Transient displacement induced in shear wave elastography: comparison between analytical results and ultrasound measurements

It is now accepted that an effective way to investigate the elastic properties of soft tissues is to generate a localized transient acoustic radiation force and to follow the associated displacements in the time/space domain. Shear waves induced by this stress field are particularly interesting in this kind of medium because they are governed by the shear elastic modulus mu, which is directly linked to the Young modulus, and spatial distribution and temporal evolution of the transient motion induced must therefore be obtained in detail. We report here a model based on the elastodynamic Green's function formalism to describe these displacements. 3D simulation of radiation force in homogenous elastic media was performed and the displacement curves computed at different radial distances for different temporal force profiles. Amplitude and duration of displacement were found to be reliable parameters to characterize the elastic properties of the medium. Experimental measurements were performed in a homogeneous agar-gelatin tissue-mimicking phantom, and two transducers were used to generate the radiation force and follow the induced displacements. Displacements obtained from different lateral locations around the applied force axis were then used to reconstruct the shear-wave propagation in a scan plane as a function of time. The experimental displacements/curves agreed with the theoretical profiles obtained by the elastodynamic Green's function formalism.     

Acoustoelasticity in soft solids: assessment of the nonlinear shear modulus with the acoustic radiation force

The assessment of viscoelastic properties of soft tissues is enjoying a growing interest in the field of medical imaging as pathologies are often correlated with a local change of stiffness. To date, advanced techniques in that field have been concentrating on the estimation of the second order elastic modulus (mu). In this paper, the nonlinear behavior of quasi-incompressible soft solids is investigated using the supersonic shear imaging technique based on the remote generation of polarized plane shear waves in tissues induced by the acoustic radiation force. Applying a theoretical approach of the strain energy in soft solid [Hamilton et al., J. Acoust. Soc. Am. 116, 41-44 (2004)], it is shown that the well-known acoustoelasticity experiment allowing the recovery of higher order elastic moduli can be greatly simplified. Experimentally, it requires measurements of the local speed of polarized plane shear waves in a statically and uniaxially stressed isotropic medium. These shear wave speed estimates are obtained by imaging the shear wave propagation in soft media with an ultrafast echographic scanner. In this situation, the uniaxial static stress induces anisotropy due to the nonlinear effects and results in a change of shear wave speed. Then the third order elastic modulus (A) is measured in agar-gelatin-based phantoms and polyvinyl alcohol based phantoms.     

A solution to diffraction biases in sonoelasticity: the acoustic impulse technique.

Several methods have been proposed to estimate the viscoelastic properties of soft biological tissues using forced low-frequency vibrations (10-500 Hz). Those methods are based on the measurement of phase velocity of the shear waves (approximately 5 m/s). It is shown in this article that the measurements of velocity as well as attenuation are subjected to biases. These biases are related to reflected waves created at boundaries, to the nonnegligible size of the piston source which causes diffraction effects and to the influence of a low-frequency compressional wave. Indeed, a theoretical analysis of the field radiated by a point source explains how mechanical vibrations of a piston generate a shear wave with a longitudinal component and how this component can interfere with a low-frequency compressional wave. However, by using a low-frequency transient excitation, these biases can be avoided. Then the precise numerical values of elasticity and viscosity can be deduced. Experiments in phantoms and beef muscles are shown. Moreover, a relative hardness imaging of a phantom composed of two media with different elasticities is presented.     

Wave anisotropy of shear viscosity and elasticity

The paper presents the theory of shear wave propagation in a “soft solid” material possessing anisotropy of elastic and dissipative properties. The theory is developed mainly for understanding the nature of the low-frequency acoustic characteristics of skeletal muscles, which carry important diagnostic information on the functional state of muscles and their pathologies. It is shown that the shear elasticity of muscles is determined by two independent moduli. The dissipative properties are determined by the fourth-rank viscosity tensor, which also has two independent components. The propagation velocity and attenuation of shear waves in muscle depend on the relative orientation of three vectors: the wave vector, the polarization vector, and the direction of muscle fiber. For one of the many experiments where attention was distinctly focused on the vector character of the wave process, it was possible to make a comparison with the theory, estimate the elasticity moduli, and obtain agreement with the angular dependence of the wave propagation velocity predicted by the theory.     

