## 3 Results
### Hemodynamic Flow Patterns in the Presence of Aortic Valve Calcification  
To better understand the impact of localized calcification on valve leaflets, we analyzed 3D velocity streamlines at various intra-cardiac time points. Figure 8 shows these streamlines for the no-calcification case under uniform flow, and Figure 9 does the same for helical flow. In both cases, the simulations were conducted over two cardiac cycles, with the displayed results taken from the second cardiac cycle. In the no-calcification case of Figure 8, the aortic valve leaflets remain closed until the pressure in the left ventricle surpasses that in the aorta, prompting the valve to open approximately at \(t=0.035\,\)s. Subsequently, the valve exhibits a fully circular orifice area where the maximum velocities occur right above the Valsalva sinus until the flow velocity reaches its peak during the systole phase. Up to this point, blood flows through the valve into the ascending aorta without noticeable resistance. Following peak systole, as the jet velocity diminishes, multiple vortices form due to flow separation at the leaflet tips and interact with the inner surface of ascending aorta. During early diastole, the aortic valve leaflets begin to close as the pressure in the left ventricle decreases below that in the aorta. This pressure difference initiates a deceleration of blood flow, and thus a loss of momentum and limited backflow towards the Valsalva sinus, aiding in the complete closure of the valve. Figure 8 shows that the spatial and temporal alignment of the Arantius nodules and coaptation lines of the no-calcification case are well coordinated and close in a timely manner. As outlined in Section 2.3, Grade 3 represents a case where only one leaflet is affected by calcification. This condition leads to spatially asymmetric blood flow as the affected leaflet undergoes unstable displacement as expected. Unlike the symmetric flow observed in the no-calcification case, this asymmetry redirects the disturbed flow toward the inner wall of the ascending aorta. As a result, the flow circulates back toward the calcified leaflet, a phenomenon evident in vectorized flow plots of Figure 10 from the late systolic and velocity streamlines plots of the early diastolic phases in Figure 8. This circulating flow pattern contributes to the earlier closure of the calcified leaflet, as presented in the early diastole plots in Figures 8 and 9. Furthermore, the magnitude of velocity in the ascending aorta reaches relatively higher values, likely attributable to the reduction in the valve orifice area. In contrast, the no-calcification case features a uniform jet flow through the orifice, whereas the presence of a constrained leaflet disrupts this flow, leading to a more chaotic jet stream. In Grade 4 of the calcified aortic valve, two leaflets exhibit significant calcium deposits on their surfaces facing the ascending aorta. Increased calcifications, mostly located on the coaptation lines, generated greater valve opening resistance, leading to a reduced orifice area and elevated jet velocity. Unlike the uniform distribution observed in the no-calcification scenario, the jet in Grade 4 becomes narrower, and the magnitude of the velocity slightly increases in the centralized flow direction. As with Grade 3, the flow displays asymmetry, primarily due to the restricted mobility of the calcified leaflets. This results in a shift of the flow toward the third uncalcified leaflet during the flow acceleration phase. More chaotic flow patterns are evident during the deceleration phase, as shown in Figure 10, where asymmetric flow contributes to the earlier closure of the calcified leaflets, further increasing flow eccentricity with a highly chaotic turbulent behavior at the ascending aorta, as is evident in the early diastole plots in of Figure 8. This deviation in flow ultimately creates a strong vortex in the ascending aorta region, leading to increased energy dissipation.