Reconstructions of shear modulus, Poisson's ratio, and density using approximate mean normal stress lambda epsilon alpha alpha as unknown

Analytical Minimization-Based Regularized Subpixel Shear-Wave Tracking for Ultrasound Elastography

Ultrasound elastography is a convenient and affordable method for imaging mechanical properties of tissue, which are often correlated with pathologies. An emerging novel elastography technique applies an external acoustic radiation force to generate a shear wave in the tissue and uses ultrasound imaging to track the shear wave. Accurate tracking of the small tissue motion is a critical step in shear-wave elastography (SWE), but it is challenging due to various sources of noise in the ultrasound data. We formulate tissue displacement estimation as an optimization problem and propose two computationally efficient approaches to estimate the displacement field. The first algorithm is referred to as dynamic programming analytic minimization (DPAM), which utilizes first-order Taylor series expansion of the highly nonlinear cost function to allow for its efficient optimization, and was previously proposed for quasistatic elastography. The second algorithm is a novel technique that utilizes second-order derivatives of the nonlinear cost function. We call the new algorithm second-order analytic minimization elastography (SESAME). We compare DPAM and SESAME to the standard normalized cross correlation (NCC) approach in the context of displacement and speed estimation of wave propagation in SWE. The results of micrometer-order displacement estimation in a uniform simulation phantom illustrate that SESAME outperforms DPAM, which in turn outperforms NCC in terms of signal-to-noise ratio (SNR) and jitter. In addition, the relative difference between true and reconstructed shear modulus (averaged over excitations at different focal depths and several scatterer realizations at each depth) is approximately 3.41%, 1.12%, and 1.01%, respectively, for NCC, DPAM, and SESAME. The performance of the proposed methods is also assessed with real data acquired using a tissue-mimicking phantom, wherein, in comparison to NCC, DPAM and SESAME improve the SNR of displacement estimates by 7.6 and 9.5 dB, respectively. Experimental results on a tissue-mimicking phantom also show that shear modulus reconstruction substantially improved with the proposed DPAM technique over NCC and with some further improvement achieved by utilizing the second-order Taylor series approximation in SESAME instead of the first-order DPAM.

Plural spectral frequency divisions for ultrasonic tissue displacement vector measurement

For a human tissue displacement vector or strain tensor measurement, the plural spectral frequency division method (PSFDM) is presented. In this report, the effectiveness of the processing is demonstrated by performing the measurements on an agar phantom interrogated by wideband ultrasonic (US) single beam scanning and compounding of steered plane wave transmissions.

Real-time regularized ultrasound elastography

This paper introduces two real-time elastography techniques based on analytic minimization (AM) of regularized cost functions. The first method (1D AM) produces axial strain and integer lateral displacement, while the second method (2D AM) produces both axial and lateral strains. The cost functions incorporate similarity of radio-frequency (RF) data intensity and displacement continuity, making both AM methods robust to small decorrelations present throughout the image. We also exploit techniques from robust statistics to make the methods resistant to large local decorrelations. We further introduce Kalman filtering for calculating the strain field from the displacement field given by the AM methods. Simulation and phantom experiments show that both methods generate strain images with high SNR, CNR and resolution. Both methods work for strains as high as 10% and run in real-time. We also present in vivo patient trials of ablation monitoring. An implementation of the 2D AM method as well as phantom and clinical RF-data can be downloaded.

Determining if the relative shear modulus or the inverse of the relative shear modulus should be imaged using axial strain ratios on agar phantoms

An axial strain image is considered to be an image of a relative shear modulus. However, through simulations of focal lesions, it was previously confirmed that reconstruction imaging of the 1-D relative shear modulus or its inverse obtained from the ratio of the axial strain in the axial direction (i.e., the axial strain ratio), with respect to a reference strain, produces larger contrast-to-noise ratios (CNRs) than that of the axial strain, although both of the reconstructions have smaller relative contrasts than that of the original shear modulus. The reference strain is appropriately chosen in the area of a homogeneous neighborhood in front of or behind the target by viewing in the B-mode and strain images. It has also been confirmed that the evaluations of CNRs of reconstructions immediately after strain measurement permit a prediction of which reconstructions should be imaged, i.e., a decision can be made without obtaining the reconstruction images, but by using the statistical evaluations of the measured strains in the focal lesion and in the surrounding region. In this study, the feasibility of using this prediction is verified through agar phantom experiments using actual measured axial strains obtained with the 2-D cross-spectrum phase-gradient method. The resulting successful prediction revealed that the assumptions of independent stationary and random axial-strain measurement errors in the focal lesion and in the surrounding region are appropriate.

