17 research outputs found
A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia
International audienceThis paper is devoted to the analysis of a mathematical model of blood cells production in the bone marrow (hematopoiesis). The model is a system of two age-structured partial differential equations. Integrating these equations over the age, we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration. This system describes the evolution of the total cell populations. By constructing a Lyapunov functional, it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium. It is also shown that the nontrivial equilibrium, the most biologically meaningful one, can become unstable via a Hopf bifurcation. Numerical simulations are carried out to illustrate the analytical results. The study maybe helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases
Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases
Hematopoiesis is a complex biological process that leads to the production
and regulation of blood cells. It is based upon differentiation of stem cells
under the action of growth factors. A mathematical approach of this process is
proposed to carry out explanation on some blood diseases, characterized by
oscillations in circulating blood cells. A system of three differential
equations with delay, corresponding to the cell cycle duration, is analyzed.
The existence of a Hopf bifurcation for a positive steady-state is obtained
through the study of an exponential polynomial characteristic equation with
delay-dependent coefficients. Numerical simulations show that long period
oscillations can be obtained in this model, corresponding to a destabilization
of the feedback regulation between blood cells and growth factors. This
stresses the localization of periodic hematological diseases in the feedback
loop
Dynamics of erythroid progenitors and erythroleukemia
International audienceThe paper is devoted to mathematical modelling of erythropoiesis, production of red blood cells in the bone marrow. We discuss intra-cellular regulatory networks which determine self-renewal and differentiation of erythroid progenitors. In the case of excessive self-renewal, immature cells can fill the bone marrow resulting in the development of leukemia. We introduce a parameter characterizing the strength of mutation. Depending on its value, leukemia will or will not develop. The simplest model of treatment of acute myeloid leukemia with chemotherapy allows us to determine the conditions of successful treatment or of its failure. We show that insufficient treatment can worsen the situation. In some cases curing may not be possible even without resistance to treatment. Modelling presented in this work is based on ordinary differential equations, reaction-diffusion systems and individual based approach
Stability Analysis of Cell Dynamics in Leukemia
Cataloged from PDF version of article.In order to better understand the dynamics of acute leukemia, and in particular to find theoretical conditions for the efficient delivery of drugs in acute myeloblastic leukemia, we investigate stability of a system modeling its cell dynamics. The overall system is a cascade connection of sub-systems consisting of distributed delays and static nonlinear feedbacks. Earlier results on local asymptotic stability are improved by the analysis of the linearized system around the positive equilibrium. For the nonlinear system, we derive stability conditions by using Popov, circle and nonlinear small gain criteria. The results are illustrated with numerical examples and simulations
Existence and asymptotics of traveling wave fronts for a coupled nonlocal diffusion and difference system with delay
In this paper, we consider a general study of a recent proposed hematopoietic stem cells model. This model is a combination of nonlocal diffusion equation and difference equation with delay. We deal with the properties of traveling waves for this system such as the existence and asymptotic behavior. By using the Schauder’s fixed point theorem combined with the method based on the construction of upper and lower solutions, we obtain the existence of traveling wave fronts for a speed c > c . The case c = c is studied by using a limit argument. We prove also that c is the critical value. We finally prove that the nonlocality increases the minimal wave speed