Inter-trabecular angle: A parameter of trabecular bone architecture in the human proximal femur that reveals underlying topological motifs

Trabecular bone is an intricate 3D network of struts and plates. Although the structure-function relations in trabecular bone have been studied since the time of Julius Wolff, controversy still exists regarding the architectural parameters responsible for its stability and resilience. We present a parameter that measures the angle between two connected trabeculae - the Inter-Trabecular Angle (ITA). We studied the ITA values derived from μCT scans of different regions of the proximal femora of 5 individuals of different age and sex. We show that the ITA angle distribution of nodes with 3 connecting trabeculae has a mean close to 120°, nodes with 4 connecting trabeculae has a mean close to 109° and nodes of higher connectivity have mean ITA values around 100°. This tendency to spread the ITAs around geometrically symmetrical motifs is highly conserved. The implication is that the ITAs are optimized such that the smallest amount of material spans the maximal 3D volume, and possibly by so doing trabecular bone might be better adapted to multidirectional loading. We also draw a parallel between trabecular bone and tensegrity structures - where lightweight, resilient and stable tetrahedron-based shapes contribute to strain redistribution amongst all the elements and to collective impact dampening.

The Inter-Trabecular Angle (ITA) is a new topological parameter of trabecular bone. The ITA characterizes the way trabeculae connect with each other at nodes, regardless of their thickness and shape. The mean ITA value of nodes with 3 trabeculae is close to 120°, of nodes with 4 trabeculae is just below 109°, and the mean ITA of nodes with 5 and more trabeculae is around 100°. Thus the connections of trabeculae trend towards adopting symmetrical shapes. This implies that trabeculae can maximally span 3D space using the minimal amount of material. We draw a parallel between this motif and the concept of tensegrity - an engineering premise to which many living creatures conform at multiple levels of organization.