# American Institute of Mathematical Sciences

June 2019, 24(6): 2417-2442. doi: 10.3934/dcdsb.2018259

## Oscillations and asymptotic convergence for a delay differential equation modeling platelet production

 1 Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne Cedex, France 2 Inria, Université de Lyon, Université Lyon 1, Institut Camille Jordan, 43 Bd. du 11 novembre 1918, F-69200 Villeurbanne Cedex, France

* Corresponding author: lois.boullu@inria.fr

Received  July 2017 Revised  April 2018 Published  October 2018

Fund Project: LB was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). Also, LB is supported by a grant of Région Rhône-Alpes and benefited of the help of the France Canada Research Fund, of the NSERC and of a support from MITACS

We analyze the existence of oscillating solutions and the asymptotic convergence for a nonlinear delay differential equation arising from the modeling of platelet production. We consider four different cell compartments corresponding to different cell maturity levels: stem cells, megakaryocytic progenitors, megakaryocytes, and platelets compartments, and the quantity of circulating thrombopoietin (TPO), a platelet regulation cytokine.

Our initial model consists in a nonlinear age-structured partial differential equation system, where each equation describes the dynamics of a single compartment. This system is reduced to a single nonlinear delay differential equation describing the dynamics of the platelet population, in which the delay accounts for a differentiation time.

After introducing the model, we prove the existence of a unique steady state for the delay differential equation. Then we determine necessary and sufficient conditions for the existence of oscillating solutions. Next we set up conditions to get local asymptotic stability and asymptotic convergence of this steady state. Finally we present a short analysis of the influence of the conditions at t < 0 on the proof for asymptotic convergence.