Spatially variant regularization of lateral displacement measurement using variance

The purpose of this work is to confirm the effectiveness of our proposed spatially variant displacement component-dependent regularization for our previously developed ultrasonic two-dimensional (2D) displacement vector measurement methods, i.e., 2D cross-spectrum phase gradient method (CSPGM), 2D autocorrelation method (AM), and 2D Doppler method (DM). Generally, the measurement accuracy of lateral displacement spatially varies and the accuracy is lower than that of axial displacement that is accurate enough. This inaccurate measurement causes an instability in a 2D shear modulus reconstruction. Thus, the spatially variant lateral displacement regularization using the lateral displacement variance will be effective in obtaining an accurate lateral strain measurement and a stable shear modulus reconstruction than a conventional spatially uniform regularization. The effectiveness is verified through agar phantom experiments. The agar phantom [60mm (height) x 100 mm (lateral width) x 40 mm (elevational width)] that has, at a depth of 10mm, a circular cylindrical inclusion (dia.=10mm) of a higher shear modulus (2.95 and 1.43 x 10(6)N/m(2), i.e., relative shear modulus, 2.06) is compressed in the axial direction from the upper surface of the phantom using a commercial linear array type transducer that has a nominal frequency of 7.5-MHz. Because a contrast-to-noise ratio (CNR) expresses the detectability of the inhomogeneous region in the lateral strain image and further has almost the same sense as that of signal-to-noise ratio (SNR) for strain measurement, the obtained results show that the proposed spatially variant lateral displacement regularization yields a more accurate lateral strain measurement as well as a higher detectability in the lateral strain image (e.g., CNRs and SNRs for 2D CSPGM, 2.36 vs 2.27 and 1.74 vs 1.71, respectively). Furthermore, the spatially variant lateral displacement regularization yields a more stable and more accurate 2D shear modulus reconstruction than the uniform regularization (however, for the regularized relative shear modulus reconstructions, slightly accurate, e.g., for 2D CSPGM, 1.51 vs 1.50). These results indicate that the spatially variant displacement component-dependent regularization will enable the 2D shear modulus reconstruction to be used as practical diagnostic and monitoring tools for the effectiveness of various noninvasive therapy techniques of soft tissue diseases (e.g., breast, liver cancers). Application of the regularization to the elevational displacement will also increase the stability of a three-dimensional (3D) reconstruction.

Comparison of contrast-to-noise ratios of axial strain, shear modulus, and inverse of shear modulus estimated by axial strain ratio

Occasionally, an axial strain image is dealt with as an image of relative shear modulus. However, we previously confirmed that the imaging of the 1-D relative shear modulus reconstruction obtained by the ratio of the axial strain in the axial direction (i.e., axial strain ratio) can yield a larger contrast-to-noise ratio (CNR) than that of the axial strain, although both the axial strain ratio and the axial strain have smaller relative contrasts than that of the original shear modulus. We also reported the imaging of the inverse of the relative shear modulus (i.e., the inverse of the axial strain ratio). In this report, the evaluations of CNRs on simulated phantoms clarify that, for local tumors, immediately after strain measurement, we can determine whether we should image the axial strain ratio or the inverse of the axial strain ratio, i.e., by using the statistical evaluations of measured strains in the tumor and the surrounding region. This also shows that the evaluations of CNRs of the strain ratio and the inverse of the strain ratio are effective. We also confirmed that reference regions should be properly set in the homogeneous neighborhood in front of or behind the target by viewing B-mode and strain images.

