Morphometry of Cerebral Arterial Bifurcations Harbouring Aneurysms: A Case-Control Study On 253 Patients

Conclusions from studies evaluating vessel dimensions and their deviations from the values resulting from the principle of minimum work (PMW) on the formation of intracranial aneurysms (IAs) are still inconclusive. The aim of our study was the morphometric analysis of cerebral arterial bifurcations harbouring aneurysms. The study comprised 147 patients with basilar artery (BA) and middle cerebral artery (MCA) aneurysms, and 106 patients constituting the control group. The following morphometric parameters were evaluated: the radii of vessels forming the bifurcation, the junction exponent, the values of the bifurcation angles (Φ 1 and Φ 2 angles between the parent vessel trunk axis and the larger or smaller branches, respectively; α angle, total bifurcation angle) and the difference between the predicted optimal and observed branch angles.


Results
The analysed parameters for internal carotid artery (ICA) bifurcations were not signi cantly different among the groups. MCA and BA bifurcation angles and the radii of the parent MCA and BA vessels with aneurysms were signi cantly higher compared to the control group. The differences between the predicted optimal and observed branch angles were signi cantly higher for BA and MCA bifurcations with aneurysms compared to the control group. The mean junction exponent for bifurcations in the circle of Willis (i.e., ICA and BA bifurcations, respectively) and MCA bifurcations with aneurysms was signi cantly lower than the theoretical optimum and not signi cantly different among the groups. In a multilevel multivariate logistic regression analysis, the branch angles and the radius from the parent vessel were signi cant independent predictors of the presence of IAs. The ROC analysis indicated that the α angle was the best performer in discriminating between aneurysmal and non-aneurysmal bifurcations.

Conclusions
The dimensions of the arteries forming the circle of Willis do not follow the PMW. The deviation from optimum bifurcation geometry for bifurcations beyond the circle of Willis (particularly a wider radius of the parent artery and a wider total bifurcation angle) may lead to the formation of IAs. Further studies are warranted to investigate the signi cance of vessel dimensions and the bifurcation angle on the magnitude of the shear stress in the walls of arterial bifurcations.

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Background Intracranial aneurysm (IA) is a disease with a complex multifactorial aetiology. The results of epidemiological and genetic studies indicate a strong involvement of genetic and environmental factors (1,2). Experimental glass model studies, in vivo studies, and computational uid dynamics (CFD) studies have shown that the increase in haemodynamic stress caused by the impact of blood ow on bifurcations is the main factor initiating the formation of IAs. It is the main factor responsible for the destructive remodelling of the arterial wall, characterised by disruption of the internal elastic lamina, the loss of medial smooth muscle cells, the reduced proliferation of smooth muscle cells, and the loss of bronectin (3,4).
Recently published data have shown that the above-mentioned vascular segments are affected by complex haemodynamic forces, such as wall shear stress (WSS), the wall shear stress gradient (WSSG), and temporal uctuations in WSS (3)(4)(5)(6)(7)(8)(9)(10). IAs usually arise at bifurcations, where the vessels are exposed to the maximum impact of WSS (5,6). The WSS experienced at a bifurcation is dependent on its geometry, including the radii of all vessels involved and the bifurcation angle (11)(12)(13). WSS is minimised when the relation between vessel diameters and bifurcation angles follows the optimality principle of minimum work (PMW) (14).
According to the PMW, the biological system expends energy to maintain circulation and metabolism, and its e ciency depends on maintaining normal continuity of blood ow with minimal energy expenditure, including saving losses resulting from the increase of WSS. Murray used the PMW to predict vessel diameters and bifurcation angles in a theoretical vascular tree that is ideal for minimising the energy essential for ensuring blood ow continuity (15,16). Murray's theoretical assumptions about the structure of the vascular network were con rmed by angiographic studies that demonstrated that the dimensions of various human blood vessels (17,18), including intracranial arteries (19), were consistent with the theoretical optimal values imposed by the PMW.
Based on the current reports, deviations in the geometry of intracranial arterial bifurcations predisposed to cerebral aneurysms, which are different from the PMW, may be a key determinant for the formation of aneurysms (14,20,21). Nevertheless, the conclusions from these reports are still inconclusive. Therefore, we planned a case-control study with the selection of patients with IAs and non-aneurysmal controls to determine selected morphometric parameters and to analyse their relationship to the PMW-derived optimal values.

