KZK Equation Research Articles

Dynamics and frequency response analysis of encapsulated microbubble under nonlinear ultrasound

Bubble dynamic behavior and frequency response of encapsulated microbubbles in nonlinear acoustic field is significant in applications such as tumor therapy, thrombolysis, tissue destruction, and ultrasonic lithotripsy. The acoustic cavitation effect includes stable cavitation and transient cavitation. The transformation from stable cavitation to transient cavitation requires a certain threshold, which is also called the transient cavitation threshold. Phospholipid-coated microbubbles are commonly used to enhance acoustic cavitation. However, the acoustic effects of different coating materials are not very clear, especially when considering the nonlinear effects caused by diffraction, scattering, and reflection during ultrasonic propagation. In this paper, the bubble dynamic behaviors and frequency responses of microbubbles under different frequencies, acoustic pressures, and viscoelastic properties of different shell materials are analyzed by coupling the Gilmore-Akulichev-Zener model with the nonlinear model of a lipid envelope and using the KZK equation to simulate the nonlinear acoustic field. At the same time, the influence of the coated material and nonlinear acoustic effects are considered. The bubble dynamic behavior and frequency response under the actually measured sound field are compared with those simulated by the KZK equation. The results show that the nonlinearity will lead the velocity of the microbubble wall to decrease, and when the pressure of ultrasound increases, the main frequency component of the microbubble oscillation increases, making the radial motion of the microbubble more violent. When the frequency changes, the closer the oscillation frequency of the microbubble is to the resonant frequency, the stronger the radial motion of the microbubble is. The coating material can change the harmonic component in the oscillation frequency. When the harmonic is close to the resonance frequency, the radial motion of the microbubble is enhanced. The elasticity of the coated material has almost no effect on the microbubble's frequency response, and the initial viscosity and surface tension of encapsulated microbubble will change the oscillation frequency distribution of encapsulated microbubble. When the initial viscosity of the coated microbubble is smaller, the subharmonic component of the microbubble oscillation increases. When the frequency of the subharmonic is closer to the resonance frequency than the main frequency, the acoustic cavitation effect is significantly enhanced. On the other hand, when the initial surface tension of the encapsulated microbubble increases, the main frequency and subharmonic component of the microbubble oscillation are enhanced, so that the acoustic cavitation effect is also enhanced. Therefore, this study can further elucidate the bubble dynamics of encapsulated microbubbles, stimulated by nonlinear ultrasound, benefiting the frequency response analysis of coated microbubbles under nonlinear acoustic fields.

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.

Spectral Shift Originating from Non-linear Ultrasonic Wave Propagation and Its Effect on Imaging Resolution

An amplitude-dependent downshift in the fundamental wave spectrum of a propagating ultrasonic pulse caused by non-linear wave propagation is described. The effects of non-linearity and the associated downshift on spatial resolution are also studied. The amounts of downshift and spatial resolution are extracted from the numerically simulated beam profile based on the KZK equation. Results for a 25-MHz transducer reveal that non-linear effects can lead to 58% additional downshift in the centre frequency of a pulse compared with a linear case with downshift caused only by attenuation. This additional downshift causes about 50% degradation in axial resolution. However, as the beam becomes narrower from the non-linear effects, the overall effect of non-linearity still leads to improved lateral resolution (≤26%). Therefore, as non-linearity increases with wave pressure, it is concluded that the increase in source pressure improves lateral resolution and degrades axial resolution.

