Computer simulation of forward wave propagation in soft tissue

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.

Endobronchial Ultrasound Image Simulation for Image-Guided Bronchoscopy

BACKGROUND/OBJECTIVE

Accurate disease diagnosis and staging are essential for patients suspected of having lung cancer. The state-of-the-art minimally invasive tools used by physicians to perform these operations are bronchoscopy, for navigating the lung airways, and endobronchial ultrasound (EBUS), for localizing suspect extraluminal cancer lesions. While new image-guided systems enable accurate bronchoscope navigation close to a lesion, no means exists for guiding the final EBUS localization of an extraluminal lesion. We propose an EBUS simulation method to assist with EBUS localization.

METHODS

The method draws on a patient's chest computed-tomography (CT) scan to model the ultrasound signal propagation through the tissue media. The method, which is suitable for simulating EBUS images for both radial-probe and convex-probe EBUS devices, entails three steps: 1) image preprocessing, which generates a 2D CT equivalent of the EBUS scan plane; 2) EBUS scan-line computation, which models ultrasound transmission to map the CT plane into a preliminary simulated EBUS image; and 3) image post-processing, which increases realism by introducing simulated EBUS imaging effects and artifacts.

RESULTS

Results show that the method produces simulated EBUS images that strongly resemble images generated live by a real device and compares favorably to an existing ultrasound simulation method. It also produces images at a rate greater than real time (i.e., 53 frames/sec). We also demonstrate a successful integration of the method into an image-guided EBUS bronchoscopy system.

CONCLUSION/SIGNIFICANCE

The method is effective and practical for procedure planning/preview and follow-on live guidance of EBUS bronchoscopy.

Effects of acoustic nonlinearity on communication performance in soft tissues

Acoustic communication has been gaining traction as an alternative communication method in nontraditional media, such as underwater or through tissue. Acoustic propagation is known to be a nonlinear phenomenon; nonlinear propagation of acoustic waves in soft tissues at biomedical frequencies and intensities has been widely demonstrated. However, the effects of acoustic nonlinearity on communication performance in biological tissues have not yet been examined. In this work, nonlinear propagation of a communication signal in soft tissues is analyzed. The relationship between communication parameters (signal amplitude, bandwidth, and center frequency) and nonlinear distortion of the communication signal propagating in soft tissues with different acoustic properties is investigated. Simulated experiments revealed that, unlike linear channels, bit error rates increase as signal amplitude and bandwidth increase. Linear and decision feedback equalizers fail to address the increased error rates. When tissue properties and transmission parameters can be estimated, receivers based on maximum likelihood sequence estimation approach the performance of an ideal receiver in an ideal additive white Gaussian noise channel.

"HIFU Beam:" A Simulator for Predicting Axially Symmetric Nonlinear Acoustic Fields Generated by Focused Transducers in a Layered Medium

"HIFU beam" is a freely available software tool that comprises a MATLAB toolbox combined with a user-friendly interface and binary executable compiled from FORTRAN source code (HIFU beam. (2021). Available: http://limu.msu.ru/node/3555?language=en). It is designed for simulating high-intensity focused ultrasound (HIFU) fields generated by single-element transducers and annular arrays with propagation in flat-layered media that mimic biological tissues. Numerical models incorporated in the simulator include evolution-type equations, either the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation or one-way Westervelt equation, for radially symmetric ultrasound beams in homogeneous and layered media with thermoviscous or power-law acoustic absorption. The software uses shock-capturing methods that allow for simulating strongly nonlinear acoustic fields with high-amplitude shocks. In this article, a general description of the software is given along with three representative simulation cases of ultrasound transducers and focusing conditions typical for therapeutic applications. The examples illustrate major nonlinear wave effects in HIFU fields including shock formation. Two examples simulate propagation in water, involving a single-element source (1-MHz frequency, 100-mm diameter, 90-mm radius of curvature) and a 16-element annular array (3-MHz frequency, 48-mm diameter, and 35-mm radius of curvature). The third example mimics the scenario of a HIFU treatment in a "water-muscle-kidney" layered medium using a source typical for abdominal HIFU applications (1.2-MHz frequency, 120-mm diameter, and radius of curvature). Linear, quasi-linear, and shock-wave exposure protocols are considered. It is intended that "HIFU beam" can be useful in teaching nonlinear acoustics; designing and characterizing high-power transducers; and developing exposure protocols for a wide range of therapeutic applications such as shock-based HIFU, boiling histotripsy, drug delivery, immunotherapy, and others.

