Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Application of the Kramers-Kronig relations to Measurements of Attenuation and Dispersion in Cancellous Bone



Kendall Waters, B K. Hoffmeister, J A. Javarone


Ultrasonic (U5) assessment of fracture risk in osteoporotic patients is based on measurements of broadband ultrasonic attenuation and speed-of-sound (505) of cancellous bone. Discrepancies in estimates of 505 can result from dispersion and frequency-dependent attenuation. An improved understanding of the relation between attenuation and dispersion in cancellous bone could benefit numerical and experimental investigations of the effects of these properties on estimates of 505. The Kramers- Kronig (K-K) dispersion relations are used for the investigation of attenuation and dispersion. We consider a finite-bandwidth form of the K-K relations that uses the U5 properties over only the experimental bandwidth. In vitro measurements of cancellous bone are 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-Iateral (ML) and superior-inferior (51) directions. Measurements were performed using a through-transmission, immersion technique. The attenuation coefficient and phase velocity were determined for each specimen. Dispersion correlated well with apparent density for ML specimens (r=-0.95) but less so for 51 specimens (r=0.61). Finite-bandwidth K-K analysis of the attenuation coefficient predicted the corresponding phase velocity. Good agreement was observed between the measured and K-K predicted dispersion (r=0.99) for all specimens.
Proceedings Title
Proc., IEEE Intl. Ultrasonics Symp.
Conference Dates
August 24-27, 2004
Conference Location
Montreal, CA, USA
Conference Title
IEEE Intl. Ultrasonics Symp.


attenuation, cancellous bone, dispersion, fracture risk, Kramers-Kronig relations, osteoporosis, ultrasound characterization


Waters, K. , Hoffmeister, B. and Javarone, J. (2004), Application of the Kramers-Kronig relations to Measurements of Attenuation and Dispersion in Cancellous Bone, Proc., IEEE Intl. Ultrasonics Symp., Montreal, CA, USA, [online], (Accessed May 30, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created August 22, 2004, Updated October 12, 2021