Is there a role for approximations to the Kramers-Kronig relations in understanding the propagation of ultrasound in bone?

Abstract

Clinical measurements of the speed of sound (SOS) usually exhibit moderately large site-to-site variations within a given individual, leading to rather wide ranges for osteoporosis, osteopenia, and normal. Consequently, the rather modest variation of phase velocity over 200 kHz to 800 kHz predicted by the approximate form of the Kramers-Kronig relations developed in our laboratory would appear to be of limited interest. However, whereas the Kramers-Kronig relations predict a systematic increase in phase velocity with frequency, laboratories in Kyoto, Paris, and Washington, D.C., each reported a systematic decrease in phase velocity with increasing frequency. We suggested that this “anomalous” (negative) decrease in velocity with frequency was actually the consequence of interference between two waves, each of which satisfied the Kramers-Kronig-predicted positive dispersion, with one wave sufficiently stronger than the other that the combination appeared to be only a single wave. Although separation of the fast and slow waves by Bayesian, Prony’s, and other methods has provided substantial insight, significant unexplained results remain and are a focus of this presentation. The author wishes to acknowledge significant contributions by former colleagues: C. Anderson, A. Bauer, G. Brandenburger, L. Bretthorst, A. Groopman, J. Hoffman, S. Handley, M. Holland, M. Hughes, E. Jaynes, J. Katz, K. Marutyan, J. Mobley, R. Norberg, M. O’Donnell, R. Trousil, and K. Waters.