Shear Modulus Reconstruction Research Articles

Imaging the subcellular viscoelastic properties of mouse oocytes

In recent years, cellular biomechanical properties have been investigated as an alternative to morphological assessments for oocyte selection in reproductive science. Despite the high relevance of cell viscoelasticity characterization, the reconstruction of spatially distributed viscoelastic parameter images in such materials remains a major challenge. Here, a framework for mapping viscoelasticity at the subcellular scale is proposed and applied to live mouse oocytes. The strategy relies on the principles of optical microelastography for imaging in combination with the overlapping subzone nonlinear inversion technique for complex-valued shear modulus reconstruction. The three-dimensional nature of the viscoelasticity equations was accommodated by applying an oocyte geometry-based 3D mechanical motion model to the measured wave field. Five domains-nucleolus, nucleus, cytoplasm, perivitelline space, and zona pellucida-could be visually differentiated in both oocyte storage and loss modulus maps, and statistically significant differences were observed between most of these domains in either property reconstruction. The method proposed herein presents excellent potential for biomechanical-based monitoring of oocyte health and complex transformations across lifespan. It also shows appreciable latitude for generalization to cells of arbitrary shape using conventional microscopy equipment.

Homogenous shear modulus reconstruction at multiple frequencies using the DICE method

The non-invasive technique of Digital Image Correlation Elastography was used to estimate the viscoelastic properties of a tissue-mimicking silicone phantom. Harmonic motion fields were generated over a frequency range of 40–90 ​Hz and measured at the phantom surface by 4 stereoscopic camera pairs. The storage modulus (G’) and the loss modulus (G”) were successfully reconstructed from the measured surface displacement data. Low variations were found for repeated measurements and the frequency dependent viscoelastic behavior was highly consistent across multiple excitation waveforms. Power-law analysis of the two viscoelastic properties provided exponent terms consistent with tissue-like materials.

A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only

The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without boundary information. We provide a general approach based on the weak definition of the stiffness-toforce operator which conduces to see the problem as a linear system. We prove that in the case of shear modulus reconstruction, we have an L 2-stability with only one measurement under minimal smoothness assumptions. This stability result is obtained though the proof that the linear operator to invert has closed range. We then describe a direct discretization which provides stable reconstructions of both isotropic and anisotropic stiffness tensors.

Enhanced complex local frequency elastography method for tumor viscoelastic shear modulus reconstruction

Background and objectivesThe Mayo Clinic provides a magnetic resonance (MR) elastography software named MRE Wave, which uses the conventional local frequency elastography (LFE) method. However, MRE Wave is unable to supply complex viscoelasticity maps for elastography. We sought to improve the local frequency estimation algorithm used in LFE, which we refer to as the Enhanced Complex Local Frequency Elastography (EC-LFE) algorithm. MethodsThe proposed algorithm uses wave equations under the hypotheses of being linear, isotropic, and locally homogeneous. Two 2D simulation models were used to investigate the accuracy and sensitivity of the EC-LFE algorithm for detecting small tumors. The corresponding statistical parameters were the relative root mean square (RMS) error and contrast-to-noise ratio (CNR). EC-LFE was investigated with two different parameter sets, one with an optimally chosen parameter ξ (EC-LFE Adj, for short) and the other with ξ = 0 (EC-LFE0). We compared the MRE Wave and the EC-LFE using series signal-to-noise (SNR) wave data. ResultsThe elasticity RMS error of the MRE Wave software was about 1%, and that of the EC-LFE0 and EC-LFE Adj were about 0.2%. The elasticity standard deviation of the MRE Wave software was about 3% of the mean value, and those of the EC-LFE0 and EC-LFE Adj were about 1% of the mean value. The elasticity CNR value of EC-LFE0 reached 1.93 times that of the MRE Wave in the region of small tumors (less than 10-point sampling). The viscosity RMS errors of the EC-LFE0 could be less than 5%. ConclusionCompared to conventional methods, the EC-LFE was more accurate and sensitive for small tumor detection and exhibited higher noise immunity. The improved algorithm output more parameters and outperformed than the MRE Wave, thereby rendering them more suitable for clinical applications.

