Simulating Focused Ultrasound Transducers Using Discrete Sources on Regular Cartesian Grids

A practical solution to improve the field of view in circular scanning-based photoacoustic tomography

One of the primary challenges in ring single-element photoacoustic tomography systems is the low image quality in areas away from the center of the ring. This is mainly due to the limited field of view (FOV) of each transducer, which in turn reduces the imaging FOV. To address this shortcoming, we have put forward a practical and straightforward solution to enhance the FOV of circular scanning-based photoacoustic tomography (CS-PAT). This is accomplished by placing transducers at different angles instead of using a single transducer placed at a normal angle to the imaging target. We also modified the ring scanner inner wall surface to significantly reduce photoacoustic reverberation. By imaging several phantoms, we show a significant improvement in the images generated by our system imaging from 4.1 to over 7 for the signal-to-noise ratio and structural similarity index increased from 41% to 70%.

ITRUSST consensus on standardised reporting for transcranial ultrasound stimulation

As transcranial ultrasound stimulation (TUS) advances as a precise, non-invasive neuromodulatory method, there is a need for consistent reporting standards to enable comparison and reproducibility across studies. To this end, the International Transcranial Ultrasonic Stimulation Safety and Standards Consortium (ITRUSST) formed a subcommittee of experts across several domains to review and suggest standardised reporting parameters for low intensity TUS, resulting in the guide presented here. The scope of the guide is limited to reporting the ultrasound aspects of a study. The guide and supplementary material provide a simple checklist covering the reporting of: (1) the transducer and drive system, (2) the drive system settings, (3) the free field acoustic parameters, (4) the pulse timing parameters, (5) in situ estimates of exposure parameters in the brain, and (6) intensity parameters. Detailed explanations for each of the parameters, including discussions on assumptions, measurements, and calculations, are also provided.

A Forward Model Incorporating Elevation-Focused Transducer Properties for 3-D Full-Waveform Inversion in Ultrasound Computed Tomography

Ultrasound computed tomography (USCT) is an emerging medical imaging modality that holds great promise for improving human health. Full-waveform inversion (FWI)-based image reconstruction methods account for the relevant wave physics to produce high spatial resolution images of the acoustic properties of the breast tissues. A practical USCT design employs a circular ring-array comprised of elevation-focused ultrasonic transducers, and volumetric imaging is achieved by translating the ring-array orthogonally to the imaging plane. In commonly deployed slice-by-slice (SBS) reconstruction approaches, the 3-D volume is reconstructed by stacking together 2-D images reconstructed for each position of the ring-array. A limitation of the SBS reconstruction approach is that it does not account for 3-D wave propagation physics and the focusing properties of the transducers, which can result in significant image artifacts and inaccuracies. To perform 3-D image reconstruction when elevation-focused transducers are employed, a numerical description of the focusing properties of the transducers should be included in the forward model. To address this, a 3-D computational model of an elevation-focused transducer is developed to enable 3-D FWI-based reconstruction methods to be deployed in ring-array-based USCT. The focusing is achieved by applying a spatially varying temporal delay to the ultrasound pulse (emitter mode) and recorded signal (receiver mode). The proposed numerical transducer model is quantitatively validated and employed in computer simulation studies that demonstrate its use in image reconstruction for ring-array USCT.

Generating Patient-Specific Acoustic Simulations for Transcranial Focused Ultrasound Procedures Based on Optical Tracking Information

