Shear Wave Motion Research Articles

Patient-specific modelling of the calf muscle under elastic compression using magnetic resonance imaging and ultrasound elastography

Compression therapy is traditionally employed to achieve a variety of therapeutic goals related to venous insufficiency but its mechanisms are neither clearly understood nor conclusively demonstrated. (Avril et al., 2010) have pointed out that the capacity of the treatment to achieve the desired medical goal is closely related to the question of transmission of pressure through the soft tissues, as a result of which biomechanical models have been developed (Dubuis et al., 2012). In these models, the prediction of pressure was made based on the rather strong assumption that the behaviour of soft tissues (muscles and adipose tissues) can be approximated as homogeneous materials. Recent developments in ultrasound imaging can be helpful to improve both our understanding of the effect of medical compression stockings (MCS) on the blood return and on the quality of our predictions. (Bercoff et al., 2004) introduced a new ultrasound based technology called Supersonic Shear Imaging (SSI), for real-time soft tissue elasticity mapping. This transient elastography technique remotely produces a low frequency shear wave inside soft tissue through a radiation force by focusing ultrasonic beams at a supersonic speed. The shear wave motion is captured at a frame rate of 5 MHz. The velocity field of the medium can be mapped an dassuming the medium to be elastic, Young's modulus can be estimated quantitatively. This non-invasive technique allows identifying the material property heterogeneities inside the different compartments of the human leg. Using this experimental technique combined with FE modeling, the present study aims at shedding a light on the influence of those heterogeneities of material properties on the pressure inside the leg under elastic compression.

Discrepancies in Reporting Tissue Material Properties

Discrepancies in Reporting Tissue Material Properties

Comb-Push Ultrasound Shear Elastography (CUSE) With Various Ultrasound Push Beams

Comb-push ultrasound shear elastography (CUSE) has recently been shown to be a fast and accurate 2-D elasticity imaging technique that can provide a full field-of-view (FOV) shear wave speed map with only one rapid data acquisition. The initial version of CUSE was termed U-CUSE because unfocused ultrasound push beams were used. In this paper, we present two new versions of CUSE-focused CUSE (F-CUSE) and marching CUSE (M-CUSE), which use focused ultrasound push beams to improve acoustic radiation force penetration and produce stronger shear waves in deep tissues (e.g., kidney and liver). F-CUSE divides transducer elements into several subgroups which transmit multiple focused ultrasound beams simultaneously. M-CUSE uses more elements for each focused push beam and laterally marches the push beams. Both F-CUSE and M-CUSE can generate comb-shaped shear wave fields that have shear wave motion at each imaging pixel location so that a full FOV 2-D shear wave speed map can be reconstructed with only one data acquisition. Homogeneous phantom experiments showed that U-CUSE, F-CUSE, and M-CUSE can all produce smooth shear wave speed maps with accurate shear wave speed estimates. An inclusion phantom experiment showed that all CUSE methods could provide good contrast between the inclusion and background with sharp boundaries while F-CUSE and M-CUSE require shorter push durations to achieve shear wave speed maps with comparable SNR to U-CUSE. A more challenging inclusion phantom experiment with a very stiff and deep inclusion shows that better shear wave penetration could be gained by using F-CUSE and M-CUSE. Finally, a shallow inclusion experiment showed that good preservations of inclusion shapes could be achieved by both U-CUSE and F-CUSE in the near field. Safety measurements showed that all safety parameters are below FDA regulatory limits for all CUSE methods. These promising results suggest that, using various push beams, CUSE is capable of reconstructing a 2-D full FOV shear elasticity map using only one push-detection data acquisition in a wide range of depths for soft tissue elasticity imaging.

Estimating material viscoelastic properties based on surface wave measurements: A comparison of techniques and modeling assumptions

Previous studies of the first author and others have focused on low audible frequency (<1 kHz) shear and surface wave motion in and on a viscoelastic material comprised of or representative of soft biological tissue. A specific case considered has been surface (Rayleigh) wave motion caused by a circular disk located on the surface and oscillating normal to it. Different approaches to identifying the type and coefficients of a viscoelastic model of the material based on these measurements have been proposed. One approach has been to optimize coefficients in an assumed viscoelastic model type to match measurements of the frequency-dependent Rayleigh wave speed. Another approach has been to optimize coefficients in an assumed viscoelastic model type to match the complex-valued frequency response function (FRF) between the excitation location and points at known radial distances from it. In the present article, the relative merits of these approaches are explored theoretically, computationally, and experimentally. It is concluded that matching the complex-valued FRF may provide a better estimate of the viscoelastic model type and parameter values; though, as the studies herein show, there are inherent limitations to identifying viscoelastic properties based on surface wave measurements.

