# Physics of Within-Tissue Wave Propagation Generated by Pulse Propagation in the Carotid Artery
## 4 Results
### Load Pressure Conditions: Shock Impulse and Traveling Impulse  
Wave propagation is illustrated in Figure 4a,b for the applied shock and traveling impulse loading conditions respectively, where the radial displacements are visualized at four different time instances. Under shock impulse loading conditions, the applied pressure load induced a local positive displacement of 0.06 mm (labeled \(\mathrm{f_{m}}\) where the index m refers to the main wave front) in the lumen at t = 0 ms, as visible in Figure 4a. Near \(\mathrm{f_{m}}\) at the same time point, there are also substantial displacements of negative magnitude (\(-0.02\) mm) generated in order to preserve volume. These are indicated with \(\mathrm{f_{z-}}\) and \(\mathrm{f_{z+}}\), where the indices z+ and z- refer to the propagation direction. The wave front \(\mathrm{f_{z-}}\) propagates backward (negative z-direction) where it is absorbed by the infinite elements. At t = 2 ms, the main wave front \(\mathrm{f_{m}}\) splits giving rise to a secondary wave front in the lumen (labeled \(\mathrm{f_{s}}\) where the index s refers to the secondary wave front) which travels ahead of \(\mathrm{f_{m}}\) and behind \(\mathrm{f_{z+}}\). Meanwhile, at the skin surface, a first perturbation is visible at z = 0 mm. At 4 ms, the wave front \(\mathrm{f_{m}}\) has traveled about \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Element Size** **[mm]** & **Total Number** **of Elements [-]** & **Shear Wave** **Velocity [m/s]** & **Total Computational** **Time [min]** \\ \hline
0.15 & 1,218,588 & 4.79 & 203 \\
0.20 & 521,078 & 4.72 & 65 \\
0.25 & 269,180 & 4.58 & 62 \\
0.30 & 149,303 & 4.51 & 41 \\
0.50 & 28,900 & 4.34 & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Averaged shear wave velocity values of backward and forward propagating waves together with their computational times are shown for the considered meshes (computational specification mentioned in Section 3.1.2). The analytical value of the shear wave velocity is 4.82 m/s. Figure 3: Comparison of theoretical and simulation results: (**a**) vector plots demonstrating normalized displacements, (**b**) z and y components of the displacement at the position z = 5 mm for depths y = 0, 5 and 10 mm. Note that simulated displacements of the finite element (FE) grid (mesh size of 0.2 mm) were interpolated to a coarser spatial grid for visualization purposes. one third of the lumen length while the wave fronts f\({}_{\text{z+}}\) and f\({}_{\text{s}}\) are completely dissipated. The skin displacement associated with f\({}_{\text{m}}\) has also traveled about one third of the model's length. At the time point of 6 ms, the maximal displacement magnitude of the main wave front has decreased to 0.01 mm at the lumen surface and to 0.003 mm at the skin surface. The main wave front is clearly reflected at the skin surface towards the lumen surface where it starts propagating as a secondary wave front along the positive z-direction (indicated with f\({}_{\text{r}}\) in Figure 4a, where the index r refers to the reflected wave front). Throughout the manuscript, we will focus on the generated positive displacements, as these have the same direction as the applied load. Figure 4b demonstrates the simulated wave propagation when a propagating 40 mmHg pressure pulse is applied to the lumen (velocity V\({}_{\text{z}}\) of 7 m/s). At the first time instance, the imposed travelling load is visible near the initial position of the lumen (z = 0 mm); however, note that the displacement magnitude (maximum of 0.03 mm) is not yet fully developed at this time point. At t = 2 ms, the induced displacement magnitude of the main wave front in the lumen is about 0.12 mm. At 4 ms, the load has travelled about half the length of the lumen, while the wave front at the skin surface is visible at about one third of the length of the model.