According to the current view on the aetiology of IAs, the haemodynamic factor, that is, mainly the magnitude of WSS and its gradient, plays a key role in the formation of aneurysms (10, 23).
The results of liquid flow in glass model studies and CFD studies have shown that the geometry of the bifurcation, including the diameter of the vessels forming the bifurcation and the bifurcation angle, plays an important role in the distribution of WSS and turbulence on the bifurcation components (12, 13, 21, 24–26). For example, Roach et al., in their glass model study, showed that when the bifurcation angle increased, the risk of turbulence at the bifurcation apex also increased, which posed a risk of endothelial damage (12). Using CFD simulations performed on parametric BA models, Tütüncü et al. showed that the change in the bifurcation angle from narrower to wider angles resulted in a significant widening of the area of accelerating WSS towards the daughter vessels (24). In turn, using glass models of the anterior communicating artery (ACoA) complex, Ujiie et al. showed that when the asymmetry of the A1 segments of both ACAs increased or the flow in one of the two A1 segments increased, a significant increase was reported in WSS (above 70 Pa) on the wall of the ACoA (13). Additionally, CFD studies of blood flow also showed that an increase in the asymmetry of the A1 segments of ACAs resulted in a significant increase in WSS (above 30 Pa) in the region of the A1 dominant/ACoA bifurcation (25).
The results of these studies are reflected in the results of observations of vascular anomalies in humans. It was clearly shown that the presence of asymmetry of the A1 segments of the ACoA complex was significantly associated with the formation of cerebral aneurysms at the junction of the dominant A1 segment and the ACoA (27–29). According to Stehbens, this is due to the increased blood flow in the vessel with a larger radius, which causes an increase in haemodynamic stress (30).
However, the conclusions from the morphometric analysis of patient-derived models of the arteries of the circle of Willis with IAs are not always so conclusive. Our study results showed that in those patients with MCA and BA aneurysms, the radii of the parent vessels were significantly larger compared to the control group. Moreover, the parent vessel diameter was also one of the independent factors associated with the occurrence of IAs. These results are consistent with our previous report in which we showed that a larger radius of the parent MCA vessel was associated with a significant increase in the volume flow rate (VFR) (31). We also showed that VFR was a factor independently associated with the formation of MCA aneurysms. Of note, the comparison of blood velocity values for the parent MCA vessel performed in this study did not show differences between those patients with MCA aneurysms and the controls (31). Therefore, the increase in the diameter of the parent vessel (and hence the increase in the cross-sectional area of the parent vessel) resulted in the increase in VFR that initiated aneurysm formation by increasing WSS at the bifurcation. However, the ROC analysis in the present study showed that the radius of the parent vessel was a poor predictor of IA formation (Table 5 and 6; Figure 2). Nevertheless, Can et al. demonstrated that the presence of BA and MCA aneurysms was significantly associated with a smaller radius of the parent vessel compared to the control group (32, 33). According to them, when the cross-sectional area of the parent vessel decreases, the blood flow velocity increases, resulting in a region of maximum haemodynamic stress at the apex of the bifurcation (32–34).
In the aetiology of IAs, next to the parent vessel diameter, the symmetry of primary bifurcation branches plays a key role. Many reports have shown that the greater the asymmetry of the branches forming the cerebral arterial bifurcation, the higher the risk of aneurysm formation (20, 21, 27, 31, 35–40). According to Zhang et al., the asymmetric bifurcation of the vessel increases the risk of aneurysm formation through the possible induction of abnormally enhanced haemodynamic stresses in the bifurcation (39, 40).
Some authors believe that the significance of the symmetry of the vessels forming a bifurcation for the formation of aneurysms cannot be considered without the theoretical assumptions of the PMW (14, 19, 20, 31). According to the PMW, continuous blood flow in the vascular system is achieved with the minimal expenditure of energy to maintain it, including saving losses resulting from the increase of WSS. According to the PMW, a balance between energy dissipation due to frictional resistance of laminar flow (shear stress) and the minimum volume of the blood and vessel wall tissue is achieved when the vessel radii are adjusted to the cube root of the volumetric flow (formula no 6) (15, 16). Therefore, from a theoretical perspective, the adjustment of a given vascular system to its energetic optimum is expressed as the junction exponent (n) in the above equation. If the radii of bifurcation vessels fulfil Murray’s formula with n = 3, the energy expenditure for circulation maintenance and the magnitude of WSS are the lowest, regardless of the bifurcation asymmetry (14).
