The circle of Willis (CoW) is a critical, arterial structure that ensures balanced, cerebral-blood supply. The fetal-type posterior cerebral artery (f-PCA) is a CoW variant that can significantly affect hemodynamics and elevate the risk of cerebrovascular diseases. This study used computational fluid dynamics simulations and a patient-specific, three-dimensional model to evaluate the hemodynamic effects of the f-PCA variants on cerebral-blood flow and key hemodynamic indices—such as time-averaged wall-shear stress (TAWSS), oscillatory shear index (OSI), pulsatility index, and resistive index. The fetal ratio (FR) is defined as the ratio of the diameter of the posterior communicating artery (PCoA) to that of the first segment (P1) of the PCA. Our findings indicate that as the FR increases, the contribution of the basilar artery to the second segment (P2) of PCA decreases significantly. Specifically, the flow rate through ipsilateral P1 decreased by 40.0% for FR = 1 and 70.9% for FR = 2, with the internal carotid artery (ICA) compensating for this reduction. Moreover, variations in f-PCA led to significant increases in TAWSS and OSI in key arterial segments (including the ipsilateral P1, PCoA, and the anterior communicating artery), which are associated with a higher risk of aneurysm initiation and growth. Under conditions of unilateral stenosis in the ipsilateral ICA, f-PCA models exhibit a more complex and pronounced impact on blood flow than models without f-PCA, emphasizing the need for detailed hemodynamic assessments in clinical evaluations and preoperative planning to mitigate the risks associated with CoW anatomical variations.

ACA

Anterior cerebral artery

ACoA

Anterior communicating artery

AsA

Ascending aorta

BA

Basilar artery

BFR

Blood flow rate

c

Contralateral

C

Compliance

CCA

Common carotid artery

CFD

Computational fluid dynamic

CoW

Circle of Willis

CS

Carotid stenosis

DsA

Descending aorta

ECA

External carotid artery

f-PCA

fetal-type posterior cerebral artery

FR

Fetal ratio

HDIs

Hemodynamic indices

i

Ipsilateral

ICA

Internal carotid artery

MCA

Middle cerebral artery

OSI

Oscillatory shear index

Pdia

Diastolic pressure

Pm

Mean blood pressure

Psys

Systolic pressure

P1

The first segment of the posterior cerebral artery

P2

The second segment of the posterior cerebral artery

PCA

Posterior cerebral artery

PCoA

Posterior communicating artery

PI

Pulsatility index

PMA

Premamillary artery

PSV

Peak systolic velocity

Rd

Distal resistance

Rp

Proximal resistance

Rt

Total resistance

RI

Resistive index

SA

Subclavian artery

TAWSS

Time-averaged wall shear stress

tCeA

Total cerebral blood flow

VA

Vertebral artery

WSS

Wall shear stress

The human brain relies heavily on its vascular system for optimal functioning. The circle of Willis (CoW) is a vital, arterial structure located at the base of the brain that ensures a balanced blood supply between the two cerebral hemispheres and between the anterior and posterior parts of the brain.1 The CoW is composed of the following arteries:2 anterior cerebral artery (ACA), internal carotid artery (ICA) at its distal tip, middle cerebral arteries (MCA), anterior communicating artery (ACoA), posterior communicating artery (PCoA), posterior cerebral artery (PCA), and basilar artery (BA). The interconnected nature of the CoW provides a crucial advantage. If an artery supplying a specific brain region is blocked, blood can be rerouted through other arteries within the circle, minimizing the risk of damage and stroke. Therefore, an incomplete CoW or any anatomical variation could increase the risk of cerebrovascular diseases. For example, Ryan et al.3 studied the magnetic resonance imaging (MRI) of 90 patients over 50 years of age, in which 73 patients had white-matter disease, and found that incomplete CoW increased the white-matter-disease burden by 58%. Zhou et al.4 showed that patients with an incomplete CoW had a higher prevalence of carotid-intraplaque hemorrhage (52.7% vs 38.5%) and hypertension (72.2% vs 55.7%) than those with a complete CoW. van Seeters et al.5 revealed a higher prevalence of ischemic stroke in patients with an incomplete anterior CoW (11/223, 4.9%) than in those with a complete CoW (19/753, 2.5%). The absence of the ACoA is often linked to cerebral aneurysms because of the altered hemodynamic stress imposed on the arterial walls.6 

Variations in the CoW anatomy are common. Based on previous anatomic studies, anatomic variations in the CoW have been observed in 22% to over 50% of the population. Ryan et al.3 presented 36/163 (22%) cases of incomplete CoW. A clinical study7 reported incomplete CoW in 101/258 (39%) patients in rural Ecuador, while the percentage was 34% for the cases studied in Egypt.8 A total of 26/50 (52%) patients had anatomical variations in the CoW.2 Ravikanth and Phillip9 showed that only 16.5% of patients presented with complete CoW; they also concluded that variants of CoW were slightly more common among females than males.

Typical anatomical variations in the CoW include the absence of the ACoA, unilateral or bilateral PCoA, fetal-type arteries, and fused vessels. Among these, fetal-type PCA (f-PCA) is a fascinating anatomical variant that influences blood-flow patterns and potentially affects neurological outcomes. The PCA is a vital component of the brain vascular system and plays a crucial role in supplying blood to various brain regions, including the occipital and inferior temporal lobes. To rephrase, the PCA plays a crucial role in functions like vision, memory, and movement. In the context of f-PCA, the P2 segment of the PCA is predominantly supplied by the ICA rather than the BA, as is normal.9,10 The prevalence of f-PCA varies considerably across populations. Ravikanth and Philip9 reported an occurrence rate of 23% (46/200), Enyedi et al.11 observed an occurrence rate of 16% (20/126), while higher percentages of 46% were reported in Ref. 12 and 40% in Ref. 13. The f-PCA is characterized by the hypoplasia or absence of the first segment (P1) of the PCA or, at least, the diameter of the P1 segment was clearly smaller than that of the PCoA at the junction, as shown in Figs. 1(b) and 1(c). Previous studies have shown that the occurrence of partial f-PCA (or hypoplasia) is more frequent than full f-PCA.9,14

FIG. 1.

Variations in the circle of Willis: (a) ordinary type; (b) and (c) fetal-type posterior cerebral artery (f-PCA). Abbreviations: BA (basilar artery), ICA (internal carotid artery), MCA (middle cerebral artery), ACA (anterior cerebral artery), PCA (posterior cerebral artery), P1 (first segment of PCA), P2 (second segment of PCA), ACoA (anterior communicating artery), and PCoA (posterior communicating artery).

FIG. 1.

Variations in the circle of Willis: (a) ordinary type; (b) and (c) fetal-type posterior cerebral artery (f-PCA). Abbreviations: BA (basilar artery), ICA (internal carotid artery), MCA (middle cerebral artery), ACA (anterior cerebral artery), PCA (posterior cerebral artery), P1 (first segment of PCA), P2 (second segment of PCA), ACoA (anterior communicating artery), and PCoA (posterior communicating artery).

Close modal

Although several studies have investigated variants of fetal-type posterior cerebral artery (f-PCA), most have focused on clinical implications, anatomical characteristics, and surgical challenges using techniques such as transcranial Doppler (TCA), MRI, and computed tomography angiography (CTA). These studies typically examined the anatomical variability of f-PCA within small populations without providing comprehensive hemodynamic analyses.10,15–21 In other words, while there is evidence that suggesting f-PCA may increase the risk of stroke,20,21 infarction,10 aneurysm rupture,21,22 and aneurysm recurrence,19 these studies primarily focus on prevalence, lacking detailed hemodynamic assessments (e.g., blood flow, pressure, wall shear stress, etc.). Chen et al.22 found that f-PCA happened in out of 25/62 (40.3%) ruptured aneurysms. When variations coexist with surgical clamping or ICA stenosis, the risk is even higher. Therefore, understanding the detailed blood distribution and flow patterns in CoWs associated with f-PCA is crucial for understanding its collateral mechanisms. This knowledge is instrumental not only in the early diagnosis but also in the preoperative planning of cerebrovascular diseases.

