The local and global geometry of trabecular bone

A bstract The organization and shape of the microstructural elements of trabecular bone govern its physical properties, are implicated in bone disease, and can serve as blueprints for biomaterial design. To devise fundamental structure-property relationships, it is essential to characterize trabecular bone from the perspective of geometry, the mathematical study of shape. Here, we used the micro-computed tomography images of 70 donors at five different sites to characterize the local and global geometry of human trabecular bone, respectively quantified by surface curvatures and Minkowski functionals. We find that curvature density maps provide sensitive shape fingerprints for bone from different sites. Contrary to a common assumption, these curvature maps also show that bone morphology does not approximate a minimal surface but exhibits a much more intricate curvature landscape. At the global (or integral) perspective, our Minkowski analysis illustrates that trabecular bone exhibits other types of anisotropy/ellipticity beyond interfacial orientation, and that anisotropy varies substantially within the trabecular structure. Moreover, we show that the Minkowski functionals unify several traditional morphometric indices. Our geometric approach to trabecular morphometry provides a fundamental language of shape that could be useful for bone failure prediction, understanding geometry-driven tissue growth, and the design of complex tissue engineering scaffolds.