Propagation of shear waves generated by a modulated finite amplitude radiation force in a viscoelastic medium

An effective way to generate localized narrow-band low-frequency shear waves within tissue noninvasively, is by the modulated radiation force, resulting from the interference of two confocal quasi-CW ultrasound beams of slightly different frequencies. By using approximate viscoelastic Green's functions, investigations of the properties of the propagated shear-field component at the fundamental modulation frequency were previously reported by our group. However, high-amplitude source excitations may be needed to increase the signal-to-noise-ratio for shear-wave detection in tissue. This paper reports a study of the generation and propagation of dynamic radiation force components at harmonics of the modulation frequency for conditions that generally correspond to diagnostic safety standards. We describe the propagation characteristics of the resulting harmonic shear waves and discuss how they depend on the parameters of nonlinearity, focusing gain, and absorption. For conditions of high viscosity (believed to be characteristic of soft tissue) and higher modulation frequencies, the approximate shear wave Green's function is inappropriate. A more exact viscoelastic Green's function is derived in k-space, and using this, it is shown that the lowpass and dispersive effects, associated with a Voigt model of tissue, are more accurately represented. Finally, it is shown how the viscoelastic properties of the propagating medium can be estimated, based on several spectral components of the shear wave spectrum.     

Supersonic shear imaging: a new technique for soft tissue elasticity mapping

Supersonic shear imaging (SSI) is a new ultrasound-based technique for real-time visualization of soft tissue viscoelastic properties. Using ultrasonic focused beams, it is possible to remotely generate mechanical vibration sources radiating low-frequency, shear waves inside tissues. Relying on this concept, SSI proposes to create such a source and make it move at a supersonic speed. In analogy with the "sonic boom" created by a supersonic aircraft, the resulting shear waves will interfere constructively along a Mach cone, creating two intense plane shear waves. These waves propagate through the medium and are progressively distorted by tissue heterogeneities. An ultrafast scanner prototype is able to both generate this supersonic source and image (5000 frames/s) the propagation of the resulting shear waves. Using inversion algorithms, the shear elasticity of medium can be mapped quantitatively from this propagation movie. The SSI enables tissue elasticity mapping in less than 20 ms, even in strongly viscous medium like breast. Modalities such as shear compounding are implementable by tilting shear waves in different directions and improving the elasticity estimation. Results validating SSI in heterogeneous phantoms are presented. The first in vivo investigations made on healthy volunteers emphasize the potential clinical applicability of SSI for breast cancer detection.     

On the effects of reflected waves in transient shear wave elastography

In recent years, novel quantitative techniques have been developed to provide noninvasive and quantitative stiffness images based on shear wave propagation. Using radiation force and ultrafast ultrasound imaging, the supersonic shear imaging technique allows one to remotely generate and follow a transient plane shear wave propagating in vivo in real time. The tissue shear modulus, i.e., its stiffness, can then be estimated from the shear wave local velocity. However, because the local shear wave velocity is estimated using a time-of- flight approach, reflected shear waves can cause artifacts in the estimated shear velocity because the incident and reflected waves propagate in opposite directions. Such effects have been reported in the literature as a potential drawback of elastography techniques based on shear wave speed, particularly in the case of high stiffness contrasts, such as in atherosclerotic plaque or stiff lesions. In this letter, we present our implementation of a simple directional filter, previously used for magnetic resonance elastography, which separates the forward- and backward-propagating waves to solve this problem. Such a directional filter could be applied to many elastography techniques based on the local estimation of shear wave speed propagation, such as acoustic radiation force imaging (ARFI), shearwave dispersion ultrasound vibrometry (SDUV), needle-based elastography, harmonic motion imaging, or crawling waves when the local propagation direction is known and high-resolution spatial and temporal data are acquired.     

Harmonic wave propagation in viscoelastic heterogeneous materials Part I: Dispersion and damping relations

An analytico-numerical method is presented to study the propagation of plane harmonic waves in infinite periodic linear viscoelastic media. Part I considers only the dispersion and attenuation of acoustical longitudinal and shear waves. To show the accuracy of the method, examples of plane harmonic wave propagation in an infinite homogeneous medium and in a periodic layered viscoelastic medium are presented. The method is then used to calculate the damping and dispersion relations for a fibre-reinforced viscoelastic composite material. The results show clearly the influence of materials' viscoelastic properties and heterogeneities on the propagation of plane harmonic waves through the media.     

