## 2 Materials and Methods
### 0D Closed-Loop Windkessel Circulatory  
To integrate the pre- and after-load circulatory systems into the 3D LV model, the outlet and inlet boundaries were linked to a 0D closed-loop circulatory circuit (Figure 2b), which was developed by [30]. The preservation of the overall blood volume was achieved by solving Equations (6)-(8) that describe the rate at which the volume changes in each compartment of the closed-loop analogue circuit.  
\[\frac{\mathrm{d}V_{LA}}{\mathrm{d}t}=Q_{ven}(t)-Q_{in}(t) \tag{6}\]  
\[\frac{\mathrm{d}V_{art}}{\mathrm{d}t}=Q_{ao}(t)-Q_{per}(t) \tag{7}\]  
\[\frac{\mathrm{d}V_{ven}}{\mathrm{d}t}=Q_{per}(t)-Q_{ven}(t) \tag{8}\]  
The inflow into the LV, represented as \(Q_{mv}\), was determined by integrating the velocity of the normal flow across the boundary of the inlet (MV annular plane). Moreover, the volume of the LV, denoted as \(V_{LV}\), was calculated by integrating the fluid domain inside the ventricle. The flow rates of blood volume within various compartments of the circulatory model were represented as follows:  
\[Q_{ao}=\begin{cases}\frac{P_{LV,pa}-P_{art}}{R_{ao}}&P_{lv,ao}\geqslant P_{art }\\ 0&P_{LV,ao}<P_{art}\end{cases} \tag{9}\]  
\[Q_{per}=\frac{P_{art}-P_{ven}}{R_{per}} \tag{10}\]  
\[Q_{ven}=\frac{P_{ven}-P_{LA}}{R_{ven}} \tag{11}\]  
The venous pressure, \(P_{ven}\), and arterial pressure, \(P_{art}\), were calculated based on the venous and arterial volumes using Equations (12) and (13). In addition, \(P_{LV,ao}\) denoted the outlet boundary pressure.  
\[P_{ven}=\frac{V_{ven}-V_{ven,0}}{C_{ven}} \tag{12}\]  
\[P_{art}=\frac{V_{art}-V_{art,0}}{C_{art}} \tag{13}\]  
The initial volumes of the venous and arterial systems are represented as \(V_{ven,0}\) and \(V_{art,0}\), respectively. The venous system's compliance is denoted as \(C_{ven}\), while the arterial system's compliance is denoted as \(C_{art}\). The pressure of the LA, \(P_{LA}\), was calculated using a prescribed function based on its volume \(V_{LA}\):  
\[P_{LA}=P_{la,0}+E_{LA}\left(V_{LA}-V_{LA,0}\right) \tag{14}\]  
where  
\[E_{LA}=E_{LA,min}+e(t)\left(E_{LA,max}-E_{LA,min}\right) \tag{15}\]  
and  
\[e(t)=\begin{cases}\frac{1}{2}\Big{(}1-\cos(\frac{\pi(t-0.74)}{0.16})\Big{)},& 0.74s<t\leqslant 0.90s\\ \frac{1}{2}\Big{(}1+\cos(\frac{\pi(t-0.90)}{0.16})\Big{)},&t>0.90s\\ \frac{1}{2}\Big{(}1+\cos(\frac{\pi(t-0.90)}{0.16})\Big{)},&0\leqslant t<0.10s \end{cases} \tag{16}\]  
The function \(e(t)\) denotes a normalized time-varying elastance function that characterizes the contraction of the LA. The parameters of the 0D closed-loop circulatory circuit,originally proposed by [30], have been utilized with certain modifications for this study (see Table 1).  
The generated 3D generic model utilized a fully coupled direct solver (PARDISO) and an automated Newton technique. In order to achieve stability in the Navier-Stokes equations, a hybrid approach using both streamline and crosswind diffusion numerical approaches is employed [31]. The simulation results were recorded at 2 ms intervals. The solution for time-dependent problems was configured with absolute and relative tolerances both set to 0.01. The mesh elements of the 3D model were created using a combination of prism, tetrahedral and pyramidal elements using COMSOL Multiphysics software (COMSOL AB, Stockholm, Sweden). To accurately capture the fluid-wall interactions near the myocardium, boundary layers consisting of two thickness elements were applied. Using the same configuration for the model, simulations were conducted using two different meshes, each with a maximum element size of 3 mm and 1.5 mm. The discrepancies in the measurements of pressure and velocity were found to be less than 5%. Consequently, our analysis led us to determine that a mesh with a maximum element size of 3 mm would be appropriate for all simulations pertaining to this 3D model.