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

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

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  • 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.

    Mathematics Subject Classification: Primary: 34K11, 34K20; Secondary: 92D25.


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  • Figure 1.  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

    Figure 2.  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

    Figure 3.  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

    Figure 4.  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 $t>t_{2n-1}$)

    Figure 5.  $ {\bf{Simplified \ model \ of \ Megakaryopoiesis}}$

    Figure 6.  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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