A time-domain fractional calculus model for shear wave parameter estimation

Abstract

A fractional calculus model that describes the effects of propagation and attenuation is introduced for shear wave parameter estimation. This fractional calculus model describes propagation with integer-order derivatives and power law atten-uation with a time-fractional derivative. This model is initially evaluated for the power law exponent <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$y=2$</tex> , which describes frequency-squared attenuation. Alignment between measured shear wave particle velocity in pig liver within the focal plane and the frequency-squared attenuation model are assessed in the time-domain, where the lack of agreement suggests that some other power law exponent is required for this pig liver data. The power law exponent <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$y = 0.9$</tex> is then evaluated within the fractional calculus model, which is evaluated with Riemann-Liouville fractional derivative. The results show that the fractional calculus model evaluated with Riemann-Liouville fractional derivative is unable to achieve alignment with the measured pig liver data, where the source of this problem is the singularity in the phase speed that is caused by the Riemann-Liouville fractional derivative. The problem is solved when the fractional derivative is instead evaluated with the Zolotarev fractional derivative, which enables excellent agreement between the measured shear wave data and the optimized waveform obtained with the proposed fractional calculus model.