Evaluation of an iterative reconstruction method for quantitative elastography

Reconstructing 3-D maps of the local viscoelastic properties using a finite-amplitude modulated radiation force

A modulated acoustic radiation force, produced by two confocal tone-burst ultrasound beams of slightly different frequencies (i.e. 2.0 MHz ± Δf/2, where Δf is the difference frequency), can be used to remotely generate modulated low-frequency (Δf ≤ 500 Hz) shear waves in attenuating media. By appropriately selecting the duration of the two beams, the energy of the generated shear waves can be concentrated around the difference frequency (i.e., Δf ± Δf/2). In this manner, neither their amplitude nor their phase information is distorted by frequency-dependent effects, thereby, enabling a more accurate reconstruction of the viscoelastic properties. Assuming a Voigt viscoelastic model, this paper describes the use of a finite-element-method model to simulate three-dimensional (3-D) shear-wave propagation in viscoelastic media containing a spherical inclusion. Nonlinear propagation is assumed for the two ultrasound beams, so that higher harmonics are developed in the force and shear spectrum. Finally, an inverse reconstruction algorithm is used to extract 3-D maps of the local shear modulus and viscosity from the simulated shear-displacement fields based on the fundamental and second-harmonic component. The quality of the reconstructed maps is evaluated using the contrast between the inclusion and the background and the contrast-to-noise ratio (CNR). It is shown that the shear modulus can be accurately reconstructed based on the fundamental component, such that the observed contrast deviates from the true contrast by a root-mean-square-error (RMSE) of only 0.38 and the CNR is greater than 30 dB. If the second-harmonic component is used, the RMSE becomes 1.54 and the corresponding CNR decreases by approximately 10-15 dB. The reconstructed shear viscosity maps based on the second harmonic are shown to be of higher quality than those based on the fundamental. The effects of noise are also investigated and a fusion operation between the two spectral components is applied to enhance the reconstruction quality. Finally, a modified shear-wave spectroscopy technique, shown to be more robust to noise, is described for the estimation of the viscoelastic properties inside and outside the spherical inclusion under conditions of increased noise.

Approaches to accommodate noisy data in the direct solution of inverse problems in incompressible plane-strain elasticity

We apply the adjoint weighted equation method (AWE) to the direct solution of inverse problems of incompressible plane strain elasticity. We show that based on untreated noisy displacements, the reconstruction of the shear modulus can be very poor. We link this poor performance to loss of coercivity of the weak form when treating problems with discontinuous coefficients. We demonstrate that by smoothing the displacements and appending a regularization term to the AWE formulation, a dramatic improvement in the reconstruction can be achieved. With these improvements, the advantages of the AWE method as a direct solution approach can be extended to a wider range of problems.

A direct vulnerable atherosclerotic plaque elasticity reconstruction method based on an original material-finite element formulation: theoretical framework

The peak cap stress (PCS) amplitude is recognized as a biomechanical predictor of vulnerable plaque (VP) rupture. However, quantifying PCS in vivo remains a challenge since the stress depends on the plaque mechanical properties. In response, an iterative material finite element (FE) elasticity reconstruction method using strain measurements has been implemented for the solution of these inverse problems. Although this approach could resolve the mechanical characterization of VPs, it suffers from major limitations since (i) it is not adapted to characterize VPs exhibiting high material discontinuities between inclusions, and (ii) does not permit real time elasticity reconstruction for clinical use. The present theoretical study was therefore designed to develop a direct material-FE algorithm for elasticity reconstruction problems which accounts for material heterogeneities. We originally modified and adapted the extended FE method (Xfem), used mainly in crack analysis, to model material heterogeneities. This new algorithm was successfully applied to six coronary lesions of patients imaged in vivo with intravascular ultrasound. The results demonstrated that the mean relative absolute errors of the reconstructed Young's moduli obtained for the arterial wall, fibrosis, necrotic core, and calcified regions of the VPs decreased from 95.3 ± 15.56%, 98.85 ± 72.42%, 103.29 ± 111.86% and 95.3 ± 10.49%, respectively, to values smaller than 2.6 × 10(-8) ± 5.7 × 10(-8)% (i.e. close to the exact solutions) when including modified-Xfem method into our direct elasticity reconstruction method.

