Interference between wave modes may contribute to the apparent negative dispersion observed in cancellous bone

Abstract

Previous work has shown that ultrasonic waves propagating through cancellous bone often exhibit a linear-with-frequency attenuation coefficient, but a decrease in phase velocity with frequency (negative dispersion) that is inconsistent with the causality-imposed Kramers–Kronig relations. In the current study, interfering wave modes similar to those observed in bone are shown to potentially contribute to the observed negative dispersion. Biot theory, the modified Biot–Attenborogh model, and experimental results are used to aid in simulating multiple-mode wave propagation through cancellous bone. Simulations entail constructing individual wave modes exhibiting a positive dispersion using plausible velocities and amplitudes, and then summing the individual modes to create mixed-mode output wave forms. Results of the simulations indicate that mixed-mode wave forms can exhibit negative dispersion when analyzed conventionally under the assumption that only one wave is present, even when the individual interfering waves exhibit positive dispersions in accordance with the Kramers–Kronig relations. Furthermore, negative dispersion is observed when little or no visual evidence of interference exists in the time-domain data. Understanding the mechanisms responsible for the observed negative dispersion could aid in determining the true material properties of cancellous bone, as opposed to the apparent properties measured using conventional data analysis techniques.

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

“…Conventional phase spectroscopy analysis performed on mixed mode signals as if only one wave were present can yield apparent negative dispersion. 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.…”

“…Previous studies have demonstrated that interfering fast wave and slow wave modes can account for the apparent negative dispersion sometimes observed in measurements of cancellous bone. [13][14][15][16][17] This apparent negative dispersion arises when conventional phase spectroscopy analysis of two overlapping waves is performed as if only one wave were present. Sometimes the presence of an additional wave is not apparent in the RF signal.…”

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

“…22,[25][26][27][28][29][30][31][32] However, many laboratories report phase velocity measurements that decrease with frequency, a phenomenon known as anomalous negative dispersion. 2,6,8,[20][21][22][33][34][35][36][37] In an effort to explain the observed phase velocity dispersion, our Laboratory proposed that conventional phase spectroscopy analysis of acquired radiofrequency data might be influenced by the presence of multiple interfering compressional wave modes.…”

“…However, in some circumstances, the two waves can strongly overlap during the time period over which the rf data are acquired, resulting in interference and difficulties in distinguishing between the two waves. 5,15,17,18,20,21 Ultrasonic measurements from many laboratories have revealed that cancellous bone exhibits a linear or nearlylinear increase in attenuation coefficient with frequency.…”

Fabric dependence of quasi-waves in anisotropic porous media

“…However, the existence of a second wave in cancellous bone has been reported in vitro Otani, 1997, 1998;Cardoso et al, 2001Cardoso et al, , 2003Mizuno et al, 2009). These two waves propagate with different velocities (Wear et al, 2005;Anderson et al, 2008Anderson et al, , 2009Wear, 2009;Nguyen et al, 2010;Wear, 2010) and have been shown to correspond to the fast and slow waves predicted by Biot's (Biot, 1941, 1955, 1956a,b, 1962a poroelastic wave propagation theory. Therefore, a poroelastic wave propagation theory is conceptually more appropriate than an equivalent media approach to characterize the properties of the porous medium.…”