Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases

Hematopoiesis is a complex biological process that leads to the production and regulation of blood cells. It is based upon differentiation of stem cells under the action of growth factors. A mathematical approach of this process is proposed to carry out explanation on some blood diseases, characterized by oscillations in circulating blood cells. A system of three differential equations with delay, corresponding to the cell cycle duration, is analyzed. The existence of a Hopf bifurcation for a positive steady-state is obtained through the study of an exponential polynomial characteristic equation with delay-dependent coefficients. Numerical simulations show that long period oscillations can be obtained in this model, corresponding to a destabilization of the feedback regulation between blood cells and growth factors. This stresses the localization of periodic hematological diseases in the feedback loop

Relaxation oscillations in a class of delay-differential equations.

We study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics. Under certain circumstances the model exhibits self-sustained oscillations, with periods which can be significantly longer than the basic cell cycle time. We show that the long periods in the oscillations occur when the cell generation rate is small, and we provide an asymptotic analysis of the model in this case. This analysis bears a close resemblance to the analysis of relaxation oscillators (such as the Van der Pol oscillator), except that in our case the slow manifold is infinite dimensional. Despite this, a fairly complete analysis of the problem is possible

A mathematical model of chronic myelogenous leukemia

Chronic Myelogenous Leukemia (CML) is one of the most common types of leukemia. It is characterized by a chronic, seemingly stable steady state, which gives rise to oscillatory instability in the hematapoietic stem cell count. There are also many cases of CML which involve oscillations about a steady state during the chronic period (called Periodic Chronic Myelogenous Leukemia). Though instabilities are found frequently in many biological systems, it is rather unusual for the stem cell count in a patient with leukemia to be nonmonotonic over time. As such, the instability in CML is of tremendous interest to mathematical biologists. A more clear understanding of the dynamics of this disease might not only help with the development of treatments or a cure to CML, but it might also be a useful aid in determining what causes instability in other oscillatory diseases such as Cyclical Neutropenia. This paper's aim is to create a mathematical model of CML which might aid us in understanding the mechanism by which the chronic phase of the disease becomes unstable and reaches the acute phase

Theory of Robustness of Irreversible Differentiation in a Stem Cell System: Chaos hypothesis

Based on extensive study of a dynamical systems model of the development of a cell society, a novel theory for stem cell differentiation and its regulation is proposed as the ``chaos hypothesis''. Two fundamental features of stem cell systems - stochastic differentiation of stem cells and the robustness of a system due to regulation of this differentiation - are found to be general properties of a system of interacting cells exhibiting chaotic intra-cellular reaction dynamics and cell division, whose presence does not depend on the detail of the model. It is found that stem cells differentiate into other cell types stochastically due to a dynamical instability caused by cell-cell interactions, in a manner described by the Isologous Diversification theory. This developmental process is shown to be stable not only with respect to molecular fluctuations but also with respect to removal of cells. With this developmental process, the irreversible loss of multipotency accompanying the change from a stem cell to a differentiated cell is shown to be characterized by a decrease in the chemical diversity in the cell and of the complexity of the cellular dynamics. The relationship between the division speed and this loss of multipotency is also discussed. Using our model, some predictions that can be tested experimentally are made for a stem cell system.Comment: 31 pages, 10 figures, submitted to Jour. Theor. Bio

Investigating the role of the experimental protocol in phenylhydrazine-induced anemia on mice recovery

Producción CientíficaProduction of red blood cells involves growth-factor mediated regulation of erythroid progenitor apoptosis and self-renewal. During severe anemia, characterized by a strong fall of the hematocrit followed by a recovery phase, these controls allow a fast recovery of the hematocrit and survival of the organism. Using a mathematical model of stress erythropoiesis and an ad hoc numerical method, we investigate the respective roles of anemia-inducing phenylhydrazine injections and physiological regulation on the organism’s recovery. By explicitly modeling the experimental protocol, we show that it mostly characterizes the fall of the hematocrit following the anemia and its severeness, while physiological process regulation mainly controls the recovery. We confront our model and our conclusions to similar experiments inducing anemia and show the model’s ability to reproduce several protocols of phenylhydrazine-induced anemia. In particular, we establish a link between phenylhydrazine effect and the severeness of the anemia.Ministerio de Economía, Industria y Competitividad (project MTM2014-56022-C2-2-P

Asymptotic methods for delay equations.

Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006