Curl-Based Finite Element Reconstruction of the Shear Modulus Without Assuming Local Homogeneity: Time Harmonic Case

In elasticity imaging, the shear modulus is obtained from measured tissue displacement data by solving an inverse problem based on the wave equation describing the tissue motion. In most inversion approaches, the wave equation is simplified using local homogeneity and incompressibility assumptions. This causes a loss of accuracy and therefore imaging artifacts in the resulting elasticity images. In this paper we present a new curl-based finite element method inversion technique that does not rely upon these simplifying assumptions. As done in previous research, we use the curl operator to eliminate the dilatational term in the wave equation, but we do not make the assumption of local homogeneity. We evaluate our approach using simulation data from a virtual tissue phantom assuming time harmonic motion and linear, isotropic, elastic behavior of the tissue. We show that our reconstruction results are superior to those obtained using previous curl-based methods with homogeneity assumption. We also show that with our approach, in the 2-D case, multi-frequency measurements provide better results than single-frequency measurements. Experimental results from magnetic resonance elastography of a CIRS elastography phantom confirm our simulation results and further demonstrate, in a quantitative and repeatable manner, that our method is accurate and robust.

An H∞ strategy for strain estimation in ultrasound elastography using biomechanical modeling constraint

The purpose of ultrasound elastography is to identify lesions by reconstructing the hardness characteristics of tissue reconstructed from ultrasound data. Conventional quasi-static ultrasound elastography is easily applied to obtain axial strain components along the compression direction, with the results inverted to represent the distribution of tissue hardness under the assumption of constant internal stresses. However, previous works of quasi-static ultrasound elastography have found it difficult to obtain the lateral and shear strain components, due to the poor lateral resolution of conventional ultrasound probes. The physical nature of the strain field is a continuous vector field, which should be fully described by the axial, lateral, and shear strain components, and the clinical value of lateral and shear strain components of deformed tissue is gradually being recognized by both engineers and clinicians. Therefore, a biomechanical-model-constrained filtering framework is proposed here for recovering a full displacement field at a high spatial resolution from the noisy ultrasound data. In our implementation, after the biomechanical model constraint is integrated into the state-space equation, both the axial and lateral displacement components can be recovered at a high spatial resolution from the noisy displacement measurements using a robust H∞ filter, which only requires knowledge of the worst-case noise levels in the measurements. All of the strain components can then be calculated by applying a gradient operator to the recovered displacement field. Numerical experiments on synthetic data demonstrated the robustness and effectiveness of our approach, and experiments on phantom data and in-vivo clinical data also produced satisfying results.

Reconstruction of elasticity: a stochastic model-based approach in ultrasound elastography

Background

The convectional strain-based algorithm has been widely utilized in clinical practice. It can only provide the information of relative information of tissue stiffness. However, the exact information of tissue stiffness should be valuable for clinical diagnosis and treatment.

Methods

In this study we propose a reconstruction strategy to recover the mechanical properties of the tissue. After the discrepancies between the biomechanical model and data are modeled as the process noise, and the biomechanical model constraint is transformed into a state space representation the reconstruction of elasticity can be accomplished through one filtering identification process, which is to recursively estimate the material properties and kinematic functions from ultrasound data according to the minimum mean square error (MMSE) criteria. In the implementation of this model-based algorithm, the linear isotropic elasticity is adopted as the biomechanical constraint. The estimation of kinematic functions (i.e., the full displacement and velocity field), and the distribution of Young’s modulus are computed simultaneously through an extended Kalman filter (EKF).

Results

In the following experiments the accuracy and robustness of this filtering framework is first evaluated on synthetic data in controlled conditions, and the performance of this framework is then evaluated in the real data collected from elastography phantom and patients using the ultrasound system. Quantitative analysis verifies that strain fields estimated by our filtering strategy are more closer to the ground truth. The distribution of Young’s modulus is also well estimated. Further, the effects of measurement noise and process noise have been investigated as well.

Conclusions

The advantage of this model-based algorithm over the conventional strain-based algorithm is its potential of providing the distribution of elasticity under a proper biomechanical model constraint. We address the model-data discrepancy and measurement noise by introducing process noise and measurement noise in our framework, and then the absolute values of Young’s modulus are estimated through the EFK in the MMSE sense. However, the initial conditions, and the mesh strategy will affect the performance, i.e., the convergence rate, and computational cost, etc.

The ultrasound elastography inverse problem and the effective criteria

The elastography (elasticity imaging) is one of the recent state-of-the-art methods for diagnosis of abnormalities in soft tissue. The idea is based on the computation of the tissue elasticity distribution. This leads to the inverse elasticity problem; in that, displacement field and boundary conditions are known, and elasticity distribution of the tissue is aimed for computation. We treat this problem by the Gauss-Newton method. This iterative method results in an ill-posed problem, and therefore, regularization schemes are required to deal with this issue. The impacts of the initial guess for tissue elasticity distribution, contrast ratio between elastic modulus of tumor and normal tissue, and noise level of the input data on the estimated solutions are investigated via two different regularization methods. The numerical results show that the accuracy and speed of convergence vary when different regularization methods are applied. Also, the semi-convergence behavior has been observed and discussed. At the end, we signify the necessity of a clever initial guess and intelligent stopping criteria for the iterations. The main purpose here is to highlight some technical factors that have an influence on elasticity image quality and diagnostic accuracy, and we have tried our best to make this article accessible for a broad audience.

