Morphological indexes and structural parameters in trabecular bone micro-models

It is well known that the mechanical behaviour of cancellous bone depends not only on its mineral content but also on the trabecular architecture, which appears to be the main responsible for trabecular bone anisotropy. In literature, the fourth order tensors that describe the elastic properties of the material (very demanding from the computational point of view) have been related to the anisotropy MIL second order fabric tensors (easily obtainable) [1-6]. Unfortunately, the formulations developed so far between volume fraction, fabric and elastic properties are bone specific and the coefficients found for one bone are not directly applicable to other bones. This is not unexpected since the normalized MIL eigenvalues Hi show a poor correlation to the stiffness components, as shown for example in Figure 1 (a), where healthy bone structures from different species are considered. However, if the normalized values of stiffness Ei are used instead of the apparent elastic moduli E_sim, correlation increases dramatically even for very different bones, like pig and rat in this case, Figure 1 (b). Starting from these considerations [6], a new approach resulted in a simple general relationship, linking volume fraction and MIL fabric tensor to the trabecular structure stiffness components. The results presented here show that the model can predict approximately 99% of the variation of the numerically computed elastic moduli for the same, healthy, pooled data set (Figure 2) and can be employed also to assess tissue degeneration due to osteoporosis or microgravity (Figure 3). Figure 1: Pooled data set from different healthy bone structures (pig and rat) - (a) Computed elastic moduli (MPa) vs. MIL normalized eigenvalues - (b) Normalized elastic moduli vs. MIL normalized eigenvalues Figure 2: Pooled data set from healthy bone structures - Predicted vs. computed elastic moduli (MPa) Figure 3: Degenerated bone structures - Predicted vs. computed elastic moduli (MPa) [1] Cowin S.C., "The relationship between the elasticity tensor and the fabric tensor", Mech. Mat., 1985, 4, 137\u2013147. [2] Van Rietbergen B. et al., "Relationships between bone morphology and bone elastic properties can be accurately quantified using high-resolution computer reconstructions", J. Orth. Res., 1998, 16, 23\u201328. [3] Kabel J. et al, "Constitutive relationships of fabric, density, and elastic properties in cancellous bone architecture", Bone, 1999, 25, 481\u2013486. [4] Homminga J. et al, "The dependence of the elastic properties of osteoporotic cancellous bone on volume fraction and fabric", J. Biomech., 2003, 36, 1461\u20131467. [5] Zysset P.K., "A review of morphology\u2013elasticity relationships in human trabecular bone: theories and experiments", J. Biomech., 2003, 36, 1469\u20131485. [6] Van Ruijven L.J. et al, "Prediction of mechanical properties of the cancellous bone of the mandibular condyle", J. Dent. Res., 2003, 82(10), 819\u2013823. [7] Cosmi F., "Morphology-based prediction of elastic properties of trabecular bone samples", Acta Bioeng. Biomech., 11(1), 2009, 3-