Simulation of the Second-Harmonic Ultrasound Field in Heterogeneous Soft Tissue Using a Mixed-Domain Method

A mixed-domain method (MDM) dubbed frequency-specific MDM (FSMDM) is introduced for the simulation of the second-harmonic ultrasound field in weakly heterogeneous media. The governing equation for the second harmonics is derived based on the quasi-linear theory. The speed of sound, nonlinear coefficient, and attenuation coefficient are all spatially varying functions in the equation. The fundamental frequency pressure field is first solved by the FSMDM and it is subsequently used as the source term for the second-harmonics equation. This equation can be again solved by the FSMDM to rapidly obtain the second-harmonic pressure field. Five 2-D cases, including one with a realistic human tissue map, are studied to systematically verify the proposed method. Results from the previously developed transient MDM are used as the benchmark solutions. Comparisons show that the two methods give similar results for all cases. More importantly, the FSMDM has a crucial advantage over the transient MDM in that it can be two orders of magnitude faster.

Soft-Tissue Aberration Correction for Histotripsy

Acoustic aberrations caused by natural heterogeneities of biological soft tissue are a substantial problem for histotripsy, a therapeutic ultrasound technique that uses acoustic cavitation to mechanically fractionate and destroy unwanted target tissue without damaging surrounding tissue. These aberrations, primarily caused by sound speed variations, result in severe defocusing of histotripsy pulses, thereby decreasing treatment efficacy. The gold standard for aberration correction (AC) is to place a hydrophone at the desired focal location to directly measure phase aberrations, which is a method that is infeasible in vivo. We hypothesized that the acoustic cavitation emission (ACE) shockwaves from the initial expansion of inertially cavitating microbubbles generated by histotripsy can be used as a point source for AC. In this study, a 500-kHz, 112-element histotripsy phased array capable of transmitting and receiving ultrasound on all channels was used to acquire ACE shockwaves. These shockwaves were first characterized optically and acoustically. It was found that the shockwave pressure increases significantly as the source changes from a single bubble to a dense cavitation cloud. The first arrival of the shockwave received by the histotripsy array was from the outer-most cavitation bubbles located closest to the histotripsy array. Hydrophone and ACE AC methods were then tested on ex vivo porcine abdominal tissue samples. Without AC, the focal pressure is reduced by 49.7% through the abdominal tissue. The hydrophone AC approach recovered 55.5% of the lost pressure. Using the ACE AC method, over 20% of the lost pressure was recovered, and the array power required to induce cavitation was reduced by approximately 31.5% compared to without AC. These results supported our hypothesis that the ACE shockwaves coupled with a histotripsy array with transmit and receive capability can be used for AC for histotripsy through soft tissue.

The nonlinear ultrasound needle pulse

Recent work has established an analytical formulation of broadband fields which extend in the axial direction and converge to a narrow concentrated line. Those unique (needle) fields have their origins in an angular spectrum configuration in which the forward propagating wavenumber of the field ( kz ) is constant across any z plane for all of the propagated frequencies. A 3 MHz-based, finite amplitude distorted simulation of such a field is considered here in a water path scenario relevant to medical imaging. That nonlinear simulation had its focal features compared to those of a comparable Gaussian beam. The results suggest that the unique convergence of the needle pulse to a narrow but extended axial line in linear propagation is also inherited by higher harmonics in nonlinear propagation. Furthermore, the linear needle field's relatively short duration focal pulses, and the asymptotic declines of its radial profiles, also hold for the associated higher harmonics. Comparisons with the Gaussian field highlight some unique and potentially productive features of needle fields.

Numerical Modeling of Ultrasound Propagation in Weakly Heterogeneous Media Using a Mixed-Domain Method

A mixed-domain method (MDM) is presented in this paper for modeling one-way linear/nonlinear wave propagation in biological tissue with arbitrary heterogeneities, in which sound speed, density, attenuation coefficients, and nonlinear coefficients are all spatial varying functions. The present method is based on solving an integral equation derived from a Westervelt-like equation. One-dimensional problems are first studied to verify the MDM and to reveal its limitations. It is shown that this method is accurate for cases with small variation of sound speed. A 2-D case is further studied with focused ultrasound beams to validate the application of the method in the medical field. Results from the MATLAB toolbox k-Wave are used as the benchmark. Normalized root-mean-square (rms) error estimated at the focus of the transducer is 0.0133 when the coarsest mesh (1/3 of the wavelength) is used in the MDM. Fundamental and second-harmonic fields throughout the considered computational domains are compared and good agreement is observed. Overall, this paper demonstrates that the MDM is a computationally efficient and accurate method when used to model wave propagation in biological tissue with relatively weak heterogeneities.

A Prototype Therapy System for Transcutaneous Application of Boiling Histotripsy

Boiling histotripsy (BH) is a method of focused ultrasound surgery that noninvasively applies millisecond-length pulses with high-amplitude shock fronts to generate liquefied lesions in tissue. Such a technique requires unique outputs compared to a focused ultrasound thermal therapy apparatus, particularly to achieve high in situ pressure levels through intervening tissue. This paper describes the design and characterization of a system capable of producing the necessary pressure to transcutaneously administer BH therapy through clinically relevant overlying tissue paths using pulses with duration up to 10 ms. A high-voltage electronic pulser was constructed to drive a 1-MHz focused ultrasound transducer to produce shock waves with amplitude capable of generating boiling within the pulse duration in tissue. The system output was characterized by numerical modeling with the 3-D Westervelt equation using boundary conditions established by acoustic holography measurements of the source field. Such simulations were found to be in agreement with directly measured focal waveforms. An existing derating method for nonlinear therapeutic fields was used to estimate in situ pressure levels at different tissue depths. The system was tested in ex vivo bovine liver samples to create BH lesions at depths up to 7 cm. Lesions were also created through excised porcine body wall (skin, adipose, and muscle) with 3-5 cm thickness. These results indicate that the system is capable of producing the necessary output for transcutaneous ablation with BH.