Standing shear waves in rubberlike layered media

Standing shear waves arising in layered media the shear modulus of which varies in a stepwise manner at the plain boundaries between the layers are considered. A general solution is obtained for the shear wave amplitudes in a resonator with an N-layer structure the lower boundary of which performs harmonic vibrations while a finite-mass plate is attached to the upper boundary. Results of calculations and measurements are presented for a resonator with a structure in which nondeformable metal layers alternate with elastic rubberlike polymer layers. It is shown that the resonance frequencies of such a resonator can be controlled by changing the number of layers and their thicknesses. It is demonstrated, both experimentally and theoretically, that, from the resonance curve of a resonator with a two-layer structure, it is possible to determine the shear modulus of one of the layers under the condition that the elasticity of the other layer is known. The method of separation into a finite number of layers is used to analyze the resonance characteristics of a one-dimensional resonator filled with a rubberlike medium the properties of which continuously vary in the direction perpendicular to the shear displacements. The choice of the number of layers depending on the type of inhomogeneity is analyzed.     

Temporal analysis of tissue displacement induced by a transient ultrasound radiation force

One of the stress sources that can be used in dynamic elastography imaging methods is the acoustic radiation force. However, displacements of the medium induced by this stress field are generally not fully understood in terms of spatial distribution and temporal evolution. A model has been developed based on the elastodynamic Green's function describing the different acoustic waves generated by focused ultrasound. The function is composed of three terms: two far-field terms, which correspond to a purely longitudinal compression wave and a purely transverse shear wave, and a coupling near-field term which has a longitudinal component and a transverse component. For propagation distances in the shear wavelength range, the predominant term is the near field term. The displacement duration corresponds to the propagation duration of the shear wave between the farthest source point and the observation point. This time therefore depends on the source size and the local shear modulus of the tissue. Evolution of the displacement/time curve profile, which is directly linked to spatial and temporal source profiles, is computed at different radial distances, for different durations of force applications and different shear elastic coefficients. Experimental results performed with an optical interferometric method in a homogeneous tissue-mimicking phantom agreed with the theoretical profiles.     

Shear and compressional wave speeds in Hertzian granular media

It is shown that the shear wave speed in a granular medium is less than that in an elastic solid of the same shear modulus-to-density ratio. Shear and compressional wave speeds are derived for granular media using a conservation of energy approach. The grains are assumed to be spherical with elastic Hertzian contacts of constant stiffness. The affine approximation is used to determine the relative displacements of grain centers, and it is also assumed that the grains are small compared to a wavelength, consistent with the effective medium approximation. Potential and kinetic energies associated with linear motion are the same as those in an elastic solid, but it is found that shear wave propagation in a granular medium involves additional energies associated with grain rotation. The partition of energies results in a reduction in the shear wave speed, relative to an elastic solid of the same shear modulus-to-density ratio. It is shown that the reduction is an inherent property of granular media, independent of any departure from the affine approximation or fluctuations in coordination number or contact stiffness. The predicted wave speed ratios are consistent with published measurements.     

The measurement of A0 and S0 lamb wave attenuation to determine the normal and shear stiffnesses of a compressively loaded interface

Guided waves in an elastic plate surrounded by air propagate with very low attenuation. This paper describes the effect on this propagation of compressively loading an elastomer with high internal damping against one surface of the elastic plate. The propagation of both A0 and S0 Lamb modes is considered. The principal effect is shown to be increased attenuation of the guided waves. This attenuation is caused by leakage of energy from the plate into the elastomer, where it is dissipated due to high viscoelastic damping. It is shown that the increase in attenuation is strongly dependent on the compressive load applied across the solid-solid interface. This interface is represented as a spring layer in a continuum model of the system. Both normal and shear stiffnesses of the interface are quantified from the attenuation of A0 and S0 Lamb waves measured at each step of the compressive loading. The normal stiffness is also measured independently by normal incidence, bulk longitudinal wave ultrasound. The resulting predictions of wave propagation behavior, such as attenuation, obtained by the model are in excellent agreement with those measured experimentally.     

