Application of the Kramers-Kronig relations to measurements of attenuation and dispersion in cancellous bone

Ultrasonic (US) assessment of fracture risk in osteoporotic patients is based on measurements of broadband US attenuation and speed of sound (SOS) of cancellous bone. Discrepancies in estimates of SOS can result from dispersion and frequency-dependent attenuation. An improved understanding of the relation between attenuation and dispersion could benefit both numerical and experimental investigations of the effects of these properties on estimates of SOS. The Kramers-Kronig (K-K) dispersion relations are used to investigate the relation between attenuation and dispersion. We consider a finite-bandwidth form of the K-K relations that uses the US properties over only the experimental bandwidth. In vitro measurements of cancellous bone were performed to validate this technique. Cancellous bone specimens were obtained from the proximal end of four bovine tibia and prepared with the axis of measurement along the medial-lateral and superior-inferior directions. Measurements were performed by use of a through-transmission, immersion technique. We observed good agreement between the measured and K-K predicted dispersion (r=0.99) for all specimens.