Purpose
The trabecular network is perceived as a collection of interconnected plate- (P) and rod-like (R) elements. Previous research has highlighted how these elements and their first-degree connectivity (P-R, R-R, P-P) influence mechanical properties of bone; yet, further work is required to better elucidate the deeply interconnected character of the trabecular network with distinct element formations conducting forces in accordance to their mechanical boundary conditions. Within this network forces act through elements: a rod or plate with a force applied to one end will transmit this force to an element connected to the other end, defining the boundary conditions for loading of each element.
To that end, this study has two aims: First, to investigate trabecular bone’s mor- phometric properties of individually segmented elements, with respect to their local boundary conditions as defined by the surrounding trabecular network, and linking them directly to bone’s overall mechanical response during loading. Second, to use this model to quantify the side-artifact, a known artifact when testing an excised specimen of trabecular bone, where vertical trabeculae lose their load-bearing capacity due to a loss of connectivity, which ultimately results in a change of the trabecular network topology.
Methods
In order to investigate loss of connectivity within the network, we used a trabecular specimen model, using cylindrical cores taken from 12 bovine femurs. Specimens were, subsequently decreased in diameter (S = 10mm, M = 14mm, L = 18mm) and tested with matching diameter platens (S, M, L). Specimens were scanned using a 40 (Scanco Medical, AG, Bassersdorf, Switzerland) and reconstructed at 40µm resolu- tion. 3D-Images were spatially decomposed using a shape conserving 3D-thinning and topological characterization algorithm to identify plate- and rod-like structures, where a sparse connectivity matrix was derived. A weighted graph was constructed using individual element bone volume, as being one major predictor of the overall mechanical performance. A Boykov-Kolmogorov maximum flow algorithm was applied to compute the effectively loaded BV/TV in accordance with the network layout. The local mechanical environment was derived from voxel-based FE-analysis (E = 15GPa, ν = 0.3), and finally ground truth was provided by conducting compres- sive mechanical testing to failure at 1% strain (Instron 8511, Norwood, MA, USA).
Results
The max-flow PR-Model successfully identified loaded elements within the trabec- ular network. Comparison of von-Mises stress derived by conventional micro-FE analysis was in good agreement. ROC analysis showed high predictive capabilities (AUC = 0.73). Max-flow through the network predicts modulus by fitting a linear power law (R² = 0.81). In comparison, prediction using conventional BV/TV results in a lower accuracy (R² = 0.72), demonstrating the ability of the Plate and Rod Network to estimate compressive elastic modulus independent of specimen size or loading boundary condition. Modulus derived from finite element analysis (EFE) was in good agreement (R² = 0.85) with the Young’s modulus from the measurements (Emeas)
Conclusions
PR-Network models are a novel approach to describe load and force transfer mecha- nisms within the trabecular network, incorporating mechanical boundary conditions within the morphological analysis, thus enabling to study intrinsic material properties of trabecular bone. Ultimately, PR-Network models may be used as an early predictor or may provide further insights in osteo-degenerative diseases.