Large Scale Parameter Estimation Problems in Frequency-Domain Elastodynamics Using an Error in Constitutive Equation Functional

This paper presents the formulation and implementation of an Error in Constitutive Equations (ECE) method suitable for large-scale inverse identification of linear elastic material properties in the context of steady-state elastodynamics. In ECE-based methods, the inverse problem is postulated as an optimization problem in which the cost functional measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses. Furthermore, in a more recent modality of this methodology introduced by Feissel and Allix (2007), referred to as the Modified ECE (MECE), the measured data is incorporated into the formulation as a quadratic penalty term. We show that a simple and efficient continuation scheme for the penalty term, suggested by the theory of quadratic penalty methods, can significantly accelerate the convergence of the MECE algorithm. Furthermore, a (block) successive over-relaxation (SOR) technique is introduced, enabling the use of existing parallel finite element codes with minimal modification to solve the coupled system of equations that arises from the optimality conditions in MECE methods. Our numerical results demonstrate that the proposed methodology can successfully reconstruct the spatial distribution of elastic material parameters from partial and noisy measurements in as few as ten iterations in a 2D example and fifty in a 3D example. We show (through numerical experiments) that the proposed continuation scheme can improve the rate of convergence of MECE methods by at least an order of magnitude versus the alternative of using a fixed penalty parameter. Furthermore, the proposed block SOR strategy coupled with existing parallel solvers produces a computationally efficient MECE method that can be used for large scale materials identification problems, as demonstrated on a 3D example involving about 400,000 unknown moduli. Finally, our numerical results suggest that the proposed MECE approach can be significantly faster than the conventional approach of L(2) minimization using quasi-Newton methods.

A regularization-free Young's modulus reconstruction algorithm for ultrasound elasticity imaging

Ultrasound elasticity imaging aims to reconstruct the distribution of elastic modulus (e.g., Young's modulus) within biological tissues, since the value of elastic modulus is often related to pathological changes. Currently, most elasticity imaging algorithms face a challenge of choosing the value of the regularization constant. We propose a more applicable algorithm without the need of any regularization. This algorithm is not only simple to use, but has a relatively high accuracy. Our method comprises of a nonrigid registration technique and tissue incompressibility assumption to estimate the two-dimensional (2D) displacement field, and finite element method (FEM) to reconstruct the Young's modulus distribution. Simulation and phantom experiments are performed to evaluate the algorithm. Simulation and phantom results showed that the proposed algorithm can reconstruct the Young's modulus with an accuracy of 63∼85%.

A new nonlinear parameter in the developed strain-to-applied strain of the soft tissues and its application in ultrasound elasticity imaging

Strain developed under quasi-static deformation has been mostly used in ultrasound elasticity imaging (UEI) to determine the stiffness change of tissues. However, the strain measure in UEI is often less sensitive to a subtle change of stiffness. This is particularly true for Crohn's disease where we have applied strain imaging to the differentiation of acutely inflamed bowel from chronically fibrotic bowel. In this study, a new nonlinear elastic parameter of the soft tissues is proposed to overcome this limit. The purpose of this study is to evaluate the newly proposed method and demonstrate its feasibility in the UEI. A nonlinear characteristic of soft tissues over a relatively large dynamic range of strain was investigated. A simplified tissue model based on a finite element (FE) analysis was integrated with a laboratory developed ultrasound radio-frequency (RF) signal synthesis program. Two-dimensional speckle tracking was applied to this model to simulate the nonlinear behavior of the strain developed in a target inclusion over the applied average strain to the surrounding tissues. A nonlinear empirical equation was formulated and optimized to best match the developed strain-to-applied strain relation obtained from the FE simulation. The proposed nonlinear equation was applied to in vivo measurements and nonlinear parameters were further empirically optimized. For an animal model, acute and chronic inflammatory bowel disease was induced in Lewis rats with trinitrobenzene sulfonic acid (TNBS)-ethanol treatments. After UEI, histopathology and direct mechanical measurements were performed on the excised tissues. The extracted nonlinear parameter from the developed strain-to-applied strain relation differentiated the three different tissue types with 1.96 ± 0.12 for normal, 1.50 ± 0.09 for the acutely inflamed and 1.03 ± 0.08 for the chronically fibrotic tissue. T-tests determined that the nonlinear parameters between normal, acutely inflamed and fibrotic tissue types were statistically significantly different (normal/ fibrotic [p = 0.0000185], normal/acutely inflamed [p = 0.0013] and fibrotic/acutely inflamed [p = 0.0029]). This technique may provide a sensitive and robust tool to assess subtle stiffness changes in tissues such as in acutely inflamed bowel wall.