Our study used simulation tools to analyze the hemodynamic characteristics of various f-PCA variants within the aorta-to-cerebral vasculature model. This approach not only enables a thorough quantitative analysis but also allows for detailed visualization of the flow dynamics. To date, there has been a significant gap in the literature regarding the use of simulations to investigate the f-PCA hemodynamic. Moreover, even existing studies that examined hemodynamic through simulations tend to focus primarily on the flow dynamics from the vertebral arteries (VA) or BA,23,24 thus providing an incomplete picture of the aorta-to-cerebral vasculature model. The aorta-to-cerebral vasculature model enables more physiologically accurate boundary conditions, reflecting the pulsatile nature of blood flow originating from the heart.

This is the first study to investigate cerebral hemodynamic in the case of f-PCA using aorta-to-cerebral vasculature models. Our research aimed to deliver a comprehensive analysis of the impact of f-PCA on cerebral blood flow and hemodynamic indices (HDIs) (such as time-averaged wall shear stress (TAWSS), oscillatory shear index (OSI), pulsatility index (PI), and resistive index (RI)), particularly focusing on areas directly affected, such as the PCA, PCoA, ACoA, premamillary artery (PMA), BA, and ICA. Additionally, we explored the influence of the growing stenosis in the unilateral ICA on the CoW in the presence of f-PCA.

A three-dimensional (3D), patient-specific model of the cardiovascular system spanning from the aorta to the cerebral vasculature was used in this study. The model employed is consistent with those used in previous studies,25–29 which were created based on CT scans taken as part of the screening-technology and outcome project in the stroke-database study.30 We only adopted the patient model referred to as P01 in Refs. 27–29 and labeled patient 3 in Refs. 25 and 26. Kang et al.29 reported similar hemodynamic-flow tendencies in the three models. As we aimed to investigate how f-PCA affects changes in hemodynamic flow around the PCA region, we chose a single representative.

The premamillary artery (PMA) is the largest primary branch of the PCoA and supplies blood to the underside of the brain. It is essential to perfuse the paramedian perforated substance, which is a brain region crucial for memory and emotional functions. Disruptions in the PMA blood flow can lead to neurological impairments such as memory deficits, emotional dysregulation, or issues with thermoregulation, often resulting from occlusion or constriction of the PMA. Typically, there is only one PMA on each side of the brain; however, some variations exist. Despite its small size, the PMA plays an important role in maintaining cerebral hemodynamics and overall brain health. Therefore, this was included in the study geometry; however, its size was determined based on dimensions from prior research owing to its limited visibility in the initial image data. The size of the PMA has been reported differently in previous studies, varying from 0.28 to 0.8 mm in diameter31–33 and from 10 to 14.6 mm in length.31,32 We selected the average—a diameter of 0.6 mm and a length of 12 mm for both sides—as these dimensions were noted as average in reference.31 The PMA was originated from the middle third of PCoA, slightly closer to the PCA side.31,32

The simulated models are shown in Fig. 2(a).

FIG. 2.

(a) Schematic of simulated model (reference—no f-PCA). (b) Mesh around the f-PCA region and near wall. (c) Boundary conditions: inlet with the pulsatile flow rate at AsA; other open-ended arteries set as Windkessel outlets with RCR information provided. Model includes AsA (ascending aorta), DsA (descending aorta), SA (subclavian artery), VA (vertebral artery), BA (basilar artery), CCA (common carotid artery), ECA (external carotid artery), ICA (internal carotid artery), MCA (middle cerebral artery), ACA (anterior cerebral artery), PCA (posterior cerebral artery), P1 (first segment of PCA), P2 (second segment of PCA), PMA (premamillary artery), ACoA (anterior communicating artery), and PCoA (posterior communicating artery); i denotes ipsilateral and c denotes contralateral side with f-PCA.

FIG. 2.

(a) Schematic of simulated model (reference—no f-PCA). (b) Mesh around the f-PCA region and near wall. (c) Boundary conditions: inlet with the pulsatile flow rate at AsA; other open-ended arteries set as Windkessel outlets with RCR information provided. Model includes AsA (ascending aorta), DsA (descending aorta), SA (subclavian artery), VA (vertebral artery), BA (basilar artery), CCA (common carotid artery), ECA (external carotid artery), ICA (internal carotid artery), MCA (middle cerebral artery), ACA (anterior cerebral artery), PCA (posterior cerebral artery), P1 (first segment of PCA), P2 (second segment of PCA), PMA (premamillary artery), ACoA (anterior communicating artery), and PCoA (posterior communicating artery); i denotes ipsilateral and c denotes contralateral side with f-PCA.

Close modal

The computational artery domain was constructed using SimVascular software. The modeling pipeline comprised the following steps: (1) Medical imaging data from the CT scans were collected from the source. (2) Vessel centerlines were then created. (3) A set of two-dimensional (2D) segmentations that involve delineating the boundaries of the arteries from the surrounding tissues were created along these paths. (4) Each group of segmentations was then lofted to form a vessel and automatically joined together to form a complete, 3D-solid model. Here, a technique was required to smooth the surface and junction between the connected vessels.

A computational mesh was generated from the 3D-solid model to finalize the modeling and was used for the simulation. Since we used the same 3D patient-specific model with the previous study,29 the final mesh was chosen to be exactly the same as that used in Ref. 29, which is a tetrahedral type and contains 7.8–8.5 × 106 elements, similar to the number mentioned in Ref. 26. The results of the mesh-sensitivity test and validation will be presented in Sec. II F. Since our interest is f-PCA, the mesh size of related arteries, i.e., PCoA, PCA (both P1 and P2 segments), and PMA, was recreated with a finer mesh, i.e., the average sizes of PCoA, and PCA were 0.22 mm, and that of PMA was 0.12 mm. The mesh near the wall was also created with a finer mesh with four prism layers, as shown in Fig. 2(b).

Van Raamt et al.20 classified f-PCA into three types: (i) partial f-PCA, if P1 is smaller than PCoA; (ii) intermediate f-PCA, if P1 and PCoA are equal or as large as each other; (iii) and full f-PCA, if P1 is absent. This classification was used to modify the model. Four models were reconstructed and compared, as shown in Fig. 3. To easily refer to these cases, the fetal ratio (FR) is defined as the ratio between the diameters of the P1 segments of the PCA and PCoA as follows:
(1)
where dPCoA is the diameter of the PCoA, and dP1 is the diameter of P1. FR < 1 indicates a non-f-PCA. Four cases were created as follows. Case 1, used as a reference (FR = 0.28), is the original model showing the complete CoW with non-f-PCA, in which the diameter ratio, PCoA:P1 = 1:3.5. Case 2 (FR = 1) refers to the transitional f-PCA, where the diameter ratio, PCoA:P1 = 1:1. Case 3 (FR = 2) refers to the partial f-PCA, where the diameter ratio, PCoA:P1 = 2:1. Case 4 (full f-PCA) refers to the fully fetal type, where P1 is absent. P1 diameter ranged from 0.95 to 3.55 mm, and PCoA ranged from 1 to 3.55 mm. In this study, we only considered unilateral f-PCA, which only occurred on the right side. The same side as the f-PCA is ipsilateral (i), and the other side is contralateral (c).
FIG. 3.

Case studies: (a) FR = 0.28 (reference), representing the original 3D patient-specific model from previous studies; (b) FR = 1, f-PCA with a PCoA/P1 diameter ratio of 1; (c) FR = 2, f-PCA with a PCoA/P1 diameter ratio of 2; (d) full f-PCA, absence of P1; (e) stenosis in the ICA with three degrees of stenosis following NASCET. Note that FR < 1 indicates CoW with no f-PCA. Both f-PCA and carotid stenosis are unilateral in the right side. Abbreviations: FR (fetal ratio, diameter ratio between PCoA and P1).

FIG. 3.

Case studies: (a) FR = 0.28 (reference), representing the original 3D patient-specific model from previous studies; (b) FR = 1, f-PCA with a PCoA/P1 diameter ratio of 1; (c) FR = 2, f-PCA with a PCoA/P1 diameter ratio of 2; (d) full f-PCA, absence of P1; (e) stenosis in the ICA with three degrees of stenosis following NASCET. Note that FR < 1 indicates CoW with no f-PCA. Both f-PCA and carotid stenosis are unilateral in the right side. Abbreviations: FR (fetal ratio, diameter ratio between PCoA and P1).

