Fast Nearfield Method Research Articles

Enabling support for Octave within the FOCUS software package

FOCUS, the “Fast Object-Oriented C+ + Simulator” (https://www.egr.msu.edu/∼fultras-web/), is a free software package that enables rapid calculations of continuous-wave and transient pressure fields generated by single transducers and phased arrays. FOCUS achieves small errors in relatively short computation times through linear memory-efficient calculations with the fast nearfield method. This capability extends to quick and accurate biomedical imaging simulations within FOCUS. Moreover, FOCUS supports some nonlinear pressure calculations, such as the continuous-wave and transient Khokhlov-Zabolotskaya-Kuznetsov (KZK) equations for circular and spherical transducers. Additionally, the Angular Spectrum Approach (ASA), a frequency-domain solution to linear computational methods, ideal for large volumetric field computations, is also included within the FOCUS package. Initial success with MATLAB motivates the creation of an Octave version that replicates the core functionalities of the original FOCUS package. GNU Octave is similar to MATLAB, but unlike MATLAB, Octave is a free software. Building and compiling FOCUS in Octave required a few minor alterations due to discrepancies in syntax and function calls. The use of Octave-compiled MEX files simplified the conversion process while introducing addition memory overhead compared to rewriting C+ + code for Oct-file structuring. The performance and capabilities of FOCUS in Octave are demonstrated and discussed.

Parallel implementations of the fast nearfield method for continuous-wave and transient pressure calculations

FOCUS, the “Fast Object-oriented C + + Simulator” (https://www.egr.msu.edu/∼fultras-web/), enables rapid calculations of continuous-wave and transient pressure fields generated by single transducers and phased arrays. FOCUS achieves small errors in relatively short computation times through memory-efficient calculations with the fast nearfield method, which converges exponentially. The fast nearfield method is also an embarrassingly parallel algorithm that supports further reductions in the computation time through hardware acceleration. Previous efforts to accelerate calculations with the fast nearfield method successfully implemented multithreading within multiple central processing unit (CPU) cores using OpenMP, and this capability is presently included with FOCUS. Prior success with OpenMP motivates ongoing efforts to further extend the functionality of FOCUS by interfacing the fast nearfield method to an NVIDIA graphics processing unit (GPU). To facilitate FOCUS calculations on a GPU, calculations with the fast nearfield method are implemented in CUDA for time-harmonic and transient pressure computations with each supported transducer shape. This combination achieves a significant reduction in the computation time compared to single-threaded execution. When evaluated on an NVIDIA 1670 GPU, the computation time is also approximately halved relative to the multithreaded OpenMP implementation. These results encourage further efforts to parallelize existing and future FOCUS routines.

A parametric evaluation of shear wave speeds estimated with the time-of-flight approach in viscoelastic media

The time-of-flight approach estimates the shear elasticity in tissue mimicking elastography phantoms and in soft tissue. The time-of-flight approach is effective in elastic phantoms, but the time-of-flight approach tends to overestimate the shear elasticity in viscoelastic phantoms and in viscoelastic soft tissues. To characterize errors in estimated parameters for different values of the shear elasticity and the shear viscosity, three-dimensional (3D) shear wave simulations are evaluated for twelve different parameter combinations. The 3D acoustic radiation force is calculated for an L7-4 transducer using the fast nearfield method and the angular spectrum approach, and then, 3D shear wave propagation in a viscoelastic medium is simulated with Green's functions for a Kelvin-Voigt model. The time-of-flight method is then evaluated within a two-dimensional plane. The results show that the accuracy of the time-of-flight method depends on the values of the shear elasticity and the shear viscosity. In particular, the error in the estimated shear elasticity increases as the shear viscosity increases, where the largest errors are observed when larger values of the shear viscosity are combined with smaller values of the shear elasticity. [Work supported in part by NIH Grants DK092255, EB023051, and EB012079.]

New phased array models for fast nearfield pressure simulations

FOCUS, the “Fast Object-oriented C + + Ultrasound Simulator,” is free MATLAB-based software that rapidly and accurately models therapeutic and diagnostic ultrasound with the fast nearfield method, time-space decomposition, and the angular-spectrum approach. FOCUS presently supports arrays of circular, rectangular, and spherically focused transducers arranged in flat planar, spherically focused, and cylindrically focused geometries. Excellent results are obtained with all of these array geometries in FOCUS for simulations of continuous-wave and transient excitations, and new array geometries are needed for B-mode simulations that are presently under development. These new array geometries also require new data structures that describe the electrical connectivity of the arrays. Efforts to develop these new features in FOCUS are underway, and results obtained with these new array geometries will be presented. Other new features for FOCUS will also be demonstrated. [Supported in part by NIH Grant R01 EB012079.]

