Abstract
Under the action of growth factors, proliferating and nonproliferating hematopoietic stem cells differentiate and divide, so as to produce blood cells. Growth factors act at different levels in the differentiation process, and we consider their action on the mortality rate (apoptosis) of the proliferating cell population. We propose a mathematical model describing the evolution of a hematopoietic stem cell population under the action of growth factors. It consists of a system of two age-structured evolution equations modelling the dynamics of the stem cell population coupled with a delay differential equation describing the evolution of the growth factor concentration. We first reduce our system of three differential equations to a system of two nonlinear differential equations with two delays and a distributed delay. We investigate some positivity and boundedness properties of the solutions, as well as the existence of steady states. We then analyze the asymptotic stability of the two steady states by studying the characteristic equation with delay-dependent coefficients obtained while linearizing our system. We obtain necessary and sufficient conditions for the global stability of the steady state describing the cell population's dying out, using a Lyapunov function, and we prove the existence of periodic solutions about the other steady state through the existence of a Hopf bifurcation.
Highlights
Hematopoietic stem cells are undifferentiated cells, located in the bone marrow, with unique capacities of differentiation and self-renewal
We model the dynamics of hematopoietic stem cells with an age structured model describing the evolution of proliferating and nonproliferating stem cells, coupled with a delay differential equation describing the production of growth factors
In the previous sections, a mathematical model of stem cells dynamics, taking into account the action of growth factors on cell proliferation. This model is based on the models of Mackey [23], Mackey and Rudnicki [28] and Belair et al [9]. It consists of a system of two age structured partial differential equations, modelling the evolution of the stem cell population, coupled with a delay differential equation describing the production of growth factors
Summary
Hematopoietic stem cells are undifferentiated cells, located in the bone marrow, with unique capacities of differentiation (the ability to produce cells committed to one blood cell lineage: white cells, red blood cells or platelets) and self-renewal In 1995 and 1998, Belair et al [9] and Mahaffy et al [30] considered an age structured system of two equations, coupled with a differential equation, modelling the dynamics of hematopoietic stem cells under the action of growth factors They assumed that the introduction rate in the proliferating phase depended on the growth factor concentration and, applying their model to the production of red blood cells (with erythropoietin as a growth factor), they managed to model normal hematopoiesis but stressed some difficulties to describe pathological cases
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