A nonlinear micropolar continuum model with diffusion–reaction equation for remodeling of bone with trabecular lattice microarchitecture

We present a nonlinear micropolar continuum model that includes a diffusion–reaction equation for the remodeling of bone with a trabecular lattice microstructure. This micropolar formulation enables the modeling of bone’s adaptive response to mechanical stimuli. The remodeling process is described as a time evolution regulated by a feedback mechanism that adjusts both the orientation of the trabecular lattice microarchitecture, thanks to the micropolar formulation, and the mechanical properties, related to key morphological features such us bone density. The junctions between trabeculae are modeled as nodal points within the micropolar framework, incorporating the following energy densities: mechanical deformation, mechano-biologic contributions, and Rayleigh-type dissipation terms that control the time evolution of remodeling. We present the weak form of the energetic formulation, which includes the diffusion–reaction equation. The cases of isotropic and orthotropic material symmetries class are deduced from the general formulation. This weak formulation is adapted to the development of a computational model based on the use of the finite element method. At the end of this paper, we discuss aspects related to validation, model complexity, and possible clinical applicability.