Comparison of fractional wave equations for power law attenuation in ultrasound and elastography

TL;DR

This paper compares various fractional wave equations used for modeling power law attenuation in ultrasound and elastography, highlighting their theoretical foundations and implications for accurate simulation beyond low frequency approximations.

Contribution

It demonstrates that fractional Szabo, power law, and fractional Laplacian wave equations are low frequency approximations of more comprehensive fractional Kelvin-Voigt and Zener wave equations, emphasizing the importance of constitutive-based models.

Findings

Fractional Szabo and related equations are low frequency approximations.

Constitutive-based fractional wave equations are preferable for non-low frequency applications.

Implications for modeling shear wave elastography are discussed.

Abstract

A set of wave equations with fractional loss operators in time and space are analyzed. It is shown that the fractional Szabo equation, the power law wave equation, and the fractional Laplacian wave equation in the causal and non-causal forms all are 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 while the former wave equations are ad hoc, heuristic equations. We show that this has consequences for use in modelling 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 preferred ones.