Benchmark problems for transcranial ultrasound simulation: Intercomparison of compressional wave models.

Computational models of acoustic wave propagation are frequently used in transcranial ultrasound therapy, for example, to calculate the intracranial pressure field or to calculate phase delays to correct for skull distortions. To allow intercomparison between the different modeling tools and techniques used by the community, an international working group was convened to formulate a set of numerical benchmarks. Here, these benchmarks are presented, along with intercomparison results. Nine different benchmarks of increasing geometric complexity are defined. These include a single-layer planar bone immersed in water, a multi-layer bone, and a whole skull. Two transducer configurations are considered (a focused bowl and a plane piston operating at 500 kHz), giving a total of 18 permutations of the benchmarks. Eleven different modeling tools are used to compute the benchmark results. The models span a wide range of numerical techniques, including the finite-difference time-domain method, angular spectrum method, pseudospectral method, boundary-element method, and spectral-element method. Good agreement is found between the models, particularly for the position, size, and magnitude of the acoustic focus within the skull. When comparing results for each model with every other model in a cross-comparison, the median values for each benchmark for the difference in focal pressure and position are less than 10% and 1 mm, respectively. The benchmark definitions, model results, and intercomparison codes are freely available to facilitate further comparisons.

Nonlinear ultrasound simulation in an axisymmetric coordinate system using a k-space pseudospectral method.

A full-wave model for nonlinear ultrasound propagation through a heterogeneous and absorbing medium in an axisymmetric coordinate system is developed. The model equations are solved using a nonstandard or k-space pseudospectral time domain method. Spatial gradients in the axial direction are calculated using the Fourier collocation spectral method, and spatial gradients in the radial direction are calculated using discrete trigonometric transforms. Time integration is performed using a k-space corrected finite difference scheme. This scheme is exact for plane waves propagating linearly in the axial direction in a homogeneous and lossless medium and significantly reduces numerical dispersion in the more general case. The implementation of the model is described, and performance benchmarks are given for a range of grid sizes. The model is validated by comparison with several analytical solutions. This includes one-dimensional absorption and nonlinearity, the pressure field generated by plane-piston and bowl transducers, and the scattering of a plane wave by a sphere. The general utility of the model is then demonstrated by simulating nonlinear transcranial ultrasound using a simplified head model.

Representing arbitrary acoustic source and sensor distributions in Fourier collocation methods.

Accurately representing acoustic source distributions is an important part of ultrasound simulation. This is challenging for grid-based collocation methods when such distributions do not coincide with the grid points, for instance when the source is a curved, two-dimensional surface embedded in a three-dimensional domain. Typically, grid points close to the source surface are defined as source points, but this can result in "staircasing" and substantial errors in the resulting acoustic fields. This paper describes a technique for accurately representing arbitrary source distributions within Fourier collocation methods. The method works by applying a discrete, band-limiting convolution operator to the continuous source distribution, after which source grid weights can be generated. This allows arbitrarily shaped sources, for example, focused bowls and circular pistons, to be defined on the grid without staircasing errors. The technique is examined through simulations of a range of ultrasound sources, and comparisons with analytical solutions show excellent accuracy and convergence rates. Extensions of the technique are also discussed, including application to initial value problems, distributed sensors, and moving sources.

Rapid calculation of acoustic fields from arbitrary continuous-wave sources.

A Green's function solution is derived for calculating the acoustic field generated by phased array transducers of arbitrary shape when driven by a single frequency continuous wave excitation with spatially varying amplitude and phase. The solution is based on the Green's function for the homogeneous wave equation expressed in the spatial frequency domain or k-space. The temporal convolution integral is solved analytically, and the remaining integrals are expressed in the form of the spatial Fourier transform. This allows the acoustic pressure for all spatial positions to be calculated in a single step using two fast Fourier transforms. The model is demonstrated through several numerical examples, including single element rectangular and spherically focused bowl transducers, and multi-element linear and hemispherical arrays.

Control of broadband optically generated ultrasound pulses using binary amplitude holograms.

In this work, the use of binary amplitude holography is investigated as a mechanism to focus broadband acoustic pulses generated by high peak-power pulsed lasers. Two algorithms are described for the calculation of the binary holograms; one using ray-tracing, and one using an optimization based on direct binary search. It is shown using numerical simulations that when a binary amplitude hologram is excited by a train of laser pulses at its design frequency, the acoustic field can be focused at a pre-determined distribution of points, including single and multiple focal points, and line and square foci. The numerical results are validated by acoustic field measurements from binary amplitude holograms, excited by a high peak-power laser.

Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method.

The simulation of nonlinear ultrasound propagation through tissue realistic media has a wide range of practical applications. However, this is a computationally difficult problem due to the large size of the computational domain compared to the acoustic wavelength. Here, the k-space pseudospectral method is used to reduce the number of grid points required per wavelength for accurate simulations. The model is based on coupled first-order acoustic equations valid for nonlinear wave propagation in heterogeneous media with power law absorption. These are derived from the equations of fluid mechanics and include a pressure-density relation that incorporates the effects of nonlinearity, power law absorption, and medium heterogeneities. The additional terms accounting for convective nonlinearity and power law absorption are expressed as spatial gradients making them efficient to numerically encode. The governing equations are then discretized using a k-space pseudospectral technique in which the spatial gradients are computed using the Fourier-collocation method. This increases the accuracy of the gradient calculation and thus relaxes the requirement for dense computational grids compared to conventional finite difference methods. The accuracy and utility of the developed model is demonstrated via several numerical experiments, including the 3D simulation of the beam pattern from a clinical ultrasound probe.

On-the-Fly Calculation of Time-Averaged Acoustic Intensity in Time-Domain Ultrasound Simulations Using a k-Space Pseudospectral Method.

This article presents a method to calculate the average acoustic intensity during ultrasound simulation using a new approach that exploits compression of intermediate results. One of the applications of high-intensity focused ultrasound (HIFU) simulations is the calculation of the thermal dose, which indicates the amount of tissue destroyed using a state-of-the-art k-space pseudospectral method. The thermal simulation is preceded by the calculation of the average intensity within the acoustic simulation. Due to the time staggering between the particle velocity and the acoustic pressure used in such simulations, the average intensity calculation is typically executed offline after the acoustic simulation consuming both disk space and time (the data can spread over terabytes). Our new approach calculates the average intensity during the acoustic simulation using the output coefficients of a new compression method which enables resolving the time staggering on-the-fly with huge disk space savings. To reduce RAM requirements, the article also presents a new 40-bit method for encoding compression complex coefficients. Experimental numerical simulations with the proposed method have shown that disk space requirements are up to 99% lower. The simulation speed was not significantly affected by the approach and the compression error did not affect the prediction accuracy of the thermal dose. From the standpoint of supercomputers, the new approach is significantly more economical. Saving computing resources increases the chances of real use of acoustic simulations in practice. The method can be applied to signals of a similar character, e.g., for electromagnetic radio waves.

Broadband All-Optical Plane-Wave Ultrasound Imaging System Based on a Fabry-Perot Scanner.

A broadband all-optical plane-wave ultrasound imaging system for high-resolution 3-D imaging of biological tissues is presented. The system is based on a planar Fabry-Perot (FP) scanner for ultrasound detection and the photoacoustic generation of ultrasound in a carbon-nanotube-polydimethylsiloxane (CNT-PDMS) composite film. The FP sensor head was coated with the CNT-PDMS film which acts as an ultrasound transmitting layer for pulse-echo imaging. Exciting the CNT-PDMS coating with nanosecond laser pulses generated monopolar plane-wave ultrasound pulses with MPa-range peak pressures and a -6-dB bandwidth of 22 MHz, which were transmitted into the target. The resulting scattered acoustic field was detected across a 15 mm ×15 mm scan area with a step size of 100 [Formula: see text] and an optically defined element size of [Formula: see text]. The -3-dB bandwidth of the sensor was 30 MHz. A 3-D image of the scatterer distribution was then recovered using a k -space reconstruction algorithm. To obtain a measure of spatial resolution, the instrument line-spread function (LSF) was measured as a function of position. At the center of the scan area, the depth-dependent lateral LSF ranged from 46 to 65 [Formula: see text] for depths between 1 and 12 mm. The vertical LSF was independent of position and measured to be [Formula: see text] over the entire field of view. To demonstrate the ability of the system to provide high-resolution 3-D images, phantoms with well-defined scattering structures of arbitrary geometry were imaged. To demonstrate its suitability for imaging biological tissues, phantoms with similar impedance mismatches, sound speed and scattering properties to those present in the tissue, and ex vivo tissue samples were imaged. Compared with conventional piezoelectric-based ultrasound scanners, this approach offers the potential for improved image quality and higher resolution for superficial tissue imaging. Since the FP scanner is capable of high-resolution 3-D photoacoustic imaging of in vivo biological tissues, the system could ultimately be developed into an instrument for dual-mode all-optical ultrasound and photoacoustic imaging.