The role of viscosity in the impulse diffraction field of elastic waves induced by the acoustic radiation force

Several ultrasound-based techniques for the estimation of soft tissue elasticity are currently being investigated. Most of them study the medium response to dynamic excitations. Such responses are usually modeled in a purely elastic medium using a Green's function solution of the motion equation. However, elasticity by itself is not necessarily a discriminant parameter for malignancy diagnosis. Modeling viscous properties of tissues could also be of great interest for tumor characterization. We report in this paper an explicit derivation of the Green's function in a viscous and elastic medium taking into account shear, bulk, and coupling waves. From this theoretical calculation, 3D simulations of mechanical waves in viscoelastic soft tissues are presented. The relevance of the viscoelastic Green's function is validated by comparing simulations with experimental data. The experiments were conducted using the supersonic shear imaging (SSI) technique which dynamically and remotely excites tissues using acoustic radiation force. We show that transient shear waves generated with SSI are modeled very precisely by the Green's function formalism. The combined influences of out-of-plane diffraction, beam shape, and shear viscosity on the shape of transient waves are carefully studied as they represent a major issue in ultrasound-based viscoelasticity imaging techniques.     

3-D FDTD simulation of shear waves for evaluation of complex modulus imaging

The Navier equation describing shear wave propagation in 3-D viscoelastic media is solved numerically with a finite differences time domain (FDTD) method. Solutions are formed in terms of transverse scatterer velocity waves and then verified via comparison to measured wave fields in heterogeneous hydrogel phantoms. The numerical algorithm is used as a tool to study the effects on complex shear modulus estimation from wave propagation in heterogeneous viscoelastic media. We used an algebraic Helmholtz inversion (AHI) technique to solve for the complex shear modulus from simulated and experimental velocity data acquired in 2-D and 3-D. Although 3-D velocity estimates are required in general, there are object geometries for which 2-D inversions provide accurate estimations of the material properties. Through simulations and experiments, we explored artifacts generated in elastic and dynamic-viscous shear modulus images related to the shear wavelength and average viscosity.     

Detection of tissue harmonic motion induced by ultrasonic radiation force using pulse-echo ultrasound and Kalman filter

A method using pulse echo ultrasound and the Kalman filter is developed for detecting submicron harmonic motion induced by ultrasonic radiation force. The method estimates the amplitude and phase of the motion at desired locations within a tissue region with high sensitivity. The harmonic motion generated by the ultrasound radiation force is expressed as extremely small oscillatory Doppler frequency shifts in the fast time (A-line) of ultrasound echoes, which are difficult to estimate. In slow time (repetitive ultrasound echoes) of the echoes, the motion also is presented as oscillatory phase shifts, from which the amplitude and phase of the harmonic motion can be estimated with the least mean squared error by Kalman filter. This technique can be used to estimate the traveling speed of a harmonic shear wave by tracking its phase changes during propagation. The shear wave propagation speed can be used to solve for the elasticity and viscosity of tissue as reported in our earlier study. Validation and in vitro experiments indicate that the method provides excellent estimations for very small (submicron) harmonic vibrations and has potential for noninvasive and quantitative stiffness measurements of tissues such as artery.     

Propagation of spontaneously actuated pulsive vibration in human heart wall and in vivo viscoelasticity estimation

Though myocardial viscoelasticity is essential in the evaluation of heart diastolic properties, it has never been noninvasively measured in vivo. By the ultrasonic measurement of the myocardial motion, we have already found that some pulsive waves are spontaneously excited by aortic-valve closure (AVC) at end-systole (T0). These waves may serve as an ideal source of the intrinsic heart sound caused by AVC. In this study, using a sparse sector scan, in which the beam directions are restricted to about 16, the pulsive waves were measured almost simultaneously at about 160 points set along the heart wall at a sufficiently high frame rate. The consecutive spatial phase distributions, obtained by the Fourier transform of the measured waves, clearly revealed wave propagation along the heart wall for the first time. The propagation time of the wave along the heart wall is very small (namely, several milliseconds) and cannot be measured by conventional equipment. Based on this phenomenon, we developed a means to measure the myocardial viscoelasticity in vivo. In this measurement, the phase velocity of the wave is determined for each frequency component. By comparing the dispersion of the phase velocity with the theoretical one of the Lamb wave (the plate flexural wave), which propagates along the viscoelastic plate (heart wall) immersed in blood, the instantaneous viscoelasticity is determined noninvasively. This is the first report of such noninvasive determination. In in vivo experiments applied to five healthy subjects, propagation of the pulsive wave was clearly visible in all subjects. For the 60-Hz component, the typical propagation speed rapidly decreased from 5 m/s just before the time of AVC (t = T0 - 8 ms) to 3 m/s at t = T0 + 10 ms. In the experiments, it was possible to determine the viscosity more precisely than the elasticity. The typical value of elasticity was about 24-30 kPa and did not change around the time of AVC. The typical transient values of viscosity decreased rapidly from 400 Pa x s at t = T0 - 8 ms to 70 Pa x s at t = T0 + 10 ms. The measured shear elasticity and viscosity in this study are comparable to those obtained for the human tissues using audio frequency in in vitro experiments reported in the literature.     

