Christophe Barrière

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.

Acoustic nonlinearity parameter measurements in solids using the collinear mixing of elastic waves

An alternative method for measuring the nonlinearity parameter β of longitudinal acoustic waves propagating in solids is presented. The method is based on the detection of the phase modulation resulting from the parametric interaction between a high frequency acoustic wave and a lower frequency acoustic pulse. Results are reported for various materials: fused quartz, duraluminum, titanium, and polystyrene.

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.

Experimental study of the acoustic radiation strain in solids

Measurements of the static displacement induced by the radiation stress associated with a longitudinal acoustic wave propagating in a solid are presented. Acoustic tone bursts were launched into fused silica and duraluminum samples. The static displacement was measured at the sample free surface with an optical interferometer. The role of the nonlinearity parameter and the variations of the dc pulse amplitude with the acoustic energy confirm results obtained by other authors. However, our conclusions on the dc pulse shape, on the influence of the tone burst duration, and of the propagation distance on the dc pulse amplitude are different.

Fourth-order shear elastic constant assessment in quasi-incompressible soft solids

In isotropic quasi-incompressible media, an expression of the elastic energy density has been developed as a function of the second-, third-, and fourth-order elastic constants (respectively μ, A, D). Thus the shear nonlinearity parameter βS depends only on these coefficients. In this letter βS is measured using finite amplitude plane shear waves in agar-gelatin based phantoms. Combining the results with recently published measurements of μ and A on the same phantoms, the fourth-order shear elastic constant D is found to be of the order of 10kPa and thus of the same order of magnitude as μ and A.

Optical measurement of large transient mechanical displacements

A simple method suitable for extracting large mechanical displacements from the phase modulation of an optical beam reflected from the moving surface is presented. In the MHz range, transient displacements larger than 1 μm have been measured with a standard heterodyne interferometer.

7B-2 Nonlinear Shear Elastic Moduli in Quasi-Incompressible Soft Solids

Dynamic elastography holds great promise for biological tissues characterization. Resulting from the radiation force induced by focused ultrasound beam, plane shear waves are generated within the medium and imaged with an ultrafast ultrasound scanner. Known as Supersonic Shear Imaging (SSI) technique, this method allows, from the measurements of shear wave velocities, to compute shear modulus (mu) maps. Beside, in order to improve tissue diagnostic, the evaluation of the nonlinear elastic moduli could be of some interest. Recently a new formulation of the nonlinear equation describing the propagation of plane shear waves in isotropic soft incompressible solids have been developed using a new expression, up to the fourth order, of the strain energy density (e): e = muI2 + A/3I3, + DI2 2. Where I2, I3 are invariants defined by Landau of the strain tensor and A, D the third and fourth order shear elastic constants. It has been shown that the nonlinearity parameter depends only on three coefficients betas = betas (mu, A, D). To date, no measurement of the parameter D have been carried out in incompressible media. In order to estimate the nonlinear parameter A, this theoretical background on soft incompressible solids is applied to the acoustoelasticity theory. Such analysis gives the variations of shear wave speed as a function of the applied stress and leads to measure both the linear shear modulus (mu) and the third order shear modulus (A). Taking advantages of the SSI technique, an acoustoelasticity experiment is performed in different incompressible soft media (agar-gelatin based phantoms). In addition, to create finite amplitude plane shear waves, the SSI technique is replaced by a vibrator applied at the surface of the phantoms. Thanks to the ultrasound ultrafast imaging system, the third harmonic component is generated by nonlinearity is measured as a function of the propagation distance. Then by comparing experiments and analytical expression of the third harmonic component given by a perturbation method, the nonlinear parameter betas is deduced. Finally, the combination of these experiments with results obtained in acoustoelasticity leads to the determination of the fourth order elastic modulus (D). First, measurements of the A modulus reveal that while the behavior of phantoms is quite close from a linear point of view, their nonlinear modulus A are quite different. Applied to acoustoelasticity, the SSI technique provides potential medical applications in in vivo conditions for nonlinear characterization of biological tissues. Second, results from the complete procedure reveal a variation of the nonlinear behavior as a function of the gelatin concentration increasing. This set of experiments provides the characterization, up to the fourth order, of the nonlinear shear elastic moduli in incompressible soft media.

Measurement of Shear Elastic Moduli in Quasi‐Incompressible Soft Solids

Recently a nonlinear equation describing the plane shear wave propagation in isotropic quasi‐incompressible media has been developed using a new expression of the strain energy density, as a function of the second, third and fourth order shear elastic constants (respectively μ, A, D) [1]. In such a case, the shear nonlinearity parameter βs depends only from these last coefficients. To date, no measurement of the parameter D have been carried out in soft solids. Using a set of two experiments, acoustoelasticity and finite amplitude shear waves, the shear elastic moduli up to the fourth order of soft solids are measured. Firstly, this theoretical background is applied to the acoustoelasticity theory, giving the variations of the shear wave speed as a function of the stress applied to the medium. From such variations, both linear (μ) and third order shear modulus (A) are deduced in agar‐gelatin phantoms. Experimentally the radiation force induced by a focused ultrasound beam is used to generate quasi‐plane l...

Optical measurements of the self-demodulated displacement and its interpretation in terms of radiation pressure.

Using a sensitive optical interferometer, the low frequency displacement nonlinearly generated by an ultrasonic tone burst propagating in a liquid is studied. Close to the source, the low frequency displacement contains a quasi-static component, which is affected by diffraction effects farther from the transducer. The experimental setup provides quantitative results, which allow the determination of the nonlinearity parameter of the liquid with a good accuracy. Such measurements are carried out in water and ethanol. Finally, the pressure associated with the low frequency displacement is discussed. Introducing the temporal mean value of the displacement, as already done in lossless solids, the noncumulative part of this second order pressure is associated with the static part of the low frequency displacement. This interpretation leads to extend the definition of the Rayleigh radiation pressure usually introduced for a continuous plane wave radiated in a confined fluid.

Nonlinearity studies in soft tissues with the supersonic shear imaging system

The ultrafast scanner has been shown to be a powerful tool to detect shear wave propagation within soft tissues in transient elastography experiments. More recently it was also used to generate shear waves thanks to the acoustic radiation pressure. This technique, the supersonic shear imaging, can easily be implemented in an acoustoelasticity experiment. Thus the association of static elastography with dynamic elastography can reveal the nonlinear properties of soft materials. Moreover, using a new theoretical approach of the strain energy in soft solid (Hamilton et al. (2003)), it is shown that the acoustoelasticity experiment can be greatly simplified. Instead of measuring shear wave speed for three different polarizations in order to completely determine the nonlinearity of standard solids, one is sufficient in soft solids to characterize the nonlinear shear elasticity.