A phase field approach for bone remodeling based on a second-gradient model

Abstract

A mechanobiological model of bone remodeling is developed involving mineralization in a moving diffuse interface separating the marrow containing the bone cells responsible for the remodeling from the newly formed bone. A scalar phase field quantifies the degree of mineralization within the interface at the level of the bone microstructure, varying continuously between the nil lower value (no mineral) and unity for the fully mineralized phase corresponding to new bone. The field equations for the mechanical, chemical, and interfacial phenomena are written under the umbrella of thermodynamics of irreversible processes. A strain gradient model is developed to account for the impact of the underlying hierarchical microstructure on the effective response of bone. Second gradient terms are motivated by the high strain and stress concentrations close to defects, both at mesoscopic and microscopic scales. The combination of the balance equations for the microforce associated to the phase field and the kinetic equations lead to the Ginzburg–Landau equation for by the phase field with a source term accounting for the dissipative microforce.