Assessment of viscous and elastic properties of sub-wavelength layered soft tissues using shear wave spectroscopy: theoretical framework and in vitro experimental validation

In elastography, quantitative imaging of soft tissue elastic properties is provided by local shear wave speed estimation. Shear wave imaging in a homogeneous medium thicker than the shear wavelength is eased by a simple relationship between shear wave speed and local shear modulus. In thin layered organs, the shear wave is guided and thus undergoes dispersive effects. This case is encountered in medical applications such as elastography of skin layers, corneas, or arterial walls. In this work, we proposed and validated shear wave spectroscopy as a method for elastic modulus quantification in such layered tissues. Shear wave dispersion curves in thin layers were obtained by finite-difference simulations and numerical solving of the boundary conditions. In addition, an analytical approximation of the dispersion equation was derived from the leaky Lamb wave theory. In vitro dispersion curves obtained from phantoms were consistent with numerical studies (deviation <1.4%). The least-mean-squares fitting of the dispersion curves enables a quantitative and accurate (error < 5% of the transverse speed) assessment of the elasticity. Dispersion curves were also found to be poorly influenced by shear viscosity. This phenomenon allows independent recovery of the shear modulus and the viscosity, using, respectively, the dispersion curve and the attenuation estimation along the propagation axis.     

Net flow of compressible viscous liquids induced by travelling waves in porous media

The influence of ultrasonic radiation on the flow of a liquid through a porous medium is analyzed. The analysis is based on a mechanism proposed by Ganiev et al. according to which ultrasonic radiation deforms the walls of the pores in the shape of travelling transversal waves. Like in peristaltic pumping, the travelling transversal wave induces a net flow of the liquid inside the pore. In this article, the wave amplitude is related to the power output of an acoustic source, while the wave speed is expressed in terms of the shear modulus of the porous medium. The viscosity as well as the compressibility of the liquid are taken into account. The Navier–Stokes equations for an axisymmetric cylindrical pore are solved by means of a perturbation analysis, in which the ratio of the wave amplitude to the radius of the pore is the small parameter. In the second-order approximation a net flow induced by the travelling wave is found. For various values of the compressibility of the liquid, the Reynolds number and the frequency of the wave, the net flow rate is calculated. The calculations disclose that the compressibility of the liquid has a strong influence on the net flow induced. Furthermore, by a comparison with the flow induced by the pressure gradient in an oil reservoir, the net flow induced by a travelling wave can not be neglected, although it is a second-order effect.     

Propagation of inhomogeneous plane waves in anisotropic viscoelastic media

A new technique is explained to study the propagation of inhomogeneous waves in a general anisotropic medium. The harmonic plane waves are considered in a viscoelastic anisotropic medium. The complex slowness vector is decomposed into propagation vector and attenuation vector for the given directions of propagation and attenuation of waves in an unbounded medium. The attenuation is further separated into the contributions from homogeneous and inhomogeneous waves. A non-dimensional inhomogeneity parameter is defined to represent the deviation of an inhomogeneous wave from its homogeneous version. Such a partition of slowness vector of a plane wave is obtained with the help of an algebraic method for solving a cubic equation and a numerical method for solving a real transcendental equation. Derived specifications enable to study the 3D propagation of inhomogeneous plane waves in a viscoelastic medium of arbitrary anisotropy. The whole procedure is wave-specific and obtains the propagation characteristics for each of the three inhomogeneous waves in the anisotropic medium. Numerical examples analyze the variations in propagation characteristics of each of the three waves with propagation direction and inhomogeneity strength.     

