Shear Wave Phase Velocity Research Articles

Method of Green’s Function for Characterization of SH Waves in Porous-Piezo Composite Structure with a Point Source

An approach of Green’s function is adopted to solve the inhomogeneous linear differential equations representing wave equations in piezo-composite materials. In particular, transference of horizontally polarised shear (SH) waves is considered in bedded structure comprising of porous-piezo electric layer lying over a heterogeneous half-space. Propagation of SH-waves is considered to be influenced by point source, situated in the heterogeneous substrate. A closed form analytical solution is obtained to establish the dispersion relation. Remarkable influence of different parameters (like elastic constant, piezoelectric constant, heterogeneity parameter, initial stress and layers thickness) on the phase and group velocity are shown graphically. Moreover, a special case of present study is shown by replacing the porous piezoelectric material with piezoelectric material. Some numerical examples are illustrated by taking the material constants of Lead Zirconate Titanate (PZT-1, PZT-5H and PZT-7) for the porous piezoelectric layer where the phase velocity of SH waves is high rather than that of piezoelectric layer.

Local Phase Velocity Based Imaging of Viscoelastic Phantoms and Tissues.

Assessment of soft tissue elasticity and viscosity is of interest in several clinical applications. In this study, we present the feasibility of the local phase velocity based imaging (LPVI) method to create images of phase velocity and viscoelastic parameters in viscoelastic tissue-mimicking materials and soft tissues. In viscoelastic materials, it is necessary to utilize wave-mode isolation using a narrow bandpass filter combined with a directional filter in order to robustly reconstruct phase velocity images with LPVI in viscoelastic media over a wide range of frequencies. A pair of sequential focused acoustic radiation force push beams, focused once on the left-hand side and once on the right-hand side of the probe, was used to produce broadband propagating shear waves. The local shear wave phase velocity is then recovered in the frequency domain for multiple frequencies, for both acquired data sets. Then, a 2-D shear wave velocity map is reconstructed by combining maps from two separate acquisitions. By testing the method on simulated data sets and performing in vitro phantom and in vivo liver tissue experiments, we show the ability of the proposed technique to generate shear wave phase velocity maps at various frequencies in viscoelastic materials. Moreover, a nonlinear least-squares problem is solved in order to locally estimate elasticity and viscosity parameters. The LPVI method with added directional and wavenumber filters can produce phase velocity images, which can be used to characterize the viscoelastic materials.

Fast Local Phase Velocity-Based Imaging: Shear Wave Particle Velocity and Displacement Motion Study.

Fast and precise noninvasive evaluation of tissue mechanical properties is of high importance in ultrasound shear wave elastography. In this study, we present an updated, faster version of the local phase velocity-based imaging (LPVI) method used to create images of local phase velocity in soft tissues. The updated LPVI implementation uses 1-D Fourier transforms in spatial dimensions separately in comparison to its original implementation. A directional filter is applied upon the shear wave field to extract the left-to-right (LR) and right-to-left (RL) propagating shear waves. A local shear wave phase velocity map is recovered based on both LR and RL waves. Finally, a 2-D shear wave velocity map is reconstructed by combining the LR and RL phase velocity maps. LPVI performance for shear wave displacement and velocity-wave motion data is examined. A study of LPVI used for only one data acquisition with multiple focused ultrasound push beams is presented. The lesion placement with respect to the pushes and whether two sequential pushes provided different results from two simultaneous radiation force pushes was investigated. The addition of white Gaussian noise to the wave motion data was also tested to examine the LPVI method's performance. Robust and accurate shear wave phase velocity maps are reconstructed using the proposed LPVI method using numerical tissue-mimicking phantoms with inclusions. Results from the numerical phantom study showed that the reconstructed, asymmetric inclusions, for various axial locations, are better preserved for shear wave particle velocity signals compared with particle displacement motion data.

Two-Point Frequency Shift Method for Shear Wave Attenuation Measurement.

