Time-domain analysis of power law attenuation in space-fractional wave equations

Ultrasound attenuation in soft tissue follows a power law as a function of the ultrasound frequency, and in medical ultrasound, power law attenuation is often described by fractional calculus models that contain one or more time- or space-fractional derivatives. For certain time-fractional models, exact and approximate time-domain Green's functions are known, but similar expressions are not available for the space-fractional models that describe power law attenuation. To address this deficiency, a numerical approach for calculating time-domain Green's functions for the Chen-Holm space-fractional wave equation and Treeby-Cox space-fractional wave equation is introduced, where challenges associated with the numerical evaluation of a highly oscillatory improper integral are addressed with the Filon integration formula combined with the Pantis method. Numerical results are computed for both of these space-fractional wave equations at different distances in breast and liver with power law exponents of 1.5 and 1.139, respectively. The results show that these two space-fractional wave equations are causal and that away from the origin, the time-domain Green's function for the Treeby-Cox space-fractional wave equation is very similar to the time-domain Green's function for the time-fractional power law wave equation.