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.     

Assessment of elastic parameters of human skin using dynamic elastography

Sonoelastography and transient elastography are two ultrasound-based techniques that facilitate noninvasive characterization of the viscoelastic properties of soft tissues by investigating their response to shear mechanical excitation. Young's modulus is the principle assessment parameter. Because it defines local tissue stiffness, it is of major interest for the medical imaging and cosmetic industries as it could replace subjective palpation by yielding local, quantitative information. In this paper, we describe a new high-resolution device capable of measuring local Young's modulus in very thin layers (1-5 mm) and devoted to the in vivo evaluation of the elastic properties of human skin. It uses an ultrasonic probe (50 MHz) for tracking the displacements induced by a 300 Hz shear wave generated by a ring surrounding the transducer. The displacements are measured using a conventional cross-correlation technique between successive ultrasonic back-scattered echoes. First, this noninvasive technique has been experimentally proven to be accurate for investigating elasticity in different skin-mimicking phantoms. Second, data were acquired in vivo on human forearms. As expected, Young's modulus was found to be higher in the dermis than in the hypodermis and other soft tissues.     

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.     

Sol-gel transition in agar-gelatin mixtures studied with transient elastography

Using the shear wave propagation in solids, the transient elastography technique has been developed to assess the elastic properties of soft tissues. Here, a new approach of transient elastography allows assessing the viscoelastic properties of soft tissues. In this paper, the method is used to follow-up the sol-gel transition of an agar-gelatin mixture noninvasively. The shear wave velocity and shear wave attenuation through the mixture were continuously monitored in the audible range of frequencies (from 50 Hz to 200 Hz). The observed changes in velocities and attenuations as a function of frequency confirmed the validity of the Voigt's model to describe the gel at its stable mechanical state. By a simple inverse problem approach, based on the one-dimensional (1-D) Helmholtz equation, the elasticity and the viscosity of such a mixture were recovered as a function of time. The results obtained are in good agreement with the literature and theoretical predictions. Overall, they demonstrate the high sensitivity of the transient elastography measurements to the rheological parameter changes in agar-gelatin mixtures during gelation.     

On the transient elastic field for an expanding spherical inclusion in 3-D solid

Summary. The three-dimensional transient elastic field of an infinite isotropic elastic medium is investigated when a phase transformation is nucleated from a point and proceeds through the crystal dynamically. The phase transformation keeps the spherical shape and expands at a speed of arbitrary time profile. This process is modeled by an expanding spherical inclusion with a spatially uniform eigenstrain. The objective of this paper is to present a general method to determine the transient displacement field for points either covered or not covered by the transformation area. This method can be applied to investigate the nucleation and expanding mechanism of phase transformation. Using a Green's function approach, an explicit procedure is presented to evaluate the 3-D displacement field when the expanding history of the spherical inclusion is given. As numerical examples, the explicit formulations are given for the transient elastic fields, when the spherical inclusion expands at a constant or an exponent damping speed with a pure dilatational eigenstrain or pure shear eigenstrain. It is found that the elastic field inside the expanding inclusion remains constant with respect to time, which is consistent with the well-known Eshelby solution for a static inclusion case.     

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.     

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.     

Numerical simulation of transient waves in a heterogeneous soil layer

The passage of elastic waves through the upper soil strata is modeled by a rather simple solution based on freely propagating waves in a continuously heterogeneous material with amplitude dependence on a single coordinate corresponding to the direction of propagation. Specifically, pseudo-dilational and pseudo-rotational waves are developed for an elastic medium with position-dependent density and thus position-dependent pressure (P) and shear (S) wavespeeds, although in a strict sense these no longer exist as such due to coupling through the material heterogeneity. Time harmonic conditions are assumed to hold at the fundamental solution level and transient signal generation is achieved through Fourier synthesis. Numerical examples for the square root of linear in the depth coordinate wavespeed profiles are used in modeling the passage of impulsively-generated signals in a soft soil deposit under two-dimensional conditions.     

Effective rheological properties of a disperse mixture of viscoelastic materials

The article deals with the problem of determining the compressive and shear moduli, and also the shear and bulk viscosity of a viscoelastic medium with spherical inclusions of some other viscoelastic material. Calculations are presented for an elastic medium with inclusions of a viscous liquid.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 50, No. 3, pp. 434–445, March, 1986.     

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.     