Model-based elastography: a survey of approaches to the inverse elasticity problem

Elastography is emerging as an imaging modality that can distinguish normal versus diseased tissues via their biomechanical properties. This paper reviews current approaches to elastography in three areas--quasi-static, harmonic and transient--and describes inversion schemes for each elastographic imaging approach. Approaches include first-order approximation methods; direct and iterative inversion schemes for linear elastic; isotropic materials and advanced reconstruction methods for recovering parameters that characterize complex mechanical behavior. The paper's objective is to document efforts to develop elastography within the framework of solving an inverse problem, so that elastography may provide reliable estimates of shear modulus and other mechanical parameters. We discuss issues that must be addressed if model-based elastography is to become the prevailing approach to quasi-static, harmonic and transient elastography: (1) developing practical techniques to transform the ill-posed problem with a well-posed one; (2) devising better forward models to capture the complex mechanical behavior of soft tissues and (3) developing better test procedures to evaluate the performance of modulus elastograms.

An H(∞) approach for elasticity properties reconstruction

PURPOSE

Quantification of object elasticity properties has significant technical implications as well as important practical applications, such as medical disease diagnosis. In general, given noisy measurements on the kinematic states of the objects from imaging data, the aim is to recover the elasticity parameters for assumed material constitutive models of the objects. The implementation is complicated caused by the large dimensionality of the parameters.

METHODS

Various versions of the least-square (LS) methods have been widely used, which, however, do not perform well under reasonably realistic levels of disturbances. Another popular strategy, based on the extended Kalman filter (EKF), is also far from optimal and subject to divergence if either the initializations are poor or the noises are not Gaussian. In this paper, the authors propose a robust system identification paradigm for the quantitative analysis of object elasticity. It is derived and extended from the H(∞) filtering principles and is particularly powerful for real-world situations where the types and levels of the disturbances are unknown.

RESULTS

Using synthetic data, the authors investigate the sensitivity of the strategies toward different types (Gaussian and Poisson) and levels of noises, as well as various initializations. The experimental results show consistently superior performance of the proposed method over the LS and EKF algorithms in reliably identifying object elastic modulus distributions.

CONCLUSIONS

Results from phase contrast imaging data of canine hearts and human MRI data are also presented, which demonstrate the power of the framework.

Investigating the impact of spatial priors on the performance of model-based IVUS elastography

This paper describes methods that provide pre-requisite information for computing circumferential stress in modulus elastograms recovered from vascular tissue-information that could help cardiologists detect life-threatening plaques and predict their propensity to rupture. The modulus recovery process is an ill-posed problem; therefore, additional information is needed to provide useful elastograms. In this work, prior geometrical information was used to impose hard or soft constraints on the reconstruction process. We conducted simulation and phantom studies to evaluate and compare modulus elastograms computed with soft and hard constraints versus those computed without any prior information. The results revealed that (1) the contrast-to-noise ratio of modulus elastograms achieved using the soft prior and hard prior reconstruction methods exceeded those computed without any prior information; (2) the soft prior and hard prior reconstruction methods could tolerate up to 8% measurement noise, and (3) the performance of soft and hard prior modulus elastograms degraded when incomplete spatial priors were employed. This work demonstrates that including spatial priors in the reconstruction process should improve the performance of model-based elastography, and the soft prior approach should enhance the robustness of the reconstruction process to errors in the geometrical information.

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

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

Automatic generation of boundary conditions using demons nonrigid image registration for use in 3-D modality-independent elastography

Modality-independent elastography (MIE) is a method of elastography that reconstructs the elastic properties of tissue using images acquired under different loading conditions and a biomechanical model. Boundary conditions are a critical input to the algorithm and are often determined by time-consuming point correspondence methods requiring manual user input. This study presents a novel method of automatically generating boundary conditions by nonrigidly registering two image sets with a demons diffusion-based registration algorithm. The use of this method was successfully performed in silico using magnetic resonance and X-ray-computed tomography image data with known boundary conditions. These preliminary results produced boundary conditions with an accuracy of up to 80% compared to the known conditions. Demons-based boundary conditions were utilized within a 3-D MIE reconstruction to determine an elasticity contrast ratio between tumor and normal tissue. Two phantom experiments were then conducted to further test the accuracy of the demons boundary conditions and the MIE reconstruction arising from the use of these conditions. Preliminary results show a reasonable characterization of the material properties on this first attempt and a significant improvement in the automation level and viability of the method.

Endoscopic ultrasound elastography

Sonoelastography is based on the knowledge that some diseases, such as cancer, lead to a change in tissue hardness. Elastography examines the elastic properties of tissues by applying a slight compression to the tissue and comparing the images obtained before and after this compression. Endoscopic ultrasonography (EUS) is today the best technique to diagnose a small pancreatic mass and to determine the histology of such lesions. However, the accuracy of EUS-FNA is around 85-90%. In this study, elastography was used to differentiate benign from malignant pancreatic masses. The bright future of the second generation of elastography, the quantitative elastography or ratio elastography, is also discussed.

