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Abstract
This paper addresses the relationships between the microscopic properties of bone and its elasticity at the millimetre scale, or mesoscale. A method is proposed to estimate the mesoscale properties of cortical bone based on a spatial distribution of acoustic properties at the microscopic scale obtained with scanning acoustic microscopy. The procedure to compute the mesoscopic stiffness tensor involves (i) the segmentation of the pores to obtain a realistic model of the porosity; (ii) the construction of a field of anisotropic elastic coefficients at the microscopic scale which reflects the heterogeneity of the bone matrix; (iii) finite element computations of mesoscopic homogenized properties. The computed mesoscopic properties compare well with available experimental data. It appears that the tissue anisotropy at the microscopic level has a major effect on the mesoscopic anisotropy and that assuming the pores filled with an incompressible fluid or, alternatively, empty, leads to significantly different mesoscopic properties.
Acknowledgements
The authors are indebted to Marc François (Université Pierre et Marie Curie-Paris6, LMT, Cachan, France) for helpful discussions on anisotropy and for providing the software SYMETRIC. Vincent Liabeuf implemented the micromechanics model used to derive the anisotropic bone properties at the microscopic level. The authors are grateful for the financial supports to: Université Pierre et Marie Curie-Paris6 (BQR funding of the project “Modélisation mécanique du tissu osseux”); and the French–German research network “Ultrasound assessment of bone strength from the tissue level to the organ level” (DFG: Grant Ra 1380/1-1; CNRS).