Close modal
Three stenosis conditions were applied to each of four cases. NASCET criteria were used to measure the degree of carotid stenosis. The following formula is used:
(2)
According to the NASCET criteria, 0% indicates the absence of stenosis, 50%–69% indicates moderate stenosis, and greater than 70% indicates severe stenosis. Stenosis was applied to the right ICA near the carotid bifurcation. Previous studies have shown that 50% stenosis in the ICA has a minor effect on the CoW in its normal form. This study aimed to investigate the combined effect of carotid stenosis (CS) and variations in f-PCA on blood flow in the CoW. Therefore, we apply two NASCET criteria: moderate and severe stenosis. The three ICA stenosis models are shown in Fig. 3(d).

The study assumed blood to be an incompressible Newtonian viscous fluid with a density of 0.001 06 g mm−3 and a constant viscosity of 0.004 g s−1 mm−1. Although blood exhibits non-Newtonian fluid, previous studies suggest that the Newtonian model can still adequately capture flow disturbances in large arteries, which minimal discrepancies in wall shear stress (WSS) distributions compared to non-Newtonian models.34–39 In addition, previous studies have indicated that the use of non-Newtonian becomes critically important for small arteries (radius < 0.1 mm),40 while for larger arteries (>6 mm) or middle-sized arteries (2   6 mm), a Newtonian model can still be a reasonable approximation.35 Many previous studies have also been simulated with the assumption of blood as Newtonian.1,19,24–28,41,42 The vessel wall was assumed to be a rigid body; this has been shown to effectively simulate realistic arterial behavior and provide affordable computational cost.26,28 Furthermore, the use of a large domain in the aorta-to-cerebral vasculature model makes it difficult to apply a fluid–structure analysis simulation.

The blood flow is solved by the Navier–Stokes equations for incompressible, Newtonian fluids. The set of governing equations consists of the Navier–Stokes equation, continuity equation, and constitutive law of Newtonian fluids as follows:
(3)
(4)
(5)
where u is the velocity vector field, p is the pressure field, ρf is the density of fluid, t is time, μf is the dynamic viscosity of blood, g is gravitational acceleration, and τ is shear stress. The governing equations were discretized using the finite element method. The pressure and velocity field are calculated using streamline-upwind/Petrov–Galerkin (SUPG) and pressure-stabilizing/Petrov–Galerkin (PSPG) methods. Time integration was performed using an implicit scheme, specifically the backward Euler method. The more details of the numerical method have been comprehensively described in previous studies.27–29 To achieve numerical stability, a numerical simulation was conducted with six cardiac cycles and a time step of 1 ms and the results of the sixth cardiac cycle were used for the analysis.
The boundary conditions are shown in Fig. 2. The inlet was set at the AsA with a pulsatile flow rate similar to that used by Olufsen et al.43 The distal ends of the other open arteries were set as outlets, as shown in Fig. 2(c). The three-element Windkessel model was used for the outlet conditions and consisted of three parameters: distal resistance (Rd), compliance (C), and proximal resistance (Rp). An electrical representation of the three parameters is shown in Fig. 2(c), and the differential equation for the 3-element Windkessel model is as follows:30,44
(6)
where Q(t) is the volumetric flow and P(t) is the pressure with respect to time. C is the capacitance, and Rt=Rd+Rp is the total resistance, which are derived as follows:
(7)
(8)
(9)
(10)
where Pm is the mean blood pressure and Q(t)¯ is the mean flow rate. Pm was calculated using the formula: Pm=(Psys+2Pdia)/3, where Psys is the systolic pressure and Pdia is the diastolic pressure. These pressures were typically assumed to be 120 and 80 mm Hg, respectively, as referenced in Ref. 25, resulting in Pm = 93.3 mm Hg. The total Q(t)¯ and total C were initially calculated from the pulsatile inlet waveform and then were evenly distributed to each outlet following the cross-sectional area rule.25,27,29,45–47 Subsequently, Rd and Rp at each outlet are calculated using Eqs. (4)–(6). Rd, Rp, and C were then iteratively tuned using the conventional procedure presented in previous studies.28,46 The RCR outlet parameters used in this study followed those in a prior study29 and are presented in Fig. 2(c). However, the model proposed by Kang et al.29 did not include PMA. Therefore, the Rt and C of PMA were initially assumed to be split from those of PCA, as follows:47 
(11)
and
(12)
where Ai, Ri, and Ci represent the total resistance at each outlet. Then, the boundary of the PMA was tuned using the procedure mentioned above.

Blood-flow rate (BFR) is a crucial metric for assessing the efficiency and adequacy of blood circulation in the cardiovascular system. The BFR is defined as follows:48 volume BFR = cross-sectional area × time-averaged velocity. The positions of the segments where the BFR was calculated, along with their average diameters, are shown in Fig. 4(a) and detailed in Table I.

FIG. 4.

Analysis in the circle of Willis: (a) cross-section where the blood flow rate is obtained. (b) Segments where the cell-average value is obtained.

FIG. 4.

Analysis in the circle of Willis: (a) cross-section where the blood flow rate is obtained. (b) Segments where the cell-average value is obtained.

Close modal
TABLE I.

The average diameter and length of arteries in Fig. 4.

Arteries cP1 cP2 iP1 iP2 cPCoA iPCoA ACoA cPMA iPMA cACA iACA cMCA iMCA BA iICA cICA
Diameter (mm)  3.6  2.6  3.9  2.1  1.0  1.0  1.0  0.6  0.6  2.3  2.3  2.9  3.0  4.0  3.7  4.4 
Length (mm)  7.9  11.4  8.6  13.1  16.6  14.6  3.0  12.0  12.0  22.0  27.1  15.7  18.2  40.3  145.4  131.1 
Arteries cP1 cP2 iP1 iP2 cPCoA iPCoA ACoA cPMA iPMA cACA iACA cMCA iMCA BA iICA cICA
Diameter (mm)  3.6  2.6  3.9  2.1  1.0  1.0  1.0  0.6  0.6  2.3  2.3  2.9  3.0  4.0  3.7  4.4 
Length (mm)  7.9  11.4  8.6  13.1  16.6  14.6  3.0  12.0  12.0  22.0  27.1  15.7  18.2  40.3  145.4  131.1 
HDIs play a crucial role in understanding the physiological dynamics of blood flow in the cardiovascular system. These indices encompass various parameters that provide insights into the forces exerted on vessel walls and the overall flow patterns within blood vessels. In this study, we analyzed two velocity-related HDIs, PI and RI, and two WSS-related HDIs, TAWSS and OSI. The PI is a measure of the pulsatility of blood flow. The RI indicates the resistance to blood flow within a vessel and ranges from 0 to 1. TAWSS is a measure of the average shear stress exerted by blood flow on the vessel wall over a cardiac cycle. The OSI quantifies directional changes in the WSS over a cardiac cycle, indicating the extent of oscillatory flow, and ranges from 0 to 0.5. These indices have been widely studied and recognized for their significance in evaluating vascular hemodynamics.49–58 Note that the values of these hemodynamic indices will be reported as cell-averaged values for each arterial segment. The segments and their corresponding lengths are illustrated in Fig. 4(b) and detailed in Table I. Their formulations are as follows:
(13)
(14)
(15)
and
(16)
where τ is the instantaneous WSS and T is the duration of the cardiac cycle.

The blood flow rates obtained from our simulations were validated against values reported in the literature, as shown in Table II. As we can see, the results of the present study fall within the range of values presented in previous studies. Additionally, Fig. 5(a) shows a strong agreement in the flow rate of the ACoA over one cardiac cycle between our study and the results reported by Kang et al.,29 further affirming the reliability of our results.

TABLE II.

Comparison of the blood flow rate (ml/min) between the present study and previous studies.