Two-dimensional shear elasticity imaging using comb-push acoustic radiation force and algebraic direct inversion of the motion differential equation

Tissue mechanical properties can be obtained by algebraic direct inversion (ADI) of the shear-wave motion differential equation, which is insensitive to reflection and geometry of the pushing beam. Shear waves normally generated by a focused ultrasound pushing beam have limited spatial extent in depth and are transient in time, leading to noisy and unstable ADI results. A comb-shape acoustic radiation force distribution can generate shear waves with longer spatial extent and time duration, facilitating more robust ADI. For 400 μs, a linear array transducer simultaneously transmitted four unfocused pushing beams (12 on elements for each pushing beam, 8 off elements between beams) into a tissue-mimicking phantom (∼1.5 kPa) with a cylindrical inclusion (∼4 kPa). The ultrasound system (Verasonics Inc.) then immediately switched to flash imaging mode (frame rate = 2 kHz, spatial resolution = 0.31 mm) to measure shear-wave displacements in a 38 mm by 30 mm domain that was used for ADI. The reconstructed 2D shear elasticity map provides accurate shear elasticity estimates (background = 1.5 kPa with 7% variance; inclusion = 3.9 kPa with 15% variance) and excellent contrast between the background and inclusion.

Calculation of shear stiffness in noise dominated magnetic resonance elastography data based on principal frequency estimation

Magnetic resonance elastography (MRE) is a non-invasive phase-contrast-based method for quantifying the shear stiffness of biological tissues. Synchronous application of a shear wave source and motion encoding gradient waveforms within the MRE pulse sequence enable visualization of the propagating shear wave throughout the medium under investigation. Encoded shear wave-induced displacements are then processed to calculate the local shear stiffness of each voxel. An important consideration in local shear stiffness estimates is that the algorithms employed typically calculate shear stiffness using relatively high signal-to-noise ratio (SNR) MRE images and have difficulties at an extremely low SNR. A new method of estimating shear stiffness based on the principal spatial frequency of the shear wave displacement map is presented. Finite element simulations were performed to assess the relative insensitivity of this approach to decreases in SNR. Additionally, ex vivo experiments were conducted on normal rat lungs to assess the robustness of this approach in low SNR biological tissue. Simulation and experimental results indicate that calculation of shear stiffness by the principal frequency method is less sensitive to extremely low SNR than previously reported MRE inversion methods but at the expense of loss of spatial information within the region of interest from which the principal frequency estimate is derived.

MR elastography of human lung parenchyma: Technical development, theoretical modeling and in vivo validation

To develop a novel MR-based method for visualizing the elastic properties of human lung parenchyma in vivo and to evaluate the ability of this method to resolve differences in parenchymal stiffness at different respiration states in healthy volunteers. A spin-echo MR Elastography (MRE) pulse sequence was developed to provide both high shear wave motion sensitivity and short TE for improved visualization of lung parenchyma. The improved motion sensitivity of this approach was modeled and tested with phantom experiments. In vivo testing was then performed on 10 healthy volunteers at the respiratory states of residual volume (RV) and total lung capacity (TLC). Shear wave propagation was visualized within the lungs of all volunteers and was processed to provide parenchymal shear stiffness maps for all 10 subjects. Density corrected stiffness values at TLC (1.83 ± 0.22 kPa) were higher than those at the RV (1.14 ± 0.14 kPa) with the difference being statistically significant (P < 0.0001). (1)H-based MR elastography can noninvasively measure the shear stiffness of human lung parenchyma in vivo and can quantitate the change in shear stiffness due to respiration. The values obtained were consistent with previously reported in vitro assessments of cadaver lungs. Further work is required to increase the flexibility of the current acquisition and to investigate the clinical potential of lung MRE.

