Abstract
The organization and shape of the microstructural elements of trabecular bone govern its physical properties, are implicated in bone disease, and serve as blueprints for biomaterial design. To devise fundamental structure-property relationships and design truly bone-mimicking biomaterials, it is essential to characterize trabecular bone structure from the perspective of geometry, the mathematical study of shape. Using micro-CT images from 70 donors at five different sites, we analyze the local and global geometry of human trabecular bone in detail, respectively by quantifying surface curvatures and Minkowski functionals. We find that curvature density maps provide distinct and 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 bone-mimicking tissue scaffolds. Statement of significanceThe architecture of trabecular bone is key in determining bone properties, and is often a starting point for the design of bone-substitutes. Despite the substantial history of bone morphometry, a fundamental characterization of trabecular bone geometry is still lacking. Therefore, we introduce a robust framework to quantify local and global trabecular bone geometry, which we apply to hundreds of micro-CT scans. Our approach relies on quantifying surface curvatures and Minkowski functionals, which are the most fundamental local and global shape quantifiers. Our results show that these shape metrics are sensitive to differences between bone types and unify traditional metrics within a single mathematical framework. This geometrical framework could also be useful to design bone-mimicking scaffolds and understand geometry-driven tissue growth.
Highlights
Many natural and man-made materials are characterized by a complex and often hierarchical spatial architecture
We focused on the more potent tensorial Minkowski functionals (MF) that can sensitively quantify the various types of intrinsic anisotropy, which is highly relevant in the study of trabecular bone [23,24]
The femoral head (FH) samples typically exhibited an apparently uniform dispersion of regions with positive and negative values of the mean curvature. Comparing this to the L2 specimen, we observed that the latter showed much more regions of highly negative mean curvature. This is the consequence of the many rod-like elements that are typically present in specimens from the lumber spine, as opposed to the more sheet-like architecture in FH specimens [5]
Summary
Many natural and man-made materials are characterized by a complex and often hierarchical spatial architecture. There is a need for a unifying, robust approach to quantitatively characterize the shape of complex materials, including trabecular bone Such a well-defined, mathematical framework is established in the realms of differential and integral geometry, providing fundamental descriptors of local and global shape. MF are versatile shape indices with strong roots in integral geometry, capable of robustly quantifying different aspects of spatial structure [2,17,18] (Fig. 1) These shape indices, which can be of scalar or tensorial nature, are fundamental in the sense that they form a basis for any other additive functional that describes the shape of a 3D body (Hadwiger-Alesker theorems) [19,20,21]
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