The Khokhlov - Zabolotskaya - Kuznetsov (KZK) equation with power law attenuation

The Khokhlov - Zabolotskaya - Kuznetsov (KZK) equation is a parabolic approximation to the Westervelt equation that models the effects of diffraction, attenuation and nonlinearity. The original form of the attenuation term in the KZK model yields frequency squared attenuation. However, for soft tissues, the attenuation follows a power law with respect to frequency with a power law exponent between 1 and 1.5. A KZK model that incorporates the effects of frequency-squared attenuation is presently included in FOCUS, the `Fast Object-oriented C++ Ultrasound Simulator´, and a nonlinear KZK simulation that accounts for power law attenuation is also needed. When the KZK equation is combined with the power law wave equation [4], a continuous wave KZK model with power law attenuation is obtained. Simulation results are evaluated in both linear lossy media and nonlinear lossy media. Simulations in the linear lossy medium are evaluated with the KZK approach and compared to results obtained with the fast nearfield method in FOCUS. The results match well in the paraxial region but differ elsewhere. The effects of nonlinearity are then included in CW KZK calculations that model the contributions from several different harmonics. The nonlinear pressure field is generated by a spherically focused transducer with an aperture radius of 7.5mm and a radius of curvature of 60mm. The peak pressure on the surface of the transducer is 1.5MPa and the fundamental frequency is 1MHz. The lossy medium is defined for brain with β = 4.3, α = 0.6 dB/cm/MHz^y, and y = 1.46, and blood with β = 4.05, α = 0.16 dB/cm/MHz^y, and y = 1.2. The number of harmonics computed in each simulation is Nmax = 20. The pressure fields for the first four harmonics are shown.