Patient population
The study included 147 patients consecutively admitted to the Department of Neurosurgery, Regional Hospital in Sosnowiec, Medical University of Silesia, Poland between June 2013 and June 2020. One hundred and fteen patients presented with unruptured middle cerebral artery (MCA) aneurysms (22 men, artery (BA) aneurysms (11 men, 22 women), aged 33 to 77 years (60 ± 11; mean ± SD), which was con rmed by three-dimensional computed tomography angiography (3D CTA). The control group consisted of 106 patients who were sex-and age-matched to both study groups, including 38 men and 98 women aged 28 to 79 years (56 ± 12; mean ± SD) in whom no pathology was found on 3D CTA.
The exclusion criteria were as follows: the presence of a central nervous system disorder other than an IA that could affect cerebral arterial blood ow (e.g., ischaemic stroke, intracerebral haemorrhage, subarachnoid haemorrhage), other severe systemic conditions (e.g., advanced malignancy, severe circulatory or multiorgan failure), the presence of haemodynamically signi cant morphological changes in the extracranial segment of the internal carotid artery (ICA), a family history of IAs and pregnancy.
Patients and controls were referred for 3D CTA after conventional computed tomography (CT) to rule out the presence of an IA or as a medical check-up of the brain for minor symptoms such as headache or vertigo.

CTA protocol
CTA was performed using a 64-channel multidetector CT scanner (GE Optima CT660, GE Healthcare, USA) with IV bolus administration of a non-ionic contrast medium (Omnipaque 350, GE Healthcare, USA). The scanning parameters included collimation of 39.38 x 0.625 mm, a spiral pitch of 0.984, a tube voltage of 120 kV, a tube amperage of 450 mA, a 0.4 second rotation time, and a slice thickness of 0.625 mm. A total of 50 mL of contrast medium followed by 30 mL of saline solution was injected into an antecubital vein at a rate of 4-4.5 mL/sec using a power injection platform (LF OptiVantage DH, USA).
CT scanning was triggered by using a Smart Prep protocol, with the region of interest placed in the common carotid artery (CCA). Image acquisition started ve seconds after the attenuation reached 100 HU. The scanning time was approximately 4.5-6.0 seconds.

Morphometric analysis of intracranial arterial bifurcations
The CTA data were transferred as DICOM les to a workstation equipped with the Mimics Innovation Suite platform (Materialise, Belgium). Mimics v.16.0 and 3-matic v.8.0 were used for image segmentation and the creation of a 3D vessel model. In all three groups of patients (with MCA aneurysms, BA aneurysms, and the controls), the following were segmented: both ICAs with their bifurcations into the MCA and the anterior cerebral artery (ACA), the parent vessel of the MCA from the artery's origin to its bifurcation and the post-bifurcation branches, and the parent vessel of the BA with the bifurcation into posterior cerebral arteries (PCAs) (Figure 1). Cases where the parent vessel of the MCA was divided into a trifurcation and cases that were not suitable for further morphometric calculations due to the poor quality of the 3D model were excluded from further morphometric analysis. In the groups of MCA and BA bifurcations with aneurysms, the aneurysm was digitally removed before the morphometric analysis using the Mimics software, leaving the parent artery and the post-bifurcation branches for further measurement. When the 3D models were nished, the vessel centreline was tted automatically to each 3D model ( Figure 1) using a computer-aided design (CAD) tool. Using the centreline, the largest curvature of the parent vessels of the MCA, ICA and BA and the largest curvatures of each of the two post-bifurcation branches were automatically calculated. The points of the largest curvature were set as close to the bifurcation as possible, but at a distance of at least 5 mm. Based on these points, the best t diameter (d 0 ) of the parent vessels of the MCA, ICA, and BA and the best t diameter of both branches (d 1 , d 2 for the larger and smaller branches, respectively) were estimated automatically.
Next, the best t diameters were used to calculate the radii of the parent vessels of the MCA, ICA, and BA (r 0 ) and the radii of both branches (r 1 and r 2 for the larger and smaller branches, respectively) using the following formula: where 'r' is the radius and 'd' is the best t diameter. The radii were used to calculate two ratios, that is, the asymmetry ratio, which was calculated using the following formula: asymmetry ratio=r 2 2 r 1 −2 (2) and the area ratio calculated according to the formula: area ratio= (r 1 2 +r 2 2 )r 0 −2 (3) Next, the centrelines with the largest curvature points were exported to 3-matic v.8.0 to measure the angles between the bifurcation components. The points of the largest curvatures of the parent vessel and the larger and smaller branches and the point of the intersection of both centrelines were used to determine the apex of the angles.
The following angle values were calculated: the α angle between the branches of MCA, ICA, and BA bifurcations, the β angle between the parent vessel of the vessel and the larger branch, and the γ angle between the parent vessel and the smaller branch ( Figure 1). Next, the angles between each of the two branches and the parent vessels of the MCA, ICA and BA axis were calculated: All morphometric measurements were performed by the same author (K. Ć-S).

