# Fractional Flow Reserve (FFR) Estimation from OCT-Based CFD Simulations: Role of Side Branches
## 3 Results and Discussion
### FFR and Flow Distribution in Idealized Artery Models
The parameters \(C_{1}\) and \(C_{2}\) are constants that depend on the dimensions of the stenosis, which will significantly increase along with stenosis severity, further leading to a large pressure drop. Based on Equation (11), the flow resistance through the stenosis can be derived as,  
\[R(q)=\frac{dP}{q}=C_{1}+C_{2}q \tag{12}\]  
For the side branch located at the upstream of the stenosis, the resistance \(R2\) is connected in a parallel configuration to the resistance in the main vessel and does not affect the FFR estimation. The FFR in both model 1 and model 2 can be calculated based on \(R1\) and \(R(q)\):  
\[FFR=\frac{V1}{V}=\frac{R1}{R1+R(q)} \tag{13}\]  
As the flow in the main vessel is governed by the inlet voltage (\(V\)) and distal resistance (\(R1\)), both models have an equal flowrate in the main vessel and an equal \(R(q)\). Hence, both models lead to the same FFR. Furthermore, in these models, the flow into the side branch is determined by the inlet voltage \(V\) and \(R2\), which are independent of \(R(q)\) and \(R1\)  
Figure 5: Simplified electrical circuit representation of the idealized models: (**a**) model 1: without side branch; (**b**) model 2: with side branch before the stenosis; (**c**) model 3: with side branch after the stenosis. These derivations agree well with the CFD-derived results for model 1 and model 2, where insignificant deviations in FFR and flowrate in the main vessel (outlet1) are observed. For model 3, when the side branch is located at the downstream after the stenosis, the distal resistances \(R1\) and \(R2\) are connected in parallel, which can be represented as a total resistance (\(Rt\)) denoted as,  
\[Rt=\frac{R1R2}{R1+R2} \tag{14}\]  
Consequently, the FFR can be calculated by Equation (15). \[FFR=\frac{V1}{V}=\frac{Rt}{R(q)+Rt}=\frac{R1}{R(q)\cdot\left(\frac{R1}{R2}+1 \right)+R1} \tag{15}\]  
In comparison to Equation (13), the term \(\frac{R1}{R2}\) in Equation (15) contributes to a decrease in the FFR value. According to [PERSON]'s law of flow distribution, the distal resistance of an artery with a diameter (\(d\)) is assumed to be proportional to \(d^{-3}\)[28]. Hence, as the main vessel has a larger diameter, the term \(\frac{R1}{R2}\) is usually less than unity. In addition, \(\frac{R1}{R2}\) decreases as the diameter of the side branch decreases and becomes negligible (<4%) if the side branch diameter is less than one third of the main vessel diameter. Moreover, it should be noted that the term \(R(q)\) depends on the flowrate, and for model 3, the flowrate across the stenosis was higher than the other models, as shown in Figure 4d. Hence, with the increase in flowrate across the stenosed region, the increased \(R(q)\) contributes to further decrease the FFR. The FFR and flow distribution results from the CFD simulations further validate these observations. Overall, these results suggested that the impact of side branch flow on FFR estimation is more critical when the stenosed region is located before side branches.