Abstract

Wolff’s law states that bone morphology evolves according to their external mechanical loading. Following this law, mathematical models using topology optimization have been developed to simulate bone shape formation, especially for trabecular bone. Less attention has been given to the bone outer shape composed of cortical bone. However, both trabecular bone and cortical bone are formed by osteoblasts and osteoclasts. Therefore, we hypothesized that the bone outer shape also adapts to the external forces, and built a mathematical model using topology optimization to understand the mechanism that generates the bone outer shape. Our model is inspired by the fish vertebra. Fish vertebra has the variety of the shapes, e.g. thick ridge structure and fine netlike structure. The differences of the shapes seem related to the fish motion, i.e. the swimming type of fish. Therefore, it is assumed that the shape of fish vertebra also evolves based on external loading conditions. Because the swimming type of fish is diverse, the model we developed can produce the structure resistant to the various movements respectively as well as explain the bone formation. We defined the time dependent reaction-diffusion equation based on the standard topology optimization. The optimization problem is supplemented with a local density penalization to mimic the local activity of osteoblasts and osteoclasts. Numerical results showed that the model can produce the various structures similar to the fish vertebra only by adjusting the few parameters of the penalization law.