Shear Wavelength Research Articles

Multi-mode multiple wave scattering in suspensions of solid particles in viscous liquids: part 1 asymptotic results

The propagation of coherent longitudinal waves in a viscous liquid containing a random distribution of spherical elastic particles is analysed using a model based on multiple scattering theory. This model, initially developed to describe the propagation of coupled longitudinal and transverse elastic waves in isotropic solid materials, takes into account wave conversions at the particle surface as well as positional correlations between particles via the pair correlation function. By modelling a Newtonian viscous liquid through complex frequency-dependent elastic moduli (i.e. using an imaginary shear modulus), the model is adapted to the case of viscous fluids and we show that conversions of longitudinal waves into transverse waves are important to take into account. We present asymptotic results from the model for case of long wavelength for the longitudinal waves, without assumptions on the shear wavelength. For high concentrations, these wave conversions are reinforced by the contribution of position correlations between the particles. In a subsequent paper, we present numerical validation of the asymptotic approximations and compare them with experimental results for solid particles in water.

Focused shear wave beam propagation through a 3D printed human rib cage

Transient elastography (TE) is used in clinical screenings for liver fibrosis. TE generates shear wave motion in the liver by vibration of a piston at the skin. The small, flat piston in current devices radiates shear waves that diverge from the liver, resulting in low SNR and high failure rates, especially in obese patients. Our group recently introduced focused shear wave beams for TE [Cormack et al., IEEE TBME (2024)], in which vibration of a concave piston generates shear waves that converge towards the focal region, thereby increasing SNR for liver stiffness estimation. However, because piston sizes needed for efficient shear wave focusing are larger than the typical intercostal space, the rib cage that lies between the skin and the liver may cause shear wave aberration and influence stiffness measurement. Here, we present measurements of broadband focused shear wave propagation in tissue-mimicking gel in which are embedded 3D printed human ribs. Both straight elliptical rods and anatomically realistic 3D printed ribs are used as aberrators. Measurements are compared to 3D simulations of shear wave beam propagation [Archer et al., JASA (2023)]. Effects of shear wave aberration by the rib cage depend on shear wavelength, piston-to-rib distance, and piston radius and curvature.

From dynamic to passive elastography, what technique used?

Elastography is a method that allows to quantify mechanical properties of biological tissues. In medical imaging, this last technique has been developed mainly in ultrasound (US) and in magnetic resonance imaging (MRI). Both imaging techniques have advantages and drawbacks depending on the organs and pathologies investigated. In this presentation we focused on two organs, where each technique has its advantage to investigate mechanical properties: brain and muscle. In brain, the preferred technique is MRI since US propagation is very complicated through the skull bone. Taking advantage of 3D acquisition, we investigated brain tumors with a new elastography approach: passive elastography. From the natural vibration, such as cardiac beating or respiration, we locally recover the natural shear wavelength that propagates giving access a stiffness mapping of the brain. Results are compared with US acquire during neurosurgery, showing a good accordance between both techniques. In muscles, US are used to recover not only stiffness but also others mechanical parameters such as viscoelastic anisotropy or shear non-linearity. Special US sequences were developed to quantify these two parameters in different physiological conditions that change in real-time. This presentation shows that depending on the investigated organ, physicists and physicians need to adapt the imaging strategy to better characterize tissues.

Transient elastography with focused shear wave beams

Transient elastography (TE) is a clinically available ultrasound elastography technique in which mechanical vibration of a small piston at the tissue surface generates a short shear wave pulse that propagates into the body. The piston doubles as an ultrasonic transducer; thus, the shear wave propagation is measured with pulse-echo ultrasound along the beam axis, yielding the wave speed and tissue stiffness. The long shear wavelength compared to piston diameter used in current devices results in low shear wave signal that decays rapidly on axis as the shear wave spreads in all directions, thus limiting the signal-to-noise ratio and imaging depth in practice. We present an overview of our new technology of focused shear wave beams, which aims to overcome these limitations by concentrating shear wave energy in TE near to the beam axis with vibration of a concave circular piston. Analytical modeling of focused shear wave generation and propagation is validated against measurements in gelatin phantoms. Results indicate that focused shear wave beams have relatively larger amplitudes and penetration depths compared to unfocused shear wave beams. Ongoing modeling and experimental efforts towards the measurement of elastic anisotropy, such as of skeletal muscle, using focused shear wave beams will be discussed.

