Elastic properties of cancellous bone in terms of elastic properties of its mineral and protein phases with application to their osteoporotic degradation

The elastic modulus of cancellous bone is derived based on the measured elastic properties of separate mineral and protein phases. Adopting the mechanics of cellular solids approach, the moduli of elasticity of cancellous, deproteinated and demineralized bone are expressed in terms of the trabecular moduli of elasticity and the corresponding density ratios using the power law expressions. The Young’s modulus of trabeculae of bone are related to the Young’s moduli of deproteinated and demineralized trabeculae through a modified mixture rule, which incorporates an appropriate weight function to account for the mineral/ protein interaction effects and the departure from the ideal mixture rule. Two expressions for the effective modulus of elasticity of cancellous bone are then derived: one in terms of the moduli of elasticity of mineral and protein trabeculae, and the other in terms of the moduli of elasticity of deproteinated and demineralized cancellous bone. The material parameters are specified from the results of compressive testing of untreated, deproteinated, and demineralized cancellous bovine femur bone. The osteoporotic decrease of the elastic moduli is then analyzed. Two evolution equations are introduced, one for the rate of loss of the mineral content of cancellous bone, and the other for the protein loss. Both losses are associated with the corresponding density and volume changes, for which appropriate equations are proposed. Based on these, and the evolution equations for morphological parameters accounting for the trabecular microarchitecture, the evolution equations are derived for the elastic moduli of deproteinated, demineralized and composite cancellous bone. A particular model of osteoporotic degradation is considered in which it is assumed that the relative ratios of the mineral and protein loss are equal to each other during the progression of osteoporosis. 2011 Elsevier Ltd. All rights reserved.