## 2 Materials and Methods
### SWE Multiphysics Model
#### 2.2.1 Pushing Sequence  
The pushing sequence in the numerical model consists of two steps: ARF generation and mechanical wave propagation (Figure 2). For the first step, the ARF applied on the PVA phantom is numerically mimicked by a volume force in combination with an interface pressure. Both types of loading act on the PVA phantom in the focal zone of the probe, extending \(\sim\)2 mm from the probe's center point in the lateral and elevation direction. The volume force acts throughout the complete thickness of the PVA phantom in this focal region, whereas the interface pressure is only active on the interfaces between phantom and water. Volume force \(b\) and interface pressure \(\pi\) are calculated based on the time-averaged acoustic intensity \(I\), of which its spatial distribution is derived by simulating acoustic probe pressures mimicking the push sequence (see Table 1) with Field II and its magnitude is scaled to 1500 W/cm\({}^{2}\)[26], as follows [21; 27]:  
\[b=\frac{2\pi I}{\rho c_{L}}, \tag{1}\]  
Figure 2: Workflow of the multiphysics platform; ARF: acoustic radiation force. \[\pi=\frac{I}{c_{1}}\left(1+R-(1-R)\frac{c_{1}}{c_{2}}\right), \tag{2}\]  
where \(\alpha\) is the attenuation coefficient [dB/cm/MHz], \(\rho\) the density of PVA [kg/m\({}^{3}\)], \(c_{L}\) the longitudinal wave speed of PVA [m/s], \(R=\left(\frac{Z_{2}-Z_{1}}{Z_{2}+Z_{1}}\right)^{2}\) the energetic reflection coefficient [-], \(Z_{1}\) and \(Z_{2}\) the acoustic impedances (\(Z_{i}=\rho_{i}c_{i}\)) [Pa\(\cdot\)s/m\({}^{3}\)], and \(c_{1}\) and \(c_{2}\) the speeds of sound in media 1 and 2 [m/s]. Material characteristics of the modeled water and PVA can be found in Table 2. The PVA's Young's modulus and viscoelastic behavior were mechanically determined on a uniaxial tensile testing machine (Instron 5944, Norwood, MA, USA), whereas its density and speed of sound were measured using the principle of Archimedes [28] and an oscilloscope respectively (for more details on all measurements, we refer to [23]). The resulting spatial distribution of the volume force in the axial-lateral plane is shown in the bottom-left panel of Figure 2. Both loads are imposed for 250 \(\upmu\)s in the numerical model. For the actual mechanical wave simulation (step II in Figure 2), the PVA phantom was modeled as one half of an ellipsoidal-shaped disk with a lateral and elevational length of 27.8 mm and 16.0 mm respectively, taking the symmetry of the imaging plane into account. For reasons of computational efficiency, we considered only half the width of the transducer in the model, and modeled structural infinite elements at the edges of the defined domain. This PVA model was meshed with 8-noded brick elements with reduced integration, leading to 355,680 elements in total. The water below and above the phantom is represented by two layers of 8-noded hexahedral acoustic elements, each with a thickness of 3.8 mm and 79,684 elements. Mechanical displacements of the PVA phantom were coupled to acoustic pressures in the water layer through a tie-constraint. The other surfaces of the modeled water were modeled to be infinite. The PVA was modeled as a viscoelastic material by assuming a 2-term Prony series model with normalized shear moduli \(g_{i}\) and relaxation times \(\tau_{i}\) as mentioned in Table 2, which are derived from a uniaxial mechanical relaxation test stretching the PVA material at 5% strain for 10 s [23]. It should be noted that the modeled viscoelasticity has a negligible influence on shear wave propagation characteristics, indicating that the actual and modeled PVA phantom have very low viscosity [23]. The water was defined as an acoustic medium in the model with bulk modulus \(K\) and density \(\rho\) as tabulated in Table 2. More details about mesh geometry, boundary conditions, material characteristics and loading can be found in [23]. The dynamic equations of motion of this numerical problem were solved by the Abaqus explicit solver and the particle velocities were extracted at a sampling rate of 40 kHz for further analysis.