Quantitative three-dimensional elasticity imaging from quasi-static deformation: a phantom study

We present a methodology to image and quantify the shear elastic modulus of three-dimensional (3D) breast tissue volumes held in compression under conditions similar to those of a clinical mammography system. Tissue phantoms are made to mimic the ultrasonic and mechanical properties of breast tissue. Stiff lesions are created in these phantoms with size and modulus contrast values, relative to the background, that are within the range of values of clinical interest. A two-dimensional ultrasound system, scanned elevationally, is used to acquire 3D images of these phantoms as they are held in compression. From two 3D ultrasound images, acquired at different compressed states, a three-dimensional displacement vector field is measured. The measured displacement field is then used to solve an inverse problem, assuming the phantom material to be an incompressible, linear elastic solid, to recover the shear modulus distribution within the imaged volume. The reconstructed values are then compared to values measured independently by direct mechanical testing.

Effective lateral modulations with applications to shear modulus reconstruction using displacement vector measurement

High accuracy in measuring target motions can be realized by combined use of our previously developed lateral Gaussian envelope cosine modulation method (LGECMM) and displacement vector measurement methods that enable simultaneous axial and lateral displacement measurements, such as the multidimensional autocorrelation method (MAM). In this paper, LGECMM is improved by using parabolic functions and Hanning windows instead of Gaussian functions in the apodization function, i.e., parabolic apodization and Hanning apodization. The new modulations enable decreases in effective aperture length (i.e., channels) and yield more accurate displacement vector measurements than LGECMM due to increased echo signal-to-noise ratio and lateral spatial resolution. That is, on the basis of a priori knowledge about ultrasound propagation using the focusing scheme and shape of the apodization function, we stopped using Fraunhofer approximation. As practical applications of the modulations, for an agar phantom that is deformed in a lateral direction, stable and accurate 2-D shear modulus reconstructions are performed using our previously developed direct inversion approach together with 2-D strain tensor measurements using MAM.

Regularization of tissue shear modulus reconstruction using strain variance

An effective setting method (that is, a method using the variances of strain tensor component measurements) is described for the properly spatially varied regularization parameters for our shear modulus reconstruction. At each position, the respective strain variances can be experimentally evaluated using plural field measurements or single field measurement, for example, when using all crosscorrelation- based methods, by using the Ziv-Zakai Lower Bound (ZZLB). The demonstrated regularization by the single field measurement using the cross-spectrum phase gradient method (MCSPGM) in experiments confirms that the use of the axial strain variance estimated by the echo signal-to-noise ratio and correlations (the combined SNRc) effectively stabilizes the 1-D reconstruction on an agar phantom and a human in vivo liver carcinoma during interstitial microwave thermal treatment. The regularization yields a spatially uniform stability in reconstruction.

Displacement vector measurement using instantaneous ultrasound signal phase - multidimensional autocorrelation and doppler methods

Two new methods of measuring a multidimensional displacement vector using an instantaneous ultrasound signal phase are described, i.e., the multidimensional autocorrelation method (MAM) and multidimensional Doppler method (MDM). A high measurement accuracy is achieved by combining either method with the lateral Gaussian envelope cosine modulation method (LGECMM) or multidirectional synthetic aperture method (MDSAM). Measurement accuracy is evaluated using simulated noisy echo data. Both methods yield accurate measurements comparable to that of our previously developed cross-spectrum phase gradient method (MCSPGM); however, they require less computational time (the order, MDM < MAM approximate, equals MCSPGM) and would provide realtime measurements. Moreover, comparisons of LGECMM and MDSAM performed by geometrical evaluations clarifies that LGECMM has potentials to yield more accurate measurements with less computational time. Both MAM and MDM can be applied to the measurement of tissue strain, blood flow, sonar data, and other target motions.