Experimental Validation of k-Wave: Nonlinear Wave Propagation in Layered, Absorbing Fluid Media.

Models of ultrasound propagation in biologically relevant media have applications in planning and verification of ultrasound therapies and computational dosimetry. To be effective, the models must be able to accurately predict both the spatial distribution and amplitude of the acoustic pressure. This requires that the models are validated in absolute terms, which for arbitrarily heterogeneous media should be performed by comparison with measurements of the acoustic field. In this article, simulations performed using the open-source k-Wave acoustics toolbox, with a measurement-based source definition, were quantitatively validated against measurements of acoustic pressure in water and layered absorbing fluid media. In water, the measured and simulated spatial-peak pressures agreed to within 3% under linear conditions and 7% under nonlinear conditions. After propagation through a planar or wedge-shaped glycerol-filled phantom, the difference in spatial-peak pressure was 8.5% and 10.7%, respectively. These differences are within or close to the expected uncertainty of the acoustic pressure measurement. The -6 dB width and length of the focus agreed to within 4% in all cases, and the focal positions were within 0.7 mm for the planar phantom and 1.2 mm for the wedge-shaped phantom. These results demonstrate that when the acoustic medium properties and geometry are well known, accurate quantitative predictions of the acoustic field can be made using k-Wave.

Full Modeling of High-Intensity Focused Ultrasound and Thermal Heating in the Kidney Using Realistic Patient Models.

OBJECTIVE: High-intensity focused ultrasound (HIFU) therapy can be used for noninvasive treatment of kidney (renal) cancer, but the clinical outcomes have been variable. In this study, the efficacy of renal HIFU therapy was studied using a nonlinear acoustic and thermal simulations in three patients. METHODS: The acoustic simulations were conducted with and without refraction in order to investigate its effect on the shape, size, and pressure distribution at the focus. The values for the attenuation, sound speed, perfusion, and thermal conductivity of the kidney were varied over the reported ranges to determine the effect of variability on heating. Furthermore, the phase aberration was studied in order to quantify the underlying phase shifts using a second order polynomial function. RESULTS: The ultrasound field intensity was found to drop on average 11.1 dB with refraction and 6.4 dB without refraction. Reflection at tissue interfaces was found to result in a loss less than 0.1 dB. Focal point splitting due to refraction significantly reduced the heating efficacy. Perfusion did not have a large effect on heating during short sonication durations. Small changes in temperature were seen with varying attenuation and thermal conductivity, but no visible changes were present with sound speed variations. The aberration study revealed an underlying trend in the spatial distribution of the phase shifts. CONCLUSION: The results show that the efficacy of HIFU therapy in the kidney could be improved with aberration correction. SIGNIFICANCE: A method is proposed by that patient specific pre-treatment calculations could be used to overcome the aberration and therefore make ultrasound treatment possible.

Full Modeling of High-Intensity Focused Ultrasound and Thermal Heating in the Kidney Using Realistic Patient Models.

OBJECTIVE: High-intensity focused ultrasound (HIFU) therapy can be used for noninvasive treatment of kidney (renal) cancer, but the clinical outcomes have been variable. In this study, the efficacy of renal HIFU therapy was studied using nonlinear acoustic and thermal simulations in three patients. METHODS: The acoustic simulations were conducted with and without refraction in order to investigate its effect on the shape, size, and pressure distribution at the focus. The values for the attenuation, sound speed, perfusion, and thermal conductivity of the kidney were varied over the reported ranges to determine the effect of variability on heating. Furthermore, the phase aberration was studied in order to quantify the underlying phase shifts using a second-order polynomial function. RESULTS: The ultrasound field intensity was found to drop on average 11.1 dB with refraction and 6.4 dB without refraction. Reflection at tissue interfaces was found to result in a loss less than 0.1 dB. Focal point splitting due to refraction significantly reduced the heating efficacy. Of all the tissue parameters, perfusion was found to affect the heating the most. Small changes in temperature were seen with varying attenuation and thermal conductivity, but no visible changes were present with sound speed variations. The aberration study revealed an underlying trend in the spatial distribution of the phase shifts. CONCLUSION: The results show that the efficacy of HIFU therapy in the kidney could be improved with aberration correction. SIGNIFICANCE: A method is proposed by which patient specific pretreatment calculations could be used to overcome the aberration and therefore make ultrasound treatment possible.

