* Corresponding author. Tel.: #1-617-627-3318; fax: #1-617-627- 3058. E-mail address: mkachano@emerald.tufts.edu (M. Kachanov). Journal of Biomechanics 33 (2000) 881}888 Impact of the porous microstructure on the overall elastic properties of the osteonal cortical bone Igor Sevostianov, Mark Kachanov* Department of Mechanical Engineering, Tufts University, 204 Anderson Hall, Medford, MA, 02155, USA Accepted 27 October 1999 Abstract Mechanical properties of bones are largely determined by their microstructure. The latter comprises a large number of diverse pores. The present paper analyzes a connection between structure of the porous space of the osteonal cortical bone and bone's overall anisotropic elastic moduli. The analysis is based on recent developments in the theory of porous materials that predict the anisotropic e!ective moduli of porous solids in terms of pores' shapes, orientations and densities. Bone's microstructure is modeled using available micrographs. The calculated anisotropic elastic constants for porous cortical bone are, mostly, in agreement with available experimental data. The in#uence of each of the pore types on the overall moduli is examined. The results of the analysis can also be used to estimate the extent of mineralization (hydroxyapatite content) if the overall porosity and the e!ective moduli are known and, vice versa, to estimate porosity from the measured moduli and the extent of mineralization.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Cortical bone; Anisotropy; Microstructure; Elastic moduli 1. Introduction Morphology of the cortical bone has been well studied and the overall elastic moduli have been measured in a number of experiments (see Martin and Burr, 1989 for review). Lang (1969, 1970) assumed that the cortical bone is transversely isotropic (the plane normal to Haversian canals being the plane of isotropy) and measured the "ve elastic constants. Katz et al. (1984) and Van Buskirk and Ashman (1981) measured ultrasonically the anisotropic moduli and showed that, in general, cortical bone pos- sesses orthotropic elastic properties, but sti!nesses in various directions normal to Haversian canals do not vary much, so that the deviation from transversal iso- tropy does not exceed 10%. Direct mechanical tests of Reilly and Burstein (1975) and Zioupos et al. (1995) also con"rmed closeness to transversal isotropy, although the measured values of elastic sti!nesses were smaller than the ones measured ultrasonically. Table 1 summarizes the data on elastic sti!nesses measured by various researchers; these data are used for comparison with the sti!nesses predicted by our method. As far as the theoretical analysis of the connection between the structure of the porous space and the overall properties of the bone is concerned, a number of relevant contributions should be mentioned. Recent review of Cowin (1999) considered the bone from the point of view of classical Biot's poroelasticity, with the emphasis on the deformation-driven #uid movements. As far as the e!ec- tive elastic properties are concerned, the following works are particularly relevant. Stech (1967) analyzed the over- all anisotropy of the cortical bone using a model contain- ing parallel cylindrical voids representing Haversian canals but did not take into account the rest of the porous space (network of canaliculi, Volkman's canals, osteocyte lacunae). Katz (1980; 1981) emphasized the hierarchical structure of the osteonal cortical bone and modeled a single osteon as a laminated circular cylinder surrounding the Haversian canal. He analyzed the im- pact on the e!ective constants of both the porous space (modeled as a set of parallel Haversian canals forming a hexagonal arrangement) and the microstructure of the dense mineralized tissue (modeled as a "ber-reinforced composite). Hogan (1992) and Crolet et al. (1993) did numerical simulations of the anisotropic moduli of the 0021-9290/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 1 - 9 2 9 0 ( 0 0 ) 0 0 0 3 1 - 2