Impact of anatomical variations of the circle of Willis on the blood flow within unruptured intracranial aneurysm

Introduction

Intracranial aneurysms (IAs) are pathological dilatations that arise in the main arteries of the circle of Willis (CoW). Spontaneous subarachnoid hemorrhage (SAH) is caused by the rupture of an aneurysm in over 80% of cases, affecting 9 in 100,000 individuals each year (1). The CoW is a complex arterial network wherein major cerebral arteries converge to form a heptagonal arrangement that provides a sufficient blood supply to the brain via collateral vascular networks and precise autoregulation systems for vascular dilation and constriction. In the CoW, the anterior cerebral arteries (ACAs) and middle cerebral arteries (MCAs) are responsible for perfusing the temporal, parietal, and frontal regions of the brain, whereas the posterior cerebral arteries (PCAs) supply blood to the brainstem, thereby maintaining normal brain function (Figure 1).

Figure 1 Schematic diagram of the circle of Willis.

Clinically, anatomical variations of the CoW are commonly observed, including arterial hypoplasia and artery absence. The probabilities of absence for the ACA-A1 segment (ACA-A1) and the PCA-P1 segment (PCA-P1) are relatively higher, at 6% and 9%, respectively (2). Anatomical variations of the CoW can lead to significant alterations in blood flow redistribution and local hemodynamics. Usually, the absence of an artery with a high blood flow can cause other arteries to compensate, which in some cases may result in undesirable flow patterns. Aneurysms often occur in conjunction with anatomical variations of the CoW; previous studies have shown that underdeveloped arterial branches and larger angles are risk factors for the occurrence of aneurysms (3-6). These studies have indicated that the occurrence of aneurysms is closely related to abnormal blood flow caused by anatomical variations. Given this, variations can equally affect the internal flow of existing aneurysms; however, there is a lack of relevant research. The role of variations in the CoW in the progression and rupture of IAs remains to be clarified.

With advancements in computational and medical imaging technologies, computational fluid dynamics (CFD) is increasingly being used to calculate the hemodynamic parameters of IAs. In the patient-specific CFD analysis of IAs, vessel and aneurysm walls are treated as rigid and only the fluid domain is modeled. From the computational results, a series of parameters that can effectively predict the risk of aneurysm rupture are statistically determined, including the following: wall shear stress (WSS), wall pressure, and flow pattern (7). However, this approach does not meet the actual elastic behavior of the vascular walls and neglects the effects of arterial wall deformation. In contrast, fluid-structure interaction (FSI) effectively simulates the interaction between fluid and solid by combining fluid dynamics, structural mechanics, and the coupling between fluids and solids, resulting in higher computational accuracy. In the comparative study of CFD and FSI, it has been found that the fluid velocity and WSS calculated using CFD are generally higher than those obtained from FSI, with the differences increasing with rises in blood pressure, reaching up to 50% (8). Currently, FSI analysis has been employed to study the effects of wall thickness, hypertension, elasticity, hyperelasticity and anisotropy of the wall, upstream artery length, and ruptured and unruptured aneurysms (9-14). Therefore, in this study, we used an FSI simulation method to conduct research and analysis on aneurysms.

This study focused on the impact of ACA-A1/PCA-P1 absence on unruptured IAs of different locations. A one-dimensional (1D) solver was used to simulate the overall blood flow distribution in the CoW. Subsequently, a three-dimensional (3D) FSI model was utilized to calculate the hemodynamic parameters under the same IA. The results were compared with those of a complete CoW, and the relative deviations in hemodynamic parameters were statistically analyzed to evaluate the impact of artery absence on the internal flow of the aneurysms. We present this article in accordance with the STROBE reporting checklist (available at https://qims.amegroups.com/article/view/10.21037/qims-2025-55/rc).

Methods

Conducting 3D FSI calculations for an entire vascular network is computationally intensive and time-consuming, whereas a 1D solver has a lower computational cost and can accurately simulate changes in blood flow conditions resulting from alterations in the vascular network topology. In order to reduce the calculation time while ensuring accuracy, a 1D solver and 3D FSI model were combined in this study, where the results of 1D calculation were converted into boundary conditions for the FSI.

1D solver

In this study, an open-source solver, the Python Network Solver (pyNS), is utilized to simulate blood flow waves (15). The vascular network is represented as a circuit made up of a series of electronic components, with each arterial segment discretized into one or more components. The specific impedance, inductance, and capacitance of these components are determined by the characteristics and dimensions of the vessels, including elasticity, length, and radius. By iteratively solving the circuit network, the instantaneous current and voltage of the electrical circuit are obtained. Finally, these current and voltage values are converted into flow and pressure.

