Simulations of nonlinear continuous wave pressure fields in FOCUS

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.

In this paper the nonlinear pulsed acoustic pressure fields from the focused square and rectangular apertures are considered. Experimental results in water of the 4D nonlinear sound field radiated from a 2.8 MHz focused (focal distance F = 80 mm) transducer of a square (20 x 20 mm) and rectangular (10 x 25 mm) geometry for various excitation levels (producing the average pressure P0 at the source equal to 0.045, 0.225 and 0.45 MPa) are presented. The measurement results are compared with the numerical calculations obtained for the same boundary conditions. The novel, developed by the second author, numerical algorithm was used to simulate the 4D nonlinear pressure field propagation from nonaxisymmetric focused acoustic sources radiating pulsed waves in the attenuating media. Our theoretical model is based on the Time-Averaged Pressure Envelope (TAPE) method, recently developed in our lab. This method employs the representation of the propagated pulsed pressure waveform as a quasi-Fourier series, i.e. as a superposition of sinusoidal wavelet- like packets with carrier frequencies equal to harmonics of the initial tone burst, and with envelopes (being functions of space coordinates and time) determined by the TAPE method. The spectrum of the propagating pressure waveform is considered to be a sum of contributions of each wavelet spectrum. Our model uses the second-order operator splitting approach with an incremental propagation scheme whereby the effects of combined diffraction and absorption are computed separately from the effects of nonlinear harmonic interactions over incremental steps. The proposed model is free from paraxial approximation, is computationally efficient and capable of predicting the 4D ultrasound field in nonlinear and attenuating media with the arbitrary absorption from pulsed, arbitrary shaped, plane and focused sources (including phased arrays with an angular beam deflection). Using a computational power of a standard PC a computation time required for the full 4D nonlinear field simulation depends on the source dimensions, radiated frequency and excitation level as well as on the medium absorption and nonlinearity strength and can vary from few minutes to few hours. Comparison between experimental and simulation results clearly show that our numerical model predicts well the structure of the nonlinear ultrasound field for all boundary conditions considered. I. INTRODUCTION The theoretical and experimental studies of the finite amplitude acoustic waves propagation in attenuating media from nonaxisymmetric sources rather rarely can be found in literature in spite of the fact that probes of the rectangular geometry (such as linear phased arrays) are commonly used in clinical practice for medical ultrasonic imaging purposes. The main reason of such situation is a lack in simpler theoretical models and in computationally efficient numerical algorithms that are able to predict accurately the nonlinear effects in 4D ultrasound fields from pulsed, arbitrarily shaped sources (plane and focused) in biological media with arbitrary frequency- dependent absorption. In recent years the only study describing the computationally efficient numerical model that is able to simulate accurately the 4D nonlinear ultrasound field in water and in biological tissues from pulsed nonaxisymmetric sources was developed by Zemp et al. (1). Their model is based on the second order operator-splitting method, proposed by Tavakkoli et al., with the modified fractional step scheme whereby the combined effects of diffraction and absorption are accounted for over half-steps and the effects of nonlinear harmonic interactions over full incremental steps. The computation of diffraction and absorption sub-steps was based on the angular spectrum technique with modified sampling method (to obtain computational savings due to larger axial propagation steps) while the computation of nonlinear steps was based on the time-domain solution to Burgers' equation. There are not reports yet describing an experimental confirmation of an agreement between the simulated nonlinear acoustic pulsed fields in water or in soft tissues from nonaxisymmetric focused sources (obtained by using the numerical model proposed) and nonlinear field from realistic probes. In this work the experimental measurement results of the 4D nonlinear field from the both square and rectangular ultrasonic focused transducer radiating the pulsed pressure wave in water are presented. The realistic beam patterns are compared with the simulation results obtained by using our novel numerical algorithm.

The tensile strength or the negative pressure required to induce cavitation in a pure liquid has been a puzzling subject. On one hand, the classical nucleation theory has met great success in predicting the nucleation rates of superheated liquids. On the other hand, most of reported experimental values of the tensile strength for different liquids are far below the prediction from the classical nucleation theory. In this study, homogeneous nucleation in liquid nitrogen and its tensile strength have been investigated. In order to carry out the measurement of the tensile strength of liquid nitrogen, different approaches for determining the pressure amplitude were studied carefully. It is shown that Raman-Nath theory, as modified by the introduction of an effective interaction length, can be used to determine the pressure amplitude in the focal plane of a focusing ultrasonic transducer. The results obtained from different diffraction orders are consistent and in good agreement with other approaches including Debye's theory and solving the KZK (Khokhlov-Zabolotskaya-Kuznetsov) equation. The results from experiments in water demonstrated that as long as the nonlinearity is not too large, the experimentally determined pressure follows closely the calculated results using either Debye's theory or the KZK equation. In addition, the light diffraction contains enough information to calculate the second-order harmonic in the sound wave. In principle, it is possible that the contribution to the acoustic wave of the higher than the second-order harmonic can be obtained. The measurement of the tensile strength was carried out in a high pressure stainless steel dewar. A High intensity ultrasonic wave was focused into a small volume of liquid nitrogen in a short time period. A probe laser beam passes through the focal region of a concave spherical transducer with small aperture angle and the transmitted light is detected with a photodiode. When the voltage on the transducer reaches a critical point, nucleation in the focal region occurs and a characteristic signal associated with the nucleation was obtained. At this moment, the pressure amplitude at the focus is calculated based on the acoustic power radiated into the liquid. In the experiment, the electrical signal on the transducer is gated at its resonance frequency with gate widths of 20 ~s to 0.2 ms and temperature range from 77 K to near 100 K. The calculated pressure amplitude is in agreement with the prediction of classical nucleation theory for the nucleation rates from 106 to lOll (bubbles/cm3

