Direct measurement of trabecular bone anisotropy using directional fractal dimension and principal axes of inertia

Abstract

Precise in vivo measurement of the trabecular bone's mechanical properties is very important for endosseous dental implant treatment and design in clinical practice. The fractal structure of trabecular bone shows directional anisotropy of the architecture, as is shown in most biological fractals. To analyze the anisotropy of the trabecular bone, the fractal geometry technique was applied to 2-dimensional plain radiographs. The power spectrum was used to calculate the fractal dimensions (FD) of the trabecular bone. The FDs calculated as a function of orientation yielded the fractal information reflecting the spatial characteristics of the trabecular bone in each direction. A polar plot of directional FDs was defined as an ellipse of inertia. The principal loading direction in a local region of the trabecular bone was determined from the minimum moment of inertia for the ellipse of FDs. The anisotropy was calculated directly as the ratio of the 2 principal moments of inertia from the ellipse. The anisotropies were measured for radiographs from the angle and incisor region of 21 human mandibles based on the principal axes of inertia and the best-fitting ellipse. The anisotropy of the angle region was significantly greater than that of the incisor region of the mandibles. The method using directional FDs as determined by the principal axis of inertia measures the anisotropy directly, using 2-dimensional plain radiographs. It can quantify the anisotropy of trabecular bone in vivo. The investigation can be applied to the analysis of the relationships between in vivo 2-dimensional parameters and 3-dimensional mechanical properties, which enables us to predict the bone mechanical properties such as strength in vivo in various regions of the mandible.