Phase-based direct average strain estimation for elastography

Alternating direction method of multipliers for displacement estimation in ultrasound strain elastography

BACKGROUND

Ultrasound strain imaging, which delineates mechanical properties to detect tissue abnormalities, involves estimating the time delay between two radio-frequency (RF) frames collected before and after tissue deformation. The existing regularized optimization-based time-delay estimation (TDE) techniques suffer from at least one of the following drawbacks: (1) The regularizer is not aligned with the tissue deformation physics due to taking only the first-order displacement derivative into account; (2) TheL 2 $L2$ -norm of the displacement derivatives, which oversmooths the estimated time-delay, is utilized as the regularizer; (3) The modulus function defined mathematically should be approximated by a smooth function to facilitate the optimization ofL 1 $L1$ -norm.

PURPOSE

Our purpose is to develop a novel TDE technique that resolves the aforementioned shortcomings of the existing algorithms.

METHODS

Herein, we propose employing the alternating direction method of multipliers (ADMM) for optimizing a novel cost function consisting ofL 2 $L2$ -norm data fidelity term andL 1 $L1$ -norm first- and second-order spatial continuity terms. ADMM empowers the proposed algorithm to use different techniques for optimizing different parts of the cost function and obtain high-contrast strain images with smooth backgrounds and sharp boundaries. We name our technique ADMM for totaL variaTion RegUlarIzation in ultrasound STrain imaging (ALTRUIST). ALTRUIST's efficacy is quantified using absolute error (AE), Structural SIMilarity (SSIM), signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and strain ratio (SR) with respect to GLUE, OVERWIND, andL 1 $L1$ -SOUL, three recently published energy-based techniques, and UMEN-Net, a state-of-the-art deep learning-based algorithm. Analysis of variance (ANOVA)-led multiple comparison tests and paired t $t$ -tests at5 % $5\%$ overall significance level were conducted to assess the statistical significance of our findings. The Bonferroni correction was taken into account in all statistical tests. Two simulated layer phantoms, three simulated resolution phantoms, one hard-inclusion simulated phantom, one multi-inclusion simulated phantom, one experimental breast phantom, and three in vivo liver cancer datasets have been used for validation experiments. We have published the ALTRUIST code at http://code.sonography.ai.

RESULTS

ALTRUIST substantially outperforms the four state-of-the-art benchmarks in all validation experiments, both qualitatively and quantitatively. ALTRUIST yields up to573 % ∗ ${573\%}^{*}$ ,41 % ∗ ${41\%}^{*}$ , and51 % ∗ ${51\%}^{*}$ SNR improvements and443 % ∗ ${443\%}^{*}$ ,53 % ∗ ${53\%}^{*}$ , and15 % ∗ ${15\%}^{*}$ CNR improvements overL 1 $L1$ -SOUL, its closest competitor, for simulated, phantom, and in vivo liver cancer datasets, respectively, where the asterisk (*) indicates statistical significance. In addition, ANOVA-led multiple comparison tests and paired t $t$ -tests indicate that ALTRUIST generally achieves statistically significant improvements over GLUE, UMEN-Net, OVERWIND, andL 1 $L1$ -SOUL in terms of AE, SSIM map, SNR, and CNR.

CONCLUSIONS

A novel ultrasonic displacement tracking algorithm named ALTRUIST has been developed. The principal novelty of ALTRUIST is incorporating ADMM for optimizing anL 1 $L1$ -norm regularization-based cost function. ALTRUIST exhibits promising performance in simulation, phantom, and in vivo experiments.

Incorporating Gradient Similarity for Robust Time Delay Estimation in Ultrasound Elastography

Energy-based ultrasound elastography techniques minimize a regularized cost function consisting of data and continuity terms to obtain local displacement estimates based on the local time-delay estimation (TDE) between radio frequency (RF) frames. The data term associated with the existing techniques takes only the amplitude similarity into account and hence is not sufficiently robust to the outlier samples present in the RF frames under consideration. This drawback creates noticeable artifacts in the strain image. To resolve this issue, we propose to formulate the data function as a linear combination of the amplitude and gradient similarity constraints. We estimate the adaptive weight concerning each similarity term following an iterative scheme. Finally, we optimize the nonlinear cost function in an efficient manner to convert the problem to a sparse system of linear equations which are solved for millions of variables. We call our technique rGLUE: robust data term in GLobal Ultrasound Elastography. rGLUE has been validated using simulation, phantom, in vivo liver, and breast datasets. In all our experiments, rGLUE substantially outperforms the recent elastography methods both visually and quantitatively. For simulated, phantom, and in vivo datasets, respectively, rGLUE achieves 107%, 18%, and 23% improvements of signal-to-noise ratio (SNR) and 61%, 19%, and 25% improvements of contrast-to-noise ratio (CNR) over global ultrasound elastography (GLUE), a recently published elastography algorithm.

