Network models for characterization of trabecular bone

Abstract

Trabecular bone is a lightweight, compliant material organized as a web of struts and rods (trabeculae) that erode with age and the onset of bone diseases like osteoporosis, leading to increased fracture risk. The traditional diagnostic marker of osteoporosis, bone mineral density (BMD), has been shown in ex vivo experiments to correlate poorly with fracture resistance when considered on its own, while structural features in conjunction with BMD can explain more of the variation in trabecular bone strength. We develop a network-based model of trabecular bone by creating graphs from micro-CT images of human bone, with weighted links representing trabeculae and nodes representing branch points. These graphs enable calculation of quantitative network metrics to characterize trabecular structure. We also create finite element models of the networks in which each link is represented by a beam, facilitating analysis of the mechanical response of the bone samples to simulated loading. We examine the structural and mechanical properties of trabecular bone at the scale of individual trabeculae (of order 0.1 mm) and at the scale of selected volumes of interest (approximately a few mm), referred to as VOIs. At the VOI scale, we find significant correlations between the stiffness of VOIs and ten different structural metrics. Individually, the volume fraction of each VOI is most strongly correlated to the stiffness of the VOI. We use multiple linear regression to identify the smallest subset of variables needed to capture the variation in stiffness. In a linear fit, we find that node degree, weighted node degree, Z-orientation, weighted Z-orientation, trabecular spacing, link length, and the number of links are the structural metrics that are most significant (p < 0.05) in capturing the variation of stiffness in trabecular networks.

Characterization of fracture in topology-optimized bioinspired networks

“…The models with stability objectives are most similar in visual resemblance to trabecular bone, and the shape of their stress distributions is also the most similar to that of bone [3]. For the models with stability objectives, however, ζ 0.001 remains much lower than for bone, which is on average approximately 0.43 [3], while this value is surpassed for C99999P00001 and C99P01.…”

“…The models with stability objectives are most similar in visual resemblance to trabecular bone, and the shape of their stress distributions is also the most similar to that of bone [3]. For the models with stability objectives, however, ζ 0.001 remains much lower than for bone, which is on average approximately 0.43 [3], while this value is surpassed for C99999P00001 and C99P01. For bone, approximately 6.7% of the total volume fraction bears less than 90% of the normalized stress [10], indicating that the stress distributions are considerably less skewed for the topology-optimized models than for bone -note, however, that the topology-optimized structures generated here are two-dimensional, while the bone volumes analyzed previously are three-dimensional.…”

“…From topology-optimized images, we generate graph models, following [3], that allow us to utilize existing graph theoretical methods to efficiently analyze the topology of networked structures. Converting a topologyoptimized structure to a graph begins with skeletonization: the "skeleton" of each image is determined by progressively thinning the image until its medial axis, a onepixel-wide line running through the center of the network, is found.…”

“…To quantify the stress distribution, we compute two metrics ζ 0.001 and σ 0.9 . ζ 0.001 is the fraction of total area with normalized stress less than or equal to 0.001, and σ 0.9 is the normalized stress value such that 90% of the total area bears stress less than or equal to this value; similar metrics were previously defined in the context of trabecular bone in [3]. Average values for ζ 0.001 and σ 0.9 are tabulated in Table I stress, followed by C99P01 at 42% (52% of beams).…”

“…This allows us to extract topological metrics that quantify the architecture of the network. This network-based method adapts the modeling approach developed by Mondal et al [3] which modeled real human trabecular bone from micro-CT images.…”