The nonlinear field produced by 1-D and 2-D imaging arrays with plane and diverging wave modes for ultrafast imaging

Combining plane and diverging wave imaging (PWI/DWI) with tissue harmonic imaging (THI) may offer improvements in image quality in applications requiring very high frame rates (ultrafast). The beam shape and magnitude of the 2nd harmonic in tissue are important design considerations. We developed a numerical model based on the KZK equation and modeled three 1-D diagnostic arrays, L11-4v linear array, P4-2v phased array, and C5-2 convex array operating in PWI/DWI. Our numerical code predicts the nonlinear field of nonaxisymmetric sources and source configurations with elevation and no azimuthal foci (plane/diverging fields), which have not been modeled before. We showed that the second harmonic produced by ultrafast THI is 2-16 dB lower than that of focused beams for the imaging arrays considered when operated at the same maximum MI. This moderate difference of the second harmonic between PWI/DWI and focused ultrasound suggests that it is feasible to combine PWI/DWI and THI. We have also investigated harmonic generation produced by diagnostic 2-D arrays for 4-D THI. From our predictions for the second harmonic field we propose beamforming approaches for 4-D cardiac THI with focused ultrasound, PWI, and DWI that would result in frame rates of 10, 81, and 1275 Hz, respectively.

Nonlinear harmonic imaging of solids

Nonlinear ultrasonics has been shown to be sensitive to small scale damage and microstructural changes in solids. However, measuring the nonlinearity parameter of a solid requires complex calibrations and point-by-point measurements using contact transducers. This poses a challenge when it comes to applying this technique for imaging. The present work explores the use of immersion based nonlinear ultrasonics to perform nonlinear harmonic imaging of solids. A solid sample immersed in liquid (water) will form a “3-layer” structures (water-solid-water) when two ultrasonics transducers are used in through transmission mode. This work will explore the analytical modeling of such a structure using the KZK equation to model the combined effect of attenuation and diffraction of the three layers. The objective of the modelling effort is to develop an inversion model to invert the acoustic nonlinearity parameter from experimental results. The nonlinear model was further validated using experimental measurements carried out on a Steel sample with localized damaged.

The new exact analytical solutions and numerical simulation of (3 + 1)-dimensional time fractional KZK equation

The KZK parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3D diffraction sound beams in dissipative media. In this paper, the exact analytical solutions of (3 + 1)-dimensional time fractional KZK equation have been constructed in the sense of modified Riemann-Liouville derivative and the (G′/G)-expansion method, the simplest equation and the fractional complex transform. As a result, some new exact analytical solutions are obtained, and the effects of diffraction, attenuation and nonlinearity are researched deeply using the obtained exact analytical solutions.

The new exact analytical solutions and numerical simulation of (3 + 1)-dimensional time fractional KZK equation

The KZK parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3D diffraction sound beams in dissipative media. In this paper, the exact analytical solutions of (3 + 1)-dimensional time fractional KZK equation have been constructed in the sense of modified Riemann-Liouville derivative and the (G′/G)-expansion method, the simplest equation and the fractional complex transform. As a result, some new exact analytical solutions are obtained, and the effects of diffraction, attenuation and nonlinearity are researched deeply using the obtained exact analytical solutions.

Development of a freely available simulator with graphical interface for modeling nonlinear focused ultrasound fields with shocks

Measurement-based modeling is gaining acceptance as a standard tool for characterizing the nonlinear fields of existing therapeutic ultrasound devices and designing new ones. Here, a freely available simulation tool is presented for modeling axially symmetric, strongly nonlinear HIFU beams with shocks in a layered propagation medium such as water and different types of tissue. Two nonlinear wave equations are included in the simulator: the KZK equation generalized to include an equivalent source boundary condition for strongly focused beams and the Westervelt equation in a nonlinear wide-angle parabolic representation. Both equations are solved in the frequency domain and permit definition of the HIFU transducer as an annular array. The geometrical parameters and power output of the array, electronic focus steering along the beam axis, and acoustic properties of the layered propagation medium can be configured via graphical interface. Visualization and output of various acoustic field parameters such as peak positive and negative pressures, shock amplitude, intensity, heat deposition rate, and total power of a beam are also provided. The simulator can be used for transducers without ideal symmetry through definition of an equivalent radially symmetric source. Widespread availability of such simulation tools will help advance standardized utilization of measurement-based modeling and facilitate the adoption of such approaches for HIFU treatment planning. [Work supported by NIH R01EB7643, R01EB025187, and RSF №14-12-00974.]

