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

Abstract

Abstract. We illustrate the appearance of oscillating solutions in delay differential equations modeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising as consequences of a destabilization of the system, for instance through a Hopf bifurcation. Models of hematopoietic stem cell dynamics are considered for their abilities to describe periodic hematological diseases, such as chronic myelogenous leukemia and cyclical neutropenia. After a review of delay models exhibiting oscillations, we focus on three examples, describing different delays: a discrete delay, a continuous distributed delay, and a state-dependent delay. In each case, we show how the system can have oscillating solutions, and we characterize these solutions in terms of periods and amplitudes.

A review of mathematical models for leukemia and lymphoma

“…Time delays have been added to account for events such as cell divisions (for instance [12], [13]) and the interactions between cancer and immune cells (for instance [14]). In [13], Adimy and Crauste present three delay differential equation (DDE) models of cycling and quiescent hematopoietic stem cells (HSCs), with constant, distributed, and state-dependent delays. These delays represent the time to complete one cell division.…”

“…All three models have been applied to studying periodic hematological diseases, which are characterized by oscillations in various blood cell populations. It is concluded that although all three models produce periodic solutions, the nature of the oscillations depends on the type of delay [13]. …”

“…The models mentioned above have been applied to several diseases within the leukemia and lymphoma families, including chronic myeloid leukemia (CML) [5, 17, 23, 26, 27, 28, 29], acute myeloid leukemia (AML) [12], T-cell lymphoma [18], and periodic hematological diseases [13, 15, 16, 30]. …”

Blood Cell Dynamics: Half of a Century of Modelling

“…Most of the published works of the beginning of this decade are in the continuity of the end of the 2000's: cyclical neutropenia treatment by G-CSF [181], [182], [71], [94], [301], CML study [139], [158], [224], [106], [48], [178], [257], AML study [275], [274], [47], granulopoiesis [262], [263], HSC study [234], [207], [277], [33], [276], [281], [218], [5], [18], [136], [74] (with two nice reviews dealing with treatments of hematological diseases [96]), [294], very few on erythropoiesis [90], [261], [68] megakaryopoiesis [269], cell fate analysis [219]. There were mathematical development of structured populations [70], [111], [67], [114] with delay equations [20], [88], [21], [15], [59], [4], …”

A comparison between the stability properties in a DDE model for leukemia and the modified fractional counterpart

“…An intermediate approach is the use of delay differential equations (DDEs) models, which exhibit more complex dynamical behaviors (see, for example [2]). …”