Alignment and calibration of dual ultrasound transducers using a wedge phantom

We present a novel method of aligning two orthogonal ultrasound transducers into a coincident scan plane. A wedge phantom design provides visual feedback to the user to facilitate alignment. Calibration provides the transformation from one transducer to the other as well as a measure of the residual error in alignment. Mean alignment error is shown to be under 1° in the rotation axes and 1 mm in translation after repeated manual alignments. The repeatability of wedge based calibration has similar results compared with N-fiducial based calibration. The accuracy of the calibration for mapping points from one transducer to the other is found to have a mean error of 1.6 mm. The dual transducer system is well suited to imaging anatomy such as the breast and may be used for spatial compounding for improving B-mode images and motion estimation compounding for improving elastography results.

Generalized anisotropic inverse mechanics for soft tissues

Elastography, which is the imaging of soft tissues on the basis of elastic modulus (or, more generally, stiffness) has become increasingly popular in the last decades and holds promise for application in many medical areas. Most of the attention has focused on inhomogeneous materials that are locally isotropic, the intent being to detect a (stiff) tumor within a (compliant) tissue. Many tissues of mechanical interest, however, are anisotropic, so a method capable of determining material anisotropy would be attractive. We present here an approach to determine the mechanical anisotropy of inhomogeneous, anisotropic tissues, by directly solving the finite element representation of the Cauchy stress balance in the tissue. The method divides the sample domain into subdomains assumed to have uniform properties and solves for the material constants in each subdomain. Two-dimensional simulated experiments on linear anisotropic inhomogeneous systems demonstrate the ability of the method, and simulated experiments on a nonlinear model demonstrate the ability of the method to capture anisotropy qualitatively even though only a linear model is used in the inverse problem. As with any inverse problem, ill-posedness is a serious concern, and multiple tests may need to be done on the same sample to determine the properties with confidence.

The performance of steady-state harmonic magnetic resonance elastography when applied to viscoelastic materials

PURPOSE

The clinical efficacy of breast elastography may be limited when the authors employ the assumption that soft tissues exhibit linear, frequency-independent isotropic mechanical properties during the recovery of shear modulus. Thus, the purpose of this research was to evaluate the degradation in performance incurred when linear-elastic MR reconstruction methods are applied to phantoms that are fabricated using viscoelastic materials.

METHODS

To develop phantoms with frequency-dependent mechanical properties, the authors measured the complex modulus of two groups of cylindrical-shaped gelatin samples over a wide frequency range (up to 1 kHz) with the established principles of time-temperature superposition (TTS). In one group of samples, the authors added varying amounts of agar (1%-4%); in the other group, the authors added varying amounts of sucrose (2.5%-20%). To study how viscosity affected the performance of the linear-elastic reconstruction method, the authors constructed an elastically heterogeneous MR phantom to simulate the case where small viscoelastic lesions were surrounded by relatively nonviscous breast tissue. The breast phantom contained four linear, viscoelastic spherical inclusions (10 mm diameter) that were embedded in normal gelatin. The authors imaged the breast phantom with a clinical prototype of a MRE system and recovered the shear-modulus distribution using the overlapping-subzone-linear-elastic image-reconstruction method. The authors compared the recovered shear modulus to that measured using the TTS method.

RESULTS

The authors demonstrated that viscoelastic phantoms could be fabricated by including sucrose in the gelation process and that small viscoelastic inclusions were visible in MR elastograms recovered using a linear-elastic MR reconstruction process; however, artifacts that degraded contrast and spatial resolution were more prominent in highly viscoelastic inclusions. The authors also established that the accuracy of the MR elastograms depended on the degree of viscosity that the inclusion exhibited.

CONCLUSIONS

The results demonstrated that reconstructing shear modulus from other constitutive laws, such as viscosity, should improve both the accuracy and quality of MR elastograms of the breast.

Ultrasound echo is related to stress and strain in tendon

The mechanical behavior of tendons has been well studied in vitro. A noninvasive method to acquire mechanical data would be highly beneficial. Elastography has been a promising method of gathering in vivo tissue mechanical behavior, but it has inherent limitations. This study presents acoustoelasticity as an alternative ultrasound-based method of measuring tendon stress and strain by reporting a relationship between ultrasonic echo intensity (B-mode ultrasound image brightness) and mechanical behavior of tendon in vitro. Porcine digital flexor tendons were cyclically loaded in a mechanical testing system while an ultrasonic echo response was recorded. We report that echo intensity closely follows the applied cyclic strain pattern in time with higher strain protocols resulting in larger echo intensity changes. We also report that echo intensity is related nonlinearly to stress and nearly linearly to strain. This indicates that ultrasonic echo intensity is related to the mechanical behavior in a loaded tissue by an acoustoelastic response, as previously described in homogeneous, nearly incompressible materials. Acoustoelasticity is therefore able to relate strain-dependent stiffness and stress to the reflected echo, even in the processed B-mode signals reflected from viscoelastic and inhomogeneous material such as tendon, and is a promising metric to acquire in vivo mechanical data noninvasively.

