Bone tissue has this extraordinary capacity to be metabolically active, continually growing and remodeling itself throughout life. Growth and remodeling are terms classically used in the literature to designate changes in the density, volume, and internal structure of the tissue under consideration. However, it is not always easy to distinguish their subtleties since they are highly interconnected. Strictly speaking, natural tissue growth is a term that primarily designates an evolution (increase or decrease) in mass resulting from adding material at constant density, a change in density at constant volume, or both simultaneously. It can occur on the surface or in the bulk of the tissue. Remodeling, on the other hand, mainly refers to an evolution in the properties of a system without a change in mass, attributable to a modification of its internal structure.

However, in the specific case of bone, the action of bone cells during the remodeling process modifying the tissue’s inner structure also results in changes in material density caused by the deposition or resorption by osteoblasts of new material on the surface of trabeculae or the walls of canals excavated by osteoclasts. These bone cells are sensitive to mechanical loads, enabling the tissue to optimize its architecture according to the mechanical stresses it undergoes. These aspects are fundamental to implantology, as osseointegration of endosseous implants plays an essential role in the success of orthopedic and dental surgical procedures. Understanding the mechanisms underlying osseointegration and predicting the evolution of bone tissue characteristics around the implant are significant challenges in biomedical research.

This work aims to develop a theoretical and numerical framework to simulate the effects of bone growth/remodeling processes and the evolution of tissue properties around an implant. The mechano-mathematical model considers the following mechanisms: (1) the evolution of mass at constant volume, and (2) the evolution of volume at constant density. The first one refers to the densification/resorption of material, and the second to the growth/atrophy of the tissue. These two phenomena are considered to be governed by local mechanical stresses induced by an external mechanical loading context. They may also coincide.

The model is written within the framework of the thermodynamics of irreversible processes and under the hypothesis of small deformations. We shall also restrict the developments to the case of an isotropic material for the sake of simplicity. However, an extension to anisotropic elastic behavior, does not pose any additional conceptual difficulties.