The popular KZK equation has been widely used to model the nonlinear acoustic wave field. However, since it is based on a parabolic approximation, it has various limitations, both at close distances from the transducer and at large radial extents from the beam axis or, in case of phased arrays, at steering angles larger than, say, 20/spl deg/. The purpose of this work is to model the nonlinear acoustic wave field without using the parabolic approximation. The model is based on the full nonlinear wave equation known as the Westervelt equation. This equation is solved numerically using an explicit finite difference scheme with fourth order centred space differences and second order centered or second order backward time differences. To check the validity of our model, numerical results for the lossless linear case were compared to analytical results. Moreover, the nonlinear beam profiles were compared to the results of a numerical solution of the KZK equation and to hydrophone measurements performed with a single element, unfocused transducer with a radius of 6.35 mm and a centre frequency of 1 MHz. We used a pulsed excitation of 6 cycles with pressure levels of 40, 220 and 500 kPa. The axial beam profiles and the lateral beam profiles at several distances were measured. For a low drive level, our numerical predictions show a good agreement with the analytic solution as well as with the measured data for all fundamental frequency beam profiles. At higher excitations, our simulations give a significant second harmonic component at 15 db below the fundamental level, which is in agreement with the measurements. The numerical data reproduces all the features at the fundamental and harmonic frequency beam profiles. For regions where the KZK equation is known to be less accurate, our model performs better. The sidelobes created at larger observation angles (above 20/spl deg/) and the nearfield variations are accurately reproduced. The performance of our model is good since it predicts accurately the harmonic field at any position with respect to the transducer. It can be used for the whole scanning sector that is required for harmonic beam profile simulations for phased arrays.

A non-linear ultrasound imaging simulation software should be capable of simulating the non-linear fields for any kind of transducer, focusing, apodization, and attenuation. At present, a major issue is the overlong simulation time of the non-linear software. An Angular Spectrum Approach (ASA) using a quasi-linear approximation for solving the Westervelt equation can simulate the second harmonic pressure at any distance. Therefore, it shortens the execution time compared with the operator splitting method. The purpose of this paper is to implement the monochromatic solution for the second harmonic component based on ASA and Field II, and to compare with results from the simulation program Abersim. A linear array transducer with a center frequency of 4 MHz and 64 active elements is used as the transmitting source. The initial plane is 5 mm away from the transducer surface, and the fundamental pressure is calculated by Field II. The second harmonic pressure in k-space along the propagating direction is calculated as an auto-convolution of the fundamental pressure multiplied by an exponential propagating coefficient. In this case, the second harmonic pressure can be calculated using ASA for any plane parallel to the initial plane. In the focal plane (elevation-lateral) at 60 mm from the transducer surface, calculated by ASA, the RMS errors for the fundamental component are 2.66% referred to Field II and 4.28% referred to Abersim. For the second harmonic component, the RMS error is 0.91% referred to Abersim.

Eighth international symposium on nonlinear acoustics

Theoretical investigations of finite amplitude wave propagation from bounded beam sources has been considered by Lee and Hamilton [J. Acoust. Soc. Am. 97, 906–917 (1995)] as well as others. These researchers have restricted their analyses to a source geometry of either an unfocused circular piston or a weakly, spherically focused circular piston. When the peak pressure does not exceed approximately 150 MPa, the Khokhlov–Zabolotskaya–Kuznetsov (KZK) nonlinear parabolic wave equation yields numerical solutions in agreement with experiments. However, little research has been reported on finite amplitude wave propagation from an array of discrete sources. Time domain simulations, based on the KZK equation, will be discussed for an eight element discrete annular array immersed in fresh and sea water. The source signal is typically a short sine wave tone burst with a center frequency of 500 kHz. The transitions from linear acoustics to nonlinear acoustics will be discussed, and comparisons between a spherically focused circular source and a properly phased array will be presented. While some data from recent experiments may be shown, the scope of this talk will be restricted to the theoretical treatment and the numerical simulations for the discrete annular array. [Work supported by ONR.] by Edson

A general solution of the lossy nonlinear KZK (Khokhlov–Zabolotskaya–Kuznetsov) equation is derived for any order harmonic of a Gaussian beam. The result shows that in a weakly focused Gaussian field all harmonics maintain a Gaussian profile in the axial direction at any distance. Moreover, the Gaussian coefficient of the nth harmonic is exactly n times as great as that of the fundamental; i.e., the beamwidth of the harmonics decreases as n−1/2.