MR elastography of soft porous tissue phantoms with varying macrostructure and pore pressure

Tissue porosity pressure mechanics play a vital role in the physiological workings and anatomical structures of soft tissue. Further understanding relationships between the pressurized state of the tissue and its function by noninvasive measurements may lead to new diagnostic methods for identifying early stages of disease pathology. The purpose of this study was to investigate the ability of Magnetic Resonance Elastography (MRE) to identify mechanics related to different pressurized states of porous soft tissue-like media. Phantoms with variations in stiffness and porous network geometries are scanned with MRE at 9.4 Tesla while pore pressure is quasi-statically loaded with 0, 50, 100, 150 mmHg gauge using air and liquid pressure. The MRE-derived shear modulus of tissue around isolated pores correlates with pore pressure. However, this was not observed for interconnected porous systems, in which the shear wavelength is much larger than the distance between neighboring pore branches. In these systems, the presented experimental results highlight the importance of a compressibility marker derived from 3D MRE acquisitions, which may be sensitive to pore pressure and pore network geometry.

Comparative Analysis of a Locally Resampling MR Elastography Reconstruction Algorithm in Liver Fibrosis.

In magnetic resonance elastography (MRE), the precision of the observed mechanical depends on the ratio between mechanical wavelength and spatial resolution. Since the mechanical wavelength may vary with actuation frequency, between patients and depending on position, a unique spatial resolution may not always generate an optimal ratio for multifrequency acquisitions, in patients with varying degrees of disease or in mechanically heterogeneous organs. To describe an MRE reconstruction algorithm that adjusts the ratio between shear wavelength and pixel size, by locally resampling the matrix of shear displacement, and to assess its performance relative to existing reconstructions in different use cases. Prospective. Four phantoms, 20 healthy volunteers (5 men, median age 34, range 20-56) and 46 patients with nonalcoholic fatty liver disease (37 men, median age 63, range 33-83). A 3 T; gradient-echo elastography sequence with 40 Hz, 60 Hz, and 80 Hz frequencies. For each algorithm, phantoms stiffness were compared against their nominal values, repeatability was calculated in healthy volunteers, and diagnostic performance in detecting advanced liver fibrosis was assessed in 46 patients. Linear regression was used to evaluate the agreement between stiffness values and phantoms stiffnesses. Bland-Altman method was used to evaluate repeatability in volunteers. The ability to diagnose advanced fibrosis was assessed by receiver operating curve analysis (with Youden index thresholds). Significance was considered at P value of 0.05. From the linear regression, the slope closest to 1 is provided by MARS (40 Hz) and k-MDEV (60H, 80 Hz). Repeatability index was best with MDEV (23%) and lowest with k-MDEV (53%). The best performance in detecting advanced fibrosis was provided by MARS at 40 Hz (area under the operating curve, AUC=0.88), k-MDEV and MARS at 60 Hz (AUC=0.91), and multimodel direct inversion (MMDI) and MARS at 80 Hz (AUC=0.90). MARS shows the best diagnostic performance to detect advanced fibrosis and the second-best results in phantoms after k-MDEV. 1. Stage 2.

An analytical solution for the dynamic tunnel–soil–shallow foundation interaction under a harmonic point load

Elastic waves excited by railway tunnels can induce excessive vibration in nearby buildings. The dynamic interaction between the tunnel, soils, and the shallow foundations of buildings can significantly affect wave transmission from the vibration source to buildings. In this study, a three-dimensional theoretical model involving structure–soil–structure interaction (SSSI) is proposed and the analytical solution is obtained. The foundation is assumed to be seated on the ground surface, and the full coupling between the soil and structures is considered through the perfect contact at the tunnel-soil and soil-foundation interfaces. The response of the foundation to an incident wave field generated by a harmonic point load in a tunnel is calculated, and the impact of the tunnel–soil–foundation interaction is evaluated. The numerical results show that the SSSI needs to be considered when the relative distance between the tunnel and the foundation is within twice the shear wavelength. The vertical compliance coefficient at the center of the foundation features a significant peak due to the existence of a tunnel at the resonance frequency of the soil-tunnel system. The existence of the foundation restrains the displacement response of the ground surface within the foundation area, while causing significant amplification of the surface displacement outside the foundation area.

