Stability of the cell dynamics in acute myeloid leukemia

Abstract

International audienceIn this paper we analyze the global asymptotic stability of the trivial solution for a multi-stage maturity acute myeloid leukemia model. By employing the positivity of the corresponding nonlinear time-delay model, where the nonlin-earity is locally Lipschitz, we establish the global exponential stability under the same conditions that are necessary for the local exponential stability. The result is derived for the multi-stage case via a novel construction of linear Lyapunov functionals. In a simpler model of hematopoiesis (without fast self-renewal) our conditions guarantee also global exponential stability with a given decay rate. Moreover, in this simpler case the analysis of the PDE model is presented via novel Lyapunov functionals for the transport equations

Analysis of a model of dormancy in cancer as a state of coexistence between tumor and healthy stem cells

“…Based on numerous studies on the modeling and the analysis of blood cell dynamics 1 (including notably the works of Mackey et al [29], [21], [12], and, Adimy et al [1], [2], and also the works of Marciniak-Czochra et al [22], Özbay et al [26], Avila et al [4], Fridman et al [14], Djema et al [6], among others), we intend to make a more general study that may cover other types of SCs and CSCs in liquid or solid tumors. For that, we revisit the model proposed in [7], which describes the coexistence of healthy and mutated hematopoietic stem cells,and is in fact valid for a wide panel of stem cells.…”

Stability of a delay system coupled to a differential-difference system describing the coexistence of ordinary and mutated hematopoietic stem cells

“…In particular, it may result in the improvement of the delivery of drugs for patients suffering from blood disorders. Not surprisingly, many authors have been interested by the modeling and the analysis of hematopoiesis, including, [19], [5], [4], [1], [2], [20], [3], [8], [13] and [24], to name but a few. In the present contribution, to obtain a coupled model of ordinary and mutated cells, we are inspired by [3] and by the new form of fast self-renewing process, [1], where a subpopulation of cells is considered to be always active in the proliferating phase.…”

On Instability and Global Asymptotic Stability of Age-structured Distributed Delay System Describing Pathological Hematopoeisis

“…Ecently, different research communities have been interested in the process of blood cell production or hematopoiesis, where mathematical models are used to perform theoretical analysis that may improve understanding of the diseases and help providing efficient treatment strategies [1], [2]. Indeed, the cell dynamics in acute myeloid leukemia were modeled and their stability studied in several papers such as [2], [3], [4], [5], [6] (and the references therein). The goal of such studies is to better understand pathological hematopoiesis and define AML therapy approaches with less side effects unlike currently used treatment relying on heavy chemotherapy [1], [9].…”

“…It is usual to use the method of characteristics [2], [4], [5], [6], [12], [19], [20], [21], [22] to transform the nonlinear transport equations describing healthy or pathological hematopoiesis into distributeddelay nonlinear systems; ( see [8]). For instance, the PDE model of hematopoiesis with a fast self-renewal presented in [6] was used in [3] to generate a nonlinear time-delay model from which the authors derived some global asymptotic and regional exponential stability. Furthermore in [15], a Lyapunov functional was used to provide sufficient conditions that make the favorable equilibrium locally exponentially stable (and to estimate its basin of attraction).…”