An orientation distribution function for trabecular bone

We describe a new method for quantifying the orientation of trabecular bone from three-dimensional images. Trabecular lattices from five human vertebrae were decomposed into individual trabecular elements, and the orientation, mass, and thickness of each element were recorded. Continuous functions that described the total mass (M(φ,θ)) and mean thickness (τ(φ,θ)) of all trabeculae as a function of orientation were derived. The results were compared with experimental measurements of the elastic modulus in three principal anatomic directions. A power law scaling relationship between the anisotropies in mass and elastic modulus was observed; the scaling exponent was 1.41 (R2 = 0.88). As expected, the preponderance of trabecular mass was oriented along the cranial-caudal direction; on average, there was 3.4 times more mass oriented vertically than horizontally. Moreover, the vertical trabeculae were 30% thicker, on average, than the horizontal trabeculae. The vertical trabecular thickness was inversely related to connectivity (R2 = 0.70; P = 0.07), suggesting a possible organization into either few, thick trabeculae or many thin trabeculae. The method, which accounts for the mechanical connectedness of the lattice, provides a rapid way to both visualize and quantify the three-dimensional organization of trabecular bone.