Studies iICA BA iPCA cPCA
Kang et al.29   202.73  ⋯  66.6 ± 7.8  73.2 ± 12 
Oktar et al.59   215.59 ± 82.82  ⋯  ⋯  ⋯ 
Hendrikse et al.60   234 ± 62  129 ± 33  ⋯  ⋯ 
Hendrikse et al. with f-PCA60   272 ± 66  91 ± 25  ⋯  ⋯ 
Amin-Hanjani et al.61   256 ± 52  138 ± 41  66 ± 15  69 ± 14 
Amin-Hanjani et al. with f-PCA61   278 ± 55  84 ± 27  ⋯  ⋯ 
Present study with no f-PCA  202.92  126.57  58.33  79.09 
Present study with f-PCA  231 ± 9.6  103.1 ± 9.59  ⋯  ⋯ 
Studies iICA BA iPCA cPCA
Kang et al.29   202.73  ⋯  66.6 ± 7.8  73.2 ± 12 
Oktar et al.59   215.59 ± 82.82  ⋯  ⋯  ⋯ 
Hendrikse et al.60   234 ± 62  129 ± 33  ⋯  ⋯ 
Hendrikse et al. with f-PCA60   272 ± 66  91 ± 25  ⋯  ⋯ 
Amin-Hanjani et al.61   256 ± 52  138 ± 41  66 ± 15  69 ± 14 
Amin-Hanjani et al. with f-PCA61   278 ± 55  84 ± 27  ⋯  ⋯ 
Present study with no f-PCA  202.92  126.57  58.33  79.09 
Present study with f-PCA  231 ± 9.6  103.1 ± 9.59  ⋯  ⋯ 
FIG 5.

(a) Mesh independent test: flow rate at iP1. (b) Comparison of flow rate at ACoA between the present study and Kang et al.29 

FIG 5.

(a) Mesh independent test: flow rate at iP1. (b) Comparison of flow rate at ACoA between the present study and Kang et al.29 

Close modal

In this study, we used the same patient-specific model as in the previous publications.27–29 The mesh size was carefully verified in these previous studies under both stenotic and non-stenotic conditions,25–29 and we applied this mesh size consistently in our study. However, as mentioned in Sec. II C, for the f-PCA cases, we maintained the same overall mesh size but used a finer mesh in the fetal regions (i.e., iPCoA and iPCA) to enhance accuracy. The mesh verification for the PCoA and PCA regions is presented in Table III and Fig. 5(a). The differences between mesh 2 and mesh 3 were minimal (<1%), so mesh 2 was selected as the final mesh. This mesh size was then applied consistently across all cases with different FRs.

TABLE III.

Mesh independence test for the f-PCA region, conducted with FR = 1. The sensitivity of the mesh was verified by analyzing systolic pressure in the iP1 and iPCoA regions.

Cases Mesh size at the fetal region (mm) No. of elements (106) iP1 iPCoA
Psys (mmHg) Δ (%) Psys (mmHg) Δ (%)
Mesh 1  0.32  7.97  115.83    114.41  ⋯ 
Mesh 2  0.22  8.02  114.48  −1.16  113.13  −1.12 
Mesh 3  0.12  8.37  115.24  0.66  114.05  0.82 
Cases Mesh size at the fetal region (mm) No. of elements (106) iP1 iPCoA
Psys (mmHg) Δ (%) Psys (mmHg) Δ (%)
Mesh 1  0.32  7.97  115.83    114.41  ⋯ 
Mesh 2  0.22  8.02  114.48  −1.16  113.13  −1.12 
Mesh 3  0.12  8.37  115.24  0.66  114.05  0.82 

In our observations and quantitative analysis, f-PCA only strongly affected the cerebral area. Therefore, despite the model covering the arteries from the heart to the brain, the results presented in this paper primarily focused on the arteries within the CoW.

1. Posterior circulation and anterior circulation of the brain

Figure 6 compares the BFR in the posterior and anterior circulations. Posterior circulation supplies blood to the back of the brain, while anterior circulation supplies blood to the front of the brain. Generally, changing FRs significantly affect the blood supply through the posterior circulation compared to the anterior circulation. However, this significant effect was primarily observed in the ipsilateral P1 (iP1), BA, and ipsilateral ICA (iICA). On the other hand, the f-PCA have a negligible effect on the MCA and ACA. Clinical data also indicated that blood flow in the MCA and ACA rarely differs significantly.13 

FIG. 6.

Effect of varying FRs on cerebral blood flow: average flow rate over a cardiac cycle for (a) posterior cerebral circulation and (b) anterior cerebral circulation. FR < 1 indicates CoW with no f-PCA.

FIG. 6.

Effect of varying FRs on cerebral blood flow: average flow rate over a cardiac cycle for (a) posterior cerebral circulation and (b) anterior cerebral circulation. FR < 1 indicates CoW with no f-PCA.

Close modal

To examine cerebrovascular dynamics, it is essential to understand the flow patterns and contributions of various arteries under different physiological and modeled conditions. Normally, the blood supply to the PCA (the P2 segment) is routed from the BA (through P1 segment). However, in patients with f-PCA variants, the predominant blood supply is shifted to the ICA (through PCoA). The f-PCA can be classified into partial and full fetal types,9,14 with each model exhibiting a different flow distribution.

Under a normal CoW (no f-PCA, i.e., FR = 0.28), the supply flow to P2 is predominantly from the BA. As FR increased, the contribution from BA decreased. Specifically, in our patient-specific model, 86% of the flow from iP1 goes to iP2 in normal CoW without f-PCA. For f-PCA models, the flow rate through iP1 decreased by 40.0% for f-PCA with FR = 1 and by 70.9% for FR = 2. This reduction is due to the smaller size of iP1 as increasing FR. However, interestingly, the flow rate at ipsilateral P2 (iP2) varies negligibly, which differs from the noticeable decrease at iP1. This is because under f-PCA, the P2 segment receives blood flow from both the BA and ICA. Therefore, when the contribution from the BA decreases (represented by the reduction in flow rate at iP1 and BA), the ICA compensates for the missing blood supply through ipsilateral PCoA (iPCoA), as evidenced by the increase in flow rate through the iICA. To facilitate understanding, Fig. 7 shows the velocity streamlines emitted from the ICA and BA under different FRs. Under the non-f-PCA condition (FR = 0.28), the supply flow to iP2 was predominantly sourced from the BA, with no flow routed through the PCoA from the ICA [Fig. 6(a)]. However, with f-PCA, an increase in the size of the iPCoA enhances the supply flow to iP2 from the iICA. As FR increases, a greater portion of the flow to iP2 is sourced from the iICA [Figs. 7(b) and 7(c)]. Furthermore, in cases of full f-PCA (i.e., with the absence of iP1), the blood flow was completely (100%) redirected from the BA to the contralateral P1 (cP1), meaning that the flow to iP2 was solely supplied by the iICA [Fig. 7(d)].

FIG. 7.

Flow distribution in CoW for (a) normal anatomical CoW and (b)–(d) variations of f-PCA. The upper row displays the streamline originating from the iICA, while the lower row shows the streamline originating from the BA. FR < 1 indicates the CoW with no f-PCA.

FIG. 7.

Flow distribution in CoW for (a) normal anatomical CoW and (b)–(d) variations of f-PCA. The upper row displays the streamline originating from the iICA, while the lower row shows the streamline originating from the BA. FR < 1 indicates the CoW with no f-PCA.

Close modal

Figure 8 shows the relationship between iP1, iP2, and iPCoA, demonstrating the distribution of blood flow to P2. FR had a significant impact on both the peak systolic velocity (PSV) and area-averaged velocity at iP1. Higher FRs tended to increase the PSV at iP1. As the size of iP1 decreased in the f-PCA model (reference > FR = 1 > FR = 2), the velocity through iP1 increased, with PSV values of 23.5, 58.4, and 74.4 cm/s, respectively [Fig. 8(a)]. The average velocity through iPCoA over one cycle is much lower than that through iP1 in the reference but gradually increases as much as iP1 in the f-PCA model with FR = 2. The PSV through iPCoA is 8.2, 18.6, and 53.4 cm/s for the no-f-PCA, f-PCA with FR = 1, and f-PCA with FR = 2 cases, respectively. However, the velocity of flow through iP2 remained similar to that in the non-f-PCA across different FRs.

FIG. 8.

Effect of varying FRs on cerebral blood flow: relationship between P1, P2, and PCoA. (a) Contour of the peak systolic velocity; (b) area-average velocity during one cardiac cycle; and (c) flow rate during one cardiac cycle. FR < 1 indicates the CoW with no f-PCA. x-axis in a and b represents time in seconds.

FIG. 8.

Effect of varying FRs on cerebral blood flow: relationship between P1, P2, and PCoA. (a) Contour of the peak systolic velocity; (b) area-average velocity during one cardiac cycle; and (c) flow rate during one cardiac cycle. FR < 1 indicates the CoW with no f-PCA. x-axis in a and b represents time in seconds.