Rapid simulations of diagnostic ultrasound with multiple-zone receive beamforming

Routines are under development in FOCUS, the “Fast Object-oriented C + + Ultrasound Simulator” (http://www.egr.msu.edu/~fultras-web), to accelerate B-mode image simulations by combining the fast nearfield method with time-space decomposition. The most recent addition to the FOCUS simulation model implements receive beamforming in multiple zones. To demonstrate the rapid convergence of these simulations in the nearfield region, simulations of a 192 element linear array with an electronically translated 64 element sub-aperture are evaluated for a transient excitation pulse with a center frequency of 3 MHz. The transducers in this simulated array are 5 mm high and 0.5133 mm wide with a 0.1 mm center to center spacing. The simulation is evaluated for a computer phantom with 100,000 scatterers. The same configuration is simulated in Field II (http://field-ii.dk), and the impulse response approach with a temporal sampling rate of 1 GHz is used as reference. Simulations are evaluated for the entire B-mode image simulated with each approach. The results show that, with sampling frequencies of 15 MHz and higher, FOCUS eliminates all of the numerical artifacts that appear in the nearfield region of the B-mode image, whereas Field II requires much higher temporal sampling frequencies to obtain similar results. [Supported in part by NIH Grant R01 EB012079.]

Improved simulations of diagnostic ultrasound with the fast nearfield method and time-space

Diagnostic ultrasound simulations are presently under development for FOCUS (http://www.egr.msu.edu/~fultras-web). To reduce the computation time without increasing the numerical error, each signal in FOCUS is calculated once, stored, and then the effects of different time delays are calculated with cubic spline interpolation. This is much more efficient than calculating the same transient signal at a scatterer repeatedly for different values of the time delay. Initially, the interpolation results were obtained from uniformly sampled signals, and now the signal start and end times are also considered. This step reduces the error in the pulse-echo calculation without significantly increasing the computation time. Simulated B-mode images were evaluated in a cyst phantom with 100 000 scatterers using this approach. Images with 50 A-lines are simulated for a linear array with 192 elements, where the translating subaperture contains 64 elements. The resulting simulated images are compared to images obtained with the same configuration in Field II (http://field-ii.dk/). An error of approximately 1% is achieved in FOCUS with a sampling frequency of 30 MHz, where Field II requires a sampling frequency of 180 MHz to reach the same error. FOCUS also reduces the simulation time by a factor of six. [Supported in part by NIH Grant R01 EB012079.]

Simulation of diagnostic ultrasound imaging with the fast nearfield method

Software for simulating diagnostic ultrasound imaging with the fast nearfield method is now available in FOCUS (http://www.egr.msu.edu/~fultras-web). The implementation of ultrasound imaging simulations in FOCUS first decomposes the excitation pulse into two shorter pulses that, when convolved together, are equivalent to the original excitation pulse. These two shorter pulses are applied to the transmit and receive apertures, reciprocity is applied to the receive aperture, and then the results of the intermediate pressure calculations are convolved to obtain the simulated pulse-echo signal. Simulation results using point scatterers are compared to Field II (http://field-ii.dk/) for simulations with a linear phased array. Results show that FOCUS achieves equal or smaller errors in less time than Field II, and the FOCUS results require much smaller sampling frequencies. The FOCUS results are achieved with an analytically equivalent algorithm that avoids aliasing problems by calculating pressures with the fast nearfield method combined with time-space decomposition. To further improve the efficiency of the imaging simulation in FOCUS, intermediate signals are computed, stored, and re-used. Ongoing efforts to incorporate new features into ultrasound imaging simulations with FOCUS will also be discussed. [This work was supported in part by NIH Grant R01 EB012079.]

Rapid Transient Pressure Field Computations in the Nearfield of Circular Transducers Using Frequency-Domain Time-Space Decomposition

The fast nearfield method, when combined with time-space decomposition, is a rapid and accurate approach for calculating transient nearfield pressures generated by ultrasound transducers. However, the standard time-space decomposition approach is only applicable to certain analytical representations of the temporal transducer surface velocity that, when applied to the fast nearfield method, are expressed as a finite sum of products of separate temporal and spatial terms. To extend time-space decomposition such that accelerated transient field simulations are enabled in the nearfield for an arbitrary transducer surface velocity, a new transient simulation method, frequency-domain time-space decomposition (FDTSD), is derived. With this method, the temporal transducer surface velocity is transformed into the frequency domain, and then each complex-valued term is processed separately. Further improvements are achieved by spectral clipping, which reduces the number of terms and the computation time. Trade-offs between speed and accuracy are established for FDTSD calculations, and pressure fields obtained with the FDTSD method for a circular transducer are compared with those obtained with Field II and the impulse response method. The FDTSD approach, when combined with the fast nearfield method and spectral clipping, consistently achieves smaller errors in less time and requires less memory than Field II or the impulse response method.

