A multiscale analytical approach for bone remodeling simulations: linking scales from collagen to trabeculae

Bone is a dynamic and hierarchical porous material whose spatial and temporal mechanical properties can vary considerably due to differences in its microstructure and due to remodeling. Hence, a multiscale analytical approach, which combines bone structural information at multiple scales to the remodeling cellular activities, could form an efficient, accurate and beneficial framework for the prognosis of changes in bone properties due to, e.g., bone diseases. In this study, an analytical formulation of bone remodeling integrated with multiscale micromechanical models is proposed to investigate the effects of structural changes at the nanometer level (collagen scale) on those at higher levels (tissue scale). Specific goals of this study are to derive a mechanical stimulus sensed by the osteocytes using a multiscale framework, to test the accuracy of the multiscale model for the prediction of bone density, and to demonstrate its multiscale capabilities by predicting changes in bone density due to changes occurring at the molecular level. At each different level, the bone composition was modeled as a two-phase material which made it possible to: (1) find a closed-form solution for the energy-based mechanical stimulus sensed by the osteocytes and (2) describe the anisotropic elastic properties at higher levels as a function of the stiffness of the elementary components (collagen, hydroxyapatite and water) at lower levels. The accuracy of the proposed multiscale model of bone remodeling was tested first by comparing the analytical bone volume fraction predictions to those obtained from the corresponding μFE-based computational model. Differences between analytical and numerical predictions were less than 1% while the computational time was drastically reduced, namely by a factor of 1 million. In a further analysis, the effects of changes in collagen and hydroxyapatite volume fractions on the bone remodeling process were simulated, and it was found that such changes considerably affect the bone density at the millimeter scale. In fact, smaller tissue density induces remodeling activities leading to finally higher overall bone density. The multiscale analytical model proposed in this study potentially provides an accurate and efficient tool for simulating patient-specific bone remodeling, which might be of importance in particular for the hip and spine, where an accurate assessment of bone micro-architecture is not possible.