Dynamic mechanical response of elastic spherical inclusions to impulsive acoustic radiation force excitation

Acoustic radiation force impulse imaging has been used clinically to study the dynamic response of lesions relative to their background material to focused, impulsive acoustic radiation force excitations through the generation of dynamic displacement field images. Dynamic displacement data are typically displayed as a set of parametric images, including displacement immediately after excitation, maximum displacement, time to peak displacement, and recovery time from peak displacement. To date, however, no definitive trends have been established between these parametric images and the tissues' mechanical properties. This work demonstrates that displacement magnitude, time to peak displacement, and recovery time are all inversely related to the Young's modulus in homogeneous elastic media. Experimentally, pulse repetition frequency during displacement tracking limits stiffness resolution using the time to peak displacement parameter. The excitation pulse duration also impacts the time to peak parameter, with longer pulses reducing the inertial effects present during impulsive excitations. Material density affects tissue dynamics, but is not expected to play a significant role in biological tissues. The presence of an elastic spherical inclusion in the imaged medium significantly alters the tissue dynamics in response to impulsive, focused acoustic radiation force excitations. Times to peak displacement for excitations within and outside an elastic inclusion are still indicative of local material stiffness; however, recovery times are altered due to the reflection and transmission of shear waves at the inclusion boundaries. These shear wave interactions cause stiffer inclusions to appear to be displaced longer than the more compliant background material. The magnitude of shear waves reflected at elastic lesion boundaries is dependent on the stiffness contrast between the inclusion and the background material, and the stiffness and size of the inclusion dictate when shear wave reflections within the lesion will interfere with one another. Jitter and bias associated with the ultrasonic displacement tracking also impact the estimation of a tissue's dynamic response to acoustic radiation force excitation.     

Monitored steady-state excitation and recovery (MSSER) radiation force imaging using viscoelastic models

Acoustic radiation force imaging methods distinguish tissue structure and composition by monitoring tissue responses to applied radiation force excitations. Although these responses are a complex, multidimensional function of the geometric and viscoelastic nature of tissue, simplified discrete biomechanical models offer meaningful insight to the physical phenomena that govern induced tissue motion. Applying Voigt and standard linear viscoelastic tissue models, we present a new radiation force technique - monitored steady-state excitation and recovery (MSSER) imaging - that tracks both steady-state displacement during prolonged force application and transient response following force cessation to estimate tissue mechanical properties such as elasticity and viscosity. In concert with shear wave elasticity imaging (SWEI) estimates for Young's modulus, MSSER methods are useful for estimating tissue mechanical properties independent of the applied force magnitude. We test our methods in gelatin phantoms and excised pig muscle, with confirmation through mechanical property measurement. Our results measured 10.6 kPa, 14.7 kPa, and 17.1 kPa (gelatin) and 122.4 kPa (pig muscle) with less than 10% error. This work demonstrates the feasibility of MSSER imaging and merits further efforts to incorporate relevant mechanical tissue models into the development of novel radiation force imaging techniques.     

Bubble-based acoustic radiation force elasticity imaging

Acoustic radiation force is applied to bubbles generated by laser-induced optical breakdown (LIOB) to study viscoelastic properties of the surrounding medium. In this investigation, femtosecond laser pulses are focused in the volume of gelatin phantoms of different concentrations to form bubbles. A two-element confocal ultrasonic transducer generates acoustic radiation force on individual bubbles while monitoring their displacement within a viscoelastic medium. Tone burst pushes of varying duration have been applied by the outer element at 1.5 MHz. The inner element receives pulse-echo recordings at 7.44 MHz before, during, and after the excitation bursts, and crosscorrelation processing is performed offline to monitor bubble position. Maximum bubble displacements are inversely related to the Young's moduli for different gel phantoms, with a maximum bubble displacement of over 200 microm in a gel phantom with a Young's modulus of 1.7 kPa. Bubble displacements scale with the applied acoustic radiation force and displacements can be normalized to correct for differences in bubble size. Exponential time constants for bubble displacement curves are independent of bubble radius and follow a decreasing trend with the Young's modulus of the surrounding medium. These results demonstrate the potential for bubble-based acoustic radiation force methods to measure tissue viscoelastic properties.     