Contrast detection in fluid-saturated media with magnetic resonance poroelastography

PURPOSE

Recent interest in the poroelastic behavior of tissues has led to the development of magnetic resonance poroelastography (MRPE) as an alternative to single-phase MR elastographic image reconstruction. In addition to the elastic parameters (i.e., Lamé's constants) commonly associated with magnetic resonance elastography (MRE), MRPE enables estimation of the time-harmonic pore-pressure field induced by external mechanical vibration.

METHODS

This study presents numerical simulations that demonstrate the sensitivity of the computed displacement and pore-pressure fields to a priori estimates of the experimentally derived model parameters. In addition, experimental data collected in three poroelastic phantoms are used to assess the quantitative accuracy of MR poroelastographic imaging through comparisons with both quasistatic and dynamic mechanical tests.

RESULTS

The results indicate hydraulic conductivity to be the dominant parameter influencing the deformation behavior of poroelastic media under conditions applied during MRE. MRPE estimation of the matrix shear modulus was bracketed by the values determined from independent quasistatic and dynamic mechanical measurements as expected, whereas the contrast ratios for embedded inclusions were quantitatively similar (10%-15% difference between the reconstructed images and the mechanical tests).

CONCLUSIONS

The findings suggest that the addition of hydraulic conductivity and a viscoelastic solid component as parameters in the reconstruction may be warranted.

A motion estimation refinement framework for real-time tissue axial strain estimation with freehand ultrasound

Ultrasound elastography has become a wellknown optional imaging method for the diagnosis of tissue abnormalities in various body parts. It images the elasticity of compliant tissues by estimating the local displacements and strains using pre- and post-compression RF echo signals. In this paper, taking the RF signal as image intensity and RF samples as pixels, we present a motion estimation framework to compute the axial tissue displacements and strains. This method takes advantage of both the block matching algorithm (BMA) and local optical flow techniques. For two frames of RF signals, coarse motion estimates are first computed using BMA. The motion estimates obtained are then used to warp the first frame toward the second one, thus making the warped frame more spatially correlated to the second one. Next, the Lucas-Kanade optical flow method is employed to compute the residual motion between the warped frame and the original second frame, with inherent sub-pixel precision. Finally, the displacements from the two steps are combined. The warp-and-refine procedure can be iterated if the residual motion is larger than a predefined empirical threshold. To test its feasibility, we first applied the method to simulated data. The results show that our method is robust to relatively large motions and is capable of generating accurate motion estimation with subsample spatial resolution. These methods have been deployed and are being tested on a commercialized ultrasound machine that previously did not have elastography functions. Quality real-time display of elastography along with freehand scanning has been accomplished. The proposed framework provides an alternative method for motion estimation with good performance, and it can potentially be improved using hardware to realize the BMA.

The place of endoscopic ultrasound in bilio-pancreatic pathology

The place of endoscopic ultrasound (EUS) in malignant pathology of the pancreas is two-fold: (1) EUS is the best examination for the diagnosis of small tumours (<3cm in diameter). Its sensitivity is greater than that of CT scan, percutaneous ultrasound or magnetic resonance imaging (MRI) and is equal to that of endoscopic retrograde cholangiopancreatography (ERCP) without sharing its invasive character; (2) EUS is also indicated in the assessment of locoregional extension of tumours judged resectable by tomodensitometric (TDM) (scanner) data. The performance of EUS seems to be greater than other imaging techniques for the diagnosis of vascular and lymph node invasion although recent studies report less good results than those of studies in 1992 to 1994, particularly for vascular involvement. Nevertheless, EUS cannot affirm the malignant or benign character of these pancreatic masses. The development over the last 20 years of linear sector-based EUS has enabled us to perform guided biopsies of such lesions. EUS-guided biopsy is today the best technique for obtaining the histology of a pancreatic mass, with a sensitivity of 85 to 87%. Furthermore, it also has a non-negligible impact on the deciding the treatment particularly in the case of adenocarcinomas (ADKP) not visible to TDM (scanners). This is currently of importance because trials are being developed of preoperative radio-chemotherapy for resectable lesions. probably in the next future, contrast-enhanced EUS (CE-EUS) and elastography will improve the results of EUS and will be necessary for a precise local staging before treatment.

