Empirical angle-dependent Biot and MBA models for acoustic anisotropy in cancellous bone

Abstract

The Biot and the modified Biot–Attenborough (MBA) models have been found useful to understand ultrasonic wave propagation in cancellous bone. However, neither of the models, as previously applied to cancellous bone, allows for the angular dependence of acoustic properties with direction. The present study aims to account for the acoustic anisotropy in cancellous bone, by introducing empirical angle-dependent input parameters, as defined for a highly oriented structure, into the Biot and the MBA models. The anisotropy of the angle-dependent Biot model is attributed to the variation in the elastic moduli of the skeletal frame with respect to the trabecular alignment. The angle-dependent MBA model employs a simple empirical way of using the parametric fit for the fast and the slow wave speeds. The angle-dependent models were used to predict both the fast and slow wave velocities as a function of propagation angle with respect to the trabecular alignment of cancellous bone. The predictions were compared with those of the Schoenberg model for anisotropy in cancellous bone and in vitro experimental measurements from the literature. The angle-dependent models successfully predicted the angular dependence of phase velocity of the fast wave with direction. The root-mean-square errors of the measured versus predicted fast wave velocities were 79.2 m s−1 (angle-dependent Biot model) and 36.1 m s−1 (angle-dependent MBA model). They also predicted the fact that the slow wave is nearly independent of propagation angle for angles about 50°, but consistently underestimated the slow wave velocity with the root-mean-square errors of 187.2 m s−1 (angle-dependent Biot model) and 240.8 m s−1 (angle-dependent MBA model). The study indicates that the angle-dependent models reasonably replicate the acoustic anisotropy in cancellous bone.