A graph model to describe the network connectivity of trabecular plates and rods.

Introduction: 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 connectivity influence the mechanical properties of bone, yet further work is required to elucidate better the deeply interconnected nature of the trabecular network with distinct element formations conducting forces per their mechanical boundary conditions. Within this network, forces act through elements: a rod or plate with force applied to one end will transmit this force to a component connected to the other end, defining the boundary conditions for the loading of each element. To that end, this study has two aims: First, to investigate the connectivity of individually segmented elements of trabecular bone with respect to their local boundary conditions as defined by the surrounding trabecular network and linking them directly to the bone's overall mechanical response during loading using a mathematical graph model of the plate and rod (PR) Network. Second, we use this model to quantify side artifacts, 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, ultimately resulting in a change of the trabecular network topology. Resuts: Connected elements derived from our model predicted apparent elastic modulus by fitting a linear regression (R 2 = 0.81). In comparison, prediction using conventional bone volume fraction results in a lower accuracy (R 2 = 0.72), demonstrating the ability of the PR Network to estimate compressive elastic modulus independent of specimen size or loading boundary condition. Discussion: PR Network models are a novel approach to describing connectivity within the trabecular network and incorporating mechanical boundary conditions within the morphological analysis, thus enabling the study of intrinsic material properties of trabecular bone. Ultimately, PR Network models may be an early predictor or provide further insights into osteo-degenerative diseases.