A continuous power law wave equation for biomedical ultrasound

Abstract

The attenuation coefficient in most biological media is a power law with respect to frequency. Measurements indicate that the power law exponent y is near or exactly one for many classes of tissue. In J. Acous. Soc. Am. [124 2861–2872 (2008)], a power law wave equation (PLWE) was proposed to model this power law dependence using Riemann-Liouville fractional derivatives. The PLWE, like most other fractional calculus models for attenuation in the time-domain, is invalid for y =1 due to a discontinuity in the phase velocity. To address this problem, a continuous power law wave equation (CPLWE) is proposed that is valid for all power law exponents between zero and two. The CPLWE utilizes the recently developed Zolotarev fractional derivative of order y, which is a nonlocal operator even for y = 1. The 3D Green's function for the CPLWE that is valid for homogeneous media is derived using stable probability density functions in the Zolotarev M-parameterization. Solutions to the CPLWE that are valid for inhomoge...