## 2 Materials and Methods
### Fluid-Structure Interaction (FSI)  
FSI is a branch of mechanics that combines computational fluid mechanics and computational solid mechanics, used to study the interaction between solid and fluid domains. During the numerical simulation of the coronary arteries, the pulsating blood flow causes periodic deformation or movement of the vessel wall, which in turn has an effect on the blood flow. The Arbitrary Lagrangian-Eulerian (ALE) is a common method for solving FSI problems, ensuring that the boundaries and interfaces between the fluid and solid domains are accurately traced during the iterative process.  
In the fluid domain, the continuity equation and momentum equation are as follows:  
\[\
abla\cdot\boldsymbol{v}=0 \tag{3}\]  
\[\rho_{f}\Big{(}\frac{\partial\boldsymbol{v}}{\partial t}+\big{(}\boldsymbol{v} -\boldsymbol{v}_{g}\big{)}\cdot\
abla\boldsymbol{v}\Big{)}=-\
abla p+\
abla \cdot T \tag{4}\]  
where the vector operator \(\
abla=i\frac{\partial}{\partial x}+j\frac{\partial}{\partial y}+k\frac{ \partial}{\partial z};\boldsymbol{v}\) is the fluid velocity vector, m/s; \(\boldsymbol{v}_{g}\) is the moving grid movement velocity, m/s; \(p\) is the blood pressure, Pa; \(\rho_{f}\) is the blood density, kg/m3; and \(T\) is the stress tensor, Pa.  
In the solid domain, the conservation equation derived from [PERSON]'s second law is as follows:  
\[\rho_{s}\frac{\partial^{2}\boldsymbol{d}_{s}}{\partial t^{2}}=\
abla\cdot \boldsymbol{\sigma}_{s} \tag{5}\]  
where \(\rho_{s}\) is the vessel wall density, kg/m3; \(\boldsymbol{d}_{s}\) is the vessel wall displacement vector, m; and \(\boldsymbol{\sigma}_{s}\) is the stress tensor, Pa.  
The governing equation for the fluid-structure interaction is:  
\[\left\{\begin{aligned} \sigma_{s}\boldsymbol{n}_{s}& =\boldsymbol{\sigma}_{f}\boldsymbol{n}_{f}\\ \boldsymbol{v}_{s}&=\boldsymbol{\sigma}_{f}\\ \boldsymbol{d}_{s}&=\boldsymbol{d}_{f}\end{aligned}\right. \tag{6}\]  
where \(\boldsymbol{n}\) is the unit normal vector; \(\boldsymbol{v}\) is the velocity vector, m/s; \(\boldsymbol{d}\) is the displacement vector, m; and \(s\) and \(f\) represent the solid domain and fluid domain, respectively.  
Numerical simulations were performed in ANSYS Workbench commercial software. The finite volume method (FVM) was used in Fluent for the fluid mechanics computations, the finite element method (FEM) was used in Transient Structural for the structural mechanics analysis, and System Coupling was used for the coupling between both. The SIMPLE  
Figure 4: Boundary conditions: velocity profile prescribed at the inlet (LM) and pressure profile imposed at the outlets (LCX, DB, and LAD).  
algorithm was used for the coupling of the pressure-velocity terms. The Second Order Implicit Euler Scheme and the Second Order Upwind Scheme were adopted for achieving the temporal and spatial discretization of the equations, respectively. For each coronary model, transient flow simulations were performed for three cardiac cycles, and the results of the last cardiac cycle were used for hemodynamic analysis in order to eliminate the effect of the flow instability at the beginning of the calculation. The total number of steps per cardiac cycle was 160, and the time step was set to 0.005 s. The solution calculation was completed in each time step until the residuals of the coupled system were less than the specified residuals. In this work, the residual was set as \(10^{-5}\).