Close modal

Similarly, the flow rate, which depends on both the velocity and cross-sectional area, varies significantly in iPCoA and iP1, whereas it differs only slightly in iP2. Figure 8(b) shows the contribution of blood flow to iP2. The average flow rate, which was closer to the average flow rate of iP2, shifted between iP1 and iPCoA as FR increased. In addition, the flow rate depends on the area. In our patient-specific model, the sizes of the two P2 segments differed, with cP2 being larger than iP2. Thus, although the averaged velocities at P2 on both sides were similar (i.e., 25.7 ± 0.86 and 25.2 ± 0.18 cm/s for iP2 and cP2, respectively), the flow rate at iP2 was lower than that at cP2.

2. Flow of the communicating arteries and premamillary arteries

The anterior and posterior cerebral circulations are interconnected through the PCoA, whereas the ACoA connects both anterior cerebral arteries. The PMA is the largest branch of PCoA. This section compares the flow rates through these arterial pathways for different FRs. Herein, the flow toward the anterior circulation through the PCoA and the flow toward the right anterior circulation through the ACoA are denoted as positive.

The flow through the contralateral PCoA (cPCoA) reached its maximum value of 0.09 ml/s in reference (FR = 0.28). As FR increased, the flow rate in the cPCoA decreased but remained relatively small compared with the changes in the iPCoA. Unlike the cPCoA, the flow through the iPCoA and ACoA shows a stronger response to increasing FR [Fig. 9(a)]. It was found that the blood flow through the iPCoA in the CoW with f-PCA is significantly higher compared to the non-f-PCA (FR = 0.28). Specifically, the flow is 721.4%, 1910.7%, and 2713.3% higher than the non-f-PCA for FR = 1, FR = 2, and full f-PCA, respectively. Meanwhile, the increase in ACoA is 20.7%, 53.9%, and 57.3% higher than the reference case for FR = 1, FR = 2, and full f-PCA, respectively. In this scenario, the increase in flow through the iPCoA and ACoA, coupled with the decrease in cPCoA flow, indicates that on the contralateral side (left side), the blood flow from the posterior to the anterior circulation is insufficient. The ACoA compensates for the flow between the two anterior circulations to balance the blood flow in the CoW. This highlights the critical importance of the iPCoA and ACoA, particularly in the absence of an iP1 segment.

FIG. 9.

(a) Effect of varying FRs on the cerebral blood flow: average flow rate over a cardiac cycle for communication arteries (i.e., PCoA and ACoA) and PMA and (b) streamline of flow into PMA. FR < 1 indicates the CoW with no f-PCA.

FIG. 9.

(a) Effect of varying FRs on the cerebral blood flow: average flow rate over a cardiac cycle for communication arteries (i.e., PCoA and ACoA) and PMA and (b) streamline of flow into PMA. FR < 1 indicates the CoW with no f-PCA.

Close modal

Because the PMA is the largest branch of the PCoA, the stability of blood flow within the PMA is also a key focus of our investigation. As shown in Fig. 9(c), in the reference case, the blood supply to the PMA originated from both the iICA and BA. This dual-source supply ensured consistent blood flow to the PMA. Notably, even with an increasing FR, the blood flow to the PMA remained remarkably stable, averaging approximately 0.022 ml/s [Fig. 9(b)]. In scenarios involving f-PCA, the blood flow to the ipsilateral region (iPMA) is predominantly shifted to being sourced by the iICA. This change in the flow distribution is also highlighted by the increased flow rate through the iPCoA. However, this shift did not significantly affect the overall blood flow to the iPMA.

It is also worth noting that the source of the blood supply to the contralateral PMA (cPMA) contrasts with that of the iPMA. On the contralateral side (left side), as FR increased, the flow supplying the cPMA gradually shifted from the contralateral ICA (cICA) to cP1, or in other words, from the BA. This is because as the size of iP1 decreases with increasing FR, there is a monitored decrease in the cPCoA flow, indicating insufficient flow from the posterior to the anterior. Consequently, the blood from the ICA was rerouted through the ACoA to compensate for the imbalanced blood flow between the two hemispheres. Hence, this affects the source of blood supply to the cPMA.

3. Change in hemodynamic indices (HDIs)

In this section, we investigate the effects of the f-PCA on certain HDIs. The data presented represent cell-averaged values for each segment. The analysis in Sec. III A 1 demonstrates that under different FRs, the flow rate and velocity show noticeable changes only in arteries directly related to the iP1 and iPCoA, that is, iP1, iP2, cP1, cP2, BA, iICA, iPCoA, AcoA, and cPCoA. Similarly, the HDIs exhibited significant variations in these arteries. In contrast, other arteries remained largely unchanged or exhibited negligible variations. Therefore, this section focuses primarily on these arteries.

a. PI and RI.

It is also worth noting that in this study, the tendencies of PI and RI were consistent with each other with increasing FR. Both indices were used to understand the resistance to flow. However, the RI uses the ratio of the difference between the systolic and diastolic velocities to the systolic velocity, whereas the PI uses the ratio of the difference between the systolic and diastolic velocities to the mean velocity. Thus, RI directly provides information on the resistance to flow, while PI represents the pulsatility of the flow. The RI values range from 0 to 1. An RI greater than 0.7 can indicate increased resistance.54,55,62 These two HDIs play a crucial role in evaluating the distal cerebrovascular resistance, noninvasive intracranial pressure, and cerebral perfusion pressure in traumatic brain injury, and hydrocephalus has been investigated in previous studies.56,57 Typically, larger arteries such as the aorta exhibit lower resistance and higher pulsatility.63,64 In contrast, cerebral arteries generally have lower PI and RI values. Prior studies have demonstrated a strong association between both PI and RI and conditions such as stenosis and stroke.28,56,63–66

As shown in Fig. 10, there was an increase in both indices for the arteries on the left side of the CoW (contralateral side) (cP1 and cP2) with increasing FR. Conversely, the right side of the CoW (ipsilateral side) (iP1 and iP2) displayed a decrease in these indices. Note that although not shown here, the other arteries in the anterior circulation (i.e., the ACA and MCA) exhibited minimal changes, with a maximum variation of less than 10%. As a result, the ACoA, which connects the two anterior circulations, undergoes a decrease in RI and PI, whereas the two PCoAs show an increase in RI and PI with increasing FR. It is also worth noting that under the f-PCA, iP1 experienced a noticeably higher RI than iPCoA, as shown in Figs. 10(b) and 10(d), which is consistent with the velocity shown in Fig. 8(a).

FIG. 10.

Effect of f-PCA on PI and RI: (a) average PI and (b) RI within each artery; (c) contour of PI in CoW; and (d) contour of RI in CoW. The values in a and b are the cell-average value of each artery segment. FR < 1 indicates the CoW with no f-PCA.

FIG. 10.

Effect of f-PCA on PI and RI: (a) average PI and (b) RI within each artery; (c) contour of PI in CoW; and (d) contour of RI in CoW. The values in a and b are the cell-average value of each artery segment. FR < 1 indicates the CoW with no f-PCA.

Close modal
b. TAWSS and OSI

TAWSS is a critical parameter in hemodynamics, representing the average shear stress exerted by blood flow on vessel walls over time. Several factors influence WSS, including blood velocity, blood viscosity, vascular geometry, etc. For instance, WSS is typically higher at bifurcation points where blood flow directly impacts the vessel walls. In this study, blood viscosity is assumed to be constant. Thus, the velocity distribution is the primary determinant of high TAWSS locations. As shown in Fig. 11, the average TAWSS showed a complex variation, in which a dramatic increase was observed at iP1, iPCoA, and ACoA. As the FR increases, the diameter of the iP1 decreases correspondingly. Figure 12(a) provides cross-sectional views of the velocity distribution at the BA–iPCA–iPCoA bifurcation. For the case with FR = 0.28, where the diameters of iP1 and BA are nearly identical (3.9 and 4.0 mm, respectively), the velocity transitions smoothly from the BA to the iP1, maintaining an equal average velocity. However, as the diameter of iP1 decreases, the velocity increases to satisfy the conservation of mass, a phenomenon often referred to as a sudden contraction problem. Moreover, due to the arterial geometry, the velocity is not uniformly distributed across the cross section, as seen in Fig. 12(a) for FR = 1, where regions of higher velocity correspond to areas of elevated TAWSS. As the FR increases, or as the size difference between the BA and iP1 becomes more pronounced, the velocity across the iP1 increases significantly, resulting in higher TAWSS. For the non-f-PCA case (FR = 0.28), the TAWSS values within iP1 ranged from 1 to 5 Pa. With a reduction in the size of the P1 segment, a substantial increase in TAWSS was observed, reaching approximately 7.3 and 10.6 Pa for FR = 1 and FR = 2, respectively. It is important to note that although the PCoA diameter remains constant across all f-PCA cases, the increased velocity associated with higher FR (as discussed in Sec. III A 1) leads to a corresponding increase in TAWSS. Similarly, while the overall velocity distribution within the ACoA does not change significantly with varying FR [Fig. 12(b)], the absolute velocity across the ACoA increases with FR, resulting in higher TAWSS values.

