PSI - Issue 79

A. Della Rocca et al. / Procedia Structural Integrity 79 (2026) 475–484

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For a comprehensive analysis of the network architecture, skeletonization was applied to the 3D structure. The structure was represented as a binary matrix, where solid voxels were assigned a value of 1 and void voxels a value of 0. The binarized voxel data underwent skeletonization using the MATLAB built-in command which iteratively thins the solid domains down to a 1-voxel-thick medial axis representation. This transformation yields the skeleton, which partially describes the structure's morphology. Following skeletonization, the skeleton points, or nodes, were topologically classified by examining the number of neighboring skeleton points within a 3×3×3 voxel cube centered on the point. This method categorized the nodes into three types (as illustrated in Fig. 3a) as follows:  End nodes (green dots), characterized by a single neighbor, representing the extremities of the trabeculae.  Intermediate nodes (black dots), characterized by two neighbors, representing the length of the trabeculae.  Branch nodes (red dots), characterized by three or more ( ≥ 3) neighbors, representing connectivity points where multiple trabeculae intersect. To minimize redundancy in the structural representation, especially at complex junctions, clustering was performed on the branch points using the density-based clustering non-parametric algorithm (DBSCAN) algorithm. This ensures that multiple closely spaced branch points, which may arise as artifacts of the thinning process, are treated as a single, representative point, resulting in a more robust and accurate network topology. Ultimately, the skeletonization and subsequent analysis allowed for the extraction of several structural descriptors, including: (i) Node densities (end, intermediate, and branch); (ii) The Euler number (a topological invariant serving as a connectivity index); (iii) Average trabecular thickness and its variance; (iv) Bone volume to total volume (BV/TV) ratios and (v) the percentile distribution of trabecular area across sectional slices. These parameters were employed to characterize the structural complexity and establish quantitative correlations with the structure's mechanical performance (see Eq. 2).

Fig. 3. (a) Skeleton structure with end points, branch points and skeleton point. (b) Skeleton structure for D and γ values

2.4. Finite element Modeling To isolate the mechanical effects of the structure's geometry, all simulations assumed isotropic, homogeneous, linear elastic material behavior. The chosen properties were based on cortical bone, which shares a similar composition with trabecular bone material. Specifically, the material was modeled with a Young's modulus (E) of 18.6 GPa and a Poisson’s ratio ( ν ) of 0.3 Cuppone et al. (2004). Since trabecula material properties are in a wide range and it is difficult to pinpoint a single value Dan et al. (2018), these cortical bone values were adopted as the best available estimates. The voxel-based spinodal structures were converted into a finite element mesh using a custom MATLAB-to ABAQUS pipeline. The base structures, initially generated as voxels, were meshed with 8-node brick elements (C3D8). The software pipeline seamlessly recognized the connectivity of these elemental blocks, simplifying the