Blood flow complies with the law of conservation of mass and momentum, expressed as Eqs. [1,2]:

∂A∂t+∂(AU)∂z=0[1]

∂U∂t+U∂U∂z+1ρ∂p∂z=fρA[2]

where t, z, ρ, and A are the time, the axial coordinate, the blood density, and the cross-section area, respectively. U represents the average axial velocity, p represents the average cross-section pressure, and f represents the viscous force per unit length. The above equations are closed by the Eqs [3,4].

p=p0+βA0A−A0[3]

β=πhE(1−σ2)[4]

where p0 is the reference pressure, A0 and h are the cross-sectional area and the wall thickness under reference pressure, respectively, E is Young’s modulus of the blood vessel, and σ is the Poisson’s ratio.

3D FSI model

Solid governing equation

The solid model of IA is regarded as a homogeneous, incompressible, linear elastic solid. The solid domain describing the arterial wall dynamics through a linear elastic model is governed by the momentum conservation equation, which can be expressed as:

ρsu˙g=∇⋅σs[5]

where ρs, σs, and u˙g are solid density, solid stress tensor, and local acceleration of the solid, respectively. In this study, the solid model density of IA is set to 1,100 kg/m³, Poisson’s ratio is 0.49, and the modulus of elasticity is 1×10⁶ Pa.

Fluid governing equation

Blood flow, as a movable domain, is governed by the incompressible Navier-Stokes equations. For FSI applications with arbitrary moving boundary constraints, the arbitrary Lagrangian-Eulerian (ALE) formulation is commonly used. The incompressible continuity equation and the ALE form of the Navier-Stokes equations can be expressed as:

∇⋅u=0[6]

ρf(∂u∂t+((u−ug)⋅∇)u)=−∇p+μ∇2u[7]

where ρf, u, and ug are the fluid density, the fluid velocity, and the moving coordinate velocity, respectively. In ALE formulation, the term (u−ug), which is the relative velocity of the fluid with respect to the moving coordinate velocity, is added to the conventional Navier-Stokes equation to account for the movement of the grid. The fluid density is set to 1,050 kg/m³, and the dynamic viscosity is 0.0035 Pas (11).

FSI interface conditions

FSI follows the fundamental principles of conservation, and the following boundary conditions are applied at the FSI interface: (I) displacements of the fluid and solid domain must be compatible; (II) tractions at these boundaries must be at equilibrium; and (III) fluid obeys the no-slip condition. These boundary conditions are expressed as:

df=ds[8]

σs⋅n^s=σf⋅n^f[9]

u=ug[10]

where d, σ, and n^ are displacements, stress tensors, and boundary normals with the subscripts f and s indicating a property of the fluid and solid, respectively.

Taking the anterior communicating artery (ACoA) aneurysm as an example, the inlet and outlet boundary conditions are shown in Figure 2A. The cross-sections of the solid domain’s inlet and outlet are constrained by setting the displacements in all directions to zero and the rotational movements around all axes to zero. No external pressure boundary condition is applied. In the ANSYS Fluent module (ANSYS, Canonsburg, PA, USA), mesh generation is performed using a hexahedral core mesh, with a maximum sieve hole size of 0.5 mm. Due to variability in the sizes of the patient IA models, the number of meshes ranges from 300,000 to 700,000. In addition, grid-independent verification of ACoA aneurysms is implemented, as demonstrated in Figure 2B. Statistics are conducted on the maximum WSS and pressure in aneurysms with different numbers of grids, and it has been found that the deviation of results is less than 3% when the number of grids is 500,000 and 1.1 million. Therefore, a grid division method of about 500,000 is adopted for calculation in this paper. Two-way coupling calculations are carried out using the Intrinsic FSI module of Fluent, and the solution is iterated within each time step until the specified residual of 10⁻³. A total of one cardiac cycle is simulated, and the data is dynamically updated at each time step. The computational accuracy of the method combining 1D and 3D models has been preliminarily verified in our previous study (6).

Figure 2 Boundary conditions and grid independent verification for the ACoA aneurysm: (A) L-ACA-A1 inlet velocity waveform and L-ACA-A2 outlet pressure waveform during one cardiac cycle; (B) maximum WSS and wall pressure in the ACoA aneurysm under varying mesh densities. ACoA, anterior communicating artery; L-ACA-A1, left-anterior cerebral artery, A1 segment; L-ACA-A2, left-anterior cerebral artery, A2 segment; WSS, wall shear stress.

