Modeling the effects of inflammation in bone fracture healing

Abstract

Abstract.A new mathematical model is presented to study the early inflammatory effects in bone healing. It consists of a system of nonlinear ordinary differential equations that represents the interactions among macrophages, mesenchymal stem cells, and osteoblasts. A qualitative analysis of the model is performed to determine the equilibria and their corresponding stability properties. A set of numerical simulations is performed to support the theoretical results. The model is also used to numerically monitor the evolution of a broken bone for different types of fractures and to explore possible treatments to accelerate bone healing by administrating anti-inflammatory drugs.

Modeling the Macrophage-Mediated Inflammation Involved in the Bone Fracture Healing Process

“…The most important effects of macrophages on bone fracture healing are observed during the inflammatory and repair phases of the healing process [11,16]. During the inflammatory phase, macrophages modulate and resolve the inflammation while, during the repair phase, macrophages provide an optimal environment for the cellular proliferation, differentiation, and tissue production.…”

“…It is further assumed that the repair process is governed by the production of m c and m b [30,37], whose final levels are used to classify the outcome of the bone healing process. Additionally, it is assumed that the debris D are proportional to the number of necrotic cells [11]. It is also assumed that unactivated macrophages M 0 do not release cytokines and do not engulf debris.…”

“…Recently, experimental and mathematical models have demonstrated that macrophages strongly regulate bone fracture healing [10,11]. Depletion of macrophages results in delayed bone formation [12].…”

“…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. To our knowledge, it was the first attempt to incorporate the macrophage interactions in the modeling of the bone fracture healing process.…”

“…In this paper, the mathematical model developed in [11] is extended, to separately incorporate the two different phenotypes of macrophages: Classically and alternatively activated macrophages, as they have distinct functions during the healing process [10,15,16]. Classically activated macrophages release high levels of pro-inflammatory cytokines, including TNF-α and IL-1β, which exhibit inhibitory and destructive properties in high concentrations [16,17].…”