Shear Modulus Distribution Research Articles

A Shear Wave Velocity Tomography System for Geotechnical Centrifuge Testing

Abstract In physical model tests using geotechnical centrifuge, it is important to evaluate soil properties during centrifugal acceleration due to the effect of stress conditions. A crosshole type shear wave velocity tomography testing system has been developed, and the distribution of maximum shear modulus in sand layers could be evaluated in the centrifuge. Piezoelectric bender elements are installed in the system as shear wave transmitters and receivers for travel time measurement. A driving and measurement system for bender element tests is prepared to be used under centrifuge operating environment. A series of tests using this system has been conducted with uniform sand samples, and the testing procedure and typical signals from bender elements are presented. Tomography inversion considering curved raypaths has been conducted using a total of 256 travel time data determined by measured signals because the first arrivals follow curved raypaths under a stress related linear variation in stiffness with depth. The maximum shear modulus variation with depth, or vertical effective stress, has been verified through comparison with resonant column test results, and the applicability of this system in general centrifuge testing has been confirmed.

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

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. 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. 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. 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.

Inverse determination of soil density and stress state using dispersion wave measurements and cone penetration tests in a non-layered soil

The determination of the void ratio and stress state distribution in undisturbed soil is still an ambitious aim which cannot be reached by sampling from bore holes. Therefore, an alternative method is proposed to determine soil density and stress state using dispersion wave measurements and cone penetration tests. Dispersion wave measurements and resonant-column tests are carried out to measure the shear wave velocity and shear modulus distribution with depth. Using finite element calculations a relationship between cone penetration resistance, stress state and void ratio is derived. From these results the void ratio distribution and the stress state can be calculated inversely. The inverse method is applied to real test data. The results of the inverse parameter determination are shown and assessed regarding the possibilities and the limitations of the presented method.

A limitation in the use of spatially stationary strain variance estimation in the regularization of shear modulus reconstruction

To yield spatially uniform stability in shear modulus reconstruction, an effective method for setting appropriate spatially varied regularization parameters using the variances of measured strain tensor components was previously reported by us. In the method, the strain variances are evaluated at each position using multiple field measurements or in real time using a single field measurement with the Ziv-Zakai lower bound (ZZLB, i.e., Vz) when using all cross-correlation-based methods. However, a stationary assumption of the displacement or strain measurement error can also be used as an alternative method of real-time reconstruction using an estimated variance (i.e., Vd or Vs). In this study, a comparison of real-time spatially variant regularizations using the above three estimated strain variances performed on an agar phantom reveals that stationary assumptions cannot be used when the target region has a large spatial variation in the shear modulus distribution (e.g., when stress concentration regions are generated in front of and behind a stiff inclusion or tumor), although the ZZLB can be used if sufficient care is taken owing to its low local correlation. When a target has a released tissue boundary, none of the estimated variances are useful. In such regions, none of the estimated variances should be used directly, because the shear modulus is not spatially smooth.

The external and internal measurement impact on shear modulus distribution within cyclic small strains in triaxal studies into cohesive soil

The paper deals with comparison of tangent moduli Gs of kaolin from Tuowice obtained from cyclic triaxial tests on the basis of external and internal reading in the small strains range (10 -5 ÷10 -3 ). The tests were carried out on a modernised test bed, enabling full saturation of specimens using the back pressure method as well as a precise internal measurement of strains by means of contactless microdisplacements sensors. The value of linearity factor L is one of adopted quality criteria for two measuring methods. Maintaining a constant deformation rate the influence of various cyclic process parameters (deviator stress amplitude - constant or variable; high or low; initial level of stress and strain, at which the unloading and reloading cycles were started; overconsolidation ratio OCR as well as cycles' number and arrangement) on the shear modulus - axial strain characteristic was studied. The obtained values of Gint and Gext (or Lint and Lext) clearly show an underestimation (even 5 times) of Gs value within the range 10 -5 ÷10 -3 when using an external measurement. In addition, the differences between Gint and Gext, which develop differently depending on specified cyclic process parameters, gradually decrease with increasing axial strains.

