Absolute stability of a system with distributed delays modeling cell dynamics in leukemia

In this paper we consider a mathematical model proposed recently by Adimy et al. (2008) for studying the cell dynamics in Acute Myelogenous Leukemia (AML). By using the circle and Popov criteria, we derive absolute stability conditions for this nonlinear system with distributed delays. Connections with the earlier results on stability of the linearized model are also made. The results are illustrated with a numerical example and simulations. I. I NTRODUCTION In a recent work, [1], Adimyet al. have proposed a mathematical model of the cell dynamics in Acute Myelogenous Leukemia (AML). This is a nonlinear system with distributed delays, consisting of several compartments connected in series. Equilibrium conditions are studied and local stability analysis for different equilibria are done in [1], where sufficient conditions for local stability are obtained using earlier results of [2]. Later, in [9] a necessary and sufficient condition is obtained for a particular choice of the distributed delay kernel for the linearized system. In the present work we study absolute stability conditions of this nonlinear system. For delay systems with uncertain nonlinearity satisfying the sector condition, primary tools of stability analysis have been the circle criterion and the Popov criterion, [7], [12], [13], see also [3], [4], [8] and their references. Here, we show that the first compartment of the mathematical model proposed in [1] can be put within the framework of these methods. For a specific form of the distributed delay kernel H. Özbay is with INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France; on leave from Dept. of Electrical & Electronics Engineering, Bilkent University, Ankara TR-06800, Turkey, e-mail: hitay@bilkent.edu. tr H. Benjelloun is with INRIA Saclay -̂Ile-de-France, 4 rue J. Monod, 91893, Orsay Cedex, France, e-mail: Houda.Benjelloun@inria.fr C. Bonnet is with INRIA Saclay -̂ Ile-de-France, 4 rue J. Monod, 91893, Orsay Cedex, France, e-mail: Catherine.Bonnet@inria.fr J. Clairambault is with INRIA Paris-Rocquencourt, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France, and INSERM team U 776 “Biological Rhythms and Cancers”, Hôpital PaulBrousse, 14 Av. Paul-Vaillant-Couturier, F94807 Villejuif Cedex, e-mail: jean.clairambault@inria.fr (studied in [9]) we derive sufficient conditions for absolute stability using Popov criterion. II. PROBLEM STATEMENT In this paper we consider the first compartment of the system introduced in [1], that is described by the following equation ẋ(t) = −δx(t)− w(t) + 2L ∫ τ 0 ef(a)w(t − a)da (1) where w(t) = x(t) β(x(t)) (2) • x is the total density of the resting cells; • δ is the death rate in the resting phase; • γ is the death rate (apoptosis) in the proliferating phase; • β(·) is the re-introduction function from the resting subpopulation into the proliferative subpopulation; • L := (1 − K), with 0 ≤ K ≤ 1, whereK is the rate of proliferating cells that divide without differentiation, i.e. that remain the first compartment; • proliferating cells can divide between the moment they enter the proliferating phase and a maximal time τ > 0 (this is the distributed time delay); • f is the mitosis (cell division) probability density; note that mitosis occurs before the age limit τ , andf(a) ≥ 0 for all a ∈ [0, τ ] and ∫ τ 0 f(a)da = 1. A block diagram representation of the system defined above is shown in Figure 1, where the subsystem represented by the transfer functionG(s) is a system with distributed delay, whose impulse response is g(t) := ef(t) for t ∈ [0, τ ] andg(t) = 0 for t > τ , i.e.,