We propose and analyze a mathematical model of hematopoietic stem cell dynamics, that takes two cell populations into account, an immature and a mature one. All cells are able to self-renew, and immature cells can be either in a proliferating or in a resting compartment. The resulting model is a system of age-structured partial differential equations, that reduces to a system of delay differential equations, with several distributed delays. We investigate the existence of positive and axial steady states for this system, and we obtain conditions for their stability. Numerically, we concentrate on the influence of variations in differentiation coefficients on the behavior of the system. In particular, we focus on applications to acute myelogenous leukemia, a cancer of white cells characterized by a quick proliferation of immature cells that invade the circulating blood. We show that a blocking of differentiation at an early stage of immature cell development can result in the over-expression of very immature cells, with respect to the mature cell population.

This 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.

Primary immune responses generate short-term effectors and long-term protective memory cells. The delineation of the genealogy linking naive, effector, and memory cells has been complicated by the lack of phenotypes discriminating effector from memory differentiation stages. Using transcriptomics and phenotypic analyses, we identify Bcl2 and Mki67 as a marker combination that enables the tracking of nascent memory cells within the effector phase. We then use a formal approach based on mathematical models describing the dynamics of population size evolution to test potential progeny links and demonstrate that most cells follow a linear naive→early effector→late effector→memory pathway. Moreover, our mathematical model allows long-term prediction of memory cell numbers from a few early experimental measurements. Our work thus provides a phenotypic means to identify effector and memory cells, as well as a mathematical framework to investigate their genealogy and to predict the outcome of immunization regimens in terms of memory cell numbers generated.

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CD8 T-cells are critical in controlling infection by intracellular pathogens. Upon encountering antigen presenting cells, T-cell receptor activation promotes the differentiation of naïve CD8 T-cells into strongly proliferating activated and effector stages. We propose a 2D-multiscale computational model to study the maturation of CD8 T-cells in a lymph node controlled by their molecular profile. A novel molecular pathway is presented and converted into an ordinary differential equation model, coupled with a cellular Potts model to describe cell-cell interactions. Key molecular players such as activated IL2 receptor and Tbet levels Computation 2014, 2 160 control the differentiation from naïve into activated and effector stages, respectively, while caspases and Fas-Fas ligand interactions control cell apoptosis. Coupling this molecular model to the cellular scale successfully reproduces qualitatively the evolution of total CD8 T-cell counts observed in mice lymph node, between Day 3 and 5.5 post-infection. Furthermore, this model allows us to make testable predictions of the evolution of the different CD8 T-cell stages.

The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.

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 and show that the distributed delay can destabilize the entire system. In particular, it is shown that a Hopf bifurcation can occur.

We propose a mathematical model describing the dynamics of a hematopoietic stem cell population. The method of characteristics reduces the age-structured model to a system of differential equations with a state-dependent delay. A detailed stability analysis is performed. A sufficient condition for the global asymptotic stability of the trivial steady state is obtained using a Lyapunov-Razumikhin function. A unique positive steady state is shown to appear through a transcritical bifurcation of the trivial steady state. The analysis of the positive steady state behavior, through the study of a first order exponential polynomial characteristic equation, concludes the existence of a Hopf bifurcation and gives criteria for stability switches. A numerical analysis confirms the results and stresses the role of each parameter involved in the system on the stability of the positive steady state.