Theoretical derivation of SNR, CNR and spatial resolution for a local adaptive strain estimator for elastography

Conventional techniques in elastography estimate the axial strain as the gradient of the displacement (time-delay) estimates obtained using cross-correlation of pre- and temporally stretched postcompression radiofrequency (RF) A-line segments. The use of a constant stretch factor for stretching the postcompression A-line is not adequate in the presence of heterogeneous targets that are commonly encountered. This led to the development of several adaptive strain estimation techniques in elastography. Yet, a theoretical framework for the image quality of adaptive strain estimation has not been established. In this work, we develop theoretical expressions for the image quality [measured in terms of the signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR) and spatial resolution] of elastograms obtained using an adaptive strain estimator developed by Alam et al. (1998). We show a linear trade-off between the SNR and axial resolution of the strain elastogram with respect to the window length used for strain estimation. The CNR shows a quadratic tradeoff with the axial resolution with respect to the window length. The SNR, CNR and axial resolution are shown to improve with the ultrasonic bandwidth.     

Noise reduction in elastograms using temporal stretching with multicompression averaging

Elastography uses estimates of the time delay (obtained by cross-correlation) to compute strain estimates in tissue due to quasistatic compression. Because the time delay estimates do not generally occur at the sampling intervals, the location of the cross-correlation peak does not give and accurate estimate of the time delay. Sampling errors in the time-delay estimate are reduced using signal interpolation techniques to obtain subsample time-delay estimates. Distortions of the echo signals due to tissue compression introduce correlation artifacts in the elastogram. These artifacts are reduced by a combination of small compressions and temporal stretching of the postcompression signal. Random noise effects in the resulting elastograms are reduced by averaging several elastograms, obtained from successive small compressions (assuming that the errors are uncorrelated). Multicompression averaging with temporal stretching is shown to increase the signal-to-noise ratio in the elastogram by an order of magnitude, without sacrificing sensitivity, resolution or dynamic range. The strain filter concept is extended in this article to theoretically characterize the performance of multicompression averaging with temporal stretching.     

The effects of digitization on the elastographic signal-to-noise ratio

In elastography, the tissue under investigation is compressed and the resulting strain is estimated from the gradient of the displacement (time-delay) estimates. The displacements are typically estimated by cross-correlating the radiofrequency (RF) ultrasound signals of the pre- and postcompressed tissue. One of the parameters used to quantify the resulting quality of the elastogram is the elastographic signal-to-noise ratio (SNR(e)). For a uniformly elastic target (a single elastic modulus), the dependence of the SNR(e) on the applied strain has a bandpass characteristic that has been termed the strain filter. Theoretical expressions for the upper bound on the strain filter were developed earlier. Yet, simulated as well as experimental strain filters derived from uniformly elastic phantoms deviate from these upper bounds. The failure to achieve the upper bounds could be partially attributed to the fact that, in both simulations and experiments, the RF signals used to compute the TDEs are sampled and quantized. Using simulated models of uniformly elastic phantoms, a study of the dependence of the strain filter on the quantization and sampling rates was performed. The results indicated that the strain filter improves with both the sampling rate and the quantization, as expected. A theoretical analysis was done to incorporate quantization as a derating factor to the strain filter.     

Wavelet denoising of displacement estimates in elastography

Wavelet shrinkage denoising of the displacement estimates to reduce noise artefacts, especially at high overlaps in elastography, is presented in this paper. Correlated errors in the displacement estimates increase dramatically with an increase in the overlap between the data segments. These increased correlated errors (due to the increased correlation or similarity between consecutive displacement estimates) generate the so-called "worm" artefact in elastography. However, increases in overlap on the order of 90% or higher are essential to improve axial resolution in elastography. The use of wavelet denoising significantly reduces errors in the displacement estimates, thereby reducing the worm artefacts, without compromising on edge (high-frequency or detail) information in the elastogram. Wavelet denoising is a term used to characterize noise rejection by thresholding the wavelet coefficients. Worm artefacts can also be reduced using a low-pass filter; however, low-pass filtering of the displacement estimates does not preserve local information such as abrupt change in slopes, causing the smoothing of edges in the elastograms. Simulation results using the analytic 2-D model of a single inclusion phantom illustrate that wavelet denoising produces elastograms with the closest correspondence to the ideal mechanical strain image. Wavelet denoising applied to experimental data obtained from an in vitro thermal lesion phantom generated using radiofrequency (RF) ablation also illustrates the improvement in the elastogram noise characteristics.     