Ultrasound shear wave elastography (SWE) is an increasingly used noninvasive modality for quantitative evaluation of tissue mechanical properties. SWE typically uses an acoustic radiation force to produce laterally propagating shear waves that are tracked in the spatial and temporal domains, in order to obtain the wave velocity. One of the ways to study the viscoelasticity is through studying the shear wave phase velocity dispersion curves. Shear wave attenuation can be also characterized in viscoelastic tissues with methods that use multiple lateral data samples. In this article, we present an alternative method for measuring the shear wave attenuation without using a rheological model two-point frequency shift (2P-FS). The technique uses information related to the amplitude spectra FS of shear waves measured at only two lateral locations. The theoretical basis for the 2P-FS is derived and validated. We examined how the first signal position and the distance between the two locations affect the shear wave attenuation estimation in the 2P-FS method. We tested this new method on digital phantom data created using the local interaction simulation approach (LISA) in viscoelastic media. Moreover, we tested data acquired from custom-made tissue-mimicking viscoelastic phantom experiments and ex vivo porcine liver measurements. We compared results from the 2P-FS method with the other two techniques used for assessing a shear wave attenuation: the FS-based method and the attenuation-measuring ultrasound shear wave elastography (AMUSE) technique. In addition, we evaluated the 2P-FS algorithm with different levels of added white Gaussian noise to the shear wave particle velocity using numerical phantoms. Tests conducted showed that the 2P-FS method gives robust results based on only two measurements and can be used to measure attenuation of viscoelastic soft tissues.

Changes in shear wave propagation within skeletal muscle during active and passive force generation

Muscle force can be generated actively through changes in neural excitation, and passively through externally imposed changes in muscle length. Disease and injury can disrupt force generation, but it can be challenging to separate passive from active contributions to these changes. Ultrasound elastography is a promising tool for characterizing the mechanical properties of muscles and the forces that they generate. Most prior work using ultrasound elastography in muscle has focused on the group velocity of shear waves, which increases with increasing muscle force. Few studies have quantified the phase velocity, which depends on the viscoelastic properties of muscle. Since passive and active forces within muscle involve different structures for force transmission, we hypothesized that measures of phase velocity could detect changes in shear wave propagation during active and passive conditions that cannot be detected when considering only group velocity. We measured phase and group velocity in the human biceps brachii during active and passive force generation and quantified the differences in estimates of shear elasticity obtained from each of these measurements. We found that measures of group velocity consistently overestimate the shear elasticity of muscle. We used a Voigt model to characterize the phase velocity and found that the estimated time constant for the Voigt model provided a way to distinguish between passive and active force generation. Our results demonstrate that shear wave elastography can be used to distinguish between passive and active force generation when it is used to characterize the phase velocity of shear waves propagating in muscle.

Shear wave propagation in a slightly compressible finitely deformed layer over a foundation with pre-stressed fibre-reinforced stratum and dry sandy viscoelastic substrate

The present paper articulates a mathematical model for the propagation of Shear wave in a layered structure comprised of slightly compressible finitely deformed layer over a foundation with pre-stressed fibre-reinforced stratum and pre-stressed dry sandy viscoelastic substrate. Method of separation of variables has been adopted as the solution technique for the problem. Closed form of dispersion equation for the propagation of shear wave in the layered structure has been deduced analytically and the obtained result has been found in agreement with the classical case. The effect of various parameters, viz. wave number, width ratio, bonding parameter, initial stress, sandiness parameter and shear viscosity on the phase velocity of shear wave has been studied through numerical computations and graphical illustrations. Some remarkable results regarding the propagation behavior of shear wave in the considered layered structure due to various affecting parameters serve as the major highlights of the present work which has significant relevance to field of geophysics, earthquake engineering and civil engineering.

