## 4 Discussions  
The graft-induced alternations in the aorta mechanical environment were never quantitatively characterized for different graft designs. The goal of this work was to delineate the role of graft shape and materials on the aortic biomechanical environments following the aortic arch graft repair. The fluid  
Figure 10: Stress distribution at t = 0.4 s along the graft location (**a**) and native tissue location (**b**). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Area (mm\({}^{2}\)) at 0.4 s** & **Area (mm\({}^{2}\)) at 0.6 s** & **V\({}_{max}\) (m/s) at 0.4 s** & **V\({}_{max}\) (m/s) at 0.6 s** \\ \hline PTFE & 447.5 & 437.5 & 0.1894 & 0.0894 \\ Cormatrix & 466.2 & 448.9 & 0.1811 & 0.0916 \\ Pulmonary & 484.1 & 461.0 & 0.1701 & 0.0924 \\ Native tissue & 510.3 & 478.0 & 0.1622 & 0.0934 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cross-sectional areas and peak velocities of B-B\({}^{\prime}\) at 0.4 s and 0.6 s.  
structure interactions of the aortic arch were characterized in terms of arterial stress, flow dynamics, and indicators associated with the initiation and progression of the vascular maladaptation [36; 37; 38] (i.e., low WSS (<0.2 Pa), high OSI (>0.2), and RRT (>10 Pa\({}^{-1}\))). The FSI computational framework has been validated by the published experimental observations [30]. It is worth noting that this study is not a clinical case study. It focused on the physical understanding of the graft design in terms of geometry and materials. It is well recognized that the geometry of the aortic arch affects the hemodynamics and thus cardiac functions [39; 40]. [PERSON] et al. [41] showed that in mice, both the aortic arch angle and cross-sectional area of the aortic arch correlated positively with the area of the plaque. Enhanced clinical outcomes were observed for an autologous vascular graft resembling the curvature of the aortic arch [42]. The crescent graft herein was proposed to better mimic the native arch shape (i.e., a smaller aortic arch angle), tapered cross-sectional area, and smooth curvature compared to the half-moon graft (Figure 2). The stress concentrations were observed at the interface between the graft and the native tissue, regardless of graft shapes. This was due to the material mismatch. The peak von Mises stress of the crescent graft was 76% smaller than the half-moon one (Figure 5). This could be explained by the local geometrical discontinuity at the interface between the graft and native tissue [43], also referred to as the geometry mismatch. The abnormal stresses have been associated with the reduced compliance of the tissue and disruption of collagen fiber structure [44; 45; 46]. The geometrical discontinuity in the half-moon graft resulted in the obvious reversed flow adjacent to the bottom border of the graft, especially at the diastolic phase. This was due to the constriction effect at the proximal border of the graft. The cross-sectional area at the proximal border of the graft discontinued the tapering of the ascending aorta, and induced a reversed tapering by implementing a half-moon graft, which is analogous to the stenosis model. The reversed flow was well-observed and explained by the adverse pressure gradient using the analogy of the stenosis case [47]. In comparison, the tapering from the ascending aorta (A-A\({}^{\prime}\)) to the aortic arch (B-B\({}^{\prime}\)) was well preserved in the crescent graft, and thus led to the minimal flow disturbance. The graft geometry was also highly related to wall movement in terms of the cross-sectional area and the mass flow rate redistributions [48]. It is interesting to observe that the difference between the PFA and the cross-sectional area for the half-moon graft was significantly larger than for that of the crescent one (Figure 7). This indicated that the flow disturbance and recirculation were more severe in the half-moon graft. The flow streamlines had the same implication as the pressure contours (details not shown for brevity).