Case presentation

The study was conducted in accordance with the Declaration of Helsinki and its subsequent amendments and approved by the Ethics Committee of the School of Mechanical Engineering and Automation of Fuzhou University (No. FZU-SMEA-24093). Four IAs cases were obtained from Emory University Open-Source Data Center (http://ecm2.mathcs.emory.edu/aneuriskweb/repository) and Fuzhou University Affiliated Provincial Hospital. The patients at Fuzhou University Affiliated Provincial Hospital provided written informed consent before participation in this study. These IAs were in different areas, including ACoA aneurysm, left-middle cerebral artery (L-MCA) aneurysm, left-internal carotid artery (L-ICA) aneurysm and left-posterior communicating artery (L-PCoA) aneurysm. The wall thickness value is set to 0.2 mm. To simplify the hemodynamic calculations, only the main arterial segments near the aneurysms are retained, resulting in the creation of four IA geometric models, as shown in Figure 3.

Figure 3 The vascular geometric models of the four IA patients: (A) ACoA aneurysm, (B) L-MCA aneurysm, (C) L-ICA aneurysm, (D) L-PCoA aneurysm. The arrow indicates the direction of blood flow entry. ACoA, anterior communicating artery; L-ICA, left-internal carotid artery; L-MCA, left-middle cerebral artery; L-PCoA, left-posterior communicating artery.

Results

Flow distribution by 1D solver

In this study, four typical artery absence scenarios [left/right-anterior cerebral artery, A1 segment (L/R-ACA-A1) absence and left/right-posterior cerebral artery, P1 segment (L/R-PCA-P1) absence] are simulated by setting the arterial diameter close to zero in the solver. The average flow results of the CoW in one cardiac cycle are shown in Figure 4. It is observed that the impact of the absence of an artery on blood flow in different segments of the CoW varies. Whether the ACA-A1 or the PCA-P1 is absent, the flow in the communicating arteries rises. Specifically, the flow in the ACoA increases nearly 40 times under ACA-A1 absence and the flow in the ipsilateral PCoA rises nearly 30 times under PCA-P1 absence. The reason for this phenomenon is that contralateral blood flow compensates through the ACoA, ensuring that the affected ACA receives an adequate blood supply under the ACA-A1 absence. Similarly, ipsilateral blood flow compensates through the PCoA, thus ensuring the blood supply to the PCA under the PCA-P1 absence. The behavior is referred to as the flow compensation effect. This underlines the important roles of the ACoA and PCoA in regulating blood flow in the CoW. A previous study has confirmed that the ACoA is an important collateral pathway if an ICA is occluded (2). Thus, although the absence of the communicating arteries may not significantly affect the overall blood flow distribution in the CoW, their importance becomes evident if other arteries are absent or undergo sudden occlusion.

Figure 4 The average flow of the arteries in the CoW over one cardiac cycle (mL/min): (A) schematic diagram of the simulated arterial network in the CoW; (B) complete CoW; (C) L-ACA-A1 absence; (D) R-ACA-A1 absence; (E) L-PCA-P1 absence; and (F) R-PCA-P1 absence. The direction of the arrows indicates the average blood flow direction, and the dashed lines represent aneurysms. CoW, circle of Willis; L-ACA-A1, left-anterior cerebral artery, A1 segment; L-PCA-P1, left-posterior cerebral artery, P1 segment; R-ACA-A1, right-anterior cerebral artery, A1 segment; R-PCA-P1, right-posterior cerebral artery, P1 segment.

FSI simulation

The calculation results, including WSS, oscillatory shear index (OSI), pressure, and von Mises stress, are presented in Tables 1-4. The values in parentheses represent the percentage deviation compared to the complete CoW.

WSS and OSI

Hemodynamics, especially the WSS, is believed to play a crucial role in the development and rupture of IAs. Based on our calculations, the impact of the absence of the arteries on WSS is the most evident, which is the primary focus of our study. As shown in Table 1, the absence of ACA-A1 leads to a 103%/147% increase in maximum WSS inside the ACoA aneurysm compared to the complete CoW. Due to the previously mentioned flow compensation effect mentioned above, the absence of ACA-A1 results in a significant flow rise of ACoA, causing highly concentrated inflow jets that arise extensive areas of abnormal high WSS. As illustrated in Figure 5, the WSS inside the IA is clearly elevated under the absence of L-ACA-A1, with values surpassing those at the vessel wall. The absence of the PCA-P1 also contributes to an increase in WSS within the ACoA aneurysm, although this effect is less pronounced.

