Two main questions are at the center of this paper. The first one concerns the choice of a rheological model in the frequency range of transient elastography, sonoelasticity or NMR elastography for soft solids (20-1000 Hz). Transient elastography experiments based on plane shear waves that propagate in an Agar-gelatin phantom or in bovine muscles enable one to quantify their viscoelastic properties. The comparison of these experimental results to the prediction of the two simplest rheological models indicate clearly that Voigt's model is the better. The second question studied in the paper deals with the feasibility of quantitative viscosity mapping using inverse problem algorithm. In the ideal situation where plane shear waves propagate in a sample, a simple inverse problem based on the Helmholtz equation correctly retrieves both elasticity and viscosity. In a more realistic situation with nonplane shear waves, this simple approach fails. Nevertheless, it is shown that quantitative viscosity mapping is still possible if one uses an appropriate inverse problem that fully takes into account diffraction in solids.

Measurement of viscoelastic properties of homogeneous soft solid using transient elastography: An inverse problem approach

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.

A solution to diffraction biases in sonoelasticity: The acoustic impulse technique

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.

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

The objective of this study is to propose a 2.5D finite/infinite element procedure for dealing with the ground vibrations induced by moving loads. Besides the two in-plane degrees of freedom (DOFs) per node conventionally used for plane strain elements, an extra DOF is introduced to account for the out-of-plane wave transmission. The profile of the half-space is divided into a near field and a semi-infinite far field. The near field containing loads and irregular structures is simulated by the finite elements, while the far field covering the soils extending to infinity by the infinite elements with due account taken of the radiation effects for moving loads. Enhanced by the automated mesh expansion procedure proposed previously by the writers, the far field impedances for all the lower frequencies are generated repetitively from the mesh created for the highest frequency considered. Finally, the accuracy of the proposed method is verified through comparison with a number of analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.

A 2.5D finite/infinite element approach for modelling visco-elastic bodies subjected to moving loads

Train-induced wave propagation in layered soils using finite/infinite element simulation

The propagation of transient waves in an elastic half-space excited by a traveling normal point load is investigated. The load is suddenly applied and then it moves rectilinearly at a constant speed along the free surface. The displacements are computed for all points of the half-space as well as for all load speeds.

The disturbance is analyzed by using multi-integral transforms and an inversion scheme based on the well-known Cagniard technique. This reduces the displacements to single integral and algebraic contributions, each of which is identified as the disturbance behind a specific wave front. The same solution is valid for all load speeds, even though the wave front geometry varies greatly, depending on the speed of the load relative to the body wave speeds. Moreover, the surface displacements are obtained from the interior ones, but only after the Rayleigh waves are computed by a separate calculation. Then, by taking advantage of the form of the exact solution, wave front expansions and Rayleigh wave approximations are computed for all load speeds.

Several other analytical results are obtained for restricted values of the load speed. In particular, when it exceeds both of the body wave speeds the steady-state displacement field is separated from the transient one and reduced to algebraic form. Also, for the limit case of zero load speed a new representation of the interior displacements for Lamb's point load problem is displayed in terms of single integrals.

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David C. Gakenheimer

Papers

Transient Excitation of an Elastic Half Space by a Point Load Traveling on the Surface