Effect of trabecular curvature on the stiffness of trabecular bone

Simplified structural models of trabecular bone have been used to model various forms of trabecular variability. The structural effects of variability of direction, length and thickness of the trabeculae have been studied using 'lattice-type' finite element models. However, many of the trabeculae are not perfectly straight, and have a small degree of curvature. The objective of this study is to quantify the influence of small curvatures of the trabeculae on the effective modulus of trabecular bone, in the principal material direction. An analytical analysis of the effect of curvature on a single trabecula is performed, utilizing the concept of cellular-solid models. Closed-form expressions are derived for the effect of curvature on the flexibility in the principal material direction. For comparison, expressions are derived for the flexibility of a straight oblique element, representing angular variability. A quantitative comparison is presented, which is dependent on the thickness of the trabeculae. It was found that small curvatures have a large effect on the stiffness of the trabecular structure. This effect is largest for thin trabeculae, and decreases for thick trabeculae. The stiffness of the trabecular structure can be reduced by a factor of up to four for thin trabeculae and up to two for thick trabeculae, even for small curvatures. The flexibility of curved elements is found to be larger than the flexibility of oblique elements with similar eccentricities. Thus it seems that curvature might play a role in determining the effective modulus of trabecular bone.