Comparison of Fractional Wave Equations for Power Law Attenuation in Ultrasound and Elastography

Abstract

A set of wave equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law wave equation and the causal fractional Laplacian wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional constitutive equations, whereas the former wave equations have been derived from the desire to model power law attenuation in applications like medical ultrasound. This has consequences for use in modeling and simulation, especially for applications that do not satisfy the low-frequency approximation, such as shear wave elastography. In such applications, the wave equations based on constitutive equations are the viable ones.

Ultrasonic attenuation and phase velocity of high-density polyethylene pipe material

“…11,14,17,[21][22][23][24][25][26] From which, it is usual to conclude that the most suitable description for attenuation of ultrasonic waves in a viscoelastic medium follows a power law,…”

“…One such example of a commonly used damping model is Kelvin-Voigt, but this loses accuracy via restriction to f 2 proportionality at ultrasonic frequencies. Causality is a necessary constraint on dispersion that couples the real and imaginary components of dispersion via the Kramers-Kronig relations, [26][27][28] …”

A multiband approach for accurate numerical simulation of frequency dependent ultrasonic wave propagation in the time domain

“…The properties of materials such as this can be described by extrapolation and inference-often from multiple decades below ultrasonic frequencies-of frequency-dependent material property parameters, such as storage and loss moduli, 3,4 provided that the modelling constraints described above do not apply. We also achieve an accurate description of ultrasonic wave propagation and scattering in the MBFE methodology without need for extrapolation across many frequency decades and the consequent inference of material properties, and without such modelling constraints as model size, found in similar FDFE models, using acoustic properties-attenuation and phase velocity.…”

Overturning of nonlinear acoustic waves in media with power-law attenuation

“…Time-domain models of wave propagation in media with power-law attenuation often employ a loss operator that is a fractional derivative with respect to either time or space [3]. The fractional derivative can be used as an ad hoc way to match measurements of power-law attenuation [4], or it can manifest as the cumulative effect of various random [5], viscoelastic [6,7], or relaxation [8] processes.…”