Myocardial elastography at both high temporal and spatial resolution for the detection of infarcts

Myocardial elastography is a novel method for noninvasively assessing regional myocardial function, with the advantages of high spatial and temporal resolution and high signal-to-noise ratio (SNR). In this paper, in-vivo experiments were performed in anesthetized normal and infarcted mice (one day after left anterior descending coronary artery [LAD] ligation) using a high-resolution (30 MHz) ultrasound system (Vevo 770, VisualSonics Inc., Toronto, ON, Canada). Radiofrequency (RF) signals of the left ventricle (LV) in longitudinal (long-axis) view and the associated electrocardiogram (ECG) were simultaneously acquired. Using a retrospective ECG gating technique, 2-D full field-of-view RF frames were acquired at an extremely high frame rate (8 kHz) that resulted in high-quality incremental displacement and strain estimation of the myocardium. The incremental results were further accumulated to obtain the cumulative displacements and strains. Two-dimensional and M-mode displacement images and strain images (elastograms), as well as displacement and strain profiles as a function of time, were compared between normal and infarcted mice. Incremental results clearly depicted cardiac events including LV contraction, LV relaxation and isovolumetric phases in both normal and infarcted mice, and also evidently indicated reduced motion and deformation in the infarcted myocardium. The elastograms indicated that the infarcted regions underwent thinning during systole rather than thickening, as in the normal case. The cumulative elastograms were found to have higher elastographic SNR (SNR(e)) than the incremental elastograms (e.g., 10.6 vs. 4.7 in a normal myocardium, and 6.0 vs. 2.4 in an infarcted myocardium). Finally, preliminary statistical results from nine normal (m = 9) and seven infarcted (n = 7) mice indicated the capability of the cumulative strain in differentiating infracted from normal myocardia. In conclusion, myocardial elastography could provide regional strain information at simultaneously high temporal (>/=0.125 ms) and spatial ( approximately 55 microm) resolution as well as high precision ( approximately 0.05 microm displacement). This technique was thus capable of accurately characterizing normal myocardial function throughout an entire cardiac cycle, at the same high resolution, and detecting and localizing myocardial infarction in vivo.     

Elastographic imaging using staggered strain estimates

Conventional techniques in elastography estimate strain as the gradient of the displacement estimates obtained through crosscorrelation of pre- and postcompression rf A-lines. In these techniques, the displacements are estimated over overlapping windows and the strains are estimated as the gradient of the displacement estimates over adjacent windows. The large amount ofnoise at high window overlaps may result in poor quality elastograms, thus restricting the applicability of conventional strain estimation techniques to low window overlaps, which, in turn, results in a small number of pixels in the image. To overcome this restriction, we propose a multistep strain estimation technique. It computes the first elastogram using nonoverlapped windows. In the next step, the data windows are shifted by a small distance (small fraction of window size) and another elastogram is produced. This is repeated until the cumulative shift equals/exceeds the window size and all the elastograms are staggered to produce the final elastogram. Simulations and experiments were performed using this technique to demonstrate significant improvement in the elastographic signal-to-noise ratio (SNRe) and the contrast-to-noise ratio (CNRe) at high window overlaps over conventional strain estimation techniques, without noticeable loss of spatial resolution. This technique might be suitable for reducing the algorithmic noise in the elastograms at high window overlaps.     

Visualisation of HIFU lesions using elastography of the human prostate in vivo: preliminary results

An imaging system was developed for prostate elastography in vivo using a transrectal ultrasound (US) probe to guide high-intensity focused US (HIFU) therapy of prostate cancer. Uniform compression was applied using a balloon, while a sector image was acquired. Strain was calculated from the gradient of the displacements obtained from the ultrasonic signal using the cross-correlation technique. Elastograms were acquired on a total of 31 patients undergoing HIFU therapy for localised prostate cancer. For two patients, only part of the prostate was treated and posttherapy magnetic resonance imaging (MRI) confirmed the size and position of the HIFU lesions seen in the elastograms as low strain areas, with a strain contrast ratio between 1.6 and 3.2. The whole prostate was treated for the next 29 patients. After treatment, the whole prostate appeared to be stiff in the elastograms and a 40% to 60% (mean 50%) decrease in average strain was observed when compared to strains measured before HIFU application. Tumours identified by biopsies and sonograms could occasionally be seen in the preoperative elastograms. Decorrelation effects occurred mainly because of low sonographic signal-to-noise ratio (SNR) and of out-of-plane motion induced by respiration.     

