Khokhlov Zabolotskaya Kuznetsov Equation Research Articles

Weakly nonlinear focused ultrasound in viscoelastic media containing multiple bubbles

To facilitate practical medical applications such as cancer treatment utilizing focused ultrasound and bubbles, a mathematical model that can describe the soft viscoelasticity of human body, the nonlinear propagation of focused ultrasound, and the nonlinear oscillations of multiple bubbles is theoretically derived and numerically solved. The Zener viscoelastic model and Keller–Miksis bubble equation, which have been used for analyses of single or few bubbles in viscoelastic liquid, are used to model the liquid containing multiple bubbles. From the theoretical analysis based on the perturbation expansion with the multiple-scales method, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation, which has been used as a mathematical model of weakly nonlinear propagation in single phase liquid, is extended to viscoelastic liquid containing multiple bubbles. The results show that liquid elasticity decreases the magnitudes of the nonlinearity, dissipation, and dispersion of ultrasound and increases the phase velocity of the ultrasound and linear natural frequency of the bubble oscillation. From the numerical calculation of resultant KZK equation, the spatial distribution of the liquid pressure fluctuation for the focused ultrasound is obtained for cases in which the liquid is water or liver tissue. In addition, frequency analysis is carried out using the fast Fourier transform, and the generation of higher harmonic components is compared for water and liver tissue. The elasticity suppresses the generation of higher harmonic components and promotes the remnant of the fundamental frequency components. This indicates that the elasticity of liquid suppresses shock wave formation in practical applications.

Ultrasonic nonlinear fields generated from transmitters with varied aperture angles

The transmitter aperture dependence of the ultrasonic nonlinear field was investigated by using Westervelt equation, Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation and spheroidal beam equation (SBE) models. 1.0-MHz focusing spherical transducers with varied half aperture angles from 10° to 40° was employed as the transmitter, the curvatures of which were fixed as 8 cm. The results show that: (1) KZK equation could simulate the acoustic distribution well for half aperture angle less than 20°; (2) the divergence between KZK equation and Westervelt equation is in proportion to the half aperture angle and surface pressure amplitude; (3) although the focus pressure was different for KZK and Westervelt equation, the shape of focal region of KZK equation was almost the same of Westervelt equation; and (4) the SBE model could simulate the acoustic distribution for a large aperture angle range.

Weakly nonlinear propagation of focused ultrasound in bubbly liquids with a thermal effect: Derivation of two cases of Khokolov–Zabolotskaya–Kuznetsoz equations

A physico-mathematical model composed of a single equation that consistently describes nonlinear focused ultrasound, bubble oscillations, and temperature fluctuations is theoretically proposed for microbubble-enhanced medical applications. The Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation that has been widely used as a simplified model for nonlinear propagation of focused ultrasound in pure liquid is extended to that in liquid containing many spherical microbubbles, by applying the method of multiple scales to the volumetric averaged basic equations for bubbly liquids. As a result, for two-dimensional and three-dimensional cases, KZK equations composed of the linear combination of nonlinear, dissipation, dispersion, and focusing terms are derived. Especially, the dissipation term depends on three factors, i.e., interfacial liquid viscosity, liquid compressibility, and thermal conductivity of gas inside bubbles; the thermal conduction is evaluated by using four types of temperature gradient models. Finally, we numerically solve the derived KZK equation and show a moderate temperature rise appropriate to medical applications.

Solitary Wave Solutions of the (3+1)-dimensional Khokhlov–Zabolotskaya–Kuznetsov Equation by using the (G'/G,1/G)-Expansion Method

In this study, the (3+1)-dimensional Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, which is a mathematical model of non-absorption and dispersion in the non-linear medium, which sheds light on the sound beam phenomenon, which has a physically important place, is examined. In order to find the exact solution of this equation, an effective and reliable method, (G^'/G,1/G)-expansion method, is used among analytical methods. The purpose of this method is to obtain more than one traveling wave solution classes depending on the conditions of the λ parameter. These classes are categorized into hyperbolic, trigonometric, complex trigonometric and rational forms. The graphics of the solitary waves represented by these successfully obtained solution classes are presented as 2-dimensional, 3-dimensional and contours. This article makes use of ready-made package programs for complex arithmetic operations and graphic drawings.