Shear modulus imaging with 2-D transient elastography

In previous work, we have shown that time-resolved 2-D transient elastography is a promising technique for characterizing the elasticity of soft tissues. It involves the measurement of the displacements induced by the propagation of low frequency (LF) pulsed shear waves in biological tissues. In this paper, we present a novel apparatus that contains a LF vibrating device surrounding a linear array of 128 ultrasonic transducers that performs ultrafast ultrasonic imaging (up to 10,000 frames/s) and that is able to follow in real time the propagation of a LF shear wave in the human body. The vibrating device is made of two rods, fixed to electromagnetic vibrators, that produce in the ultrasonic image area a large amplitude shear wave. The geometry has been chosen both to enhance the sensitivity and to create a quasi linear shear wave front in the imaging plane. An inversion algorithm is used to recover the shear modulus map from the spatio-temporal data, and the first experimental results obtained from tissue-equivalent materials are presented.     

Monitoring of thermal therapy based on shear modulus changes: I. shear wave thermometry

The clinical applicability of high-intensity focused ultrasound (HIFU) for noninvasive therapy is today 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 the 2-D mapping of temperature changes during HIFU treatments. This new concept of shear wave thermometry is experimentally implemented here using conventional ultrasonic imaging probes. HIFU treatment and monitoring were, respectively, performed using a confocal setup consisting of a 2.5-MHz single-element transducer focused at 30 mm on ex vivo samples and an 8-MHz ultrasound diagnostic probe. Thermocouple measurements and ultrasound-based thermometry were used as a gold standard technique and were combined with SWI on the same device. The SWI sequences consisted of 2 successive shear waves induced at different lateral positions. Each wave was created using 100-μs pushing beams 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%. Elasticity and temperature mapping was achieved every 3 s, leading to realtime monitoring of the treatment. Tissue stiffness was found to decrease in the focal zone for temperatures up to 43°C. Ultrasound-based temperature estimation was highly correlated to stiffness variation maps (r2 = 0.91 to 0.97). A reversible calibration phase of the changes of elasticity with temperature can be made locally using sighting shots. This calibration process allows for the derivation of temperature maps from shear wave imaging. Compared with conventional ultrasound-based approaches, shear wave thermometry is found to be much more robust to motion artifacts.     

Mode II Macrocrack Initiation in Orthotropic Composite Viscoelastic Plate

Safe loads and initiation time for a straight macrocrack in viscoelastic orthotropic material that is intended to model a fiber composite plate under shear loads is investigated. The composite material is modeled by viscoelastic orthotropic medium. Determination of expression for crack shear displacement as function of time is based on the corresponding elastic solution and the method of operator continued fractions. Initiation time is obtained as a solution of integral equation for the incubation period. Numerical calculations are given for mode II macrocrack initiation.     

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.     

Some results on finite amplitude elastic waves propagating in rotating media

Summary. Two questions related to elastic motions are raised and addressed. First: in which theoretical framework can the equations of motion be written for an elastic half-space put into uniform rotation? It is seen that nonlinear finite elasticity provides such a framework for incompressible solids. Second: how can finite amplitude exact solutions be generated? It is seen that for some finite amplitude transverse waves in rotating incompressible elastic solids with general shear response the solutions are obtained by reduction of the equations of motion to a system of ordinary differential equations equivalent to the system governing the central motion problem of classical mechanics. In the special case of circularly-polarized harmonic progressive waves, the dispersion equation is solved in closed form for a variety of shear responses, including nonlinear models for rubberlike and soft biological tissues. A fruitful analogy with the motion of a nonlinear string is pointed out.     

Dynamics of uniaxial tension of a viscoelastic strain-hardening body in a system with one degree of freedom. Part 1. Prescribed motion

The behavior of an open mechanical dissipative system formed by a viscoelastic hardening body and an elastic working element used for the energy transfer between the testing machine and the deformed body is described by a third-order dynamic differential equation with controlling parameters that depend on the reduced mass and stiffness of the system, its viscous resistance, the degree of strain hardening, the type of the stressed state of the body, and the dissipation of energy in a viscous ambient medium. We analyze the dynamics of uniaxial tension of the deformed body below and above its elasticity limit for the case where the forces induced in the process of motion are determined by the kinematics of the testing machine with prescribed motion. We establish the dynamic nature of the nonlinear section of the tensile stress-strain diagram beyond the elasticity limit of the viscoelastic body corresponding to the so-called “nonlinear elasticity”. It is shown that the appearance of this section is connected with a transient relaxation process. Upon the termination of this process, the forces acting in the system are determined by the viscous flow of the body corresponding to its yield limit. Above the elasticity limit of the body, we observe the formation of a bistable state of the system caused by changes in the controlling parameters and lag effects and leading to its macroscopic acoustic activity. I. N. Frantsevich Institute for Problems in Material Science, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 16–27, July–August, 1998.     

