The fundamental principle of CFD analysis involves discretizing a fluid domain into control volumes and solving the governing equations that describe fluid motion. These governing equations encompass the continuity equation and the Navier-Stokes equation, which are derived from the conservation laws of mass and momentum. The continuity equation and the incompressible Navier-Stokes equation are expressed as follows:

Continuity equation: $\frac{\partial \rho }{\partial t}+▽·\left(\rho \overrightarrow{\nu }\right)=S_m$

Incompressible Navier-Stokes equation: $\rho \frac{D\overrightarrow{\nu }}{Dt}=-▽P+\mu {▽}^2\overrightarrow{\nu }+\rho \overrightarrow{g}$

where $\overrightarrow{\nu }$ and $\overrightarrow{g}$ represent the velocity and gravity vectors, respectively. ρ, μ, and P denote density, viscosity, and pressure, respectively. The general process of CFD analysis is depicted in Fig. 1. First, a 3-dimensional (3D) model of a cerebral artery is reconstructed using digital imaging and communications in medicine. Subsequently, the 3D model undergoes manipulation for CFD analysis. Next, meshing and the application of boundary conditions, such as inlet and outlet specifications, are performed on the 3D model. Finally, hemodynamic parameters such as velocity and wall shear stress (WSS) are computed.

CFD analysis proves to be an invaluable tool for quantitatively assessing blood flow within cerebral arteries and aneurysms. The velocity obtained through CFD analysis can be utilized in diverse applications pertaining to cerebral aneurysms. Fig. 2 demonstrates an example of velocity evaluation in the circle of Willis and a cerebral aneurysm.

Several assumptions were premised for the CFD analysis introduced for the hemodynamics of the intracranial vessels. Blood flow was assumed to be Newtonian for the convenience of simulation in the CFD analysis. In addition, the blood vessel was assumed to be a rigid wall in the CFD analysis. Therefore, the effects of the wall thickness and material properties of the blood vessels on the formation, growth, and rupture of cerebral aneurysms were excluded, and only the effect of blood flow was considered. However, since a cerebral aneurysm is a morphological change in blood vessels, it cannot be assumed that blood vessels do not change.

In addition to these intrinsic limitations of CFD analysis, it is not patient-specific. To analyze natural phenomena, the material properties of the target must be identified. However, the material properties of blood vessels used in previous studies were not patient-specific and were obtained from a cohort of healthy patients. Each patient had different material properties and vessel thicknesses. Furthermore, the velocity, viscosity, and blood pressure corresponding to the boundary conditions for the CFD analysis had different values for each patient. To date, studies have assumed that the boundary conditions are uniformly the same value for each study.

CFD analysis is based on several assumptions, which can be an intrinsic limitation of this analysis method. Nevertheless, CFD analysis is the underlying method for hemodynamic analysis, and we need to study the most important parameters in the CFD analysis of cerebral aneurysms.

WSS is among the most important hemodynamic parameters in CFD analysis and is a frictional force from the blood flow tangential to the arterial lumen. The relative difference in velocity between two parallel objects creates shear stress [13]. In the normal range (1.5–2.5 Pa) of WSS, endothelial function is regulated [14]. However, if the WSS is outside the normal range, histological changes related to the aneurysm may occur. Although WSS is conventionally associated with the natural history (formation, growth, and rupture) of cerebral aneurysms, the effect of the WSS on each natural history remains controversial.

Several studies utilizing animal models have identified a high WSS as an important parameter in aneurysm formation [15-21]. In addition, Can and Du [22] reported a strong positive correlation between elevated WSS and the location of aneurysm formation in their systematic review and meta-analysis. The authors included 19 studies that investigated WSS using CFD for geometrical models of intracranial aneurysms and found that high WSS was associated with formation and low WSS was associated with rupture of intracranial aneurysms [22].

Unlike the consensus that aneurysm formation occurs in regions with high WSS, the exact role of WSS in the growth and rupture of intracranial aneurysms is controversial. It is unclear whether a high or low WSS plays a principal role in growth and rupture. There are “high” and “low” WSS theories, both of which explain why the hemodynamic environment within the aneurysm interacts with the cellular elements of the aneurysm wall, resulting in further weakening. However, the differences revolve around the mechanisms that cause the weakening of the weakening [21].

There are several hemodynamic parameters in CFD analysis. Previous hemodynamic studies define several hemodynamic parameters associated with natural history of cerebral aneurysms. Among the various hemodynamic parameters, oscillatory shear index (OSI) and WSS gradient (WSSG) have demonstrated their role in aneurysm rupture risk analysis.

OSI is a dimensionless parameter that indicates how the direction of WSS changes at a specific location during a cardiac cycle.

Previous studies reported that a higher OSI was observed in ruptured than in unruptured aneurysms, or high OSI corresponded to the rupture point [27]. They suggested that low WSS and high OSI predict rupture risk of cerebral aneurysms [27, 28].

WSSG is defined as the spatial derivative of WSS along the direction of flow. It can be thought of as the change in WSS along the length of the vessel [14]. Accelerating flow creates a positive WSSG, while decelerating flow creates a negative WSSG. Meng et al. [28] explained that the formation of cerebral aneurysms occurs in regions exposed to high WSS with a positive WSSG in the mural cell-mediated pathway theory.

As shown in the defintions presented above, both OSI and WSSG are secondary parameters derived from the concept of WSS. Cho et al. [2] defined a novel parameter, the so-called combined hemodynamic parameter, associated with aneurysmal rupture. Similarly, it was derived from the perception of WSS.

A cerebral aneurysm is a structural change that occurs when the shape of blood vessels changes due to the influence of blood flow. However, because CFD analysis assumes rigid blood vessel walls, the effects of the blood vessels are ignored. In contrast, fluid-structure interaction (FSI) analysis assumes that the blood vessel is not rigid but deformable; therefore, the effects of the wall thickness and mechanical properties of the aneurysms can be evaluated in a more realistic manner. In FSI analysis, using the pressure induced by the blood flow calculated from CFD analysis, the deformation of a cerebral artery can be evaluated by calculating the strain (Fig. 3). Strain represents how much a cerebral artery is stretched. Higher strain indicates that the blood vessel is more severely stretched. FSI analysis involves complex computational techniques, and the results depend on the thickness or elastic modulus of the vessel, which is difficult to obtain using common diagnostic imaging devices [34]. Nevertheless, several hemodynamic studies have used FSI analysis [35-37].

Cho et al. [35] suggested in FSI analysis that strain was more effective than WSS in predicting the rupture risk of cerebral aneurysms. In addition, unlike WSS, which shows inconsistency as it pregresses from formation to growth and rupture, high strain continues to be involved in the natural history of cerebral aneurysms. Kim et al. [36] reported that strain has an important role in the formation of cerebral aneurysms.

Despite overcoming several intrinsic limitations of CFD analysis, FSI analysis still has some limitations. The material properties and wall thickness of the cerebral artery are greatly influenced by the deformation of the cerebral artery. However, because material properties and wall thicknesses vary from person to person, it is difficult to determine the material properties and wall thicknesses of individuals. Although there have been efforts to study the material properties and wall thickness of the cerebral artery, there is still a need for further research in this area.