Figure 5 WSS and pressure distribution of ACoA aneurysm at peak systole. (A,G) Complete CoW; (B,H) L-ACA-A1 absence; (C,I) L-PCA-P1 absence. OSI distribution within one cardiac cycle. (D) Complete CoW; (E) L-ACA-A1 absence; (F) L-PCA-P1 absence. The red arrow indicates the size of the point parameter. ACoA, anterior communicating artery; CoW, circle of Willis; L-ACA-A1, left-anterior cerebral artery, A1 segment; L-PCA-P1, left-posterior cerebral artery, P1 segment; OSI, oscillatory shear index; R-ACA-A1, right-anterior cerebral artery, A1 segment; WSS, wall shear stress.

OSI serves as an important index for measuring the degree of changes in the direction of blood flow in the tumor and is closely related to the progression of aneurysms. When ACA-A1 is absent, the maximum OSI value within ACoA aneurysm increases by 19% or more compared to that in a complete CoW (Table 2), indicating that in ACA-A1 absence, WSS not only significantly increases in value but also exhibits a more pronounced amplitude in directional changes. In addition, we found that the locations where high OSI (0.38) occurs correspond to regions of low WSS (7.23 Pa) far from the flow impact regions, as shown in Figure 5B,5E.

For MCA aneurysms, no significant impact of artery absence on WSS is observed. Notably, the WSS in MCA aneurysms is considerably higher than that in other aneurysms, with a maximum value of approximately 400 Pa.

For L-ICA and L-PCoA aneurysms, the mean WSS decreases by 40% and 39%, respectively, in the absence of L-ACA-A1, whereas it increases by 44% and 11%, respectively, in the absence of R-ACA-A1. The reason for this is that the absence of ACA-A1 causes contrasting effects on the flow velocity in the contralateral and ipsilateral ICA, leading to an increase on the contralateral side and a decrease on the ipsilateral side. As shown in Table 5, the variation of WSS within the aneurysm is well correlated with the variation of velocity in the parent artery. Under conditions of abnormally high and low flow velocity, the IAs located on the ICA are influenced by the flow field, thus altering the distribution of WSS. Furthermore, the absence of PCA-P1 increases the mean WSS in the ipsilateral ICA and PCoA aneurysms by 20–25%, with almost no effect on the contralateral side.

Pressure

Hypertension has been considered an independent risk factor for IAs rupture. Localized maximum pressure rise on the wall caused by abnormal blood flow can lead to sudden rupture of the aneurysm. It has been confirmed that the high pressure areas coincide with the rupture point (16). In this study, it was found that the maximum pressure in the ACoA aneurysm increased by 0.90 kPa/0.95 kPa in the absence of L/R-ACA-A1 (Table 3). The reason can be explained by the fact that the concentrated jet flow for ACA-A1 absence results in a substantial rise in the pressure of the impacted region. Figure 5H shows that the maximum pressure is located around the dome of the aneurysm. Compared with the absence of L-PCA-P1, the pressure increase in the ACoA aneurysm is more pronounced in the absence of L-ACA-A1, as the absence of PCA-P1 does not exert a significant influence on the flow field within the anterior circulation. Regardless of whether the CoW is complete, the wall pressure of the L-MCA aneurysm is the highest among all the aneurysms considered, reaching up to 26 kPa.

Von Mises stresses

The stress state of a solid material is crucial for predicting and analyzing its failure. Von Mises stress offers a valuable tool by condensing a complex stress field into a single, comprehensive value, thereby facilitating our ability to predict the damage to aneurysmal walls and evaluate the rupture risks. Based on the results presented in Table 4, the absence of an artery does not significantly increase the von Mises stress within the aneurysm, with an increase of less than 5%. The stress state of the aneurysm wall depends on both the structural geometry and the loading conditions. On the one hand, the mechanical response of the wall depends on the local morphology, more specifically, the wall curvature, with larger values at curved wall portions. In this study, the geometric structure of an aneurysm remained unchanged regardless of whether the CoW is complete or not. On the other hand, the load condition on the aneurysm wall (i.e., wall pressure) changed only minimally (see Table 3).

Regardless of whether the CoW is complete, the von Mises stress of the MCA aneurysm was the highest among all the aneurysms considered at approximately 0.9 MPa. As shown in Figure 6, the maximum von Mises stress occurred at the aneurysmal neck, rather than at the dome or the parent artery. This is not surprising since the wall pressure of the MCA aneurysm was the highest among all the aneurysms considered and the curvature at the aneurysmal neck was the maximal. Considering both above factors, the maximum von Mises stress occurred at the neck of the MCA aneurysm.