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.

Magnetic resonance poroelastography: an algorithm for estimating the mechanical properties of fluid-saturated soft tissues

Magnetic resonance poroelastography (MRPE) is introduced as an alternative to single-phase model-based elastographic reconstruction methods. A 3-D finite element poroelastic inversion algorithm was developed to recover the mechanical properties of fluid-saturated tissues. The performance of this algorithm was assessed through a variety of numerical experiments, using synthetic data to probe its stability and sensitivity to the relevant model parameters. Preliminary results suggest the algorithm is robust in the presence of noise and capable of producing accurate assessments of the underlying mechanical properties in simulated phantoms. Furthermore, a 3-D time-harmonic motion field was recorded for a poroelastic phantom containing a single cylindrical inclusion and used to assess the feasibility of MRPE image reconstruction from experimental data. The elastograms obtained from the proposed poroelastic algorithm demonstrate significant improvement over linearly elastic MRE images generated using the same data. In addition, MRPE offers the opportunity to estimate the time-harmonic pressure field resulting from tissue excitation, highlighting the potential for its application in the diagnosis and monitoring of disease processes associated with changes in interstitial pressure.

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.

Calculating tissue shear modulus and pressure by 2D Log-Elastographic methods

Shear modulus imaging, often called elastography, enables detection and characterization of tissue abnormalities. In this paper the data is two displacement components obtained from successive MR or ultrasound data sets acquired while the tissue is excited mechanically. A 2D plane strain elastic model is assumed to govern the 2D displacement, u. The shear modulus, μ, is unknown and whether or not the first Lamé parameter, λ, is known the pressure p = λ∇ · u which is present in the plane strain model cannot be measured and is unreliably computed from measured data and can be shown to be an order one quantity in the units kPa. So here we present a 2D Log-Elastographic inverse algorithm that: (1) simultaneously reconstructs the shear modulus, μ, and p, which together satisfy a first order partial differential equation system, with the goal of imaging μ; (2) controls potential exponential growth in the numerical error; and (3) reliably reconstructs the quantity p in the inverse algorithm as compared to the same quantity computed with a forward algorithm. This work generalizes the Log-Elastographic algorithm in [20] which uses one displacement component, is derived assuming the component satisfies the wave equation, and is tested on synthetic data computed with the wave equation model. The 2D Log-Elastographic algorithm is tested on 2D synthetic data and 2Din-vivo data from Mayo Clinic. We also exhibit examples to show that the 2D Log-Elastographic algorithm improves the quality of the recovered images as compared to the Log-Elastographic and Direct Inversion algorithms.

Contrast-enhanced endoscopic ultrasound and elastosonoendoscopy

Until recently, there was no contrast harmonic imaging technique available for EUS examination. Second-generation US contrast agents produce harmonic signals at lower acoustic powers and, therefore, are suitable for EUS imaging at low acoustic powers. CE-EUS could provide a contribution to the differential diagnosis between a primary pancreatic carcinoma, chronic pancreatitis and a pancreatic metastasis, and therefore can have a decisive influence on the selection of appropriate therapeutic strategies (follow-up, chemotherapy or surgery, for example). However, histology remains the standard in the differential diagnosis of pancreatic tumours. Regarding lymph nodes, CE-EUS cannot replace EUS-guided fine-needle aspiration. Elastography examines the elastic properties of tissues by applying a slight compression to the tissue and comparing an image obtained before and after this compression. EUS elastography is a new application in the field of endosonography and seems to be able to differentiate fibrous and benign tissue from malignant lesions. While our results are very encouraging, further research will be needed to further define the place of this new technique and should be aimed at further defining criteria for accurate elastography as well as subsequently assessing the technique using multiple operators in a blinded setting. EUS-guided sonoelastography has the potential for further guiding the diagnosis and therapy of gastrointestinal-related tumours.

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.

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.

Linear and nonlinear elasticity imaging of soft tissue in vivo: demonstration of feasibility

We establish the feasibility of imaging the linear and nonlinear elastic properties of soft tissue using ultrasound. We report results for breast tissue where it is conjectured that these properties may be used to discern malignant tumors from benign tumors. We consider and compare three different quantities that describe nonlinear behavior, including the variation of strain distribution with overall strain, the variation of the secant modulus with overall applied strain and finally the distribution of the nonlinear parameter in a fully nonlinear hyperelastic model of the breast tissue.