MEFM For Exact Solutions Of The (3+1) Dimensional KZK Equation and (3+1) Dimensional JM Equation

In this study, we have applied the modified exp -expansion function method (MEFM) to obtain the exact travelling wave solutions for the (3+1) dimensional Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation and (3+1)-dimensional Jimbo-Miwa (JM) equation. Dark soliton solution, dark-bright soliton solution, hyperbolic function solution and trigonometric function solution of the (3+1) dimensional KZK equation and (3+1)-dimensional JM equation have been found by using this method. After that, we have scratched the 2D and 3D graphs for all exact solutions obtained in this study by using Wolfram Mathematica 9. Thus, the graphical simulations openly show force of this method.

Determination of the Ultrasonic Non-linearity Parameter B/A versus Frequency for Water

For tissue characterization, it is desirable to determine B/A using high-frequency transducers. Moreover, an accurate estimate of B/A at elevated frequencies (or at least the assumption of frequency independence of B/A) is required to evaluate the safety of high-frequency systems. However, common finite-amplitude approaches become increasingly inaccurate at high frequencies. In this article, a practical variation of the finite-amplitude method is proposed which combines experiments with numerical simulations of the Khokhlov–Zabolotskaya–Kuznetsov equation and can be used at elevated frequencies. The results at low frequencies show that the proposed approach is accurate with lower uncertainties compared with the finite-amplitude method because it avoids assumptions and approximations. The measured values of B/A versus frequency for water at 2.25–20 MHz show that there is no statistically significant variation in B/A values with frequency, and therefore the assumption of frequency independence of B/A is realistic.

Codimension two Lie invariant solutions of the modified Khokhlov–Zabolotskaya–Kuznetsov equation

Invariant solutions for the modified Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation are obtained by using classical Lie symmetries. The complete set of local point symmetries is established for modified KZK equation governing the propagation of finite amplitude. An optimal set of two‐dimensional inequivalent subalgebras of the maximal Lie invariance algebra is constructed. The optimality among the subalgebras of Lie algebra is proved in a constructive manner by using rank of coefficient matrix of general two‐dimensional element and successive application of adjoint actions. On the basis of these subalgebras, we carry out group invariant reductions and compute exact solutions for different classes of subalgebras in an optimal system. Mathematical and physical behaviors of different invariant solutions are shown graphically demonstrating that classical Lie symmetries are capable of solving the modified KZK equation.

Theoretical Improvement of a KZK equation for focused ultrasound in bubbly liquids with thermal effects

Toward medical applications such as HIFU treatment, weakly nonlinear evolution of focused ultrasound in initially quiescent bubbly liquids is theoretically investigated with a special attention to the consideration of a thermal effect. Although our group (Kanagawa, JASA, 2015) was derived a Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation for quasi-plane progressive ultrasound with initially nonuniform number density of microbubbles, thermal effects such as thermal conduction were completely neglected. In this paper, we modify our previous KZK equation by incorporating thermal effects, i.e., the thermal conduction and thermodynamics inside bubble, and the liquid viscous damping in bulk liquid. By using the method of multiple scales to basic equations for bubbly liquids introducing an energy equation proposed by Prosperetti (JFM, 1991), a revised KZK equation incorporating all the thermal effects was derived. Although a newly discovered dissipation term without differentiation corresponds to the thermal conduction, the coefficient of original dissipation term was expressed by the liquid viscosity and liquid compressibility (i.e., acoustic radiation damping). Hence, the physics of dissipation by thermal conduction differs that by liquid viscosity and compressibility. Furthermore, we emphasize that a thermal effect contributes the nonlinear effect, that is, nonlinear coefficient include the ratio of specific heats of the gas inside the bubble.

Temperature Dependence of Tissue Thermal Parameters Should Be Considered in the Thermal Lesion Prediction in High-Intensity Focused Ultrasound Surgery

