Abstract
A new mathematical model is presented to study the effects of macrophages on the bone fracture healing process. The model consists of a system of nonlinear ordinary differential equations that represents the interactions among classically and alternatively activated macrophages, mesenchymal stem cells, osteoblasts, and pro- and anti-inflammatory cytokines. A qualitative analysis of the model is performed to determine the equilibria and their corresponding stability properties. Numerical simulations are also presented to support the theoretical results, and to monitor the evolution of a broken bone for different types of fractures under various medical interventions. The model can be used to guide clinical experiments and to explore possible medical treatments that accelerate the bone fracture healing process, either by surgical interventions or drug administrations.
Highlights
Bone fractures are becoming a serious worldwide problem, due to their high frequency and surgical complications
The only source of activated macrophages, M1 and M2, is M0. Even though both phenotypes of activated macrophages have the ability to release both pro- and anti-inflammatory cytokines, it is assumed that only M1 deliver c1 and M2 deliver c2, as those are the major cytokines for each phenotype [38]
The bone fracture healing process for humans involves the same cells, cytokines, and qualitative dynamics, differing only in the number of cells, concentrations, and the length of time it takes for a full recovery [24]
Summary
Bone fractures are becoming a serious worldwide problem, due to their high frequency and surgical complications. In [11], a mathematical model, based on the interactions among macrophages, MSCs, and osteoblasts, was developed to study the regulatory effects of two generic pro- and anti-inflammatory cytokines during the early stages of bone fracture healing. Alternatively activated macrophages are characterized by the secretion of anti-inflammatory cytokines, such as IL-10, which increase their phagocytic activities, mitigate the inflammatory responses, promote growth, and accelerate fracture healing [2,3,15,16] This extension leads to a more realistic model, by incorporating the different phagocytic rates and the separate production of the pro- and anti-inflammatory cytokines by the two types of macrophages [15,18].
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