Comparison of the Khokhlov–Zabolotskaya–Kuznetsov equation and Fourier-continuation for modeling high-intensity focused ultrasound beams.

Abstract

High-intensity focused ultrasound (HIFU) employs acoustic amplitudes that are high enough that nonlinear propagation effects are important in the evolution of the sound field. A common model for HIFU beams is the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation which accounts for nonlinearity and absorption of sound beams. The KZK equation models diffraction using the parabolic or paraxial approximation. For many HIFU sources the source aperture is similar to the focal length and the paraxial approximation may not be appropriate. Here we compare results obtained using the “Texas code” a time-domain numerical solution to the KZK equation to a new method of solving nonlinear differential equations: the Fourier-continuation (FC) method. Here the FC method solves the underlying fluid dynamics equations and does not invoke the paraxial approximation. Results were obtained for a 1.1-MHz focused HIFU source. The medium parameters were taken to either be hyperviscous water or to match human liver. For a low focusing gain transducer (focal length 50λ and radius 10λ) the KZK and FC models showed excellent agreement through the focal region. As the source radius was increased, discrepancies started to appear and for a radius of 30λ, the KZK model did not capture diffraction of the HIFU source accurately. [Work supported by NSF 0835795.]