FIG. 11.

Effect of f-PCA on TAWSS and OSI: (a) average TAWSS and (b) OSI within each artery; (c) contour of TAWSS in CoW and (d) contour of OSI in CoW. The values in (a) and (b) are the cell-average value of each arterial segment. FR < 1 indicates the CoW with no f-PCA.

FIG. 11.

Effect of f-PCA on TAWSS and OSI: (a) average TAWSS and (b) OSI within each artery; (c) contour of TAWSS in CoW and (d) contour of OSI in CoW. The values in (a) and (b) are the cell-average value of each arterial segment. FR < 1 indicates the CoW with no f-PCA.

Close modal
FIG. 12.

Cross-sectional views of the velocity distribution at (a) the BA—PCA—PCoA bifurcation and (b) the ACoA.

FIG. 12.

Cross-sectional views of the velocity distribution at (a) the BA—PCA—PCoA bifurcation and (b) the ACoA.

Close modal

In addition, Fig. 10(c) shows the contours of TAWSS for different FRs. As can be seen, the contour again indicates that the effect of f-PCA on the TAWSS only occurs in the area associated with f-PCA. These arteries have a high potential for aneurysm initiation.67–70 Further discussion is provided in Sec. IV B.

The average OSI values are presented in Fig. 11(b). OSI measures the directional changes in WSS throughout a cardiac cycle, reflecting the degree of oscillatory flow. The OSI ranges from 0 to 0.5, with values below 0.1 generally considered low, while values above 0.3 are typically regarded as high and clinically concerning.71–73 Elevated OSI (near 0.5) indicates significant oscillatory or reversing flow, commonly observed in areas of disturbed flow, such as bifurcations, curved arteries, or regions downstream of stenoses. High OSI is linked to endothelial dysfunction, which promotes atherogenesis and increases the risk of plaque development and vascular disease.74 As shown in Figs. 11(b) and 11(d), f-PCA has a negligible impact on the overall OSI distribution. Note that, in Fig. 11(b), although the iPCoA shows a noticeable decrease in OSI, the values remain below 0.1, which is considered low and unlikely to adversely affect the vascular system.

1. Cerebral blood flow

As we can see, with different f-PCA, only a few arteries show variation. Therefore, we again focused only on the influence of the growing stenosis in the right ICA on these arteries. Herein, by quantitative analysis, we realized that the same tendencies and increase/decrease rates were observed in most arteries, except for the arteries related to the f-PCA and communicating arteries. Figure 13 illustrates the progressive changes in blood flow for cP1, iP1, iP2, cPCoA, iPCoA, and ACoA across the three models (FR = 0.28, FR = 1, and FR = 2). The models with f-PCA exhibited a more complex effect on blood flow than the non-f-PCA model did. Under conditions with no stenosis in the right ICA, as explained in Sec. III A, the arteries on the ipsilateral side (iP1, iP2, etc.) and the cPCoA exhibited a decrease in flow rate with increasing FR, whereas the contralateral side (cP1, cP2, etc.) and the two communicating arteries, iPCoA and ACoA, showed an increase with increasing FR. These results were consistent under moderate CS (50%).

FIG. 13.

Effect of stenosis in the right ICA on the blood flow rate for different FRs at cP1, iP1, iP2, cPCoA, iPCoA, and ACoA. (a) Average flow rate over a cardiac cycle. (b) Flow rate ratio is calculated using the flow rate under 0% stenosis as reference. FR < 1 indicates CoW with no f-PCA. x-axis represents the stenosis degree in %.

FIG. 13.

Effect of stenosis in the right ICA on the blood flow rate for different FRs at cP1, iP1, iP2, cPCoA, iPCoA, and ACoA. (a) Average flow rate over a cardiac cycle. (b) Flow rate ratio is calculated using the flow rate under 0% stenosis as reference. FR < 1 indicates CoW with no f-PCA. x-axis represents the stenosis degree in %.

Close modal

However, it is important to note that the f-PCA models experienced a stronger effect from ipsilateral CS under moderate CS conditions than the non-f-PCA model, particularly at iPCoA and iP1. Under severe CS conditions (75%), the effects became more complex because of the combined impact of the ipsilateral f-PCA and CS (both on the right side).

a. iP1 and cP1

These two PCA segments receive blood flow from the BA. Under 75% CS, with FR = 1 (equal sized iPCoA and iP1), the flow rate through iP1 increases and that through cP1 decreases. This occurs because, with the iPCoA size equal to that of iP1, this pathway becomes crucial for providing blood from the posterior circulation to the anterior circulation (when the flow from the iICA drops due to CS). Therefore, to maintain a balanced blood flow in the CoW, more blood is routed to iP1 so that it can supply blood to the iPCoA. This explanation is supported by the streamlines shown in Fig. 14.

FIG. 14.

Flow distribution in the CoW under 75% stenosis in the right ICA for f-PCA with FR = 1 and FR = 2. (a) Streamlines from iICA, (b) streamlines from BA, and (c) streamlines from cICA.

FIG. 14.

Flow distribution in the CoW under 75% stenosis in the right ICA for f-PCA with FR = 1 and FR = 2. (a) Streamlines from iICA, (b) streamlines from BA, and (c) streamlines from cICA.

Close modal
b. iP2

Furthermore, because the blood flow in iP2 is sourced by both the iICA and BA under variations in f-PCA, a higher FR and a higher degree of CS result in a reduced flow to iP2.

c. Communicating arteries

Previous studies have shown the importance of communicating arteries during CS progression in normal CoWs.24,26,28,75 Our non-f-PCA model corroborates this tendency, showing a dramatic increase in the flow rates in the iPCoA and ACoA under severe CS conditions (75%). Because the CS is located on the right side, the communicating artery on the left contralateral side (cPCoA) exhibits minimal influence during CS progression. The changes in blood flow within the communicating arteries of the CoW became more complex in the presence of f-PCA.

First, the ACoA shows a similar tendency and rate of increase in both the f-PCA models and the non-f-PCA model. This is because, as previously observed, variations in the f-PCA have a minimal effect on the arteries in the anterior circulation connected by the ACoA. Consequently, the ACoA exhibited a similar increase rate in the models with and without f-PCA.

Second, as explained in the previous paragraph, cPCoA exhibited minimal changes during CS progression in the non-f-PCA model; this conclusion also applies to the model with f-PCA. It is noteworthy that f-PCA with FR = 1 showed a noticeable increase at 75% CS; however, this increase was much smaller than that of iPCoA and ACoA.

Third, the flow rate at the iPCoA increased under severe CS conditions in both the non-f-PCA case and f-PCA with FR = 1. This underscores the importance of communicating arteries in severe CS conditions. However, we observed a reduction in the flow rate for the f-PCA with FR = 2. This reduction was due to the decreased size of iP1, resulting in a reduced flow rate at iP1, which normally supplies the iPCoA. In this situation, the iPCoA must receive blood flow from the iICA to provide sufficient blood for iP2. However, under severe CS conditions (75%) in the iICA, the blood flow through the iICA significantly decreases, causing a reduction in the flow rate at the iPCoA.

2. Overall change in PI, RI, TAWSS, and OSI during CS progression

Previous studies have comprehensively investigated the effect of CS on the same (3D-specific) patient27–29 without f-PCA. They showed that, overall, the HDIs showed a negligible change during moderate CS (i.e., 50%).28 In this section, we discuss the effects of different f-PCA on the HDIs. Our aim was to understand the effects of CS on the CoW with f-PCA.

