Bayesian estimation of the underlying bone properties from mixed fast and slow mode ultrasonic signals

Abstract

We recently proposed that the observed apparent negative dispersion in bone can arise from the interference between fast wave and slow wave modes, each exhibiting positive dispersion [Marutyan et al., J. Acoust. Soc. Am. 120, EL55–EL61 (2006)]. In the current study, we applied Bayesian probability theory to solve the inverse problem: extracting the underlying properties of bone. Simulated mixed mode signals were analyzed using Bayesian probability. The calculations were implemented using the Markov chain Monte Carlo with simulated annealing to draw samples from the marginal posterior probability for each parameter.

Determining attenuation properties of interfering fast and slow ultrasonic waves in cancellous bone

“…14,15,17 Previous work has demonstrated, however, that when decomposed, the fast wave and slow wave each exhibit positive dispersion. 19,27 Negative dispersion has been measured not only in trabecular bone [29][30][31][32][33] and cortical bone 34 in vitro but also in trabecular bone-mimicking phantoms. 35,36 Several models have been proposed that predict negative dispersion including multilayer models, 31,37 multiple scattering models, 38 and independent scattering models.…”

“…Therefore, the ability to separate the fast waves and slow waves is of considerable importance. Bayesian probability theory is one method for determining the individual properties of the interfering wave modes, 19,27 but in principle any method that can isolate the two waves could be applied. Wear, for example, recently demonstrated that the modified least squares Prony's method was also able to decompose a mixed-mode signal and yield accurate estimates of its ultrasonic properties.…”

“…7, respectively]. 18 More detailed explanations of Bayesian parameter estimation can be found in Anderson et al, 19 Bretthorst et al, 26 and Marutyan et al 27 The values of the six parameters used in the propagation model above are given in Table I. Once these six parameters were known, the individual fast and slow waves could be generated for each sample thickness d, which was varied from 6 to 15 mm in 1 mm increments.…”

Inverse problems in cancellous bone: Estimation of the ultrasonic properties of fast and slow waves using Bayesian probability theory

“…20,21 Solving the inverse problem-that is, reconstructing the ultrasonic properties of the interfering fast and slow waves-could provide more reliable information about the medium under study. Studies undertaken by Sebaa et al, 38 our Laboratory, [39][40][41][42] and Wear 43 have addressed various ways of addressing inverse problems in the ultrasonic investigation of cancellous bone to accomplish these goals. The objective of the current study is to extend and enhance our Laboratory's proposed technique of using Bayesian probability theory to recover the properties of individual interfering waves in data acquired on bone and bone-mimicking phantoms.…”

Negative dispersion in bone: The role of interference in measurements of the apparent phase velocity of two temporally overlapping signals

“…[7][8][9] In previous work we proposed an alternative explanation of the observed negative dispersion and presented the results of numerical simulations demonstrating that negative dispersion can result from the interference of two propagating modes, each of which exhibits a positive dispersion, consistent with the Kramers-Kronig predictions. 10,11 The goal of the present study is to demonstrate this negative dispersion experimentally using the simplest example of a phantom that is capable of producing two such interfering waves. Most ultrasonic experimental data are acquired using phase sensitive piezoelectric devices.…”