Quantitative elasticity imaging: what can and cannot be inferred from strain images

Density imaging using a multiple-frequency DBIM approach

Current inverse scattering methods for quantitative density imaging have limitations that keep them from practical experimental implementations. In this work, an improved approach, termed the multiple-frequency distorted Born iterative method (MF-DBIM) algorithm, was developed for imaging density variations. The MF-DBIM approach consists of inverting the wave equation by solving for a single function that depends on both sound speed and density variations at multiple frequencies. Density information was isolated by using a linear combination of the reconstructed single-frequency profiles. Reconstructions of targets using MF-DBIM from simulated data were compared with reconstructions using methods currently available in the literature, i.e., the dual-frequency DBIM (DF-DBIM) and T-matrix approaches. Useful density reconstructions, i.e., root mean square errors (RMSEs) less than 30%, were obtained with MF-DBIM even with 2% Gaussian noise in the simulated data and using frequency ranges spanning less than an order of magnitude. Therefore, the MFDBIM approach outperformed both the DF-DBIM method (which has problems converging with noise even an order of magnitude smaller) and the T-matrix method (which requires a ka factor close to unity to achieve convergence). However, the convergence of all the density imaging algorithms was compromised when imaging targets with object functions exhibiting high spatial frequency content.

On the potential of a new IVUS elasticity modulus imaging approach for detecting vulnerable atherosclerotic coronary plaques: in vitro vessel phantom study

Peak cap stress amplitude is recognized as a good indicator of vulnerable plaque (VP) rupture. However, such stress evaluation strongly relies on a precise, but still lacking, knowledge of the mechanical properties exhibited by the plaque components. As a first response to this limitation, our group recently developed, in a previous theoretical study, an original approach, called iMOD (imaging modulography), which reconstructs elasticity maps (or modulograms) of atheroma plaques from the estimation of strain fields. In the present in vitro experimental study, conducted on polyvinyl alcohol cryogel arterial phantoms, we investigate the benefit of coupling the iMOD procedure with the acquisition of intravascular ultrasound (IVUS) measurements for detection of VP. Our results show that the combined iMOD-IVUS strategy: (1) successfully detected and quantified soft inclusion contours with high positive predictive and sensitivity values of 89.7 ± 3.9% and 81.5 ± 8.8%, respectively, (2) estimated reasonably cap thicknesses larger than ∼300 µm, but underestimated thinner caps, and (3) quantified satisfactorily Young's modulus of hard medium (mean value of 109.7 ± 23.7 kPa instead of 145.4 ± 31.8 kPa), but overestimated the stiffness of soft inclusions (mean Young`s moduli of 31.4 ± 9.7 kPa instead of 17.6 ± 3.4 kPa). All together, these results demonstrate a promising benefit of the new iMOD-IVUS clinical imaging method for in vivo VP detection.

Shear wave speed recovery in sonoelastography using crawling wave data

The crawling wave experiment, in which two harmonic sources oscillate at different but nearby frequencies, is a development in sonoelastography that allows real-time imaging of propagating shear wave interference patterns. Previously the crawling wave speed was recovered and used as an indicator of shear stiffness; however, it is shown in this paper that the crawling wave speed image can have artifacts that do not represent a change in stiffness. In this paper, the locations and shapes of some of the artifacts are exhibited. In addition, a differential equation is established that enables imaging of the shear wave speed, which is a quantity strongly correlated with shear stiffness change. The full algorithm is as follows: (1) extract the crawling wave phase from the spectral variance data; (2) calculate the crawling wave phase wave speed; (3) solve a first-order PDE for the phase of the wave emanating from one of the sources; and (4) compute and image the shear wave speed on a grid in the image plane.

An inverse problem approach for elasticity imaging through vibroacoustics

A methodology for estimating the spatial distribution of elastic moduli using the steady-state dynamic response of solids immersed in fluids is presented. The technique relies on the ensuing acoustic field from a remotely excited solid to inversely estimate the spatial distribution of Young's modulus of biological structures (e.g., breast tissue). This work proposes the use of Gaussian radial basis functions (GRBF) to represent the spatial variation of elastic moduli. GRBF are shown to possess the advantage of representing smooth functions with quasi-compact support and can efficiently represent elastic moduli distributions such as those that occur in soft biological tissue in the presence of unhealthy tissue (e.g., tumors and calcifications). The direct problem consists of a coupled acoustic-structure interaction boundary-value problem solved in the frequency domain using the finite element method. The inverse problem is cast as an optimization problem in which the error functional is defined as a measure of discrepancy between an experimentally measured response and a finite element representation of the system. Nongradient based optimization algorithms are used to solve the resulting optimization problem. The feasibility of the proposed approach is demonstrated through a series of simulations and an experiment. For comparison purposes, the surface velocity response was also used for the inverse characterization as the measured response in place of the acoustic pressure.