Inverse coefficient problem of the variable properties reconstruction for the viscoelastic cylindrical waveguide

AbstractAn effective technique for inverse coefficient problem solving of the shear modulus reconstruction for both inhomogeneous elastic and viscoelastic cylindrical waveguide is proposed. The identification is carried out using the displacement field data measured on the finite part of the outer boundary of the cylindrical waveguide in the torsional vibrations mode. An iterative scheme for solving the inverse problem is developed. At each iteration step, the correction to the shear modulus distribution function is calculated using the Fredholm integral equation of the first kind with a smooth kernel. Numerical results are obtained for both direct and inverse problems for different laws of the shear modulus variation.

Does group velocity always reflect elastic modulus in shear wave elastography?

.Dynamic elastography is an attractive method to evaluate tissue biomechanical properties. Recently, it was extended from US- and MR-based modalities to optical ones, such as optical coherence tomography for three-dimensional (3-D) imaging of propagating mechanical waves in subsurface regions of soft tissues, such as the eye. The measured group velocity is often used to convert wave speed maps into 3-D images of the elastic modulus distribution based on the assumption of bulk shear waves. However, the specific geometry of OCE measurements in bounded materials such as the cornea and skin calls into question elasticity reconstruction assuming a simple relationship between group velocity and shear modulus. We show that in layered media the bulk shear wave assumption results in highly underestimated shear modulus reconstructions and significant structural artifacts in modulus images. We urge the OCE community to be careful in using the group velocity to evaluate tissue elasticity and to focus on developing robust reconstruction methods to accurately reconstruct images of the shear elastic modulus in bounded media.

A Size-Dependent Cost Function to Solve the Inverse Elasticity Problem

Characterizing nonhomogeneous elastic property distribution of solids is of great significance in various engineering fields. In this paper, we observe that the solution to the inverse problem utilizing the standard optimization-based inverse approach is sensitive to the sizes of inclusions. The standard optimization-based inverse approach minimizes a cost function, containing the absolute error between the measured and computed displacements in L2 norm. To address this issue, we propose a novel inverse scheme to characterize nonhomogeneous shear modulus distribution of solids. In this novel method, the cost function is modified, and is dependent on the size of the inclusions. A number of simulated experiments are performed, and demonstrate that the proposed approach is capable of improving the shear modulus contrast in inclusions and reducing the size sensitivity. Furthermore, a theoretical analysis is conducted to validate what we have observed in simulated experiments. This theoretical analysis reveals that what we have observed in the simulated experiments is not induced by the numerical issues Instead, the size sensitivity issue is induced by regularization. The findings of this work encourage us to propose new cost functions for the optimization-based inverse approach to improve the quality of the shear modulus reconstruction.

Ultrasound shear wave simulation of wave propagation at oblique angles.

Shear wave elasticity imaging (SWEI) has been used to measure the local tissue elasticity. The local tissue shear modulus can be reconstructed from the displacement field of shear waves using an algebraic Helmholtz inversion (AHI) equation or a time-of-flight (TOF)-based algorithm. The shear waves, which are generated by successive focusing of ultrasonic beams at different depths, propagate at oblique angles rather than along the lateral position. The wave propagation at oblique angles can result in bias in shear modulus reconstruction using the AHI equation or the TOF-based algorithm. In this study, the effect of wave propagation at oblique angles on the tissue shear modulus reconstruction was investigated using in silico finite element (FE) simulation. An FE elastic tissue with a hard inclusion model was designed. The shear waves with propagation angles of 0°, 5°, and 10° were applied to the model. The shear modulus and the percentage error in the model were computed using the AHI equation and the TOF-based algorithm at each propagation angle from 0° to 10°. For the AHI equation, the percentage error was 0% at propagation angles of 0° and 5°, and 1% at a propagation angle of 10° in the inclusion. In the surrounding tissue, the percentage error was 0% at propagation angles of 0°, 5°, and 10°. For the TOF-based algorithm, the percentage error was 0% at propagation angles of 0° and 5°, and 40% at a propagation angle of 10° in the inclusion. In the surrounding tissue, the percentage error was 0% at propagation angles of 0° and 5°, and 35% at a propagation angle of 10° in the inclusion. Therefore, whereas the TOF-based algorithm produced critical bias in shear modulus reconstruction by the shear wave propagation at oblique angles, the AHI equation was not affected by the propagation.

