Comparison of temporal and spectral scattering methods using acoustically large breast models derived from magnetic resonance images.

Accurate and efficient modeling of ultrasound propagation through realistic tissue models is important to many aspects of clinical ultrasound imaging. Simplified problems with known solutions are often used to study and validate numerical methods. Greater confidence in a time-domain k-space method and a frequency-domain fast multipole method is established in this paper by analyzing results for realistic models of the human breast. Models of breast tissue were produced by segmenting magnetic resonance images of ex vivo specimens into seven distinct tissue types. After confirming with histologic analysis by pathologists that the model structures mimicked in vivo breast, the tissue types were mapped to variations in sound speed and acoustic absorption. Calculations of acoustic scattering by the resulting model were performed on massively parallel supercomputer clusters using parallel implementations of the k-space method and the fast multipole method. The efficient use of these resources was confirmed by parallel efficiency and scalability studies using large-scale, realistic tissue models. Comparisons between the temporal and spectral results were performed in representative planes by Fourier transforming the temporal results. An RMS field error less than 3% throughout the model volume confirms the accuracy of the methods for modeling ultrasound propagation through human breast.

Acoustic scattering by arbitrary distributions of disjoint, homogeneous cylinders or spheres.

A T-matrix formulation is presented to compute acoustic scattering from arbitrary, disjoint distributions of cylinders or spheres, each with arbitrary, uniform acoustic properties. The generalized approach exploits the similarities in these scattering problems to present a single system of equations that is easily specialized to cylindrical or spherical scatterers. By employing field expansions based on orthogonal harmonic functions, continuity of pressure and normal particle velocity are directly enforced at each scatterer using diagonal, analytic expressions to eliminate the need for integral equations. The effect of a cylinder or sphere that encloses all other scatterers is simulated with an outer iterative procedure that decouples the inner-object solution from the effect of the enclosing object to improve computational efficiency when interactions among the interior objects are significant. Numerical results establish the validity and efficiency of the outer iteration procedure for nested objects. Two- and three-dimensional methods that employ this outer iteration are used to measure and characterize the accuracy of two-dimensional approximations to three-dimensional scattering of elevation-focused beams.

A mesh-free approach to acoustic scattering from multiple spheres nested inside a large sphere by using diagonal translation operators.

A multiple-scattering approach is presented to compute the solution of the Helmholtz equation when a number of spherical scatterers are nested in the interior of an acoustically large enclosing sphere. The solution is represented in terms of partial-wave expansions, and a linear system of equations is derived to enforce continuity of pressure and normal particle velocity across all material interfaces. This approach yields high-order accuracy and avoids some of the difficulties encountered when using integral equations that apply to surfaces of arbitrary shape. Calculations are accelerated by using diagonal translation operators to compute the interactions between spheres when the operators are numerically stable. Numerical results are presented to demonstrate the accuracy and efficiency of the method.

Scattering calculation and image reconstruction using elevation-focused beams.

Pressure scattered by cylindrical and spherical objects with elevation-focused illumination and reception has been analytically calculated, and corresponding cross sections have been reconstructed with a two-dimensional algorithm. Elevation focusing was used to elucidate constraints on quantitative imaging of three-dimensional objects with two-dimensional algorithms. Focused illumination and reception are represented by angular spectra of plane waves that were efficiently computed using a Fourier interpolation method to maintain the same angles for all temporal frequencies. Reconstructions were formed using an eigenfunction method with multiple frequencies, phase compensation, and iteration. The results show that the scattered pressure reduces to a two-dimensional expression, and two-dimensional algorithms are applicable when the region of a three-dimensional object within an elevation-focused beam is approximately constant in elevation. The results also show that energy scattered out of the reception aperture by objects contained within the focused beam can result in the reconstructed values of attenuation slope being greater than true values at the boundary of the object. Reconstructed sound speed images, however, appear to be relatively unaffected by the loss in scattered energy. The broad conclusion that can be drawn from these results is that two-dimensional reconstructions require compensation to account for uncaptured three-dimensional scattering.

A k-space method for acoustic propagation using coupled first-order equations in three dimensions.

