# Using Autoregressive with Exogenous Input Models to Study Pulsatile Flows
## 2 FSI Simulation of a Simplified 2D Leaflet Structure
### FSI Simulation Results and Parametric Study  
Results obtained in the FSI simulations are shown in Figure 2 by representing nine stages describing the following situations:  
* Stage 1. Initial motionless stage with zero-flow conditions.
* Stage 2. Once flow rate \(\hat{G}\) starts to increase from Stage 1 to 2, the leaflets are forced to open as a consequence of a negative pressure difference (\(P_{1}-P_{0}<0\)).
* Stage 3. The flow rate and \(S_{v}\) reach their maximum values.
* Stage 4. The pressure difference is relatively zero, allowing the leaflets to remain opened.
* Stage 5. The flow rate starts to decrease.
* Stage 6. The increment on the pressure difference (\(P_{1}-P_{0}>0\)) causes a reduction in \(S_{v}\).
* Stage 7. Leaflet closure only occurs when there is no significant downstream flow rate, producing a pressure difference that is slightly positive.
* Stages 8-9. The last stages show that once there is no flow conditions, the pressure difference and opening area values become zero. This condition does not necessarily mean that the leaflets remain static. Once they are fully closed, they begin to oscillate until the forward stage begins once again.  
The design of the input curve avoids a back-flow behavior in the leaflets; consequently, this curve was designed to only provide cases with correct functionality. Despite the slight differences produced by the 2D simplification, previous results on the FSI simulation of leaflets [15] showed a qualitative agreement with the aforementioned experimental case [42]. Finally, in order to observe the behavior of the leaflets under different conditions, a parametric study was carried out. The variations consisted of changing the elastic modulus (\(E\)) in the solid domain and the period \(T\) of the input signal \(U_{f}^{*}(t^{*})\) in the fluid domain. The proposal was to have data of 45 different combinations, retrieved from five variations in \(E\) and nine variations in \(T\). All combinations shared the same values of \(\rho_{f}\), \(\mu\), \(\
u\), \(\rho_{s}\), \(d\), and \(l\) and possessed almost the same volume \(V\). The variations for \(E\) were chosen considering three different elastic conditions [46]. The first one is for situations where the leaflet material is particularly hard, with \(S_{v}/S<0.5\). The second considers the opposite, when the leaflet material is extremely soft, resulting in \(S_{v}/S>0.7\). The last condition considers an intermediate elastic behavior, leading to \(0.5\leq S_{v}/S\leq 0.7\). From a previos work [42], we considered an initial \(E=2.15\) MPa, two smaller values \(E=0.1\) MPa and \(E=1\) MPa, and two larger values \(E=3.16\) MPa and \(E=10\) MPa, in order to represent the three aforementioned conditions. More information regarding the mesh-size selection for both the fluid and solid domains, as well as a structural analysis for the leaflet model to obtain the initial conditions for the FSI simulations can be found in the Supplementary Material.  
Figure 2: **(a)** Each stage represents the results of different stages of the FSI simulation using \(U_{max}=0.0964\) m/s, \(T=1.5015\) s, and \(E=2.15\) MPa. **(b)** This figure indicates the information (red marks) of the previous nine stages measured for \(\hat{G}\), \(\Delta P\), and \(S_{v}\) as a function of \(t^{*}\). The maximum values obtained are \(\hat{G}_{max}\)=0.1446 \(\times\) 10\({}^{-3}\) m\({}^{3}\)/s, \(|P_{min}|\)=163.697 Pa, and \(S=0.03\) m.  
Regarding the variations in \(T\), we initially tested one of the experimental periods proposed where \(T=1.502\) s. Additionally, two more values were chosen from here: \(T=1\) s and \(T=3.0\) s. Finally, in order to have nine variations, intermediate values were selected. With these forty-five combinations, we conducted their respective FSI simulations, and the results showed the same behavior as the curves presented in Figure 2. The frequencies, and the corresponding periods, selected for this study are presented in Table 2.