Second-Order Ultrasound Elastography With L1-Norm Spatial Regularization

Time delay estimation (TDE) between two radio-frequency (RF) frames is one of the major steps of quasi-static ultrasound elastography, which detects tissue pathology by estimating its mechanical properties. Regularized optimization-based techniques, a prominent class of TDE algorithms, optimize a nonlinear energy functional consisting of data constancy and spatial continuity constraints to obtain the displacement and strain maps between the time-series frames under consideration. The existing optimization-based TDE methods often consider the L2 -norm of displacement derivatives to construct the regularizer. However, such a formulation over-penalizes the displacement irregularity and poses two major issues to the estimated strain field. First, the boundaries between different tissues are blurred. Second, the visual contrast between the target and the background is suboptimal. To resolve these issues, herein, we propose a novel TDE algorithm where instead of L2 -, L1 -norms of both first- and second-order displacement derivatives are taken into account to devise the continuity functional. We handle the non-differentiability of L1 -norm by smoothing the absolute value function's sharp corner and optimize the resulting cost function in an iterative manner. We call our technique Second-Order Ultrasound eLastography (SOUL) with the L1 -norm spatial regularization ( L1 -SOUL). In terms of both sharpness and visual contrast, L1 -SOUL substantially outperforms GLobal Ultrasound Elastography (GLUE), tOtal Variation rEgulaRization and WINDow-based time delay estimation (OVERWIND), and SOUL, three recently published TDE algorithms in all validation experiments performed in this study. In cases of simulated, phantom, and in vivo datasets, respectively, L1 -SOUL achieves 67.8%, 46.81%, and 117.35% improvements of contrast-to-noise ratio (CNR) over SOUL. The L1 -SOUL code can be downloaded from http://code.sonography.ai.

Combining First- and Second-Order Continuity Constraints in Ultrasound Elastography

Ultrasound elastography is a prominent noninvasive medical imaging technique that estimates tissue elastic properties to detect abnormalities in an organ. A common approximation to tissue elastic modulus is tissue strain induced after mechanical stimulation. To compute tissue strain, ultrasound radio frequency (RF) data can be processed using energy-based algorithms. These algorithms suffer from ill-posedness to tackle. A continuity constraint along with the data amplitude similarity is imposed to obtain a unique solution to the time-delay estimation (TDE) problem. Existing energy-based methods exploit the first-order spatial derivative of the displacement field to construct a regularizer. This first-order regularization scheme alone is not fully consistent with the mechanics of tissue deformation while perturbed with an external force. As a consequence, state-of-the-art techniques suffer from two crucial drawbacks. First, the strain map is not sufficiently smooth in uniform tissue regions. Second, the edges of the hard or soft inclusions are not well-defined in the image. Herein, we address these issues by formulating a novel regularizer taking both first- and second-order derivatives of the displacement field into account. The second-order constraint, which is the principal novelty of this work, contributes both to background continuity and edge sharpness by suppressing spurious noisy edges and enhancing strong boundaries. We name the proposed technique: Second-Order Ultrasound eLastography (SOUL). Comparative assessment of qualitative and quantitative results shows that SOUL substantially outperforms three recently developed TDE algorithms called Hybrid, GLUE, and MPWC-Net++. SOUL yields 27.72%, 62.56%, and 81.37% improvements of the signal-to-noise ratio (SNR) and 72.35%, 54.03%, and 65.17% improvements of the contrast-to-noise ratio (CNR) over GLUE with data pertaining to simulation, phantom, and in vivo tissue, respectively. The SOUL code can be downloaded from code.sonography.ai.

