# A Coupled Multiscale Approach to Modeling Aortic Valve Mechanics in Health and Disease
## 3 Results
### Fe\({}^{2}\) Result: Healthy Valve  
Simulations were tested on an in-house 32-node cluster, Iron, and run on 60 nodes of the 8109 cluster SAVIO ([[http://research-it.berkeley.edu/services/high-performance-computing](http://research-it.berkeley.edu/services/high-performance-computing)]([http://research-it.berkeley.edu/services/high-performance-computing](http://research-it.berkeley.edu/services/high-performance-computing)) accessed on 1 September 2021). Each simulation ran for approximately 20 h on the latter. The deformed shapes of the AV leaflet simulated under the five steps of the loading protocol defined in Table 1 can be seen in Figure 11. We computed an average Jacobian in all the elements throughout the loading of \(J=\det\mathbf{F}\approx 1\), within 1.5%, indicating the desired quasi-incompressible behavior. _Valvular interstitial cell aspect ratio (VICAR)_: the VICAR plots are computed from the RVEs at every Gauss point in the elements along the symmetry plane: a total of \(144\times 8=1152\) RVEs. The average was reported, and where shown, error bars represent standard deviation. Note that the authors of the experimental data did not specify whether their error bars represented one standard deviation or one standard error. Throughout this discussion, a two-tailed \(t\)-test was used to test significant differences in means. We used the term _significant_ to indicate that the \(p\)-value is \(p<0.05\) for the \(t\)-test, i.e., the difference in means is statistically significant. Figure 10: Convergence study for the macroscale mesh. Each curve represents displacements (in the belly region of the leaflet) at the load steps in Table 1. The mesh size represented by the middle point was chosen for computational considerations. Simulated VICAR ratios for \(\sigma_{\theta}=5^{\circ}\) and the volume ratio VR = 0.01 were calculated for different transvalvular pressures (see Figure 12). In the figure, the red curve indicates \"apparent\" VICAR and the blue curve is the actual VIC aspect ratio (see Figure 7 and the corresponding discussion). The black curve is experimental data from [5]. We then examined the effect of varying the parameters \(\sigma_{\theta}\) and the volume ratio (VR) on the \"apparent\" VICAR (see Figure 13). We only observed a _significant_ difference in the \"apparent\" VICAR at \(\sigma_{\theta}=20^{\circ}\) vs. \(\sigma_{\theta}=\{5^{\circ},10^{\circ}\}\), with a remarkable decrease in the former. No _significant_ variation was found when varying VR over two orders of magnitude. We then briefly looked at the initial cusp curvature defined by (19). Figure 14 demonstrates the variation of the VICAR with respect to \(y_{0}\), as a fraction of the size parameter \(r\) (e.g., an offset of \(10\%\implies y_{0}=0.1r\)). _Significant_ but small differences are observed throughout. We note that for large values of \(y_{0}\), convergence issues in the FE problem were encountered. Finally, Figure 15 demonstrates the VICAR in each layer. _Significant_ differences were observed everywhere except between the spongiosa and ventricularis at low pressures. The fibrosa demonstrates a larger VICAR, consistent with experimental findings. Figure 11: Leaflet deformation under the load protocol defined in Table 1. Contours are the 11 Almansi strain component (non-dimensional) in the global laboratory frame. Figure 12: VICAR result as an average of 1156 RVEs measured along the radial direction in the center of the leaflet. One standard deviation’s error bars are shown. Red curve indicates “apparent” VIC aspect ratio as measured _a la_ Section 2.2.2 and a blue curve is the actual VIC aspect ratio. The black curve is the experimental data from [PERSON] [5] with undefined error bars. Figure 13: The effect of the VIC orientation and size on the “apparent” aspect ratio.