A previously described two-dimensional k-space method for large-scale calculation of acoustic wave propagation in tissues is extended to three dimensions. The three-dimensional method contains all of the two-dimensional method features that allow accurate and stable calculation of propagation. These features are spectral calculation of spatial derivatives, temporal correction that produces exact propagation in a homogeneous medium, staggered spatial and temporal grids, and a perfectly matched boundary layer. Spectral evaluation of spatial derivatives is accomplished using a fast Fourier transform in three dimensions. This computational bottleneck requires all-to-all communication; execution time in a parallel implementation is therefore sensitive to node interconnect latency and bandwidth. Accuracy of the three-dimensional method is evaluated through comparisons with exact solutions for media having spherical inhomogeneities. Large-scale calculations in three dimensions were performed by distributing the nearly 50 variables per voxel that are used to implement the method over a cluster of computers. Two computer clusters used to evaluate method accuracy are compared. Comparisons of k-space calculations with exact methods including absorption highlight the need to model accurately the medium dispersion relationships, especially in large-scale media. Accurately modeled media allow the k-space method to calculate acoustic propagation in tissues over hundreds of wavelengths.

Born iterative reconstruction using perturbed-phase field estimates.

A method of image reconstruction from scattering measurements for use in ultrasonic imaging is presented. The method employs distorted-wave Born iteration but does not require using a forward-problem solver or solving large systems of equations. These calculations are avoided by limiting intermediate estimates of medium variations to smooth functions in which the propagated fields can be approximated by phase perturbations derived from variations in a geometric path along rays. The reconstruction itself is formed by a modification of the filtered-backpropagation formula that includes correction terms to account for propagation through an estimated background. Numerical studies that validate the method for parameter ranges of interest in medical applications are presented. The efficiency of this method offers the possibility of real-time imaging from scattering measurements.

An eigenfunction method for reconstruction of large-scale and high-contrast objects.

A multiple-frequency inverse scattering method that uses eigenfunctions of a scattering operator is extended to image large-scale and high-contrast objects. The extension uses an estimate of the scattering object to form the difference between the scattering by the object and the scattering by the estimate of the object. The scattering potential defined by this difference is expanded in a basis of products of acoustic fields. These fields are defined by eigenfunctions of the scattering operator associated with the estimate. In the case of scattering objects for which the estimate is radial, symmetries in the expressions used to reconstruct the scattering potential greatly reduce the amount of computation. The range of parameters over which the reconstruction method works well is illustrated using calculated scattering by different objects. The method is applied to experimental data from a 48-mm diameter scattering object with tissue-like properties. The image reconstructed from measurements has, relative to a conventional B-scan formed using a low f-number at the same center frequency, significantly higher resolution and less speckle, implying that small, high-contrast structures can be demonstrated clearly using the extended method.

A ring transducer system for medical ultrasound research.

An ultrasonic ring transducer system has been developed for experimental studies of scattering and imaging. The transducer consists of 2048 rectangular elements with a 2.5-MHz center frequency, a 67% -6 dB bandwidth, and a 0.23-mm pitch arranged in a 150-mm-diameter ring with a 25-mm elevation. At the center frequency, the element size is 0.30lambda x 42lambda and the pitch is 0.38lambda. The system has 128 parallel transmit channels, 16 parallel receive channels, a 2048:128 transmit multiplexer, a 2048:16 receive multiplexer, independently programmable transmit waveforms with 8-bit resolution, and receive amplifiers with time variable gain independently programmable over a 40-dB range. Receive signals are sampled at 20 MHz with 12-bit resolution. Arbitrary transmit and receive apertures can be synthesized. Calibration software minimizes system nonidealities caused by noncircularity of the ring and element-to-element response differences. Application software enables the system to be used by specification of high-level parameters in control files from which low-level hardware-dependent parameters are derived by specialized code. Use of the system is illustrated by producing focused and steered beams, synthesizing a spatially limited plane wave, measuring angular scattering, and forming b-scan images.

Reduction of variance in spectral estimates for correction of ultrasonic aberration.