Optical tracking is a real-time transducer positioning method for transcranial focused ultrasound (tFUS) procedures, but the predicted focus from optical tracking typically does not incorporate subject-specific skull information. Acoustic simulations can estimate the pressure field when propagating through the cranium but rely on accurately replicating the positioning of the transducer and skull in a simulated space. Here, we develop and characterize the accuracy of a workflow that creates simulation grids based on optical tracking information in a neuronavigated phantom with and without transmission through an ex vivo skull cap. The software pipeline could replicate the geometry of the tFUS procedure within the limits of the optical tracking system (transcranial target registration error (TRE): 3.9 ± 0.7 mm). The simulated focus and the free-field focus predicted by optical tracking had low Euclidean distance errors of 0.5±0.1 and 1.2±0.4 mm for phantom and skull cap, respectively, and some skull-specific effects were captured by the simulation. However, the TRE of simulation informed by optical tracking was 4.6±0.2, which is as large or greater than the focal spot size used by many tFUS systems. By updating the position of the transducer using the original TRE offset, we reduced the simulated TRE to 1.1 ± 0.4 mm. Our study describes a software pipeline for treatment planning, evaluates its accuracy, and demonstrates an approach using MR-acoustic radiation force imaging as a method to improve dosimetry. Overall, our software pipeline helps estimate acoustic exposure, and our study highlights the need for image feedback to increase the accuracy of tFUS dosimetry.

Multivariable-incorporating super-resolution residual network for transcranial focused ultrasound simulation

BACKGROUND AND OBJECTIVE

Transcranial focused ultrasound (tFUS) has emerged as a new non-invasive brain stimulation (NIBS) modality, with its exquisite ability to reach deep brain areas at a high spatial resolution. Accurate placement of an acoustic focus to a target region of the brain is crucial during tFUS treatment; however, the distortion of acoustic wave propagation through the intact skull casts challenges. High-resolution numerical simulation allows for monitoring of the acoustic pressure field in the cranium but also demands extensive computational loads. In this study, we adopt a super-resolution residual network technique based on a deep convolution to enhance the prediction quality of the FUS acoustic pressure field in the targeted brain regions.

METHODS

The training dataset was acquired by numerical simulations performed at low-(1.0 mm) and high-resolutions (0.5mm) on three ex vivo human calvariae. Five different super-resolution (SR) network models were trained by using a multivariable dataset in 3D, which incorporated information on the acoustic pressure field, wave velocity, and localized skull computed tomography (CT) images.

RESULTS

The accuracy of 80.87±4.50% in predicting the focal volume with a substantial improvement of 86.91% in computational cost compared to the conventional high-resolution numerical simulation was achieved. The results suggest that the method can greatly reduce the simulation time without sacrificing accuracy and improve the accuracy further with the use of additional inputs.

CONCLUSIONS

In this research, we developed multivariable-incorporating SR neural networks for transcranial focused ultrasound simulation. Our super-resolution technique may contribute to promoting the safety and efficacy of tFUS-mediated NIBS by providing on-site feedback information on the intracranial pressure field to the operator.

Full-Waveform Inversion for Breast Ultrasound Tomography Using Line-Shape Modeled Elements

OBJECTIVE

The objective of the work described here was to incorporate the spatial shapes of the transducer elements into the framework of the full-waveform inversion.

METHODS

An element is treated as its cross-section in the 2-D imaging plane, that is, a line segment. The elements are not simply modeled as a set of point sources on their surface to avoid staircasing artifacts. By use of the Fourier collocation method, an element is spatially represented as the discrete convolution between its spatial distribution and a band-limited delta function. The excitation pulses on the emitters and recorded signals on the receivers are then weighted based on the discrete convolution results. Digital and physical experiments are implemented to validate the method.

DISCUSSION

It is meaningful to model the shapes of the elements if their spatial sizes are similar to or larger than the acoustic wavelengths. It should, however, be noted that because this article focuses on 2-D imaging, the inter-plane effects are not considered.

CONCLUSION

The approach helps reduce the root mean square errors and increase the structural similarity of the reconstructed images. It also helps to improve the stability of convergence and to accelerate the convergence speed.

Pulse-Echo Technique to Compensate for Laminate Membrane Transmission Loss in Phantom-Based Ultrasonic Attenuation Coefficient Measurements

OBJECTIVES

Accurately measuring the attenuation coefficient (AC) of reference phantoms is critical in clinical applications of quantitative ultrasound. Phantom AC measurement requires proper compensation of membrane transmission loss. Conventional methods require separate membrane samples to obtain membrane transmission loss. Unfortunately, separate membrane samples are often unavailable. A pulse-echo approach is proposed herein to compensate for membrane transmission loss without requiring separate membrane samples.