This study considers the temperature-dependent thermal parameters (specific heat capacity, thermal diffusivity and thermal conductivity) used when predicting the temperature rise of tissue exposed to high-intensity focused ultrasound (HIFU). Numerical analysis was performed using the Khokhlov–Zabolotskaya–Kuznetsov equation coupled with a bioheat transfer function. The thermal parameters were set as the functions of temperature using experimental data. The results revealed that, for liver tissue exposed to HIFU with a focal intensity of 3000 W/cm2 for 10 s, the predicted focal temperature rise was 23% lower and the thermal lesion area 41% smaller than those predicted without considering the temperature dependence. The prediction was validated by experimental observations on thermal lesions visualized in a tissue-mimicking phantom. The present results suggest that temperature-dependent thermal parameters should be considered in the prediction of HIFU-induced temperature rise to avoid lowering ultrasonic output settings for HIFU surgery. The aim of the present study was to investigate how significantly the temperature dependence of the thermal parameters affects the thermal dose imposed on the tissue by a typical clinical HIFU device. A numerical simulation was performed using a thermo-acoustic algorithm coupling the non-linear Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation (Meaney et al. 1998; Filonenko and Khokhlova 2001) and a bio-heat transfer (BHT) equation (Pennes 1948). Thermal parameters of liver tissue were modeled in the present study as functions of temperature and were incorporated into the BHT equation to compensate for the variations in thermal parameters with temperature. Experimental validation was achieved by comparing the predictions with the thermal lesions formed in the tissue-mimicking phantoms.

Numerical and Experimental Studies on Inclined Incidence Parametric Sound Propagation

The Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation has been widely used in the simulation and calculation of nonlinear sound fields. However, the accuracy of KZK equation reduced due to the deflection of the direction of the sound beam when the sound beam is inclined incidence. In this paper, an equivalent sound source model is proposed to make the calculation direction of KZK calculation model consistent with the sound propagation direction after acoustic refraction, so as to improve the accuracy of sound field calculation under the inclined incident conditions. The theoretical research and pool experiment verify the feasibility and effectiveness of the proposed method.

Wide-Angle Parabolic Approximation for Modeling High-Intensity Fields from Strongly Focused Ultrasound Transducers

A novel numerical algorithm based on the wide-angle parabolic approximation is developed for modeling linear and nonlinear fields generated by axially symmetric ultrasound transducers. An example of a strongly focused single-element transducer is used to compare the results of ultrasound field simulations based on the Westervelt equation, Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation with differently modified boundary condition, and nonlinear wide-angle parabolic equation. It is demonstrated that having a computational speed comparable to modeling the KZK equation, the use of wide-angle parabolic approximation makes it possible to obtain solutions for highly focused ultrasound beams that are closer in accuracy to solutions based on the Westervelt equation.

Prediction of HIFU Propagation in a Dispersive Medium via Khokhlov–Zabolotskaya–Kuznetsov Model Combined with a Fractional Order Derivative

High intensity focused ultrasound (HIFU) has been proven to be promising in non-invasive therapies, in which precise prediction of the focused ultrasound field is crucial for its accurate and safe application. Although the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation has been widely used in the calculation of the nonlinear acoustic field of HIFU, some deviations still exist when it comes to dispersive medium. This problem also exists as an obstacle to the Westervelt model and the Spherical Beam Equation. Considering that the KZK equation is the most prevalent model in HIFU applications due to its accurate and simple simulation algorithms, there is an urgent need to improve its performance in dispersive medium. In this work, a modified KZK (mKZK) equation derived from a fractional order derivative is proposed to calculate the nonlinear acoustic field in a dispersive medium. By correcting the power index in the attenuation term, this model is capable of providing improved prediction accuracy, especially in the axial position of the focal area. Simulation results using the obtained model were further compared with the experimental results from a gel phantom. Good agreements were found, indicating the applicability of the proposed model. The findings of this work will be helpful in making more accurate treatment plans for HIFU therapies, as well as facilitating the application of ultrasound in acoustic hyperthermia therapy.

Investigation of thermomechanical motion in a nanomechanical resonator based on optical intensity mapping

In the present study, the acoustic field of a high intensity focused ultrasound (HIFU) transducer in water was measured by using a commercially available needle hydrophone intended for HIFU use. To validate the results of hydrophone measurements, numerical simulations of HIFU fields were performed by integrating the axisymmetric Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation from the frequency-domain perspective with the help of a MATLAB-based software package developed for HIFU simulation. Quantitative values for the focal waveforms, the peak pressures, and the size of the focal spot were obtained in various regimes of linear, quasilinear, and nonlinear propagation up to the source pressure levels when the shock front was formed in the waveform. The numerical results with the HIFU simulator solving the KZK equation were compared with the experimental data and found to be in good agreement. This confirms that the numerical simulation based on the KZK equation is capable of capturing the nonlinear pressure field of therapeutic HIFU transducers well enough to make it suitable for HIFU treatment planning.

