2D numerical simulations of ultrasound propagation in random anisotropic media: Occurrence of two longitudinal waves in bone-like structures

The ultrasonic characterization of bone biomechanical properties can improve bone fracture risk assessment with a noninvasive, non-ionizing and relatively inexpensive method. These methods appear to have significant potential and are currently underutilized because the physics of ultrasound propagation in bone is still poorly understood. Some phenomena experimentally observed in trabecular bone remain indeed unclear, such as the possible propagation of two longitudinal waves with different velocities. This study focuses on determining the conditions necessary to develop two waves on a simple model, and in particular on the influence of guided modes in their occurrence. Numerical simulations were conducted using a finite difference (FDTD) tool that can simulate the propagation of elastic waves in heterogeneous anisotropic media with fine control. Binary random media with fully controlled elasticity and anisotropy were designed by randomly distributing elliptic-shaped patterns on 2D maps with an orientation ensuring global anisotropy. The material properties were defined at all points in space. The coherent wave was calculated by averaging over a large number of realizations of the propagation field. The two waves, which have been reportedly observed in the literature under certain conditions, were observed. Four conditions were found necessary for their occurrence: i) a given porosity range, ii) a propagation parallel to the ellipses, iii) the possibility of the solid matrix to support shear waves and iv) long propagation paths in the solid matrix, ensured by connectivity of the ellipses. These results suggest that the occurrence of the two waves observed in trabecular bone could be attributed to the existence of modes guided by the trabeculae. Important bone micro-architecture parameters such as anisotropy and connectivity could potentially be retrieved from ultrasonic measurements, improving the evaluation of fracture risk.