Numerical study of a confocal ultrasonic setup for cavitation creation

Acoustic cavitation has found a wide range of applications in the last few decades. For potential applications involving cavitation, the acoustic characteristics of a confocal ultrasonic setup are studied: two high-intensity focused ultrasound transducers are mounted so that their focal points overlap. A mathematical simulator is developed that takes into account nonlinear propagation, absorption, and diffraction. Each one of these physical effects is solved in the frequency domain for successive planes. Comparing the confocal setup with equivalent single transducer setups, it is shown that, with the confocal configuration, nonlinear distortion of the waveform is reduced, resulting in a greater peak rarefactional pressure and a lower peak positive pressure. Furthermore, additional features are investigated for confocal configurations such as a greater spatial stability for the focal point, which can be maintained while increasing the pressure level, and a focal region consisting of interference acting as an acoustic trap.

Design of HIFU Transducers for Generating Specified Nonlinear Ultrasound Fields

Various clinical applications of high-intensity focused ultrasound have different requirements for the pressure levels and degree of nonlinear waveform distortion at the focus. The goal of this paper is to determine transducer design parameters that produce either a specified shock amplitude in the focal waveform or specified peak pressures while still maintaining quasi-linear conditions at the focus. Multiparametric nonlinear modeling based on the Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation with an equivalent source boundary condition was employed. Peak pressures, shock amplitudes at the focus, and corresponding source outputs were determined for different transducer geometries and levels of nonlinear distortion. The results are presented in terms of the parameters of an equivalent single-element spherically shaped transducer. The accuracy of the method and its applicability to cases of strongly focused transducers were validated by comparing the KZK modeling data with measurements and nonlinear full diffraction simulations for a single-element source and arrays with 7 and 256 elements. The results provide look-up data for evaluating nonlinear distortions at the focus of existing therapeutic systems as well as for guiding the design of new transducers that generate specified nonlinear fields.

Optimization of contrast resolution by genetic algorithm in ultrasound tissue harmonic imaging

The development of ultrasound imaging techniques such as pulse inversion has improved tissue harmonic imaging. Nevertheless, no recommendation has been made to date for the design of the waveform transmitted through the medium being explored. Our aim was therefore to find automatically the optimal "imaging" wave which maximized the contrast resolution without a priori information. To overcome assumption regarding the waveform, a genetic algorithm investigated the medium thanks to the transmission of stochastic "explorer" waves. Moreover, these stochastic signals could be constrained by the type of generator available (bipolar or arbitrary). To implement it, we changed the current pulse inversion imaging system by including feedback. Thus the method optimized the contrast resolution by adaptively selecting the samples of the excitation. In simulation, we benchmarked the contrast effectiveness of the best found transmitted stochastic commands and the usual fixed-frequency command. The optimization method converged quickly after around 300 iterations in the same optimal area. These results were confirmed experimentally. In the experimental case, the contrast resolution measured on a radiofrequency line could be improved by 6% with a bipolar generator and it could still increase by 15% with an arbitrary waveform generator.

A framework for simulating ultrasound imaging based on first order nonlinear pressure-velocity relations

An ultrasound imaging framework modeled with the first order nonlinear pressure-velocity relations (NPVR) and implemented by a half-time staggered solution and pseudospectral method is presented in this paper. The framework is capable of simulating linear and nonlinear ultrasound propagation and reflections in a heterogeneous medium with different sound speeds and densities. It can be initialized with arbitrary focus, excitation and apodization for multiple individual channels in both 2D and 3D spatial fields. The simulated channel data can be generated using this framework, and ultrasound image can be obtained by beamforming the simulated channel data. Various results simulated by different algorithms are illustrated for comparisons. The root mean square (RMS) errors for each compared pulses are calculated. The linear propagation is validated by an angular spectrum approach (ASA) with a RMS error of 3% at the focal point for a 2D field, and Field II with RMS errors of 0.8% and 1.5% at the electronic and the elevation focuses for 3D fields, respectively. The accuracy for the NPVR based nonlinear propagation is investigated by comparing with the Abersim simulation for pulsed fields and with the nonlinear ASA for monochromatic fields. The RMS errors of the nonlinear pulses calculated by the NPVR and Abersim are respectively 2.4%, 7.4%, 17.6% and 36.6% corresponding to initial pressure amplitudes of 50kPa, 200kPa, 500kPa and 1MPa at the transducer. By increasing the sampling frequency for the strong nonlinearity, the RMS error for 1MPa initial pressure amplitude is reduced from 36.6% to 27.3%.