Simplified Green’s function for surface waves in quasi-incompressible elastic plates with application to elastography

Surface wave elastography is a growing method to estimate the elasticity in soft solids. It is particularly useful in the case of agrifoods like meat, cheese, or fruits because it does not require major infrastructure or large equipment and could be developed in portable devices. However, estimating the shear elastic properties from surface wave measurements is not straightforward. The shear wavelength in those materials is cm sized for the excitation frequencies usually employed in elastography (∼102 Hz), and the size of samples is comparable to it. Thus, the surface wave speed is frequency dependent with no direct relation to the shear wave speed. In this work we propose a simplified Green’s function for soft solid elastic plates which allows to retrieve the shear elasticity from near field measurements. The model is compared with experimental results obtained in agar–gelatin phantoms and food samples (cheese and bovine liver). The results show a good overall agreement although improvements can be achieved by incorporating diffraction and viscosity to the model.

An improved 25‐point finite‐difference scheme for frequency‐domain elastic wave modelling

ABSTRACTFrequency‐domain finite‐difference modelling is widely used in exploration geophysics. However, it involves solving a large system of linear algebraic equations, which may cause huge computational costs, especially for large‐scale models. To address this issue, we have proposed an improved 25‐point finite‐difference scheme for wavefield modelling of two‐dimensional frequency‐domain elastic wave equations. The biggest difference between the improved 25‐point scheme with other 25‐point schemes is the finite‐difference formulas for spatial derivatives. The proposed 25‐point scheme applies to equal and unequal grid intervals, and its optimization coefficients depend on the grid‐spacing ratio and Poisson's ratio. The dispersion analysis indicates that within the phase velocity errors of 1% and 2%, the improved 25‐point scheme only requires approximately 2.3 and 2.2 grid points per shear wavelength, which has greater accuracy than the existing schemes. To eliminate artificial boundary reflections, we apply the perfectly matched layer boundary conditions to the model edges. Several numerical examples are presented to prove the validity of the improved 25‐point scheme.

An affine generalized optimal scheme with improved free-surface expression using adaptive strategy for frequency-domain elastic wave equation

Effective frequency-domain numerical schemes are central for forward modeling and inversion of the elastic wave equation. The rotated optimal nine-point scheme is a highly utilized finite-difference (FD) numerical scheme. This scheme makes a weighted average of the derivative terms of the elastic wave equations in both the original and the rotated coordinate systems. In comparison with the classical nine-point scheme, it can simulate S-waves better and has higher accuracy at nearly the same computational cost. Nevertheless, this scheme limits the rotation angle to 45°; thus, the grid sampling intervals in the x- and z-directions need to be equal. Otherwise, the grid points will not lie on the axes, which dramatically complicates the scheme. Affine coordinate systems do not constrain axes to be perpendicular to each other, providing enhanced flexibility. Based on the affine coordinate transformations, we developed a new affine generalized optimal nine-point scheme. At the free surface, we applied the improved free-surface expression with an adaptive parameter-modified strategy. The new optimal scheme had no restriction that the rotation angle must be 45°. Dispersion analysis shows that our scheme can effectively reduce the required number of grid points per shear wavelength for both equal and unequal sampling intervals compared to the classical nine-point scheme. Moreover, this reduction improves with the increase of Poisson’s ratio. Three numerical examples demonstrate that our scheme can provide more accurate results than the classical nine-point scheme in terms of both the internal and free-surface grid points.

Low-frequency (105 Hz) shear modulus of nanosuspension

The low-frequency shear wave propagation in a suspension of silica nanoparticles in a polyethylsiloxane liquid was studied. The shear wave length was measured on ultrasonic interferometer, and it is equal to 55 μm. The value of the tangent of the mechanical losses angle is determined to be 0.18. These parameters were used to calculate the shear modulus of the investigated colloidal suspension; its value is 0.15‒105 Pa. The results obtained are in quite agreement with the data obtained by another way of the acoustic resonance method.

Influence of gas expansion on the propagation of a premixed flame in a spatially periodic shear flow

It has been previously demonstrated that thermal gas expansion might have a role in boundary layer flashback of premixed turbulent flames [Gruber et al., J Fluid Mech 2012], inducing local flow-reversal in the boundary layer's low-velocity streaks on the reactants’ side of the flame and facilitating its upstream propagation. We perform a two-dimensional numerical investigation of the interaction between a periodic shear flow and a laminar premixed flame. The periodic shear is a simplified model for the oncoming prolonged streamwise velocity streaks with alternating regions of high and low velocities found in turbulent boundary layers in the vicinity of the walls. The parametric study focuses on the amplitude and wavelength of the periodic shear flow and on the gas expansion ratio (unburnt-to-burnt density ratio). With the increase of the amplitudes of the periodic shear flow and of the gas expansion, the curved flame velocity increases monotonically. The flame velocity dependence on the periodic shear wavelength is non-monotonic, which is consistent with previous theoretical studies of curved premixed flame velocity. The flame shape that is initially formed by the oncoming periodic shear appears to be metastable. At a later stage of the flame propagation, the flame shape transforms into the stationary one dominated by the Darrieus-Landau instability.

