## 4 Discussion  
Instead of focusing on patient-specific model analysis, we aimed to evaluate different angulations in proximal AAA necks. Thus, an idealized geometry was used, and the AAA diameter providing an average repairing threshold with an average probability of rupture was used. Notably, the risk of adverse events was higher when the aortic neck angulation became severe (>60\({}^{\circ}\)) [37]. The three-dimensional model of an idealized AAA  
Figure 11: TAWSS according to (**a**) interval (**A–F**) in cardiac cycle and (**b**) proximal neck angulation. OSI according to (**c**) interval (**A–F**) in cardiac cycle and (**d**) proximal neck angulation.  
Figure 10: Peak WSS according to proximal neck angulation.  
was used to perform CFD simulations and visualize the effect of severe proximal neck angulation on the hemodynamic parameters, including flow velocity, pressure, and WSS distributions. In addition, CFD simulations of idealized AAA were performed to assess time-dependent hemodynamics. Furthermore, the analytical approach to a fully defined pulsatile velocity waveform was determined to better explain the predicted flow pattern in the abdominal aorta.  
The influence of the proximal neck angulation on flow inside the aneurysm sac clearly shows complex recirculation and flow impingement, which alters blood flow patterns in both angulated and non-angulated proximal necks. In contrast, when the proximal neck is straight, blood flow inside the aneurysm sac is laminar, with a very small area of recirculation [38]. We also evaluated the changes in hemodynamic parameters for four neck angulations of idealized AAA models. The simulation results indicate that increasing the proximal neck angulation increases recirculation in the aneurysm sac, especially at late systole and diastole. [PERSON] et al. used simple idealized angulated neck AAA models and found an inlet recirculation zone that is dominant during the diastole [24]. Furthermore, they found that the vortex is retained during this phase.  
We investigated the effect of proximal angulations for idealized AAA models on the blood flow and WSS for severe neck angles (>60\({}^{\circ}\)), which are scarce in the literature. [PERSON] et al. computationally demonstrated that peak WSS and TAWSS in the AAA sac increased for proximal neck angulation above 30\({}^{\circ}\)[23]. Furthermore, the peak WSS at an inlet angle of 40\({}^{\circ}\) was higher than that at an inlet angle of 20\({}^{\circ}\) during the peak systole [24]. This finding agrees with our results, which show that increasing the neck angulation leads to a higher WSS peak. Moreover, our results show that an angulated neck angle >60\({}^{\circ}\) induces flow turbulence and irregularity in blood flow streamlines. This can change the WSS and its distribution, potentially weakening the artery wall [16; 26]. However, previous studies report inconsistent findings on the thrombus deposition and WSS and OSI sites that induce a high risk of rupture [39].  
Our study has several limitations, including the simplifying assumptions of the AAA geometry and other parameters. Specifically, we neglected intricate blood vessels and small branches. In addition, a laminar model with a second-order implicit method for transient formulations was developed for the cardiac cycle (i.e., systole and diastole). The aortic wall was also assumed to be rigid, potentially underestimating the wall stress in CFD simulations. Moreover, the outlet boundary condition was set to zero-gauge pressure, which can be increased by applying the outflow pressure boundary condition. We also assumed blood to be a Newtonian fluid with a specific hematocrit value, which may be unrealistic for various disease conditions. Finally, the iliac bifurcation angle was neglected as a complement to the proximal neck angulation in the study of hemodynamic changes using the AAA model.