Abstract
We propose a mathematical model to describe the evolution of hematopoietic stem cells (HSCs) and stromal cells in considering the bi-directional interaction between them. Cancerous cells are also taken into account in our model. HSCs are structured by a continuous phenotype characterising the population heterogeneity in a way relevant to the question at stake while stromal cells are structured by another continuous phenotype representing their capacity of support to HSCs. We then analyse the model in the framework of adaptive dynamics. More precisely, we study single Dirac mass steady states, their linear stability and we investigate the role of parameters in the model on the nature of the evolutionary stable distributions (ESDs) such as monomorphism, dimorphism and the uniqueness properties. We also study the dominant phenotypes by an asymptotic approach and we obtain the equation for dominant phenotypes. Numerical simulations are employed to illustrate our analytical results. In particular, we represent the case of the invasion of malignant cells as well as the case of co-existence of cancerous cells and healthy HSCs.
Highlights
Hematopoietic stem cells (HSCs), developing in the bone marrow, are immature cells that are precursors of all lineages of blood cells: red blood cells, white blood cells and megacaryocytes
We introduce a mathematical model for the interaction between hematopoietic stem cells (HSCs) and stromal cells with the aim to better understand the nature of the dialogue between them as well as their dynamics
We have introduced a mathematical model for the interaction between hematopoietic stem cells and their support cells
Summary
Hematopoietic stem cells (HSCs), developing in the bone marrow, are immature cells that are (the earliest in development) precursors of all lineages of blood cells: red blood cells, white blood cells and megacaryocytes (whose fragmentation gives rise to platelets). Mackey [18], inspired by Burns and Tannock [6] and Lajtha [15], have introduced a first mathematical model of the form of a system of delay differential equations for the dynamics of HSCs where the populations are divided into two groups (proliferating cells and quiescent cells) and the time delay corresponds to the proliferating phase duration. Our mathematical model and some notions in the framework of adaptive dynamics, in particular, evolutionary stable distributions (ESDs) are given in the remaining part of this section
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