Comparisons of inverse and forward problem approaches to elastography

We present a full-wave inversion approach with total variation regularization for elastography. The proposed method is based on the minimization of an error in constitutive equations functional augmented with a least squares data misfit term referred to as MECE for “modified error in constitutive equations.” The main theme of this paper is to demonstrate several key strengths of the proposed method on experimental data. In addition, some illustrative examples are provided where the proposed method is compared with a common shear wave elastography (SWE) approach. To this end, ultrasonically tracked displacement data from an acoustic radiation force (ARF) pulse are used in different phantoms including phantom with layered inclusion and triangle inclusion. The results indicate that the MECE approach can produce accurate shear modulus reconstructions in comparison with SWE, especially around the sharp edges in the layered and triangle inclusions. We compare shear modulus reconstruction using MECE and SWE with original inclusion shapes using two-dimensional normalized zero mean cross correlation, edge preservation index and dice coefficient similarity index. [Work supported by NIH Grant R01 CA174723.]

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

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

A complex elastographic hyperbolic solver (CEHS) to recover frequency dependent complex shear moduli in viscoelastic models utilizing one or more displacement data-sets

In this paper, we present a linear marching scheme to recover frequency-dependent complex shear moduli in viscoelastic models utilizing two sets of single component displacement data. The proposed method is designed to provide stable and accurate estimation of the tissue viscoelastic stiffness parameters by solving a first-order complex partial differential equation. To control the exponential growth of the numerical error resulting from one of the complex coefficients in the inverse equation, a modified upwind discretization is utilized on the first-order derivative terms of the target parameter. The algorithm is fully stablized when: (1) carefully chosen multiple data-sets are combined to eliminate the remaining complex coefficient that contributes to exponential error growth; and (2) a modified Tikhonov regularization is applied to the inversion method. We obtain the stability result in the norm so that the numerical scheme is convergent at fractional 1 / 2 order. Its performance is compared with the performance of the Algebraic Inversion Model previously investigated. We present shear modulus reconstructions from synthetic data, from laboratory phantom data and match frequency-dependent complex moduli from phantom data to several viscoelastic models. Since we have previously presented phase wave speed images from interference patterns, we exhibit those images here for comparison.

Modified error in constitutive equations (MECE) approach for ultrasound elastography.

A partial differential equation-constrained optimization approach is presented for reconstructing mechanical properties (e.g., elastic moduli). The proposed method is based on the minimization of an error in constitutive equations functional augmented with a least squares data misfit term referred to as MECE for "modified error in constitutive equations." The main theme of this paper is to demonstrate several key strengths of the proposed method on experimental data. In addition, some illustrative examples are provided where the proposed method is compared with a common shear wave elastography (SWE) approach. To this end, both synthetic data, generated with transient finite element simulations, as well as ultrasonically tracked displacement data from an acoustic radiation force (ARF) experiment are used in a standard elasticity phantom. The results indicate that the MECE approach can produce accurate shear modulus reconstructions with significantly less bias than SWE.

Mechanics Based Tomography: A Preliminary Feasibility Study.

We present a non-destructive approach to sense inclusion objects embedded in a solid medium remotely from force sensors applied to the medium and boundary displacements that could be measured via a digital image correlation system using a set of cameras. We provide a rationale and strategy to uniquely identify the heterogeneous sample composition based on stiffness (here, shear modulus) maps. The feasibility of this inversion scheme is tested with simulated experiments that could have clinical relevance in diagnostic imaging (e.g., tumor detection) or could be applied to engineering materials. No assumptions are made on the shape or stiffness quantity of the inclusions. We observe that the novel inversion method using solely boundary displacements and force measurements performs well in recovering the heterogeneous material/tissue composition that consists of one and two stiff inclusions embedded in a softer background material. Furthermore, the target shear modulus value for the stiffer inclusion region is underestimated and the inclusion size is overestimated when incomplete boundary displacements on some part of the boundary are utilized. For displacements measured on the entire boundary, the shear modulus reconstruction improves significantly. Additionally, we observe that with increasing number of displacement data sets utilized in solving the inverse problem, the quality of the mapped shear moduli improves. We also analyze the sensitivity of the shear modulus maps on the noise level varied between 0.1% and 5% white Gaussian noise in the boundary displacements, force and corresponding displacement indentation. Finally, a sensitivity analysis of the recovered shear moduli to the depth, stiffness and the shape of the stiff inclusion is performed. We conclude that this approach has potential as a novel imaging modality and refer to it as Mechanics Based Tomography (MBT).