Ultrasound-based relative elastic modulus imaging for visualizing thermal ablation zones in a porcine model

The feasibility of using ultrasound-based elastic modulus imaging to visualize thermal ablation zones in an in vivo porcine model is reported. Elastic modulus images of soft tissues are estimated as an inverse optimization problem. Ultrasonically measured displacement data are utilized as inputs to determine an elastic modulus distribution that provides the best match to this displacement field. A total of 14 in vivo thermal ablation zones were investigated in this study. To determine the accuracy of delineation of each thermal ablation zone using elastic modulus imaging, the dimensions (lengths of long and short axes) and the area of each thermal ablation zone obtained from an elastic modulus image were compared to the corresponding gross pathology photograph of the same ablation zone. Comparison of elastic modulus imaging measurements and gross pathology measurements showed high correlation with respect to the area of thermal ablation zones (Pearson coefficient = 0.950 and p < 0.0001). The radiological-pathological correlation was slightly lower (correlation = 0.853, p < 0.0001) for strain imaging among these 14 in vivo ablation zones. We also found that, on average, elastic modulus imaging can more accurately depict thermal ablation zones, when compared to strain imaging (14.7% versus 22.3% absolute percent error in area measurements, respectively). Furthermore, elastic modulus imaging also provides higher (more than a factor of 2) contrast-to-noise ratios for evaluating these thermal ablation zones than those on corresponding strain images, thereby reducing inter-observer variability. Our preliminary results suggest that elastic modulus imaging might potentially enhance the ability to visualize thermal ablation zones, thereby improving assessment of ablative therapies.

Quantitative ultrasonic elastography for gel dosimetry

Advanced radiation techniques such as intensity-modulated radiotherapy (IMRT) for complex geometries in which targets are close to organs at risk have been introduced in radiation therapy, creating a need for procedures that allow easy three-dimensional (3-D) measurement of dose for verification purposes. Polymer gels that change their material properties when irradiated have been suggested for such use. For example, the change in their magnetic properties has been thoroughly investigated with magnetic resonance imaging (MRI). Also, we have previously shown that the mechanical stiffness, i.e., Young's modulus, of these gels changes with dose. This finding prompted us to assess whether we can image a radiation-induced stiffness distribution with quantitative ultrasound elastography and whether the stiffness distribution is correlated with the dose distribution. A methacrylic-acid-based gel was loaded with scatterers to create an ultrasound echoic signal. It was irradiated to create a rod-like region of increased stiffness with a 10 x 10 mm(2) cross-section. The gel block was compressed in a frame that restricted the movement of the gel to planes orthogonal to the long axis of the irradiated region and ultrasonic echo data were acquired in the central plane during compression. This simplified irradiation pattern and experimental set-up were designed to approximate plane-strain conditions and was chosen for proof of concept. The movement of the gel was tracked from ultrasound images of a different compressional state using cross-correlation, enabling a displacement map to be created. The shear modulus was reconstructed using an inverse algorithm. The role of the magnitude of the regularization parameter in the inverse problem and the boundary conditions in influencing the spatial distribution of stiffness and, thus, final dose contrast was investigated through parametric studies. These parameters were adjusted using prior knowledge about the stiffness in parts of the material, e.g., the background was not irradiated and therefore its stiffness was homogeneous. It was observed that a suitable choice for these reconstruction parameters was essential for a quantitative application of stiffness measurement such as dosimetry. The dose contrast and distribution found with the optimal parameters were close to those obtained with MRI. Initial results reported in this article are encouraging and indicate that with ongoing refinement of ultrasound elastography techniques and accompanying inverse algorithms, this approach could play an important role in gel dosimetry.

Theoretical limitations of the elastic wave equation inversion for tissue elastography

This article examines the theoretical limitations of the local inversion techniques for the measurement of the tissue elasticity. Most of these techniques are based on the estimation of the phase speed or the algebraic inversion of a one-dimensional wave equation. To analyze these techniques, the wave equation in an elastic continuum is revisited. It is proven that in an infinite medium, harmonic shear waves can travel at any phase speed greater than the classically known shear wave speed, mu/rho, by demonstrating this for a special case with cylindrical symmetry. Hence in addition to the mechanical properties of the tissue, the phase speed depends on the geometry of the wave as well. The elastic waves in an infinite cylindrical rod are studied. It is proven that multiple phase speeds can coexist for a harmonic wave at a single frequency. This shows that the phase speed depends not only on the mechanical properties of the tissue but also on its shape. The final conclusion is that the only way to avoid theoretical artifacts in the elastograms obtained by the local inversion techniques is to use the shear wave equation as expressed in the curl of the displacements, i.e., the rotations, for the inversion.

Young's modulus reconstruction for radio-frequency ablation electrode-induced displacement fields: a feasibility study

Radio-frequency (RF) ablation is a minimally invasive treatment for tumors in various abdominal organs. It is effective if good tumor localization and intraprocedural monitoring can be done. In this paper, we investigate the feasibility of using an ultrasound-based Young's modulus reconstruction algorithm to image an ablated region whose stiffness is elevated due to tissue coagulation. To obtain controllable tissue deformations for abdominal organs during and/or intermediately after the RF ablation, the proposed modulus imaging method is specifically designed for using tissue deformation fields induced by the RF electrode. We have developed a new scheme under which the reconstruction problem is simplified to a 2-D problem. Based on this scheme, an iterative Young's modulus reconstruction technique with edge-preserving regularization was developed to estimate the Young's modulus distribution. The method was tested in experiments using a tissue-mimicking phantom and on ex vivo bovine liver tissues. Our preliminary results suggest that high contrast modulus images can be successfully reconstructed. In both experiments, the geometries of the reconstructed modulus images of thermal ablation zones match well with the phantom design and the gross pathology image, respectively.

Elastography method for reconstruction of nonlinear breast tissue properties

Elastography is developed as a quantitative approach to imaging linear elastic properties of tissues to detect suspicious tumors. In this paper a nonlinear elastography method is introduced for reconstruction of complex breast tissue properties. The elastic parameters are estimated by optimally minimizing the difference between the computed forces and experimental measures. A nonlinear adjoint method is derived to calculate the gradient of the objective function, which significantly enhances the numerical efficiency and stability. Simulations are conducted on a three-dimensional heterogeneous breast phantom extracting from real imaging including fatty tissue, glandular tissue, and tumors. An exponential-form of nonlinear material model is applied. The effect of noise is taken into account. Results demonstrate that the proposed nonlinear method opens the door toward nonlinear elastography and provides guidelines for future development and clinical application in breast cancer study.

