## 2 Material and Methods
### Numerical Simulations  
CFD analysis of the four patient-specific cases were carried out in a dynamic regime. ANSYS Fluent (ANSYS Inc, Canonsburg, USA) was used to solve the Navier-Stokes governing equations of the fluid. The CT images belonging to the mid-diastolic phase of the cardiac cycle (equivalent to 60%) were used to build the numerical models of the four cases.  
Following sensitivity mesh-analysis, a volumetric grid with tetrahedral and hexahedral elements was generated for all the models. Final meshes counted 2,362,103 elements for the chicken wing model, 3,964,963 for the cauliflower model, 1,932,011 for the windsock model and 4,218,847 for cactus model. Hexahedral elements were used along the wall of the models, where a 5-layers inflation was generated.  
To take into account for the opening and closure actions of the valve during the cardiac cycle, a transient velocity outflow condition was assigned to the MV orifices. Following this strategy, the closure period of the MV, i.e., the ventricular systole, was simulated when a null velocity occurred, while the non-zero velocity values corresponded to the passage of the blood flow during the ventricular diastole, i.e., when the valve is open. A velocity profile representative of the atrial fibrillation (AF) condition was imposed at the MV section [18], while a transient pressure profile was assigned at the PVs (Figure 2). The pressure profile was generated based on a custom script taking into account for the MV velocity and the physiological pressure range of the LA (5-13 mmHg) during the cardiac cycle. The wall was assumed as rigid with no-slip condition.  
Figure 1: Example of volumetric CT segmentation with axial (**a**), sagittal (**b**) and coronal (**c**) planes and resulted 3D model (**d**). Identification of the area of the ostium (\(A_{o}\)) and of the centerline \(C(s)\) from \(P_{o}\), where \(P_{t}\) (**e**).  
Figure 2: CFD model of LA with LAA, PVs and MV, including inflow pressure boundaries (marked in red) and velocity outflow boundary (marked in blue) (**a**). PVs inflow pressure and MV outflow velocity curves during cardiac cycle (**b**).  
The blood was treated as Newtonian and incompressible, as widely accepted to model blood in cardiovascular problems [8; 9; 10; 11; 14; 19; 20] with a density of 1060 kg/m\({}^{3}\) and a viscosity of 0.0035 Pa s. A laminar model was implemented based on the calculation of the Reynolds number [11].  
All the simulations were run for four cardiac cycles, with each cardiac cycle lasting 0.8 s, to allow the full development of the flow. The time domain was discretized with a time step of 0.005 s. The hemodynamics analysis was performed in the last cycle to avoid the inclusion of nonlinear start-up effects.  
CFD simulations were evaluated in terms of velocity fields. Velocity and pressure patterns were visually analysed to assess the correct implementation of the atrial flow direction. In particular, velocity fields were analyzed in the LAA portion of the four cases. Three planes of analysis were created, corresponding to the ostium plane, a middle plane and a distal plane, and the mean through-plane velocity was computed.