# Fractional Flow Reserve (FFR) Estimation from OCT-Based CFD Simulations: Role of Side Branches
## 2 Material and Methods
### Boundary Conditions  
A static pressure and a zero gradient boundary conditions were specified at the inlet, which are described by Equations (5) and (6).  
\[P=P_{in} \tag{5}\]\[\frac{dU}{dn}=0 \tag{6}\]  
For the idealized models, \(P_{in}\) was defined as 90 mm Hg. For the patient-specific models, the pressure was measured using the guide catheter. In Equation (6), \(n\) denotes the normal vector at the boundary. The artery wall was considered as stationary, and a zero-wall velocity boundary condition was imposed at the artery wall.  
\[U_{wall}=0 \tag{7}\]  
At each outlet, a resistance-based boundary condition was imposed to mimic the coronary microvascular resistance (CMVR) downstream from each outlet. Such a boundary condition computes the outlet pressure (\(P_{out}\)) based on the flowrate (\(Q_{out}\)) and the defined resistance (\(R_{out}\)).  
\[P=P_{out}=Q_{out}R_{out} \tag{8}\]  
An estimation of the appropriate resistance (\(R_{out}\)) value at each outlet is required for an accurate computation of FFR. However, the direct measurement of the resistance of the microvascular bed at each outlet is not available and difficult to measure. Hence, to estimate the CMVR at each outlet, the structured tree model approach suggested by [PERSON] et al. (2010) was used. This approach has been previously employed by several studies to simulate coronary blood flow dynamics and FFR (21,22) and has shown agreement with corresponding perfusion resistance measurements (22). This model suggests an asymmetric, fractal-like vascular tree structure based on morphological laws of vascular branching. The root of the vascular tree originates from an outlet and bifurcates into daughter vessels with diameters scaled by factors of \(\alpha\) and \(\beta\). Each branch length is derived from a length/diameter ratio (\(\gamma\)). The branching is terminated when a vessel diameter is less than a minimum diameter (\(d_{min}\)), which represents the arteriolar diameter of the vascular bed (20).  
For a branch segment of diameter (\(d_{i}\)), the corresponding resistance (\(R_{i}\)) is derived as (23),  
\[R_{i}=\frac{128\mu\gamma}{\pi d_{i}^{3}} \tag{9}\]  
In addition, considering the parallel branching structure, the resistance (\(R\)) at any bifurcation is calculated as,  
\[\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}} \tag{10}\]  
where the resistances of the daughter vessels are denoted by \(R_{1}\) and \(R_{2}\). Subsequently, considering [PERSON]'s law and treating each branch as circuit elements, the total resistance of the vascular tree can be calculated. In the current study, the following parameters were used: (\(\alpha=0.9\), \(\beta=0.5\), \(\gamma=25\)(24), \(D_{min}=50\)(20)). To simulate the hyperemic conditions, the calculated resistance is multiplied by a factor of 0.24, which corresponds to the experimental observation with intravenous administration of adenosine 140 \(\mu\)g/kg/min (25).  
For the patient-specific models, the resistances at each side branch were determined using the structured tree approach, while the resistance at the distal end of the main vessel was determined to match the pressure wire measurement. The same distal resistance value was imposed when the model was considered without side branches.