Accounting for the Spatial Observation Window in the 2-D Fourier Transform Analysis of Shear Wave Attenuation

Abstract

Recent measurements of shear wave propagation in viscoelastic materials have been analyzed by constructing the two-dimensional Fourier transform (2DFT) of the shear wave signal and measuring the phase velocity c(ω) and attenuation α(ω) from the peak location and full width at half maximum (FWHM) of the 2DFT signal at discrete frequencies. However, when the shear wave is observed over a finite spatial range, the 2DFT signal is a convolution of the true signal and the observation window, and measurements using the FWHM can give biased results. In this study, we describe a method to account for the size of the spatial observation window using a model of the 2DFT signal and a nonlinear, least squares fitting procedure to determine c(ω) and α(ω). Results from the analysis of finite element simulation data are in agreement with c(ω) and α(ω) calculated from the material parameters used in the simulation. Results obtained in a viscoelastic phantom demonstrate that the measured attenuation is independent of the observation window and are in agreement with measurements of c(ω) and α(ω) obtained using the previously described progressive phase and exponential decay analysis.

Characterization of Viscoelastic Materials Using Group Shear Wave Speeds

“…Experimental measurements were analyzed using a material model with shear wave attenuation described as a linear function of frequency with proportionality factor α 0 , αfalse(ωfalse)=α0|ω|.This model was chosen because previous observations of shear wave propagation in viscoelastic phantoms indicate that the shear wave attenuation is approximately linear in frequency, see for example, Fig. 4 of Rouze, et al [17]. The model (33) differs from the Voigt material model (10) used in a preliminary investigation [24] of material characterization using group shear wave speeds.…”

“…The model (33) differs from the Voigt material model (10) used in a preliminary investigation [24] of material characterization using group shear wave speeds. In particular, using (4), the low frequency dependence of the Voigt model attenuation is quadratic, αVoigtfalse(ωfalse)~η2μ0ρμ00.2emω2, and does not agree with the observed frequency dependence in [17]. Further discussion of the choice of material model is included in Sec.…”

“…First, as described by Chen, et al [4] and Deffieux, et al [13], the phase of the asymptotic Hankel function (9) varies linearly with the coordinate r , so a linear fit of phase vs. position can be used to measure the wavenumber k and phase velocity c ( ω ) = ω / k. Similarly, the attenuation α ( ω ) can be measured using the exponential decay with position in (9) [14, 15], or by using the Hankel function dependence in (8) [9]. A second method to measure the phase velocity and attenuation is by constructing the two-dimensional Fourier transform (2DFT) of the spatial-temporal shear wave signal and to measure the phase velocity and attenuation from the peak location and width of the 2DFT signal at each specific temporal frequency [8, 16, 17]. It is important to realize that both of these analysis methods depend on the specific form of the solution (5), and that this solution does not account for the size and shape of an ARFI excitation which are known to give biased results for the phase velocity [8] and group shear wave speed [18].…”

Robust Phase Velocity Dispersion Estimation of Viscoelastic Materials Used for Medical Applications Based on the Multiple Signal Classification Method

“…However, adjusting the distance range of the data, i.e., the distance from the source and the distance traveled, can cause changes to the dispersion. This has been observed in practice [50] but not systematically explained yet.…”

“…The acoustic radiation force push produced by array transducers is often approximated as a cylindrical source as opposed to a plane wave [4]. As a result, it has become more common, particularly when trying to solve for the shear wave attenuation, a correction is applied to account for the geometric attenuation produced by the cylindrical wave source [25], [47]–[50]. In the reports by Rouze, et al, and Nenadic, et al, this correction does not yield a large change in the wave velocity dispersion.…”