Relative shear modulus reconstruction for visualization with no geometrical artifact

We previously reported several useful shear modulus reconstruction methods obtained from motion or static equations and reference shear moduli that yield an absolute reconstruction of shear modulus, density or inertia using the finite difference method (FDM) or finite element method (FEM) with strain tensor measurement (i.e., methods referred to as Methods A to C and F). Methods A to C are particularly useful for three-dimensional (3D) shear modulus reconstruction in an incompressible case or in the case of an inhomogeneous Poisson’s ratio by using the mean normal stress as an unknown, whereas Method F using a typical Poisson’s ratio is particularly useful when the reconstruction dimensionality is low or when the Poisson’s ratio is homogeneous. For instance, when using FDM, although Method B permits the absolute reconstruction of a mean normal stress by using a reference mean normal stress, Methods A and C yield biased mean normal stress reconstructions using no reference value and a quasi-reference value (e.g., zero), respectively. In this report, we describe our newly developed Methods D and E obtained from Methods A to C and F, which yield a unique relative shear modulus reconstruction with no geometrical artifact. In Method D, no reference shear modulus is used with Methods A to C and F, whereas in Method E, an arbitrary finite value (e.g., unity) is used with Methods A to C and F as a quasi-reference shear modulus at one or more arbitrary points (i.e., a quasi-reference point) or in an arbitrary region (i.e., a quasi-reference region) in the region of interest. That is, both Method D and Method E permit the visualization of the shear modulus distribution without using an absolute reference value (e.g., no reference material or typical value), which is particularly useful when dealing with deeply located tissues such as the liver and heart tissues. Methods D and E use iterative methods such as the conjugate gradient method to solve algebraic equations. As confirmed from simulated phantom deformation data, each version of Method E results in a higher convergence speed and reconstruction accuracy than the corresponding Method D. Although several versions of Method D using FEM are practically useful, Method E obtainable should be used for such relative reconstructions rather than the corresponding version of Method D, in combination with Methods A to C and F.

Efficient implementations of a pseudodynamical stochastic filtering strategy for static elastography

A computationally efficient pseudodynamical filtering setup is established for elasticity imaging (i.e., reconstruction of shear modulus distribution) in soft-tissue organs given statically recorded and partially measured displacement data. Unlike a regularized quasi-Newton method (QNM) that needs inversion of ill-conditioned matrices, the authors explore pseudodynamic extended and ensemble Kalman filters (PD-EKF and PD-EnKF) that use a parsimonious representation of states and bypass explicit regularization by recursion over pseudotime. Numerical experiments with QNM and the two filters suggest that the PD-EnKF is the most robust performer as it exhibits no sensitivity to process noise covariance and yields good reconstruction even with small ensemble sizes.

A stabilized forward elasticity finite element formulation yields a stable and convergent inverse elasticity solution.

“Elastography” refers to the procedure of using ultrasound to image tissue deformation. Images showing tissue deformation are sensitive to distributions of tissue mechanical properties. The mechanical properties themselves may be quantified by solving an inverse elasticity problem. An iterative approach to solving the inverse problem can be formulated by repeated solutions of the forward problem. That is, the shear modulus distribution sought is that which predicts a displacement field most consistent with the measured displacement field. Here we show that given plane displacement measurements for plane stress elasticity, the continuous elastic equilibrium equation uniquely determines the modulus distribution. On the other hand, we shall also demonstrate that the discrete elasticity equations from standard FEM discretization does not, even at the limit of infinite mesh refinement. We diagnose the problem as an underenforcement of the elasticity equations. With this knowledge, we have been able to design new forward elasticity FEM formulations that provide provably convergent forward and inverse elasticity solutions.

The number distribution of complex shear modulus of single cells measured by atomic force microscopy

The viscoelastic properties of a large number of mouse fibroblast NIH3T3 cells ( n≃130) were investigated by combining atomic force microscopy (AFM) with a microarray technique. In the experiments, the cells were arranged and cultured in the wells of a microarray substrate, and a force modulation mode experiment was used to measure the complex shear modulus, G*, of individual cells in a frequency range 0.5–200 Hz. The frequency dependence of G* of the cells exhibited a power-law behavior and similar frequency dependencies have been observed in several cell types cultured on flat substrates. This indicated that the NIH3T3 cells cultured in the wells of a microarray have analogous structural organization to those cells cultured on flat substrates. The number distribution of both the storage and loss moduli of G* fitted well to a log-normal distribution function, whereas the power-law exponent estimated by a power-law structural damping model showed a normal distribution function. These results showed that combining AFM with a microarray technique was a suitable approach for investigating the statistics of rheological properties of living cells without the requirement of cell surface modification.

