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.

💡 Research Summary

The paper presents a mathematically rigorous framework for describing hematopoiesis—the production and regulation of blood cells—by explicitly incorporating the role of growth factors and the intrinsic time delay associated with the cell‑cycle. The authors formulate a system of three coupled delay differential equations (DDEs) whose state variables represent the populations of hematopoietic stem cells, progenitor cells, and mature blood cells. The nonlinear interaction terms model the feedback loop in which mature cells secrete growth factors that stimulate differentiation of stem cells, while the same growth factors are down‑regulated by the concentration of mature cells, creating a negative feedback mechanism.

A crucial biological feature captured by the model is the cell‑cycle duration, denoted τ, which represents the time required for a progenitor cell to complete division and for the resulting mature cell to enter the circulation. This delay is introduced as a discrete lag in the DDEs, reflecting experimental observations that the turnover time of blood cells is on the order of days. Linearizing the system around a positive steady‑state yields a characteristic equation of exponential‑polynomial form:

\