Combined bone internal and external remodeling based on Eshelby stress

Models for the internal and external bone remodeling are developed in the framework of the thermodynamics of irreversible processes. Internal remodeling is essentially accounted for by an evolution of the internal bone density versus the trace of the Eshelby stress. The growing surface is endowed with a specific mechanical behavior elaborated from a surface potential, depending upon the elastic surface strain and the external normal vector. The surface remodeling velocity is related to the driving force for growth identified as the surface divergence of Eshelby stress. The developed formalism is applied to simulate bone internal and external remodeling for 2D geometries, showing the influence of external mechanical stimuli on the evolution of the external shape of bone. Simulations of the evolution of the shape and density of bone structures are carried out successively at mesoscopic, microscopic and macroscopic levels, the former representing cellular level simulations of the trabecular unit cell. At the macroscopic level, a linear elastic constitutive law for trabecular bone is derived, relying on the discrete homogenization approach, whereby the continuum parameters such as stiffness evolve with morphology as remodeling occurs within bone. The developed numerical platform is able to simulate combined bone internal and external remodeling at both trabecular and macroscopic scales.