Vulnerable atherosclerotic plaque elasticity reconstruction based on a segmentation-driven optimization procedure using strain measurements: theoretical framework

It is now recognized that prediction of the vulnerable coronary plaque rupture requires not only an accurate quantification of fibrous cap thickness and necrotic core morphology but also a precise knowledge of the mechanical properties of plaque components. Indeed, such knowledge would allow a precise evaluation of the peak cap-stress amplitude, which is known to be a good biomechanical predictor of plaque rupture. Several studies have been performed to reconstruct a Young's modulus map from strain elastograms. It seems that the main issue for improving such methods does not rely on the optimization algorithm itself, but rather on preconditioning requiring the best estimation of the plaque components' contours. The present theoretical study was therefore designed to develop: 1) a preconditioning model to extract the plaque morphology in order to initiate the optimization process, and 2) an approach combining a dynamic segmentation method with an optimization procedure to highlight the modulogram of the atherosclerotic plaque. This methodology, based on the continuum mechanics theory prescribing the strain field, was successfully applied to seven intravascular ultrasound coronary lesion morphologies. The reconstructed cap thickness, necrotic core area, calcium area, and the Young's moduli of the calcium, necrotic core, and fibrosis were obtained with mean relative errors of 12%, 4% and 1%, 43%, 32%, and 2%, respectively.

Quasi-Static Ultrasound Elastography

Elastography is a new imaging modality where elastic tissue parameters related to the structural organization of normal and pathological tissues are imaged. Basic principles underlying the quasi-static elastography concept and principles are addressed. The rationale for elastographic imaging is reinforced using data on elastic properties of normal and abnormal soft tissues. The several orders of magnitude difference between the elastic modulus of normal and abnormal tissues which is the primary contrast mechanism in elastographic imaging underlines the probability of success with this imaging modality. Recent advances enabling the clinical practice of elastographic imaging in real-time on clinical ultrasound systems is also discussed.In quasi-static elastography, radiofrequency echo signals acquired before and after a small (about 1%) of applied deformation are correlated to estimate tissue displacements. Local tissue displacement vector estimates between small segments of the pre- and post-deformation signals are estimated and the corresponding strain distribution imaged. Elastographic imaging techniques are based on the hypothesis that soft tissues deform more than stiffer tissue, and these differences can be quantified in images of the tissue strain tensor or the Young's modulus.Clinical applications of quasi-static elastography have mushroomed over the last decade, with the most commonly imaged areas being the breast, prostate, thyroid, cardiac, treatment monitoring of ablation procedures and vascular imaging applications.

Estimation of nonlinear mechanical properties of vascular tissues via elastography

A new method is proposed for estimation of nonlinear elastic properties of soft tissues. The proposed approach involves a combination of nonlinear finite element methods with a genetic algorithm for estimating tissue stiffness profile. A multipoint scheme is introduced that satisfies the uniqueness condition, improves the estimation performance, and reduces the sensitivity to image noise. The utility of the proposed techniques is demonstrated using optical coherence tomography (OCT) images. The approach is, however, applicable to other imaging systems and modalities, as well, provided a reliable image registration scheme. The proposed algorithm is applied to realistic (2D) and idealized (3D) arterial plaque models, and proves promising for the estimation of intra-plaque distribution of nonlinear material properties.

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.

Stable, intelligible ultrasonic strain imaging

BACKGROUND: Freehand quasistatic strain imaging can reveal qualitative information about tissue stiffness with good spatial accuracy. Clinical trials, however, repeatedly cite instability and variable signal-to-noise ratio as significant drawbacks. METHODS: This study investigates three post-processing strategies for quasistatic strain imaging. Normalisation divides the strain by an estimate of the stress field, the intention being to reduce sensitivity to variable applied stress. Persistence aims to improve the signal-to-noise ratio by time-averaging multiple frames. The persistence scheme presented in this article operates at the pixel level, weighting each frame's contribution by an estimate of the strain precision. Precision-based display presents the clinician with an image in which regions of indeterminate strain are obscured behind a colour wash. This is achieved using estimates of strain precision that are faithfully propagated through the various stages of signal processing. RESULTS AND DISCUSSION: The post-processing strategy is evaluated qualitatively on scans of a breast biopsy phantom and in vivo head and neck examinations. Strain images processed in this manner are observed to benefit from improved stability and signal-to-noise ratio. There are, however, limitations. In unusual though conceivable circumstances, the normalisation procedure might suppress genuine stiffness variations evident in the unprocessed strain images. In different circumstances, the raw strain images might fail to capture significant stiffness variations, a situation that no amount of post-processing can improve. CONCLUSION: The clinical utility of freehand quasistatic strain imaging can be improved by normalisation, precision-weighted pixel-level persistence and precision-based display. The resulting images are stable and generally exhibit a better signal-to-noise ratio than any of the original, unprocessed strain images.

