Comparison of Mechanical Quantities as Bone Remodeling Stimuli

Abstract

In this chapter, different trabecular-level mechanical quantities are compared for determining their appropriateness as bone remodeling stimuli. Distribution functions of the mechanical quantities were evaluated by using digital image-based finite element models of rat vertebral bodies subject to physiological loading conditions. Strain energy density (SED) and von Mises equivalent stress were considered as local mechanical quantities, while SED integration and stress non-uniformity were considered as integral mechanical quantities. The analysis demonstrated that the mechanical quantities were non-uniformly distributed over the trabecular surface owing to the three-dimensionally complex trabecular structure. The distribution patterns of the four mechanical quantities were compared in terms of the skewness of their distribution functions. The results support the notion that the integral formalism, proposed for bone remodeling stimuli on the basis of the process of bone cells to sense mechanical stimuli, corresponds to trabecular structural adaptation to its mechanical environment.

This Chapter was adapted from Tsubota and Adachi (2006) with permission from The Japan Society of Mechanical Engineers.

Appendix: A Remodeling Equilibrium Around Mean Stimulus

The rate of the trabecular surface movement $\dot{M}$ was assumed to be a simple linear function of the mechanical stimulus S, as shown in Eq. (10.5). If C and S ^ref do not depend on time and are non-site specific, the extent of the net change in the total bone mass ${\dot{M}}^{\mathrm{total}}$ is expressed by the rate coefficient C, the reference stimulus S ^ref, the mean stimulus $\overline{S}$ in the trabecular bone region, and the total trabecular surface area ∫dA, as:

$$ {\displaystyle \begin{array}{l}{\dot{M}}^{\mathrm{total}}=\int \dot{M}\mathrm{d}A\\ {}\kern3em =\int C\left(S-{S}^{\mathrm{ref}}\right)\mathrm{d}A\\ {}\kern3em =C\left(\int S\mathrm{d}A-\int {S}^{\mathrm{ref}}\mathrm{d}A\right)\\ {}\kern3em =C\left(\overline{S}-{S}^{\mathrm{ref}}\right)\int \mathrm{d}A\kern1em \left(\because \int S\mathrm{d}A=\overline{S}\int \mathrm{d}A\right)\end{array}} $$

(10.8)

Because ${\dot{M}}^{\mathrm{total}}$ should be zero in a normal trabecular bone in the remodeling equilibrium , Eq. (10.8) indicates that the reference stimulus S ^ref is equivalent to the mean stimulus $\overline{S}$ of a normal trabecular bone.