Ultrasound propagation in clusters of elliptic (two-dimensional) or ellipsoidal (three-dimensional) scatterers randomly distributed in a fluid is investigated numerically. The essential motivation for the present work is to gain a better understanding of ultrasound propagation in trabecular bone. Bone microstructure exhibits structural anisotropy and multiple wave scattering. Some phenomena remain partially unexplained, such as the propagation of two longitudinal waves. The objective of this study was to shed more light on the occurrence of these two waves, using finite-difference simulations on a model medium simpler than bone. Slabs of anisotropic, scattering media were randomly generated. The coherent wave was obtained through spatial and ensemble-averaging of the transmitted wavefields. When varying relevant medium parameters, four of them appeared to play a significant role for the observation of two waves: (i) the solid fraction, (ii) the direction of propagation relatively to the scatterers orientation, (iii) the ability of scatterers to support shear waves, and (iv) a continuity of the solid matrix along the propagation. These observations are consistent with the hypothesis that fast waves are guided by the locally plate/bar-like solid matrix. If confirmed, this interpretation could significantly help developing approaches for a better understanding of trabecular bone micro-architecture using ultrasound.

A better understanding of the mechanisms of ultrasound propagation in trabecular bone could considerably help improving the quantitative ultrasonic techniques commonly used to assess bone quality. Some phenomena experimentally observed in trabecular bone remain poorly understood, such as the possible propagation of two compressional waves with different velocities. In this study, elastic wave propagation has been simulated using a finite-difference time-domain method in two-dimensions. Trabecular bone was modeled by a binary random medium with fully controlled elasticity and anisotropy. To do so, elliptic-shaped patterns were randomly distributed on two-dimensional (2-D) maps with an orientation ensuring global anisotropy. The coherent wave was obtained by averaging over a large number of random maps. Several conditions for the observation of the two waves have been identified: (i) The propagation has to occur in a direction parallel to the main orientation of the medium. (ii) some of the elliptic pattern elements had to be connected, which suggests the importance of a percolation threshold. (iii) It is necessary to take into account shear waves in the solid phase. This suggests that bone microarchitecture parameters (anisotropy and connectivity) could be retrieved from ultrasonic measurements, improving the evaluation of fracture risk.