Viscoelastic characterization of soft tissue from dynamic finite element models

An iterative solution to the inverse problem of elasticity and viscosity is proposed in this paper. A new dynamic finite element model that is consistent with known rheological models has been derived to account for the viscoelastic changes in soft tissue. The model assumes known lumped masses at the nodes, and comprises two vectors of elasticity and viscosity parameters that depend on the material elasticity and viscosity distribution, respectively. Using this deformation model and the observed dynamic data for harmonic excitation, the inverse problem is solved to reconstruct the viscosity and elasticity in the medium by using a Gauss-Newton-based approach. As in other inverse problems, previous knowledge of the parameters on the boundaries of the medium is necessary to assure uniqueness and convergence and to obtain an accurate map of the viscoelastic properties. The sensitivity of the solutions to noise, model and boundary conditions has been studied through numerical simulations. Experimental results are also presented. The viscosity and elasticity of a gelatin-based phantom with inclusion of known properties have been reconstructed and have been shown to be close to the values obtained using standard rheometry.

An inverse method for imaging the local elasticity of atherosclerotic coronary plaques

The rupture of thin-cap fibroatheroma (TCFA) plaques is a major cause of acute coronary events. A TCFA has a trombogenic soft lipid core, shielded from the blood stream by a thin, possibly inflamed, stiff cap. The majority of atherosclerotic plaques resemble a TCFA in terms of overall structural composition, but have a more complex, heterogeneous morphology. An assessment of the material distribution is vital for quantifying the plaque's mechanical stability and for determining the effect of plaque-stabilizing pharmaceutical agents. We describe a new automated inverse elasticity method, intravascular ultrasound (IVUS) modulography, which is capable of reconstructing a heterogeneous Young's modulus distribution. The elastogram (i.e., spatial strain distribution) of the plaque is the input for the method, and is measured using the clinically available technique, IVUS elastography. Our method incorporates a novel divide-and-conquer strategy, allowing the reconstruction of TCFAs as well as heterogeneous plaques with localized regions of soft, weakened tissue. The method was applied to ex vivo elastograms, which were simulated from the cross sections of postmortem human coronary plaques. To demonstrate the clinical feasibility of the method, measured elastograms from human atherosclerotic coronary arteries were analyzed. One elastogram was measured in vitro; the other, in vivo. The method approximated the true Young's modulus distribution of all simulated plaques, while the in vitro reconstruction was in agreement with histology. In conclusion, the IVUS modulography in combination with the IVUS elastography has strong potential to become an all-encompassing modality for detecting plaques, for assessing the information related to their rupture-proneness, and for imaging their heterogeneous elastic material composition.

An intelligent interface for freehand strain imaging

We present a new, intelligent interface for freehand strain imaging, which has been designed to support clinical trials investigating the potential of ultrasonic strain imaging for diagnostic purposes across a broad range of target pathologies. The aim with this interface is to make scanning easier and to help clinicians learn the necessary scanning technique quickly, by providing real time feedback indicating the quality of the strain data as they are produced. The methods require a pixel-level indicator of estimation precision, which can be calculated in-line with strain estimation. This is exploited in novel approaches to normalisation, persistence and display. The effect of each component is indicated in the results with examples from in vitro and in vivo scanning. As well as providing real-time feedback, the images are easier to interpret because data at unacceptably low signal-to-noise ratios do not reach the display. Additionally, the level of noise in the displayed images is actually reduced compared with other methods that use the same strain estimates with the same level of persistence. The interface also considerably reduces the difficulty in producing volumes of strain data from freehand three-dimensional scans.

Regularization for ultrasonic measurements of tissue displacement vector and strain tensor

In this report, we present the regularization results of displacement vector measurement by the multidimensional cross-spectrum phase gradient method (MCSPGM), multidimensional autocorrelation method (MAM), and multidimensional Doppler method (MDM) without lateral modulation. In addition to the results of spatially variant regularization in displacement vector measurement using displacement variances, the results of displacement component-dependent regularization are presented. That is, according to the measurement accuracy of displacement components, the regularization is properly applied to the respective displacement components. For instance, only the regularization on the lateral displacement is effective. That is, for the lateral nonmodulation case, the accuracies and stabilities of lateral/elevational displacement measurements significantly increase. In conjunction, the convergence speed of phase matching also increases. The demonstrated measurements of the displacement vector distributions in experiments using inhomogeneous shear modulus agar phantoms confirm that displacement-componentdependent regularization enables better strain tensor measurement and shear modulus reconstruction from the viewpoints of accuracy and stability. The contrast-to-noise ratio (CNR) is useful to set regularization parameters properly for the displacement regularization, whereas the CNR is useless for shear modulus regularization in the sense that the accuracy of reconstruction value cannot be evaluated.

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.

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).

Ultrasonic axial strain measurement for lateral tissue deformation

An axial strain measurement technique and a one-dimensional (1D) shear modulus reconstruction technique were developed previously by using the multidimensional radio-frequency (RF) echo phase matching method and an axial strain ratio, respectively. In this study, these techniques have been applied using a conventional ultrasound (US) imaging system with the constraint that the tissue deforms predominantly in the lateral direction, orthogonal to the ultrasound beam axis. Conventionally, axial strain measurement and axial strain ratio are used when the tissue deforms predominantly in the axial direction by extracorporeally applied pressure or vibration. In any case, an axial strain can be accurately measured using the multidimensional RF-echo phase matching method; such a measurement will be useful and will enable a simple reconstruction in a lateral deformation case. The usefulness of these techniques is demonstrated by agar phantom experiments; several problems related to their use in cases such as accuracy, bias error and detectability of inhomogeneity (contrast-to-noise ratio) are addressed. These techniques will be effective for deep regions-of-interest (ROIs) such as liver tissues, which are inaccessible from the body surface and normally deformed by heart motion or pulsation. Moreover, such techniques will enable the evaluation of the elasticity of various tissues under normal motion such as the arm and leg muscles during exercise.

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.

