Decomposition of interfering ultrasonic waves in bone and bone-mimicking materials

Cancellous bone is a lattice-like arrangement of solid trabeculae surrounded by soft bone marrow. When interrogated with ultrasonic waves, the complex architecture gives rise to two longitudinal modes known as a fast and a slow wave. Depending on experimental conditions and the ultrasonic characteristics of the bone sample under investigation, the two waves may strongly overlap in the ultrasounic data. Analyzing such data conventionally, as if only one wave were present, can potentially mask or alter bone quality parameters that are commonly used in clinical sonometry. In this study, ultrasonic data were acquired on a bovine femur condyle specimen and on a plastic bone-mimicking phantom constructed from Lucite and Lexan blocks. The acquired data were used as inputs to a program that implements a Bayesian calculation to model the ultrasound signal as two interfering plane waves and then estimates the ultrasonic parameters of the fast and slow waves. The calculations were carried out using Markov chain Monte Carlo (MCMC) with simulated annealing to approximate the Bayesian calculations. The models showed good agreement with the acquired data from both the bone and bone-mimicking phantom. The sound velocities that maximized the joint posterior probability for the model of the bone-mimicking fantom were 2765 m/s (fast wave) and 2193 m/s (slow wave), values that are moderately close to the true values for Lucite (2734 m/s) and Lexan (2185 m/s). These parameter estimates provide more reliable estimates of the material properties of the plastic bone-mimicking phantom than conventional analysis. The mean ± standard deviation velocities in the bone sample at a representative site were 2036±4m/s (fast wave) and 1511±1m/s (slow wave).