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

Functional differential equations with unbounded delay in extrapolation spaces

International audienceWe study the existence, regularity and stability of solutions for nonlinear partial neutral functional differential equations with unbounded de-lay and a Hille-Yosida operator on a Banach space X. We consider two non-linear perturbations: the first one is a function taking its values in X and the second one is a function belonging to a space larger than X, an extrapolated space. We use the extrapolation techniques to prove the existence and regu-larity of solutions and we establish a linearization principle for the stability of the equilibria of our equation

Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics

We study a mathematical model describing the dynamics of a pluripotent stem cell population involved in the blood production process in the bone marrow. This model is a differential equation with a time delay. The delay describes the cell cycle duration and is uniformly distributed on an interval. We obtain stability conditions independent of the delay. We also show that the distributed delay can destabilize the entire system. In particularly, it is shown that Hopf bifurcations can occur

Modelling and Asymptotic Stability of a Growth Factor-Dependent Stem Cells Dynamics Model with Distributed Delay

21 pagesInternational audienceUnder 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

Bifurcation dans un modèle non-linéaire de production du sang

Les éditeurs des Rencontres du Non-Linéaires ont autorisé la publication du fichier éditeur sur HAL.International audienceNous étudions un modèle mathématique de production des cellules du sang décrit par une équation différentielle non-linéaire. La prise en compte de la longueur du cycle cellulaire introduit un retard naturel dans le modèle. Les échanges cellulaires entre la phase de prolifération et la phase de repos sont caractérisés par la présence d'un terme non-linéaire dans l'équation (fonction de Hill). Le modèle que nous étudions tient compte du caractère non constant du cycle cellulaire. Dans ce cas, nous obtenons un résultat de stabilité globale en utilisant une fonction de Lyapunov et nous parvenons aussi à montrer qu'une bifurcation de Hopf survient inévitablement dans le processus de production du sang. L'existence de solutions périodiques nous permet de décrire un grand nombre de maladies hématologiques, caractérisées par des oscillations de toutes les cellules sanguines (en particulier, la leucémie myéloïde chronique périodique)

Age-structured model of hematopoiesis dynamics with growth factor-dependent coefficients

International audienceWe propose and analyze an age-structured partial differential model for hematopoietic stem cell dynamics, in which proliferation, differentiation and apoptosis are regulated by growth factor concentrations. By integrating the age-structured system over the age and using the characteristics method, we reduce it to a delay differential system. We investigate the existence and stability of the steady states of the reduced delay differential system. By constructing a Lyapunov function, the trivial steady state, describing cell's dying out, is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state, the most biologically meaningful one, is analyzed using the characteristic equation. This study may be helpful in understanding the uncontrolled proliferation of blood cells in some hematological disorders

Un mod\`ele non-lin\'eaire de prolif\'eration cellulaire : extinction des cellules et invariance

This paper analyses a nonlinear age-maturity structured system which arises as a model of the blood cellular production in the bone marrow. The resulting model is a nonlinear first-order partial differential equation in which there is a distributed temporal delay and a retardation in the maturation variable. We prove that uniqueness of solutions depends only on small maturity cells (stem cells) and we give a result of invariance

Bohr-Neugebauer type theorem for some partial neutral functional differential equations

21 pagesIn this work, we study the existence of almost periodic solutions for some partial neutral functional differential equations. Using the variation of constants formulaand the spectral decomposition of the phase space developed in [6], we prove that the existence of an almost periodic solution is equivalent to the existence of a bounded solution on $\mathbb{R}^+$