Global Ultrasound Elastography in Spatial and Temporal Domains

In this paper, a novel computationally efficient quasi-static ultrasound elastography technique is introduced by optimizing an energy function. Unlike conventional elastography techniques, three radio frequency (RF) frames are considered to devise a nonlinear cost function consisting of data intensity similarity term, spatial regularization terms and, most importantly, temporal continuity terms. We optimize the aforesaid cost function efficiently to obtain the time-delay estimation (TDE) of all samples between the first two and last two frames of ultrasound images simultaneously, and spatially differentiate the TDE to generate axial strain map. A novelty in our spatial and temporal regularizations is that they adaptively change based on the data, which leads to substantial improvements in TDE. We handle the computational complexity resulting from incorporation of all samples from all three frames by converting our optimization problem to a sparse linear system of equations. Consideration of both spatial and temporal continuity makes the algorithm more robust to signal decorrelation than the previous algorithms. We name the proposed method GUEST: Global Ultrasound Elastography in Spatial and Temporal directions. We validated our technique with simulation, experimental phantom, and in vivo liver data and compared the results with two recently proposed TDE methods. In all the experiments, GUEST substantially outperforms other techniques in terms of signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and strain ratio (SR) of the strain images.

Estimation of Strain Elastography from Ultrasound Radio-Frequency Data by Utilizing Analytic Gradient of the Similarity Metric

Most strain imaging techniques follow a pipeline strategy: in the first step, tissue displacement is estimated from radio-frequency (RF) frames, and in the second step, a spatial derivative operation is applied. There are two main issues that arise from this framework. First, the gradient operation amplifies noise, and therefore, smoothing techniques have to be adopted. Second, strain estimation does not exploit the original RF data. It rather relies solely on the noisy displacement field. In this paper, a novel technique is proposed that utilizes both the displacement field and the RF frames to accurately obtain the strain estimates. The normalized cross correlation (NCC) metric between two corresponding windows around the samples of the pre- and post-compressed images is employed to generate a dissimilarity measurement. The derivative of NCC with respect to the strain is analytically derived using the chain rule. This allows an efficient minimization of the dissimilarity metric with respect to the strain using the gradient descent optimization technique. The effectiveness of the proposed method is investigated through simulation data, phantom experiments, and in vivo patient data. The experimental results show that exploiting the information in RF data significantly improves the strain estimates.

Speckle tracking and speckle content based composite strain imaging for solid and fluid filled lesions

Strain imaging though for solid lesions provides an effective way for determining their pathologic condition by displaying the tissue stiffness contrast, for fluid filled lesions such an imaging is yet an open problem. In this paper, we propose a novel speckle content based strain imaging technique for visualization and classification of fluid filled lesions in elastography after automatic identification of the presence of fluid filled lesions. Speckle content based strain, defined as a function of speckle density based on the relationship between strain and speckle density, gives an indirect strain value for fluid filled lesions. To measure the speckle density of the fluid filled lesions, two new criteria based on oscillation count of the windowed radio frequency signal and local variance of the normalized B-mode image are used. An improved speckle tracking technique is also proposed for strain imaging of the solid lesions and background. A wavelet-based integration technique is then proposed for combining the strain images from these two techniques for visualizing both the solid and fluid filled lesions from a common framework. The final output of our algorithm is a high quality composite strain image which can effectively visualize both solid and fluid filled breast lesions in addition to the speckle content of the fluid filled lesions for their discrimination. The performance of our algorithm is evaluated using the in vivo patient data and compared with recently reported techniques. The results show that both the solid and fluid filled lesions can be better visualized using our technique and the fluid filled lesions can be classified with good accuracy.

Comparison of windowing effects on elastography images: Simulation, phantom and in vivo studies

In this paper, we have evaluated the use of smooth windows for ultrasound elastography. In ultrasound elastography, local tissue strain is estimated using operations such as cross-correlation on local segments of RF data. In this process, local data segments are selected by multiplying the RF data by a rectangular window. Such data truncation causes non-ideal spectral behavior, which can be mitigated by using smooth windows. Accordingly, we hypothesize that the use of smooth windows may improve the elastographic signal-to-noise ratio (SNRe) and contrast-to-noise ratio (CNRe) of strain images. The effects of using smooth windows have not been fully characterized for time-domain strain estimators. Thus, we have compared the elastographic performance of rectangular, Hanning, Gaussian, and Chebyshev windows used in conjunction with cross-correlation based algorithm and adaptive stretching algorithm using finite element method (FEM) simulation, experimental phantom, and in vivo data. Smooth windows are found to improve the SNRe by up to 3.94 for FEM data and by up to 1.76 for phantom data which represent 76% and 60.52% improvements, respectively. CNRe improves by up to 12.23 for FEM simulated data and by up to 4.28 for phantom data which represent 213.07% and 248.2% improvements, respectively. Mean structural similarity (MSSIM) was used for assessing the image perceptual quality and smooth windows improved it by up to 0.22 (85.98% improvement) for simulated data. We have evaluated these parameters at 1-6% applied strains for the experimental phantom and at 1%, 2%, 4%, 6%, 8%, and 12% applied strains for FEM simulation. We observed a maximum deterioration in axial resolution of 0.375 mm (which is on the order of the wavelength, 0.3mm) due to smooth windows. "Salt-and-pepper" noise from false-peak errors has also been reduced. Smooth windows increased the lesion-to-background contrast (by increasing the CNRe by 213.07%) of a low contrast lesion (10-dB). For the in vivo cases, use of smooth windows resulted in better depiction of lesions, which is important for lesion classification. In this work, we have used an ATL Ultramark 9 scanner with an L10-5 (7.5 MHz) probe for the phantom experiment and a Sonix SP500 scanner with an L14-5/38 probe (10 MHz) for in vivo data collection.