Modeling of wave propagation for medical ultrasound: a review

Numerical modeling of medical ultrasound has advanced tremendously in the past two decades. This opens up a great number of opportunities for medical ultrasound and associated technologies. Numerous new governing equations and algorithms have emerged and been applied to studying various medical ultrasound applications, including ultrasound imaging, photo-acoustic imaging, and therapeutic ultrasound. In addition, thanks to the rapid development of computers, modeling acoustic wave propagation in three-dimensional, large-scale domains has become a reality. This article will provide an indepth literature and technical review of recent progress on numerical modeling of medical ultrasound. Future challenges will also be discussed.

An improved wave-vector frequency-domain method for nonlinear wave modeling

In this paper, a recently developed wave-vector frequency-domain method for nonlinear wave modeling is improved and verified by numerical simulations and underwater experiments. Higher order numeric schemes are proposed that significantly increase the modeling accuracy, thereby allowing for a larger step size and shorter computation time. The improved algorithms replace the left-point Riemann sum in the original algorithm by the trapezoidal or Simpson's integration. Plane waves and a phased array were first studied to numerically validate the model. It is shown that the left-point Riemann sum, trapezoidal, and Simpson's integration have first-, second-, and third-order global accuracy, respectively. A highly focused therapeutic transducer was then used for experimental verifications. Short high-intensity pulses were generated. 2-D scans were conducted at a prefocal plane, which were later used as the input to the numerical model to predict the acoustic field at other planes. Good agreement is observed between simulations and experiments.

High-order pseudo-Gaussian scalar acoustical beams

Exact solutions of the scalar Helmholtz equation describing tightly spherically-focused beams are introduced without any approximations using the complex source point method in spherical coordinates. The generalized solutions, valid for any integer degree n and order m, describe high-order pseudo-Gaussian vortex, intermediate (vortex), hollow (nonvortex), and trigonometric (non-vortex) beams having an arbitrary beam waist w0. A very useful property of these beams is the efficient and fast computational modeling of tightly focused or quasi-collimated wave-fronts depending on the dimensionless waist parameter kw0, where k is the wave number of the acoustical radiation. Examples that illustrate hollow vortex and non-vortex beams are provided, and numerical simulations for the magnitude, isosurface, and phase plots of the pressure wave field of higher-order quasi-Gaussian beams are evaluated with particular emphasis on kw0 for strongly (kw0 = 3) to weakly focused (i.e., quasi-collimated) beams (kw0 = 7). Potential applications are in beam-forming design, imaging, particle sizing and manipulation in acoustical tweezers, and phenomena related to scattering, radiation force, and torque.

Multi-line transmission in medical imaging using the second-harmonic signal

The emergence of three-dimensional imaging in the field of medical ultrasound imaging has greatly increased the number of transmissions needed to insonify a whole volume. With a large number of transmissions comes a low image frame rate. When using classical transmission techniques, as in two-dimensional imaging, the frame rate becomes unacceptably low, prompting the use of alternative transmission patterns that require less time. One alternative is to use a multi-line transmission (MLT) technique which consists of transmitting several pulses simultaneously in different directions. Perturbations appear when acquiring and beamforming the signal in the direction of one pulse because of the pulses sent in other directions. The edge waves from the pulses transmitted in a different direction add to the signal transmitted in the direction of interest, resulting in artifacts in the final image. Taking advantage of the nonlinear propagation of sound in tissue, the second-harmonic signal can be used with the MLT technique. The image obtained using the second-harmonic signal, compared with an image obtained using the fundamental signal, should have reduced artifacts coming from other pulses transmitted simultaneously. Simulations, backed up by experiments imaging a wire target and an in vivo left ventricle, confirm that the hypothesis is valid. In the studied case, the perturbations appear as an increase in the signal level around the main echo of a point scatterer. When using the fundamental signal, the measured amplitude level of the perturbations was approximately -40 dB compared with the maximum signal amplitude (-27 dB in vivo), whereas it was around -60 dB (-45 dB in vivo) for the second-harmonic signal. The MLT technique encounters limitations in the very near field where the pulses overlap and the perturbation level also increases for images with strong speckle and low contrast.

Methods for reverberation suppression utilizing dual frequency band imaging

Reverberations impair the contrast resolution of diagnostic ultrasound images. Tissue harmonic imaging is a common method to reduce these artifacts, but does not remove all reverberations. Dual frequency band imaging (DBI), utilizing a low frequency pulse which manipulates propagation of the high frequency imaging pulse, has been proposed earlier for reverberation suppression. This article adds two different methods for reverberation suppression with DBI: the delay corrected subtraction (DCS) and the first order content weighting (FOCW) method. Both methods utilize the propagation delay of the imaging pulse of two transmissions with alternating manipulation pressure to extract information about its depth of first scattering. FOCW further utilizes this information to estimate the content of first order scattering in the received signal. Initial evaluation is presented where both methods are applied to simulated and in vivo data. Both methods yield visual and measurable substantial improvement in image contrast. Comparing DCS with FOCW, DCS produces sharper images and retains more details while FOCW achieves best suppression levels and, thus, highest image contrast. The measured improvement in contrast ranges from 8 to 27 dB for DCS and from 4 dB up to the dynamic range for FOCW.