Microvessel rupture induced by high-intensity therapeutic ultrasound\u2014a study of parameter sensitivity in a simple in vivo model

BackgroundSafety analyses of transcranial therapeutic ultrasound procedures require knowledge of the dependence of the rupture probability and rupture time upon sonication parameters. As previous vessel-rupture studies have concentrated on a specific set of exposure conditions, there is a need for more comprehensive parametric studies.MethodsProbability of rupture and rupture times were measured by exposing the large blood vessel of a live earthworm to high-intensity focused ultrasound pulse trains of various characteristics. Pressures generated by the ultrasound transducers were estimated through numerical solutions to the KZK (Khokhlov-Zabolotskaya-Kuznetsov) equation. Three ultrasound frequencies (1.1, 2.5, and 3.3 MHz) were considered, as were three pulse repetition frequencies (1, 3, and 10 Hz), and two duty factors (0.0001, 0.001). The pressures produced ranged from 4 to 18 MPa. Exposures of up to 10 min in duration were employed. Trials were repeated an average of 11 times.ResultsNo trends as a function of pulse repetition rate were identifiable, for either probability of rupture or rupture time. Rupture time was found to be a strong function of duty factor at the lower pressures; at 1.1 MHz the rupture time was an order of magnitude lower for the 0.001 duty factor than the 0.0001. At moderate pressures, the difference between the duty factors was less, and there was essentially no difference between duty factors at the highest pressure. Probability of rupture was not found to be a strong function of duty factor. Rupture thresholds were about 4 MPa for the 1.1 MHz frequency, 7 MPa at 3.3 MHz, and 11 MPa for the 2.5 MHz, though the pressure value at 2.5 MHz frequency will likely be reduced when steep-angle corrections are accounted for in the KZK model used to estimate pressures. Mechanical index provided a better collapse of the data (less separation of the curves pertaining to the different frequencies) than peak negative pressure, for both probability of rupture and rupture time.ConclusionThe results provide a database with which investigations in more complex animal models can be compared, potentially establishing trends by which bioeffects in human vessels can be estimated.

Numerical evaluation of absorbing boundary layers for the transient Khokhlov–Zabolotskaya–Kuznetsov Equation

FOCUS, the “Fast Object-Oriented C + + Ultrasound Simulator” (http://www.egr.msu.edu/~fultras-web), simulates nonlinear ultrasound propagation in the time-domain by numerically evaluating the transient Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation. KZK simulations in FOCUS previously required a computational grid with a large radial distance relative to the aperture radius to reduce the effect of reflections from the boundary. To decrease the size of the grid required for these calculations, an absorbing boundary layer derived from the power law wave equation provides an alternative to the stretched coordinate system in a perfectly matched layer. Simulations of the linear pressure fields generated by a spherically focused transducer with an aperture radius of 1.5 cm and a radius of curvature of 6 cm are evaluated for a short pulse with a center frequency of 1 MHz and a peak surface pressure of 0.5 MPa. Numerical results for linear KZK simulations with and without the absorbing boundary layer are compared to an on-axis analytical solution. Results of nonlinear KZK simulations with and without the PML are also evaluated, and all of these show that the absorbing layer effectively attenuates the wavefronts that reach the boundary of the computational grid. [This work was supported in part by NIH Grant R01 EB012079.]

Influence of Temperature-Dependent Thermal Parameters on Temperature Elevation of Tissue Exposed to High-Intensity Focused Ultrasound: Numerical Simulation

High-intensity focused ultrasound (HIFU) has been used successfully as a non-invasive modality in treating solid tumors. The temperature rise HIFU irradiation causes in a tissue depends on the thermal properties of the tissue. This study was motivated by our observation that the thermal properties of a tissue vary significantly with temperature (Guntur SR, Lee KI, Paeng DG, Coleman AJ, Choi MJ. Ultrasound Med Biol 2013;39:1771–1784). This research investigated how significantly the alteration of tissue thermal parameters, in the ranges of values measured at 25°C–90°C, affects prediction of the temperature elevation of tissue under the same HIFU exposure. The numerical simulation was performed by coupling a non-linear Khokhlov–Zabolotskaya–Kuznetsov equation with a bio-heat transfer function. In the conventional method of prediction, the thermal parameters were set as constants measured at room temperature (25°C). This study compared the conventional prediction with those predicted with different thermal parameters measured at the various temperatures up to 90°C. The results indicated that the conventional method significantly overestimated the rise in focal temperature in the liver tissue exposed to a clinical HIFU field, compared with the prediction made using thermal parameters measured at temperatures that cause thermal denaturation. This finding suggests that temperature-dependent thermal parameters should be considered in predicting the temperature rise in a tissue to avoid use of an insufficient thermal dose in treatment planning for HIFU surgery.