METHODS

The proposed method consists of the following steps. First, the insertion loss, caused by phantom attenuation and membrane transmission loss, is measured. Second, the membrane reflection coefficient is measured. Third, the unknown acoustic parameters of the membrane and phantom material are estimated by fitting theoretical reflection coefficient to the measured one. Finally, the fitted parameters are used to estimate membrane transmission loss and phantom AC. The proposed method was validated through k-Wave simulations and phantom experiments. Experimental AC measurements were repeated on 5 distinct phantoms by 2 operators to assess the repeatability and reproducibility of the proposed method. Five transducers were used to cover a broad bandwidth (0.7-16 MHz).

RESULTS

The acquired AC in the simulations had a maximum error of 0.06 dB/cm-MHz for simulated phantom AC values ranging from 0.5 to 1 dB/cm-MHz. The acquired AC in the experiments had a maximum error of 0.045 dB/cm-MHz for phantom AC values ranging from 0.28 to 1.48 dB/cm-MHz. Good repeatability and cross-operator reproducibility were observed with a mean coefficient of variation below 0.054.

CONCLUSION

The proposed method simplifies phantom AC measurement while providing satisfactory accuracy and precision.

Focal depth localization for highly focused transducers in isotropic materials

Focusing equations aim to define the point in a solid at which a transducer beam will reach a minimum cross section. The most commonly used focusing equation relies on a small angle assumption that inherently excludes sharply focused transducers with significant curvature. In this article, a revised focusing equation is proposed for normal and oblique incidence through a fluid-solid interface. The closed-form equation is derived using ray tracing approaches similar to the conventional expression but circumvents the paraxial approximation, extending the applicability to sharply focused probes. Both conventional and modified focusing equations are compared through normal and oblique incidence ray diagrams, and the proximity to the computationally derived geometric focus is explored. The proposed modification to the focusing equation generally results in a closer approximation to the geometric focus, a smaller beam cross section, and a greater time convergence when compared to the conventional focusing equation.

A Comparative Feasibility Study for Transcranial Extracorporeal Shock Wave Therapy

The potential beneficial regenerative and stimulatory extracorporeal shock wave therapy (ESWT) applications to the central nervous system have garnered interest in recent years. Treatment zones for these indications are acoustically shielded by bones, which heavily impact generated sound fields. We present the results of high-resolution tissue-realistic simulations, comparing the viability of different ESWT applicators in their use for transcranial applications. The performances of electrohydraulic, electromagnetic, and piezoelectric transducers for key reflector geometries are compared. Based on density information obtained from CT imaging of the head, we utilized the non-linear wave propagation toolset Matlab k-Wave to obtain spatial therapeutic sound field geometries and waveforms. In order to understand the reliability of results on the appropriate modeling of the skull, three different bone attenuation models were compared. We find that all currently clinically ESWT applicator technologies show significant retention of peak pressures and energies past the bone barrier. Electromagnetic transducers maintain a significantly higher energy flux density compared to other technologies while low focusing strength piezoelectric applicators have the weakest transmissions. Attenuation estimates provide insights into sound field degradation and energy losses, indicating that effective transcranial therapies can readily be attained with current applicators. Furthermore, the presented approach will allow for future targeted in silico development and the design of applicators and therapy plans to ultimately improve therapeutic outcomes.

Discrete Approximations of Acoustic Source Distributions

The modeling of source a distributions of finite spatial extent in ultrasound and medical imaging applications is a problem of longstanding interest. In time-domain methods, such as the finite-difference time-domain or pseudospectral approaches, one requirement is the representation of such distributions over a grid, normally Cartesian. Various artifacts, including staircasing errors, can arise. In this short contribution, the problem of the representation of distribution over a grid is framed as an optimization problem in the Fourier domain over a preselected set of grid points, thus maintaining control over computational cost and allowing the fine-tuning of the optimization to the wavenumber range of interest for a particular numerical method. Numerical results are presented in the important special case of the spherical cap or bowl source.

