This chapter describes material optimization models and their application to the simulation of the bone adaptation process due to mechanical loading.

In 1892 Julius Wolff ^[1] described the dependency of bone on applied loads in his "Law of Bone Remodelling": The internal bone architecture depends on stresses and the trabeculae are aligned with principal stress directions. Following Wolff observations this chapter presents models for bone adaptation/remodelling based on material optimization models. Bone is assumed as a cellular material with variable relative density. The changes on bone density and orientation are obtained by the minimization of a cost function accounting for both the structural stiffness and the biological cost associated with metabolic maintenance of the bone tissue. The effective bone properties are obtained by an asymptotic homogenization method. These homogenized properties are computed for the cellular material obtained by the period repetition of a given base cell, thus with a given material symmetry, or alternatively solving a local material optimization problem in order to introduce the material symmetry as a variable The models described are analyzed and their relations and analogies with other models presented in the literature are discussed.

^[1]Wolff J., Das Gesetz der Transformation der knochen, (Hirchwild, Berlin, 1892). Translated as: The Law of Bone Remodelling, (Springer-Verlag, Berlin, 1986).