Figure 6 Von Mises stress distribution of four aneurysms at peak systole under complete CoW: (A) ACoA aneurysm; (B) L-MCA aneurysm; (C) L-ICA aneurysm; (D) L-PCoA aneurysm. ACoA, anterior communicating artery; CoW, circle of Willis; L-ICA, left-internal carotid artery; L-MCA, left-middle cerebral artery; L-PCoA, left-posterior communicating artery.

Discussion

Aberrant hemodynamics can disrupt the equilibrium of vascular homeostasis and drive destructive remodeling to cause the progression and rupture of IAs. In this paper, the impact of artery absence on the hemodynamics and stress parameters of IAs is evaluated quantitatively by a 1D solver and 3D FSI model. Although predicting the rupture of IAs remains challenging due to its complex mechanisms, the extent of the impact on IAs can be assessed by comparing the calculations of a complete CoW with those of a defective CoW. Among all the parameters we considered, the variation in the WSS was the most pronounced. The flow compensation effect caused by the absence of arteries can result in abnormally high flow in certain vessels, leading to an increase in WSS as the flow rises. Based on our simulation results, in the absence of ACA-A1, the maximum WSS within the ACoA aneurysm increased by more than 100% compared to the complete CoW. In addition, WSS increased by 44% and 11%, respectively, in the contralateral ICA and PCoA aneurysms, whereas it decreased by 40% and 39%, respectively, in the ipsilateral ICA and PCoA aneurysms. A previous study has already demonstrated that the unusually high WSS can trigger mural-cell-mediated destructive remodeling, yet unusually low WSS can trigger inflammatory-cell-mediated destructive remodeling (17). Thereby, the absence of ACA-A1 may drive different mechanistic pathways in the bilateral ICA and PCoA aneurysms growth and rupture. The absence of PCA-P1 has a less significant impact on the global blood flow of the CoW compared to the absence of ACA-A1, but it can still lead to an increase of over 20% in WSS within the ipsilateral ICA and PCoA aneurysms.

Besides WSS, other factors (e.g., OSI, wall pressure and stress) also associate with IA ruptures. Under the conditions of ACA-A1 absence, hemodynamic analysis of ACoA aneurysms revealed a characteristic pattern where regions distal to the direct flow impingement zone simultaneously exhibit reduced WSS and elevated OSI. This hemodynamic environment mechanistically aligns with established pathological pathways, as low WSS in combination with high OSI is recognized to activate inflammatory-cell-mediated signaling cascades. The observed hemodynamic profile likely initiates a self-amplifying degradation mechanism within the aneurysm wall, wherein the triggered inflammatory response progressively compromises structural integrity (18,19). This pathophysiological cycle accelerates the aneurysmal wall remodeling process, ultimately driving the pathological progression toward rupture through sustained matrix degradation and biomechanical weakening of the vascular tissue.

Our computational analysis further demonstrated the distinct biomechanical distributions within IAs, with peak pressures primarily concentrated in the dome and neck regions, whereas the maximum von Mises stress was particularly localized to the aneurysmal neck. Notably, this study adopted a simplified assumption of uniform wall thickness for the aneurysmal sac. Although this approach facilitates computational modeling, it may overestimate the von Mises stress values in the neck region. These findings underscore the importance of integrating high-resolution, patient-specific wall thickness measurements into future hemodynamic models to enhance the predictive accuracy of rupture risk assessments (20-22). Finally, based on the observed hemodynamic characteristics, we hypothesize that the anatomical absence of the ACA-A1 or PCA-P1 segment may represent a novel morphological risk factor, potentially leading to aneurysm development and rupture through sustained hemodynamic damage and inadequate vascular remodeling.

Although an obvious impact of artery absence on the MCA aneurysm was not observed in our results, comparisons of the parameters within IAs of various locations reveal that the MCA aneurysm experiences highest WSS, wall pressure and stress. This can be explained by the fact that the MCA aneurysm in this study is located at the bifurcation of arteries (see Figure 3B). Therefore, MCA aneurysms on arterial bifurcation may continuously be at a high risk of rupture.

Clinically, the management of incidentally discovered aneurysms often presents significant challenges in treatment decision-making (23,24). Generally, clinicians develop treatment plans for IA based on individual patient factors, such as age, overall health status, medical history, signs and symptoms, tolerance to specific medications, and the morphology of the IA (25-27). Some untreated aneurysms may remain stable for many years, whereas others may continue to grow and rupture unexpectedly. Therefore, assessing the risk of aneurysm rupture is crucial. On the one hand, IAs with a very high risk or those showing signs of further development require timely surgical intervention. On the other hand, conservative treatment approaches should be adopted for IAs with a low risk, as surgical treatment can sometimes lead to complications that may exceed the risks associated with aneurysm rupture itself. In summary, many researchers have investigated ruptured and unruptured aneurysms using various methods to identify parameters that can accurately predict aneurysm rupture or risk factors associated with IA rupture (28-30).

