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.     

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.     

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.     

Narrowband shear wave generation by a Finite-Amplitude radiation force: The fundamental component

A highly localized source of low-frequency shear waves can be created by the modulated radiation force resulting from two intersecting quasi-continuous-wave ultrasound beams of slightly different frequencies. In contrast to most other radiation force-based methods, these shear waves can be narrowband. Consequently, different frequency-dependent effects will not significantly affect their spectrum as they propagate within a viscoelastic medium, thereby enabling the viscoelastic shear properties of the medium to be determined at any given modulation frequency. This can be achieved by tracking the shear wave phase delay and change in amplitude over a specific distance. In this paper we explore the properties of short duration (dynamic) low-frequency shear wave propagation and study how the shear displacement field depends on the excitation conditions. Our investigations make use of the approximate Green's functions for viscoelastic media, and the evolution of such waves is studied in the spatiotemporal domain from a theoretical perspective. Although nonlinearities are included in our confocal source model, just the properties of the fundamental shear component are examined in this paper. We examine how the shear wave propagation is affected by the shear viscosity, the coupling wave, the spatial distribution of the force, the shear speed, and the duration of the modulated wave. A method is proposed for estimating the shear viscosity of a viscoelastic medium. In addition, it is shown how the Voigt model paremeters can be extracted from the frequency-dependent speed and attenuation.     

Application of 1-D transient elastography for the shear modulus assessment of thin-layered soft tissue: comparison with supersonic shear imaging technique

Elasticity estimation of thin-layered soft tissues has gained increasing interest propelled by medical applications like skin, corneal, or arterial wall shear modulus assessment. In this work, the authors propose one-dimensional transient elastography (1DTE) for the shear modulus assessment of thin-layered soft tissue. Experiments on three phantoms with different elasticities and plate thicknesses were performed. First, using 1DTE, the shear wave speed dispersion curve inside the plate was obtained and validated with finite difference simulation. No dispersive effects were observed and the shear wave speed was directly retrieved from time-of-flight measurements. Second, the supersonic shear imaging (SSI) technique (considered to be a gold standard) was performed. For the SSI technique, the propagating wave inside the plate is guided as a Lamb wave. Experimental SSI dispersion curves were compared with finite difference simulation and fitted using a generalized Lamb model to retrieve the plate bulk shear wave speed. Although they are based on totally different mechanical sources and induce completely different diffraction patterns for the shear wave propagation, the 1DTE and SSI techniques resulted in similar shear wave speed estimations. The main advantage of the 1DTE technique is that bulk shear wave speed can be directly retrieved without requiring a dispersion model.     

Shear elasticity probe for soft tissues with 1-D transient elastography

Important tissue parameters such as elasticity can be deduced from the study of the propagation of low frequency shear waves. A new method for measuring the shear velocity in soft tissues is presented in this paper. Unlike conventional transient elastography in which the ultrasonic transducer and the low frequency vibrator are two separated parts, the new method relies on a probe that associates the vibrator and the transducer, which is built on the axis of the vibrator. This setup is easy to use. The low frequency shear wave is driven by the transducer itself that acts as a piston while it is used in pulse echo mode to acquire ultrasonic lines. The results obtained with the new method are in good agreement with those obtained with the conventional one.     

Elastic stress transmission in cellular systems—Analysis of wave propagation

A closely packed array of thin-walled rings constitutes an idealisation of a cellular structure. Elastic waves propagating through such structures must do so via the ring (cell) walls. A theoretical investigation into the propagation of elastic stresses in thin-walled circular rings is undertaken to examine the nature of wave transmission. Three modes of motion, corresponding to shear, extensional and flexural waves, are established and their respective velocities defined by a cubic characteristic equation. The results show that all three waves are dispersive. By neglecting extension of the centroidal axis and rotary inertia, explicit approximate solutions can be obtained for flexural waves. Employment of Love's approach for extensional waves [Love AEH. A treatise on the mathematical theory of elasticity, 4th ed. New York: Dover Publications; 1944. p. 452–3] enables approximate solutions for shear waves to be derived. The three resulting approximate solutions exhibit good agreement with the exact solutions of the characteristic equation over a wide range of wavelengths. The effects of material property, ring wall thickness and ring diameter on the three wave modes are discussed, and the results point to flexural waves as the dominant means of elastic energy transmission in such cellular structures. Wave velocities corresponding to different frequency components determined from experimental results are compared with theoretical predictions of group velocity for flexural waves and good correlation between experimental data and theory affirms this conclusion.     

