KZK Model Research Articles

Existence of Classical Solutions for a Class of the Khokhlov-Zabolotskaya-Kuznetsov Type Equations

In medical sciences, during medical exploration and diagnosis of tissues or in medical imaging, we often use mathematical models to answer questions related to these examinations. Among these models, the nonlinear partial differential equation of the Khokhlov--Zabolotskaya--Kuznetsov type (abbreviated as the KZK equation) is of proven interest in ultrasound acoustics problems. This mathematical model describes the nonlinear propagation of a sound pulse of finite amplitude in a thermo-viscous medium. The equation is obtained by combining the conservation of mass equation, the conservation of momentum equation and the equations of state. It should be noted that for this equation little mathematical analysis is reserved. This equation takes into account three combined effects: the diffraction of the wave, the absorption of energy and the nonlinearity of the medium in which the wave propagates. KZK-type equation introduced in this paper is a modified version of the KZK model known in acoustics. We study a class of~the~Khokhlov--Zabolotskaya--Kuznetsov type equations for the existence of global classical solutions. We~give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results, we propose a new approach based on recent theoretical results.

Numerical approaches to simulating nonlinear ultrasound fields generated by diagnostic-type transducers

Two theoretical approaches for simulating nonlinear focused ultrasound fields generated by a diagnostic convex array are compared. The first model is based on the three-dimensional Westervelt equation and describes the full structure of the array field with high accuracy. However, it requires great computational resources and is technically difficult. The second model is based on an axially symmetric form of the parabolic KZK equation for estimating the strength of nonlinear effects in the focal region of a beam, which reduces the computational time by a factor of several hundreds. To establish the boundary conditions to the KZK model, the radius and the focal length of a circular piston source are defined such that the simulated field on the beam axis in the linear case fits the real structure of the field in the focal region. It is shown that the parabolic model can be used to accurately describe the spatial and temporal structure of the field generated by a diagnostic transducer in the focal region of the beam along its axis and in the plane of the beam’s electronic focusing.

Two reduced-order approaches for characterizing the acoustic output of high-power medical ultrasound transducers

An approach that combines numerical modeling with measurements is gaining acceptance for field characterization in medical ultrasound. The general method is suitable for accurate simulation of the fields radiated by therapeutic transducers in both water and tissue. Here, three characterization methods are compared. Simulations based on the 3D Westervelt equation with a boundary condition determined from acoustic holography measurements are used as the most accurate benchmark method. Two simplified methods are based on an axially symmetric nonlinear parabolic formulation, either the KZK equation or its wide-angle extension. Various approaches for setting a boundary condition to the parabolic models are presented and discussed. Simulation results obtained with the proposed methods are compared for a typical therapeutic array and a strongly focused single-element transducer, with validation measurements recorded by a fiber optic hydrophone at the focus at increasing acoustic outputs. It is shown that the wide-angle parabolic model is more accurate than the KZK model in governing diffraction effects in the nearfields of the focused beams. However, both methods give accurate results in the focal zone, even at very high outputs when shocks are present. [Work supported by and RSF 14-12-00974, P01 DK043881, NIH EB7643, and NSBRI through NASA 9-58.]

Modeling shock-wave fields generated by a diagnostic-type transducer

New applications of ultrasound imaging may benefit from increased in situ pressure levels. However strong nonlinear propagation effects have not been studied in detail for fields generated by diagnostic transducers. Here we compare two modeling approaches for predicting shock formation in fields generated by a curvilinear imaging probe (C5-2). The field was simulated in two ways: 1) a 3D full-diffraction model based on the Westervelt equation, with a boundary condition defined on a cylindrical surface; 2) an axially symmetric parabolic model based on the KZK equation to define a flat, circular equivalent source. For both models, boundary conditions are adjusted to match low-power axial pressure measurements in the focal lobe of the beam. Simulations and measurements were performed for operation of 40, 64, and 128 central probe elements, and a wide range of clinically relevant output levels was considered. Comparison of focal waveforms shows that the KZK model can predict the amplitudes of fully developed shocks within 5% for 64 and 128 active elements, and within 15% for 40 elements. The full 3D calculations provide better agreement with experiments (within 3%) but require significantly more computational resources. [Work supported by RSF 14-12-00974 and a scholarship of the president of Russia.]

