# Physics of Within-Tissue Wave Propagation Generated by Pulse Propagation in the Carotid Artery
## 4 Results
### Patient-Specific Carotid Artery  
The chosen mesh for the patient-specific carotid artery model consisted of 2,095,200 elements (average maximum edge length 0.40 mm and average minimum edge length 0.15). Simulated shear waves propagated with a velocity of 4.72 m/s (\(-\)2.07% deviation compared to the theoretical value of 4.82 m/s). Figure 8 shows the radial displacement inside the patient-specific CA model for four time instances, when imposing a travelling pressure impulse (V\({}_{\mathrm{z}}\) of 7 m/s) inside the arterial lumen. At the time 2 ms, the traveling pressure is localized near the proximal end of the artery (z = 0 mm) and the induced displacement has a maximum amplitude of 0.03 mm. At 4 ms, the induced lumen displacement due to the applied load is visible at the position z = 20 mm and the wave has almost reached the skin level. When the pulse passes through the bifurcation (time 6 and 8 ms), the wave propagation evolves into a more complex displacement pattern: secondary perturbations are generated and the wave propagation direction from lumen to skin surface has a quasi-horizontal trajectory due to the bending of the external carotid artery (ECA). Maximum displacement magnitude is then 0.10 mm at the lumen level and 0.06 mm at the skin level. Figure 9 shows the TOF plots of the out-of-plane displacement (y direction) along the centerline distance at the level of the top of the lumen and along the artery long axis for the skin of the patient-specific CA model. The travelling wave front at the lumen surface is composed of a wave front which changes  
Figure 8: Radial displacement depicting wave propagation at four time instances for the traveling impulse with a velocity of 7 m/s. Note that a cylindrical coordinate system is used in this figure to obtain a more intuitive visual representation of wave propagation. Figure 7: TOF plot at the skin surface for the CCA with depth d of **(a)** 5 mm and **(b)** 15 mm when imposing a traveling pulse (V\({}_{\mathrm{z}}\) of 4, 7 and 10 m/s). The propagation velocities of the main (V\({}_{\mathrm{1}}\)) and secondary (V\({}_{\mathrm{2}}\)) wave front are marked by slope of the dashed and solid lines, respectively. speed along the length of the model due the geometrical changes of the arterial upper wall compared to the centerline of the artery. Three different speeds were calculated for the main wave front, i.e., 7.09, 10.1 and 14.5 m/s, as reported in Figure 9. At the skin surface, the main wave front has a speed of 6.78 m/s, which is in good agreement with the imposed V\({}_{\mathrm{z}}\) (-3.1%). A secondary wave front (maximum amplitude of 0.03 mm compared to 0.06 mm for the main wave front) with propagation velocity of 9.79 m/s was also observed. As with the CCA model, the temporal profile of the y-displacement at locations z = 20 and 30 mm is shown in the right panel of Figure 9, where the red markers indicate the wave trajectory determined using the Radon transform. Both displacement signals show a pattern in which the main displacement peak, representing the main wave front, is easy identifiable. Figure 10 shows the spatio-temporal displacement plots at the skin surface when considering traveling pressure pulse velocities V\({}_{\mathrm{z}}\) of 4 and 10 m/s. The main wave front propagated with a speed V\({}_{\mathrm{1}}\) of 5.88 m/s (47.0%) and 13.6 m/s (36.0%) for imposed V\({}_{\mathrm{z}}\) of 4 and 10 m/s, respectively. Secondary wave fronts were also observed, as indicated on the top of each plot in Figure 10.  
Comparing the results of all imposed traveling pulses in Figure 10, it is clear that the maximum amplitudes of the main wave front at the skin surface (indicated with speed V\({}_{\mathrm{1}}\)) were 0.05 mm for V\({}_{\mathrm{z}}\) of 4 and 10 m/s and 0.06 mm for V\({}_{\mathrm{z}}\) of 7 m/s, whereas the maximal amplitude of the secondary wave fronts (indicated with speed V\({}_{\mathrm{2}}\)) was 0.03 mm for V\({}_{\mathrm{z}}\) of 4, 7 and 0.04 mm for V\({}_{\mathrm{z}}\) of 10 m/s.