Evaluation of Robustness of S-Transform Based Phase Velocity Estimation in Viscoelastic Phantoms and Renal Transplants

Ultrasound shear wave elastography (SWE) methods are being used to differentiate healthy versus diseased tissue on the basis of their viscoelastic mechanical properties. Tissue viscoelasticity is often studied by analyzing shear wave phase velocity dispersion curves, which is the variation of phase velocity with frequency or wavelength. Recently, a unique approach using a generalized Stockwell transformation (GST-SFK) was proposed for the calculation of dispersion curves in viscoelastic media over expanded frequency band. In this work, the method’s robustness was evaluated on data from five custom-made viscoelastic tissue-mimicking phantoms and sixty in vivo renal transplants. For each phantom, 15 shear wave motion data acquisitions were taken, while 10–13 acquisitions were acquired for renal transplants measured in the renal cortex. For each data-set mean and standard deviation (SD) of estimated phase velocity dispersion curves were studied. In addition, the viscoelastic parameters of the Zener model were examined, which were preceded by a convergence analysis. For viscoelastic phantoms scanned with a research ultrasound scanner, and for the in vivo renal transplants scanned with a clinical scanner, the decisive advantage of the GST-SFK method over the standard two-dimensional Fourier transform (2D-FT) method was shown. The GST-SFK method provided dispersion curve estimates with lower SD over a wider frequency band in comparison to the 2D-FT method. These advantages are relevant to the analysis of the mechanical properties of tissues in clinical practice to discriminate healthy from diseased tissue.

where ρ stands for the density and ω is an angular frequency, i.e., ω = 2πf .In order to estimate µ 1 and µ 2 parameters, V s (ω) was estimated using a nonlinear least-squares problem (NLSQ) in a form Equations ( S3) and ( 8), in the main manuscript, were numerically solved using the MATLAB solver lsqcurvefit.The KV and Zener fits were done using two sets of frequency ranges: a short one (with reduced variability and SD) that one would select based on the phase velocity curves estimated using the 2D-FT method, and an extended frequency range.The short frequency range used for the KV fit and the TM phantoms was: 150-400 Hz for Phantom I, 150-1400 Hz for Phantom II, 150-1400 Hz for Phantom III, 150-700 Hz for Phantom IV, and 150-600 Hz for Phantom V.The extended frequency range of 150-1800 Hz for the TM phantoms was used.
Three frequency ranges were used for renal transplants, i.e.: • Case 1: a fixed frequency range of 200-450 Hz, where all groups (except D for 2D-FT) had a coefficient of variation (CV) < 30%; • Case 2: a fixed frequency range of 200-900 Hz; • Case 3: frequency range starting from 200 Hz up to the maximum frequency for which CV < 30% for a given subject group and given approach.The coefficient of variation (CV) was defined as CV = SD M EAN • 100% and shows the degree of variation relative to the sample mean.

II. TM PHANTOMS
Shear wave particle velocity motion data for custom-made TM viscoelastic phantoms were examined.Fig. S1a shows shear wave spatiotemporal data measured for five different TM phantoms.The results for a single acquisition, randomly selected, were displayed, as an example.From Fig. S1a it can be seen that the shear wave velocity is the lowest for Phantom I, and the highest for Phantom V.
Figures S1b and S1c present the frequency-wavenumber (f-k) distribution, also known as k-space which shows the distribution of shear wave energy, reconstructed based on the 2D-FT (Fig. S1b), and GST-SFK (Fig. S1c) methods.The f-k maps were normalized by wavenumber maxima in the frequency direction to highlight the differences between the two methods.Using the f-k maps, phase velocity reconstructions were estimated in Figs.S1d and S1e.The phase velocity maps have superimposed markers corresponding to the maximum peaks of the phase velocity.These results were calculated for the experimental, custom-made TM viscoelastic phantoms I-V, for a randomly selected data acquisitions, which correspond to the shear wave particle velocity motion data shown in Fig. S1a.
Considerable differences can be observed between the results for the 2D-FT method, and the GST-SFK approach.The f-k main energy distributions presented in Figs.S1b and S1c, which determine the shear wave propagation mode in the material, starts diffuse for higher frequencies, i.e. above 500 Hz for Phantom I, and above 1000 Hz for Phantoms II-V.This effect was also seen in the phase velocity reconstructions in Figs.S1d and S1e, where increased magnitude decay for 2D-FT is observed, for above frequencies.
Figures S2 and S4 present the mean phase velocity dispersion curves (depicted as gray dots) obtained from measurements using the 2D-FT (top row) and GST-SFK (bottom row) approaches.Fitted analytical phase velocity curves calculated using the Zener and Kelvin-Voigt (KV) viscoelastic models are overlaid on the data, considering three different frequency ranges.The convergence analysis of the Zener and KV models can be observed in Figs.S3 and S5, respectively.
Comparing the fits of the Zener and KV models, it is evident that the KV model provides reliable fitting for shorter frequency ranges in comparison to the Zener model.This observation is supported by the convergence analysis (low NoR values correspond to good curve fit), which highlights the stability of the KV model parameters in these frequency ranges.The KV model's suitability for shorter frequency ranges indicates that it accurately captures the rheological behavior of the data within that limited frequency range.Furthermore, it is worth noting that the mean phase velocity curves obtained using the 2D-FT method have a reduced usable bandwidth for the rheological model fit compared to the GST-SFK approach.This limitation suggests that the available frequency range for fitting the rheological models is narrower when employing the 2D-FT method.The convergence analysis supports the finding that the KV model is reliable for shorter frequency ranges, while the Zener model provides a more appropriate representation for a wider frequency range.
Figure S6 shows box plots calculated for estimated Kelvin-Voigt parameters, for the GST-SFK and 2D-FT methods.The short and long frequency ranges were used for calculations.