Size-dependent effective dynamic properties of unidirectional nanocomposites with interface energy effects

This article studies the size effect on wave propagation characteristics of plane longitudinal and transverse elastic waves in a two-phase nanocomposite consisting of transversely isotropic and unidirectionally oriented identical cylindrical nanofibers embedded in a transversely isotropic homogeneous matrix. The surface elasticity theory is employed to incorporate the interfacial stress effects. The effect of random distribution of nanofibers in the composite medium is taken into account via a generalized self-consistent multiple scattering model. The phase velocities and attenuations of longitudinal and shear waves along with the associated dynamic effective elastic constants are calculated for a wide range of frequencies and fiber concentrations. The numerical results reveal that interface elasticity at nanometer length scales can significantly alter the overall dynamic mechanical properties of nanofiber-reinforced composites. Limiting cases are considered and excellent agreements with solutions available in the literature have been obtained.     

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.     

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.     

Acoustic waves in bounded anisotropic media: theorems, estimations, and computations

This paper discusses some basic achievements in theoretical studies on acoustic wave propagation along boundaries in anisotropic solids. In particular, the following issues are reviewed: existence theorems for subsonic surface and interface waves, leaky surface acoustic waves (SAW) and their relation to "supersonic" SAWs and fast exceptional bulk waves, the resonance reflection of bulk waves in the vicinity of leaky wave branches. General conclusions are illustrated by numerical examples.     

Monitoring of thermal therapy based on shear modulus changes: II. Shear wave imaging of thermal lesions

The clinical applicability of high-intensity focused ultrasound (HIFU) for noninvasive therapy is currently hampered by the lack of robust and real-time monitoring of tissue damage during treatment. The goal of this study is to show that the estimation of local tissue elasticity from shear wave imaging (SWI) can lead to a precise mapping of the lesion. HIFU treatment and monitoring were respectively performed using a confocal setup consisting of a 2.5-MHz single element transducer focused at 34 mm on ex vivo samples and an 8-MHz ultrasound diagnostic probe. Ultrasound-based strain imaging was combined with shear wave imaging on the same device. The SWI sequences consisted of 2 successive shear waves induced at different lateral positions. Each wave was created with pushing beams of 100 μs at 3 depths. The shear wave propagation was acquired at 17,000 frames/s, from which the elasticity map was recovered. HIFU sonications were interleaved with fast imaging acquisitions, allowing a duty cycle of more than 90%. Thus, elasticity and strain mapping was achieved every 3 s, leading to real-time monitoring of the treatment. When thermal damage occurs, tissue stiffness was found to increase up to 4-fold and strain imaging showed strong shrinkages that blur the temperature information. We show that strain imaging elastograms are not easy to interpret for accurate lesion characterization, but SWI provides a quantitative mapping of the thermal lesion. Moreover, the concept of shear wave thermometry (SWT) developed in the companion paper allows mapping temperature with the same method. Combined SWT and shear wave imaging can map the lesion stiffening and temperature outside the lesion, which could be used to predict the eventual lesion growth by thermal dose calculation. Finally, SWI is shown to be robust to motion and reliable in vivo on sheep muscle.     

1-D elasticity assessment in soft solids from shear wave correlation: the time-reversal approach

One-channel time-reversal (TR) experiments allow focalization of waves in reverberant cavities. According to the Rayleigh criterion, the focal spot width is directly related to the wavelength and therefore depends on the mechanical properties of the medium. Thus, the general idea of this work is to extract quantitative estimations of these mechanical properties using a time-reversal approach based on cross-correlations of the wave field. An external source creates mechanical waves in the audible frequency range. One component of the vectorial field is measured along a line as function of time with signal processing developed in the field of 1-D elastography. The shear wavelength information is deduced from these mechanical waves using spatiotemporal correlations and interpreted in the frame of the time-reversal symmetry. The impact of wave attenuation in soft solids is reduced using a spatial average of the correlation field. The result is shown to be suitable for global elasticity estimation. The advantage is that the technique is almost independent of the source kind, shape, and time excitation function. This robustness as regard to shear wave source allows translation of this technique to applications in the medical field, including deep or moving organs.     