Errors in phase velocity estimation owing to the method used for shear wave waveform phase extraction

In ultrasound elastography, the shear wave phase velocity, cs, is often used to calculate the stiffness, which can indicate whether a biological tissue has an abnormality. The shear wave phase velocity can also be used to compute shear wave dispersion, which can be related to viscosity or guided wave effects. The most common methods used for extracting the phase from waveform vibration to estimate cs are the Kalman filter (KF), the fast Fourier Transform (FFT), and the Levenberg–Marquardt Algorithm (LMA). Currently, there are no studies in the literature comparing the performance of these methods; hence, this study aims to show the error in estimating the phase velocity obtained with the mentioned methods and to compare them through computational simulations and experiments. In the computational simulations, the absolute value of mean and the standard deviation of relative error of cs were evaluated as a function of signal-to-noise ratio, number of cycles of the shear wave, pulse repetition frequency, and initial phase. The experimental setup used an ultrasonic benchtop system, with a 2.104-MHz frequency transducer, TV, to generate acoustic radiation force at a plastic sphere and consequently induce shear wave propagation in a 7% gelatin phantom. Another ultrasonic benchtop system, with a 4.89-MHz frequency transducer, was used on pulse-echo mode to monitor the shear wave propagation at different positions within the phantom. The computational simulation results demonstrated that LMA and KF demonstrated the best performance in extracting the phase employed to estimate the shear wave phase velocity. In some situations, the cs values obtained by means of KF, FFT, and LMA methods in the experimental results presented a statistical difference, but the highest differences between the mean of cs was smaller than 0.03 m s−1.

РАСПРОСТРАНЕНИЕ ВОЛНЫ РЭЛЕЯ ВДОЛЬ ГРАНИЦЫ ПОЛУПРОСТРАНСТВА, ОПИСЫВАЕМОГО УПРОЩЕННОЙ МОДЕЛЬЮ КОССЕРА

We consider a simplified (reduced) dynamic model of a Cosserat medium, which occupies an intermediate position between the classical dynamic theory of elasticity and the proper Cosserat medium model, which has asymmetry in the stress tensor and the presence of moment stresses. In contrast to the latter, in the simplified model, three of the six elastic constants are zero and, as a result, there is no moment stress tensor. In the two-dimensional formulation for the model of a reduced medium, the problem of the propagation of an elastic surface wave along the half-space boundary was solved. The solution of the equations was described as the sum of the scalar and vector potentials, and only one component of the vector potential is nonzero. It is shown that such a wave, in contrast to the classical surface Rayleigh wave, has a dispersion. In the plane “phase velocity-frequency” for such waves there are two dispersion branches: the lower (acoustic) and upper (optical). With increasing frequency, the phase velocity of the wave related to the lower dispersion branch decreases. The phase velocity of the wave related to the upper dispersion branch increases with increasing frequency. The phase velocity of the surface wave in the entire frequency range exceeds the phase velocity of the bulk shear wave. The stresses and displacements arising in the zone of propagation of the surface wave are calculated.

RSNA/QIBA efforts to standardize shear wave speed as a biomarker for liver fibrosis staging

Commercial shear elasticity systems have been available to clinicians for several years for characterizing liver stiffness. The Radiological Society of North America (RSNA) established a Quantitative Imaging Biomarker Alliance (QIBA) committee on Ultrasound Shear Wave Speed to help achieve better consistency between the shear wave speed and elastic modulus values reported by different manufacturer systems. An inter-laboratory study of shear wave speed estimation was conducted at academic, industry and clinical laboratories using commercial ultrasound and magnetic resonance elastography systems in elastic (Phase I) and viscoelastic (Phase II) homogeneous, isotropic tissue-mimicking phantoms. In the elastic phantoms, shear wave speeds from different systems were within ±5% of the grand mean of all system measurements. Different appraisers and replicate measurements at a given site were not sources of significant variability in shear wave speed measurements. In the viscoelastic phantoms, shear wave speeds from different systems were within ±15% of the grand mean of all systems, with the stiffest phantoms at the deepest focal depth (7 cm) having the greatest measurement variance. Magnetic resonance elastography measurements at 140 Hz in the viscoelastic phantoms were consistent to within ±5% of thosemade with the ultrasound shear elasticity systems, while measurements made at 60 Hz had a mean bias of -2.6% relative to ultrasound systems. A method for reporting group shear wave velocities using both displacement and velocity data, along with shear wave phase velocity analysis, has been developed as a common calibration approach for the phantoms. An open-source repository of research scanner shear wave sequences and processing code has been created to allow manufacturers to have a calibration standard to assess compliance with the QIBA US shear wave speed profile and to achieve more consistent results to facilitate more rapid clinical adoption of this technology to evaluate liver fibrosis.

