Extracting fast and slow wave velocities and attenuations from experimental measurements of cancellous bone using Bayesian probability theory

The consensus among many laboratories is that the attenuation coefficient of cancellous bone exhibits an approximately linear-with-frequency dependence. In the majority of cases, the phase velocity decreases with frequency. This negative dispersion appears to be inconsistent with the causality-imposed Kramers-Kronig (KK) relations for media with a linear-with-frequency attenuation coefficient. The porous structure of cancellous bone can support two compressional waves, known as a fast wave and a slow wave, that can overlap in time. Our laboratory in St. Louis has sought to explain the observed negative dispersion as an artifact of analyzing rf data containing two interfering waves as if only one wave were present. In this study, the inverse problem of how to recover the individual fast and slow waves from interference data was addressed. Waves transmitted through bone samples were analyzed using Bayesian probability theory to recover the individual properties of the fast and slow waves. Data at nine independent sites were acquired in Paris on a bovine femur condyle sample using broadband 500 kHz center frequency transducers. Each rf line served as input to a Bayesian analysis program. In the Bayesian calculation, ultrasonic wave propagation through cancellous bone was modeled as the superposition of two plane waves characterized by a linear-with-frequency attenuation coefficient and a logarithmic-withfrequency increasing phase velocity. The calculation employed Markov chain Monte Carlo (MCMC) to obtain estimates of the joint posterior probability for all parameters in the model. In all cases where the data processed by conventional means exhibited negative dispersion, two waves with positive dispersions were recovered with Bayesian analysis. The mean ± SD fast and slow wave velocities for the nine sites analyzed were (2072 ± 43) m/s and (1518 ± 22) m/s, respectively. The mean ± SD slopes of the attenuation coefficients were (17.3 ± 9.9) dB/cm/MHz and (10.8 ± 5.1) dB/cm/MHz for the fast and slow waves, respectively. Many complicating factors, including phase cancellation at the face of a piezoelectric receiver and diffraction effects, are not explicitly accounted for in the present model. Nevertheless, the Bayesian models proved to be a reliable method for recovering fast and slow waves from data that yielded negative dispersions when processed as if a single wave were present.