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

Abstract

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 (µ) 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): 2 2 3 2 3 DI I A µI e + + = . Where I 2 , I 3 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 S = S (µ, 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 (µ) 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 S 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.

Third- and fourth-order constants of incompressible soft solids and the acousto-elastic effect

“…The thin straight lines correspond to the linear part of the ρv 2 -e curves. Here the initial shear modulus is µ = 6.6 kPa, the thirdorder Landau constant is A = −37.7 kPa, and the coefficient of non-linearity is ν = 3.5, in line with the experimental results of Renier et al [6].…”

“…Finally, the results can easily be expanded in terms of the elongation e = λ − 1. Here we find that (6) and we do not need to specify the higher-order terms for our purpose, which is to establish relationships in the form of Eq. ( 2).…”

“…We conclude this section with an example taken from experimental investigations. Renier et al [6] use a small elongation (compression) to deduce a linear ρv 2 -σ relationship from experimental data on Agar-Gelatin based phantoms: they conclude that for their 5% Gelatin sample, µ ≃ 6.6 kPa and A ≃ −37.7 kPa. Then they use finite-amplitude shear waves to estimate the coefficient of non-linearity: they find ν ≃ 3.5 (see also the combination of Ref.…”

“…where E is the Green strain tensor and µ, A, and D are second, third-, and fourth-order elasticity constants, respectively, at least 16 articles [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] have studied the dynamics of those solids.…”

Shear waves elastography for assessment of human Achilles tendon's biomechanical properties: an experimental study

“…At last, at high deformation, strain and stress are no more negligible, and the shear wave propagation is modified due to the inner stress of the tendon ("acoustoelasticity theory"). Shear nonlinear parameters intervene and, although they are not easily measurable, they should be taken into account (Renier et al, 2007). Helfenstein-Didier et al (2016), in their recent comparison of conventional SWE and new dispersion technique, found a high significant correlation (r=0.84).…”