Fractal ladder models and power law wave equations.
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.
Kelly
James F
JF
Department of Applied Mathematics, Naval Postgraduate School, Monterey, California 93943, USA. jfkelly@nps.edu
McGough
Robert J
RJ
D000162:Acoustics; D000465:Algorithms; D000818:Animals; D002468:Cell Physiological Phenomena / Q000502:physiology; D004548:Elasticity; D017709:Fractals; D016014:Linear Models; D008322:Mammals; D008954:Models, Biological; D017711:Nonlinear Dynamics; D013314:Stress, Mechanical; D014783:Viscosity