Robert C. Waag

Correction of ultrasonic wavefront distortion using backpropagation and a reference waveform method for time‐shift compensation

A model is introduced to describe ultrasonic pulse amplitude and shape distortion as well as arrival time fluctuation produced by propagation through specimens of human abdominal wall. In the model, amplitude and shape distortion develops as the wavefront propagates in a uniform medium after passing through a phase screen that only causes time shifts. This distortion is compensated by a backpropagation of the wavefront using the angular spectrum method. The compensation employed waveforms emitted by a pointlike source and measured after propagation through the tissue. The waveforms were first corrected for geometric path and then were backpropagated over a sequence of increasing distances. At each distance, a waveform similarity factor was calculated to find the backpropagation distance at which the waveforms were most similar. A new method was devised to estimate pulse arrival time for geometric correction as well as to perform time-shift compensation. The method adaptively derives a reference waveform that is then cross correlated with all the waveforms in the aperture to obtain a surface of arrival times. The surface was smoothed iteratively to remove outlying points due to waveform distortion. The mean (+/- s.d.) of the waveform similarity factor for 14 specimens was found to be 0.938 (+/- 0.025) initially. After backpropagation of waveforms to the distance of maximum waveform similarity for each specimen, the waveform similarity factor improved to 0.967 (+/- 0.015). The corresponding energy level fluctuation in the wavefront was 4.2 (+/- 0.4) dB initially and became 3.3 (+/- 0.3) dB after backpropagation. For wavefronts focused at 180 mm, the -30 dB mean (+/- s.d.) effective radius of the focus was 4.2 (+/- 1.2) mm with time-shift compensation in the aperture and became 2.5 (+/- 0.5) mm with backpropagation followed by time-shift compensation. These results indicate that a phase screen placed some distance away from the aperture is an improved model for the description of wavefront distortion produced by human abdominal wall and that wavefront backpropagation followed by time-shift estimation and compensation is an effective method to compensate for such distortion.

Measurements of ultrasonic pulse arrival time and energy level variations produced by propagation through abdominal wall

Ultrasonic pulse arrival time and energy level variations introduced by propagation through human abdominal wall specimens have been measured. A hemispheric transducer transmitted an ultrasonic pulse that was detected by a linear array transducer after propagation through an abdominal wall section. The array was translated in the elevation direction to collect data over a two-dimensional aperture. Differences in arrival time and energy level between the measured waveforms and calculated references that account for geometric delay and spreading were found. Plots of waveforms compensated for geometric path, maps of time delay differences and energy level fluctuations, and statistics derived from these for water paths and tissue paths characterize the measurement system and describe the time delay differences and energy level fluctuations caused by 14 different human abdominal wall specimens. Repeated measurements using the same specimens show that individual tissue path measurements are reproducible, the results depend on specimen position, and frozen storage of a specimen for three months does not appear to alter the time delay differences and energy level fluctuations produced by the specimen. Comparison of measurements at room and body temperature indicates that appreciably higher time delay differences occur at body temperature while energy level fluctuations and time delay difference patterns are less affected. For the 14 different abdominal wall specimens, the rms time delay differences and energy level fluctuations have average values of 43.0 ns and 3.30 dB, respectively, and the associated correlation lengths of the time delay differences and energy level fluctuations are 7.90 and 2.28 mm, respectively. The spatial patterns of time delay difference and energy level fluctuation in the reception plane appear largely uncorrelated, although some background variations in energy level fluctuation are similar to features in time delay difference maps. The results provide important new information about the variety and range of ultrasonic wave front arrival and energy variations caused by transmission through abdominal wall.

Simulation of ultrasonic pulse propagation through the abdominal wall

Ultrasonic pulse propagation through the abdominal wall has been simulated using a model for two-dimensional propagation through anatomically realistic tissue cross sections. The time-domain equations for wave propagation in a medium of variable sound speed and density were discretized to obtain a set of coupled finite-difference equations. These difference equations were solved numerically using a two-step MacCormack scheme that is fourth-order accurate in space and second-order accurate in time. The inhomogeneous tissue of the abdominal wall was represented by two-dimensional matrices of sound speed and density values. These values were determined by processing scanned images of abdominal wall cross sections stained to identify connective tissue, muscle, and fat, each of which was assumed to have a constant sound speed and density. The computational configuration was chosen to simulate that of wavefront distortion measurements performed on the same specimens. Qualitative agreement was found between those measurements and the results of the present computations, indicating that the computational model correctly depicts the salient characteristics of ultrasonic wavefront distortion in vivo. However, quantitative agreement was limited by the two-dimensionality of the computation and the absence of detailed tissue microstructure. Calculations performed using an asymptotic straight-ray approximation showed good agreement with time-shift aberrations predicted by the full-wave method, but did not explain the amplitude fluctuations and waveform distortion found in the experiments and the full-wave calculations. Visualization of computed wave propagation within tissue cross sections suggests that amplitude fluctuations and waveform distortion observed in ultrasonic propagation through the abdominal wall are associated with scattering from internal inhomogeneities such as septa within the subcutaneous fat. These observations, as well as statistical analysis of computed and observed amplitude fluctuations, suggest that weak fluctuation models do not fully describe ultrasonic wavefront distortion caused by the abdominal wall.