Figure 15 shows the progressive changes in the average PI, RI, and TAWSS on the right (ipsilateral) and left (contralateral) sides of the CoW. Under CS progression with the combined effect of ipsilateral f-PCA, PI and RI showed a noticeable reduction on the right side of the CoW but small variations on the left side of the CoW. It is also worth noting that the f-PCA experienced a stronger decrease than the non-f-PCA (FR = 0.28). This conclusion is also valid for the variation of TAWSS; however, TAWSS expresses a dramatic increase, not a decrease, like PI and RI. These variations indicated that the HDIs of the f-PCA model were more strongly influenced by CS than those of the model without f-PCA.

FIG. 15.

Changes in PI, RI, and TAWSS during stenosis progression in the right ICA for different FRs. (a) Average values on the right side of CoW (i.e., iACA, iMCA, iP1, and iP2). (b) Average values on the left side of CoW (i.e., cACA, cMCA, cP1, and cP2). Values are averages for each arterial segment. FR < 1 indicates CoW with no f-PCA. x-axis represents the stenosis degree in %.

FIG. 15.

Changes in PI, RI, and TAWSS during stenosis progression in the right ICA for different FRs. (a) Average values on the right side of CoW (i.e., iACA, iMCA, iP1, and iP2). (b) Average values on the left side of CoW (i.e., cACA, cMCA, cP1, and cP2). Values are averages for each arterial segment. FR < 1 indicates CoW with no f-PCA. x-axis represents the stenosis degree in %.

Close modal

Figure 16 shows the contours of the TAWSS and OSI, along with a comparison of the maximum TAWSS values for each f-PCA at different degrees of CS. With higher degrees of CS and FR, the maximum TAWSS of the ACoA increased considerably, reaching values of 47.1 and 164.9 at 75% CS for the f-PCA with FR = 2 and full f-PCA, respectively. The TAWSS of iP1 also showed a dramatic increase with increasing FR at 75% CS. Notably, at the bifurcation between the iPCoA and iICA, a significant reduction in TAWSS was observed at 75% CS for the f-PCA model with FR = 2, whereas the OSI remained high. Although the overall distribution of the OSI was not strongly affected, high OSI values frequently occurred at the bifurcations and curvatures of the arteries. Consequently, these locations exhibited a significant increase in the OSI during CS progression. A comprehensive discussion of how these values affect the cerebrovascular disease is provided in Sec. IV B.

FIG. 16.

Effect on TAWSS and OSI during stenosis progression in the right ICA for different FRs. (a) Contour of TAWSS, (b) Contour of OSI, and (c) Maximum TAWSS. FR < 1 indicates CoW with no f-PCA.

FIG. 16.

Effect on TAWSS and OSI during stenosis progression in the right ICA for different FRs. (a) Contour of TAWSS, (b) Contour of OSI, and (c) Maximum TAWSS. FR < 1 indicates CoW with no f-PCA.

Close modal

Overall, the cases with f-PCA show a more pronounced effect of the ipsilateral CS than the cases with non-f-PCA. This indicates that f-PCA exacerbates the hemodynamic impact of CS, highlighting the need for careful consideration of anatomical variations.

The hemodynamic behavior of cerebral arteries, particularly in the context of anatomical variations, such as f-PCA, plays a pivotal role in understanding the progression and risks of cerebrovascular diseases. f-PCA significantly alters the blood flow distribution within the CoW, affecting key hemodynamic indices such as TAWSS and OSI. These changes are crucial for identifying areas of high hemodynamic stress that are often associated with an increased risk of aneurysm formation and rupture.

The impact of CS further complicates the hemodynamics of the CoW, especially in the presence of f-PCA. As stenosis progresses in the right ICA, the roles of the PCoA and ACoA become more critical, leading to significant increases in the TAWSS and localized changes in the OSI. Understanding these hemodynamic responses is essential to evaluate the risk of cerebrovascular events and plan appropriate clinical interventions. This study provides insights into the detailed hemodynamic effects observed under varying f-PCA and CS conditions.

When carotid ligation occurs, the P1 segments of the PCA serve as primary collateral pathways.76 Furthermore, in a normal CoW configuration, the blood flow through the PCoA is very small, as confirmed by previous studies.1,29,77 In the case of a CoW with f-PCA, the P1 segments are typically smaller and the PCoA is typically larger, which can disrupt the blood flow balance in the CoW. Section III B 1 discussed the changes in these arteries under different variations of f-PCA during CS progression. In summary, Fig. 17 illustrates the total cerebral blood flow at different FRs and CS progression. The total blood flow rate of the cerebral blood supply was defined as follows:
(17)
FIG. 17.

Effect of stenosis progression in the right ICA on the total cerebral blood flow corresponding to different FRs. FR < 1 indicates the CoW with no f-PCA.

FIG. 17.

Effect of stenosis progression in the right ICA on the total cerebral blood flow corresponding to different FRs. FR < 1 indicates the CoW with no f-PCA.

Close modal

Notably, tCeA through the CoW remained relatively stable in the no-CS condition, suggesting that f-PCA does not significantly affect the overall blood supply to the brain. Specifically, compared to non-f-PCA, tCeA decreased by only 0.35% and 0.49% for FR = 1 and FR = 2, respectively. Regardless of whether f-PCA or non-f-PCA is present, moderate stenosis had an insignificant effect on tCeA. However, in cases of severe stenosis, tCeA was lowest for f-PCA with FR = 2. Specifically, for non-f-PCA, tCeA decreased by 0.79% and 6.18% with CS at 50% and 75%, respectively. For FR = 1, these decreases were 0.20% and 4.77%, and for FR = 2, they were 0.73% and 8.58%. In conclusion, stenosis in the ipsilateral ICA had a more severe impact on blood flow in the CoW with partial f-PCA.

WSS is an important hemodynamic factor. Researchers have extensively investigated the intricate relationship between WSS and cerebrovascular diseases such as aneurysms or atherosclerosis.49,52,58,78–81 An aneurysm is a balloon-like weakening or bulging of an arterial section. Brain aneurysms are also known intracranial aneurysms. Aneurysms can rupture and cause life-threatening hemorrhagic stroke. Researchers have hypothesized that various hemodynamic factors, such as flow patterns and WSS, play a role in the aneurysm process,82 with findings indicating that a high WSS may instigate aneurysm formation, whereas a low WSS can facilitate the growth of existing aneurysms.81 Because most aneurysms are treated when they have already been diagnosed/formed, it is difficult to observe their initial growth. Many patients with brain injuries due to accidents have a high risk of developing aneurysms because of the brain–blood-flow dysfunction. Therefore, we can predict and prevent aneurysm formation based on WSS.

The location and structure of aneurysms in rats and monkeys are similar to those of human cerebral aneurysms.76,83,84 Kondo et al.76 and Fukuda et al.85 studied rat cerebral aneurysms and suggested that the aneurysm often grows at the apex of the arterial bifurcation owing to an increase in the WSS, which causes arterial wall degeneration. Steiger et al.86 concluded that the hemodynamic stress (typical >= 5 Pa) raises the aneurysm formulation, while the rupture of aneurysm often occurs at the dome of aneurysm where the WSS is lowest. They also indicated that in aneurysms, the maximum WSS is commonly observed at the neck of the aneurysm, which is also the conclusion of Refs. 87 and 88. For example, for MCA aneurysms, the maximum WSS is 14.39 Pa and the average WSS is 3.64 Pa, whereas it is only approximately 1.64 Pa within the aneurysm region.88 Many other studies have also indicated the importance of evaluating the WSS in the formation/initiation and growth/development of aneurysm.89–91 

This region had a low WSS or disturbed flow, which is noteworthy. While a high WSS is commonly associated with the formation and initiation of aneurysms, a low WSS has been extensively studied for its contribution to the initiation and progression of atherogenesis plaque.38,79,92,93 Atherosclerosis is a condition in which fatty deposits build up on the inner walls of arteries, narrowing the arteries and reducing blood flow.

