Abstract

Abstract

Several experimental studies derived relationships between density and macroscale material properties of trabecular bone, taking the form E=αρ^β, where E is Young's modulus, ρ is density, and α and β are constants. Classical structural mechanics demonstrates β can vary between 1 (behaviour of the trabecular lattice is dominated by the axial stiffness of individual trabeculae) and 3 (behaviour is dominated by the bending stiffness of individual trabeculae). The ratio between rods (round trabeculae characterised by radius) and plates (flat trabeculae characterised by thickness) is also believed to govern the macroscale material properties of trabecular bone. To assess feasible ranges of α and β for trabecular bone, and their dependence on rod to plate ratio, 25 virtual samples of trabecular bone were generated as Voronoi lattices. Each 8×8×8mm sample was composed of 320 randomly generated Voronoi cells forming a foam like structure. Edges formed the rod network. Faces formed the plate network. Tissue level Young's modulus was set to 18,000MPa. Relative density was varied: 0.05, 0.1, 0.15, 0.2, 0.25. Rod to plate ratio was varied: 100:0, 75:25, 50:50, 25:75, 0:100. Macroscale Young's modulus was averaged in three orthotropic directions and used to find α and β. Around 14,000 3-noded quadratic beam elements represented rods, with average length of 0.63mm, and around 42,000 8-noded quadratic shell elements represented plates, with average area of 0.10mm². Results for α and β were 3274 and 1.463 for 100% rods, 3646 and 1.067 for 50:50 rods and plates, and 4981 and 1.062 for 100% plates, showing the presence of plates improves the stiffness characteristics of trabecular bone. Work investigating the impact of element based geometry optimisation is ongoing. The work has important implications for the onset of conditions including osteoporosis and osteoarthritis, as well as those designing 3D printed scaffolds and implants.