Time‐shift compensation of ultrasonic pulse focus degradation using least‐mean‐square error estimates of arrival time

Focus degradation produced by abdominal wall has been compensated using a least-mean-square error estimate of arrival time. The compensation was performed on data from measurements of ultrasonic pulses from a curved transducer that emits a hemispheric wave and simulates a point source. The pulse waveforms were measured in a two-dimensional aperture after propagation through a water path and after propagation through 14 different specimens of human abdominal wall. Time histories of the virtual point source were reconstructed by removing the time delays produced by geometric path differences and also removing time shifts produced by propagation inhomogeneities in the case of compensation, finding the complex amplitudes of the Fourier harmonics across the aperture, calculating the Fraunhofer diffraction pattern of each harmonic, and summing the patterns. This process used a least-mean-square error solution for the relative delay expressed in terms of the arrival time differences between neighboring points and included an algorithm to determine arrival time differences when correlation based estimates were unsatisfactory due to dissimilarity of neighboring waveforms. Comparisons of reconstructed time histories in the image plane show that the -10-dB effective radius of the focus for reception through abdominal wall without compensation for inhomogeneities averaged 48% greater than the corresponding average effective radius for ideal waveforms, while time-shift compensation reduced the average -10-dB effective radius to a value that is only 4% greater than for reception of ideal waveforms. The comparisons also indicate that the average ratio of energy outside an ellipsoid defined by the -10-dB effective widths to the energy inside that ellipsoid is 1.81 for uncompensated tissue path data and that time-shift compensation reduced this average to 0.93, while the corresponding average for ideal waveforms was found to be 0.35. These results show that time-shift compensation yields a significant improvement over the uncompensated case although other factors must be considered to achieve an ideal diffraction limited focus.

Focusing and imaging using eigenfunctions of the scattering operator

An inverse scattering method that uses eigenfunctions of the scattering operator is presented. This approach provides a unified framework that encompasses eigenfunction methods of focusing and quantitative image reconstruction in arbitrary media. Scattered acoustic fields are described using a compact, normal operator. The eigenfunctions of this operator are shown to correspond to the far-field patterns of source distributions that are directly proportional to the position-dependent contrast of a scattering object. Conversely, the eigenfunctions of the scattering operator specify incident-wave patterns that focus on these effective source distributions. These focusing properties are employed in a new inverse scattering method that represents unknown scattering media using products of numerically calculated fields of eigenfunctions. A regularized solution to the nonlinear inverse scattering problem is shown to result from combinations of these products, so that the products comprise a natural basis for efficient and accurate reconstructions of unknown inhomogeneities. The corresponding linearized problem is solved analytically, resulting in a simple formula for the low-pass-filtered scattering potential. The linear formula is analytically equivalent to known filtered-backpropagation formulas for Born inversion, and, at least in the case of small scattering objects, has advantages of computational simplicity and efficiency. A similarly efficient and simple formula is derived for the nonlinear problem in which the total acoustic pressure can be determined based on an estimate of the medium. Computational results illustrate focusing of eigenfunctions on discrete and distributed scattering media, quantitative imaging of inhomogeneous media using products of retransmitted eigenfunctions, inverse scattering in an inhomogeneous background medium, and reconstructions for data corrupted by noise.