Group versus Phase Velocity of Shear Waves in Soft Tissues.

Across the varieties of waves that have been studied in physics, it is well established that group velocities can be significantly greater than or less than phase velocities measured within comparable frequency bands, depending on the particular mechanisms involved. The distinction between group and phase velocities is important in elastography, because diagnoses are made based on shear wave speed estimations from a variety of techniques. We review the general definitions of group and phase velocity and examine their specific relations within an important general class of rheological models. For the class of tissues and materials exhibiting power law dispersion, group velocity is significantly greater than phase velocity, and simple expressions are shown to interrelate the commonly measured parameters. Examples are given from phantoms and tissues.

Rheological determinants for simultaneous staging of hepatic fibrosis and inflammation in patients with chronic liver disease.

The purpose of this study is to investigate the use of fundamental rheological parameters as quantified by MR elastography (MRE) to measure liver fibrosis and inflammation simultaneously in humans. MRE was performed on 45 patients at 3T using a vibration frequency of 56Hz. Fibrosis and inflammation scores were obtained from liver biopsies. Biomechanical properties were quantified in terms of complex shear modulus G* as well as shear wave phase velocity c and shear wave attenuation α. A rheological fractional derivative order model was used to investigate the linear dependence of the free model parameters (dispersion slope y, intrinsic speed c0 , and intrinsic relaxation time τ) on histopathology. Leave-one-out cross-validation was then utilized to demonstrate the effectiveness of the model. The intrinsic speed c0 increases with hepatic fibrosis, while an increased relaxation time τ is reflective of more inflammation of the liver parenchyma. The dispersion slope y does not depend either on fibrosis or on inflammation. The proposed rheological model, given this specific parameterization, establishes the functional dependences of biomechanical parameters on histological fibrosis and inflammation. The leave-one-out cross-validation demonstrates that the model allows identification, from the MRE measurements, of the histology scores when grouped into low-/high-grade fibrosis and low-/high-grade inflammation with significance levels of P=0.0004 (fibrosis) and P=0.035 (inflammation). The functional dependences of intrinsic speed and relaxation time on fibrosis and inflammation, respectively, shed new light onto the impact hepatic pathological changes on liver tissue biomechanics in humans. The dispersion slope y appears to represent a structural parameter of liver parenchyma not impacted by the severity of fibrosis/inflammation present in this patient cohort. This specific parametrization of the well-established rheological fractional order model is valuable for the clinical assessment of both fibrosis and inflammation scores, going beyond the capability of the plain shear modulus measurement commonly used for MRE.

Temperature dependent of viscoelasticity measurement on fat emulsion phantom using acoustic radiation force elasticity imaging method.

During the past two decades, tissue elasticity has been extensively studied and has been used in clinical disease diagnosis. But biological soft tissues are viscoelastic in nature. Therefore, they should be simultaneously characterized in terms of elasticity and viscosity. In addition, the mechanical properties of soft tissues are temperature dependent. However, how the temperature influences the shear wave dispersion and the viscoelasticity of soft tissue are still unclear. The aim of this study is to compare viscoelasticity of fat emulsion phantom with different temperature using acoustic radiation force elasticity imaging method. In our experiment, we produced four proportions of ultrasonic phantom by adding fat emulsion gelatin. Through adjusting the component of the fat emulsion, we change the viscoelasticity of the ultrasonic phantom. We used verasonics system to gather data and voigt model to fit the elasticity and viscosity value of the ultrasonic phantom we made. The influence of temperature to the ultrasonic phantom also measured in our study. The results show that the addition of fat emulsion to the phantom can increase the viscosity of the phantom, and the shear wave phase velocity decreases gradually at each frequency with the temperature increases, which provides a new material for the production of viscoelastic phantom.