Citation: Loïs Boullu, Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. Oscillations and asymptotic convergence for a delay differential equation modeling platelet production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2417-2442. doi: 10.3934/dcdsb.2018259
##### References:
 [1] R. Apostu and M. C. Mackey, Understanding cyclical thrombocytopenia: A mathematical modeling approach, Journal of Theoretical Biology, 251 (2008), 297-316. doi: 10.1016/j.jtbi.2007.11.029. [2] J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two delay models for erythropoiesis, Math. Biosciences, 128 (1995), 317-346. [3] L. Berezansky, E. Braverman and L. Idels, Mackey-Glass model of hematopoiesis with monotone feedback revisited, Applied Mathematics and Computation, 219 (2013), 4892-4907. doi: 10.1016/j.amc.2012.10.052. [4] 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. [5] C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: Ⅱ. Cyclical neutropenia, Journal of Theoretical Biology, 237 (2005), 133-146. doi: 10.1016/j.jtbi.2005.03.034. [6] F. Crauste, Stability and Hopf bifurcation for a first-order delay differential equation with distributed delay, in Complex Time-Delay Systems (ed. F. M. Atay), Understanding Complex Systems, Springer, Berlin, 2010,263-296. doi: 10.1007/978-3-642-02329-3_8. [7] A. de Graaf, Thrombopoietin and hematopoietic stem cells, Cell Cycle, 10 (2011), 1582-1589. doi: 10.4161/cc.10.10.15619. [8] V. R. Deutsch and A. Tomer, Advances in megakaryocytopoiesis and thrombopoiesis: From bench to bedside, British Journal of Haematology, 161 (2013), 778-793. doi: 10.1111/bjh.12328. [9] J. Eller, I. Gyori, M. Zollei and F. Krizsa, Modelling thrombopoiesis regulation - Ⅰ Model description and simulation results, Comput. Math. Appl., 14 (1987), 841-848. doi: 10.1016/0898-1221(87)90233-1. [10] K. Gopalsamy, M. R. S. Kulenovic and G. Ladas, Oscillations and global attractivity in models of hematopoiesis, Journal of Dynamics and Differential Equations, 2 (1990), 117-132. doi: 10.1007/BF01057415. [11] K. Gopalsamy, S. I. Trofimchuk and N. R. Bantsur, A note on global attractivity in models of hematopoiesis, Ukrainian Mathematical Journal, 50 (1998), 3-12. doi: 10.1007/BF02514684. [12] I. Gyori and G. E. Ladas, Oscillation Theory of Delay Differential Equations: With Applications, Oxford mathematical monographs, Oxford University Press, 1991. doi: 10.1086/418288. [13] I. Gyori and S. I. Trofimchuk, On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation, Nonlinear Analysis: Theory, Methods & Applications, 48 (2002), 1033-1042. doi: 10.1016/S0362-546X(00)00232-7. [14] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer New York, 1993. doi: 10.1007/978-1-4612-4342-7. [15] A. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54. [16] K. Kaushansky, The molecular mechanisms that control thrombopoiesis, Journal of Clinical Investigation, 115 (2005), 3339-3347. doi: 10.1172/JCI26674. [17] K. Kaushansky, S. Lok, R. D. Holly, V. C. Broudy, N. Lin, M. C. Bailey, J. W. Forstrom, M. M. Buddle, P. J. Oort, F. S. Hagen, G. J. Roth, T. Papayannopoulou and D. C. Foster, Promotion of megakaryocyte progenitor expansion and differentiation by the c-Mpl ligand thrombopoietin, Nature, 369 (1994), 568-571. doi: 10.1038/369568a0. [18] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, no. 191 in Mathematics in science and engineering, Academic Press, 1993. [19] I. Kubiaczyk and S. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Mathematical and Computer Modelling, 35 (2002), 295-301. doi: 10.1016/S0895-7177(01)00166-2. [20] M. Kulenovic and G. Ladas, Linearized oscillations in population dynamics, Bulletin of Mathematical Biology, 49 (1987), 615-627. doi: 10.1007/BF02460139. [21] G. P. Langlois, M. Craig, A. R. Humphries, M. C. Mackey, J. M. Mahaffy, J. Bélair, T. Moulin, S. R. Sinclair and L. Wang, Normal and pathological dynamics of platelets in humans, Journal of Mathematical Biology, 75 (2017), 1411-1462. doi: 10.1007/s00285-017-1125-6. [22] J.-W. Li and S. S. Cheng, Remarks on a set of sufficient conditions for global attractivity in a model of hematopoiesis, Computers & Mathematics with Applications, 59 (2010), 2751-2755. doi: 10.1016/j.camwa.2010.01.043. [23] M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. [24] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956. [25] J. Mallet-Paret, Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315. doi: 10.1016/0022-0396(88)90157-X. [26] J. J. M. Oliveira, Asymptotic Stability for Population Models and Neural Networks with Delays, Ph.D thesis, Universidade de Lisboa, 2008. [27] L. Pang, M. J. Weiss and M. Poncz, Megakaryocyte biology and related disorders, The Journal of Clinical Investigation, 115 (2005), 3332-3338. doi: 10.1172/JCI26720. [28] S. R. Patel, The biogenesis of platelets from megakaryocyte proplatelets, Journal of Clinical Investigation, 115 (2005), 3348-3354. doi: 10.1172/JCI26891. [29] M. Santillan, J. M. Mahaffy, J. Bélair and M. C. Mackey, Regulation of platelet production: The normal response to perturbation and cyclical platelet disease, Journal of Theoretical Biology, 206 (2000), 585-603. doi: 10.1006/jtbi.2000.2149. [30] A. Schmitt, J. Guichard, J. M. Masse, N. Debili and E. M. Cramer, Of mice and men: Comparison of the ultrastructure of megakaryocytes and platelets, Experimental Hematology, 29 (2001), 1295-1302. doi: 10.1016/S0301-472X(01)00733-0. [31] R. Stoffel, A. Wiestner and R. C. Skoda, Thrombopoietin in thrombocytopenic mice: Evidence against regulation at the mRNA level and for a direct regulatory role of platelets, Blood, 87 (1996), 567-573. [32] J. L. Swinburne and C. Mackey, Cyclical thrombocytopenia: Characterization by spectral analysis and a review, Journal of Theoretical Medecine, 2 (2000), 81-91. doi: 10.1080/10273660008833039. [33] H.-O. Walther, The 2-dimensional attractor of $x'(t) = -μ x(t) + f(x(t-1))$, Memoirs of the American Mathematical Society, 113 (1995), vi+76 pp. doi: 10.1090/memo/0544. [34] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of a system of red blood cells, Mat. Stos., 6 (1976), 23-40. [35] Q. Wen, B. Goldenson and J. D. Crispino, Normal and malignant megakaryopoiesis, Expert Reviews in Molecular Medicine, 13 (2011), e32. doi: 10.1017/S1462399411002043. [36] H. E. Wichmann, M. D. Gerhardts, H. Spechtmeyer and R. Gross, A mathematical model of thrombopoiesis in rats, Cell and Tissue Kinetics, 12 (1979), 551-567. doi: 10.1111/j.1365-2184.1979.tb00176.x. [37] M. Yu and A. B. Cantor, Megakaryopoiesis and Thrombopoiesis: An Update on Cytokines and Lineage Surface Markers, in Platelets and Megakaryocytes (eds. J. M. Gibbins and M. P. Mahaut-Smith), vol. 788, Springer New York, 2011, 291-303. doi: 10.1007/978-1-61779-307-3_20. [38] A. Zaghrout, A. Ammar and M. M. A. El-Sheikh, Oscillations and global attractivity in delay differential equations of population dynamics, Applied Mathematics and Computation, 77 (1996), 195-204. doi: 10.1016/S0096-3003(95)00213-8.

