A Model for Megakaryopoiesis with State-Dependent Delay

Abstract

We analyze the stability of a system of differential equations with a threshold defined delay arising from a model for platelet production. We consider a maturity-structured population of megakaryocyte progenitors and an age-structured population of platelets, where the cytokine thrombopoietin (TPO) increases the maturation rate of progenitors. Using the quasi-steady-state approximation for TPO dynamics and the method of characteristics, partial differential equations are reduced to a system of two differential equations with a state-dependent delay accounting for the variable maturation rate. We start by introducing the model and proving the positivity and boundedness of the solutions. Then we use a change of variables to obtain an equivalent system of two differential equations with a constant delay, from which we prove existence and uniqueness of the solution. As linearization around the unique positive steady state yields a transcendental characteristic equation of third degree, we introduce the main result, a new framework for stability analysis on models with fixed delays. This framework is then used to describe the stability of the megakaryopoiesis with respect to its parameters. Finally, with parameters being obtained and estimated from data, we give an example in which oscillations appear when the death rate of progenitors is increased 10-fold.

Stability analysis of an equation with two delays and application to the production of platelets

“…The assumption of a regulation of platelet count relying on variable megakaryocyte production was recently studied by Boullu et al [6]. Using a non constant maturation rate for megakaryoblasts (cells from which megakaryocytes originate), this lead to a system of two delay-differential equations.…”

“…The objective of this paper is to study in detail the local stability of the equilibria of (1.1). Then, (1.1) is used as a simplification of the model presented in [6] to study the impact of an increased destruction of platelets on the stability, through the increase of the random death rate γ as well as through the decrease of survival time τ 2 . Indeed, both the increase of γ and the decrease of τ 2 may be induced by the presence of platelet-specific antibodies [12,19,22].…”

“…For B = 0, we divide one equality by the other to obtain 6) and when ω, A and τ are known, B is given by…”

“…Because the expressions of a ij involve C and D, we need to fix g such that the point (τ, B) corresponds to the crossing of the lower stability boundary seen on Figure 2 which corresponds to subcase VI of the "complex" case in Kuznetsov numbering scheme (see [17] p. 363). The Maple commands are available online 6 . Figure 4.1 represents the different phase portraits for the system (4.8) associated with different sectors of the (µ 1 , µ 2 ) plane.…”

“…As mentioned in the introduction, we now use the results obtained above on a model describing the production of platelets, in order to study the role of the two mechanisms for destruction (random and deterministic). This model may be seen as a simplification of the one given in [6]. Indeed, we represent the regulation by the platelet number not as a continuous action on the maturity speed of megakaryocyte progenitors, but more directly by a regulation of the rate of production of new megakaryocytes.…”