Simulation Study on Local and Integral Mechanical Quantities at Single Trabecular Level as Candidates of Remodeling Stimuli

Abstract

Mechanical quantities at the microstructural level play important roles as stimuli of trabecular bone remodeling by which cancellous bone maintains and adapts its trabecular structure to the mechanical environment. In this study, distribution functions of mechanical quantities on trabecular surfaces were estimated using digital image-based finite element models of five specimens of rat vertebral bodies under physiological loading conditions. As the representative quantities that have been used as mechanical stimuli in the remodeling rate equation, strain energy density (SED) and von Mises equivalent stress were considered for the mechanical quantities at the local point (namely, local mechanical quantities), and SED integration and stress nonuniformity were considered for the quantities that are integrated in space (namely, integral mechanical quantities). As a result of finite element analysis, it was demonstrated that these mechanical quantities were nonuniformly distributed over the trabecular surface due to the three-dimensionally complicated trabecular structure, even though only simple external loading was applied to the vertebral body. The magnitude of the skewness of the distribution function was calculated in order to compare the distribution patterns of the four mechanical quantities. The skewness for SED and equivalent stress in all the loading cases were larger than those for SED integration and stress nonuniformity, respectively, except that in the loading of axial compression, the skewness for equivalent stress was smaller than that for stress nonuniformity. It was also revealed that the skewness varied with changes in the external loading conditions, where changes in the mean and the standard deviation of the skewness for SED integration and stress nonuniformity were smaller than those for SED and equivalent stress. The results support the understanding that the concept of the integral formulae proposed for the bone remodeling stimulus corresponds not only to the actual biological system but also to the observed phenomenon of trabecular structural adaptation to the mechanical environment.