Dynamics of Bone Remodeling by Using Mathematical Model Under ABC Time-Fractional Derivative

Bone remodeling is a dynamic biological process that preserves bone strength and structure through the coordinated actions of osteoblasts, osteoclasts, osteocytes, and bone mass density. Traditional models based on ordinary differential equations often fail to capture the memory-dependent nature of these interactions. In this study, we propose a novel mathematical model of bone remodeling using the Atangana–Baleanu–Caputo fractional derivative, which accounts for the non-local and hereditary characteristics of biological systems. The model introduces fractional-order dynamics into a previously established ODE framework while maintaining the intrinsic symmetry between bone-forming and bone-resorbing mechanisms, as well as the balance mediated by porosity-related feedback. We establish the existence, uniqueness, and positivity of solutions, and analyze the equilibrium points and their global stability using a Lyapunov function. Numerical simulations under various fractional orders demonstrate symmetric convergence toward equilibrium across all biological variables. The results confirm that fractional-order modeling provides a more accurate and balanced representation of bone remodeling and reveal the underlying symmetry in the regulation of bone tissue. This work contributes to the growing use of fractional calculus in modeling physiological processes and highlights the importance of symmetry in both mathematical structure and biological behavior.