Ultrasound Tissue Harmonic Imaging based on Wavelet Correlation Analysis

In this paper, an ultrasound tissue harmonic imaging method based on wavelet transform has been proposed. Wavelet transform is utilized to analyze the correlation between mother wavelet and the received echoes. In order to obtain a better correlation, a new mother wavelet named ‘Tissue wavelet’ is constructed according to the KZK equation. The proposed method was validated by experiments performed on an ultrasound flow phantom. Results showed that the contrast and axial resolution have been improved effectively.

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 of the Nonlinear Field Produced by Diagnostic Ultrasound Arrays in Plane Wave Mode

Tissue harmonic imaging (THI) improves images and reduces clutter in diagnostic ultrasound. Plane wave imaging (PWI), enables high frame rates facilitating such innovations as shear wave elastography and new Doppler capabilities. Nonlinear PWI may offer improvements in aberration and reverberation artifacts of synthetic imaging and 3D THI. Our objective is to model the nonlinear field of ultrasound arrays in PWI. Such a field has not been modeled before. A numerical solution of the KZK equation was used to model diagnostic ultrasound arrays. In PWI, the full aperture is used without focusing in the azimuthal direction while a fixed lens is used in elevation. We considered the L9-3 (Philips) array (38.2x9mm), elevation focus 20 mm, operating at 3.5MHz. The coefficient of nonlinearity was 5.5 and attenuation 0.3dB/cm-MHz. Our numerical solution was in agreement with Field II when considering linear propagation (differences ∼ 5%). The second harmonic was maximized in the azimuthal plane around 20mm due to the influence of the elevation focus. For MI=1.5 the second harmonic component was within 12dB of the fundamental but decreased with depth after 20mm. The 2nd harmonic of PWI mode was within 8dB of that from a conventional focused field. The nonlinear field of a diagnostic ultrasound array operating in PWI mode was successfully modeled. Combining PWI and nonlinear propagation will be beneficial in synthetic imaging and 3D THI, where frame rate and image quality is important.

Parametric array for tissue harmonic imaging

More than 50% of abdominal scans are performed on technically difficult patients due to worldwide increase of obesity. Tissue harmonic imaging has shown some improvements on obese patients but the problem remains unsolved. The use of dual frequency pulses allows the generation of sum and difference frequencies in tissue. We hypothesize that the difference frequency (Δf, parametric array) can survive larger penetration depths and increase SNR while maintaining the spatial characteristics of the primary wave. Our objective was to optimize the parameters for efficient generation of the parametric array and study its spatial characteristics for imaging obese patients. Numerical simulations using KZK equation and measurements in tissue-like media were performed to study parametric arrays. Pulses with two frequencies (f1, f2) were considered where 0.1f1≤Δf≤f1. Pulse inversion was utilized to isolate even harmonics, including Δf. The amplitude of Δf was maximized at Δf = f1. The axial resolution is not seriously compromised by the lower frequency of Δf when using PI. For 0.5 dB/cm attenuation difference between Δf and 2f1, Δf echoes from 20 cm deep were greater by 10 dB. Application of the parametric array on larger technically difficult patients would offer similar benefits as those in sonar applications.

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.]

Layer contributions to the nonlinear acoustic radiation from stratified media

This study presents the thorough investigation of the second harmonic generation scenario in a three fluid layer system. An emphasis is on the evaluation of the nonlinear parameter B/A in each layer from remote measurements. A theoretical approach of the propagation of a finite amplitude acoustic wave in a multilayered medium is developed. In the frame of the KZK equation, the weak nonlinearity of the media, attenuation and diffraction effects are computed for the fundamental and second harmonic waves propagating back and forth in each of the layers of the system. The model uses a gaussian expansion to describe the beam propagation in order to quantitatively evaluate the contribution of each part of the system (layers and interfaces) to its nonlinearity. The model is validated through measurements on a water/aluminum/water system. Transmission as well as reflection configurations are studied. Good agreement is found between the theoretical results and the experimental data. The analysis of the second harmonic field sources measured by the transducers from outside the stratified medium highlights the factors that favor the cumulative effects.