Unmitigated numerical solution to the diffraction term in the parabolic nonlinear ultrasound wave equation

Various numerical algorithms have been developed to solve the Khokhlov-Kuznetsov-Zabolotskaya (KZK) parabolic nonlinear wave equation. In this work, a generalized time-domain numerical algorithm is proposed to solve the diffraction term of the KZK equation. This algorithm solves the transverse Laplacian operator of the KZK equation in three-dimensional (3D) Cartesian coordinates using a finite-difference method based on the five-point implicit backward finite difference and the five-point Crank-Nicolson finite difference discretization techniques. This leads to a more uniform discretization of the Laplacian operator which in turn results in fewer calculation gridding nodes without compromising accuracy in the diffraction term. In addition, a new empirical algorithm based on the LU decomposition technique is proposed to solve the system of linear equations obtained from this discretization. The proposed empirical algorithm improves the calculation speed and memory usage, while the order of computational complexity remains linear in calculation of the diffraction term in the KZK equation. For evaluating the accuracy of the proposed algorithm, two previously published algorithms are used as comparison references: the conventional 2D Texas code and its generalization for 3D geometries. The results show that the accuracy/efficiency performance of the proposed algorithm is comparable with the established time-domain methods.

Characterization of a multi-element clinical HIFU system using acoustic holography and nonlinear modeling

High-intensity focused ultrasound (HIFU) is a treatment modality that relies on the delivery of acoustic energy to remote tissue sites to induce thermal and/or mechanical tissue ablation. To ensure the safety and efficacy of this medical technology, standard approaches are needed for accurately characterizing the acoustic pressures generated by clinical ultrasound sources under operating conditions. Characterization of HIFU fields is complicated by nonlinear wave propagation and the complexity of phased-array transducers. Previous work has described aspects of an approach that combines measurements and modeling, and here we demonstrate this approach for a clinical phased-array transducer. First, low amplitude hydrophone measurements were performed in water over a scan plane between the array and the focus. Second, these measurements were used to holographically reconstruct the surface vibrations of the transducer and to set a boundary condition for a 3-D acoustic propagation model. Finally, nonlinear simulations of the acoustic field were carried out over a range of source power levels. Simulation results were compared with pressure waveforms measured directly by hydrophone at both low and high power levels, demonstrating that details of the acoustic field, including shock formation, are quantitatively predicted.

Computation of nonlinear ultrasound fields using a linearized contrast source method

Nonlinear ultrasound is important in medical diagnostics because imaging of the higher harmonics improves resolution and reduces scattering artifacts. Second harmonic imaging is currently standard, and higher harmonic imaging is under investigation. The efficient development of novel imaging modalities and equipment requires accurate simulations of nonlinear wave fields in large volumes of realistic (lossy, inhomogeneous) media. The Iterative Nonlinear Contrast Source (INCS) method has been developed to deal with spatiotemporal domains measuring hundreds of wavelengths and periods. This full wave method considers the nonlinear term of the Westervelt equation as a nonlinear contrast source, and solves the equivalent integral equation via the Neumann iterative solution. Recently, the method has been extended with a contrast source that accounts for spatially varying attenuation. The current paper addresses the problem that the Neumann iterative solution converges badly for strong contrast sources. The remedy is linearization of the nonlinear contrast source, combined with application of more advanced methods for solving the resulting integral equation. Numerical results show that linearization in combination with a Bi-Conjugate Gradient Stabilized method allows the INCS method to deal with fairly strong, inhomogeneous attenuation, while the error due to the linearization can be eliminated by restarting the iterative scheme.

Investigation of an angular spectrum approach for pulsed ultrasound fields

An Angular Spectrum Approach (ASA) is formulated and employed to simulate linear pulsed ultrasound fields for high bandwidth signals. A geometrically focused piston transducer is used as the acoustic source. Signals are cross-correlated to find the true sound speed during the measurement to make the simulated and measured pulses in phase for comparisons. The calculated sound speed in the measurement is varied between 1487.45 m/s and 1487.75 m/s by using different initial values in the ASA simulation. Results from the pulsed ASA simulations using both Field II simulated and hydrophone measured acoustic sources are compared to the Field II simulated and hydrophone measured pulses, respectively. The total relative root mean square (RMS) errors of the pulsed ASA are investigated by using different time-point, zero-padding factors, spatial sampling interval and temporal sampling frequency in the simulation. Optimal parameters for the ASA are found in the simulation. The RMS error of the ASA simulation is reduced from 10.9% to 2.4% for the optimal parameters when comparing to Field II simulations. The comparison between the ASA calculated and measured pulses are illustrated and the corresponding RMS error is 25.4%.

