Time-domain analysis of power law attenuation in space-fractional wave equations

Ultrasound attenuation in soft tissue follows a power law as a function of the ultrasound frequency, and in medical ultrasound, power law attenuation is often described by fractional calculus models that contain one or more time- or space-fractional derivatives. For certain time-fractional models, e...

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Bibliographic Details
Published in:The Journal of the Acoustical Society of America 2018-07, Vol.144 (1), p.467-477
Main Authors: Zhao, Xiaofeng, McGough, Robert J.
Format: Article
Language:English
Subjects:
Acoustics
Algorithms
Biomedical Acoustics
Computer Simulation
Electromagnetic Phenomena
Fourier Analysis
Humans
Models, Theoretical
Numerical Analysis, Computer-Assisted
Sound
Time Factors
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Summary:Ultrasound attenuation in soft tissue follows a power law as a function of the ultrasound frequency, and in medical ultrasound, power law attenuation is often described by fractional calculus models that contain one or more time- or space-fractional derivatives. For certain time-fractional models, exact and approximate time-domain Green's functions are known, but similar expressions are not available for the space-fractional models that describe power law attenuation. To address this deficiency, a numerical approach for calculating time-domain Green's functions for the Chen–Holm space-fractional wave equation and Treeby–Cox space-fractional wave equation is introduced, where challenges associated with the numerical evaluation of a highly oscillatory improper integral are addressed with the Filon integration formula combined with the Pantis method. Numerical results are computed for both of these space-fractional wave equations at different distances in breast and liver with power law exponents of 1.5 and 1.139, respectively. The results show that these two space-fractional wave equations are causal and that away from the origin, the time-domain Green's function for the Treeby–Cox space-fractional wave equation is very similar to the time-domain Green's function for the time-fractional power law wave equation.
ISSN:0001-4966
1520-8524
DOI:10.1121/1.5047670