The goal of this research was to assist clinicians in better predicting the likelihood of IA rupture and making more reliable treatment decisions. From a hemodynamic perspective, the absence of either the ACA-A1 or PCA-P1 has a negative impact on existing aneurysms to some extent. Therefore, for patients diagnosed with IAs, it is recommended to focus on evaluating the integrity of the CoW by computed tomography angiography/magnetic resonance angiography/digital subtraction angiography (CTA/MRA/DSA), especially for the absence of ACA-A1 and PCA-P1. This may help to prevent unexpected rupture of the aneurysm and bleeding. Comprehensive risk stratification for IAs must extend beyond evaluating aneurysm-specific parameters and conventional risk factors, such as the size, regularity of geometric shapes, and aspect ratio (the ratio of the height of the aneurysm to that of the neck) (31,32). The anatomical absence of ACA-A1/PCA-P1 segments should be systematically classified as a clinically validated independent risk factor. Therefore, patients harboring these vascular variants require automated escalation in rupture risk stratification to reflect their heightened vulnerability to hemodynamic instability and pathological remodeling.

Conclusions

In this study, the hemodynamic and stress parameters inside four patient-specific IAs with artery absence in CoW were analyzed by a combination of a 1D solver and 3D FSI model. According to the results, the variation of pressure and von Mises stress is less than 6% with the artery (ACA-A1/PCA-P1) absence, whereas the WSS varies significantly according to the location of aneurysm. In the absence of ACA-A1, the maximum WSS within the ACoA aneurysm increases by more than 100% compared to the complete CoW. In addition, WSS increases in the contralateral ICA and PCoA aneurysms, whereas it decreases in the ipsilateral ICA and PCoA aneurysms. The absence of PCA-P1 has a less significant impact on the global blood flow of the CoW compared to the absence of ACA-A1, but it still leads to an increase in WSS within the ipsilateral ICA and PCoA aneurysms. Thus, the absence of ACA-A1 or PCA-P1 may serve as a rupture risk factor of IAs.

The limitations of this study include its retrospective design and small sample size. Therefore, we must emphasize the need for additional validation of larger cohorts to confirm observed trends. The assumption of uniform wall thickness adopted in this study may ignore the real stress concentration caused by irregular wall thickness. This limitation highlights the need for more precise wall thickness measurements in future studies to improve the reliability of the computed stress distributions.

In our model, simplified assumptions are used for inflow and outflow boundary conditions, which may affect the simulation results to some extent. Although these assumptions can reflect the physiological state to a degree, they are not fully equivalent to the actual complex hemodynamic environment. We will explore more complex boundary condition settings in follow-up studies to further improve the clinical relevance of the model.

We acknowledge that there are some challenges in the comprehensive experimental validation of 1D–3D integrated models, mainly due to the lack of sufficient experimental data comparison. We look forward to improving the accuracy and reliability of the model in the future by comparing the direction with experimental data (such as with 4D MRI data) and further numerical validation.

Acknowledgments

None.

Footnote

Reporting Checklist: The authors have completed the STROBE reporting checklist. Available at https://qims.amegroups.com/article/view/10.21037/qims-2025-55/rc

Data Sharing Statement: Available at https://qims.amegroups.com/article/view/10.21037/qims-2025-55/dss

Funding: This work was supported by the Foundation of Fujian Finance Department (No. 2024881).

Conflicts of Interest: All authors have completed the ICMJE uniform disclosure form (available at https://qims.amegroups.com/article/view/10.21037/qims-2025-55/coif). The authors have no conflicts of interest to declare.

Ethical Statement: The authors are accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. The study was conducted in accordance with the Declaration of Helsinki and its subsequent amendments. Approval of all ethical and experimental procedures and protocols was granted by the Ethics Committee of the School of Mechanical Engineering and Automation of Fuzhou University (No. FZU-SMEA-24093). All patients provided written informed consent before participation in this study.

Open Access Statement: This is an Open Access article distributed in accordance with the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International License (CC BY-NC-ND 4.0), which permits the non-commercial replication and distribution of the article with the strict proviso that no changes or edits are made and the original work is properly cited (including links to both the formal publication through the relevant DOI and the license). See: https://creativecommons.org/licenses/by-nc-nd/4.0/.