Strain estimation by a Fourier Series-based extrema tracking algorithm for elastography

In this paper, a new strain estimator using extrema tracking based on Fourier Series expansion (ETBFS) is proposed for ultrasonic elastography. In this method, the extremum is determined by solving an equation constructed by obtaining the first order derivative of the Fourier Series expansion and setting it to zero. Unlike other tracking algorithms, the ETBFS method can locate the extrema of radio frequency (RF) signals exactly between two adjacent sampling points and achieve a sub-sample accuracy without additional explicit interpolation. The correspondence between the located extrema in the pre- and post-compressed RF signal segments are constructed with a fine matching technique, with which the displacements and strains are estimated. Experimental results on a finite-element-modeling (FEM) simulation phantom show that the new proposed method can provide a more accurate displacement estimation than the standard cross-correlation (CC)-based method and the scale-invariant keypoints tracking (SIKT) algorithm. Moreover, performance analysis in terms of elastographic signal-to-noise ratio (SNRe), elastographic contrast-to-noise ratio (CNRe) and the real-versus-estimated strain error (RESE) also indicate that the dynamic range of the strain filter and its sensitivity can be improved with this new method.

A class of kernel based real-time elastography algorithms

In this paper, a novel real-time kernel-based and gradient-based Phase Root Seeking (PRS) algorithm for ultrasound elastography is proposed. The signal-to-noise ratio of the strain image resulting from this method is improved by minimizing the cross-correlation discrepancy between the pre- and post-compression radio frequency signals with an adaptive temporal stretching method and employing built-in smoothing through an exponentially weighted neighborhood kernel in the displacement calculation. Unlike conventional PRS algorithms, displacement due to tissue compression is estimated from the root of the weighted average of the zero-lag cross-correlation phases of the pair of corresponding analytic pre- and post-compression windows in the neighborhood kernel. In addition to the proposed one, the other time- and frequency-domain elastography algorithms (Ara et al., 2013; Hussain et al., 2012; Hasan et al., 2012) proposed by our group are also implemented in real-time using Java where the computations are serially executed or parallely executed in multiple processors with efficient memory management. Simulation results using finite element modeling simulation phantom show that the proposed method significantly improves the strain image quality in terms of elastographic signal-to-noise ratio (SNRe), elastographic contrast-to-noise ratio (CNRe) and mean structural similarity (MSSIM) for strains as high as 4% as compared to other reported techniques in the literature. Strain images obtained for the experimental phantom as well as in vivo breast data of malignant or benign masses also show the efficacy of our proposed method over the other reported techniques in the literature.

Hybrid algorithm for elastography to visualize both solid and fluid-filled lesions

We propose a novel strain estimation technique that can produce reliable strain images of both solid and fluid-filled lesions. In our method, a kernel-based correlation coefficient technique and a speckle tracking-based strain estimation technique are combined into a single algorithm. The elegance of our algorithm is that fluid-filled lesions are first automatically identified by three selection criteria, and strain in those parts is estimated using the kernel-based correlation coefficient technique. Strain where fluid-filled lesions have not been detected is estimated using a speckle tracking-based algorithm, and then these two estimates are merged to form the final image. Any speckle tracking algorithm can be used in our proposed technique; however, we used a modified version of the direct average spectral strain estimation technique to describe our algorithm. We modified the direct average spectral strain estimation algorithm to track smaller strain variation and to facilitate strain calculation from multiple frames. We describe the performance of our proposed hybrid algorithm using in vivo patient data. Both the solid and fluid-filled lesions are clearly visible in the strain images produced by our proposed approach and are of better quality in terms of contrast-to-noise ratio and border sharpness than the strain images generated by other reported techniques. We also validate the performance of our proposed multiframe technique using experimental phantom data and in vivo patient data. The results reveal that the quality of the strain image can be improved using the multiframe technique compared with its dual-frame counterpart.