Implementation of a perfectly matched layer in nonlinear continuous wave ultrasound simulations

FOCUS, the "Fast Object-Oriented C + + Ultrasound Simulator" (http://www.egr.msu.edu/~fultras-web), simulates nonlinear ultrasound propagation by numerically evaluating the Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation. For continuous-wave excitations, KZK simulations in FOCUS previously required that the simulations extend over large radial distances relative to the aperture radius, which reduced the effect of reflections from the boundary on the main beam. To reduce the size of the grid required for these calculations, a perfectly matched layer (PML) was recently added to the KZK simulation routines in FOCUS. Simulations of the linear pressure fields generated by a spherically focused transducer with an aperture radius of 1.5 cm and a radius of curvature of 6cm are evaluated for a peak surface pressure of 0.5 MPa and a 1 MHz fundamental frequency. Results of linear KZK simulations with and without the PML are compared to an analytical solution of the linear KZK equation on-axis, and the results show that simulations without the PML require a radial boundary that is at least seven times the aperture radius, whereas the PML enables accurate simulations for a radial boundary that is only two times the aperture radius. [This work was supported in part by NIH Grant R01 EB012079.]

Application of the split-step Padé approach to nonlinear field predictions

We herein propose a new theoretical approach for analyzing the nonlinear propagation of directive sound beams emitted from a planar piston source with a circular aperture. The proposed approach relies on the split-step Padé approximation, which is an efficient method for obtaining wide-angle one-way wave equations, especially in underwater acoustics. Despite including only two Padé terms in the expansion, the theory was applicable to a beam angle of up to ±40° relative to the main propagation direction, the angle of which is approximately twice that of the Khokhlov–Zabolotskaya–Kuznetsov equation, which is based on parabolic approximation. In order to demonstrate the effectiveness of the newly proposed theoretical approach, we performed an experiment using an airborne ultrasonic emitter with a circular aperture of 7.5cm in radius. We drove the emitter powerfully at a 36-kHz and 40-kHz bi-frequency signal and measured the beam patterns of the primary and secondary waves, such as parametric sounds within wide propagation angles. Excellent agreement between measured data and the corresponding numerical simulations supports the validity of the proposed model equations and the computational methods for their numerical solutions.

Numerical simulation of parametric sound generation and its application to length-limited sound beam

In this study we propose a simulation model for predicting the nonlinear sound propagation of ultrasound beams over a distance of a few hundred wavelengths, and we estimate the beam profile of a parametric array. Using the finite-difference time-domain method based on the Yee algorithm with operator splitting, axisymmetric nonlinear propagation was simulated on the basis of equations for a compressible viscous fluid. The simulation of harmonic generation agreed with the solutions of the Khokhlov–Zabolotskaya–Kuznetsov equation around the sound axis except near the sound source. As an application of the model, we estimated the profiles of length-limited parametric sound beams, which are generated by a pair of parametric sound sources with controlled amplitudes and phases. The simulation indicated a sound beam with a narrow truncated array length and a width of about one-quarter to half that of regular a parametric beam. This result confirms that the control of sound source conditions changes the shape of a parametric beam and can be used to form a torch like low-frequency sound beam.

A Comparative Analysis of Preprocessing Methods for the Parametric Loudspeaker Based on the Khokhlov–Zabolotskaya–Kuznetsov Equation for Speech Reproduction

Based on the Berktay's farfield solution, various preprocessing methods were proposed to reduce the distortion of the highly directional audible signal in the parametric loudspeaker. However, the Berktay's farfield solution is an approximated model of nonlinear acoustic propagation. To determine the effectiveness of these methods, we analyze various preprocessing methods theoretically for directional speech reproduction using the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation, which provides a more accurate model of nonlinear acoustic propagation. In order to reduce the distortion effectively in the parametric loudspeaker with these preprocessing methods, the initial sound pressure level of the carrier frequency is found to be less than 132 dB according to the KZK equation. Unlike the Berktay' farfield solution that results in a +12 dB/octave gain slope, different gain slopes are derived using the KZK equation and appropriate equalizers are proposed to improve the frequency response of the parametric loudspeaker. The optimal preprocessing method for directional speech reproduction is established based on the KZK equation, which has a relatively flat frequency response of the desired speech signal and the best total harmonic distortion performance.