Adaptive dynamics of hematopoietic stem cells and their supporting stroma: a model and mathematical analysis

Thanh Nam Nguyen; Jean Clairambault; Thierry Jaffredo; Benoît Perthame; Delphine Salort; Thanh Nam Nguyen; Jean Clairambault; Thierry Jaffredo; Benoît Perthame; Delphine Salort

doi:10.3934/mbe.2019243

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4818-4845. doi: 10.3934/mbe.2019243

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Adaptive dynamics of hematopoietic stem cells and their supporting stroma: a model and mathematical analysis

1.
Sorbonne Université, Université Paris-Diderot, CNRS, INRIA, Laboratoire Jacques-Louis Lions, F-75005 Paris, France
2.
Sorbonne Université, Institut de biologie Paris-Seine (IBPS), UMR 7238 CNRS Laboratoire de Biologie Computationnelle et Quantitative, F-75005 Paris, France
3.
Sorbonne Université, Institut de biologie Paris-Seine (IBPS), CNRS UMR7622, Inserm U1156, Laboratoire de biologie du développement, F-75005 Paris, France

Received: 22 December 2018 Accepted: 07 May 2019 Published: 29 May 2019

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.
- adaptive cell population dynamics,
- hematopoietic stem cells,
- stromal cells,
- leukemic stem cells,
- Dirac concentrations,
- asymptotic methods
Citation: Thanh Nam Nguyen, Jean Clairambault, Thierry Jaffredo, Benoît Perthame, Delphine Salort. Adaptive dynamics of hematopoietic stem cells and their supporting stroma: a model and mathematical analysis[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4818-4845. doi: 10.3934/mbe.2019243

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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.

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