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.

An efficient quantitative ultrasonic elastography based on variational h‐refinement

AbstractQuantitative elastography is a method to visualise a stiffness distribution. It is motivated by the observation that changes in mechanical properties of soft tissue mostly include important diagnostic information. With an ultrasound–elastography system, the displacement field can be calculated from the pre– and post–deformation image. Using the assumption that the material is elastic, isotropic and nearly incompressible, the distribution of shear modulus is determined by solving an inverse problem. While common approaches use a constant mesh, a variational h–refinement is in the focus of this work. In doing so, the efficiency and accuracy of the determination can be increased. The presented results are generated by using numerical simulations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Ultrasonic strain imaging and reconstructive elastography for biological tissue

Mechanical properties of biological tissue represent important diagnostic information and are of histological and pathological relevance. In order to obtain non-invasively mechanical properties of tissue, we developed a real-time strain imaging system for clinical applications. The output data of this system also allow an inverse elastography approach leading to the spatial distribution of the relative elastic modulus of tissue. The internal displacement field of biological tissue is determined using the above mentioned strain imaging system by applying quasi-static compression to the considered tissue. Axial displacements are calculated by comparing echo signal sets obtained prior to and immediately following less than 0.1% compression, using the fast root seeking technique. Strain images representing mechanical tissue properties in a non-quantitative manner are displayed in real-time mode. For additional quantitative imaging, the stiffness distribution is calculated from the displacement field assuming the investigated material to be elastic, isotropic, and nearly incompressible. Different inverse problem approaches for calculating the shear modulus distribution using the internal displacement field have been implemented and compared. The results of an ongoing clinical study with more than 200 patients show, that our real-time strain imaging system is able to differentiate malignant and benign tissue areas in the prostate with a high degree of accuracy (sensitivity = 76% and specificity = 89%). The reconstruction approaches applied to the strain image data deliver quantitative tissue information and seem promising for an additional differential diagnosis of lesions in biological tissue. Our real-time system has the potential of improving diagnosis of prostate and breast cancer.

Interseismic strain accumulation: Spin‐up, cycle invariance, and irregular rupture sequences

Using models of infinite length strike‐slip faults in an elastic layer above linear viscoelastic regions, we investigate interseismic deformation. In the models we investigate, interseismic strain accumulation on mature faults is the result of the cumulative effects of all previous ruptures and is independent of the fault loading conditions. The time for a fault to spin‐up to a mature state depends on the rheologies and the fault loading conditions. After the model has spun‐up, the temporal variation of shear stresses is determined by the fault slip rate and model rheologies. The change in stress during spin‐up depends on the slip rate, rheologies, and fault loading conditions but is independent of the magnitude of the initial stress. Over enough cycles such that the cumulative deformation is block‐like, the average mature interseismic velocities are equal to the interseismic velocities of an elastic model with the same geometry and distribution of shear moduli. In a model that has spun‐up with the fault rupturing periodically, the cumulative deformation is block‐like at the end of each seismic cycle, and the interseismic deformation is cycle‐invariant (i.e., the same in all cycles). When the fault ruptures nonperiodically, the fault spins up to a mature state that is the same as if the fault had ruptured periodically with the mean slip rate. When the fault slip rate within each cycle varies, the interseismic deformation evolves toward the cycle‐invariant deformation determined by the most recent fault slip rate. Around a fault whose slip rate has been faster (slower) than average, interseismic velocities are larger (smaller) than the cycle‐invariant velocities and increase (decrease) from cycle to cycle.

2261: Multidimensional reconstruction/imaging of shear modulus distribution of living soft tissues

2261: Multidimensional reconstruction/imaging of shear modulus distribution of living soft tissues

2133: Real-time reconstruction/imaging of shear modulus distribution by calculating strain ratio

2133: Real-time reconstruction/imaging of shear modulus distribution by calculating strain ratio

Three‐dimensional ultrasound image registration and shear elastic modulus reconstruction