Calculation of predicted optimal morphometric parameters
According to the PMW, the adjustment of a given vascular system to its theoretical optimum is expressed as the junction exponent (n): r 0 n = r 1 n + r 2 n (6) where the optimal value of the junction exponent for an energetically ideal vessel bifurcation equals 3.
The junction exponents for all MCA, ICA, and BA bifurcations were obtained with an online calculator available at http://www.wolframalpha.com.
The optimal angles between the axis of the parent vessels of the MCA, ICA and BA and the larger and smaller post-bifurcation branches (ϕ 1 , ϕ 2 , respectively) as well as the total bifurcation angle (ϕ 1 + ϕ 2 ) were predicted using four PMW-derived optimality rules according to the following equations (22): cosϕ 2 =(r 0 4 +r 2 4 -r 1 4 )(2r 0 2 r 2 2 ) −1 (8) cos(ϕ 1 + ϕ 2 )= (r 0 4 -r 1 4 -r 2 4 )(2r 1 2 r 2 2 ) −1 (9) Standard protocol approvals, registrations and patient consent The study's protocol was approved by the Institutional Review Board at the Medical University of Silesia in Katowice, Poland and all procedures were carried out in accordance with the relevant guidelines and regulations. Each patient was informed about the purpose and course of the research and asked to give their informed consent to participate in the project.

Statistical analysis
Normal distribution of the study variables was veri ed with the Shapiro-Wilk test. The results were presented as means and standard deviations (SD). The means with normal distribution were compared using the one-way ANOVA, including post-hoc analysis with the Tukey test for unequal sample sizes, and the means with skewed distributions were compared using the Kruskal-Wallis one-way ANOVA with multiple comparisons. All morphometric and haemodynamic parameters that showed signi cant intergroup differences were subjected to logistic regression analysis with a stepwise addition mode. The potential predictors of IA formation were identi ed using univariate analysis. Based on the univariate analysis, the variables with p-values < 0.1 (except those correlated with one another) were included in the multivariate logistic regression model to identify the independent predictors of aneurysm formation. The results were presented as odds ratios (ORs) and 95% con dence intervals (CIs). The independent predictors of aneurysm formation were subjected to receiver operating characteristic (ROC) analysis to identify their cut-off values with optimal sensitivity and speci city. The Youden index was used to determine the optimal cut-off point. The results were considered statistically signi cant for p-values < 0.05. Statistical analyses were performed with Statistica v.13.3 (Tibco Software Inc.).

