## 3 Results
### Quantification of the Helicity  
The results so far show that helicity, in general, is considered to have favorable physiological effects as it promotes efficient transportation of blood, reduces flow disruptions, and helps prevent the accumulation of blood stasis behind the leaflets and aortic sinuses. In Section 2.5, six helicity-based bulk flow descriptors are presented. The helicity descriptors for both uncalcified and calcified aortic valve scenarios, under both uniform and helical flow conditions, are summarized in Table 2. These values, obtained from the Valsalva sinus and ascending aorta, were integrated over the second period. The helicity-baseddescriptors offer valuable insights into the intricate flow changes and helical characteristics of flow topology induced by aortic valve calcifications.  
Figure 18: Distribution of RRT [Pa\({}^{-1}\)] across leaflets for varying grades of calcification and inflow types (uniform and helical).  
\(H_{k}\). For uniform flow, the \(h_{1}\) descriptor suggests that the severity of calcification directly correlates with an increase in helical flow. A zero \(h_{1}\) value signifies reflectional symmetry in the flow. While a \(h_{1}\) value of \(-0.204\) for the no-calcification case indicates a minor left-handed helical flow, this case is the closest to having reflectional symmetry among all the considered scenarios. Interestingly, the Grade 4 case has a left-handed flow trend, whereas Grades 3, 5, and 6 exhibit right-handed helical flows. In the context of helical inflow, \(h_{1}\) values see a dramatic rise due to the inherent characteristics of the flow patterns. However, as the calcification grade increases, \(h_{1}\) values tend to decrease. This suggests that higher grades of calcification disturb the flow, thereby reducing its helicity. Specifically, the lowest \(h_{1}\) value for Grade 4 indicates a shift toward a left-handed helical flow. The second descriptor, \(h_{2}\), shows the integration of the absolute value of \(H_{k}\), reflecting the total helical flow in the domain without considering its direction. The \(h_{2}\) values rise along with the severity of calcification, implying that helical flow structures are intensified due to the increased stiffness of the valve leaflets and asymmetric flow conditions. Notably, the elevated \(h_{2}\) values for Grades 3 and 4, when compared to Grade 5, highlight the impact of localized calcification on one or two leaflets. The third descriptor, \(h_{3}\), ranges from \(-1\) to 1 and indicates the presence of left-handed (negative) or right-handed (positive) helical structures within the domain. The descriptor \(h_{4}\) emphasizes the balance between opposing rotational structures and disregards the major rotation direction. The descriptors \(h_{3}\) and \(h_{4}\) tend toward zero as the calcification grade increases, suggesting that the natural helical flows set by inflow conditions become increasingly disrupted, resulting in less coherent vortices. The fifth descriptor, \(h_{5}\), computes the proportion between the volumes of \(V_{id}\) and \(V_{im}\), while the sixth, \(h_{6}\), calculates the ratio of \(H_{k}\) values across the same volumes. Values of \(h_{5}\) exceeding one for the uncalcified and Grades 3-5 cases suggest a higher volume occupied by the dominant rotational direction. Conversely, lower \(h_{6}\) values for these grades indicate a lower rotational intensity. In contrast, Grade 6 shows a higher volume occupied by the minor rotational direction, as indicated by \(h_{5}\), while \(h_{6}\) reveals higher intensity in the dominant rotational direction. Overall, these findings strongly indicate that each case of aortic valve calcification generates a distinct helical flow tendency, which is significantly amplified by the presence of the helical inflow pattern, with likely implications for the progression of cardiovascular diseases.