A variance reduction factor is defined to describe the rate of convergence and accuracy of spectra estimated from overlapping ultrasonic scattering volumes when the scattering is from a spatially uncorrelated medium. Assuming that the individual volumes are localized by a spherically symmetric Gaussian window and that centers of the volumes are located on orbits of an icosahedral rotation group, the factor is minimized by adjusting the weight and radius of each orbit. Conditions necessary for the application of the variance reduction method, particularly for statistical estimation of aberration, are examined. The smallest possible value of the factor is found by allowing an unlimited number of centers constrained only to be within a ball rather than on icosahedral orbits. Computations using orbits formed by icosahedral vertices, face centers, and edge midpoints with a constraint radius limited to a small multiple of the Gaussian width show that a significant reduction of variance can be achieved from a small number of centers in the confined volume and that this reduction is nearly the maximum obtainable from an unlimited number of centers in the same volume.

Statistical estimation of ultrasonic propagation path parameters for aberration correction.

Parameters in a linear filter model for ultrasonic propagation are found using statistical estimation. The model uses an inhomogeneous-medium Green's function that is decomposed into a homogeneous-transmission term and a path-dependent aberration term. Power and cross-power spectra of random-medium scattering are estimated over the frequency band of the transmit-receive system by using closely situated scattering volumes. The frequency-domain magnitude of the aberration is obtained from a normalization of the power spectrum. The corresponding phase is reconstructed from cross-power spectra of subaperture signals at adjacent receive positions by a recursion. The subapertures constrain the receive sensitivity pattern to eliminate measurement system phase contributions. The recursion uses a Laplacian-based algorithm to obtain phase from phase differences. Pulse-echo waveforms were acquired from a point reflector and a tissue-like scattering phantom through a tissue-mimicking aberration path from neighboring volumes having essentially the same aberration path. Propagation path aberration parameters calculated from the measurements of random scattering through the aberration phantom agree with corresponding parameters calculated for the same aberrator and array position by using echoes from the point reflector. The results indicate the approach describes, in addition to time shifts, waveform amplitude and shape changes produced by propagation through distributed aberration under realistic conditions.

Examples of design curves for multirow arrays used with time-shift compensation.

The effect of 1.75-D array element height on time-shift compensation is analyzed using a two-dimensional array system and a distributed aberration phantom. The analysis uses hydrophone measurements of transmit beams. The results indicate that, for imaging at 3 MHz through aberration representative of the abdominal wall, similar focus compensation performance can be expected from arrays with element pitches less than or equal to 3.0 mm.

Estimation and compensation of ultrasonic wavefront distortion using a blind system identification method.

A common random input filter model is described for estimation and correction of wavefront aberration in ultrasonic b-scan imaging. In the model, aberration between the focus and the transducer elements is represented by the response of a linear filter bank to a common random signal. The response of each filter in the bank is found using a two-level extension of an existing subspace method for blind system identification. The receive waveforms are compensated using an inverse filter, and the transmit waveforms are predistorted using time reversal. To test the model, experiments were conducted using a two-dimensional array system to obtain echoes from a point reflector and from a random medium in each case through an aberrator. The aberrator is a phantom that mimics wavefront distortion produced by human abdominal wall, and the random medium is made to mimic ultrasonic characteristics of human liver. The results indicate the method can improve both the transmit and the receive focus and can outperform time-shift estimation and compensation as well as the method of backpropagation followed by timeshift estimation and compensation.

Comparisons of lesion detectability in ultrasound images acquired using time-shift compensation and spatial compounding.

The effects of aberration, time-shift compensation, and spatial compounding on the discrimination of positive-contrast lesions in ultrasound b-scan images are investigated using a two-dimensional (2-D) array system and tissue-mimicking phantoms. Images were acquired within an 8.8 x 12-mm2 field of view centered on one of four statistically similar 4-mm diameter spherical lesions. Each lesion was imaged in four planes offset by successive 45 degree rotations about the central scan line. Images of the lesions were acquired using conventional geometric focusing through a water path, geometric focusing through a 35-mm thick distributed aberration phantom, and time-shift compensated transmit and receive focusing through the aberration phantom. The views of each lesion were averaged to form sets of water path, aberrated, and time-shift compensated 4:1 compound images and 16:1 compound images. The contrast ratio and detectability index of each image were computed to assess lesion differentiation. In the presence of aberration representative of breast or abdominal wall tissue, time-shift compensation provided statistically significant improvements of contrast ratio but did not consistently affect the detectability index, and spatial compounding significantly increased the detectability index but did not alter the contrast ratio. Time-shift compensation and spatial compounding thus provide complementary benefits to lesion detection.

