Catheline S, Gennisson J L, Fink M
Laboratoire Ondes et Acoustique, ESPCI, Université Paris VII, U.M.R. C.N.R.S. 7587, 10 rue Vauquelin, 75231 Paris cedex 05, France.
J Acoust Soc Am. 2003 Dec;114(6 Pt 1):3087-91. doi: 10.1121/1.1610457.
Transient elastography is a powerful tool to measure the speed of low-frequency shear waves in soft tissues and thus to determine the second-order elastic modulus mu (or the Young's modulus E). In this paper, it is shown how transient elastography can also achieve the measurement of the nonlinear third-order elastic moduli of an Agar-gelatin-based phantom. This method requires speed measurements of polarized elastic waves measured in a statically stressed isotropic medium. A static uniaxial stress induces a hexagonal anisotropy (transverse isotropy) in solids. In the special case of uniaxially stressed isotropic media, the anisotropy is not caused by linear elastic coefficients but by the third-order nonlinear elastic constants, and the medium recovers its isotropic properties as soon as the uniaxial stress disappears. It has already been shown how transient elastography can measure the elastic (second-order) moduli in a media with transverse isotropy such as muscles. Consequently this method, based on the measurement of the speed variations of a low-frequency (50-Hz) polarized shear strain waves as a function of the applied stress, allows one to measure the Landau moduli A, B, C that completely describe the third-order nonlinearity. The several orders of magnitude found among these three constants can be justified from the theoretical expression of the internal energy.
瞬时弹性成像技术是一种强大的工具，可用于测量软组织中低频剪切波的速度，从而确定二阶弹性模量μ（或杨氏模量E）。本文展示了瞬时弹性成像技术如何还能实现对基于琼脂-明胶的体模的非线性三阶弹性模量的测量。该方法需要测量在静态应力各向同性介质中传播的极化弹性波的速度。静态单轴应力会在固体中引起六方各向异性（横向各向同性）。在单轴应力各向同性介质的特殊情况下，各向异性不是由线性弹性系数引起的，而是由三阶非线性弹性常数引起的，并且一旦单轴应力消失，介质就会恢复其各向同性特性。已经证明了瞬时弹性成像技术如何能够测量具有横向各向同性的介质（如肌肉）中的弹性（二阶）模量。因此，基于测量低频（50赫兹）极化剪切应变波的速度变化作为所施加应力的函数的这种方法，能够测量完全描述三阶非线性的朗道模量A、B、C。这三个常数之间发现的几个数量级差异可以从内能的理论表达式得到解释。
