Bone fracture healing with magnesium: a computational model of immune modulation and regenerative dynamics in union and non-union cases

Magnesium (Mg²⁺)-based biomaterials have gained attention due to their biodegradability and ability to influence both the inflammatory response and osteogenic activity during bone regeneration. However, how these effects depend on the concentration and delivery profile of Mg²⁺ and how they vary across successful (union) and impaired (non-union) conditions, have not been systematically characterized. To address this, we built upon an established model in the literature and incorporated Mg²⁺-responsive mechanisms into a system of nonlinear differential equations representing the dynamics of macrophage polarization, cytokine signaling, mesenchymal stem cell behavior, and bone tissue formation. Fracture healing was simulated across a range of Mg²⁺ concentrations (0-20 mM) and delivery profiles (constant, burst, delayed) under both union and non-union conditions. The results revealed that healing is supported within a specific concentration range (1-5 mM), where Mg²⁺ promotes polarization toward regenerative, anti-inflammatory M₂ macrophage polarization, enhances progenitor cell activity, and accelerates tissue regeneration. Higher concentrations (≥10 mM) led to unfavorable immune responses and impaired cell function, though delayed-release profiles mitigated these effects while preserving the regenerative potential. Importantly, under non-union conditions, appropriate Mg²⁺ dosing and delayed delivery restored healing dynamics and initiated bone formation, demonstrating therapeutic promise in impaired bone healing scenarios. The model predictions aligned well with reported in vivo data, supporting the validity of the simulated outcomes. In summary, the in silico Mg²⁺ model offers a mechanistic framework to understand how local Mg²⁺ concentration and delivery influence the coordination between immune modulation and tissue repair during bone healing.