It is widely recognized that tissue pathologies often change biomechanical properties. For instance, neoplastic tissue is typically highly vascularized, contains abnormal concentrations of extracellular proteins (i.e., collagen, proteoglycans) and has a high interstitial fluid pressure compared to most normal tissues. The aim of this work is to develop and evaluate an ultrasound technique to quantitatively measure and image the mechanical properties of soft tissues in three dimensions. The intended application of our work is in the detection and diagnosis of breast cancer and other soft tissue pathologies. The specific goal of this project is the accurate measurement of the elastic shear modulus distribution of a tissue‐mimicking ultrasound phantom, using a 3‐D ultrasound imaging system. This requires the design and characterization of algorithms to provide three‐dimensional motion estimates from ultrasound images and to solve the three‐dimensional inverse problem to recover shear elastic modulus. Modulus images, reconstructed using our algorithms, are presented and compared to values determined by independent mechanical testing. The effect of the boundary conditions and regularization methods on the reconstructed modulus images and the uniqueness of the solution is also discussed.

Shear Modulus Identification in Soft Tissue‐like Materials

AbstractElastography is a method motivated by the observation that changes in tissue elasticity are generally correlated with pathological phenomena. By applying quasi‐static compression to the considered tissue‐like object, components of the internal displacement field can be determined with an ultrasound‐elastography system. Based on the assumption that the investigated material is elastic, isotropic, and nearly incompressible, the stiffness distribution can be calculated from the displacement field in the framework of inverse problems. In this paper, existing procedures for calculating the shear modulus distribution with knowledge of internal displacement field are presented. The methods are compared with respect to the stability of algorithms using numerical simulations and ultrasound measurements. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Internal Friction Study of a Composite with a Negative Stiffness Constituent

Composites with negative stiffness constituents can exhibit material properties that exceed conventional bounds. Composites with VO2as negative stiffness inclusions and tin as the stabilizing matrix were prepared via powder metallurgy. Specimens were tested over a range of temperature in torsion using broadband viscoelastic spectroscopy. Composites processed via powder metallurgy exhibited internal friction anomalies over a broad range of temperatures, in contrast to the single, sharp anomalies reported previously from cast specimens. The detailed material behavior encompassed a variety of responses, which were also dependent on the number of thermal cycles. Composite theory predictions assuming a distribution of negative shear moduli can account for peak broadening.

The exploration of the lunar interior using torsional oscillations

The diagnostic capabilities of the torsional oscillations for probing the structure of the interiors of the Moon are investigated. Models with no core, a liquid core, and a solid core are considered. The profiles of compressional and shear wave velocities, V P and V S, for the lunar interior estimated by Bills and Ferrari (J. Geophys. Res. 82 (1977) 1306), Goins et al. (J. Geophys. Res. 86 (1981b) 5061) and Nakamura (J. Geophys. Res. 8 (1983) 677) from the Apollo lunar seismic network are used. For all these models the periods of torsional oscillations for fundamental modes ℓ=2–100 and four overtones have been calculated. The derivatives of the dimensionless eigenfrequency with respect to the dimensionless shear modulus and density are calculated and tabulated for use. These data can be used for the determination of corrections to the model density and shear modulus distributions due to their small change. The damping of torsional oscillations is also studied. Several trial distributions of the dissipative function Q as a function of radius are considered. It is shown, that the torsional modes with ℓ⩾7, n=0 can be recorded. These modes provide information on the external layers down to about 500 km depth.

Elasticity imaging with time-resolved pulsed elastography

Time-resolved pulsed elastography is a promising technique for imaging the shear modulus of soft tissues. It is based on the investigation of a low-frequency transient shear wave using an ultrafast ultrasonic imaging system (up to 10<th>000 frames/s) composed of 128 channels sampled at 50 MHz and having 2 Mbytes. The system is connected to a linear array of transducers. Displacements induced by the propagating shear wave are measured using cross correlation of the ultrasonic signals. A low-frequency vibrating device was designed. The linear array of transducers is placed between two rods fixed to electromagnetic vibrators. The rods are either orthogonal or parallel to the active surface of the transducer array. The low-frequency shear waves are sent using the rods which are placed in such a way that the lobes of the induced shear waves superimpose in front of the transducer array. Large displacements are observed in the tissues which makes deeper investigations possible. Movies of the low-frequency (50–200 Hz) shear wave propagation through soft tissues can be used either to estimate the shear modulus distribution in the medium by direct local inversion, or to localize visually lesions of unexpected elasticity. In vivo measurements in human breast will be presented and discussed.