Nonlinear elastodynamics in micro-inhomogeneous solids observed by head-wave based dynamic acoustoelastic testing

Dynamic acoustoelastic testing provides a more complete insight into the acoustic nonlinearity exhibited by micro-inhomogeneous media like granular and cracked materials. This method consists of measuring time of flight and energy modulations of pulsed ultrasonic waves induced by a low-frequency standing wave. Here pulsed ultrasonic head waves were employed to assess elastic and dissipative nonlinearities in a region near the surface of a solid. Synchronization of the ultrasound pulse sequence with the low-frequency excitation provided instantaneous variations in the elastic modulus and the attenuation as functions of the instantaneous low-frequency strain. Weak quadratic elastic nonlinearity and no dissipative nonlinearity were detected in duralumin. In limestone, distinction between tensile and compressive behaviors revealed an asymmetry in the acoustic nonlinearity and hysteresis in both the elastic modulus and the attenuation variations. Measured nonlinear acoustical parameters are in good agreement with values obtained by different techniques. Reversible acoustically induced conditioning modified the acoustic nonlinearity both quantitatively and qualitatively. It reduced tension-compression asymmetry, suggesting a nonequilibrium modification of the sources of acoustic nonlinearity. Additionally to the metrology of the acoustic nonlinearity, head wave based dynamic acoustoelastic testing may be a useful tool to monitor changes in the microstructure or the accumulation of damage in solids.     

Flexural edge waves in semi-infinite elastic plates

This paper presents a detailed analysis of the dispersion for flexural edge waves in semi-infinite isotropic elastic plates. A solution to the dynamic equations of motion is constructed by the superposition of two partial solutions, each providing zero shear stresses at the plate faces. A dispersion equation is expressed via the determinant of an infinite system of linear algebraic equations. The system is reduced to a finite one by taking into account the asymptotic behaviour of unknown coefficients. The accuracy of the solution is confirmed by a good agreement with the available experimental data and by a proper satisfaction of the prescribed boundary conditions.A detailed analysis of dispersion properties for the edge wave and corresponding displacements at various frequencies is carried out. In addition to the well-known results it is shown that the plate height does not influence the existence of the edge wave at high frequencies and, as the frequency increases, the phase velocity of the edge wave in a semi-infinite plate asymptotically approaches the velocity of an edge wave in a right-angled wedge. The performed analysis allows evaluating the plate theories such as the Kirchhoff theory or other refined plate theories developed for modeling edge waves in semi-infinite elastic plates at low frequencies.     

Scattering of a Rayleigh surface acoustic wave by a small-size inhomogeneity in a solid half-space

We use the Born approximation of the perturbation method to solve the problem of scattering of a harmonic Rayleigh surface acoustic wave by a weak-contrast inhomogeneity that is small compared with the wavelength and is located in a solid half-space near its boundary. The material of the inhomogeneity differs from the material of the half-space only in its density. The Rayleigh wave incident on the inhomogeneity is excited by a monochromatic surface force source acting normally to the half-space boundary. We derive expressions for the displacement fields in the scattered spherical compressional and shear (SV- and SH-polarized) waves. Scattering of the Rayleigh wave into a Rayleigh wave is studied in detail. We find expressions for the vertical and horizontal components of the displacement vector in the scattered Rayleigh wave as well as its radiated power. It is shown that the field of the scattered surface wave is mainly formed by vertical oscillations of the inhomogeneity in the field of the incident wave. In this case, the radiated power for the scattered Rayleigh wave formed by vertical motion of the inhomogeneity in the incident-wave field depends on the depth of the inhomogeneity as the fourth power of the function describing the well-known depth dependence of the vertical displacements in the Rayleigh surface wave. Correspondingly, the dependence of the radiated power for the scattered Rayleigh wave formed by horizontal motion of the inhomogeneity depends on its location depth as the fourth power of the depth dependence of the horizontal displacements in the Rayleigh surface wave. We perform calculations of the ratio between the powers of the scattered and incident Rayleigh waves for different ratios between the velocities of the compressional and shear waves in a solid. It is shown that the radiated power for the scattered surface wave decreases sharply with increasing depth of the subsurface-inhomogeneity location. Thus, the scattering of a Rayleigh wave into a Rayleigh wave is fairly efficient only when the location depth of the inhomogeneity does not exceed about one-third of the wavelength of the shear wave in an elastic medium.     