Spatially variant regularization for tissue strain measurement and shear modulus reconstruction

PURPOSE

Regarding the regularization of the displacement vector measurement and shear modulus reconstruction, we propose to properly set the regularization parameters, i.e., to use spatially variant regularization parameters at each point in the region of interest, because the measurement accuracies of the displacements and strains vary spatially.

METHOD

As the measurement accuracies of the strains can be evaluated using the correlation coefficient when using the cross-spectrum phase gradient method, preliminarily the regularization parameters were set proportional to the reciprocal of a power of the correlation coefficient.

RESULTS AND CONCLUSION

Such a regularization scheme realizes the spatially uniform stabilities of the strain measurement and shear modulus reconstruction. The effectiveness of this method was verified by showing the regularized results of the axial strain measurement and of one-dimensional (1-D) shear modulus reconstruction obtained in vivo from a human liver carcinoma (that was treated by interstitial microwave coagulation therapy) as well as by a 1-D shear modulus reconstruction obtained using an agar phantom.

[STEAM-sequence with multi-echo-readout for static magnetic resonance elastography]

In static magnetic resonance elastography, the elasticity of an object is determined by measuring the internal displacement between two compression states. To reduce signal loss during the long time delay between application of external deformation and the static compression state, a STEAM sequence with a long mixing time is used This results in long scan times. The aim of this work was the development of a STEAM sequence with a multi-echo-readout, which allows the reduction of scan time and number of necessary external deformations. This new sequence was compared to the standard STEAM sequence on an agarose gel phantom with a hard inclusion. In addition, the elasticity of thermal tissue lesions was investigated, which were induced using high-intensity focused ultrasound (HIFU). During a given measurement time, more acquisitions per image can be taken using the multi-echo-readout. As a result the signal-to-noise ratio is increased and errors in the data become clearly smaller. Drawbacks of. the new sequence are its higher signal loss due to T2-decay and its greater sensitivity against ghosting artefacts caused by k-space segmentation. During the investigation of the thermally-induced lesions, a clear contrast in elasticity between normal tissue and the treated region was observed. Taking advantage of the greater accuracy of the new STEAM sequence, it was shown, that this contrast is significantly larger than the one in conventional MR parameters.

A parallelizable real-time motion tracking algorithm with applications to ultrasonic strain imaging

Ultrasound-based mechanical strain imaging systems utilize signals from conventional diagnostic ultrasound systems to image tissue elasticity contrast that provides new diagnostically valuable information. Previous works (Hall et al 2003 Ultrasound Med. Biol. 29 427, Zhu and Hall 2002 Ultrason. Imaging 24 161) demonstrated that uniaxial deformation with minimal elevation motion is preferred for breast strain imaging and real-time strain image feedback to operators is important to accomplish this goal. The work reported here enhances the real-time speckle tracking algorithm with two significant modifications. One fundamental change is that the proposed algorithm is a column-based algorithm (a column is defined by a line of data parallel to the ultrasound beam direction, i.e. an A-line), as opposed to a row-based algorithm (a row is defined by a line of data perpendicular to the ultrasound beam direction). Then, displacement estimates from its adjacent columns provide good guidance for motion tracking in a significantly reduced search region to reduce computational cost. Consequently, the process of displacement estimation can be naturally split into at least two separated tasks, computed in parallel, propagating outward from the center of the region of interest (ROI). The proposed algorithm has been implemented and optimized in a Windows system as a stand-alone ANSI C++ program. Results of preliminary tests, using numerical and tissue-mimicking phantoms, and in vivo tissue data, suggest that high contrast strain images can be consistently obtained with frame rates (10 frames s(-1)) that exceed our previous methods.

Performance analysis of steady-state harmonic elastography

Shear modulus estimation can be confounded by the ill-posed nature of the inverse elasticity problem. In this paper, we report the results of experiments conducted on simulated and gelatin phantoms to investigate the effect of various parameters (i.e., regularization, spatial filtering and the subzone generation process) associated with shear modulus reconstruction on the statistical accuracy (mean squared error), and image quality (i.e., contrast and spatial resolution) of the recovered mechanical properties. The results indicate several interesting observations. Firstly, the intrinsic spatial resolution of magnetic resonance elastography (MRE) is dependent on both regularization and spatial filtering. Secondly, the elastographic contrast-to-noise ratio (CNR(e)) increases with increasing regularization and spatial filtering, but it was not affected by the zoning parameters (i.e., the subzones and the extent of the overlap). Thirdly, the statistical accuracy (MSE) of the recovered property improved with increasing regularization, and spatial filtering weight, but the size of the subdomains and their overlap had no significant effect.

Computational elastography from standard ultrasound image sequences by global trust region optimization

A new approach is proposed to estimate the spatial distribution of shear modulus of tissues in-vivo. An image sequence is acquired using a standard medical ultrasound scanner while varying the force applied to the handle. The elastic properties are then recovered simultaneously with the inter-frame displacement fields using a computational procedure based on finite element modeling and trust region constrained optimization. No assumption about boundary conditions is needed. The optimization procedure is global, taking advantage of all available images. The algorithm was tested on phantom, as well as on real clinical images.