Measurement and correction of ultrasonic pulse distortion produced by the human breast

Ultrasonic wavefront distortion produced by transmission through breast tissue specimens was measured in a two-dimensional aperture. Differences in arrival time and energy level between the measured waveforms and references that account for geometric delay and spreading were calculated. Also calculated was a waveform similarity factor that is decreased from 1.0 by changes in waveform shape. For nine different breast specimens, the arrival time fluctuations had an average (+/- s.d.) rms value of 66.8 (+/- 12.6) ns and an associated correlation length of 4.3 (+/- 1.1) mm, while the energy level fluctuations had an average rms value of 5.0 (+/- 0.5) dB and a correlation length of 3.4 (+/- 0.8) mm. The corresponding waveform similarity factor was 0.910 (+/- 0.023). The effect of the wavefront distortion on focusing and the ability of time-shift compensation to remove the distortion were evaluated by comparing parameters such as the -30-dB effective radius, the -10-dB peripheral energy ratio, and the level at which the effective radius departs from an ideal by 10% for the focus obtained without compensation, with time-shift estimation and compensation in the aperture, and with time-shift estimation and compensation performed after backpropagation. For the nine specimens, the average -10-dB peripheral energy ratio of the focused beams fell from 3.82 (+/- 1.83) for the uncompensated data to 0.96 (+/- 0.18) with time-shift compensation in the aperture and to 0.63 (+/- 0.07) with time-shift compensation after backpropagation. The average -30-dB effective radius and average 10% deviation level were 4.5 (+/- 0.8) mm and -19.2 (+/- 3.5) dB, respectively, for compensation in the aperture and 3.2 (+/- 0.7) mm and -22.8 (+/- 2.8) dB, respectively, for compensation after backpropagation. The corresponding radius for the uncompensated data was not meaningful because the dynamic range of the focus was generally less than 30 dB in the elevation direction, while the average 10% deviation level for the uncompensated data was -4.9 (+/- 4.1) dB. The results indicate that wavefront distortion produced by breast significantly degrades ultrasonic focus in the low MHz frequency range and that much of this degradation can be eliminated using wavefront backpropagation and time-shift compensation.

In-vivo measurements of ultrasound attenuation in normal or diseased liver

Ultrasonic attenuation coefficients of liver have been derived from echoes received by a modified commercial B-scan imaging instrument. Values have been measured from selected regions within liver scans of 59 individuals, of which 15 cases were presumed normal (based on medical histories), and the remainder were involved with diffuse liver disease such as alcoholic cirrhosis, chemotherapy toxicity, chronic hepatitis, and liver metastases. Medical histories on most individuals include the results of serum liver function enzymes, conventional B-scan examinations, and exposure to drugs and alcohol. The results of CT abdominal scans (N = 13) and/or liver biopsy (N = 12) were also available. The results show that normal attenuation values for human liver are 0.054 +/- 0.009 Np/cm-MHz (0.47 dB/cm-MHz) with a frequency dependence of fn, where n = 1.05 +/- 0.25, in agreement with in vitro studies of mammalian liver. In diffuse liver disease, no relationship was found between the attenuation coefficient and the results of CT or conventional ultrasonic examination. A trend towards higher attenuation with increased fibrosis and fat, as graded from liver biopsies, was noted, but the results were generally not statistically significant. However, a significant correlation was found between high values of attenuation and abnormal liver function tests. High attenuation is also found with ingestion of alcohol, chemotherapeutic agents, and steroids, all of which may affect liver composition.

Ultrasound Imaging: Waves, Signals, and Signal Processing

This article reviews Ultrasound Imaging: Waves, Signals, and Signal Processing by Bjorn A. J. Angelsen , Norway, 2000. 1416 pp.

A Review of Tissue Characterization from Ultrasonic Scattering

490.00. ISB: 82-995811-0-9, ISBN: 82-995811-1-7

The effect of abdominal wall morphology on ultrasonic pulse distortion. Part I. Measurements

Ultrasonic scattering is related to tissue architecture by a model which expresses the scattered wave pressure as the product of a frequency-dependent factor and the three-dimensional Fourier transform of a spatially windowed scattering function. The scattering function is the sum of the variations in compressibility and the variations in density, the latter being weighted by the cosine of the scattering angle. An analogous Fourier transform relation describes the average scattered intensity in terms of the correlation of the scattering function between points in a scattering volume. These Fourier transforms permit different scattering measurements to be related via trajectories in wavespace. Backscatter measurements made on blood, eye, liver, spleen, brain, and heart show the feasibility of differentiating tissues and determining spacing. Ultrasonic measurements of random medium model properties have demonstrated that average differential scattering cross sections per unit volume can be found accurately and precisely, and that scattering cross sections per unit volume can be used to make quantitative statements about changes in scattering properties. Measurements of calf liver ultrasonic differential and total scattering cross sections show that calf liver is a weak scatterer with nearly one-half of the total scattered power occurring between scattering angles of 25-45°. Measurements of scattering by fixed pig and human liver show qualitative agreement with predictions of scattering from architecture observed optically. Results presently accumulated indicate the promise of scattering measurements for tissue characterization, but extensive additional in vitro and in vivo development is required before their clinical utility for diagnosis is known.