Limitations of the Caputo time-fractional wave equation when applied to shear waves in pig liver

Abstract

When applied to compressional waves, the Caputo time-fractional wave equation describes power law attenuation in soft tissue. However, when applied to shear waves, the Caputo wave equation produces some unexpected behavior. To demonstrate examples of the time-domain waveforms generated by this fractional calculus model, the Caputo wave equation is numerically evaluated for multiple orders of the time-fractional derivative between 0 and 1, for an extended range of values for the relaxation time, and for a fixed value of the shear wave speed extracted from ex vivo pig liver. The computed waveforms are then compared to the measured shear wave particle velocities obtained from ex vivo pig liver. These comparisons reveal that the full-width at half maximum (FWHM) of the positive component of the main shear wave particle velocity waveform measured in pig liver is at least an order of magnitude greater than the FWHM obtained from the Caputo wave equation for all parameter combinations evaluated. These preliminary results indicate that the Caputo wave equation produces time-domain waveforms that are inconsistent with measured shear wave data in pig liver and that further efforts are required to establish more effective fractional calculus models for shear waves in soft tissue.