References

1. Neifert SN, Chapman EK, Martini ML, Shuman WH, Schupper AJ, Oermann EK, Mocco J, Macdonald RL. Aneurysmal Subarachnoid Hemorrhage: the Last Decade. Transl Stroke Res 2021;12:428-46. [Crossref] [PubMed]
2. Alastruey J, Parker KH, Peiró J, Byrd SM, Sherwin SJ. Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J Biomech 2007;40:1794-805. [Crossref] [PubMed]
3. Nam SW, Choi S, Cheong Y, Kim YH, Park HK. Evaluation of aneurysm-associated wall shear stress related to morphological variations of circle of Willis using a microfluidic device. J Biomech 2015;48:348-53. [Crossref] [PubMed]
4. Bor AS, Velthuis BK, Majoie CB, Rinkel GJ. Configuration of intracranial arteries and development of aneurysms: a follow-up study. Neurology 2008;70:700-5. [Crossref] [PubMed]
5. Greving JP, Wermer MJ, Brown RD Jr, Morita A, Juvela S, Yonekura M, Ishibashi T, Torner JC, Nakayama T, Rinkel GJ, Algra A. Development of the PHASES score for prediction of risk of rupture of intracranial aneurysms: a pooled analysis of six prospective cohort studies. Lancet Neurol 2014;13:59-66. [Crossref] [PubMed]
6. Zheng R, Han Q, Hong W, Yi X, He B, Liu Y. Hemodynamic characteristics and mechanism for intracranial aneurysms initiation with the circle of Willis anomaly. Comput Methods Biomech Biomed Engin 2024;27:727-35. [Crossref] [PubMed]
7. Kimura H, Osaki S, Hayashi K, Taniguchi M, Fujita Y, Seta T, Tomiyama A, Sasayama T, Kohmura E. Newly Identified Hemodynamic Parameter to Predict Thin-Walled Regions of Unruptured Cerebral Aneurysms Using Computational Fluid Dynamics Analysis. World Neurosurg 2021;152:e377-86. [Crossref] [PubMed]
8. Sun H, Sze K, Chow K, Tsang AC. A comparative study on computational fluid dynamic, fluid-structure interaction and static structural analyses of cerebral aneurysm. Engineering Applications of Computational Fluid Mechanics 2022;16:262-78.
9. Valencia A, Burdiles P, Ignat M, Mura J, Bravo E, Rivera R, Sordo J. Fluid structural analysis of human cerebral aneurysm using their own wall mechanical properties. Comput Math Methods Med 2013;2013:293128. [Crossref] [PubMed]
10. Sun H, Sze K, Tang AYS, Tsang AC, Yu ACH, Chow KW. Effects of aspect ratio, wall thickness and hypertension in the patient-specific computational modeling of cerebral aneurysms using fluid-structure interaction analysis. Engineering Applications of Computational Fluid Mechanics 2019;13:229-44.
11. Lee CJ, Zhang Y, Takao H, Murayama Y, Qian Y. A fluid-structure interaction study using patient-specific ruptured and unruptured aneurysm: the effect of aneurysm morphology, hypertension and elasticity. J Biomech 2013;46:2402-10. [Crossref] [PubMed]
12. Torii R, Oshima M, Kobayashi T, Takagi K, Tezduyar TE. Fluid–structure interaction modeling of a patient-specific cerebral aneurysm: influence of structural modeling. Computational Mechanics 2008;43:151-9.
13. Lee CJ, Zhang Y, Takao H, Murayama Y, Qian Y. The influence of elastic upstream artery length on fluid-structure interaction modeling: a comparative study using patient-specific cerebral aneurysm. Med Eng Phys 2013;35:1377-84. [Crossref] [PubMed]
14. Gao B, Ding H, Ren Y, Bai D, Wu Z. Study of Typical Ruptured and Unruptured Intracranial Aneurysms Based on Fluid-Structure Interaction. World Neurosurg 2023;175:e115-28. [Crossref] [PubMed]
15. Manini S, Antiga L, Botti L, Remuzzi A. pyNS: an open-source framework for 0D haemodynamic modelling. Ann Biomed Eng 2015;43:1461-73. [Crossref] [PubMed]
16. Suzuki T, Stapleton CJ, Koch MJ, Tanaka K, Fujimura S, Suzuki T, Yanagisawa T, Yamamoto M, Fujii Y, Murayama Y, Patel AB. Decreased wall shear stress at high-pressure areas predicts the rupture point in ruptured intracranial aneurysms. J Neurosurg 2020;132:1116-22. [Crossref] [PubMed]