PSF dedicated to estimation of displacement vectors for tissue elasticity imaging with ultrasound

This paper investigates a new approach devoted to displacement vector estimation in ultrasound imaging. The main idea is to adapt the image formation to a given displacement estimation method to increase the precision of the estimation. The displacement is identified as the zero crossing of the phase of the complex cross-correlation between signals extracted from the lateral direction of the ultrasound RF image. For precise displacement estimation, a linearity of the phase slope is needed as well as a high phase slope. Consequently, a particular point spread function (PSF) dedicated to this estimator is designed. This PSF, showing oscillations in the lateral direction, leads to synthesis of lateral RF signals. The estimation is included in a 2-D displacement vector estimation method. The improvement of this approach is evaluated quantitatively by simulation studies. A comparison with a speckle tracking technique is also presented. The lateral oscillations improve both the speckle tracking estimation and our 2-D estimation method. Using our dedicated images, the precision of the estimation is improved by reducing the standard deviation of the lateral displacement error by a factor of 2 for speckle tracking and more than 3 with our method compared to using conventional images. Our method performs 7 times better than speckle tracking. Experimentally, the improvement in the case of a pure lateral translation reaches a factor of 7. Finally, the experimental feasibility of the 2-D displacement vector estimation is demonstrated on data acquired from a Cryogel phantom.

Finite element analysis of tissue deformation with a radiofrequency ablation electrode for strain imaging

Recent studies have shown that radiofrequency (RF) electrode displacement or deformation-based strain imaging can be used as an alternate imaging modality to monitor and to evaluate ablative therapies for liver tumors. This paper describes a biomechanical model used to study RF electrode deformation-based strain imaging, in conjunction with a simulated medical ultrasound linear array transducer. The computer simulations reported here are important steps toward understanding this biomechanical system in vivo, thus providing a basis for improving system design, including the motion tracking algorithm and image guidance for performing RF electrode displacement-strain imaging in vivo.

Elasticity reconstruction for ultrasound elastography using a radial compression: an inverse approach

To reduce the inherent mechanical artifacts in the strain images, many groups have investigated solutions to the inverse problem in elastography. However, in prostate elastography or intravascular elastography where the compression direction is radial, the inverse problem has not been studied thoroughly. In this paper, an iterative approach is proposed to reconstruct tissue elasticity for ultrasound elastography using a radial compression. The method is based upon the stress-strain relations in the polar coordinates. Computer simulations in an intravascular model are performed to illustrate the feasibility of this method in reducing the mechanical artifacts of the strain images. The reconstructed elasticity error and the contrast-transfer efficiency (CTE) as a function of the iteration number show that the inverse approach converges with a few iterations.

Towards an acoustic model-based poroelastic imaging method: II. experimental investigation

Soft biological tissue contains mobile fluid. The volume fraction of this fluid and the ease with which it may be displaced through the tissue could be of diagnostic significance and may also have consequences for the validity with which strain images can be interpreted according to the traditional idealizations of elastography. In a previous paper, under the assumption of frictionless boundary conditions, the spatio-temporal behavior of the strain field inside a compressed cylindrical poroelastic sample was predicted (Berry et al. 2006). In this current paper, experimental evidence is provided to confirm these predictions. Finite element modeling was first used to extend the previous predictions to allow for the existence of contact friction between the sample and the compressor plates. Elastographic techniques were then applied to image the time-evolution of the strain inside cylindrical samples of tofu (a suitable poroelastic material) during sustained unconfined compression. The observed experimental strain behavior was found to be consistent with the theoretical predictions. In particular, every sample studied confirmed that reduced values of radial strain advance with time from the curved cylindrical surface inwards towards the axis of symmetry. Furthermore, by fitting the predictions of an analytical model to a time sequence of strain images, parametric images of two quantities, each related to one or more of three poroelastic material constants were produced. The two parametric images depicted the Poisson's ratio (nu(s)) of the solid matrix and the product of the aggregate modulus (H(A)) of the solid matrix with the permeability (k) of the solid matrix to the pore fluid. The means of the pixel values in these images, nu(s) = 0.088 (standard deviation 0.023) and H(A)k = 1.449 (standard deviation 0.269) x 10(-7) m(2) s(-1), were in agreement with values derived from previously published data for tofu (Righetti et al. 2005). The results provide the first experimental detection of the fluid-flow-induced characteristic diffusion-like behavior of the strain in a compressed poroelastic material and allow parameters related to the above material constants to be determined. We conclude that it may eventually be possible to use strain data to detect and measure characteristics of diffusely distributed mobile fluid in tissue spaces that are too small to be imaged directly.

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

As a differential diagnosis technique for living soft tissues, we are developing ultrasonic-strain-measurement-based shear modulus reconstruction methods. Previously, we reported three-dimensional (3-D) and 2-D reconstruction methods utilizing a typical Poisson's ratio very close to 0.5 (nearly-incompressible). However, because a decrease in the accuracy of the reconstructed value was confirmed to be due to the difference between the original value and the set value, we proposed 3-D and 2-D methods of reconstructing Poisson's ratio as well. Furthermore, we proposed methods of reconstructing density and dealing with dynamic deformation. However, due to tissue incompressibility, the reconstructions of shear modulus, Poisson's ratio, and density became unstable. In this report, to obtain stable, unique reconstructions, we describe a new reconstruction method using mean normal stress approximated by the product of one of Lame's constants X and volume strain epsilon alpha alpha as an unknown. Regularization is simultaneously applied to the respective distributions to decrease the instability of the reconstructions due to measurement errors of the deformation. This method also enables stable, unique reconstructions of shear modulus and density under the condition that the mean normal stress remains unknown. We also verify the effectiveness of this method through 3-D simulations, while showing erroneous artifacts occurring when 2-D and 1-D reconstructions are performed.

Early biomechanical changes in lower extremity vein grafts—distinct temporal phases of remodeling and wall stiffness

BACKGROUND

The geometric and biomechanical changes that contribute to vein graft remodeling are not well established. We sought to measure patterns of adaptation in lower extremity vein grafts and assess their correlation with clinical outcomes.

