Analytical and numerical approximations for the lossy on-axis impulse response of a circular piston

Abstract

In biological tissues, the frequency dependence of attenuation and speed of sound for ultrasound propagation is described by the power law wave equation, which is a partial differential equation with fractional time derivatives. As demonstrated previously, the time domain Green's function for the power law wave equation combined with the Rayleigh-Sommerfeld integral is an effective reference for calculations of the lossy on-axis impulse response of a circular piston. Using the result obtained from this reference, two different approximations to the lossy on-axis impulse response are evaluated. The first approximation is an analytical expression that is proportional to the difference between two cumulative distribution functions for maximally skewed stable probability densities. The second approximation numerically convolves the lossless impulse response with a maximally skewed stable probability density function. The results show that both approximations achieve relatively small errors. Furthermore, the analytical approximation provides an excellent estimate for the arrival time of the lossy impulse response, whereas the departure time of the lossy impulse response is more difficult to characterize due to the heavy tail of the maximally skewed stable probability density function. Both approximations are rapidly calculated with the STABLE toolbox. [supported in part by NIH Grant R01 EB012079.]