## 2 Materials and Methods
### LV Mechanics and Fluid Formulation  
The transverse isotropic hyperelastic model proposed by [PERSON] and [PERSON] (1970) was utilized to describe the passive reaction of the LV (see Equation (1)).  
\[\psi=\underbrace{\frac{a_{i}}{2b_{i}}\ exp(b(I_{1}-3))}_{\psi_{iso}}+\underbrace{ \frac{a_{f}}{2b_{f}}\ exp(b_{f}(I_{4f}-1)^{2}-1)}_{\psi_{f}}+\underbrace{\frac {1}{2}\ k(J-1)ln(J)))}_{\psi_{vol}} \tag{1}\]  
The strain energy function \(\psi\) can be divided into three components: \(\psi_{iso}\) for isotropic behavior, \(\psi_{f}\) for fiber behavior, and \(\psi_{vol}\) for volumetric behavior. Additionally, the first isochoric invariant of the right Cauchy-Green tensor, \(\mathbf{C}\), is denoted by \(I_{1}\). The fourth isochoric invariant, \(I_{4f}\), accounts for transverse isotropy and is defined as \(\vec{F}\cdot(\mathbf{C}\vec{F})\). Here, \(\mathbf{F}\) represents the orientation of the local fiber unit vector. The determinant of the deformation gradient tensor is symbolized as \(J\) in the \(\psi_{vol}\) component. It should be emphasized that cardiac fibers have an impact on the overall strain-energy function, \(\psi\), only when the value of \(I_{4f}>1\)([PERSON], 2018). Small portion of LV apex, as shown in Figure 1a, were designated to have isotropic characteristics. Therefore, only the strain-energy associated with isotropic deformation, denoted as \(\psi_{iso}\), was applied to this segment. The parameters utilized in \(\psi\) are adopted from ([PERSON], 2018), where the isotropic parameters, \(a_{i}\) and \(b_{i}\), are set to 2.280 KPa and 9.726, respectively. \(\psi_{f}\), \(a_{f}\) and \(b_{f}\) in \(\psi_{f}\) are set to 1.685 KPa and 15.779, respectively. \(k\) in the volumetric part is set to 250 KPa.  
The active stress associated with muscular contraction, denoted as \(T_{A}\) (Equation (3)) and integrated into the second Piola-Kirchhoff stress tensor, \(\mathbf{S}\), as follows:  
\[T_{A}=K_{TA}\begin{cases}e^{[-(6t-1.8)^{4}]}&t<0.28s\\ &\\ e^{[-(1.8t-0.504)^{2}]}&t\geqslant 0.28s\end{cases} \tag{2}\]  
\[\mathbf{T}=\frac{\partial\psi}{\partial\mathbf{E}}+T_{A}[\frac{\vec{S}\otimes \vec{S}}{2.5}+\frac{\vec{N}\otimes\vec{N}}{2.5}+\vec{F}\otimes\vec{F}] \tag{3}\]  
where \(\mathbf{E}\) in Equation (3) corresponds to the Green-Lagrange strain tensor, \(K_{TA}\) in Equation (2) denotes the amplitude of contractile strength and was set to 100 kPa ([PERSON], 2018; [PERSON], 2018). The ventricular systole begins at a time of 0.28 s. Based on the results of ([PERSON], 2018), it is significant to notice that the active stress \(K_{TA}\) contributes only 40% in \(\vec{S}\) and \(\vec{N}\) compared to the active tension in the \(\vec{F}\), as shown in Equation (3). The velocity and pressure of the blood flow were determined by solving the incompressible Navier-Stokes equation (Equation 4) and the mass continuity equation (Equation (5)), respectively.  
\[\rho_{f}\frac{\partial\vec{u}_{f}}{\partial t}+\rho_{f}(\vec{u}_{f}\cdot\
abla _{x})\vec{u}_{f}=\
abla_{x}\cdot(-p\mathbf{I}+\mu_{f}(\
abla_{x}\vec{u}_{f}+( \
abla_{x}\vec{u}_{f})^{T})) \tag{4}\]  
\[\
abla_{x}\vec{u}_{f}=0 \tag{5}\]  
The velocity of the fluid in the LV cavity is represented by \(\vec{u}_{f}\), while the pressure is denoted by the symbol \(p\). In Equations (4) and (5), the symbol \(l\) denotes the identity matrix, while \(\
abla_{x}\) represents the spatial gradient, defined as \(\
abla_{x}=(\frac{\partial}{\partial x},\frac{\partial}{\partial y})^{T}\). The blood's density and viscosity, represented as \(\rho_{f}\) and \(\mu_{f}\), respectively, were set to 1060 kg/m\({}^{3}\) and \(4.71\times 10^{-3}\) Pa s. The two-way coupling framework accurately captured the interaction between blood and LV structure. More precisely, the velocity of the fluid, denoted as \(\vec{u}_{f}\), was made equal to \(\frac{\partial\vec{u}_{f}}{\partial t}\) (velocity at the LV structure surfaces). The Yeoh smoothing method was utilized to determine the displacement of the deformed mesh, thus facilitating a more consistent deformation of the mesh.