Aberration compensation of an ultrasound imaging instrument with a reduced number of channels.

Focusing and imaging qualities of an ultrasound imaging system that uses aberration correction were experimentally investigated as functions of the number of parallel channels. Front-end electronics that consolidate signals from multiple physical elements can be used to lower hardware and computational costs by reducing the number of parallel channels. However, the signals from sparse arrays of synthetic elements yield poorer aberration estimates. In this study, aberration estimates derived from synthetic arrays of varying element sizes are evaluated by comparing compensated receive focuses, compensated transmit focuses, and compensated b-scan images of a point target and a cyst phantom. An array of 80 x 80 physical elements with a pitch of 0.6 x 0.6 mm was used for all of the experiments and the aberration was produced by a phantom selected to mimic propagation through abdominal wall. The results show that aberration correction derived from synthetic arrays with pitches that have a diagonal length smaller than 70% of the correlation length of the aberration yield focuses and images of approximately the same quality. This connection between correlation length of the aberration and synthetic element size provides a guideline for determining the number of parallel channels that are required when designing imaging systems that employ aberration correction.

Reduced-Rank Approximations to the Far-Field Transform in the Gridded Fast Multipole Method.

The fast multipole method (FMM) has been shown to have a reduced computational dependence on the size of finest-level groups of elements when the elements are positioned on a regular grid and FFT convolution is used to represent neighboring interactions. However, transformations between plane-wave expansions used for FMM interactions and pressure distributions used for neighboring interactions remain significant contributors to the cost of FMM computations when finest-level groups are large. The transformation operators, which are forward and inverse Fourier transforms with the wave space confined to the unit sphere, are smooth and well approximated using reduced-rank decompositions that further reduce the computational dependence of the FMM on finest-level group size. The adaptive cross approximation (ACA) is selected to represent the forward and adjoint far-field transformation operators required by the FMM. However, the actual error of the ACA is found to be greater than that predicted using traditional estimates, and the ACA generally performs worse than the approximation resulting from a truncated singular-value decomposition (SVD). To overcome these issues while avoiding the cost of a full-scale SVD, the ACA is employed with more stringent accuracy demands and recompressed using a reduced, truncated SVD. The results show a greatly reduced approximation error that performs comparably to the full-scale truncated SVD without degrading the asymptotic computational efficiency associated with ACA matrix assembly.

The Fast Multipole Method and Fourier Convolution for the Solution of Acoustic Scattering on Regular Volumetric Grids.

The fast multipole method (FMM) is applied to the solution of large-scale, three-dimensional acoustic scattering problems involving inhomogeneous objects defined on a regular grid. The grid arrangement is especially well suited to applications in which the scattering geometry is not known a priori and is reconstructed on a regular grid using iterative inverse scattering algorithms or other imaging techniques. The regular structure of unknown scattering elements facilitates a dramatic reduction in the amount of storage and computation required for the FMM, both of which scale linearly with the number of scattering elements. In particular, the use of fast Fourier transforms to compute Green's function convolutions required for neighboring interactions lowers the often-significant cost of finest-level FMM computations and helps mitigate the dependence of FMM cost on finest-level box size. Numerical results demonstrate the efficiency of the composite method as the number of scattering elements in each finest-level box is increased.

A model of distributed phase aberration for deblurring phase estimated from scattering.