A comparative study of full-wave inversion and local time-of-flight approaches in elastography

Most of the current techniques for elastography rely on a pointwise measurement of the induced shear waves. A common assumption in these methods is plane wave propagation in an unbounded domain and local homogeneity of the medium. Because of these simplifying assumptions, complex wave patterns, boundary conditions, and interfaces can present significant challenges to these methods. On the other hand, general nonlinear optimization approaches with PDE constraints relax the underlying assumptions of planar waves and unbounded domains and, hence, can handle very general conditions. However, this generality usually leads to a higher computationally expense as compared to time-of-flight methods. Therefore, the added computational expense and higher complexity of these optimization approaches needs to be justified in the context of elastography. In this work, we present a recent study that compares shear modulus reconstructions obtained with a PDE-constrained optimization approach with a conventional shear wave...

3D Ultrasound Shear Wave Imaging Simulation of Heterogeneous Tissue

Shear wave imaging can quantitatively measure tissue elasticity in vivo using a 2D algebraic Helmholtz inversion (AHI) equation. The 2D AHI equation reconstructs tissue elasticity by assuming that the shear wave diffraction in the elevational direction (out-of-plane direction) is not substantial. However, the application of 2D AHI equation is limited in assessing heterogeneous tissues, as the shear wave diffraction in the elevational direction critically affects the shear modulus reconstruction. A 3D AHI equation can improve the accuracy of shear modulus reconstruction for heterogeneous tissues. In this study, a finite element (FE) based shear wave simulation was performed on an early stage of cancer and an advanced cancer tissue model. The shear modulus of FE model computed by the proposed 3D AHI equation was compared to the 2D AHI equation. The 3D AHI equation substantially improved accuracy of shear modulus computation especially for the early stage of cancer model. The results from the shear modulus of inclusion by the 3D AHI equation produced the same as the input value of 150 kPa and two times lower variation than the shear modulus obtained by the 2D AHI equation.

Estimating the non-homogeneous elastic modulus distribution from surface deformations

We present a novel approach to solve the inverse problem in finite elasticity for the non-homogeneous shear modulus distribution solely from known surface deformation fields. The inverse problem is posed as a constrained optimization problem under regularization and solved utilizing the adjoint equations. Hypothetical “measured” surface displacement fields are created, by inducing indentations on the exterior of the specimen. These surface displacement fields are used to test the inverse strategy on a problem domain consisting of a stiff circular inclusion in a softer homogeneous background. We observe that the shear modulus reconstruction as well as the shape of the circular inclusion improves with an increasing number of surface displacement fields. Furthermore, with increasing noise level in the surface displacement field, the contrast of the reconstructions decreases.

A coupled subsample displacement estimation method for ultrasound-based strain elastography

Obtaining accurate displacement estimates along both axial (parallel to the acoustic beam) and lateral (perpendicular to the beam) directions is an important task for several clinical applications such as shear strain imaging, modulus reconstruction and temperature imaging, where a full description of the two or three-dimensional (2D/3D) deformation field is required. In this study we propose an improved speckle tracking algorithm where axial and lateral motion estimations are simultaneously performed to enhance motion tracking accuracy. More specifically, using conventional ultrasound echo data, this algorithm first finds an iso-contour in the vicinity of the peak correlation between two segments of the pre- and post-deformation ultrasound radiofrequency echo data. The algorithm then attempts to find the center of the iso-contour of the correlation function that corresponds to the unknown (sub-sample) motion vector between these two segments of echo data.This algorithm has been tested using computer-simulated data, studies with a tissue-mimicking phantom, and in vivo breast lesion data. Computer simulation results show that the method improves the accuracy of both lateral and axial tracking. Such improvements are more significant when the deformation is small or along the lateral direction. Results from the tissue-mimicking phantom study are consistent with findings observed in computer simulations. Using in vivo breast lesion data we found that, compared to the 2D quadratic subsample displacement estimation methods, higher quality axial strain and shear strain images (e.g. 18.6% improvement in contrast-to-noise ratio for shear strain images) can be obtained for large deformations (up to 5% frame-to-frame and 15% local strains) in a multi-compression technique. Our initial results demonstrated that this conceptually and computationally simple method could improve the image quality of ultrasound-based strain elastography with current clinical equipment.

