## 2 Materials and Methods
### Post-Processing
#### 2.3.2 Shear Modulus Estimation  
As our previous work [23] has shown that dispersive shear wave propagation patterns arose in the studied setting due to geometry, two different shear modulus estimation techniques were applied on the mechanical and (real and virtual) imaged wave propagation, i.e., a time-of-flight (TOF) method--implemented in commercial SWE systems and used for non-dispersive media--and a phase velocity analysis--used for dispersive media. The real and virtual imaging acquisitions were pre-processed by averaging the axial velocities over 0.6 mm axial depth and temporally up-sampling the slow time domain by a factor 10.  
For the TOF method, the shear wave's position was tracked by searching the maximal axial velocity for every lateral spatial location as a function of time and fitting a linear model to estimate the shear wave velocity (goodness of fit should be equal to or larger than 0.95) [23; 36]. In general, to make the most complete use of the measured data and to increase the reliability of the fit, axial velocity data acquired from all probe elements should be taken into account during this linear fitting procedure. This is true for large isotropic homogeneous elastic media, but usually data from the probe's edge elements is discarded due to low signal-to-noise ratio and/or high attenuation of the propagating shear wave in the measurement. Even though the studied PVA setting is isotropic, homogeneous and low viscous, the left ventricular geometry induces dispersive shear wave features in the SWE-acquisitions which affect the tracked shear wave's position as a function of time. To investigate the effect of this observation on the results of the TOF method, we altered the number of data points taken into account during the TOF fitting procedure: the shear wave speed was estimated by rejecting 5 and 20 data points from the probe's edge elements for each shear wave. The shear modulus \(\mu\) can then be derived from this wave speed \(c_{T}\) by assuming an isotropic bulky elastic material with density \(\rho\) and applying the following formula:  
\[\mu=\rho c_{T}^{2} \tag{4}\]  
For the phase velocity analysis, measured or simulated dispersion characteristics were derived by taking the 2D Fast Fourier Transform (FFT) of the axial velocity wave propagation pattern as a function of lateral space and slow time at a specific depth [37]. Subsequently, the wavenumber \(k\) with the maximal Fourier energy is tracked at each frequency \(f\) in order to identify the main excited mode. Phase velocity \(c_{\varphi}\) as a function of frequency \(f\) is found through \(c_{\varphi}=(2\pi f)/k\). The shear modulus is then estimated by fitting a theoretical model in a least squares manner to the obtained dispersion curve. Neglecting the ventricular curvature [38] and PVA's viscoelasticity, and assuming that the main excited mode is the first antisymmetric mode (A0) [17], we minimized the difference between the theoretical A0 dispersion curve of a plate in water and the extracted dispersion characteristics over a frequency range spanning from 0.2 kHz up to maximally 2 kHz, dependent on the considered acquisition [37; 39]. Only fits giving a standard deviation less than 0.6 kPa for the shear modulus estimate were considered.  
Both procedures were repeated for multiple depths across the phantom's thickness (\(n=10\)). For further details on both shear modulus estimation techniques, we refer to [23].