Characterization of Pressure Fields of Focused Transducers at TÜBİTAK UME

Field radiated by HIFU (High Intensity Focused Ultrasound) has been investigated by measuring its pressure field and mapping in 2-D and 3-D. A new ultrasound pressure measurement system has been designed and constructed at TÜBİTAK UME (The Scientific and Technological Research Council of Turkey, the National Metrology Institute). System consists of a water tank, positioning system, measurement devices and a controlling program. The hydrophone was attached to a 3-axis, computer-controlled positioning system for alignment with the ultrasound source. The signal was captured and analyzed by the commercially available LabVIEW 8.1 software. The measurements of the ultrasound field were carried out with a needle hydrophone. For each waveform, p, p+ and p-pressures have been calculated. Wave behaviors produced by the KZK model and from experiments look like similar in general. In p, p+, p- the focal point, zero point after the primary peak (focus) and extremum points in the near field well match.

파라메트릭 배열을 이용한 해저지층 탐사 알고리즘

In this paper, we propose an threshold-based Schur algorithm for estimating the media characteristics of sub-bottom multi-layers by using the signal generated by a parametric array transducer. We use the KZK model to generate a parametric array signal, and use the proposed threshold-based Schur algorithm for estimating the reflection coefficients of multiple sea bottom layers. Using computer simulation, we verify that the difference frequency component generated by the KZK model prevails over the signals of primary frequencies at long range. For the simulation, we use the transmit signal generated by the KZK and the reflected signal obtained from a lattice filter model for the seawater and sub-bottom of multi-level non-homogeneous layers. Through the simulation, we verify that the proposed threshold-based Schur algorithm can give much more accurate and efficient estimates of the reflection coefficients than methods using received signal, matched filter output signal, and normal Schur algorithm output.

Nonlinear focal shift beyond the geometrical focus in moderately focused acoustic beams

The phenomenon of the displacement of the position along the axis of the pressure, intensity, and radiation force maxima of focused acoustic beams under increasing driving voltages (nonlinear focal shift) is studied for the case of a moderately focused beam. The theoretical and experimental results show the existence of this shift along the axis when the initial pressure in the transducer increases until the acoustic field reaches the fully developed nonlinear regime of propagation. Experimental data show that at high amplitudes and for moderate focusing, the position of the on-axis pressure maximum and radiation force maximum can surpass the geometrical focal length. On the contrary, the on-axis pressure minimum approaches the transducer under increasing driving voltages, increasing the distance between the positive and negative peak pressure in the beam. These results are in agreement with numerical KZK model predictions and the existed data of other authors and can be explained according to the effect of self-refraction characteristic of the nonlinear regime of propagation.

Full‐diffraction and parabolic axisymmetric numerical models to characterize nonlinear ultrasound fields of two‐dimensional therapeutic arrays.

Numerical modeling has been shown to be an effective tool to characterize nonlinear pressure fields for single‐element HIFU transducers, but it has not yet been applied for the much more complex three‐dimensional (3‐D) fields generated by therapeutic phased arrays. In this work, two approaches are presented to simulate nonlinear effects in the field of a 256‐element focused array. A new full‐diffraction approach includes rigorous 3‐D simulations of the nonlinear wave equation with a boundary condition given at the elements of the array. A second simpler approach is based on the KZK model and a focused piston source as the boundary condition. The effective aperture and initial pressure of the piston source are set by matching linear simulations of the two models in the focal region. It is shown that as output power is increased, agreement in the focal waveforms of the two simulations, even when shocks were present, is maintained up to very high power outputs of the array. These results demonstrate the feas...

Computational aspects in high intensity ultrasonic surgery planning

Therapeutic ultrasound treatment planning is discussed and computational aspects regarding it are reviewed. Nonlinear ultrasound simulations were solved with a combined frequency domain Rayleigh and KZK model. Ultrasonic simulations were combined with thermal simulations and were used to compute heating of muscle tissue in vivo for four different focused ultrasound transducers. The simulations were compared with measurements and good agreement was found for large F-number transducers. However, at F # 1.9 the simulated rate of temperature rise was approximately a factor of 2 higher than the measured ones. The power levels used with the F # 1 transducer were too low to show any nonlinearity. The simulations were used to investigate the importance of nonlinarities generated in the coupling water, and also the importance of including skin in the simulations. Ignoring either of these in the model would lead to larger errors. Most notably, the nonlinearities generated in the water can enhance the focal temperature by more than 100%. The simulations also demonstrated that pulsed high power sonications may provide an opportunity to significantly (up to a factor of 3) reduce the treatment time. In conclusion, nonlinear propagation can play an important role in shaping the energy distribution during a focused ultrasound treatment and it should not be ignored in planning. However, the current simulation methods are accurate only with relatively large F-numbers and better models need to be developed for sharply focused transducers.