Suitability of poroelastic and viscoelastic mechanical models for high and low frequency MR elastography.

Descriptions of the structure of brain tissue as a porous cellular matrix support application of a poroelastic (PE) mechanical model which includes both solid and fluid phases. However, the majority of brain magnetic resonance elastography (MRE) studies use a single phase viscoelastic (VE) model to describe brain tissue behavior, in part due to availability of relatively simple direct inversion strategies for mechanical property estimation. A notable exception is low frequency intrinsic actuation MRE, where PE mechanical properties are imaged with a nonlinear inversion algorithm. This paper investigates the effect of model choice at each end of the spectrum of in vivo human brain actuation frequencies. Repeat MRE examinations of the brains of healthy volunteers were used to compare image quality and repeatability for each inversion model for both 50 Hz externally produced motion and ≈1 Hz intrinsic motions. Additionally, realistic simulated MRE data were generated with both VE and PE finite element solvers to investigate the effect of inappropriate model choice for ideal VE and PE materials. In vivo, MRE data revealed that VE inversions appear more representative of anatomical structure and quantitatively repeatable for 50 Hz induced motions, whereas PE inversion produces better results at 1 Hz. Reasonable VE approximations of PE materials can be derived by equating the equivalent wave velocities for the two models, provided that the timescale of fluid equilibration is not similar to the period of actuation. An approximation of the equilibration time for human brain reveals that this condition is violated at 1 Hz but not at 50 Hz. Additionally, simulation experiments when using the "wrong" model for the inversion demonstrated reasonable shear modulus reconstructions at 50 Hz, whereas cross-model inversions at 1 Hz were poor quality. Attenuation parameters were sensitive to changes in the forward model at both frequencies, however, no spatial information was recovered because the mechanisms of VE and PE attenuation are different. VE inversions are simpler with fewer unknown properties and may be sufficient to capture the mechanical behavior of PE brain tissue at higher actuation frequencies. However, accurate modeling of the fluid phase is required to produce useful mechanical property images at the lower frequencies of intrinsic brain motions.

The variance of quantitative estimates in shear wave imaging: Theory and experiments

In this paper, we investigate the relationship between the estimated shear modulus produced in shear wave imaging and the acquisition parameters. Using the framework of estimation theory and the Cramer-Rao lower bound applied both to the estimation of the velocity field variance and to the estimation of the shear wave travel time, we can derive the analytical formulation of the shear modulus variance σ(2)(μ) using relevant physical parameters such as the shear wave frequency, bandwidth, and ultrasonic parameters. This variance corresponds to the reproducibility of shear modulus reconstruction for a deterministic, quasi-homogeneous, and purely elastic medium. We thus consider the shear wave propagation as a deterministic process which is then corrupted during its observation by electronic noise and speckle decorrelation caused by shearing. A good correlation was found between analytical, numerical, and experimental results, which indicates that this formulation is well suited to understand the parameters' influence in those cases. The analytical formula stresses the importance of high-frequency and wideband shear waves for good estimation. Stiffer media are more difficult to assess reliably with identical acquisition signal-to-noise ratios, and a tradeoff between the reconstruction resolution of the shear modulus maps and the shear modulus variance is demonstrated. We then propose to use this formulation as a physical ground for a pixel-based quality measure that could be helpful for improving the reconstruction of real-time shear modulus maps for clinical applications.

Ultrasonic agar phantom experiment for comparison of the measurement accuracy of tissue elasticity obtained by displacement vector measurement using lateral modulation with multidimensional autocorrelation and Doppler methods and corresponding one-dimensional methods

Comparison of the measurement accuracy of elasticity was performed through agar phantom experiments, ie, two-dimensional (2D) strain tensor components and 2D shear modulus reconstruction obtained by previously developed multidimensional displacement vector measure- ment methods on the basis of autocorrelation and Doppler methods (2D AM and 2D DM) and the corresponding 1D methods. The multidimensional methods yield more accurate and stable results than the corresponding 1D methods. As is shown, however, such 1D methods can also be used when the echo signal-to-noise ratio is high, eg, obtainable by previously developed lateral modulation with parabolic apodizations.