Mean-intercept anisotropy analysis of porous media. I. Analytic formulae for anisotropic Boolean models

The local and global geometry of trabecular bone

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 SIGNIFICANCE: The 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.

Reconstructing porous structures from FIB-SEM image data: Optimizing sampling scheme and image processing

Nano-porous materials can be imaged spatially by focused ion beam scanning electron microscopy (FIB-SEM). This method generates a stack of SEM images that has to be segmented (or reconstructed) to serve as basis for structural characterization. To this end, we apply two state-of-the-art algorithms. We study the influence of the original image's voxel size on estimates of morphological characteristics and effective permeabilities. Special attention is paid to analyzing anisotropies due to the FIB-SEM typical anisotropic sampling. Quantitative comparison of morphological descriptors and flow properties of reconstructed data is enabled by the use of synthetic FIB-SEM sets for which a ground truth is available. Moreover, in that case, reconstruction parameters can be chosen optimally, too.