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

Previous Article
DCDS-B Home
This Issue
Next Article
Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time

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

Mostafa Adimy ^1, , Abdennasser Chekroun ^1, and Tarik-Mohamed Touaoula ^2,

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

Abstract
Related Papers

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.

Keywords: Age-structured partial differential equations, hematopoietic stem cells., cell dynamic, delay differential-difference system, stability switch, Lyapunov function, Hopf bifurcation.

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

Citation: Mostafa Adimy, Abdennasser Chekroun, Tarik-Mohamed Touaoula. Age-structured and delay differential-difference model of hematopoietic stem cell dynamics. Discrete & 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. 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, 54 (2003), 1469. 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. 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. 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. 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.
[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. 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. 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. 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. 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. 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. 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. 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, (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. 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.
[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. 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. 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.
[21]	P. Vegh, J. Winckler and F. Melchers, Long-term "in vitro'' proliferating mouse hematopoietic progenitor cell lines,, Immunology Letters, 130 (2010), 32. 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.

show all references

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. 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, 54 (2003), 1469. 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. 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. 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. 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.
[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. 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. 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. 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. 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. 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. 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. 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, (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. 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.
[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. 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. 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.
[21]	P. Vegh, J. Winckler and F. Melchers, Long-term "in vitro'' proliferating mouse hematopoietic progenitor cell lines,, Immunology Letters, 130 (2010), 32. 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.

[1]	Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385
[2]	Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[3]	Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[4]	Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
[5]	Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577
[6]	Shaoli Wang, Jianhong Wu, Libin Rong. A note on the global properties of an age-structured viral dynamic model with multiple target cell populations. Mathematical Biosciences & Engineering, 2017, 14 (3) : 805-820. doi: 10.3934/mbe.2017044
[7]	Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295
[8]	Piotr Gwiazda, Karolina Kropielnicka, Anna Marciniak-Czochra. The Escalator Boxcar Train method for a system of age-structured equations. Networks & Heterogeneous Media, 2016, 11 (1) : 123-143. doi: 10.3934/nhm.2016.11.123
[9]	R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147
[10]	Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
[11]	Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[12]	Andrey B. Muravnik. On the Cauchy problem for differential-difference parabolic equations with high-order nonlocal terms of general kind. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 541-561. doi: 10.3934/dcds.2006.16.541
[13]	Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112
[14]	Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355
[15]	Lesia V. Baranovska. Pursuit differential-difference games with pure time-lag. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1021-1031. doi: 10.3934/dcdsb.2019004
[16]	Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[17]	Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[18]	Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control & Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231
[19]	Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111
[20]	Yicang Zhou, Zhien Ma. Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences & Engineering, 2009, 6 (2) : 409-425. doi: 10.3934/mbe.2009.6.409

2017 Impact Factor: 0.972

American Institute of Mathematical Sciences