A relationship between non-exponential stress relaxation and delayed elasticity in the viscoelastic process in amorphous solids: Illustration on a chalcogenide glass

Inorganic glasses are viscoelastic materials since they exhibit, below as well as above their glass transition temperature, a viscoelastic deformation under stress, which can be decomposed into a sum of an elastic part, an inelastic (or viscous) part and a delayed elastic part. The delayed elastic part is responsible for the non-linear primary creep stage observed during creep tests. During a stress relaxation test, the strain, imposed, is initially fully elastic, but is transformed, as the stress relaxes, into an inelastic and a delayed elastic strains. For linear viscoelastic materials, if the stress relaxation function can be fitted by a stretched exponential function, the evolution of each part of the strain can be predicted using the Boltzmann superposition principle. We develop here the equations of these evolutions, and we illustrate their accuracy by comparing them with experimental evolutions measured on GeSe₉ glass fibers. We illustrate also, by simple equations, the relationship between any kind of relaxation function based on additive contribution of different relaxation processes and the delayed elastic contribution to stress relaxation: the delayed elasticity is directly correlated to the dispersion of relaxations times of the processes involved during relaxation.     

Modeling of transient ultrasonic wave propagation using the distributed point

Transient ultrasonic waves in a fluid medium generated by a flat circular and a point-focused transducer of finite size are modeled by the distributed point source method (DPSM). DPSM is a Green's-function-based semi-analytical mesh-free technique which is modified here to incorporate the transient loading from a finite-sized acoustic transducer. Conventional DPSM solves acoustic problems in steady-state frequency domain. Here, DPSM is extended to the time domain without the fast Fourier transform (FFT) but using the Green's function in the time domain. This modified method is denoted t-DPSM. Harmonic point sources of DPSM are replaced by time-dependent point sources in t-DPSM. Generated t-DPSM results are compared with the finite element (FE) results for both focused and flat circular transducers. The developed method is used to solve the transient problem of wave scattering by an air bubble in a fluid as the bubble is moved horizontally or vertically from the focal point of the focused transducer. The received energy signal is compared for different eccentricities.     

Characterizing the viscoelastic properties of thin hydrogel-based constructs for tissue engineering applications.

We present a novel indentation method for characterizing the viscoelastic properties of alginate and agarose hydrogel based constructs, which are often used as a model system of soft biological tissues. A sensitive long working distance microscope was used for measuring the time-dependent deformation of the thin circular hydrogel membranes under a constant load. The deformation of the constructs was measured laterally. The elastic modulus as a function of time can be determined by a large deformation theory based on Mooney-Rivlin elasticity. A viscoelastic theory, Zener model, was applied to correlate the time-dependent deformation of the constructs with various gel concentrations, and the creep parameters can therefore be quantitatively estimated. The value of Young's modulus was shown to increase in proportion with gel concentration. This finding is consistent with other publications. Our results also showed the great capability of using the technique to measure gels with incorporated corneal stromal cells. This study demonstrates a novel and convenient technique to measure mechanical properties of hydrogel in a non-destructive, online and real-time fashion. Thus this novel technique can become a valuable tool for soft tissue engineering.     

On the dynamic interaction between a penny-shaped crack and an expanding spherical inclusion in 3-D solid

Three-dimensional stress investigation on the interaction between a penny-shaped crack and an expanding spherical inclusion in an infinite 3-D medium is studied in this paper. The spherical transformation area (the inclusion) expands in a self-similar way. By using the superposition principle, the original physical problem is decomposed into two sub-problems. The transient elastic filed of the medium with an expanding spherical inclusion is derived with the dynamic Green's function. A time domain boundary integral equation method (BIEM) is then adopted to solve the current problem. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. Numerical examples for the Mode I stress intensity factor are presented to assess the dynamic effect of the expanding inclusion.