Fast simulation of non-linear pulsed ultrasound fields using an angular spectrum approach

A fast non-linear pulsed ultrasound field simulation is presented. It is implemented based on an angular spectrum approach (ASA), which analytically solves the non-linear wave equation. The ASA solution to the Westervelt equation is derived in detail. The calculation speed is significantly increased compared to a numerical solution using an operator splitting method (OSM). The ASA has been modified and extended to pulsed non-linear ultrasound fields in combination with Field II, where any array transducer with arbitrary geometry, excitation, focusing and apodization can be simulated. The accuracy of the non-linear ASA is compared to the non-linear simulation program - Abersim, which is a numerical solution to the Burgers equation based on the OSM. Simulations are performed for a linear array transducer with 64 active elements, focus at 40mm, and excitation by a 2-cycle sine wave with a center frequency of 5MHz. The speed is increased approximately by a factor of 140 and the calculation time is 12min with a standard PC, when simulating the second harmonic pulse at the focal point. For the second harmonic point spread function the full width error is 1.5% at -6dB and 6.4% at -12dB compared to Abersim.

A k-space method for moderately nonlinear wave propagation

A k-space method for moderately nonlinear wave propagation in absorptive media is presented. The Westervelt equation is first transferred into k-space via Fourier transformation, and is solved by a modified wave-vector time-domain scheme. The present approach is not limited to forward propagation or parabolic approximation. One- and two-dimensional problems are investigated to verify the method by comparing results to analytic solutions and finite-difference time-domain (FDTD) method. It is found that to obtain accurate results in homogeneous media, the grid size can be as little as two points per wavelength, and for a moderately nonlinear problem, the Courant-Friedrichs-Lewy number can be as large as 0.4. Through comparisons with the conventional FDTD method, the k-space method for nonlinear wave propagation is shown here to be computationally more efficient and accurate. The k-space method is then employed to study three-dimensional nonlinear wave propagation through the skull, which shows that a relatively accurate focusing can be achieved in the brain at a high frequency by sending a low frequency from the transducer. Finally, implementations of the k-space method using a single graphics processing unit shows that it required about one-seventh the computation time of a single-core CPU calculation.

Fast simulation of second harmonic ultrasound field using a quasi-linear method

Nonlinear propagation of sound has been exploited in the last 15 years in medical ultrasound imaging through tissue harmonic imaging (THI). THI creates an image by filtering the received ultrasound echo around the second harmonic frequency band. This technique produces images of enhanced quality due to reduced body wall reverberation, lower perturbations from off-axis echoes, and multiple scattering of reduced amplitude. In order to optimize the image quality it is essential to be able to predict the amplitude level and spatial distribution of the propagating ultrasound pulse. A method based on the quasi-linear approximation has been developed to quickly provide an estimate of the ultrasound pulse. This method does not need to propagate the pulse stepwise from the source plane to the desired depth; it directly computes a transverse profile at any depth from the definitions of the transducer and the pulse. The computation handles three spatial dimensions which allows for any transducer geometry. A comparison of pulse forms, transverse profiles, as well as axial profiles obtained by this method and state-of-the-art simulators, the KZKTexas code, and Abersim, shows a satisfactory match. The computation time for the quasi-linear method is also smaller than the time required by the other methods.

Transmit beamforming for optimal second-harmonic generation

A simulation study of transmit ultrasound beams from several transducer configurations is conducted to compare second-harmonic imaging at 3.5 MHz and 11 MHz. Second- harmonic generation and the ability to suppress near field echoes are compared. Each transducer configuration is defined by a chosen f-number and focal depth, and the transmit pressure is estimated to not exceed a mechanical index of 1.2. The medium resembles homogeneous muscle tissue with nonlinear elasticity and power-law attenuation. To improve computational efficiency, the KZK equation is utilized, and all transducers are circular-symmetric. Previous literature shows that second-harmonic generation is proportional to the square of the transmit pressure, and that transducer configurations with different transmit frequencies, but equal aperture and focal depth in terms of wavelengths, generate identical second-harmonic fields in terms of shape. Results verify this for a medium with attenuation f1. For attenuation f1.1, deviations are found, and the high frequency subsequently performs worse than the low frequency. The results suggest that high frequencies are less able to suppress near-field echoes in the presence of a heterogeneous body wall than low frequencies.

