Fractal ladder models and power law wave equations
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Authors
Kelly, James F.
McGough, Robert J.
Subjects
Advisors
Date of Issue
2009-10
Date
October 2009
Publisher
Acoustical Society of America (ASA)
Language
Abstract
The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100 MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the Kramers–Kronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters.
Type
Article
Description
The article of record as published may be found at http://dx.doi.org/10.1121/1.3204304
Series/Report No
Department
Applied Mathematics
Organization
Identifiers
NPS Report Number
Sponsors
NIH Grant
No. 1R21 CA121235 J.F.K.
NRC Postdoctoral Associateship program.
NRC Postdoctoral Associateship program.
Funder
NIH Grant
No. 1R21 CA121235 J.F.K.
NRC Postdoctoral Associateship program.
NRC Postdoctoral Associateship program.
Format
Citation
Fractal ladder models and power law wave equations; James F. Kelly, Robert J. McGough; J Acoust Soc Am. 2009 Oct; 126(4): 2072–2081. doi: 10.1121/1.3204304; PMCID: PMC2771060; 327
Distribution Statement
Rights
This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.