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 (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.