## 2 Methods  
### Multiscale Modeling Approach  
The multiscale model in this study was composed of two components: (1) the three dimensional (3D) patient-specific ACoAA model with anterior cerebral circulation; and (2) the efferent arteries of the 3D model were coupled with zero-dimensional (0D) models that represent the distal resistance and compliance of the cerebral vascular system. The patient-specific 3D model of anterior cerebral circulation with an ACoAA was reconstructed based on computed tomography angiography (CTA) images of a patient with ACoAA. Different degrees of stenosis, including severe stenosis (75%), moderate stenosis (50%) and mild stenosis (15%), were artificially modeled and inserted into the left ICA (LICA) of the 3D model to represent the revascularization progress, and the non-stenoic model served as the post-operative control (Figure 1). Figure 1: Models of the anterior cerebral circulation with anterior communicating artery aneurysm (ACoAA): (**a**) severe stenosis (75%); (**b**) moderate stenosis model (50%); (**c**) mild stenosis model (15%); and (**d**) non-stenoic model. According to the North American Symptomatic Carotid Endarterectomy Trial (NASCET) [26], the definition of stenosis degree was given as Equation (1). \[\text{S}=\left(1-\frac{d_{s}}{d_{n}}\right)\times 100\% \tag{1}\]  
where S is the stenosis degree, \(d_{s}\) is the ICA lumen diameter of the stenosis segment and \(d_{n}\) is the ICA lumen diameter of the normal segment. To compute the \(d_{n}\), the cross-sectional area of the distal joints of stenosis models and the patient-specific model were calculated in MIMICS software (Materialise Inc., Leuven, Belgium) first, then the \(d_{n}\) could be calculated from the equation of the circle area. The blood flow in arteries was assumed to be incompressible Newtonian fluid. The 3D flow domain was governed by the mass conservation equation (Equation (2)) and unsteady Navier-Stokes equation (Equation (3))  
\[\
abla\cdot V=0 \tag{2}\]  
\[\rho\left(\frac{\partial V}{\partial t}+\left(V\cdot\
abla V\right)\right)=- \
abla p+\mu\
abla^{2}V \tag{3}\]  
where \(\rho\) is the blood density, \(V\) is the velocity vector, \(p\) is the pressure and \(\mu\) is the dynamic viscosity. The arterial wall was assumed as rigid for the 3D model and no-slip wall boundary conditions were imposed at the wall boundary. The dynamic viscosity and density of the blood were 0.004 Pa-s and 1060 kg/cm\({}^{3}\), respectively. Due to the limited resolution of clinical imaging techniques and restricted computational resources, 3D modeling of the entire cerebral circulation is still a challenge. Alternatively, the 0D Windkessel model has been extensively utilized in hemodynamics studies of the cerebral vascular system. In the 0D model, a set of lumped parameters were set to represent the arterial tree distal to their 3D parental vessel. The lumped parameters including proximal resistance (\(R_{p}\)), compliance (C) and distal resistance (\(R_{d}\)), reflect the proximal viscous resistance, arterial wall elasticity and the resistance of the capillaries and venous circulation, respectively. Equation (4) gives the differential description of the 0D model. \[Q_{in}=C\frac{d}{dt}\left(P_{in}-\left(Q_{in}R_{p}+P_{c}\right)\right)+\frac{ P_{in}-\left(Q_{in}R_{p}+P_{out}\right)}{R_{d}} \tag{4}\]  
where \(Q_{in}\), \(P_{in}\) and \(P_{out}\) are the proximal flow rate, proximal pressure and terminal pressure of the 0D model, respectively. Table 1 listed the values of lumped parameters. To set matching conditions in the stenotic cases, the volumetric flow rates at the inlets were adjusted based on the stenosis degree to maintain the physiological pressure level. The pressure was maintained in all the cases studied. Figure 2 illustrates the scheme of boundary conditions.