Measurement and simulation of steered acoustic fields generated by a multielement array for therapeutic ultrasound

Modelling of fields generated by therapeutic ultrasound arrays can be prone to errors arising from differences from nominal transducer parameters, and variations in relative outputs of array elements when driven under different conditions, especially when simulating steered fields. Here, the effect of element size, element positions, relative source pressure variations, and electrical crosstalk on the accuracy of modelling pressure fields generated by a 555 kHz 32-element ultrasonic array were investigated. For this transducer, errors in pressure amplitude and focal position were respectively reduced from 20% to 4% and 3.3 mm to 1.5 mm using crosstalk prediction, and experimentally determined positions.

Quantitative prediction of the extent of pelvic tumour ablation by magnetic resonance-guided high intensity focused ultrasound

BACKGROUND

Patient suitability for magnetic resonance-guided high intensity focused ultrasound (MRgHIFU) therapy of pelvic tumors is currently assessed by visual estimation of the proportion of tumor that can be reached by the device's focus (coverage). Since it is important to assess whether enough energy reaches the tumor to achieve ablation, a methodology for estimating the proportion of the tumor that can be ablated (treatability) was developed. Predicted treatability was compared against clinically achieved thermal ablation.

METHODS

MR Dixon sequence images of five patients with recurrent gynecological tumors were acquired during their treatment. Acousto-thermal simulations were performed using k-Wave for three exposure points (the deepest and shallowest reachable focal points within the tumor, identified from tumor coverage analysis, and a point halfway in-between) per patient. Interpolation between the resulting simulated ablated tissue volumes was used to estimate the maximum treatable depth and hence, tumor treatability. Predicted treatability was compared both to predicted tumor coverage and to the clinically treated tumor volume. The intended and simulated volumes and positions of ablated tissues were compared.

RESULTS

Predicted treatability was less than coverage by 52% (range: 31-78%) of the tumor volume. Predicted and clinical treatability differed by 9% (range: 1-25%) of tumor volume. Ablated tissue volume and position varied with beam path length through tissue.

CONCLUSION

Tumor coverage overestimated patient suitability for MRgHIFU therapy. Employing patient-specific simulations improved treatability assessment. Patient treatability assessment using simulations is feasible.

Modeling continuous source distributions in wave-based virtual acoustics

All acoustic sources are of finite spatial extent. In volumetric wave-based simulation approaches (including, e.g., the finite difference time domain method among many others), a direct approach is to represent such continuous source distributions in terms of a collection of point-like sources at grid locations. Such a representation requires interpolation over the grid and leads to common staircasing effects, particularly under rotation or translation of the distribution. In this article, a different representation is shown, based on a spherical harmonic representation of a given distribution. The source itself is decoupled from any particular arrangement of grid points, and is compactly represented as a series of filter responses used to drive a canonical set of source terms, each activating a given spherical harmonic directivity pattern. Such filter responses are derived for a variety of commonly encountered distributions. Simulation results are presented, illustrating various features of such a representation, including convergence, behaviour under rotation, the extension to the time varying case, and differences in computational cost relative to standard grid-based source representations.

Superposition method for modelling boundaries between media in viscoelastic finite difference time domain simulations

Finite-difference time domain (FDTD) techniques are widely used to model the propagation of viscoelastic waves through complex and heterogeneous structures. However, in the specific case of media mixing liquid and solid, attempts to model continuous media onto a Cartesian grid produces errors when the liquid-solid interface between different media do not align precisely with the Cartesian grid. The increase in spatial resolution required to eliminate this grid staircasing effect can be computationally prohibitive. Here, a modification to the Virieux staggered-grid FDTD scheme called the superposition method is presented. This method is intended to reduce this staircasing effect while keeping a manageable computational time. The method was validated by comparing low-spatial-resolution simulations against simulations with sufficiently high resolution to provide reasonably accurate results at any incident angle. The comparison of the root-mean-square of the stress amplitude maps showed that the amplitude of artifactual waves could be reduced by several orders of magnitude when compared to the Virieux staggered-grid FDTD method and that the superposition method helped to significantly reduce the staircasing effect in FDTD simulations.

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.