Noise performance and signal-to-noise ratio of shear strain elastograms

In this paper, we develop a theoretical expression for the signal-to-noise ratio (SNR) of shear strain elastograms. The previously-developed ideas for the axial strain filter (ASF) and lateral strain filter (LSF) are extended to define the concept of the shear strain filter (SSF). Some of our theoretical results are verified using simulations and phantom experiments. The results indicate that the signal-to-noise ratio of shear-strain elastograms (SNRsse) improves with increasing shear strain and with improvements in system parameters such as the sonographic signal-to-noise ratio (SNRs) beamwidth, center frequency and fractional bandwidth. The results also indicate that the amount of axial strain present along with the shear strain is an important parameter that determines the upper bound on SNRsse. The SNRsse will be higher in the absence of additional deformation due to axial strain.     

Strain estimation in phase-sensitive optical coherence elastography

We present a theoretical framework for strain estimation in optical coherence elastography (OCE), based on a statistical analysis of displacement measurements obtained from a mechanically loaded sample. We define strain sensitivity, signal-to-noise ratio and dynamic range, and derive estimates of strain using three methods: finite difference, ordinary least squares and weighted least squares, the latter implemented for the first time in OCE. We compare theoretical predictions with experimental results and demonstrate a ~12 dB improvement in strain sensitivity using weighted least squares compared to finite difference strain estimation and a ~4 dB improvement over ordinary least squares strain estimation. We present strain images (i.e., elastograms) of tissue-mimicking phantoms and excised porcine airway, demonstrating in each case clear contrast based on the sample's elasticity.     

Estimating axial and lateral strain using a synthetic aperture elastographic imaging system

Model-based elastography is an emerging technique with clinical applications in imaging vascular tissues, guiding minimally invasive therapies and diagnosing breast and prostate cancers. Its usage is limited because ultrasound can measure only the axial component of displacement with high precision. The goal of this study was to assess the effect of lateral sampling frequency, lateral beam-width and the number of active transmission elements on the quality of axial and lateral strain elastograms. Elastographic imaging was performed on gelatin-based phantoms with a modified commercial ultrasound scanner. Three groups of radio-frequency (RF) echo frames were reconstructed from fully synthetic aperture data. In the first group, all 128 transmission elements (corresponding to a lateral beamwidth of 0.22 mm at the center of the field of view) were used to reconstruct RF echo frames with A-line densities that varied from 6.4 lines/mm to 51.2 lines/mm. In the second group, the size of the aperture was varied to produce RF echo frames with lateral beamwidths ranging from 0.22 mm to 0.43 mm and a fixed A-line density of 25.6 lines/mm. In the third group, sparse arrays with varying number of active transmission elements (from 2 to 128) were used to reconstruct RF echo frames, whose A-line density and lateral beamwidth were fixed to 25.6 lines/mm and 0.22 mm, respectively. Applying a two-dimensional (2-D) displacement estimator to the pre- and post-deformed RF echo frames produced displacement elastograms. Axial and lateral strain elastograms were computed from displacement elastograms with a least squares strain estimator. The quality of axial and lateral strain elastograms improved with increasing applied strain and A-line density but decreased with increasing lateral beamwidth and deteriorated as the number of active transmission elements in the sparse arrays were reduced. This work demonstrated that the variance incurred when estimating the lateral component of displacement was reduced considerably when elastography was performed with a synthetic aperture ultrasound imaging system. Satisfactory axial and lateral strain elastograms were produced using a sparse array with as few as 16 active transmission elements.     

