A mathematical model of multistage hematopoietic cell lineages

Abstract

We investigate a mathematical model of blood cell production in the bone marrow (hematopoiesis). The model describes both the evolution of primitive hematopoietic stem cellsand the maturation of these cells as they differentiate to form the three types of blood cells (red blood cells, white cells and platelets). The primitive hematopoietic stem cells and the first generations of each line (progenitors) are able to self-renew, and can be either in a proliferating or in a resting phase (G0-phase). These properties are gradually lost while cells become more and more mature. The three types of progenitors and mature cells are coupled to each other via their common origin in primitive hematopoietic stem cells compartment. Peripheral control loops of primitive hematopoietic stem cells and progenitors as well as a local autoregulatory loop are considered in the model. The resulting system is composed by eleven age-structured partial differential equations. To analyze this model, we don’t take into account cell age-dependence of coefficients, that prevents a usual reduction of the structured system to an unstructured delay differential system. We investigate some fundamental properties of the solutions of this system, such as boundedness and positivity. We study the existence of stationary solutions: trivial, axial and positive steady states. Then we give conditions for the local asymptotic stability of the trivial steady state and by using a Lyapunov function, we obtain a sufficient condition for its global asymptotic stability. In some particular cases, we analyze the local asymptotic stability of the positive steady state by using the characteristic equation. Finally, by numerical simulations, we illustrate our results and we show that a change in the duration of cell cycle can cause oscillations. This can be related to observations of some periodical hematological disease such as chronic myelogenous leukemia, cyclical neutropenia, cyclical thrombocytopenia, etc.