Correction of aberration in ultrasound imaging uses the response of a point reflector or its equivalent to characterize the aberration. Because a point reflector is usually unavailable, its equivalent is obtained using statistical methods, such as processing reflections from multiple focal regions in a random medium. However, the validity of methods that use reflections from multiple points is limited to isoplanatic patches for which the aberration is essentially the same. In this study, aberration is modeled by an offset phase screen to relax the isoplanatic restriction. Methods are developed to determine the depth and phase of the screen and to use the model for compensation of aberration as the beam is steered. Use of the model to enhance the performance of the noted statistical estimation procedure is also described. Experimental results obtained with tissue-mimicking phantoms that implement different models and produce different amounts of aberration are presented to show the efficacy of these methods. The improvement in b-scan resolution realized with the model is illustrated. The results show that the isoplanatic patch assumption for estimation of aberration can be relaxed and that propagation-path characteristics and aberration estimation are closely related.

Large-scale propagation of ultrasound in a 3-D breast model based on high-resolution MRI data.

A 40 x 35 x 25-mm(3) specimen of human breast consisting mostly of fat and connective tissue was imaged using a 3-T magnetic resonance scanner. The resolutions in the image plane and in the orthogonal direction were 130 microm and 150 microm, respectively. Initial processing to prepare the data for segmentation consisted of contrast inversion, interpolation, and noise reduction. Noise reduction used a multilevel bidirectional median filter to preserve edges. The volume of data was segmented into regions of fat and connective tissue by using a combination of local and global thresholding. Local thresholding was performed to preserve fine detail, while global thresholding was performed to minimize the interclass variance between voxels classified as background and voxels classified as object. After smoothing the data to avoid aliasing artifacts, the segmented data volume was visualized using isosurfaces. The isosurfaces were enhanced using transparency, lighting, shading, reflectance, and animation. Computations of pulse propagation through the model illustrate its utility for the study of ultrasound aberration. The results show the feasibility of using the described combination of methods to demonstrate tissue morphology in a form that provides insight about the way ultrasound beams are aberrated in three dimensions by tissue.

Spatial coherence analysis applied to aberration correction using a two-dimensional array system.

Complex degree of coherence functions are computed using synthetic and measured ultrasound data to demonstrate noteworthy aspects of coherence analysis in the context of aberration correction. Coherence functions calculated from synthetic data illustrate the importance of proper normalization of the constituent cross-correlation integrals when weak elements and receiver directivity are significant factors. The synthetic data also show that a spike can occur at the zero-lag position of the coherence function when the signal-to-noise ratio is reduced by element directivity near the edges of a large aperture. The latter observation is confirmed by experimental data acquired through tissue-mimicking distributed aberration phantoms using a low f-number two-dimensional array system. The coherence of data acquired at neighboring elements is not changed by time-shift compensation of transmit and receive focusing, but time-shift compensation does improve the coherence of echoes measured over larger separations. The resulting increase in coherence widths evaluated at levels between 0.2 and 0.5 is correlated with narrower -10 dB and -20 dB effective widths in focuses visualized using single-transmit images. Iterative focus compensation methods may benefit from aberration estimation algorithms that take advantage of these longer-range correlations in random-scattering waveforms.

A k-space method for coupled first-order acoustic propagation equations.

A k-space method for large-scale simulation of ultrasonic pulse propagation is presented. The present method, which solves the coupled first-order differential equations for wave propagation in inhomogeneous media, is derived in a simple form analogous to previous finite-difference methods with staggered spatial and temporal grids. Like k-space methods based on second-order wave equations, the present method is exact for homogeneous media, unconditionally stable for "slow" [c(r) < or = c0] media, and highly accurate for general weakly scattering media. In addition, unlike previous k-space methods, the form of the method allows straightforward inclusion of relaxation absorption and perfectly matched layer (PML) nonreflecting boundary conditions. Numerical examples illustrate the capabilities of the present k-space method. For weakly inhomogeneous media, accurate results are obtained using coarser temporal and spatial steps than possible with comparable finite-difference and pseudospectral methods. The low dispersion of the k-space method allows accurate representation of frequency-dependent attenuation and phase velocity associated with relaxation absorption. A technique for reduction of Gibbs phenomenon artifacts, in which compressibility and exponentially scaled density functions are smoothed by half-band filtering, is introduced. When employed together with this smoothing technique, the k-space method provides high accuracy for media including discontinuities, high-contrast inhomogeneities, and scattering structures smaller than the spatial grid resolution.