# Numerical Investigation of the Relationship between
## 3 Results
### Optimization of Anastomosis Angle
#### 3.1.3 Wall Shear Stress  
Wall shear stress (WSS) is the measure of the force exerted by a fluid motion on a solid boundary and vice versa toward the direction of the local tangent plane. The amount of shear stress generated on blood vessel walls is critical for endothelial homeostasis, so WSS is viewed as an essential predictor or indicator of cardiovascular diseases [40]. Despite the consensus that WSS is important for endothelial homeostasis, two opposing claims exist about the optimal WSS for bypass grafts. One hypothesis claims that high WSS must be sought because low WSS values cause stenosis due to the intimal thickening of vessel walls and occlusion of the blood vessel lumen due to plaque formation [3; 41]. Meanwhile, the other hypothesis suggests that higher WSS may have detrimental effects on bypass graft performance as it is linked to endothelial damage that can result in thrombosis and lesions [4; 5; 6], and possibly, intimal thickening as well [7]. To avoid these complications, we optimized the anastomosis angle of the bypass grafts by selecting the angle that would yield WSS values that are closer to normal, physiological WSS levels (0.1-7 Pa) [8; 9]. Figure 8 shows the spatial average of WSS according to the anastomosis angle. The values reported were computed over the whole wall of the host artery. The results indicated a positive linear relationship between the anastomosis angle and the spatial average values of WSS (r\({}^{2}\) = 0.9923). This relationship may be explained by both the bulk motion and the secondary flow components of the fluid. As observed in the axial velocity contour maps (Figure 6), higher axial velocities near the walls and higher velocity gradients were observed for larger anastomosis angles which translate to increased shear stress [42]. Furthermore, the swirl motion represented by the secondary velocity also generated higher shear stress in the artery walls due to the tangential velocity vector [39]. This seemed to suggest that increasing the anastomosis angle between the graft and artery may have the beneficial effect of reducing the tendency of the graft to induce intimal thickening by increasing the WSS. However, this presented a problem wherein an increased anastomosis angle resulted in the development of pathologically high WSS. The colored areas in Figure 9 show the regions of the unrolled artery affected by pathologically high and low WSS. These areas were only located near the anastomosis region. Hence, only pertinent regions were shown instead of the entire length of the host artery. Significant effects were observed when looking at the relationship between the anastomosis angle and areas with abnormally high WSS (>7 Pa). The areas affected by abnormally high WSS were situated in the region of the artery that directly receives blood flow from the graft outlet. Based on the axial velocity profile, upon flow separation, high-velocity fluid streams were accelerated by centrifugal force toward this region, justifying why they were exclusively found in this area. In Figure 10, large decreases in areas of abnormally high WSS were observed whenever the anastomosis angle was decreased, even  
Figure 7: Secondary velocity contours and cross-flow streamlines 50 mm from the anastomosis toe. being completely eliminated at a 30\({}^{\circ}\) angle, suggesting the effectiveness of using smaller anastomosis angles. The anastomosis angle seems to have no significant effects when looking at areas affected by pathologically low WSS. The smallest and largest areas affected by pathologically low WSS had a percentage difference of only 5.3%, so no significant improvement would be observed whichever angle is used. Moreover, the areas affected by pathologically low shear stresses were located in the bypassed region where little to no blood is expected to flow. Improvement within this area is of little use in optimizing the performance of bypass grafts. Figure 8: Spatial average of WSS over the whole host artery walls. Figure 9: Areas on the host artery (unrolled) affected by abnormal WSS.