j.automatica.2008.10.024. [14] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer New York, 1993. doi: 10.1007/978-1-4612-4342-7. [15] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Academic Press, San Diego, 1993. [16] J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, Journal of Theoretical Biology, 270 (2011), 143-153. doi: 10.1016/j.jtbi.2010.11.024. [17] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis., Blood, 51 (1978), 941-956. [18] 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. [19] L. Pujo-Menjouet and M. C. Mackey, Contribution to the study of periodic chronic myelogenous leukemia, C. R. Biol., 327 (2004), 235-244, URL http://linkinghub.elsevier.com/retrieve/pii/S1631069104000253. doi: 10.1016/j.crvi.2003.05.004. [20] H. Takizawa, R. R. Regoes, C. S. Boddupalli, S. Bonhoeffer and M. G. Manz, Dynamic variation in cycling of hematopoietic stem cells in steady state and inflammation, J. Exp. Med., 208 (2011), 273-284, URL http://jem.rupress.org/content/208/2/273.full. [21] P. Vegh, J. Winckler and F. Melchers, Long-term "in vitro'' proliferating mouse hematopoietic progenitor cell lines, Immunology Letters, 130 (2010), 32-35, URL http://www.sciencedirect.com/science/article/pii/S0165247810000544. doi: 10.1016/j.imlet.2010.02.001. [22] A. Wilson, E. Laurenti, G. Oser, R. C. van der Wath, W. Blanco-Bose, M. Jaworski, S. Offner, C. F. Dunant, L. Eshkind, E. Bockamp, P. Lió, H. R. MacDonald and A. Trumpp, Hematopoietic stem cells reversibly switch from dormancy to self-renewal during homeostasis and repair, Cell, 135 (2008), 1118-1129, URL http://www.sciencedirect.com/science/article/pii/S009286740801386X.
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