17. Meng H, Tutino VM, Xiang J, Siddiqui A. High WSS or low WSS? Complex interactions of hemodynamics with intracranial aneurysm initiation, growth, and rupture: toward a unifying hypothesis. AJNR Am J Neuroradiol 2014;35:1254-62. [Crossref] [PubMed]
18. Wang H, Balzani D, Vedula V, Uhlmann K, Varnik F. On the Potential Self-Amplification of Aneurysms Due to Tissue Degradation and Blood Flow Revealed From FSI Simulations. Front Physiol 2021;12:785780. [Crossref] [PubMed]
19. Wang H, Uhlmann K, Vedula V, Balzani D, Varnik F. Fluid-structure interaction simulation of tissue degradation and its effects on intra-aneurysm hemodynamics. Biomech Model Mechanobiol 2022;21:671-83. [Crossref] [PubMed]
20. Blankena R, Kleinloog R, Verweij BH, van Ooij P, Ten Haken B, Luijten PR, Rinkel GJ, Zwanenburg JJ. Thinner Regions of Intracranial Aneurysm Wall Correlate with Regions of Higher Wall Shear Stress: A 7T MRI Study. AJNR Am J Neuroradiol 2016;37:1310-7. [Crossref] [PubMed]
21. Tobe Y, Yagi T, Kawamura K, Suto K, Sawada Y, Hayashi Y, Yoshida H, Nishitani K, Okada Y, Kitahara S, Umezu M. Three-dimensional wall-thickness distributions of unruptured intracranial aneurysms characterized by micro-computed tomography. Biomech Model Mechanobiol 2024;23:1229-40. [Crossref] [PubMed]
22. Bolem S, Valeti C, Thankom Philip N, Sudhir BJ, Patnaik BSV. Patient-specific arterial wall generation for intracranial aneurysms with a variable and a near realistic vessel wall thickness for FSI studies. Med Eng Phys 2024;130:104211. [Crossref] [PubMed]
23. Shen Z, Zhao Y, Gu X, Fang J, Yang J, Li T, Fan B. Systematic review of treatment for unruptured intracranial aneurysms: clipping vs coiling. Turkish Neurosurgery 2024;34:377-87. [Crossref] [PubMed]
24. Tataranu LG, Munteanu O, Kamel A, Gheorghita KL, Rizea RE. Advancements in Brain Aneurysm Management: Integrating Neuroanatomy, Physiopathology, and Neurosurgical Techniques. Medicina (Kaunas) 2024.
25. Lauzier DC, Huguenard AL, Srienc AI, Cler SJ, Osbun JW, Chatterjee AR, Vellimana AK, Kansagra AP, Derdeyn CP, Cross DT, Moran CJ. A review of technological innovations leading to modern endovascular brain aneurysm treatment. Front Neurol 2023;14:1156887. [Crossref] [PubMed]
26. Macdonald RL, Schweizer TA. Spontaneous subarachnoid haemorrhage. Lancet 2017;389:655-66. [Crossref] [PubMed]
27. Deshmukh AS, Priola SM, Katsanos AH, Scalia G, Costa Alves A, Srivastava A, Hawkes C. The Management of Intracranial Aneurysms: Current Trends and Future Directions. Neurol Int 2024;16:74-94. [Crossref] [PubMed]
28. Bruneau DA, Valen-Sendstad K, Steinman DA. Onset and nature of flow-induced vibrations in cerebral aneurysms via fluid-structure interaction simulations. Biomech Model Mechanobiol 2023;22:761-71. [Crossref] [PubMed]
29. Varble N, Trylesinski G, Xiang J, Snyder K, Meng H. Identification of vortex structures in a cohort of 204 intracranial aneurysms. J R Soc Interface 2017;14:20170021. [Crossref] [PubMed]
30. Wang GX, Liu LL, Yang Y, Wen L, Duan CM, Yin JB, Zhang D. Risk factors for the progression of unruptured intracranial aneurysms in patients followed by CT/MR angiography. Quant Imaging Med Surg 2021;11:4115-24. [Crossref] [PubMed]
31. Wang X, Huang X. Risk factors and predictive indicators of rupture in cerebral aneurysms. Front Physiol 2024;15:1454016. [Crossref] [PubMed]
32. Suzuki T, Takao H, Rapaka S, Fujimura S, Ioan Nita C, Uchiyama Y, Ohno H, Otani K, Dahmani C, Mihalef V, Sharma P, Mohamed A, Redel T, Ishibashi T, Yamamoto M, Murayama Y. Rupture Risk of Small Unruptured Intracranial Aneurysms in Japanese Adults. Stroke 2020;51:641-3. [Crossref] [PubMed]