Principal Strain Vascular Elastography: Simulation and Preliminary Clinical Evaluation

It is difficult to produce reliable polar strain elastograms (radial and circumferential) because the center of the carotid artery is typically unknown. Principal strain imaging can overcome this limitation, but suboptimal lateral displacement estimates make this an impractical approach for visualizing mechanical properties within the carotid artery. We hypothesized that compounded plane wave imaging can minimize this problem. To test this hypothesis, we performed (i) simulations with vessels of varying morphology and mechanical behavior (i.e., isotropic and transversely isotropic), and (ii) a pilot study with 10 healthy volunteers. The accuracy of principal and polar strain (computed using knowledge of the precise vessel center) elastograms varied between 7% and 17%. In both types of elastograms, strain concentrated at the junction between the fibrous cap and the vessel wall, and the strain magnitude decreased with increasing fibrous cap thickness. Elastograms of healthy volunteers were consistent with those of transversely isotropic homogeneous vessels; they were spatially asymmetric, a trend that was common to both principal and polar strains. No significant differences were observed in the mean strain recovered from principal and polar strains (p > 0.05). This investigation indicates that principal strain elastograms measured with compounding plane wave imaging overcome the problems incurred when polar strain elastograms are computed with imprecise estimates of the vessel center.     

Spatial Compounding Technique to Obtain Rotation Elastogram: A Feasibility Study

The perception of stiffness and slipperiness of a breast mass on palpation is used by physicians to assess the level of suspicion of a lesion as being malignant or benign. However, most current ultrasound elastography imaging methods provide only stiffness-related information. There is no existing approach that provides information about the local rigid body rotation undergone by only a loosely bonded, asymmetrically oriented lesion subjected to a small quasi-static compression. The inherent poor lateral resolution in ultrasound imaging poses a limitation in estimating the local rigid body rotation. Several techniques have been reported in the literature to improve the lateral resolution in ultrasound imaging, and among them is spatial compounding. In this study, we explore the feasibility of obtaining better-quality rotation elastograms with spatial compounding through simulations using Field II and experiments on tissue-mimicking phantoms. The phantom was subjected to axial compression (～1%–2%) from the top, and the angular axial and lateral displacement estimates were obtained using a multilevel 2-D displacement tracking algorithm at different insonification angles. A rotation elastogram (RE) was obtained by taking half of the difference between the lateral gradient of the axial displacement estimates and the axial gradient of the lateral displacement estimates. Contrast-to-noise ratio was used to quantify the improvements in quality of RE. Contrast-to-noise ratio values were calculated by varying the maximum steering angle and the incremental angle, and its effects on RE quality were evaluated. Both simulation and experimental results corroborated and indicated a significant improvement in the quality of RE using compounding technique.     

Improvement of elastographic displacement estimation using a two-step cross-correlation method

The cross-correlation algorithm used to compute the local strain components for elastographic imaging requires a minimum radio-frequency data segment length of around 10 wavelengths to obtain accurate and precise strain estimates with a reasonable signal-to-noise ratio. Shorter radio-frequency data segments generally introduce increased estimation errors as the information content in the data segment reduces. However, shorter data segments and increased overlaps are essential to improve the axial resolution in the strain image. In this paper, we propose a two-step cross-correlation technique that enables the use of window lengths on the order of a single wavelength to provide displacement and strain estimates with similar noise properties as those obtained with a 10 wavelength window. The first processing step utilizes a window length on the order of 10 wavelengths to obtain coarse displacement estimates between the pre- and post-compression radio frequency data frames. This coarse displacement is then interpolated and utilized as the initial guess-estimate for the second cross-correlation processing step using the smaller window. This step utilizes a single wavelength window to improve the axial resolution in strain estimation, without significantly compromising the noise properties of the image. Simulation and experimental results show that the signal-to-noise and contrast-to-noise ratio estimates improve significantly at the smaller window lengths with the two-step processing when compared with the use of a similar sized window in the currently utilized single window method.     