Investigation of the repeatability and reproducibility of hydrophone measurements of medical ultrasound fields

Accurate measurements of acoustic pressure are required for characterisation of ultrasonic transducers and for experimental validation of models of ultrasound propagation. Errors in measured pressure can arise from a variety of sources, including variations in the properties of the source and measurement equipment, calibration uncertainty, and processing of measured data. In this study, the repeatability of measurements made with four probe and membrane hydrophones was examined. The pressures measured by these hydrophones in three different ultrasound fields, with both linear and nonlinear, pulsed and steady state driving conditions, were compared to assess the reproducibility of measurements. The coefficient of variation of the focal peak positive pressure was less than 2% for all hydrophones across five repeated measurements. When comparing hydrophones, pressures measured in a spherically focused 1.1 MHz field were within 7% for all except 1 case, and within 10% for a broadband 5 MHz pulse from a diagnostic linear array. Larger differences of up to 55% were observed between measurements of a tightly focused 3.3 MHz field, which were reduced for some hydrophones by the application of spatial averaging corrections. Overall, the major source of these differences was spatial averaging and uncertainty in the complex frequency response of the hydrophones.

Acoustic Field Characterization of Medical Array Transducers Based on Unfocused Transmits and Single-Plane Hydrophone Measurements

Medical ultrasonic arrays are typically characterized in controlled water baths using measurements by a hydrophone, which can be translated with a positioning stage. Characterization of 3D acoustic fields conventionally requires measurements at each spatial location, which is tedious and time-consuming, and may be prohibitive given limitations of experimental setup (e.g., the bath and stage) and measurement equipment (i.e., the hydrophone). Moreover, with the development of new ultrasound sequences and modalities, multiple measurements are often required to characterize each imaging mode to ensure performance and clinical safety. Acoustic holography allows efficient characterization of source transducer fields based on single plane measurements. In this work, we explore the applicability of a re-radiation method based on the Rayleigh–Sommerfeld integral to medical imaging array characterization. We show that source fields can be reconstructed at single crystal level at wavelength resolution, based on far-field measurements. This is herein presented for three practical application scenarios: for identifying faulty transducer elements; for characterizing acoustic safety parameters in focused ultrasound sequences from 2D planar measurements; and for estimating arbitrary focused fields based on calibration from an unfocused sound field and software beamforming. The results experimentally show that the acquired pressure fields closely match those estimated using our technique.

Validation of hybrid angular spectrum acoustic and thermal modelling in phantoms

In focused ultrasound (FUS) thermal ablation of diseased tissue, acoustic beam and thermal simulations enable treatment planning and optimization. In this study, a treatment-planning methodology that uses the hybrid angular spectrum (HAS) method and the Pennes' bioheat equation (PBHE) is experimentally validated in homogeneous tissue-mimicking phantoms. Simulated three-dimensional temperature profiles are compared to volumetric MR thermometry imaging (MRTI) of FUS sonications in the phantoms, whose acoustic and thermal properties are independently measured. Additionally, Monte Carlo (MC) uncertainty analysis is performed to quantify the effect of tissue property uncertainties on simulation results. The mean error between simulated and experimental spatiotemporal peak temperature rise was +0.33°C (+6.9%). Despite this error, the experimental temperature rise fell within the expected uncertainty of the simulation, as determined by the MC analysis. The average errors of the simulated transverse and longitudinal full width half maximum (FWHM) of the profiles were -1.9% and 7.5%, respectively. A linear regression and local sensitivity analysis revealed that simulated temperature amplitude is more sensitive to uncertainties in simulation inputs than in the profile width and shape. Acoustic power, acoustic attenuation and thermal conductivity had the greatest impact on peak temperature rise uncertainty; thermal conductivity and volumetric heat capacity had the greatest impact on FWHM uncertainty. This study validates that using the HAS and PBHE method can adequately predict temperature profiles from single sonications in homogeneous media. Further, it informs the need to accurately measure or predict patient-specific properties for improved treatment planning of ablative FUS surgeries.