Theoretical Analysis of Shear Wave Interference Patterns by Means of Dynamic Acoustic Radiation Forces

Acoustic radiation forces associated with high intensity focused ultrasound stimulate shear wave propagation allowing shear wave speed and shear viscosity estimation of tissue structures. As wave speeds are meters per second, real time displacement tracking over an extend field-of-view using ultrasound is problematic due to very high frame rate requirements. However, two spatially separated dynamic external sources can stimulate shear wave motion leading to shear wave interference patterns. Advantages are shear waves can be imaged at lower frame rates and local interference pattern spatial properties reflect tissue's viscoelastic properties. Here a theoretical analysis of shear wave interference patterns by means of dynamic acoustic radiation forces is detailed. Using a viscoelastic Green's function analysis, tissue motion due to a pair of focused ultrasound beams and associated radiation forces are presented. Overall, this paper theoretically demonstrates shear wave interference patterns can be stimulated using dynamic acoustic radiation forces and tracked using conventional ultrasound imaging.

Rayleigh–Lamb wave propagation on a fractional order viscoelastic plate

A previous study of the authors published in this journal focused on mechanical wave motion in a viscoelastic material representative of biological tissue [Meral et al., J. Acoust. Soc. Am. 126, 3278-3285 (2009)]. Compression, shear and surface wave motion in and on a viscoelastic halfspace excited by surface and sub-surface sources were considered. It was shown that a fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time, resulted in closer agreement with experiment as compared with conventional (integer order) models, such as those of Voigt and Zener. In the present study, this analysis is extended to another configuration and wave type: out-of-plane response of a viscoelastic plate to harmonic anti-symmetric Lamb wave excitation. Theoretical solutions are compared with experimental measurements for a polymeric tissue mimicking phantom material. As in the previous configurations the fractional order modeling assumption improves the match between theory and experiment over a wider frequency range. Experimental complexities in the present study and the reliability of the different approaches for quantifying the shear viscoelastic properties of the material are discussed.

Surface response of a fractional order viscoelastic halfspace to surface and subsurface sources

Previous studies by the second author published in this journal focused on low audible frequency (40-400 Hz) shear and surface wave motion in and on a viscoelastic material representative of biological tissue. Specific cases considered were that of surface wave motion on a halfspace caused by a finite rigid circular disk located on the surface and oscillating normal to it [Royston et al., J. Acoust. Soc. Am. 106, 3678-3686 (1999)] and compression, shear, and surface wave motion in a halfspace generated by a subsurface finite dipole [Royston et al., J. Acoust. Soc. Am. 113, 1109-1121 (2003)]. In both studies, a Voigt model of viscoelasticity was assumed in the theoretical treatment, which resulted in agreement between theoretical predictions and experimental measurements over a limited frequency range. In the present article, the linear viscoelastic assumption in these two prior works is revisited to consider a (still linear) fractional order Voigt model, where the rate-dependent damping component that is dependent on the first derivative of time is replaced with a component that is dependent on a fractional derivative of time. It is shown that in both excitation source configurations, the fractional order Voigt model assumption improves the match of theory to experiment over a wider frequency range (in some cases up to the measured range of 700 Hz).

Wave propagation through elastic granular and granular auxetic materials

AbstractThe wave propagation through a material with a granular micro‐structure is investigated. The simplest such medium is the granular medium of equal spheres. The theory demonstrates that for long wavelengths the micro‐structured material exhibits modes of wave motion that are also found for a traditional medium (that is, a medium that satisfies simple continuum stress equilibrium laws and no consideration for particle spin and moment of inertia of the particles is accounted for). However, an extra shear wave and rotational oscillatory particle motion are also found. The limit of a traditional material is obtained by letting tangential contact stiffnesses vanish. Towards higher wave numbers the micro‐structure reveals itself in the shear wave speeds by displaying strongly anomalous behaviour compared to a traditional isotropic, elastic medium. An expansion in terms of structural sums (these are sums over the contact interaction weighed with a string of branch vectors) is explored. The lowest order non‐trivial terms (second order structural sums) yield basic behaviour, but a higher order expansion (including fourth order sums) shows a more pronounced anomaly for wave numbers approaching the inverse particle diameter. Yet higher order terms capture more information than can possibly be present in a granular medium and it is senseless to examine these. The theory is relevant to granular materials and granular auxetics. In the latter case the tangential contact stiffness needs to exceed the normal stiffness, which is not possible for ordinary solid contacts, but can be achieved with special arrangements.