Influence of Stress on Propagation of Shear Wave in Piezoelectric-Piezoelectric (PE-PE) Composite Layered Structure

Shear waves (SH) propagation in piezoelectric composite under the influence of initial stress is investigated analytically and numerically. The dispersion equation of shear waves propagation in direction normal to the layering is obtained in presence of initial stress. Numerical solutions were obtained for evaluating the effect of stress on dimensionless frequency and phase velocity. The effect of stress on stop band is discussed in this study. It can be concluded from the results that initial stress has significant effect on propagation characteristics of shear waves. The variation of initial stress has small effect on the phase velocity of shear waves. This study provides insight for development of piezoelectric composite structure under the influence of initial stress.

Shear waves in magneto-elastic transversely isotropic (MTI) layer bonded between two heterogeneous elastic media

ABSTRACTThis paper studies shear wave propagation in magneto-elastic transversely isotropic material, sandwiched between a layer and a half-space of heterogeneous elastic materials. Elastic constants of the layer and half-space are assumed to vary in a parabolic form with depth. Whittaker’s functions and variable separable techniques have been employed to calculate the interior deformations; consequently, we obtain a general dispersion relation for shear wave. Effects of various affecting parameters on phase velocity of shear wave are considered through some numerical examples. In addition, a comparative study has been carried out for three examples of sandwiched layer, namely Beryl, Magnesium and isotropic.

Analysis of the transient surface wave propagation in soft-solid elastic plates.

In dynamic elastography, the goal is to estimate the Young's modulus from audio-frequency wave propagation in soft-tissues. Within this frequency range, the shear wavelength is centimeter-sized while the compressional wavelength is meter-sized. Thus, the experimental data are usually collected in the near-field of the source. Near-field effects have been widely studied for bulk wave propagation. However, the near- and transient-fields of surface and guided waves have received less attention. In this work, the transient surface displacement field in soft-solid elastic plates in vacuum is analyzed. Due to the high Poisson's ratio, mode conversion has special characteristics in soft-solids. They are analyzed through this work where it is shown that the transient-field over the surface can be interpreted by tracing a few reflections. The authors show the existence of a critical distance needed for the formation of Rayleigh-Lamb modes. Below this distance, only direct surface waves propagate without contribution from reflected waves. Thus, the dispersion curve differs from that predicted by Rayleigh-Lamb modes. Instead, the authors propose a model based on the interference of surface waves, which agree with the experimental data. In addition, the conditions needed in order to retrieve the shear wave phase velocity from the surface field are given.

Designing two-dimensional metamaterials of controlled static and dynamic properties

In the current work, we elaborate two-dimensional metamaterials of controlled anisotropy. To that scope, we employ diamond and octagon-shaped planar lattices with and without inner links. Using a dedicated homogenization technique, we derive closed-form expressions for the lattice’s effective mechanical properties. We analyse the effect of the lattice’s configuration on the metamaterial’s effective static properties, identifying configurations with mechanical attributes desirable for morphing, biomedical and mechanical engineering applications. We thereafter compute the lattice’s wave propagation characteristics, deriving a link between the metamaterials’ static and dynamic properties. In particular, we analyse the longitudinal and shear wave phase velocity dependence on the lattice’s geometric configuration. Thereupon, we identify architectural arrangements for which the phase velocity vanishes in certain propagation directions, exhibiting wave propagation isolation characteristics. We demonstrate that the detected isolation features can systematically arise for lattice architectural designs that yield highly anisotropic static properties (thus high material moduli ratios) and anti-auxetic material behaviours (thus non-negative Poisson’s ratio values).

Generation and measurement of shear waves in micellar fluids using ultrasound-based methods

Wormlike micellar fluids are viscoelastic and their mechanical properties have been evaluated with mechanically generated shear waves combined with optical detection as well as rheological testing. In this work, we describe the use of acoustic radiation force and ultrafast ultrasound imaging to generate shear waves and measure their propagation, respectively. This shear wave elastography (SWE) method has been used to measure the viscoelastic mechanical properties in tissue-mimicking phantoms and soft tissues. We tested micellar fluids made from cetrimonium bromide (CTAB) and sodium salicylate (NaSAL) with a 5:3 ratio with different concentrations (100, 200, 300, and 400 mM). We fit an extended Maxwell model to the rheological test data (0.001-15.91Hz) and used the model parameters to calculate the shear wave phase velocity in the range of the SWE data (100-500 Hz). The phase velocities calculated from the rheology testing compared well with the SWE results. The mean absolute error was less than 0.02 m/s for all the micellar fluids tested. The SWE method is nondestructive and can be used for characterization of the viscoelasticity of micellar fluids, which could be used as a model for biological tissues. [This work was supported in part by grant R01DK092255.]