Results
Assessment of the statistical signi cance of the differences among the mean values of the morphometric variables for the ICA in the study groups Table 1 shows a comparison of the morphometric variables, that is, the α, Φ 1 , and Φ 2 angles, the differences between the predicted and observed values for the above angles, the radii (r 0 , r 1 , r 2 ), the asymmetry ratio, the area ratio, and the junction exponent for the ICA among the patient groups (i.e., those with diagnosed MCA aneurysms, BA aneurysms and the controls). No statistically signi cant differences were found in the mean values of the above variables among the study groups. NS Continuous values are expressed as means ± SD. An, aneurysm; R ICA, the right internal carotid artery; L ICA, the left internal carotid artery; α angle, total bifurcation angle; Φ1 angle, the angle between the axis of the parent vessel and the larger branch; Φ2 angle, the angle between the axis of the parent vessel and the smaller branch; r0, the radius for the parent vessel; r1, the radius for the largest branch; r2, the radius for the smallest branch; ϕ 1 + ϕ 2 , total predicted optimal angle; ϕ 1 , predicted optimal angle between the parent vessel and the largest branch; ϕ 2 , predicted optimal angle between the parent vessel and the smallest branch.
Assessment of the statistical signi cance of the differences among the mean values of the morphometric variables for the MCA in the study groups The results of the comparison of the morphometric variables for the MCA among the groups of patients with MCA aneurysms, BA aneurysms, and the controls are given in Table 2. The post-hoc analysis showed that the α, Φ 1 , and Φ 2 angles in the MCA bifs w/An group were signi cantly higher compared to the corresponding angles in the MCA bifs w/o An group and the control group. However, the α and Φ 2 angles in the same bifurcation group were also signi cantly higher compared to the α and Φ 2 angles of the R MCA and the L MCA in those patients with BA aneurysms.  The comparison of the differences between the predicted and observed values for the α, Φ 1 , and Φ 2 angles for the MCA among the study groups showed that they were signi cantly higher for the α angle in the MCA w/An group compared to all the other MCA bifurcations (i.e., MCA w/o An, MCA bifurcation in the groups of patients with BA aneurysms and the controls). However, for the Φ 1 and Φ 2 angles, they were signi cantly higher compared to the MCA w/o An group and the control group.
Assessment of the statistical signi cance of the differences among the mean values of the morphometric variables for the BA in the study groups The results of the comparison of the morphometric variables for the BA among those patients with MCA aneurysms, BA aneurysms and the controls are given in Table 3. The post-hoc analysis showed that the α, Φ1, and Φ2 angles, and the r 0 and r 1 radii in the BA w/An group were signi cantly higher compared to the variables in the other groups of BA bifurcations. Comparison between the observed and predicted values of the α, Φ 1 , and Φ 2 angles Table 4 shows the comparison of the observed values of the α, Φ 1 , and Φ 2 angles with their predicted values. In all study groups (i.e., the groups of patients with MCA aneurysms, BA aneurysms, and the controls), the observed values of the above angles were signi cantly higher for all arterial bifurcations (i.e., ICA, MCA, and BA) when compared to their predicted values (Table 4). Univariate and multivariate analyses The univariate logistic regression analysis identi ed the r 0 , the junction exponent, the area ratio, the asymmetry ratio, the Φ 1 , Φ 2 , and α angles, and the Φ 1 , Φ 2 , and α angles classi ed in tertiles as signi cant prognostic factors for the formation of aneurysms (Table 5). For legends, see Table 1.
Disregarding the correlated parameter and considering the parameters relevant to the univariate model, two multivariate logistic regression models were constructed. The rst model (A) included only the tertiles of the Φ 1 and Φ 2 angles, and the r 0 and r 1 radii, while the second model (B) included the tertiles of the α angle, and the r 1 and r 0 radii. The nal results of both analyses are given in Table 6.

Predictors of aneurysm formation -the ROC analysis
The ROC curves for all the independent predictors for aneurysm formation are given in Figure 2

Discussion
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 ow 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)(25)(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 signi cant 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 ow in one of the two A1 segments increased, a signi cant increase was reported in WSS (above 70 Pa) on the wall of the ACoA (13). Additionally, CFD studies of blood ow also showed that an increase in the asymmetry of the A1 segments of ACAs resulted in a signi cant increase in WSS (above 30 Pa) in the region of the A1 dominant/ACoA bifurcation (25).
The results of these studies are re ected 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 signi cantly associated with the formation of cerebral aneurysms at the junction of the dominant A1 segment and the ACoA (27)(28)(29). According to Stehbens, this is due to the increased blood ow 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 signi cantly 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 signi cant increase in the volume ow 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 signi cantly associated with a smaller radius of the parent vessel compared to the control group (32,33). According to them, when the crosssectional area of the parent vessel decreases, the blood ow velocity increases, resulting in a region of maximum haemodynamic stress at the apex of the bifurcation (32)(33)(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)(36)(37)(38)(39)(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 signi cance 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 ow 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 ow (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 ow (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 ful l 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 signi cantly 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 signi cantly 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 ndings 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) signi cantly deviating from 3 in all study groups (Table 1 and  Most imaging-based studies (e.g., 3D rotational angiography, MRA, CTA) have shown that a wide bifurcation angle constitutes a signi cant 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)(43)(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 signi cantly 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 signi cantly 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 signi cant 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.

Limitations
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 The study's protocol was approved by the Institutional Review Board at the Medical University of Silesia in Katowice, Poland and all procedures were carried out in accordance with the relevant guidelines and regulations. Each patient was informed about the purpose and course of the research and asked to give their informed consent to participate in the project.

Consent for publication
Not applicable.

Availability of data and materials
Fully anonymized data not published within this article will be made available by request from any quali ed investigator following the EU General Data Protection Regulation.

Competing interests
The authors declare that they have no competing interests.

Funding
This research received no speci c grant from any funding agency in the public, commercial or not-forpro t sectors.
Authors' contributions WK conceptualised and supervised the study. KĆS, WK acquired the data. WK, KĆS, EK analysed data. AH conducted the statistical analysis. WK and KĆS wrote the manuscript. WK, WW and PŁ edited the manuscript. VA, vertebral artery.