Natural shear wave imaging using vocal tract vibrations: Introducing vocal passive elastography (V-PE) to thyroid elasticity mapping

We present a shear wave elastography (SWE) approach for thyroid characterization that is inspired by passive elastography, which extracts elasticity from the natural vibrations in living tissues that are caused by cardiac motion, blood pulsatility, and muscle activity. On thyroid, this physiological noise is mainly due to the carotid pulsation, which is in the 1–10 Hz bandwidth and is located right next to the gland. In order to decrease the shear wavelength and increase the signal to noise ratio, we propose to create in the thyroid a complex shear wave field by using natural vocal tract vibrations. The nature of the sound can be easily modified to be narrow or broad band, with small or large amplitude. Using correlation-based algorithm and a sound sustained at 150 Hz, we have developed an innovative technique using ultrasound, allowing us to compute a 2D shear wave velocity map, superposed onto a B-mode ultrasound image of a volunteer's thyroid. Using our vocal passive elastography method, shear wave velocity was measured at every point within a mask surrounding the thyroid with a pixel resolution of 150 × 150 μm2. The mean shear wave speed value measured is 3.2 m/s, taking its value from 0.7 m/s to 8.8 m/s. The values obtained were in good agreement with comparative shear wave elastography (SWE) measurements.

Scattering coefficients for a sphere in a visco-acoustic medium for arbitrary partial wave order

Analytical solutions are reported for the scattering coefficients of a solid elastic sphere suspended in a viscous fluid for arbitrary partial wave order. Expressions are derived for incident compressional and shear wave modes, taking into account the viscosity of the surrounding fluid and resultant wave mode conversion. The long compressional wavelength limit is employed to simplify the derivation, whereas no restriction is placed on the shear wavelength in the fluid compared to the particle dimension. The analytical approximations are compared with numerical results obtained from matrix inversion of the boundary equations and agree within the validity domain of the solutions.

Super-resolution limit of shear-wave elastography

When a wave field is measured within a propagative medium, it is widely accepted that the resulting image resolution depends on the measuring point density, and no longer on the wavelength. Indeed, in situ measurements allow the near-field details needed for super-resolution to be retrieved. Rarely studied in elastography, this is supported here by experiments. A passive elastography imaging of two inclusions in a tissue mimicking phantom is shown with a resolution down to 1/45 of a shear wavelength.

Numerical modeling of elastic wave in frequency-domain by using staggered grid fourth-order finite-difference scheme

Simulation of elastic wave propagation is an important method for oil and gas exploration. Accuracy and efficiency of elastic wave simulation in complex geological environment are always the focus issue. In order to improve the accuracy and efficient in numerical modeling of elastic modeling, a staggered grid fourth-order finite-difference scheme of modeling elastic wave in frequency-domain is developed, which can provide stable numerical solution with fewer number of grid points per wavelength. The method is implemented on first-order velocity-stress equation and a parsimonious spatial staggered-grid with fourth-order approximation of the first-order derivative operator. Numerical tests show that the accuracy of the fourth-order staggered-grid stencil is superior to that of the mixed-grid and other conventional finite difference stencils, especially in terms of shear-wave phase velocity. Measures of mass averaging acceleration and optimization of finite difference coefficients are taken to improve the accuracy of numerical results. Meanwhile, the numerical accuracy of the finite difference scheme can be further improved by enlarging the mass averaging area at the price of expanding the bandwidth of the impedance matrix that results in the reduction of the number of grid points to 3 per shear wavelength and computer storage requirement in simulation of practical models. In our scheme, the phase velocities of compressional and shear wave are insensitive to Poisson's ratio that does not occur to conventional finite difference scheme in most cases, and also the elastic wave modeling can degenerate to acoustic case automatically when the medium is pure fluid or gas. Furthermore, the staggered grid scheme developed in this study is suitable for modeling waves propagating in media with coupling fluid-solid interfaces that are not resolved very well for previous finite difference method. Cited as : Ma, C., Gao, Y., Lu, C. Numerical modeling of elastic wave in frequency-domain by using staggered grid fourth-order fifinite-difference scheme. Advances in Geo-Energy Research, 2019, 3(4): 410-423, doi: 10.26804/ager.2019.04.08

