Thèse de doctorat en Physique
Soutenue le 02-07-2014
à Paris 11 , dans le cadre de Ecole doctorale Sciences et Technologies de l'Information, des Télécommunications et des Systèmes (Orsay, Essonne) , en partenariat avec Laboratoire des signaux et systèmes (L2S) (Gif-sur-Yvette, Essonne) (laboratoire) .
Le président du jury était Hugues Mounier.
Acute Myeloid Leukemia : Modelling and Stability Analysis
Acute Myeloid Leukemia (AML) is a cancer of white cells characterized by a quick proliferation of immature cells, that invade the circulating blood and become more present than mature blood cells. This thesis is devoted to the study of two mathematical models of AML. In the first model studied, the cell dynamics are represented by PDE’s for the phases G₀, G₁, S, G₂ and M. We also consider a new phase called Ğ₀, between the exit of the M phase and the beginning of the G₁ phase, which models the fast self-renewal effect of cancerous cells. Then, by analyzing the solutions of these PDE’s, the model has been transformed into a form of two coupled nonlinear systems involving distributed delays. An equilibrium analysis is done, the characteristic equation for the linearized system is obtained and a stability analysis is performed. The second model that we propose deals with a coupled model for healthy and cancerous cells dynamics in AML consisting of two stages of maturation for cancerous cells and three stages of maturation for healthy cells. The cell dynamics are modelled by nonlinear partial differential equations. Applying the method of characteristics enable us to reduce the PDE model to a nonlinear distributed delay system. For an equilibrium point of interest, necessary and sufficient conditions of local asymptotic stability are given. Finally, we derive stability conditions for both mathematical models by using a Lyapunov approach for the systems of PDEs that describe the cell dynamics.
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