The context of high or low WSS also varies depending on the object and its geometry. We followed these criteria from a previous review.80 The authors divided the WSS characteristics into three regimes. Baseline WSS, which refers to the WSS range in straight arteries (in their paper, it ranges from 1.5 to 2.5 Pa). At the bifurcation, when blood flows from the parent artery to daughter arteries, it often appears as a recirculation zone or disturbed flow owing to the abrupt geometrical change at the bifurcation area, resulting in a low WSS in this region. This low-WSS region also occurs in the area after stenosis, where blood flow encounters a sudden narrowing area. This phenomenon is referred to as vena contracta in the context of fluid dynamics. Furthermore, at the apex (or tip) of the bifurcation, where blood flow impinges, the WSS is often higher than the baseline WSS (referred to as a high WSS). In their study, the WSS values for each range were as follows: <1 Pa (low), 1.5 − 2.5 Pa (baseline), and >3 Pa (high) WSS. In our study, we used the non-f-PCA case and bifurcation in BA−PCA to define these three ranges, as shown in Fig. 18. In this study, we define any WSS > 5 Pa as a high WSS and any WSS < 1 Pa as a low WSS. The high WSS range was also similar with that presented in Refs. 86 and 94.

FIG. 18.

Description of high WSS (>5 Pa) and low WSS (<1 Pa) regions. (a) Streamline and contour of TAWSS at the BA−PCA bifurcation: low WSS occurs at the recirculation zone, while high WSS is observed at the apex/tip of bifurcation due to the flow impingement. (b) Streamline and contour of TAWSS across the ICA stenosis: high WSS is observed at the throat of stenosis area, followed immediately by the low WSS region.

FIG. 18.

Description of high WSS (>5 Pa) and low WSS (<1 Pa) regions. (a) Streamline and contour of TAWSS at the BA−PCA bifurcation: low WSS occurs at the recirculation zone, while high WSS is observed at the apex/tip of bifurcation due to the flow impingement. (b) Streamline and contour of TAWSS across the ICA stenosis: high WSS is observed at the throat of stenosis area, followed immediately by the low WSS region.

Close modal

In this study, we investigated two important WSS-related HDIs, TAWSS and OSI. Notably, a low TAWSS often coincides with a high OSI, particularly at the vascular bifurcations and curvatures where the flow is inherently disturbed. The combination of a low TAWSS and high OSI is especially concerning because it has been linked to an increased risk of aneurysm rupture78,95 and promotes plaque formation.79,93

Figure 16 shows that as FR increases, the maximum TAWSS increases significantly at iP1, iPCoA, and ACoA. These arteries are considered to be related to the aneurysm occurrences. The locations of maximum TAWSS show that they are all at bifurcations: BA−PCA (P1), ICA−PCoA, P1−PCoA, and ACoA−ACA. Winkler et al.96 reported four types of basilar aneurysms, of which three have aneurysm occurrences at the point of bifurcation between the basilar artery and the PCA (P1). Aneurysms in the communicating arteries (ACoA and PCoA) are the most common types of intracranial aneurysm.67–70 Although aneurysms can occur at various locations within the ACoA,67,97 true aneurysms of the PCoA are considered rare. Most aneurysms of PCoA arise at the bifurcation of ICA and PCoA.70,98 For example, high TAWSS and increased velocity at the bifurcation between the ipsilateral PCoA and the ICA in the f-PCA model (with FR = 2) indicated a potential risk of aneurysm formation at this bifurcation, as depicted in Fig. 19. A previous study99 has also concluded that the larger PCoAs have a higher potential of PCoA aneurysm formation.

FIG. 19.

The velocity streamline across the bifurcation of ICA-PCoA. FR < 1 indicates the CoW with no f-PCA.

FIG. 19.

The velocity streamline across the bifurcation of ICA-PCoA. FR < 1 indicates the CoW with no f-PCA.

Close modal

An ICA aneurysm is the most common cerebral aneurysm and is often observed at the bifurcation between the ICA and the ACA or MCA. However, in our study, f-PCA did not seem to affect the hemodynamics of the ACA and MCA, in which there were minor changes in TAWSS and OSI with increasing FR.

Aneurysms have also been reported in the P2 segments of the PCA.100,101 Aneurysms in the PCA are considered rare; however, ruptured posterior circulation is correlated with a higher mortality risk than anterior circulation.102,103 Our study observed that the increase in the area of high TAWSS in the P2—with increasing FR—was slightly further from the point of bifurcation between P1 and the PCoA. This high-TAWSS region spreads even further to the PCoA side in the full f-PCA case.

This study provides a comprehensive analysis of the hemodynamic effects of f-PCA variants on the CoW using CFD simulations. The findings revealed significant quantitative changes in cerebral blood flow and hemodynamic indices, emphasizing the impact of the f-PCA on cerebral circulation.

Variations in f-PCA led to substantial changes in blood flow distribution within the CoW, with the ipsilateral side being more significantly impacted than the contralateral side. As the fetal ratio (FR) increased, there was a notable reduction in the contribution of the basilar artery to the P2 segment of the PCA, with a 40.0% reduction at FR = 1 and a 70.9% reduction at FR = 2. This reduction is compensated by ICA. Additionally, this study also highlights the increased importance of communicating arteries in the CoW with f-PCA. For our patient-specific model, the flow through the ipsilateral PCoA increased significantly, being 7.2 times higher for FR = 1 and 19.1 times higher for FR = 2 compared to the reference case.

The presence of f-PCA exacerbated the hemodynamic disturbances caused by ipsilateral carotid stenosis, particularly under severe stenosis conditions, leading to more pronounced impacts on blood flow and WSS. Under severe carotid stenosis (75%), the maximum TAWSS values showed dramatic increases, reaching up to 164.9 Pa in the ACoA for full f-PCA. Additionally, there are significant increases in TAWSS and OSI in key arterial segments due to f-PCA variants, especially under conditions of severe stenosis.

Our study's findings have significant clinical implications. The identification of elevated TAWSS and OSI in patients with f-PCA suggests an increased risk of aneurysm initiation and growth in specific arterial segments. Furthermore, the detailed flow dynamics from our simulations provide valuable insights for surgical planning, particularly by identifying potential hemodynamic challenges. For instance, we observed that increased FR substantially elevates blood flow through the PCoA, indicating that during surgical procedures such as PCoA aneurysm clipping, obstructing this flow could severely compromise the blood supply to the PCA, leading to potential arterial dysfunction. In summary, our study offers a comprehensive analysis of the hemodynamic changes associated with f-PCA variations and carotid stenosis, enhancing the understanding of the associated risks. This knowledge can assist clinicians in identifying high-risk patients, improving diagnostic accuracy, and informing more effective pre-operative planning.

Although this study provides valuable insights into the hemodynamic effects of f-PCA variants on the CoW, it has several limitations that should be considered. First, the simulations were based on patient-specific 3D models derived from a limited dataset, which may not fully capture the diversity of anatomical variations present in the general population. This limitation may have affected the generalizability of the findings to all individuals with the f-PCA variant. However, previous studies18,19,38 have demonstrated that the tendencies of blood flow and hemodynamic indices observed in our model (with non-f-PCA) are consistent with those derived from two other patient-specific models. This consistency suggests that, despite being based on a single patient, our results are capable of supporting general conclusions about the effects of f-PCA on cerebral blood flow. For future research, we aim to include a larger sample of patient-specific models and to further analyze the impact of this fetal-type on cerebral blood flow, particularly in the context of surgical treatment for PCoA aneurysms. Second, the study assumed that blood behaves as a Newtonian fluid and that the vessel walls are rigid. In reality, blood exhibits non-Newtonian properties, and the arterial walls are elastic and can deform under pressure. These assumptions may lead to simplifications that do not entirely reflect the complex hemodynamics of the cerebral vasculature. Previous studies38,39 have found that the Newtonian model can be a sufficient tool for assessing OSI and capturing flow disturbances. However, they also suggest that for a more accurate assessment, particularly in regions with low TAWSS values (like those associated with atherosclerotic plaques), more appropriate viscosity models might be necessary. Additionally, this study focused on unilateral carotid stenosis and f-PCA. Thus, it does not explore the effects of bilateral stenosis that may coexist with unilateral f-PCA variants and bilateral f-PCA variants.

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A5A1022977). This work was also supported by a Korea University Grant.

The authors have no conflicts to disclose.

Thi Thanh Giang Le: Data curation (lead); Formal analysis (lead); Investigation (equal); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead). Sangwon Ryu: Conceptualization (supporting); Methodology (equal); Software (supporting); Writing – original draft (supporting). Jung Jae Yoon: Methodology (supporting); Software (supporting); Writing – original draft (supporting). Taekkyun Nam: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal). Jaiyoung Ryu: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

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