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##### References:
 [1] R. Apostu and M. C. Mackey, Understanding cyclical thrombocytopenia: A mathematical modeling approach, Journal of Theoretical Biology, 251 (2008), 297-316. doi: 10.1016/j.jtbi.2007.11.029. [2] J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two delay models for erythropoiesis, Math. Biosciences, 128 (1995), 317-346. [3] L. Berezansky, E. Braverman and L. Idels, Mackey-Glass model of hematopoiesis with monotone feedback revisited, Applied Mathematics and Computation, 219 (2013), 4892-4907. doi: 10.1016/j.amc.2012.10.052. [4] 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. [5] C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: Ⅱ. Cyclical neutropenia, Journal of Theoretical Biology, 237 (2005), 133-146. doi: 10.1016/j.jtbi.2005.03.034. [6] F. Crauste, Stability and Hopf bifurcation for a first-order delay differential equation with distributed delay, in Complex Time-Delay Systems (ed. F. M. Atay), Understanding Complex Systems, Springer, Berlin, 2010,263-296. doi: 10.1007/978-3-642-02329-3_8. [7] A. de Graaf, Thrombopoietin and hematopoietic stem cells, Cell Cycle, 10 (2011), 1582-1589. doi: 10.4161/cc.10.10.15619. [8] V. R. Deutsch and A. Tomer, Advances in megakaryocytopoiesis and thrombopoiesis: From bench to bedside, British Journal of Haematology, 161 (2013), 778-793. doi: 10.1111/bjh.12328. [9] J. Eller, I. Gyori, M. Zollei and F. Krizsa, Modelling thrombopoiesis regulation - Ⅰ Model description and simulation results, Comput. Math. Appl., 14 (1987), 841-848. doi: 10.1016/0898-1221(87)90233-1. [10] K. Gopalsamy, M. R. S. Kulenovic and G. Ladas, Oscillations and global attractivity in models of hematopoiesis, Journal of Dynamics and Differential Equations, 2 (1990), 117-132. doi: 10.1007/BF01057415. [11] K. Gopalsamy, S. I. Trofimchuk and N. R. Bantsur, A note on global attractivity in models of hematopoiesis, Ukrainian Mathematical Journal, 50 (1998), 3-12. doi: 10.1007/BF02514684. [12] I. Gyori and G. E. Ladas, Oscillation Theory of Delay Differential Equations: With Applications, Oxford mathematical monographs, Oxford University Press, 1991. doi: 10.1086/418288. [13] I. Gyori and S. I. Trofimchuk, On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation, Nonlinear Analysis: Theory, Methods & Applications, 48 (2002), 1033-1042. doi: 10.1016/S0362-546X(00)00232-7. [14] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer New York, 1993. doi: 10.1007/978-1-4612-4342-7. [15] A. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations, Differential Equations Dynam. Systems, 11 (2003), 33-54. [16] K. Kaushansky, The molecular mechanisms that control thrombopoiesis, Journal of Clinical Investigation, 115 (2005), 3339-3347. doi: 10.1172/JCI26674. [17] K. Kaushansky, S. Lok, R. D. Holly, V. C. Broudy, N. Lin, M. C. Bailey, J. W. Forstrom, M. M. Buddle, P. J. Oort, F. S. Hagen, G. J. Roth, T. Papayannopoulou and D. C. Foster, Promotion of megakaryocyte progenitor expansion and differentiation by the c-Mpl ligand thrombopoietin, Nature, 369 (1994), 568-571. doi: 10.1038/369568a0. [18] Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, no. 191 in Mathematics in science and engineering, Academic Press, 1993. [19] I. Kubiaczyk and S. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Mathematical and Computer Modelling, 35 (2002), 295-301. doi: 10.1016/S0895-7177(01)00166-2. [20] M. Kulenovic and G. Ladas, Linearized oscillations in population dynamics, Bulletin of Mathematical Biology, 49 (1987), 615-627. doi: 10.1007/BF02460139. [21] G. P. Langlois, M. Craig, A. R. Humphries, M. C. Mackey, J. M. Mahaffy, J. Bélair, T. Moulin, S. R. Sinclair and L. Wang, Normal and pathological dynamics of platelets in humans, Journal of Mathematical Biology, 75 (2017), 1411-1462. doi: 10.1007/s00285-017-1125-6. [22] J.