Predicting the compressive mechanical behavior of bone.

The principal objectives of this study were to determine the mathematical dependency of the compressive mechanical properties of human bone on several commonly used measures of bone composition, and to assess variations in this dependency based upon the composition range spanned by the data. Destructive mechanical tests were conducted along the superior-inferior axis of 496 cubic specimens of human trabecular and cortical bone from five male donors (ages 46-84 yr), including specimens from lumbar vertebrae and femoral metaphyses and diaphyses. There was over a 3000-fold variation in strength (S, ultimate stress) and over a 20,000-fold variation in stiffness (E, elastic modulus) over the range of apparent dry density (rho a = 0.05-1.89 g cm-3), apparent ash density (rho alpha = 0.03-1.22 g cm-3) and mineral content (alpha = 17.4-66.2%) examined. Both linear and power models produced very high correlations (R2 > 0.81) between mechanical properties and bone composition, but the linear models resulted in a much greater percent deviation (PD) of the predicted dependent variable with respect to the measured value, in comparison to power models. The best correlations were obtained using rho alpha as the only independent variable: S (MPa) = 117 rho alpha 1.93 +/- 0.04 (R2 = 0.969, PD = 29.9, E (GPa) = 10.5 rho alpha 2.57 +/-0.04 (R2 = 0.965, PD = 46.7). Power models of bone stiffness and strength, incorporating only low density data (rho alpha < 0.2 g cm-3, rho a < 0.3), were characterized by approximately squared exponents and these models underestimated the stiffness (five-fold) and overestimated the strength (two-fold) for higher density data, which were characterized by exponents greater than two. Using a subset of the data based upon an apparent dry density range of 0.22 < rho a < 1.89 g cm-3, it was possible to obtain a mathematical relationship in which bone stiffness and strength were precisely proportional to the cube and square, respectively, of the apparent dry density. These results indicate that the mathematical dependency of bone compressive mechanical properties on composition is closely dependent upon the density and mineral content range examined and, in terms of a single compositional measure, is best predicted by apparent ash density expressed as a power function.