Fundamental and second-harmonic ultrasound field computation of inhomogeneous nonlinear medium with a generalized angular spectrum method

The simulation of nonlinear propagation of ultrasound waves is typically based on the Kuznetsov-Zabolotskaya- Khokhlov equation. A set of simulators has been proposed in the literature but none of them takes into account a possible spatial 3-D variation of the nonlinear parameter in the investigated medium. This paper proposes a generalization of the angular spectrum method (GASM) including the spatial variation of the nonlinear parameter. The proposed method computes the evolution of the fundamental and second-harmonic waves in four dimensions (spatial 3-D and time). For validation purposes, the one-way fields produced by the GASM are first compared with those produced by established reference simulators and with experimental one-way fields in media with a homogeneous nonlinear parameter. The same simulations are repeated for media having an axial variation of the nonlinear parameter. The mean errors estimated in the focal region are less than 4.0% for the fundamental and 5.4% for the second harmonic in all cases. Finally, the fundamental and second-harmonic fields simulated for media having nonlinear parameter variations in the axial, lateral, and elevation directions, which cannot be simulated with other currently available methods, are presented. The new approach is also shown to yield a reduction in computation time by a factor of 13 with respect to the standard nonlinear simulator.

Sources of image degradation in fundamental and harmonic ultrasound imaging: a nonlinear, full-wave, simulation study

A full-wave equation that describes nonlinear propagation in a heterogeneous attenuating medium is solved numerically with finite differences in the time domain. This numerical method is used to simulate propagation of a diagnostic ultrasound pulse through a measured representation of the human abdomen with heterogeneities in speed of sound, attenuation, density, and nonlinearity. Conventional delay-and-sum beamforming is used to generate point spread functions (PSFs) that display the effects of these heterogeneities. For the particular imaging configuration that is modeled, these PSFs reveal that the primary source of degradation in fundamental imaging is due to reverberation from near-field structures. Compared with fundamental imaging, reverberation clutter in harmonic imaging is 27.1 dB lower. Simulated tissue with uniform velocity but unchanged impedance characteristics indicates that for harmonic imaging, the primary source of degradation is phase aberration.

SIMULATION OF THREE-DIMENSIONAL NONLINEAR FIELDS OF ULTRASOUND THERAPEUTIC ARRAYS

A novel numerical model was developed to simulate three-dimensional nonlinear fields generated by high intensity focused ultrasound (HIFU) arrays. The model is based on the solution to the Westervelt equation; the developed algorithm makes it possible to model nonlinear pressure fields of periodic waves in the presence of shock fronts localized near the focus. The role of nonlinear effects in a focused beam of a two-dimensional array was investigated in a numerical experiment in water. The array consisting of 256 elements and intensity range on the array elements of up to 10 W/cm(2) was considered. The results of simulations have shown that for characteristic intensity outputs of modern HIFU arrays, nonlinear effects play an important role and shock fronts develop in the pressure waveforms at the focus.

Verification of the Westervelt equation for focused transducers

This study investigates the validity of the Westervelt equation for focused transducers. The angular spectrum method is employed to analyze the second-harmonic acoustic field under the weakly nonlinear approximation. Although it is well known that the Westervelt equation is accurate for the case of quasi-plane waves, the present work demonstrates accurate solution for the highly focused case of a spherically-curved ultrasound transducer having an aperture angle of 80°. It is further found that the solution error is inversely dependent on the nonlinearity coefficient.

Sources of image degradation in fundamental and harmonic ultrasound imaging using nonlinear, full-wave simulations

A full-wave equation that describes nonlinear propagation in a heterogeneous attenuating medium is solved numerically with finite differences in the time domain (FDTD). This numerical method is used to simulate propagation of a diagnostic ultrasound pulse through a measured representation of the human abdomen with heterogeneities in speed of sound, attenuation, density, and nonlinearity. Conventional delay-andsum beamforming is used to generate point spread functions (PSF) that display the effects of these heterogeneities. For the particular imaging configuration that is modeled, these PSFs reveal that the primary source of degradation in fundamental imaging is reverberation from near-field structures. Reverberation clutter in the harmonic PSF is 26 dB higher than the fundamental PSF. An artificial medium with uniform velocity but unchanged impedance characteristics indicates that for the fundamental PSF, the primary source of degradation is phase aberration. An ultrasound image is created in silico using the same physical and algorithmic process used in an ultrasound scanner: a series of pulses are transmitted through heterogeneous scattering tissue and the received echoes are used in a delay-and-sum beamforming algorithm to generate images. These beamformed images are compared with images obtained from convolution of the PSF with a scatterer field to demonstrate that a very large portion of the PSF must be used to accurately represent the clutter observed in conventional imaging.

Evaluation of a wave-vector-frequency-domain method for nonlinear wave propagation

A wave-vector-frequency-domain method is presented to describe one-directional forward or backward acoustic wave propagation in a nonlinear homogeneous medium. Starting from a frequency-domain representation of the second-order nonlinear acoustic wave equation, an implicit solution for the nonlinear term is proposed by employing the Green's function. Its approximation, which is more suitable for numerical implementation, is used. An error study is carried out to test the efficiency of the model by comparing the results with the Fubini solution. It is shown that the error grows as the propagation distance and step-size increase. However, for the specific case tested, even at a step size as large as one wavelength, sufficient accuracy for plane-wave propagation is observed. A two-dimensional steered transducer problem is explored to verify the nonlinear acoustic field directional independence of the model. A three-dimensional single-element transducer problem is solved to verify the forward model by comparing it with an existing nonlinear wave propagation code. Finally, backward-projection behavior is examined. The sound field over a plane in an absorptive medium is backward projected to the source and compared with the initial field, where good agreement is observed.