Our study showed that the values of indices that determine the symmetry of the MCA and BA bifurcations were not statistically significantly different among the study groups (Table 2 and 3). Nevertheless, in the MCA bifurcations with aneurysms group, the value of the junction exponent (n) was significantly lower than in the other groups (Table 2). This means that the vascular dimensions of the MCA bifurcation with an aneurysm do not follow the PMW, which could result in higher haemodynamic stress in MCA bifurcations and the formation of an aneurysm. These results are in line with the findings of other authors who also reported deviations in the value of the junction exponent (n) from n = 3 in bifurcations with aneurysms (14, 20, 21). Of note, the other two bifurcations (i.e., ICA and BA) were characterised by the values of the junction exponent (n) significantly deviating from 3 in all study groups (Table 1 and 3). This finding is in line with the observations of Ingebrigtsen et al., who found that the values of the junction exponent (n) were significantly lower for ICA and BA bifurcations (both with and without an aneurysm).
According to Ingebrigtsen et al., the PMW establishes strict functional relations between volumetric flow, flow velocity, and the vessel dimensions and bifurcation angles of a typical vascular tree in which there is no communication between the bifurcation branches. However, blood flow through the circle of Willis is the combination of flow from three vessels (both ICAs and the BA) further communicated through the communicating arteries. Therefore, the unique anatomy of the circle of Willis, significantly different from the normal branching nature addressed by the optimality principle, results in the fact that the arterial bifurcations of the circle of Willis do not follow the PMW. According to Ingebrigtsen et al., bifurcations in the circle of Willis may be consistent with the optimality principles that have not yet been determined (41). While the MCA bifurcation and other bifurcations of cerebral arteries beyond the circle of Willis would follow the optimality principle of minimum work, the formation of aneurysms would be associated with deviations from the optimal bifurcation geometry (41).
Most imaging-based studies (e.g., 3D rotational angiography, MRA, CTA) have shown that a wide bifurcation angle constitutes a significant risk factor for IA formation. Those studies included the bifurcations which were generally considered to be predisposed for aneurysm development: ACoA complex (27, 37, 40, 42–44), the BA (24, 33, 38, 40), and the MCA (31, 32, 39, 40, 45).
We also found that BA and MCA bifurcation angles in those patients with BA and MCA aneurysms were significantly higher than the other BA and MCA bifurcation angles used for the comparison.
Furthermore, the total bifurcation angle was the best predictor for the risk assessment for cerebral aneurysm formation (univariate and multivariate analyses, Figure 2). So far, only a few studies that evaluated the effect of the bifurcation angle on the magnitude of shear stress at vessel bifurcations using CFD simulations have shown that an increase in the total bifurcation angle results in abnormally enhanced haemodynamic stresses at the arterial bifurcations (24, 26, 42).
On the other hand, we found that the value of the total angles of ICA, BA, and MCA bifurcations without an aneurysm was significantly different from the values predicted by the PMW. Nevertheless, the differences between the predicted and observed values in the bifurcation groups were greatest in the groups of MCA and BA bifurcations with an IA compared to BA and MCA bifurcations without an aneurysm. These results are in line with Ingebrigtsen et al. who analysed 107 BA, ICA, and MCA bifurcations with and without aneurysms and found significant differences among groups with respect to the mean bifurcation angles and the mean differences between the predicted optimal and observed angles (41). The above discrepancies become understandable in light of the results of Zamir and Bigelow (22, 46), who reported that even considerable deviations from the optimal angles could result in a relatively low (2–5%) increase in energy cost.
Our study has several limitations. First, given the retrospective nature of our study, we cannot conclude that wider bifurcation angles preceded the formation of aneurysms because the formation of aneurysms may have altered the bifurcation morphology. Second, the study may also have suffered from selection bias. Although participants were recruited prospectively, some patients with aneurysms that were not detected on CT because of their small size may have been inadvertently excluded. Third, since only three selected bifurcations in the circle of Willis were analysed, further studies are warranted to evaluate the morphology of a large number of bifurcations in the circle of Willis and other bifurcations of intracranial arteries beyond the circle of Willis. Fourth, further studies using CFD techniques are needed to assess the changes in shear stress values that could verify the relationship between the deviations in the dimensions of bifurcation vessels and bifurcation angle values and aneurysm formation.