A Mathematical Model to Study the Fundamental Functions of Phagocytes and Inflammatory Cytokines During the Bone Fracture Healing Process
A mathematical model is presented to study the effects of phagocytes and inflammatory cytokines on bone fracture healing during the early stages of the process. The model incorporates the interactions among macrophages, mesenchymal stem cells, osteoblasts, inflammatory cytokines, and the cartilage and bone extracellular matrices. The resulting system of nonlinear ordinary differential equations is studied analytically and numerically. The stability analysis revealed that the excessive accumulation of phagocytes and inflammatory cytokines at the injury site can lead to unsuccessful fracture healing, while the numerical simulations showed that optimal healing depends on the abilities of phagocytes to efficiently engulf debris. A variety of numerical simulations are also presented to monitor the healing of a broken bone under different biological conditions, suggesting multiple possible ways to guide clinical experiments and factors that can be manipulated to achieve optimal outcomes.