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Hématologie

Modelling the dynamics of normal and pathological haematopoiesis Volume 14, issue 5, Septembre-Octobre 2008

Illustrations

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To be able to predict quantitatively onco-haematological treatment outcomes, it is essential to accurately describe the qualitative dynamics of blood cell production, i.e. the time evolution properties for the cell populations — equilibrium, extinction, oscillation or unlimited growth. Here, we review and present examples of several cell population mathematical models for haematopoiesis that played an important role in the understanding of this problem. Thirty years ago, one of the first deterministic mathematical models of haematopoiesis was proposed in which a stem cell population is divided into a proliferating and a quiescent fraction [1]. Four parameters describe the cell cycle: the rate of entry into proliferation, the duration of the proliferative phase and the rate of loss (apoptosis) for the proliferative phase and for the quiescent phase (differentiation). It was shown that an increase in the rate of apoptosis was sufficient to explain the observed oscillations in reticulocyte numbers in haemolytic anaemia. Subsequent models were developed to explain particular dynamics in other blood diseases (reviewed in [5]). Despite early success, new problems, like those arising in onco-haematology, called for different modelling frameworks. Three current approaches are reviewed here. Stochastic models use the probabilistic nature of the fate of blood cells — death, differentiation, mutation — to characterize the structure of the haematopoietic system [16,17]. These models are useful to study the dynamics of rare events (mutations) occurring in small stem cell populations [19]. Spatial models of the bone marrow are being developed to help understanding the role of cell movement and competition for space. These spatial models can be reaction-diffusion equations [28] or multi-agent, individual-centered models [33]. Compartment models address specific questions related to haematotoxicity of various drugs, in the sense that they do not aim at explaining the origin or the dynamics of the diseases but at quantitatively predicting therapeutic outcomes of these drugs [20,21]. We chose to develop four examples of applications of mathematical models in haematopoiesis: periodic physiological haematopoiesis; chronic myeloid leukaemia; spatial competition within the bone marrow; and feedback regulation and self-renewal in the erythroid lineage after stress. These examples have been chosen amongst many others because they appear to be significant illustrations of the different models listed in the review.