An iterative method for the computation of nonlinear, wide-angle, pulsed acoustic fields of medical diagnostic transducers

The development and optimization of medical ultrasound transducers and imaging modalities require a computational method that accurately predicts the nonlinear acoustic pressure field. A prospective method should provide the wide-angle, pulsed field emitted by an arbitrary planar source distribution and propagating in a three-dimensional, large scale domain holding a nonlinear acoustic medium. In this paper, a method is presented that is free of any assumed wavefield directionality. The nonlinear acoustic wave equation is solved by treating the nonlinear term as a contrast source. This formulation leads to an iterative scheme that involves the repetitive solution of a linear wave problem through Green's function method. It is shown that accurate field predictions may be obtained within a few iterations. Moreover, by employing a dedicated numerical convolution technique, the method allows for a discretization down to two points per wavelength or period of the highest frequency of interest. The performance of the method is evaluated through a number of nonlinear field predictions for pulsed transducers with various geometries. The results demonstrate the directional independence of the method. Moreover, comparison with results from several existing methods shows that the method accurately predicts the nonlinear field for weak to moderate nonlinearity.

Transmit beams adapted to reverberation noise suppression using dual-frequency SURF imaging

A method that uses dual-frequency pulse complexes of widely separated frequency bands to suppress noise caused by multiple scattering or multiple reflections in medical ultrasound imaging is presented. The excitation pulse complexes are transmitted to generate a second order ultrasound field (SURF) imaging synthetic transmit beam. This beam has reduced amplitude near the transducer, which illustrates the multiple scattering suppression ability of the imaging method. Field simulations solving a nonlinear wave equation are used to calculate SURF imaging beams, which are compared with beams for pulse inversion (PI) and fundamental imaging. In addition, a combined SURF and PI beam generation is described and compared with the beams mentioned above. A quality ratio, relating the energy within the near-field to that within the imaging region, is defined and used to score the multiple scattering and multiple reflection suppression abilities when imaging with the different beams. The realized combined SURF-PI beam scores highest, followed by SURF, PI (that score equally well), and the fundamental. The amplitude in the imaging region and therefore also the SNR is highest for the fundamental followed by SURF, PI, and SURF-PI. The work hence indicates that when substituting PI for SURF, one may trade increased SNR into use of increased imaging frequencies without loss of multiple scattering and multiple reflection noise suppression.

Modeling the propagation of nonlinear three-dimensional acoustic beams in inhomogeneous media

A three-dimensional model of the forward propagation of nonlinear sound beams in inhomogeneous media, a generalized Khokhlov-Zabolotskaya-Kuznetsov equation, is described. The Texas time-domain code (which accounts for paraxial diffraction, nonlinearity, thermoviscous absorption, and absorption and dispersion associated with multiple relaxation processes) was extended to solve for the propagation of nonlinear beams for the case where all medium properties vary in space. The code was validated with measurements of the nonlinear acoustic field generated by a phased array transducer operating at 2.5 MHz in water. A nonuniform layer of gel was employed to create an inhomogeneous medium. There was good agreement between the code and measurements in capturing the shift in the pressure distribution of both the fundamental and second harmonic due to the gel layer. The results indicate that the numerical tool described here is appropriate for propagation of nonlinear sound beams through weakly inhomogeneous media.

Aberration in nonlinear acoustic wave propagation

Theory and simulations are presented indicating that imaging at the second-harmonic frequency does not solve the problem of ultrasonic wave aberration. The nonlinearity of acoustic wave propagation in biological tissue is routinely exploited in medical imaging because the improved contrast resolution leads to better image quality in many applications. The major sources of acoustic noise in ultrasound images are aberration and multiple reflections between the transducer and tissue structures (reverberations), both of which are the result of spatial variations in the acoustic properties of the tissue. These variations mainly occur close to the body surface, i.e., the body wall. As a result, the nonlinearly generated, second harmonic is believed to alleviate both reverberation and aberration because it is assumed that the second harmonic is mainly generated after the body wall. However, in the case of aberration, the second harmonic is generated by an aberrated source. Thus the second harmonic experiences considerable aberration at all depths, originating from this source. The results in this paper show that the second harmonic experiences similar aberration as its generating source, the first harmonic.

Algorithm for the nonlinear propagation of acoustic beams from phased arrays and nonplanar sources

A multistage algorithm is presented to allow for the convenient and accurate computation of the finite amplitude-distorted acoustic fields produced by phased arrays and nonplanar sources. The algorithm also allows for non-normal (angled) beam computations.