Mathematical modelling of trabecular bone structure: the evaluation of analytical and quantified surface to volume relationships in the femoral head and iliac crest

The three-dimensional architecture of trabecular bone has structural trends related to physical function as described by Wolff's law. Mathematical modelling provides a means of analysing these structures through the use of simplified representations. A single measure of mineralized bone volume per unit volume of structure (Vv) and the surface area of mineralized bone per unit volume of structure (Sv) does not identify a particular architecture in any detail; the way in which Sv changes in relation to Vv does provide this information as the structure remodels. A series of structures using the elements of plates and rods were created. The rates of change of Sv with respect to Vv for trabecular structures give insight into differences in such models. Structures in the femoral head and iliac crest were analysed by power curve regression. In the principal compressive region, just above the medial cortex, advanced osteoarthritis was associated with a preferential loss of rods from the normal trabecular structure, resulting in a more plate-like architecture. The iliac crest remodelling that takes place in the osteoporotic appears to be the result of a generalised bone loss with some of the thinner elements of the structure being removed completely, resulting in an increase in unit cell dimension. The consequence of changing unit cell size has a major impact on surface availability for osteoblastic and osteoclastic activity. The simple plate model as a basis for the stereological analysis of trabecular structures is therefore limited because of the mixed plate and rod nature of trabecular architecture.