Optimization of ultrasound radiation force source for shear wave dispersion measurements.

The objective of this study was to analyze and optimize the ultrasound radiation force source for creating shear waves for quantitative elasticity and viscosity measurements. A simulation model based on work by Bercoff et al., [IEEE Trans. Ultrason. Ferroelectr. Freq. Contr. 51(11), 1523–34 (2004)] was utilized to examine the shear wave propagation from a spatiotemporal impulsive force and the simulated radiation force generated from a linear array transducer. Parameters such as aperture size, F-number of the transducer, and medium shear elasticity and viscosity were varied to analyze how the shear wave amplitude and shape changes. The wave attenuation and group and phase velocities were evaluated. The aperture size and the F-number of the transducer affected the spatial distribution of the propagating shear wave significantly. The influence of the coupling term between the bulk and shear waves was analyzed spatially and temporally by evaluating parameters such as shear wave amplitude, wave attenuation, and the group and phase velocities. This simulation model provides insight for making shear wave propagation measurements from the perspective of optimizing the spatial and temporal characteristics of induced shear wave motion from an applied radiation force. [This work was supported in part by Grant EB002167 from NIH.]

Shear wave dispersion ultrasound vibrometry of gel dosimetry phantoms.

Shear wave dispersion ultrasound vibrometry (SDUV) has been introduced to use the dispersive nature of shear wave speed to locally estimate the material properties or tissue. Shear waves were created using a mechanical shaker and pulsed radiation force. Using the information from the phase one can estimate the shear velocity. A theoretical Voigt model allows us to find, at different frequencies (100–400 Hz), the viscosity and shear elasticity of the polymer gel dosimeter. The aim of this work was to perform the SDUV and mechanical vibrometry techniques to study the viscoelastic properties of absorbed dose phantoms. The experiments were performed using five phantoms of different absorbed doses varying 10 up to 50 Gy, while a nonirradiated phantom was kept as a control. The SDUV technique was performed using a pulsed radiation force to vibrate the gel provide by a confocal transducer with a focal depth of 70 mm, nominal frequency of 3 MHz. For the mechanical vibrometry and SDUV experiments, shear wave motion measurements were made with a commercial ultrasound machine with a research using a 10 MHz linear array transducer. The quantitative elasticity and viscosity results for these two experiments were compared.

Rayleigh–Lamb wave propagation on a fractional order viscoelastic plate.

Previous studies of the authors published in JASA focused on shear and surface wave motion in and on a viscoelastic material representative of biological tissue. Specific cases considered were that of surface wave motion on a half‐space caused by a finite rigid circular disk located on the surface and oscillating normal to it and compression, and shear and surface wave motion in a half‐space generated by a subsurface finite dipole. In both situations, recently it has been shown that viscoelasticity models based on fractional order derivatives result in closer agreement with experiment as compared to conventional models, such as those of Voigt and Zener. In the present study, this comparative analysis is extended to another configuration and wave‐type, that of out‐of‐plane response of a circular viscoelastic plate to harmonic anti‐symmetric excitation. The Rayleigh–Lamb equation is solved using complex wave numbers based on conventional and fractional order viscoelastic models for the medium. The new theoretical solutions are compared with experimental measurements. The underlying physical motivation for the use of a fractional viscoelastic assumption and the potential practical benefits, such as improved medical diagnostics, are discussed. [Work supported by NIH Grants EB004885, EB008373, and EB007537.]

Model of a strong discontinuity for the equations of spatial long waves propagating in a free-boundary shear flow

The long-wave equations describing three-dimensional shear wave motion of a free-surface ideal fluid are rearranged to a special form and used to describe discontinuous solutions. Relations at the discontinuity front are derived, and stability conditions for the discontinuity are formulated. The problem of determining the flow parameters behind the discontinuity front from known parameters before the front and specified velocity of motion of the front are investigated.

