Mathematical approaches to bone reformation phenomena and numerical simulations

Bone remodeling is metabolism of the bone through repetition of the resorption by osteoclasts and formation by osteoblasts. Osteoblasts produce inorganic calcium phosphate, which is converted to hydroxyapatite, and organic matrix consisting mainly of type I collagen, and then they deposit new bone to the part of the bone resorbed by osteoclasts. Osteoclasts dissociate calcium by secreting acid and degrade organic components by releasing lysosomal enzymes. Moreover, osteocytes in the bone play an important role in sensing various physical loads and conveying signals to activate osteoblasts. These three kinds of cells are linked to each other and perform the bone remodeling. Appropriate parameters representing the states of the bone and marrow are introduced and a mathematical model describing the bone remodeling phenomena is presented. The model involves an interface equation which determines the surface of the bone. The associated discrete model is formulated and its stable solvability is verified. Results of numerical simulations on a computer aided design system are visualized and then compared to clinical bone data. This work may be applied to medical science and in particular to dentistry.