METHODS

We conducted a prospective, longitudinal study of patients undergoing infrainguinal reconstruction with autogenous conduit. In addition to standard duplex surveillance, lumen diameter (of a defined index segment of the conduit) and pulse wave velocity (PWV) were assessed by ultrasound imaging at surgery and at 1, 3, and 6 months postoperatively. Graft dimensions and wall stiffness were correlated with clinical outcomes.

RESULTS

There were 92 patients and 96 limbs in this study. On average, vein graft lumen diameter increased during the first month of implantation from 0.37 +/- .01 cm to 0.45 +/- 0.02 cm (mean +/- SEM; P = .002), representing a relative change of +21.6% (median +/- 14%; range, -31 to +67%) during this period. Of the entire cohort, 72% of grafts demonstrated appreciable dilation of the index segment during the first month. Index segment lumen diameter did not change appreciably beyond 1 month, with the notable exception of arm vein conduits, which showed continued tendency to dilate. PWV increased during the first 6 months (17.2 +/- 1.2 m/s to 23.2 +/- 2.4 m/s; P = .008), reflecting a nearly 40% increase in conduit stiffness (2.0 +/- .6 Mdynes/cm to 3.3 +/- .8 Mdynes/cm, P = .01). The greatest relative increase (25%) in PWV occurred from months 1 to 3. Loss of primary patency occurred in 24 cases (19 revisions, 5 occlusions), with a mean reintervention time of 7.6 months. Grafts that demonstrated early positive remodeling (lumen dilatation) had a trend of increased primary patency (P = .08, log rank). Among the grafts that failed, a trend was noted toward greater wall stiffness at 1 month, 2.7 vs 1.5 Mdynes (P = .08).

CONCLUSION

Vein graft remodeling appears to involve at least two distinct temporal phases. Outward remodeling of the lumen occurs early, and wall stiffness changes occur in a more delayed fashion. Early outward remodeling may be important for successful vein graft adaptation.

Shear modulus reconstruction in dynamic elastography: time harmonic case

This paper presents a direct inversion approach for reconstructing the elastic shear modulus in soft tissue from dynamic measurements of the interior displacement field during time harmonic excitation. The tissue is assumed to obey the equations of nearly incompressible, linear, isotropic elasto-dynamics in harmonic motion. A finite element discretization of the governing equations is used as a basis, and a procedure is outlined to eliminate the need for boundary conditions in the inverse problem. The hydrostatic stress (pressure) is also reconstructed in the process, and the effect of neglecting this term in the governing equations, which is common practice, is considered. The approach does not require iterations and can be performed on sub-regions of the domain resulting in a computationally efficient method. A sensitivity study is performed to investigate the detectability of abnormal regions of different size and shear modulus contrast from the background. The algorithm is tested on simulated data on a two-dimensional domain, where the data are generated on a very fine mesh to get a near exact solution, then downsampled to a coarser mesh that is similar to the spatial discretization of actual data, and noise is added. Results showing the effect of the hydrostatic stress term and noise are presented. A reconstruction using MR measured experimental data involving a tissue-mimicking phantom is also shown to demonstrate the algorithm.

Towards an acoustic model-based poroelastic imaging method: I. Theoretical foundation

The ultrasonic measurement and imaging of tissue elasticity is currently under wide investigation and development as a clinical tool for the assessment of a broad range of diseases, but little account in this field has yet been taken of the fact that soft tissue is porous and contains mobile fluid. The ability to squeeze fluid out of tissue may have implications for conventional elasticity imaging, and may present opportunities for new investigative tools. When a homogeneous, isotropic, fluid-saturated poroelastic material with a linearly elastic solid phase and incompressible solid and fluid constituents is subjected to stress, the behaviour of the induced internal strain field is influenced by three material constants: the Young's modulus (E(s)) and Poisson's ratio (nu(s)) of the solid matrix and the permeability (k) of the solid matrix to the pore fluid. New analytical expressions were derived and used to model the time-dependent behaviour of the strain field inside simulated homogeneous cylindrical samples of such a poroelastic material undergoing sustained unconfined compression. A model-based reconstruction technique was developed to produce images of parameters related to the poroelastic material constants (E(s), nu(s), k) from a comparison of the measured and predicted time-dependent spatially varying radial strain. Tests of the method using simulated noisy strain data showed that it is capable of producing three unique parametric images: an image of the Poisson's ratio of the solid matrix, an image of the axial strain (which was not time-dependent subsequent to the application of the compression) and an image representing the product of the aggregate modulus E(s)(1-nu(s))/(1+nu(s))(1-2nu(s)) of the solid matrix and the permeability of the solid matrix to the pore fluid. The analytical expressions were further used to numerically validate a finite element model and to clarify previous work on poroelastography.

Enhancing the performance of model-based elastography by incorporating additional a priori information in the modulus image reconstruction process

Model-based elastography is fraught with problems owing to the ill-posed nature of the inverse elasticity problem. To overcome this limitation, we have recently developed a novel inversion scheme that incorporates a priori information concerning the mechanical properties of the underlying tissue structures, and the variance incurred during displacement estimation in the modulus image reconstruction process. The information was procured by employing standard strain imaging methodology, and introduced in the reconstruction process through the generalized Tikhonov approach. In this paper, we report the results of experiments conducted on gelatin phantoms to evaluate the performance of modulus elastograms computed with the generalized Tikhonov (GTK) estimation criterion relative to those computed by employing the un-weighted least-squares estimation criterion, the weighted least-squares estimation criterion and the standard Tikhonov method (i.e., the generalized Tikhonov method with no modulus prior). The results indicate that modulus elastograms computed with the generalized Tikhonov approach had superior elastographic contrast discrimination and contrast recovery. In addition, image reconstruction was more resilient to structural decorrelation noise when additional constraints were imposed on the reconstruction process through the GTK method.

