# A Coupled Multiscale Approach to Modeling Aortic Valve Mechanics in Health and Disease
## 2 Materials and Methods
### Implementation of FE\({}^{2}\) Approach to Aortic Valve Leaflet Tissue
#### 2.2.2 Microscale Model (RVE)  
The geometric representation of the RVE is shown in Figure 6 via the mesh. Mixed-formulation four-node tetrahedral elements were used in an unstructured mesh generated with the open source package Iso2 mesh [40]. A coarse mesh (Figure 6 right) was chosen for computational efficiency. The VIC is approximated as an ellipsoid, with aspect ratios (in CRT coordinates): \(C/T=1.8\), \(C/R=1.3\), and a major axis length of 9.6 \(\upmu\)m. The VIC volume ratio (VR) is approximated from [PERSON] et al. [5] but allowed to vary. We assume that the cell behaves as a Neohookean material, i.e., (20) with \(C_{1m}=C_{2m}=C_{1f}=C_{2f}=0\), \(\mu=400\) Pa and [4]\(K=2.2\) MPa and is perfectly bonded with the ECM. The ECM material is given by [PERSON] et al. [39]:  
\[\psi=C_{1m}\big{\{} \exp\big{[}C_{2m}(I_{1}-3)\big{]}-1\big{\}}+\sum_{i=1}^{n_{f}} \frac{C_{1f}}{2C_{2f}}\bigg{\{} \exp\big{[}C_{2f}(j_{4}^{i}-1)_{+}^{3}\big{]}-1\bigg{\}} \tag{20}\] \[+c_{1}(I_{1}-3)+c_{2}(J^{2}-1)+c_{3}\ln(J).\]  
with the parameters defined in Table 2. We used Taylor boundary conditions; i.e., the deformation gradient is imposed everywhere in the domain. Let \(\mathbf{F}_{t}^{k}\) be the deformation gradient at time \(t\) and iteration \(k\) passed to an RVE (\(\Omega\)). Then, the displacement field is:  
\[\mathbf{u}_{t}^{k}(\mathbf{X})=(\mathbf{F}_{t}^{k}-\mathbf{I})\mathbf{X},\ \ \ \forall\mathbf{X}\in\Omega. \tag{21}\]  
In the spirit of the experiments, we fixed the macroscale model around the edges (with the exception of the symmetry boundary conditions) and applied a uniform pressure \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & \(\mathbf{C}_{1m}\) **(Pa)** & \(\mathbf{C}_{2m}\) **(-)** & \(\mathbf{C}_{1f}\) **(Pa)** & \(\mathbf{C}_{2f}\) **(-)** & \(\mathbf{\sigma}_{f}\) **(\({}^{\circ}\))** \\ \hline Fibrosa & \(4.72\times 10^{-2}\) & 6.7 & 16.31 & 43.19 & 0.64 \\ \hline Ventricularis & 0.25 & 0.39 & 1.51 & 3.63 & 9.71 \\ \hline Spongiosa & 0 & 0 & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of calibrated model parameters (see (20)). \(\mu=1.62\) kPa and \(K=2.2\) MPa. Figure 6: **Right**: Low-fidelity RVE mesh used for FE\({}^{2}\) computation; **Left**: High-fidelity RVE mesh used for post-processing calculation. Both meshes enforce the same volume ratio. Meshes generated and visualized with Iso2 mesh [40]. loading (Table 1) on the aortic side (with the curvature). Note that the large deformation necessitates the use of convecting \"follower\" pressure loads [33]. We computed the VICAR by tracking the deformation of nodes at the ends of the major and minor axes (circumferential and transmural directions, respectively) of the VIC during the loading. To compute the \"apparent\" aspect ratio, we tracked nodes at the ends of the major and minor axes of the elliptical projection of the ellipsoidal VIC on a cutting plane (Figure 7), similar to the procedure followed in the experiments. The cutting plane is a subset of the circumferential-transmural plane and convects with the deformed body to maintain the same reference configuration. The initial cutting plane is defined with a random (normal) perturbation from the center of the ellipsoid. Let \(\mathcal{P}_{0}\) define the reference cutting plane, then:  
\[\mathcal{P}_{t}=\mathbf{\varphi}_{t}(\mathcal{P}_{0}). \tag{22}\]  
We computed the VICAR in a post-processing step of the analysis. The deformation gradients at each quadrature point in space and time were applied to a higher fidelity RVE mesh (see Figure 6). Furthermore, we randomly perturbed the location and orientation of the VIC inside the RVE for each quadrature point. We assumed that the major axis of the ellipsoid is randomly oriented in a normal cone about the circumferential direction. Let:  
\[\mathbf{\mathcal{E}}=[\mathbf{a}_{c},\mathbf{a}_{r},\mathbf{a}_{t}], \tag{23}\]  
represent the axes of the ellipsoid with \(\mathbf{\mathcal{E}}^{T}\mathbf{\mathcal{E}}=\mathbf{\mathcal{E}}\mathbf{\mathcal{E}}^{T}= \mathbf{I}\).