A mixture-like model for tumor-immune system interactions.

We introduce a mathematical model based on mixture theory intended to describe the tumor-immune system interactions within the tumor microenvironment. The equations account for the geometry of the tumor expansion, and the displacement of the immune cells, driven by diffusion and chemotactic mechanisms. They also take into account the constraints in terms of nutrient and oxygen supply. The numerical investigations analyze the impact of the different modeling assumptions and parameters. Depending on the parameters, the model can reproduce elimination, equilibrium or escape phases and it identifies a critical role of oxygen/nutrient supply in shaping the tumor growth. In addition, antitumor immune cells are key factors in controlling tumor growth, maintaining an equilibrium while protumor cells favor escape and tumor expansion.

Multistage hematopoietic stem cell regulation in the mouse: A combined biological and mathematical approach.

We have reconciled steady-state and stress hematopoiesis in a single mathematical model based on murine in vivo experiments and with a focus on hematopoietic stem and progenitor cells. A phenylhydrazine stress was first applied to mice. A reduced cell number in each progenitor compartment was evidenced during the next 7 days through a drastic level of differentiation without proliferation, followed by a huge proliferative response in all compartments including long-term hematopoietic stem cells, before a return to normal levels. Data analysis led to the addition to the 6-compartment model, of time-dependent regulation that depended indirectly on the compartment sizes. The resulting model was finely calibrated using a stochastic optimization algorithm and could reproduce biological data in silico when applied to different stress conditions (bleeding, chemotherapy, HSC depletion). In conclusion, our multi-step and time-dependent model of immature hematopoiesis provides new avenues to a better understanding of both normal and pathological hematopoiesis.

Model-Based Assessment of the Role of Uneven Partitioning of Molecular Content on Heterogeneity and Regulation of Differentiation in CD8 T-Cell Immune Responses.

Activation of naive CD8 T-cells can lead to the generation of multiple effector and memory subsets. Multiple parameters associated with activation conditions are involved in generating this diversity that is associated with heterogeneous molecular contents of activated cells. Although naive cell polarisation upon antigenic stimulation and the resulting asymmetric division are known to be a major source of heterogeneity and cell fate regulation, the consequences of stochastic uneven partitioning of molecular content upon subsequent divisions remain unclear yet. Here we aim at studying the impact of uneven partitioning on molecular-content heterogeneity and then on the immune response dynamics at the cellular level. To do so, we introduce a multiscale mathematical model of the CD8 T-cell immune response in the lymph node. In the model, cells are described as agents evolving and interacting in a 2D environment while a set of differential equations, embedded in each cell, models the regulation of intra and extracellular proteins involved in cell differentiation. Based on the analysis of in silico data at the single cell level, we show that immune response dynamics can be explained by the molecular-content heterogeneity generated by uneven partitioning at cell division. In particular, uneven partitioning acts as a regulator of cell differentiation and induces the emergence of two coexisting sub-populations of cells exhibiting antagonistic fates. We show that the degree of unevenness of molecular partitioning, along all cell divisions, affects the outcome of the immune response and can promote the generation of memory cells.

Existence and stability of periodic solutions of an impulsive differential equation and application to CD8 T-cell differentiation.

Unequal partitioning of the molecular content at cell division has been shown to be a source of heterogeneity in a cell population. We propose to model this phenomenon with the help of a scalar, nonlinear impulsive differential equation (IDE). To study the effect of molecular partitioning at cell division on the effector/memory cell-fate decision in a CD8 T-cell lineage, we study an IDE describing the concentration of the protein Tbet in a CD8 T-cell, where impulses are associated to cell division. We discuss how the degree of asymmetry of molecular partitioning can affect the process of cell differentiation and the phenotypical heterogeneity of a cell population. We show that a moderate degree of asymmetry is necessary and sufficient to observe irreversible differentiation. We consider, in a second part, a general autonomous IDE with fixed times of impulse and a specific form of impulse function. We establish properties of the solutions of that equation, most of them obtained under the hypothesis that impulses occur periodically. In particular, we show how to investigate the existence of periodic solutions and their stability by studying the flow of an autonomous differential equation. Then we apply those properties to prove the results presented in the first part.