On the differences between two-dimensional and three-dimensional simulations for assessing elastographic image quality: a simulation study

In this work, we introduced an elastographic simulation framework, which estimates upper bounds on elastographic image quality by accounting for three-dimensional (3D) tissue motion and the 3D nature of the ultrasound beam. For the boundary conditions and the range of applied strains considered in this study, it was observed that for applied strains smaller than 0.7%, fast two-dimensional (2D) simulations and 3D simulations predicted similar upper bounds on elastographic signal-to-noise (SNR(e)) and contrast-to-noise ratios (CNR(e)); however, for applied strains greater than 0.7%, the predictions by 2D simulations grossly overestimated the achievable results when compared with upper bound results from 3D simulations. It was also found that linear increments in the elevational-to-lateral beamwidth ratio (beam ratio) resulted in nonlinear degradation in the achievable upper bounds on elastographic signal-to-noise ratio. For the modulus contrast ratio of ten between the target and the background, the peak difference in the prediction of contrast-to-noise by 2D and 3D simulations was approximately 10 dB, whereas, for modulus contrast ratio of 1.5, the peak difference increased to approximately 30 dB. No significant difference was observed between the spatial resolution predicted by 2D and 3D simulations; however, increase in beam ratio resulted in decrease in target detectability, especially at lower modulus contrast ratios.     

Improvements in elastographic contrast-to-noise ratio using spatial-angular compounding

Spatial-angular compounding is a new technique developed for improving the signal-to-noise ratio (SNR) in elastography. Under this method, elastograms of a region-of-interest (ROI) are obtained from a spatially weighted average of local strain estimated along different insonification angles. In this article, we investigate the improvements in the strain contrast and contrast-to-noise ratio (CNR) of the spatially compounded elastograms. Spatial angular compounding is also applied and evaluated in conjunction with global temporal stretching. Quantitative experimental results obtained using a single-inclusion tissue-mimicking phantom demonstrate that the strain contrast reduces slightly but the CNR improves by around 8 to 13 dB. We also present experimental spatial angular compounding results obtained from an in vitro thermal lesion in canine liver tissue embedded in a gelatin phantom that demonstrate the improved visual characteristics (due to the improved CNR) of the compound elastogram. The experimental results provide guidelines for the practical range of maximum insonification angles and estimates of the optimum angular increment.     

Myocardial fiber shortening in the circumferential direction produces left ventricular wall thickening during contraction

When one bends the elbow by shortening of the biceps, a knot of muscle is observed in his or her upper arm, indicating that muscle shortening is converted to muscle standing in the perpendicular direction due to the incompressibility of skeletal muscle. A similar mechanism may work in the thickening process of the left ventricular (LV) wall. Although myocardial fibers of the left ventricle shorten by about 20% along the fiber direction when they contract, thickening of the LV wall during contraction often exceeds 50%. Thus, the aim of the present study was to clarify the mechanism by which myocardial fiber shortening produces such remarkable thickening of the LV wall. We hypothesized that myocardial fiber shortening in the circumferential direction causes myocardial transformation perpendicular to the fiber direction, thereby producing LV wall thickening. We evaluated this hypothesis using an incompressible model of the LV wall. In 15 healthy male volunteers (38±13 years), we calculated theoretical peak thickening values of the inner and outer LV wall layers and compared them with directly measured peak thickening values using Doppler strain imaging at the corresponding areas. The theoretical peak thickening and directly measured peak thickening were >60% in the LV inner layer. The theoretical peak thickening was correlated with the directly measured peak thickening in the inner (r=0.75, p<0.05) and outer (r=0.61, p<0.05) layers. We conclude that shortening of LV circumferential myocardial fiber and incompressibility of myocardium produce LV wall thickening during contraction.     

A general solution for catheter position effects for strain estimation in intravascular elastography

Intravascular ultrasound (US) elastography reveals the elastic properties of vascular tissue and plaque. However, misalignment of the US catheter in the vessel lumen can cause incorrect strain estimation in intravascular US elastography caused by strain projection artifacts. In this paper, we present a general theoretical solution where the impact of catheter eccentricity, tilt and noncoplanar errors on the strain estimates are derived. Appropriate corrections to strain estimates can then be applied with prior knowledge of the catheter position information to reduce the strain projection artifacts. Simulations using a frequency-domain-based algorithm that models intravascular US imaging before and after a specified deformation are presented. The simulations are used to verify the theoretical derivations for two displacement situations (linear and nonlinear) under intraluminal pressure, with and without stress decay. The linear displacement case demonstrates that the correction factor is dependent only on the angle between the US beam and the cross-sectional plane of the vessel. For the nonlinear displacement case, where a l/r stress decay in the displacement is modeled, the correction factor becomes a more complicated function of the azimuthal angle.     