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 modeling of nonlinear (NL) ultrasound (US) beam propagation plays a key role in designing state-of-art medical US systems due to the improvements in image quality coming from the use of second harmonic overtones. Describing the combined effects of diffraction, absorption and nonlinearity, the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation is the most commonly used model to address this simulation problem. This work introduces an algorithm capable of providing good estimations of pressure profiles at arbitrary depth without requiring stepping techniques. Our proposal consists in a perturbative approach to the solution of the KZK equation cast in a discrete Fourier-Bessel domain applicable to axisymmetric propagation from circular transducers. The resulting algorithm is capable of simulating NL US fields with significant computational savings and an accuracy at least comparable to other standard approaches to the same problem. Comparing our algorithm to the output of a publicly available NL KZK solver, significant cost reductions and performance improvements were recorded.

Theoretical analysis and some experiments are performed on nonlinearly generated harmonic components in bounded sound beams emitted from a rectangular aperture source. The Khokhlov–Zabolotskaya–Kuznetsov equation, which takes account of nonlinearity, dissipation, and diffraction effects in the beams, is numerically solved by means of the alternating direction implicit difference method. Using a planar source of size 24×44 cm, axial sound pressures and beam patterns of the first three harmonics are measured in air for initially sinusoidal ultrasounds of 25- and 30-kHz frequency, and are compared with the theory. They are in relatively good agreement. Deformation of the source face from circular to rectangular shape results in the unclear appearance of pressure peaks and dips with propagation. Within the framework of these studies, the harmonic pressure levels in the far field are almost the same as from a circular aperture source with equal face area and equal initial pressure, independent of the source levels.

The measurement of the acoustic pressure of a planar ultrasonic wave by light diffraction is well established. The ability to do similar measurements in the case of spherical waves has been doubted. However, we show that the range of validity can be extended to the focal region of a spherical concave piezoelectric transducer. Light is passed through the focal plane of a spherical concave transducer and is diffracted as a result of the variation in the index of refraction. The peak pressure can be calculated from the diffraction intensity by making the following simplification. We assume that in the focal plane the ultrasound can be approximated by a profiled planar wave, which in turn can be modeled by a wave of constant amplitude and effective width. The experimental results for moderate-pressure amplitudes in water compare favorably with the calculations using the Khokhlov–Zabolotskaya–Kuznetsov equation, which incorporates both nonlinearity and diffraction effects of the acoustic field.

The Khokhlov–Zabolotskaya–Kuznetsov equation

Results of a systematic investigation of a high intensity sound beam from a plane circular transducer are presented for a large range of excitation levels. The ratio of the shock formation distance to the Rayleigh distance ranges from 1.7 to 0.06. A detailed description of the experimental setup is given. Problems encountered when calibrating highly intense and highly directive sound sources are discussed, and a novel method of calibration is proposed. Experimental results for the case of cw excitation are presented and compared with a simulation using the Bergen Code, thus allowing to assert the usefulness of the model (Khokhlov–Zabolotskaya–Kuznetsov equation) and of the numerical algorithm (spectral decomposition and finite difference method of the transformed beam equation) for high intensity sound beams. A new effect, nonlinear defocusing of the beam, is uncovered.

Pulsed focused ultrasound (pFUS) uses short acoustic pulses delivered at low duty cycle and moderate intensity to noninvasively apply mechanical stress or introduce disruption to tissue. Ultrasound-guided pFUS has primarily been used for inducing cavitation at the focus, with or without contrast agents, to promote drug delivery to tumors. When applied in tandem with contrast agents, pFUS is often administered using an ultrasound imaging probe, which has a small footprint and does not require a large acoustic window. The use of nonlinear pFUS without contrast agents was recently shown to be beneficial for localized tissue disruption, but required higher ultrasound pressure levels than a conventional ultrasound imaging probe could produce. In this work, we present the design of a compact dual-use 1-MHz transducer for ultrasound-guided pFUS without contrast agents. Nonlinear pressure fields that could be generated by the probe, under realistic power input, were simulated using the Westervelt equation. In water, fully developed shocks of 42-MPa amplitude and peak negative pressure of 8 MPa were predicted to form at the focus at 458-W acoustic power or 35% of the maximum reachable power of the transducer. In absorptive soft tissue, fully developed shocks formed at higher power (760 W or 58% of the maximum reachable power) with the shock amplitude of 33 MPa and peak negative pressure of 7.5 MPa. The electronic focus-steering capabilities of the array were evaluated and found to be sufficient to cover a target with dimensions of 19 mm in axial direction and 44 mm in transversal direction.