The Effect of Tissue Physiological Variability on Transurethral Ultrasound Therapy of the Prostate.

Therapeutic ultrasound is an investigational modality which could potentially be used for minimally invasive treatment of prostate cancer. Computational simulations were used to study the effect of natural physiological variations in tissue parameters on the efficacy of therapeutic ultrasound treatment in the prostate. The simulations were conducted on a clinical ultrasound therapy system using patient computed tomography (CT) data. The values of attenuation, perfusion, specific heat capacity and thermal conductivity were changed within their biological ranges to determine their effect on peak temperature andthermal dose volume. Increased attenuation was found to have the biggest effect on peak temperature with a 6.9%rise. The smallest effect was seen with perfusion with ±0.2% variation in peak temperature. Thermal dose was mostly affected by specific heat capacity which showed a 20.7% increase in volume with reduced heat capacity. Thermal conductivity had the smallest effect on thermal dose with up to 2.1% increase in the volume with reduced thermal conductivity. These results can be used to estimate the interpatient variation during the therapeutic ultrasound treatment of the prostate.

Transurethral ultrasound therapy of the prostate in the presence of calcifications: A simulation study.

PURPOSE: Transurethral ultrasound therapy is an investigational treatment modality which could potentially be used for the localized treatment of prostate cancer. One of the limiting factors of this therapy is prostatic calcifications. These attenuate and reflect ultrasound and thus reduce the efficacy of the heating. The aim of this study is to investigate how prostatic calcifications affect therapeutic efficacy, and to identify the best sonication strategy when calcifications are present. METHODS: Realistic computational models were used on clinical patient data in order to simulate different therapeutic situations with naturally occurring calcifications as well as artificial calcifications of different sizes (1-10 mm) and distances (5-15 mm). Furthermore, different sonication strategies were tested in order to deliver therapy to the untreated tissue regions behind the calcifications. RESULTS: The presence of calcifications in front of the ultrasound field was found to increase the peak pressure by 100% on average while the maximum temperature only rose by 9% during a 20-s sonication. Losses in ultrasound energy were due to the relatively large acoustic impedance mismatch between the prostate tissue and the calcifications (1.63 vs 3.20 MRayl) and high attenuation coefficient (0.78 vs 2.64 dB/MHz1.1 /cm), which together left untreated tissue regions behind the calcifications. In addition, elevated temperatures were seen in the region between the transducer and the calcifications. Lower sonication frequencies (1-4 MHz) were not able to penetrate through the calcifications effectively, but longer sonication durations (20-60 s) with selective transducer elements were effective in treating the tissue regions behind the calcifications. CONCLUSIONS: Prostatic calcifications limit the reach of therapeutic ultrasound treatment due to reflections and attenuation. The tissue regions behind the calcifications can possibly be treated using longer sonication durations combined with proper transducer element selection. However, caution should be taken with calcifications located close to sensitive organs such as the urethra, bladder neck, or rectal wall.

Nonlinear 3-D simulation of high-intensity focused ultrasound therapy in the Kidney.

Kidney cancer is a severe disease which can be treated non-invasively using high-intensity focused ultrasound (HIFU) therapy. However, tissue in front of the transducer and the deep location of kidney can cause significant losses to the efficiency of the treatment. The effect of attenuation, refraction and reflection due to different tissue types on HIFU therapy of the kidney was studied using a nonlinear ultrasound simulation model. The geometry of the tissue was derived from a computed tomography (CT) dataset of a patient which had been segmented for water, bone, soft tissue, fat and kidney. The combined effect of inhomogeneous attenuation and soundspeed was found to result in an 11.0 dB drop in spatial peak-temporal average (SPTA) intensity in the kidney compared to pure water. The simulation without refraction effects showed a 6.3 dB decrease indicating that both attenuation and refraction contribute to the loss in focal intensity. The losses due to reflections at soft tissue interfaces were less than 0.1 dB. Focal point shifting due to refraction effects resulted in -1.3, 2.6 and 1.3 mm displacements in x-, y- and z-directions respectively. Furthermore, focal point splitting into several smaller subvolumes was observed. The total volume of the secondary focal points was approximately 46% of the largest primary focal point. This could potentially lead to undesired heating outside the target location and longer therapy times.