Power spectral strain estimators in elastography

Elastography can produce quality strain images in vitro and in vivo. Standard elastography uses a coherent cross-correlation technique to estimate tissue displacement and tissue strain using a subsequent gradient operator. Although coherent estimation methods generally have the advantage of being highly accurate and precise, even relatively small undesired motions are likely to cause enough signal decorrelation to produce significant degradation of the elastogram. For elastography to become more universally practical in such applications as hand-held, intravascular and abdominal imaging, the limitations associated with coherent strain estimation methods that require tissue and system stability, must be overcome. In this paper, we propose the use of a spectral-shift method that uses a centroid shift estimate to measure local strain directly. Furthermore, we also show theoretically that a spectral bandwidth method can also provide a direct strain estimation. We demonstrate that strain estimation using the spectral-shift technique is moderately less precise, but far more robust than the cross-correlation method. A theoretical analysis, simulations and experimental results are used to illustrate the properties associated with this method.     

Correction for simultaneous catheter eccentricity and tilt in intravascular elastography

Intravascular elastography can provide significant new information about the elastic properties of vascular tissue and plaque, useful for the diagnosis of disease and appropriate selection of interventional methods. Knowledge of the plaque composition, vulnerability and its elastic properties can assist the clinician in selecting appropriate interventional techniques. However, several noise sources have to be addressed to obtain quality intravascular elastograms. Misalignment of the vessel lumen and the ultrasound beam can produce erroneous strain estimates in elastography. Errors in the strain estimate are introduced due to the eccentricity and tilt of the intravascular transducer within the vessel lumen. Previous work in this area has provided theoretical expressions for the correction of eccentricity and tilt errors when they occur independent of each other. However, under most imaging conditions, both eccentricity and tilt errors are simultaneously present. In this paper, we extend the theoretical correction factor by accounting for the influence of both of these errors occurring simultaneously in the positioning of the catheter within the vessel lumen.     

Performance Evaluation of Methods for Two-Dimensional Displacement and Strain Estimation Using Ultrasound Radio Frequency Data

In elastography, several methods for 2-D strain imaging have been introduced, based on both raw frequency (RF) data and speckle-tracking. Although the precision and lesion detectability of axial strain imaging in terms of elastographic signal-to-noise ratio (SNRe) and elastographic contrast-to-noise ratio (CNRe) have been reported extensively, analysis of lateral precision is still lacking. In this paper, the performance of different 2-D correlation RF- and envelope-based strain estimation methods was evaluated using simulation data and phantom experiments. Besides window size and interpolation methods for subsample displacement estimation, the influence of recorrelation techniques was examined. Precision and contrast of the measured displacements and strains were assessed using the difference between modeled and measured displacements, SNRe and CNRe. In general, a 2-D coarse-to-fine displacement estimation method is favored, using envelope data for window sizes exceeding the theoretical upper bound for strain estimation. Using 2-D windows of RF data resulted in better displacement estimates for both the axial and lateral direction than 1-D RF-based or envelope-based techniques. Obtaining subsample lateral displacement estimates by fitting a predefined shape through the cross-correlation function (CCF) yielded results similar to those obtained with up-sampling of RF data in the lateral direction. Using a CCF model was favored because of the decreased computation time. Local aligning and stretching of the windows (recorrelation) resulted in an increase of 2–17 and 6–7 dB in SNRe for axial and lateral strain estimates, respectively, over a range of strains (0.5 to 5.0%). For a simulated inhomogeneous phantom (2.0% applied strain), the measured axial and lateral SNRes were 29.2 and 20.2 dB, whereas the CNRes were 50.2 dB and 31.5 dB, respectively. For the experimental data, lower SNRe (axial: 28.5 dB; lateral: 17.5 dB) and CNRe (axial: 39.3 dB; lateral: 31 dB) were found. In conclusion, a coarse-to-fine approach is favored using RF data on a fine scale. The use of 2D parabolic interpolation is favored to obtain subsample displacement estimates. Recorrelation techniques, such as local aligning and stretching, increase SNRe and CNRe in both directions. (E-mail: r.lopata@cukz.umcn.nl)