# Physics of Within-Tissue Wave Propagation Generated by Pulse Propagation in the Carotid Artery
## 5 Discussion
Despite this limitation, the simulation results showed that the chosen numerical settings made it possible to reproduce a shear wave propagation speed with a deviation of \(-2.0\%\) from the analytical value, whereas Rayleigh wave speed propagation deviated only \(0.9\%\). Additionally, a good match between the spatial and temporal displacement characteristics of simulation and theory was observed (see Figure 3). The geometry of the patient-specific model introduced a higher degree of complexity which considerably complicated the test of the computational settings. Only shear waves were simulated in this set-up, resulting in a shear wave speed propagation that differed by \(-2.0\%\) from the analytical value. Consecutively, a parametric study was set up to study the impact of different arterial pressure loads and geometrical conditions on the induced skin displacement in a controlled environment. Tracked displacement patterns showed the coexistence of a primary (higher amplitude) and secondary wave front travelling with different velocities both at the boundary of the lumen and the skin surface. Figure 11 summarizes the estimated wave velocities for the primary (V\({}_{1}\)) and secondary (V\({}_{2}\)) wave fronts at the skin level for all simulated cases. Initially, a localized, non-propagating short pressure impulse was used to induce free wave propagation in the soft-tissue mimicking material surrounding the CCA model. Tracked displacement patterns in the CCA model showed a main wave front propagating with a velocity V\({}_{1}\) of 4.66 m/s at the lumen surface and a similar wave speed was found for the first secondary wave front at the skin level (V\({}_{2}\) = 4.79 m/s). These values approached the analytical value of shear (4.82 m/s) and Rayleigh (4.59 m/s) wave speed, however, Figures 4 and 5 also clearly showed a complex wave pattern with multiple wave fronts due to phenomena such as wave recombination, refraction and reflection, and therefore the relation between the observed wave patterns and the ideal free wave propagation modes is unclear. Additionally, it should be noted that we are only analyzing one displacement component in the TOF-plots (component relevant for LDV-analyses), whereas the traveling waves will have displacement components in multiple directions. When considering a traveling impulse (wave speeds varying within the physiological range 4-10 m/s), it can be observed that for imposed propagation speeds of 7 and 10 m/s, all tested conditions in the CCA model yielded an estimated propagation speed V\({}_{1}\) close to the imposed value (differences \(-\)5.0 to \(-\)0.4; Figure 11). However, the lower imposed wave speed of 4 m/s is overestimated at the skin surface by 4%-33.5% in the model, in which the deviation increased with increasing arterial depth. It should be noted that the selected model's geometry and boundary conditions play a role in reaching a steady state regime for wave pattern development at the skin surface. However, since the tracked skin speeds for the higher imposed PWV corresponded fairly well with the imposed value for all considered arterial depths, we assume that the present transition phenomena in current modeling environment are  
Figure 11: Summary of the estimated V\({}_{1}\) and V\({}_{2}\) values at the skin surface for all the simulations. The dotted line represents the reference. minimal. As such, these results suggest that observed changes in wave velocity at the skin level may not fully reflect the effective changes in the arterial pulse wave velocity, as shown in Figure 11.  
For the patient-specific CA model, the estimated lumen wave speed was composed of a split main wave front characterized by different velocities along the length of the model, as shown in Figure 9. This split wave front is caused by the differences in length and orientation of the upper wall compared to that of the centerline at x = 0 mm.