Identification of material properties of orthotropic elastic cylinders immersed in fluid using vibroacoustic techniques

A numerical study is presented to show the potential for using vibroacoustic-based experiments to identify elastic material properties of orthotropic cylindrical vessels immersed in fluids. Sensitivity analyses and a simulated inverse problem are shown to quantify the potential for material characterization through the use of acoustic emissions. For comparison purposes, the analyses are also shown with the normal component of the velocity at the surface of the cylinder as the measured response in place of the acoustic pressure. The simulated experiment consisted of an orthotropic cylinder immersed in water with an impact force applied to the surface of the cylinder. The material parameters of the cylinder considered in the analyses were the circumferential and longitudinal elastic moduli, and the in-plane shear modulus. The velocity response is shown to provide sufficient information for characterizing all three moduli from a single experiment. Alternatively, the acoustic pressure response is shown to provide sufficient information for characterizing only the two elastic moduli from a single experiment. The analyses show that the acoustic pressure response does not have sufficient sensitivity to the in-plane shear modulus for characterization purposes.

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.

Noninvasive Young's modulus evaluation of tissues surrounding pulsatile vessels using ultrasound Doppler measurement

This paper presents an indirect approach to estimating the mechanical properties of tissues surrounding the arterial vessels using ultrasound (US) Doppler measurements combined with an inverse problem-solving method. The geometry of the structure and the dynamic behavior of the inner fluid are first evaluated using a novel dual-beam US system. A numerical phantom associated with a parametric finite element simulator that calculates the hydrodynamic pressure and the displacement on the walls' boundaries is then built. The simulation results are iteratively compared to the US measurement results to deduce the value of the unknown parameters, i.e., the Young's modulus and the pressure resulting from the downstream load. The feasibility of the proposed approach was experimentally tested in vitro using a phantom composed of a latex tube surrounded by a cryogel tissue-mimicking material.

Transient elastography using impulsive ultrasound radiation force: a preliminary comparison with surface palpation elastography

The use of impulsive acoustic radiation force for transient strain imaging was investigated and compared with conventional elastography. A series of experiments were performed to evaluate the performances of the technique on gelatine phantoms containing inclusions and to determine a range of applications where radiation force elastography may be useful compared with static elastography. Slip boundaries and cylindrical inclusions of varying elastic modulus were placed in background materials. A focused ultrasound transducer was used to apply localised radiation force to a small volume of tissue mimic (100 mm3) for durations of 8 ms. A conventional real-time ultrasound imaging probe was used to obtain radio- frequency echo signals. The resulting strains were mapped using ultrasound correlation-based methods. The instantaneous strain immediately following cessation of the radiation force was observed at depth within homogeneous gels and within stiff inclusions. The highly localised and transient strain that is produced at depth permits the sensing of variations in tissue elastic properties that are difficult to detect with conventional elastography, due to greater independence from boundary conditions. In particular, radiation force elastograms were more homogeneous in the background and within the inclusions and displayed a superior contrast-transfer-efficiency, particularly for regions that had negative modulus contrast or that were disconnected from the background or the anterior medium by a low friction boundary.

A combined FEM/genetic algorithm for vascular soft tissue elasticity estimation

Tissue elasticity reconstruction is a parameter estimation effort combining imaging, elastography, and computational modeling to build maps of soft tissue mechanical properties. One application is in the characterization of atherosclerotic plaques in diseased arteries, wherein the distribution of elastic properties is required for stress analysis and plaque stability assessment. In this paper, a computational scheme is proposed for elasticity reconstruction in soft tissues, combining finite element modeling (FEM) for mechanical analysis of soft tissues and a genetic algorithm (GA) for parameter estimation. With a model reduction of the discrete elasticity values into lumped material regions, namely the plaque constituents, a robust, adaptive strategy can be used to solve inverse elasticity problems involving complex and inhomogeneous solution spaces. An advantage of utilizing a GA is its insistence on global convergence. The algorithm is easily implemented and adaptable to more complex material models and geometries. It is meant to provide either accurate initial guesses of low-resolution elasticity values in a multi-resolution scheme or as a replacement for failing traditional elasticity estimation efforts.

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.

Elastic modulus imaging: some exact solutions of the compressible elastography inverse problem

We consider several inverse problems motivated by elastography. Given the (possibly transient) displacement field measured everywhere in an isotropic, compressible, linear elastic solid, and given density rho, determine the Lamé parameters lambda and mu. We consider several special cases of this problem: (a) for mu known a priori, lambda is determined by a single deformation field up to a constant. (b) Conversely, for lambda known a priori, mu is determined by a single deformation field up to a constant. This includes as a special case that for which the term [see text]. (c) Finally, if neither lambda nor mu is known a priori, but Poisson's ratio nu is known, then mu and lambda are determined by a single deformation field up to a constant. This includes as a special case plane stress deformations of an incompressible material. Exact analytical solutions valid for 2D, 3D and transient deformations are given for all cases in terms of quadratures. These are used to show that the inverse problem for mu based on the compressible elasticity equations is unstable in the limit lambda --> infinity. Finally, we use the exact solutions as a basis to compute non-trivial modulus distributions in a simulated example.