Viscoelastic properties of normal rat liver measured by ultrasound elastography: Comparison with oscillatory rheometry.

Ultrasound elastography has been widely used to measure liver stiffness. However, the accuracy of liver viscoelasticity obtained by ultrasound elastography has not been well established. To assess the accuracy of ultrasound elastography for measuring liver viscoelasticity and compare to conventional rheometry methods. In addition, to determine if combining these two methods could delineate the rheological behavior of liver over a wide range of frequencies. The phase velocities of shear waves were measured in livers over a frequency range from 100 to 400 Hz using the ultrasound elastography method of shearwave dispersion ultrasound vibrometry (SDUV), while the complex shear moduli were obtained by rheometry over a frequency range of 1 to 30 Hz. Three rheological models, Maxwell, Voigt, and Zener, were fit to the measured data obtained from the two separate methods and from the combination of the two methods. The elasticity measured by SDUV was in good agreement with that of rheometry. However, the viscosity measured by SDUV was significantly different from that of rheometry. The results indicate that the high frequency components of the dispersive data play a much more important role in determining the dispersive pattern or the viscous value than the low frequency components. It was found that the Maxwell model is not as appropriate as the Voigt and Zener models for describing the rheological behavior of liver.

Dynamic mechanical analysis to assess viscoelasticity of liver tissue in a rat model of nonalcoholic fatty liver disease

Nonalcoholic fatty liver disease (NAFLD) is the most common liver disorder in both developed and developing countries. A noninvasive method of detecting early stage NAFLD and distinguishing non-alcoholic steatohepatitis (NASH) from simple steatosis (SS) would be useful. The over-accumulation of fat in hepatocytes alters the physical microstructure and chemical contents of the liver tissue. This study included dynamic mechanical analysis (DMA) testing on liver samples from a rat model of NAFLD to determine whether the tissue shows any significant changes in viscoelasticity due to the histological changes. Liver steatosis was induced in 57 rats by gavage feeding of a high fat emulsion; 12 rats received a standard diet only and served as controls. Each rat provided 2 or 3 samples for DMA tests. The shear modulus and loss modulus were measured at 9 frequency points evenly-spaced in the range from 1Hz to 41Hz. The phase velocity of shear wave was calculated from the measured modulus. Multivariate T2 test was used to assess the significance of intra-group difference. The results showed significant changes (p < 0.05) in storage modulus in livers with moderate to severe (S2 to S4) steatosis in comparison with livers without steatosis (S0), while the loss modulus demonstrated significant changes earlier in stage S1, indicating that fat accumulation affects the mechanical properties of liver, particularly viscosity. However, no significant differences were observed between the steatosis grades. These results also suggest that mild inflammation may affect the mechanical properties, which requires further verification. These findings provide new information about the mechanical properties of livers with NAFLD in low frequency range and suggest that it is possible to distinguish normal livers from livers with NAFLD.

Iterative solution to bulk wave propagation in polycrystalline materials.

This article reevaluates two foundational models for bulk ultrasonic wave propagation in polycrystals. A decoupling of real and imaginary parts of the effective wave number permits a simple iterative method to obtain longitudinal and shear wave attenuation constants and phase velocity relations. The zeroth-order solution is that of Weaver [J. Mech. Phys. Solids 38, 55-86 (1990)]. Continued iteration converges to the unified theory solution of Stanke and Kino [J. Acoust. Soc. Am. 75, 665-681 (1984)]. The converged solution is valid for all frequencies. The iterative method mitigates the need to solve a nonlinear, complex-valued system of equations, which makes the models more robust and accessible to researchers. An analysis of the variation between the solutions is conducted and is shown to be proportional to the degree of inhomogeneity in the polycrystal.