Surface waves in fiber-reinforced anisotropic general viscoelastic media of higher orders with voids,rotation, and electromagnetic field

In this article, we investigated an effect of external magnetic field on the propagation of surface waves in a perfect electrically conducting fiber-reinforced anisotropic general viscoelastic media of rational and higher orders with voids in a rotating medium. The general surface wave speed is derived to investigate effect of electromagnetic field and rotation on surface waves in the presence of voids and viscosity. Boundary conditions are applied to obtain the secular equation for generalized types of waves. Particular cases of Stoneley, Love and Rayleigh waves are derived. The results obtained are more general in the sense that some earlier published results are obtained from our result as special cases. In the absence of voids, the results for viscoelasticity of order zero are in good agreement with the fiber-reinforced materials. Also by neglecting the reinforced elastic parameters, the results reduce to a well-known isotropic medium. It is observed that surface waves cannot propagate in a strong initially applied electromagnetic field and rotation. Numerical results for particular cases have been obtained and displayed graphically. The results indicate that the effects of voids, anisotropic, fiber-reinforcement, rotation and electromagnetic field are very pronounced and applicable for the phenomena.     

Formulation and validation of Berenger's PML absorbing boundary forthe FDTD simulation of acoustic scattering

Berenger's perfectly matched layer (PML) absorbing boundary condition for electromagnetic (EM) waves is derived to absorb 2-D and 3-D acoustic waves in finite difference time domain (FDTD) simulation of acoustic wave propagation and scattering. A PML medium suitable for acoustic waves is constructed. Plane wave propagation in the PML medium is solved for both 2-D and 3-D cases and explicit FDTD boundary conditions are derived. The equations show that a matched PML medium is a perfect simulation of free space in that a plane wave does not change its direction of propagation or its speed when it propagates from free space into a matched PML medium. FDTD simulation of a pulsed point source propagating in two dimensions is carried out to test the performance of the PML boundary for acoustic waves. Results show that an eight layer PML boundary condition reduces the reflected error 40 dB over Mur's second order boundary condition     

Propagation of torsional waves in a viscoelastic layer over an inhomogeneous half space

This paper presents a theoretical study on propagation of torsional surface waves in a homogeneous viscoelastic isotropic layer with Voigt type viscosity over an inhomogeneous isotropic infinite half space. The non-homogeneity in half space is assumed to arise due to exponential variation in shear modulus and density. A closed-form solution has been obtained for the displacement in the layer as well as for a infinite half space. The dispersion and absorption relations for an torsional wave under the assumed geometry have been found. Numerical results are presented for propagation characteristics in terms of a number of non-dimensionalized parameters and have been produced graphically. This study investigates the effect of various parameters, namely non-homogeneity parameter, internal friction, the layer width and complex wave number on dissipation function and phase velocity of the torsional wave. Results in some special cases are also compared with existing solutions available from analytical methods, which show a close agreement.     

Laser-Generated Rayleigh Waves Propagating in Transparent Viscoelastic Adhesive Coating/Metal Substrate Systems

We have established numerical models for simulating laser-generated Rayleigh waves in coating/substrate systems by a finite element method and investigated the propagation characteristics of Rayleigh waves in systems concerning the viscoelasticity and transparency of adhesive coatings. In this way, we have studied the influence of the mechanical properties of the coating, such as the elastic moduli, viscoelastic moduli, coating thickness, transparency, and coating material, on the propagation characteristics of the Rayleigh waves. The results show that the propagation characteristics of the Rayleigh waves can be divided into low- and high-frequency parts. The high-frequency propagation characteristics of the Rayleigh wave are closely related to the properties of the adhesive coating.     

Shearwave dispersion ultrasound vibrometry (SDUV) for measuring tissue elasticity and viscosity

Characterization of tissue elasticity (stiffness) and viscosity has important medical applications because these properties are closely related to pathological changes. Quantitative measurement is more suitable than qualitative measurement (i.e., mapping with a relative scale) of tissue viscoelasticity for diagnosis of diffuse diseases where abnormality is not confined to a local region and there is no normal background tissue to provide contrast. Shearwave dispersion ultrasound vibrometry (SDUV) uses shear wave propagation speed measured in tissue at multiple frequencies (typically in the range of hundreds of Hertz) to solve quantitatively for both tissue elasticity and viscosity. A shear wave is stimulated within the tissue by an ultrasound push beam and monitored by a separate ultrasound detect beam. The phase difference of the shear wave between 2 locations along its propagation path is used to calculate shear wave speed within the tissue. In vitro SDUV measurements along and across bovine striated muscle fibers show results of tissue elasticity and viscosity close to literature values. An intermittent pulse sequence is developed to allow one array transducer for both push and detect function. Feasibility of this pulse sequence is demonstrated by in vivo SDUV measurements in swine liver using a dual transducer prototype simulating the operation of a single array transducer.