Equivalent-Source Acoustic Holography for Projecting Measured Ultrasound Fields Through Complex Media

Holographic projections of experimental ultrasound measurements generally use the angular spectrum method or Rayleigh integral, where the measured data are imposed as a Dirichlet boundary condition. In contrast, full-wave models, which can account for more complex wave behavior, often use interior mass or velocity sources to introduce acoustic energy into the simulation. Here, a method to generate an equivalent interior source that reproduces the measurement data is proposed based on gradient-based optimization. The equivalent-source can then be used with full-wave models (for example, the open-source k-Wave toolbox) to compute holographic projections through complex media including nonlinearity and heterogeneous material properties. Numerical and experimental results using both time-domain and continuous-wave sources are used to demonstrate the accuracy of the approach.

Multi-resolution simulation of focused ultrasound propagation through ovine skull from a single-element transducer

Transcranial focused ultrasound (tFUS) is emerging as a non-invasive brain stimulation modality. Complicated interactions between acoustic pressure waves and osseous tissue introduce many challenges in the accurate targeting of an acoustic focus through the cranium. Image-guidance accompanied by a numerical simulation is desired to predict the intracranial acoustic propagation through the skull; however, such simulations typically demand heavy computation, which warrants an expedited processing method to provide on-site feedback for the user in guiding the acoustic focus to a particular brain region. In this paper, we present a multi-resolution simulation method based on the finite-difference time-domain formulation to model the transcranial propagation of acoustic waves from a single-element transducer (250 kHz). The multi-resolution approach improved computational efficiency by providing the flexibility in adjusting the spatial resolution. The simulation was also accelerated by utilizing parallelized computation through the graphic processing unit. To evaluate the accuracy of the method, we measured the actual acoustic fields through ex vivo sheep skulls with different sonication incident angles. The measured acoustic fields were compared to the simulation results in terms of focal location, dimensions, and pressure levels. The computational efficiency of the presented method was also assessed by comparing simulation speeds at various combinations of resolution grid settings. The multi-resolution grids consisting of 0.5 and 1.0 mm resolutions gave acceptable accuracy (under 3 mm in terms of focal position and dimension, less than 5% difference in peak pressure ratio) with a speed compatible with semi real-time user feedback (within 30 s). The proposed multi-resolution approach may serve as a novel tool for simulation-based guidance for tFUS applications.

Rapid Spatial Mapping of Focused Ultrasound Fields Using a Planar Fabry-Pérot Sensor

Measurement of high acoustic pressures is necessary in order to fully characterize clinical high-intensity focused ultrasound (HIFU) fields, and for accurate validation of computational models of ultrasound propagation. However, many existing measurement devices are unable to withstand the extreme pressures generated in these fields, and those that can often exhibit low sensitivity. Here, a planar Fabry-Pérot interferometer with hard dielectric mirrors and spacer was designed, fabricated, and characterized, and its suitability for measurement of nonlinear focused ultrasound fields was investigated. The noise equivalent pressure (NEP) of the scanning system scaled with the adjustable pressure detection range between 49 kPa for pressures up to 8 MPa and 152 kPa for measurements up to 25 MPa, over a 125 MHz measurement bandwidth. Measurements of the frequency response of the sensor showed that it varied by less than 3 dB in the range 1-62 MHz. The effective element size of the sensor was 65 and waveforms were acquired at a rate of 200 Hz. The device was used to measure the acoustic pressure in the field of a 1.1 MHz single-element spherically focused bowl transducer. Measurements of the acoustic field at low pressures compared well with measurements made using a Polyvinylidene difluoride needle hydrophone. At high pressures, the measured peak focal pressures agreed well with the focal pressure modeled using the Khokhlov-Zabolotskaya-Kuznetsov equation. Maximum peak positive pressures of 25 MPa and peak negative pressures of 12 MPa were measured, and planar field scans were acquired in scan times on the order of 1 min. The properties of the sensor and scanning system are well suited to measurement of nonlinear focused ultrasound fields, in both the focal region and the low-pressure peripheral regions. The fast acquisition speed of the system and its low NEP are advantageous, and with further development of the sensor, it has potential in application to HIFU metrology.

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.