Frequency domain calculations of pulse-echo ultrasound signals with the fast nearfield method

Simulations of pulse-echo signals transmitted and received by an ultrasound phased array are evaluated in the frequency domain using FOCUS, and the results are compared with the output of Field II. The frequency domain simulation approach in FOCUS calculates the fast Fourier transform of the input signal, evaluates the individual frequency components of the transmit and receive pressure transfer functions using the fast nearfield method, and then computes the inverse fast Fourier transform of the product to obtain a numerical representation of the received time signal. With this approach, synthetic aperture signals are simulated in Matlab for each pair of transmit and receive elements for three transmit elements and sixteen receive elements. Each element is excited with a 3 cycle Hanning-weighted input pulse with a 3 MHz center frequency. The temporal sampling of the computed signal in FOCUS is 12 MHz. To achieve approximately the same accuracy, Field II requires a temporal sampling of 4 GHz. Examples of ...

Fast ultrasound beam prediction for linear and regular two-dimensional arrays

Real-time beam predictions are highly desirable for the patient-specific computations required in ultrasound therapy guidance and treatment planning. To address the longstanding issue of the computational burden associated with calculating the acoustic field in large volumes, we use graphics processing unit (GPU) computing to accelerate the computation of monochromatic pressure fields for therapeutic ultrasound arrays. In our strategy, we start with acceleration of field computations for single rectangular pistons, and then we explore fast calculations for arrays of rectangular pistons. For single-piston calculations, we employ the fast near-field method (FNM) to accurately and efficiently estimate the complex near-field wave patterns for rectangular pistons in homogeneous media. The FNM is compared with the Rayleigh-Sommerfeld method (RSM) for the number of abscissas required in the respective numerical integrations to achieve 1%, 0.1%, and 0.01% accuracy in the field calculations. Next, algorithms are described for accelerated computation of beam patterns for two different ultrasound transducer arrays: regular 1-D linear arrays and regular 2-D linear arrays. For the array types considered, the algorithm is split into two parts: 1) the computation of the field from one piston, and 2) the computation of a piston-array beam pattern based on a pre-computed field from one piston. It is shown that the process of calculating an array beam pattern is equivalent to the convolution of the single-piston field with the complex weights associated with an array of pistons. Our results show that the algorithms for computing monochromatic fields from linear and regularly spaced arrays can benefit greatly from GPU computing hardware, exceeding the performance of an expensive CPU by more than 100 times using an inexpensive GPU board. For a single rectangular piston, the FNM method facilitates volumetric computations with 0.01% accuracy at rates better than 30 ns per field point. Furthermore, we demonstrate array calculation speeds of up to 11.5 X 10(9) field-points per piston per second (0.087 ns per field point per piston) for a 512-piston linear array. Beam volumes containing 256(3) field points are calculated within 1 s for 1-D and 2-D arrays containing 512 and 20(2) pistons, respectively, thus facilitating future real-time thermal dose predictions.

Fast MATLAB-based ultrasound simulation software: Transient calculations.

The FOCUS software package quickly calculates transient pressure fields generated by ultrasound phased arrays. This software is useful for visualizing pressure wave propagation in the nearfield region, and the results are directly applicable to beamforming evaluations and phased array design for diagnostic and therapeutic applications. FOCUS includes support for transient pressure calculations with single transducers, including flat circular and rectangular pistons as well as spherically focused shells, and arrays of flat circular and rectangular transducers. Routines to calculate time delays for focusing the arrays are also included with the software. Transient calculations are performed with a combination of the fast nearfield method and time space decomposition, which achieves rapid convergence to reduce the numerical error while significantly reducing the computation time. Within the FOCUS software, time space decomposition is implemented with a simplified approach that decreases the overhead and reduces the computational complexity. The advantages of FOCUS over competing approaches include short computation times and small numerical errors. Examples of the supported element and array geometries will be shown, and sample codes will be provided. Transient pressure fields computed with FOCUS and with FIELD II are compared, and results show that, for the same peak error value, FOCUS is several times faster for calculations in the nearfield region.

Fast MATLAB-based ultrasound simulation software: Time harmonic calculations.