III. IN VIVO RENAL TRANSPLANTS
The experimental in vivo renal transplant data were investigated using the GST-SFK approach and the 2D-FT method for shear wave phase velocity estimation, for clinical applications.Results for these two methods were compared and evaluated.Four groups of subjects were examined as discussed in the main manuscript, and summarized in Table S1.Spatiotemporal shear wave particle velocity signals for three subjects from each group were presented in Figs.S7a, S11a, S15a, and S19a, for healthy subjects, subjects with inflammation and no IFTA, subjects with IFTA, but no inflammation, and subjects with IFTA and inflammation, respectively.
The two-dimensional, normalized by wavenumber maxima f-k distribution maps, as well as, two-dimensional phase velocity results, with marked maxima of the phase velocity, were shown in Figs.S7b-S7e, S11b-S11e, S15b-S15e, and S19b-S19e, for all four groups of subjects, A-D, respectively.The differences between the two methods can be distinguished.The f-k distribution maps for the 2D-FT method have disturbance for the main shear wave particle velocity signal which increases with frequency.This, of course, translates into phase velocity maps which are unreliable at higher frequencies, starting at approximately 400 Hz, depending on the subject under consideration.In turn, the GST-SFK method gives much better robustness for corresponding subjects.The f-k distribution maps display the shear wave energy distribution in a stable manner over at least twice the frequency range compared to the 2D-FT.The phase velocity maps from GST-SFK are more homogeneous and have a higher magnitude as a function of frequency compared to 2D-FT, from which the phase velocity dispersion curves for the main shear wave mode can be extracted.
Similar as for the TM viscoelastic phantoms, examples of the fitted analytical phase velocity curves calculated using the Zener and KV models are shown in Figs.S8, S12, S16, and S20.Results for the convergence analysis for the Zener viscoelastic model were shown in Figs.S9, S13, S17, and S21 for subjects A-D, respectively.Alike convergence results, however obtained for the KV model, were summarized in Figs.S10, S14, S18, and S22.
The analysis of the data revealed that the E 2 parameter exhibited the highest variation across the entire frequency range that was tested.Specifically, for shorter frequency ranges (e.g., <600 Hz for A1, <400 Hz for B1, etc.), the E 2 parameter showed significantly elevated values, suggesting a similarity between the behavior of the Zener model and the KV model.In these cases, the KV model also appeared to be suitable, as evidenced by the very low NoR (<0.5 m 2 /s 2 ) values observed within these frequency ranges.This finding is supported by the convergence analysis conducted specifically for the KV model.
However, when considering a wider frequency range for the GST-SFK approach, the KV model is no longer applicable, and the Zener model becomes more appropriate.Over this broader range, all three parameters of the Zener model demonstrate stabilization (indicating convergence) and exhibit low NoR values (often <1 m 2 /s 2 ).This suggests that the Zener model better captures the rheological properties of the data in this extended frequency range.
Figure S23 shows box plots calculated for estimated KV parameters, which can be compared with the Zener model in Fig. 10, in the main manuscript.
Fig. S1: (a) Spatiotemporal shear wave particle velocity signals.The frequency-wavenumber (f-k) distribution reconstructed based on the (b) 2D-FT, and (c) GST-SFK methods.The f-k maps are normalized by wavenumber maxima in the frequency direction.Phase velocity reconstructions based on the (d) 2D-FT, and (e) GST-SFK methods, for shear wave motion measurements.The phase velocity maps have superimposed markers corresponding to the maximum peaks of the phase velocity.Results were calculated for the experimental, custom-made tissue-mimicking (TM) viscoelastic phantoms I-V, for a randomly selected data acquisitions.
Fig. S23: Box plots calculated for estimated Kelvin-Voigt parameters (a) shear modulus, µ 1 , (b) viscosity, µ 2 , and (c) the norm of residuals, NoR, for GST-SFK and 2D-FT methods.White circles represent mean values, whereas a solid line within the box corresponds to a median value.A fixed frequency range of 200-450 Hz was used for the KV fit in the top row, for both techniques, where all groups (except D for 2D-FT) had CV < 30% (Case 1, top row).The middle row presents the KV fit for frequency range of 200-900 Hz (Case 2, middle row).The bottom row shows the KV fit for frequency range starting from 200 Hz up to the maximum frequency for which CV < 30% for a given subject group and given approach (Case 3, bottom row).Results are presented for the in vivo renal transplant data, for subject groups A-D.All groups consisted of 15 subjects each.

TABLE S1 :
In vivo renal transplant data divided into four groups based on the inflammation and Interstitial Fibrosis and Tubular Atrophy (IFTA) presence.Group A corresponds to healthy subjects.All groups consisted of 15 subjects each.