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.     

Generation of SH-type waves in a poroelastic layer sandwiched between prestressed orthotropic and nonhomogeneous mantles

The present work is concerned with a detailed illustration on the study of horizontally polarized shear waves (SH-type) propagation in a prestressed fluid saturated anisotropic porous layer sandwiched by prestressed orthotropic medium and nonhomogeneous mantles. The frequency equation for the assumed model is derived and their medium characteristics, such as porosity, prestress, anisotropy, and nonhomogeneity, are discussed. Numerical treatment is given to analyze these effects on phase velocities of SH-type waves and is plotted in various graphs. The parametric study divulges that the magnitude of wave velocities increases with the increase of nonhomogeneity parameter and prestress parameter.     

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.     

Diffraction field of a low frequency vibrator in soft tissues using transient elastography

For the last 10 years, interest has grown in low frequency shear waves that propagate in the human body. However, the generation of shear waves by acoustic vibrators is a relatively complex problem, and the directivity patterns of shear waves produced by the usual vibrators are more complicated than those obtained for longitudinal ultrasonic transducers. To extract shear modulus parameters from the shear wave propagation in soft tissues, it is important to understand and to optimize the directivity pattern of shear wave vibrators. This paper is devoted to a careful study of the theoretical and the experimental directivity pattern produced by a point source in soft tissues. Both theoretical and experimental measurements show that the directivity pattern of a point source vibrator presents two very strong lobes for an angle around 35 degrees . This paper also points out the impact of the near field in the problem of shear wave generation.     

A new features on S-waves propagation in a nonhomogeneous anisotropic incompressible medium under influence of gravity field and initial stress with and without electromagnetic field and rotation

The present article is concerned with the investigation of the propagation of shear waves in a nonhomogeneous anisotropic incompressible medium under the effect of the electromagnetic field, gravity field, rotation, and initial stress taking into account a comparison between presence and absence of magnetic field, initial stress, and rotation. Analytical analysis reveals that the velocity of propagation of the shear waves depends upon the direction of propagation, the anisotropy, magnetic field, rotation, gravity field, nonhomogeneity of the medium, and the initial stress. The frequency equation that determines the velocity of the shear waves has been obtained. Some special cases are also deduced from the present investigation. In fact, these equations are an agreement with the corresponding classical results when the medium is isotropic. Numerical results have been given and illustrated graphically in each case considered. The results indicate that the effects of gravity field, initial stress, magnetic field, electric field, anisotropy, and rotation are very pronounced. Also, the absence of initial stress, magnetic field, and rotation tends to increasing of the S-waves velocity compared with presence of them.     

The investigation of the nonlocal longitudinal stress waves with modified couple stress theory

In the present work, the propagation of longitudinal stress waves is investigated using a modified couple stress theory. The analysis of wave motion is based on a Love rod model including the effects of lateral deformation. The present analysis also considers the effect of shear stress components. By applying Hamilton??s principle, the explicit nonlocal elasticity solution is obtained, and the effects of shear stress and length scale parameter are discussed.     

Methods for solving one-dimensional boundary-value problems in a nonlinear elastic medium

Summary One-dimensional problems connected with a mechanical exposure to the boundary of a nonlinear elastic half-space, which leads to a constantly accelerated motion of this boundary, are considered. The value , equal to the square root of the ratio between the velocity of the boundary motion and the velocity of longitudinal wave propagation in a linear elastic medium, is used as the value characterizing the intensity of this exposure. It is shown that as a result of such an exposure shock waves of small or finite amplitude may propagate in the half-space. The asymptotic matching principle and the ray method are used as methods of solution. The merits and demerits of each method are analyzed. It has been inferred that the matched asymptotic method can be applied to waves of small amplitude, and the ray method is usable when investigating the propagation of shock waves not only of small amplitude, but of finite amplitude as well if the time of a consideration of the shock process is not long. The results obtained by the two methods for shock waves of small amplitude are in close agreement. It has been demonstrated that the ray method is adaptable for solving more intricate boundary-value problems resulting in the propagation of several shock waves of finite amplitude at a time. The problem connected with the constantly accelerated motion of one of the boundaries of an initially deformed elastic layer provides an example.