Experiments and molecular-dynamics simulation of elastic waves in a plasma crystal radiated from a small dipole source

The radiation of elastic waves from a localized source is observed experimentally in a two-dimensional plasma crystal. An initial shear stress applied by a laser forms a small dipole source. The emerging complex wave pattern is shown to consist of outgoing compressional and shear wave pulses. Subsequent structures are identified as inward-going waves due to the finite size of the source region, which reappear on the opposite side. The compressional wave forms a trailing wave train due to strong dispersion, while the nondispersive shear wave evolves into a vortex-antivortex pair on either side. The experiments are compared with a molecular-dynamics simulation.     

Sonoelastographic imaging of interference patterns for estimation of shear velocity distribution in biomaterials

The authors have recently demonstrated the shear wave interference patterns created by two coherent vibration sources imaged with the vibration sonoelastography technique. If the two sources vibrate at slightly different frequencies omega and omega+deltaomega, respectively, the interference patterns move at an apparent velocity of (deltaomega/2omega)upsilon(shear), where upsilon(shear) is the shear wave speed. We name the moving interference patterns "crawling waves." In this paper, we extend the techniques to inspect biomaterials with nonuniform stiffness distributions. A relationship between the local crawling wave speed and the local shear wave velocity is derived. In addition, a modified technique is proposed whereby only one shear wave source propagates shear waves into the medium at the frequency omega. The ultrasound probe is externally vibrated at the frequency omega-deltaomega. The resulting field estimated by the ultrasound (US) scanner is proven to be an exact representation of the propagating shear wave field. The authors name the apparent wave motion "holography waves." Real-time video sequences of both types of waves are acquired on various inhomogeneous elastic media. The distribution of the crawling/holographic wave speeds are estimated. The estimated wave speeds correlate with the stiffness distributions.     

Dispersion of interface waves in sediments with power-law shear speed profiles. I. Exact and approximate analytical results.

In the upper tens of meters of ocean bottom, unconsolidated marine sediments consisting of clay, silt, or fine sand with high porosity are "almost incompressible" in the sense that the shear wave velocity is much smaller than the compressional wave velocity. The shear velocity has very large gradients close to the ocean floor leading to strong coupling of compressional and shear waves in such "soft" sediments. The weak compressibility opens an avenue for developing a theory of elastic wave propagation in continuously stratified soft sediments that fully accounts for the coupling. Elastic waves in soft sediments consist of "fast" waves propagating with velocities close to the compressional velocity and "slow" waves propagating with velocities on the order of the shear velocity. For the slow waves, the theory predicts the existence of surface waves at the ocean-sediment boundary. In the important special case of the power-law depth-dependence of shear rigidity, phase and group velocities of the interface waves are shown to scale as a certain power of frequency. An explicit, exact solution was obtained for the surface waves in sediments characterized by constant density and a linear increase of shear rigidity with depth, that is, for the case of shear speed proportional to the square root of the depth below the sediment-water interface. Asymptotic and perturbation techniques were used to extend the result to more general environments. Theoretical dispersion relations agreed well with numerical simulations and available experimental data and, as demonstrated in a companion paper [D. M. F. Chapman and O. A. Godin, J. Acoust. Soc. Am 110, 1908 (2001)] led to a simple and robust inversion of interface wave travel times for shear velocity profiles in the sediment.     

