The effect of porosity on the elastic properties of dry long cortical bone and ultrasound propagation

When osteoporosis occurs, the long cortical bone shows increased porosity. The effect of porosity on ultrasonic body wave and axial ultrasonic guided wave propagation in long cortical bone has not been fully investigated. In this paper, in order to evaluate the effects of partial porosity on the elastic properties and the ultrasonic wave propagation in dry long cortical bone, the relation between partial porosity and the effective anisotropic elastic modulus of dry cortical bone has been established using stiffness contribution tensor and Maxwell homogenization scheme. The anisotropic porous long cortical bone model has been constructed, the acoustic field is analytically solved, and the effects of different levels of partial porosities on the elastic properties of long cortical bone are studied. The effects of porosity on the ultrasonic body wave propagation velocity, and guided wave dispersion characteristics are carefully investigated. The results show that the elastic moduli $C33d$⁠, $C44d$⁠, the body waves (including longitudinal wave, SH wave and SV wave) propagating along the vertical direction, longitudinal guided wave modes L₁ and L₂ in the low frequency plateau region are more sensitive to the canalicular porosity; However, the elastic modulus $C11d$⁠, $C66d$⁠, the body waves (including longitudinal wave, SH wave) propagating along the horizontal direction (along the isotropic plane of long cortical bone) are more sensitive to Haversian porosity. Flexural guided wave modes are sensitive to both types of porosity. This paper provides a theoretical basis for nondestructive evaluation of osteoporosis using ultrasonic waves.