-W. Li and S. S. Cheng, Remarks on a set of sufficient conditions for global attractivity in a model of hematopoiesis, Computers & Mathematics with Applications, 59 (2010), 2751-2755. doi: 10.1016/j.camwa.2010.01.043. [23] M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. [24] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956. [25] J. Mallet-Paret, Morse decompositions for delay-differential equations, Journal of Differential Equations, 72 (1988), 270-315. doi: 10.1016/0022-0396(88)90157-X. [26] J. J. M. Oliveira, Asymptotic Stability for Population Models and Neural Networks with Delays, Ph.D thesis, Universidade de Lisboa, 2008. [27] L. Pang, M. J. Weiss and M. Poncz, Megakaryocyte biology and related disorders, The Journal of Clinical Investigation, 115 (2005), 3332-3338. doi: 10.1172/JCI26720. [28] S. R. Patel, The biogenesis of platelets from megakaryocyte proplatelets, Journal of Clinical Investigation, 115 (2005), 3348-3354. doi: 10.1172/JCI26891. [29] M. Santillan, J. M. Mahaffy, J. Bélair and M. C. Mackey, Regulation of platelet production: The normal response to perturbation and cyclical platelet disease, Journal of Theoretical Biology, 206 (2000), 585-603. doi: 10.1006/jtbi.2000.2149. [30] A. Schmitt, J. Guichard, J. M. Masse, N. Debili and E. M. Cramer, Of mice and men: Comparison of the ultrastructure of megakaryocytes and platelets, Experimental Hematology, 29 (2001), 1295-1302. doi: 10.1016/S0301-472X(01)00733-0. [31] R. Stoffel, A. Wiestner and R. C. Skoda, Thrombopoietin in thrombocytopenic mice: Evidence against regulation at the mRNA level and for a direct regulatory role of platelets, Blood, 87 (1996), 567-573. [32] J. L. Swinburne and C. Mackey, Cyclical thrombocytopenia: Characterization by spectral analysis and a review, Journal of Theoretical Medecine, 2 (2000), 81-91. doi: 10.1080/10273660008833039. [33] H.-O. Walther, The 2-dimensional attractor of $x'(t) = -μ x(t) + f(x(t-1))$, Memoirs of the American Mathematical Society, 113 (1995), vi+76 pp. doi: 10.1090/memo/0544. [34] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of a system of red blood cells, Mat. Stos., 6 (1976), 23-40. [35] Q. Wen, B. Goldenson and J. D. Crispino, Normal and malignant megakaryopoiesis, Expert Reviews in Molecular Medicine, 13 (2011), e32. doi: 10.1017/S1462399411002043. [36] H. E. Wichmann, M. D. Gerhardts, H. Spechtmeyer and R. Gross, A mathematical model of thrombopoiesis in rats, Cell and Tissue Kinetics, 12 (1979), 551-567. doi: 10.1111/j.1365-2184.1979.tb00176.x. [37] M. Yu and A. B. Cantor, Megakaryopoiesis and Thrombopoiesis: An Update on Cytokines and Lineage Surface Markers, in Platelets and Megakaryocytes (eds. J. M. Gibbins and M. P. Mahaut-Smith), vol. 788, Springer New York, 2011, 291-303. doi: 10.1007/978-1-61779-307-3_20. [38] A. Zaghrout, A. Ammar and M. M. A. El-Sheikh, Oscillations and global attractivity in delay differential equations of population dynamics, Applied Mathematics and Computation, 77 (1996), 195-204. doi: 10.1016/S0096-3003(95)00213-8.
Model of Megakaryopoiesis. The linear differentiation process, starting from HSC and ending with platelets, is positively regulated by TPO. The quantity of TPO is in turn modulated by the number of platelets: the more platelets, the less circulating TPO
Oscillations appear when $\alpha_A$ increases. As $\alpha_A$ (the maximum number of platelets that a megakaryocyte can shed, see Equation (6)) increases, $R = rqe^{r\left( \gamma +p\right) }-\frac{1}{e}$ becomes positive and $x$ (blue) starts to oscillate around $x^*$ (dashed red). Black marks are placed where $x(t)$ goes through $x^*$. (A) $\alpha_A = 5000, R = -0.0492$ and there are no oscillations. (C) $\alpha_A = 10000, R = 7.6863$ and there are oscillations. (B) $\alpha_A = 20000, R = 83$ and there are oscillations
Solutions of (18) with or without low initial slope. (Top) The solution goes through $x(t) = x^*$ after $t = r$, it meets the low initial slope criterion. (Bottom) The solution goes through $x(t) = x^*$ before $t = r$, it does not meet the low initial slope criterion
An example of sequences $(y_n)_{n\in \mathbb{N}}$ and $(z_n)_{n\in \mathbb{N}}$. The decreasing (resp. increasing) sequence $(z_n)_{n\in \mathbb{N}}$ (resp. $(y_n)_{n\in \mathbb{N}}$) bounds $x(t)$ for $t>t_{2n}^*$ (resp. for $t>t_{2n-1}$)
${\bf{Simplified \ model \ of \ Megakaryopoiesis}}$
Initial slope and initial conditions. Four solutions of the equation (44) (blue) where different initial conditions lead to different relative position for $\tau$, $r$ (dashed green) and the time $t_0$ when $P(t)$ crosses $P^*$ (dashed red).
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