Transient elastography in anisotropic medium: application to the measurement of slow and fast shear wave speeds in muscles

From the measurement of a low frequency (50-150 Hz) shear wave speed, transient elastography evaluates the Young's modulus in isotropic soft tissues. In this paper, it is shown that a rod source can generate a low frequency polarized shear strain waves. Consequently this technique allows to study anisotropic medium such as muscle. The evidence of the polarization of low frequency shear strain waves is supported by both numeric simulations and experiments. The numeric simulations are based on theoretical Green's functions in isotropic and anisotropic media (hexagonal system). The experiments in vitro led on beef muscle proves the pertinent of this simple anisotropic pattern. Results in vivo on man biceps shows the existence of slow and fast shear waves as predicted by theory.     

Nonlinear shear wave interaction in soft solids

This paper describes nonlinear shear wave experiments conducted in soft solids with transient elastography technique. The nonlinear solutions that theoretically account for plane and nonplane shear wave propagation are compared with experimental results. It is observed that the cubic nonlinearity implied in high amplitude transverse waves at f(0)=100 Hz results in the generation of odd harmonics 3f(0), 5f(0). In the case of the nonlinear interaction between two transverse waves at frequencies f(1) and f(2), the resulting harmonics are f(i)+/-2f(j)(i,j=1,2). Experimental data are compared to numerical solutions of the modified Burgers equation, allowing an estimation of the nonlinear parameter relative to shear waves. The definition of this combination of elastic moduli (up to fourth order) can be obtained using an energy development adapted to soft solid. In the more complex situation of nonplane shear waves, the quadratic nonlinearity gives rise to more usual harmonics, at sum and difference frequencies, f(i)+/-f(j). All components of the field have to be taken into account.     

Nonlinear surface waves in soft, weakly compressible elastic media

Nonlinear surface waves in soft, weakly compressible elastic media are investigated theoretically, with a focus on propagation in tissue-like media. The model is obtained as a limiting case of the theory developed by Zabolotskaya [J. Acoust. Soc. Am. 91, 2569-2575 (1992)] for nonlinear surface waves in arbitrary isotropic elastic media, and it is consistent with the results obtained by Fu and Devenish [Q. J. Mech. Appl. Math. 49, 65-80 (1996)] for incompressible isotropic elastic media. In particular, the quadratic nonlinearity is found to be independent of the third-order elastic constants of the medium, and it is inversely proportional to the shear modulus. The Gol'dberg number characterizing the degree of waveform distortion due to quadratic nonlinearity is proportional to the square root of the shear modulus and inversely proportional to the shear viscosity. Simulations are presented for propagation in tissue-like media.     

Nonlinear Longitudinal Strain Waves in a Solid Exposed to Pulsed Laser Radiation

The propagation of longitudinal strain waves in a solid with quadratic nonlinearity of elastic continuum was studied in the context of a model that takes into account the joint dynamics of elastic displacements in the medium and the concentration of the laser-induced point defects. The input equations of the problem are reformulated in terms of only the total displacements of the medium points. In this case, the presence of structural defects manifests itself in the emergence of a delayed response of the system to the propagation of the strain-related perturbations, which is characteristic of media with relaxation or memory. The model equations describing the nonlinear displacement wave were derived with allowance made for the values of the relaxation parameter. The influence of the generation, relaxation, and the strain-induced drift of defects and the flexoelectricity on the propagation of this wave was analyzed. It is shown that, for short relaxation times of defects, the strain can propagate in the form of both shock fronts and solitary waves (solitons). Exact solutions depending on the type of relation between the coefficients in the equation and describing both the shock-wave structures and the evolution of solitary waves are presented. In the case of longer relaxation times, shock waves do not form and the strain wave propagates only in the form of solitary waves or a train of solitons. The contributions of the finiteness of the defect-recombination rate and the flexoelectricity to linear elastic moduli and spatial dispersion are determined.     

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

Localization of a transverse elastic wave in a semibounded acoustic superlattice of ferrimagnetic and superconducting layers: 3. Leaky surface modes and associated resonances

In terms of the effective medium method, conditions for the existence of leaky shear surface acoustic waves in a semibounded fine-layered magnetic superlattice consisting of ferrimagnetic and superconducting layers are determined. On this basis, the possibility of a resonance interaction between a surface elastic SH wave propagating in the magnetic superlattice and a shear bulk wave propagating in the adjacent nonmagnetic medium is investigated.