Three-dimensional simulations of ultrasonic axial transmission velocity measurement on cortical bone models

The ultrasonic axial transmission technique, used to assess cortical shells of long bones, is investigated using numerical simulations based on a three-dimensional (3D) finite difference code. We focus our interest on the effects of 3D cortical bone geometry (curvature, cortical thickness), anisotropy, and microporosity on speed of sound (SOS) measurements for different frequencies in the MHz range. We first show that SOS values measured on tubular cortical shells are identical to those measured on cortical plates of equal thickness. Anisotropy of cortical bone is then shown to have a major impact on SOS measurement as a function of cortical thickness. The range of SOS values measured on anisotropic bone is half the range found when bone is considered isotropic. Dependence of thickness occurs for cortical shell thinner than 0.5 x lambda(bone) in anisotropic bone (lambda(bone): wavelength in bone), whereas it occurs for cortical shell thinner than lambda(bone) when anisotropy is neglected. Sensitivity of SOS along the bone axis to intracortical microporosity is shown to be approximately -20 m s(-1) per percent of porosity. Using homogenized porous bone, we finally show that the cortical depth that contributes to lateral wave SOS measurement is approximately 1-1.5 mm for frequencies ranging from 500 kHz to 2 MHz under classical in vivo measurement conditions.