Spectral bearing capacity analysis of strip footings under pseudo-dynamic excitation

ABSTRACT Spectral seismic bearing capacity of strip foundations is evaluated by the limit equilibrium method (LEM). In this study, the pseudo-dynamic approach is followed to take into account seismic accelerations. The formulation of spectral pseudo-dynamic bearing capacity of strip foundations is presented for the Coulomb failure mechanism. The results are reported in the form of joint bearing capacity considering the contributions of cohesion, surcharge, and soil weight, simultaneously. A parametric study is conducted to evaluate the influence of soil friction angle, interface wall friction angle, dimensionless ratio of cohesion, depth factor, primary and shear wavelength values, amplification factor as well as the frequency on the seismic bearing capacity factor. The results of this study are shown in the form of design aids and tabular for the sake of simplicity to be used by engineers. The consistency of the results with those reported in literature shows the ability and efficiency of the methodology presented herein. It is shown that both the seismic loading characteristics and shear strength parameters affect the bearing capacity of shallow footings.

Appraising Kirchhoff approximation theory for the scattering of elastic shear waves by randomly rough defects

Rapid and accurate methods, based on the Kirchhoff approximation (KA), are developed to evaluate the scattering of shear waves by rough defects and quantify the accuracy of this approximation. Defect roughness has a strong effect on the reflection of ultrasound, and every rough defect has a different surface, so standard methods of assessing the sensitivity of inspection based on smooth defects are necessarily limited. Accurately resolving rough cracks in non-destructive evaluation (NDE) inspections often requires shear waves since they have higher sensitivity to surface roughness than longitudinal waves. KA models are attractive, since they are rapid to deploy, however they are an approximation and it is important to determine the range of validity for the scattering of ultrasonic shear waves; this range is found here. Good agreement between KA and high fidelity finite element simulations is obtained for a range of incident/scattering angles, and the limits of validity for KA are found to be much stricter than for longitudinal wave incidence; as the correlation length of rough surfaces is reduced to the order of the incident shear wavelength, a combination of multiple scattering and surface wave mode conversion leads to KA predictions diverging from those of the true diffuse scattered fields.

Evaluation of shear wave dispersion in hepatic viscoelastic models including fibrous structure

In diagnosis of chronic hepatitis, shear wave elastography is utilized for the evaluation of fibrosis progression by estimating the viscoelasticity from shear wave speeds. The dispersion slope method is studied to evaluate the viscosity and fibrosis stage. However, when the shear wave length is a little larger than the size of fibrous structure in liver fibrosis tissues, shear wave propagates repeating reflection or refraction and influences the dispersion slope. We simulate shear wave propagation in a model with a fibrous structure and compare the dispersion slope with a homogeneous model in order to investigate the effect of the fibrous structure. As a result, the dispersion slope is dependent on the viscosity in a high frequency range but affected by the fibrous structure in a low frequency range. Results show that the fibrous structure influences contents of low frequencies. Using different frequency ranges to estimate the dispersion slope can help more precise viscoelasticity analysis.

Waveguide effects and implications for cardiac magnetic resonance elastography: A finite element study.

Magnetic resonance elastography (MRE) is increasingly being applied to thin or small structures in which wave propagation is dominated by waveguide effects, which can substantially bias stiffness results with common processing approaches. The purpose of this work was to investigate the importance of such biases and artifacts on MRE inversion results in: (i) various idealized 2D and 3D geometries with one or more dimensions that are small relative to the shear wavelength; and (ii) a realistic cardiac geometry. Finite element models were created using simple 2D geometries as well as a simplified and a realistic 3D cardiac geometry, and simulated displacements acquired by MRE from harmonic excitations from 60 to 220Hz across a range of frequencies. The displacement wave fields were inverted with direct inversion of the Helmholtz equation with and without the application of bandpass filtering and/or the curl operator to the displacement field. In all geometries considered, and at all frequencies considered, strong biases and artifacts were present in inversion results when the curl operator was not applied. Bandpass filtering without the curl was not sufficient to yield accurate recovery. In the 3D geometries, strong biases and artifacts were present in 2D inversions even when the curl was applied, while only 3D inversions with application of the curl yielded accurate recovery of the complex shear modulus. These results establish that taking the curl of the wave field and performing a full 3D inversion are both necessary steps for accurate estimation of the shear modulus both in simple thin-walled or small structures and in a realistic cardiac geometry when using simple inversions that neglect the hydrostatic pressure term. In practice, sufficient wave amplitude, signal-to-noise ratio, and resolution will be required to achieve accurate results.