## 2 Materials and Methods
### Geometric Model  
The geometric model employed in this study is based on the native aortic valve without calcification, adapted from the computer-aided design (CAD) framework proposed by [PERSON] [31]. The model consists of an ascending aorta represented as a straight cylindrical tube illustrated in Figure 1, with dimensions as follows: a height of 0.116 m, an inlet diameter of 0.025 m [32], and a wall thickness of 0.001 m. The aortic valve, without calcification, was modeled with a uniform thickness of 0.0005 m and a height of 0.015 m.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Part** & **Material Model** & **Parameters** \\ \hline Aortic valve & Neo-Hookean, \(\mu=140\,\mathrm{kPa}\) & \(\lambda=17\,\mathrm{MPa}\) \\ Calcifications & Linear elastic, \(E=2\,\mathrm{MPa}\) & \(\
u=0.45\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties for the calcified aortic valve models.  
Based on clinical works in the literature, the calcification formation follows specific patterns. [PERSON] et al. [13] conducted research on the formation process of calcium buildup and determined that it is not a random occurrence. According to their findings, calcium deposits tend to form along the coaptation line of the cusps, a phenomenon known as the coaptation pattern. Alternatively, they may spread along the attachment zone in the abdominal area, referred to as the radial pattern. According to the physiological anatomy report of [PERSON] et al. [6], calcification occurs regularly, progressing from the cusps toward the free edge of the leaflet. [PERSON] et al. [33] conducted a study in which they provided a classification system for assessing the severity of calcification, which can be categorized into several grades, as outlined below:  
* Grade 1 corresponds to the absence of calcification;
* Grade 2 associates with specific regions displaying elevated reflectivity (calcification) but no zones of intense calcification;
* Grade 3 refers to a substantial enhancement in one leaflet's reflectivity, with other leaflets showing comparable or less pronounced changes than those in Grade 2;
* Grade 4 characterizes by a significant increase in reflectivity in two leaflets, with the third leaflet exhibiting equivalent or lesser modifications than those noted in Grade 2;
* Grade 5 signifies a moderate enhancement in reflectivity across all leaflets;
* Grade 6 corresponds to a severe increase in reflectivity throughout all leaflets.  
As Grade 2 calcification has a lower impact compared to the other grades, we did not include this grade in our simulations, and the calcified leaflet geometries were designed for Grades 3-6. As mentioned before, the main objective of our work was to analyze the influence of 3D calcification patterns on the hemodynamic behavior of aortic valve. For this goal, we adopted primary geometries of calcific deposits from the work of [PERSON] et al. [34] and combined them with the reference geometry of the native aortic valve. Model geometries were developed based on the various calcification patterns of several grades reported in the work of [33]. It should be noted that Grade 6 represents the most severe  
Figure 1: Simulation geometry across various grades of calcification.  
case, where the deposits are present on both the aortic and ventricular sides of the leaflets, as shown in Figure 1.