Microscopic magnetic resonance elastography (μMRE)

Magnetic resonance elastography (MRE) was extended to the microscopic scale to image low-frequency acoustic shear waves (typically less than 1 kHz) in soft gels and soft biological tissues with high spatial resolution (34 micromx34 micromx500 microm). Microscopic MRE (microMRE) was applied to agarose gel phantoms, frog oocytes, and tissue-engineered adipogenic and osteogenic constructs. Analysis of the low-amplitude shear wave pattern in the samples allowed the material stiffness and viscous loss properties (complex shear stiffness) to be identified with high spatial resolution. microMRE experiments were conducted at 11.74 T in a 56-mm vertical bore magnet with a 10 mm diameterx75 mm length cylindrical space available for the elastography imaging system. The acoustic signals were generated at 550-585 Hz using a piezoelectric transducer and high capacitive loading amplifier. Shear wave motion was applied in synchrony with the MR pulse sequence. The field of view (FOV) ranged from 4 to 14 mm for a typical slice thickness of 0.5 mm. Increasing the agarose gel concentration resulted in an increase in shear elasticity and shear viscosity. Shear wave motion propagated through the frog oocyte nucleus, enabling the measurement of its shear stiffness, and in vitro shear wave images displayed contrast between adipogenic and osteogenic tissue-engineered constructs. Further development of microMRE should enable its use in characterizing stiffer materials (e.g., polymers, composites, articular cartilage) and assessing with high resolution the mechanical properties of developing tissues.

Microscopic dynamic magnetic resonance elastography

Microscopic magnetic resonance elastography (uMRE) is a high resolution imaging technique for measuring the viscoelastic properties of small synthetic and biological samples. Mechanical shear waves, typically with amplitudes of less than 100 μm and frequencies of 500–600 Hz, are induced using a piezoelectric oscillator directly coupled to the region of interest. By using multiple phase offsets and motion encoding gradients we acquire data that allows the generation of images that depict shear wave motion and the calculation of local values of the tissue viscoelastic properties. Recent MRE investigations are increasingly being conducted at higher spatial resolution to establish histological correlations between elasticity maps and tissue structures; such microscopic MRE studies require stronger static fields, stronger magnetic field gradients, higher performance RF coils, and more compact, higher frequency mechanical actuators. Microscopic MRE experiments were conducted at 11.74 T in a 54 mm diameter vertical bore magnet with a 10 mm diameter×25 mm length cylindrical space available for imaging. The field-of-view ranged from 4 to 14 mm. The study was conducted on agarose gel phantoms of different concentrations ranging from 025%–1% w. Different biological samples, including frog oocytes and tissue engineered constructs, were also tested.

Power transmission in L-shaped plates including flexural and in-plane vibration

In this paper, power flow propagation in plates connected in an L-joint is investigated in both the low and high frequency ranges. An exact solution is derived to describe the flexural, in-plane longitudinal and in-plane shear wave motion in the plates. The coupled plates are simply supported along two parallel sides, and free at the other two ends. A point force is used to generate flexural wave motion only. The flexural wave coefficients are determined from the boundary conditions, continuity equations at the driving force locations, and continuity equations at the corner junction of the plates. Structural intensity expressions are used to examine the structural noise transmission in the low and high frequency ranges. The contributions from the individual wave types are also examined.

Shear stiffness estimation using intravoxel phase dispersion in magnetic resonance elastography.

Dynamic MR elastography (MRE) is a phase-contrast technique in which the periodic shear motion of an object is encoded as variations in the phase of the reconstructed images. An alternative MRE method is presented whereby waves are depicted as intensity variations in the magnitude images due to intravoxel phase dispersion (IVPD). A theoretical framework is developed to model how the IVPD magnitude data are related to the underlying shear wave motion, and how they can be used to estimate shear stiffness. The results are shown in a series of phantom experiments to demonstrate that IVPD MRE complements phase-contrast MRE.

Propagation of shear waves in piezoelectric “superlattice-substrate” structures

The results of a computer simulation of the dispersion relations for the propagation of shear waves in piezoelectric “superlattice-substrate” structures are analyzed. The superlattice consists of a finite number of layers and is made up of materials with 6mm symmetry. The dispersion relations are obtained using a formulation for periodic hamiltonian systems. This approach makes it possible to account for the anisotropy, the piezoelectric interaction of the mechanical and electric fields, and an arbitrary number of layers in the superlattice. Numerical results are presented for CdS-ZnO layers. Selective spatial localization of acoustic modes is demonstrated for different spectral regions. The effects of the ordering of the superlattice layers, of their number, and of the boundary conditions on the dispersion spectra and on the form of the shear wave motion are examined.