A library of fast C++ routines is created for MATLAB simulations of time harmonic pressures generated by ultrasound phased arrays. These programs are included in the recently released FOCUS software package, where FOCUS is an acronym for “fast object-oriented C++ ultrasound simulator.” FOCUS is intended for large scale phased array simulations, characterizations of beamforming strategies, and analysis of new phased array designs. This software supports nearfield simulations of individual flat circular and rectangular pistons and focused spherical shells. Routines for pressure calculations with flat and curved phased array geometries consisting of circular and rectangular elements are also included. Within the FOCUS program, time harmonic simulations are performed with a combination of the fast nearfield method and the angular spectrum approach. Comparisons of errors and computation times with competing programs are evaluated for single element and phased array calculations, and results show that, for the same peak error value, FOCUS achieves an order of magnitude or more reduction in the computation time in large phased array simulations.

Optimal simulations of ultrasonic fields produced by large thermal therapy arrays using the angular spectrum approach.

The angular spectrum approach is evaluated for the simulation of focused ultrasound fields produced by large thermal therapy arrays. For an input pressure or normal particle velocity distribution in a plane, the angular spectrum approach rapidly computes the output pressure field in a three dimensional volume. To determine the optimal combination of simulation parameters for angular spectrum calculations, the effect of the size, location, and the numerical accuracy of the input plane on the computed output pressure is evaluated. Simulation results demonstrate that angular spectrum calculations performed with an input pressure plane are more accurate than calculations with an input velocity plane. Results also indicate that when the input pressure plane is slightly larger than the array aperture and is located approximately one wavelength from the array, angular spectrum simulations have very small numerical errors for two dimensional planar arrays. Furthermore, the root mean squared error from angular spectrum simulations asymptotically approaches a nonzero lower limit as the error in the input plane decreases. Overall, the angular spectrum approach is an accurate and robust method for thermal therapy simulations of large ultrasound phased arrays when the input pressure plane is computed with the fast nearfield method and an optimal combination of input parameters.

A 2D fast near-field method for calculating near-field pressures generated by apodized rectangular pistons.

Analytical two-dimensional (2D) integral expressions are derived for fast calculations of time-harmonic and transient near-field pressures generated by apodized rectangular pistons. These 2D expressions represent an extension of the fast near-field method (FNM) for uniformly excited pistons. After subdividing the rectangular piston into smaller rectangles, the pressure produced by each of the smaller rectangles is calculated using the uniformly excited FNM expression for a rectangular piston, and the total pressure generated by an apodized rectangular piston is the superposition of the pressures produced by all of the subdivided rectangles. By exchanging summation variables and performing integration by parts, a 2D apodized FNM expression is obtained, and the resulting expression eliminates the numerical singularities that are otherwise present in numerical models of pressure fields generated by apodized rectangular pistons. A simplified time space decomposition method is also described, and this method further reduces the computation time for transient pressure fields. The results are compared with the Rayleigh-Sommerfeld integral and the FIELD II program for a rectangular source with each side equal to four wavelengths. For time-harmonic calculations with a 0.1 normalized root mean square error (NRMSE), the apodized FNM is 4.14 times faster than the Rayleigh-Sommerfeld integral and 59.43 times faster than the FIELD II program, and for a 0.01 NRMSE, the apodized FNM is 12.50 times faster than the Rayleigh-Sommerfeld integral and 155.06 times faster than the FIELD II program. For transient calculations with a 0.1 NRMSE, the apodized FNM is 2.31 times faster than the Rayleigh-Sommerfeld integral and 4.66 times faster than the FIELD II program, and for a 0.01 NRMSE, the apodized FNM is 11.90 times faster than the Rayleigh-Sommerfeld integral and 24.04 times faster than the FIELD II program. Thus, the 2D apodized FNM is ideal for fast pressure calculations and for accurate reference calculations in the near-field region.

A fast near-field method for calculations of time-harmonic and transient pressures produced by triangular pistons

Analytical expressions are demonstrated for fast calculations of time-harmonic and transient near-field pressures generated by triangular pistons. These fast expressions remove singularities from the impulse response, thereby reducing the computation time and the peak numerical error with a general formula that describes the near-field pressure produced by any triangular piston geometry. The time-domain expressions are further accelerated by a time-space decomposition approach that analytically separates the spatial and temporal components of the numerically computed transient pressure. Applied to a Hanning-weighted input pulse, time-space decomposition converts each spatio-temporal integral into six spatial integral evaluations at each field point. Time-harmonic and transient calculations are evaluated for an equilateral triangle with sides equal to four wavelengths, and the resulting errors are compared to pressures obtained with exact and approximate implementations of the impulse response method. The results show that the fast near-field method achieves smaller maximum errors and is consistently faster than the impulse response and methods that approximate the impulse response.