New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods

In this paper, new exact solutions of the time fractional KdV–Khokhlov–Zabolotskaya–Kuznetsov (KdV–KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann–Liouville derivative is used to convert the nonlinear time fractional KdV–KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV–KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV–KZK equation.

Characterization of medical ultrasound fields using modeling with a boundary condition obtained from measurements

Numerical modeling is becoming an important metrological tool for accurate characterization of ultrasound fields generated by medical transducers in water and in situ. While acoustic propagation equations are well established, setting a boundary condition relevant to the experiment still remains a critical problem. Two methods differing in complexity are described to address this problem. The first method comprises 3D simulations of the Westervelt equation with a boundary condition determined from acoustic holography measurements. The second, simpler method utilizes simulations based either on the KZK or Westervelt equation with an equivalent source boundary condition obtained by matching linear simulations to low-amplitude beam profiles along and transverse to the axis of the source. Calculations with both methods are compared to fiber optic hydrophone measurements of 2D phased therapeutic arrays, a 128-element diagnostic probe, and strongly focused single-element transducers. It is shown that the 3D Westervelt model with holographic boundary condition can accurately simulate the entire nonlinear ultrasound field. The simplified methods based on an equivalent source are shown to give accurate results in the focal zone of the transducers, even when shocks are present in the focal waveform. [Work supported by NIH EB7643, EB016118, DK043881, and RSF 14-12-00974.]

Significance of accurate diffraction corrections for the second harmonic wave in determining the acoustic nonlinearity parameter

The accurate measurement of acoustic nonlinearity parameter β for fluids or solids generally requires making corrections for diffraction effects due to finite size geometry of transmitter and receiver. These effects are well known in linear acoustics, while those for second harmonic waves have not been well addressed and therefore not properly considered in previous studies. In this work, we explicitly define the attenuation and diffraction corrections using the multi-Gaussian beam (MGB) equations which were developed from the quasilinear solutions of the KZK equation. The effects of making these corrections are examined through the simulation of β determination in water. Diffraction corrections are found to have more significant effects than attenuation corrections, and the β values of water can be estimated experimentally with less than 5% errors when the exact second harmonic diffraction corrections are used together with the negligible attenuation correction effects on the basis of linear frequency dependence between attenuation coefficients, α2 ≃ 2α1.

The effect of KZK pressure equation on the sonoluminescence in water and fat tissues

The effect of the produced light flashes from sonoluminescence (SL) on the fat tissue and water is studied. By using KZK equation as an essential equation for calculating the thermal source in bio-liquids, the effective bubble parameters in quasi-adiabatic model are calculated and compared in these systems. It is noticed that the temperature and the intensity for fat tissue are about 30% and 38% less than the ones for water respectively. These results are almost in good agreement with the only experimental measurement denoting less SL temperature in bio-liquids which present more suitable condition for using SL in such applications.

Harmonic generation with a dual frequency pulse.

Nonlinear imaging was implemented in commercial ultrasound systems over the last 15 years offering major advantages in many clinical applications. In this work, pulsing schemes coupled with a dual frequency pulse are presented. The pulsing schemes considered were pulse inversion, power modulation, and power modulated pulse inversion. The pulse contains a fundamental frequency f and a specified amount of its second harmonic 2f. The advantages and limitations of this method were evaluated with both acoustic measurements of harmonic generation and theoretical simulations based on the KZK equation. The use of two frequencies in a pulse results in the generation of the sum and difference frequency components in addition to the other harmonic components. While with single frequency pulses, only power modulation and power modulated pulse inversion contained odd harmonic components, with the dual frequency pulse, pulse inversion now also contains odd harmonic components.