Periodic oscillations in leukopoiesis models with two delays

Abstract

The term leukopoiesis describes processes leading to the production and regulation of white blood cells. It is based on stem cells differentiation and may exhibit abnormalities resulting in severe diseases, such as cyclical neutropenia and leukemias. We consider a nonlinear system of two equations, describing the evolution of a stem cell population and the resulting white blood cell population. Two delays appear in this model to describe the cell cycle duration of the stem cell population and the time required to produce white blood cells. We establish sufficient conditions for the asymptotic stability of the unique nontrivial positive steady state of the model by analyzing roots of a second degree exponential polynomial characteristic equation with delay-dependent coefficients. We also prove the existence of a Hopf bifurcation which leads to periodic solutions. Numerical simulations of the model with parameter values reported in the literature demonstrate that periodic oscillations (with short and long periods) agree with observations of cyclical neutropenia in patients.

Cell Modelling of Hematopoiesis

“…The most recent models in blood diseases in the literature are the so-called maturity-agetime structured models consisting of a system of partial differential equations describing instability of the cell population facing aplastic anemia [3,4,13,14]. Some other deterministic models using either systems of differential equations or age structured systems deal with cyclical neutropenia [5,6,16,17], thrombocytopenia [25,26], or chronic myelogenous leukemia [1,2,10,22,23,24]. It is important here to notice that most of the continuous models deal with density only while our approach in the paper is to consider a population of individual cells.…”

Periodic Solutions in a Mathematical Model for the Treatment of Chronic Myelogenous Leukemia

“…The treatment of CML with drugs active only on the mature neutrophil cells line was modeled using the model of lekopoiesis introduced in [5] and studied in more detail in [4] and also in [7] where a complete model of hematopoiesis is given with evolution of differentiated cells taken into account. Since the treatment is supposed to act only on mature white blood cells, this is the only equation that was considered for mature cells evolution.…”

Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling

“…Bernard et al [21] used the Mackey's model to study the existence of oscillations in cyclic neutropenia. Adimy et al [3][4][5][6][7][8][9][10][11][12] analyzed various versions of Mackey's model and investigated the effect of perturbations of cell cycle duration, the rate of apoptosis, differentiation rate, and reintroduction rate from the quiescent compartment to the proliferating one on the behavior of the cell population.…”

“…Several modifications of Mackey's model have been proposed, accounting for different nonlinearities [31,32] or more discrete delays [11,27,28] in order to describe the different hematopoietic lineages. Complexity increases in such models, yet they can roughly be analyzed in the same way.…”