Lossy propagation of the Gulfstream Quiet Spike

The Gulfstream Quiet Spike is a telescoping nose extension with multiple segments of decreasing diameter intended to reduce the outdoor loudness of a sonic boom. This work examines the influence of the atmosphere on a propagated sonic boom signature for this device. First, atmospheric data, averaged by season, has been collected as a function of latitude to reasonably bound the range of meteorological conditions. These atmospheric profiles are subsequently input to nonlinear, lossy propagation code that computes the primary sonic boom carpet for a supersonic aircraft in steady, level flight. This code is an augmented implementation of the KZK model equation for one-dimensional propagation. The confluence of shocks generated by the Quiet Spike remains distinct and do not coalesce, as intended per the design. The amplitudes, rise times and spacing of the shocks from the individual spike segments are strongly governed by dispersion and attenuation. Loudness levels calculated from the resulting ground signatures show an observable change of over 4 dB at certain azimuths due to the various atmospheric conditions. Based on these results, minor design changes are recommended to improve the acoustic performance of the Gulfstream Quiet Spike.

Khokhlov–Zabolotskaya–Kuznetsov type equation: nonlinear acoustics in heterogeneous media

The KZK type equation introduced in this Note differs from the traditional form of the KZK model known in acoustics by the assumptions on the nonlinear term. For this modified form, a global existence and uniqueness result is established for the case of non-constant coefficients. Afterwards the asymptotic behaviour of the solution of the KZK type equation with rapidly oscillating coefficients is studied. To cite this article: I. Kostin, G. Panasenko, C. R. Mecanique 334 (2006).

Investigation of superharmonic sound propagation and imaging in biological tissues in vitro

This article presents both theoretical and experimental studies on the superharmonic generation and its imaging in biological tissues. A superharmonic component is defined as a summation of the third-, fourth-, and fifth-order harmonics. A superharmonic signal is produced using an 8-mm-diam, 2.5-MHz planar piston source that is excited by eight-cycle, 2.5-MHz tone bursts. Axial and lateral field distributions of the superharmonic component and the second harmonic are first calculated based on the nonlinear KZK model and then compared with those experimentally determined at two different source pressures of 0.5 and 1 MPa. Results indicate that the amplitude of the superharmonic component can exceed that of the second harmonic, depending on the axial distance and the fundamental pressure amplitude. Also, the 3-dB beamwidth of the superharmonic component is about 23% narrower than that of the second harmonic. Additional experiments are performed in vitro using liver and fatty tissues in transmission mode and produced two-dimensional images using the fundamental, the second harmonic, and the superharmonic signals. Although the clinical applicability of this work still needs to be assessed, these results indicate that the superharmonic image quality is better than that of the other two images.

A new time-domain approach for nonlinear wave propagation: Comparison with the KZK equation approach in the case of an unfocused cw beam

A time-domain numerical model for simulating acoustic pulse focusing in soft tissue has already been presented. In this model, the main effects responsible for finite-amplitude wave propagation, i.e., diffraction, frequency-dependent attenuation, and nonlinearity, were taken into account. Here, a comparison between the results of this model with those of the KZK model is presented. Using this model, the acoustic pressure field of a plane circular transducer was simulated in water for a large range of cw-mode input excitation levels. The on- and off-axis pressure-time waveforms and their corresponding harmonic components were calculated using this model and the KZK model, and very good agreement was obtained between the two models. Furthermore, these results were verified by previously published experimental measurements. At present, all implementations of the KZK model are limited to planar or weakly focused sources. This study, along with this group’s previous work, reveals the validity of this finite-amplitude wave propagation model for a wide degree of source focusing (from a planar to a highly focused source), and in cw as well as PW modes. It is also shown that the computation time for this model is much less than for the KZK, using the same input parameters.