Tomography-based 3-D anisotropic elastography using boundary measurements

While ultrasound- and magnetic resonance-based elastography techniques have proved to be powerful biomedical imaging tools, most approaches assume isotropic material properties. In this paper, a general framework is developed for tomography-based anisotropic elastography. An anatomically well-motivated piece-wise homogeneous model is proposed to represent a class of biological objects consisting of different regions. With established tomography modality, static displacements are measured on the entire external and internal boundaries, and the force distribution is recorded on part of the external surface. A principle is proposed to identify the anisotropic elastic moduli of the constituent regions with the obtained boundary measurements. The reconstruction procedure is optimization-based with minimizing an objective function that measures the difference between the predicted and observed displacements. Analytic gradients of the objective function with respect to the elastic moduli are calculated using an adjoint method, and are utilized to significantly improve the numerical efficiency. Simulations are performed to identify the elastic moduli in a breast phantom consisting of soft tissue and a hard tumor. For isotropic phantom, one set of the boundary measurements enables unique reconstruction results for the tissue and tumor. For anisotropic phantom, however, multiple sets of the measurements corresponding to different deformation modes become necessary.

Comparative evaluation of strain-based and model-based modulus elastography

Elastography based on strain imaging currently endures mechanical artefacts and limited contrast transfer efficiency. Solving the inverse elasticity problem (IEP) should obviate these difficulties; however, this approach to elastography is often fraught with problems because of the ill-posed nature of the IEP. The aim of the present study was to determine how the quality of modulus elastograms computed by solving the IEP compared with those produced using standard strain imaging methodology. Strain-based modulus elastograms (i.e., modulus elastograms computed by simply inverting strain elastograms based on the assumption of stress uniformity) and model-based modulus elastograms (i.e., modulus elastograms computed by solving the IEP) were computed from a common cohort of simulated and gelatin-based phantoms that contained inclusions of varying size and modulus contrast. The ensuing elastograms were evaluated by employing the contrast-to-noise ratio (CNR(e)) and the contrast transfer efficiency (CTE(e)) performance metrics. The results demonstrated that, at a fixed spatial resolution, the CNR(e) of strain-based modulus elastograms was statistically equivalent to those computed by solving the IEP. At low modulus contrast, the CTE(e) of both elastographic imaging approaches was comparable; however, at high modulus, the CTE(e) of model-based modulus elastograms was superior.

Assessment of vulnerable plaque composition by matching the deformation of a parametric plaque model to measured plaque deformation

Intravascular ultrasound (IVUS) elastography visualizes local radial strain of arteries in so-called elastograms to detect rupture-prone plaques. However, due to the unknown arterial stress distribution these elastograms cannot be directly interpreted as a morphology and material composition image. To overcome this limitation we have developed a method that reconstructs a Young's modulus image from an elastogram. This method is especially suited for thin-cap fibroatheromas (TCFAs), i.e., plaques with a media region containing a lipid pool covered by a cap. Reconstruction is done by a minimization algorithm that matches the strain image output, calculated with a parametric finite element model (PFEM) representation of a TCFA, to an elastogram by iteratively updating the PFEM geometry and material parameters. These geometry parameters delineate the TCFA media, lipid pool and cap regions by circles. The material parameter for each region is a Young's modulus, EM, EL, and EC, respectively. The method was successfully tested on computer-simulated TCFAs (n = 2), one defined by circles, the other by tracing TCFA histology, and additionally on a physical phantom (n = 1) having a stiff wall (measured EM = 16.8 kPa) with an eccentric soft region (measured EL = 4.2 kPa). Finally, it was applied on human coronary plaques in vitro (n = 1) and in vivo (n = 1). The corresponding simulated and measured elastograms of these plaques showed radial strain values from 0% up to 2% at a pressure differential of 20, 20, 1, 20, and 1 mmHg respectively. The used/reconstructed Young's moduli [kPa] were for the circular plaque EL = 50/66, EM = 1500/1484, EC = 2000/2047, for the traced plaque EL = 25/1, EM = 1000/1148, EC = 1500/1491, for the phantom EL = 4.2/4 kPa, EM = 16.8/16, for the in vitro plaque EL = n.a./29, EM = n.a./647, EC = n.a./1784 kPa and for the in vivo plaque EL = n.a./2, EM = n.a./188, Ec = n.a./188 kPa.

Evaluation of the adjoint equation based algorithm for elasticity imaging

Recently a new adjoint equation based iterative method was proposed for evaluating the spatial distribution of the elastic modulus of tissue based on the knowledge of its displacement field under a deformation. In this method the original problem was reformulated as a minimization problem, and a gradient-based optimization algorithm was used to solve it. Significant computational savings were realized by utilizing the solution of the adjoint elasticity equations in calculating the gradient. In this paper, we examine the performance of this method with regard to measures which we believe will impact its eventual clinical use. In particular, we evaluate its abilities to (1) resolve geometrically the complex regions of elevated stiffness; (2) to handle noise levels inherent in typical instrumentation; and (3) to generate three-dimensional elasticity images. For our tests we utilize both synthetic and experimental displacement data, and consider both qualitative and quantitative measures of performance. We conclude that the method is robust and accurate, and a good candidate for clinical application because of its computational speed and efficiency.