Shear modulus reconstruction by ultrasonically measured strain ratio

PURPOSE

In addition to a description of our three previously developed one-dimensional (1D) methods from the viewpoint of shear modulus reconstruction using the strain ratio, two new methods for stabilizing the 1D methods are described, together with their limitations. As confirmed using human in vivo breast tissues, method 1 for evaluating the strain ratio itself is useful when the measurement accuracy of the strain distribution is high. However, because tissues having high shear moduli, such as scirrhous carcinoma, often form singular points/regions, both methods 2 and 3 using the strain ratio (initial estimate) and a regularization method are effective for realizing a unique, stable, useful shear modulus reconstruction. Because method 3 carries out implicit integration only at singular points/regions, whereas method 2 carries out implicit integration throughout the region of interest (ROI), the smaller number of singular points enables more rapid shear modulus reconstruction by method 3 than by method 2. Like method 1, method 3 is also useful when the measurement accuracy of the strain distribution is high. However, when evaluating strain distribution in an ROI with a high spatial resolution to obtain a shear modulus reconstruction having a high spatial resolution, shear modulus reconstructions obtained by methods 1, 2, and 3 often become laterally unstable due to the instability and low accuracy of the strains in the reference regions (reference strains), i.e., regularization in methods 2 and 3 cannot reduce the instability in the initial estimate.

METHODS

To cope with this instability, (i) the reconstruction obtained by calculating the strain ratio should be low-pass filtered; for breast tissues, in particular, the reconstruction of the inverse shear modulus should be low-pass filtered, not the reconstruction of the shear modulus. (ii) Otherwise, when using homogeneous regions as a reference, such as a block of reference material, fatty tissue, or parenchyma, evaluation of the reference strains with a low spatial resolution is effective.

RESULTS

Although such evaluation yields a stable reconstruction with a high spatial resolution compared with that obtained by the low-pass filtering of the strain ratio, we confirmed through simulations that, when reducing artifacts due to a 1D reconstruction of the shear modulus, the evaluation yields a low-accuracy reconstruction value of inhomogeneity. In contrast, in such a case the low-pass filtering of the strain ratio yields a more accurate reconstruction value.

CONCLUSION

All the above-mentioned methods using the strain ratio realize real-time shear modulus reconstruction and should be selected appropriately in conventional ultrasonic imaging equipment by considering the application of the reconstruction (i.e., in accordance with the measurement accuracy of the strains and the occurrence of artifacts).

Effective shear modulus reconstruction obtained with approximate mean normal stress remaining unknown

We previously reported Methods A and B for reconstructing tissue shear modulus and density using mean normal stress as an unknown. The use of Method A enables us to obtain such reconstructions with the mean normal stress remaining unknown by using an iterative method to solve algebraic equations. However, Method A results in a low convergence speed and a low reconstruction accuracy compared with Method B that enables a reconstruction of mean normal stress together. Thus, in this report, we describe a new, rapid and accurate method, Method C, that enables the reconstructions of shear modulus and density in real time with a higher accuracy than Method A. In Method A, no reference mean normal stress is used. In Method C, an arbitrary finite value is used as a quasireference mean normal stress at an arbitrary point (i.e., a quasireference point) or an arbitrary region (i.e., a quasireference region) in the region of interest on the basis of the fact that the gradient operator implemented on the mean normal stress becomes positive-definite. When a quasireference region can be realized, Method C enables such reconstructions with a high accuracy and a high convergence speed similar to Method B. The effectiveness of Method C was verified using simulated phantom deformation data. Method C must be used instead of Method A as a practical method, in combination with Method B.

Reconstruction of thermal property distributions of tissue phantoms from temperature measurements—thermal conductivity, thermal capacity and thermal diffusivity

We report robust noninvasive techniques for reconstructing the thermal properties of living tissues, such as thermal conductivity, thermal capacity and thermal diffusivity, for the diagnosis, monitoring and planning of thermal treatments. Internal temperature distributions can be measured using ultrasonic imaging or magnetic resonance imaging. Provided that the reference thermal properties are given in the region of interest as initial conditions, by solving bioheat transfer equations as simultaneous first-order partial differential equations having temperature distributions as inhomogeneous coefficients, we can determine thermal property distributions. A novel regularized numerical solution is also presented to realize useful, unique, stable reconstructions of the thermal property distributions. To verify the feasibility of the numerical solution, simulations and ultrasonic phantom experiments are conducted. The reconstruction of perfusion by blood flow and thermal source/sink by this approach is also addressed.