Some of the most compelling mathematical and physical models for the attenuation and dispersion of medical ultrasound in soft tissue are based on fractional calculus. Several of these models utilize fractional derivatives with respect to the temporal variables, and a few others define fractional derivatives with respect to the spatial variables. Of these fractional calculus models, the power law wave equation is particularly appealing due to several unique features. For example, the power law wave equation exactly describes power law attenuation of the form a(f) = a_0 |f|^y for all frequencies f, where a_0 is the attenuation constant and y is the power law exponent. In contrast, all of the other fractional calculus models only approximately describe power law attenuation over a limited range of frequencies. More importantly, the power law wave equation is the only fractional calculus model applied to medical ultrasound that has an exact time domain Green's function. The exact time domain Green's function for the power law wave equation enables the derivation of expressions that simultaneously describe the effects of diffraction, power law attenuation, and dispersion. Other useful properties also follow from the time domain Green's function for the power law wave equation.
This talk will introduce fractional calculus in the context of medical ultrasound, specifically focusing on the power law wave equation. The source of the fractional derivative in the context of power law attenuation will be shown, and then the power law wave equation and the exact time domain Green's function of the power law wave equation will be established. From this result, other useful time-domain expressions will be obtained. In addition, causal and noncausal models for power law attenuation will be compared, and a stochastic model that provides the underpinnings of a new explanation for the power law attenuation of ultrasound in soft tissue will also be presented.
Robert McGough is an Associate Professor in the Department of Electrical and Computer Engineering at Michigan State University. Prof. McGough obtained his B.E. in Electrical Engineering (summa cum laude) from Vanderbilt University, his M.S. in Electrical Engineering from the University of Illinois Urbana-Champaign, and his Ph.D. in Electrical Engineering from the University of Michigan. Prior to his arrival at Michigan State, he was an Assistant Research Professor in the Department of Radiation Oncology at Duke University Medical Center. His current research interests include rapid linear modeling of pressure fields generated by phased arrays for therapeutic and diagnostic applications, fractional calculus models of attenuation and dispersion, nonlinear ultrasound, shear wave imaging, phased array beamforming, and phased array transducer fabrication, all in the context of medical ultrasound. He is a member the Acoustical Society of America, IEEE, and the Society for Thermal Medicine. He is presently the Chair of the Biomedical Acoustics Technical Committee for the Acoustical Society of America.