# A Coupled Multiscale Approach to Modeling Aortic Valve Mechanics in Health and Disease
## 4 Discussion
### Summary and Outlook for Future Extensions  
In this work, we demonstrated the feasibility of using computational homogenization, or FE\({}^{2}\), for modeling the coupled, multiscale behavior of aortic valve (AV) tissue. The natural extension is to model the full aortic valve geometry with the time-resolved dynamics of the AV solid and fluid mechanics. We argue that multiscale modeling is necessary for understanding AV behavior and the method we presented provides a feasible way of achieving a fully coupled multiscale analysis for aortic valves. Furthermore, the RVEs can be used to develop cellular-driven models.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline Load (mmHg) & 1 & 2 & 4 & 60 & 90 \\ \hline Circumferential Strain Ratio & 0.96 & 1.19 & 1.32 & 1.26 & 1.21 \\ \hline Radial Strain Ratio & \(-\)0.39 & 0.14 & 0.31 & 0.42 & 0.41 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Average ratio of the circumferential and radial Almansi strains in the belly region seen in a leaflet with a calcific nodule relative to values seen in a healthy leaflet.  
We further utilized the multiscale AV leaflet model to study three calcification topologies as idealized representations of different stages of calcific aortic stenosis progression: early-stage nodules, a partial arc, and a mature arc pattern. In the former, we saw that the presence of the calcification nodules led to a larger circumferential Almansi strain and VIC aspect ratio that presumably drives further calcification growth in a positive-feedback loop manner. Furthermore, we noticed that the \"apparent\" aspect ratio, as measured by slicing a section of the tissue and observing the 2D aspect ratio of the cell cross-section, did not necessarily exhibit the true aspect ratio.  
In the more advanced calcification cases, we observed the disturbed biomechanics of the leaflet which presumably results in stenotic behavior. We observed a lower aspect ratio consistent with a natural \"saturation\" of the calcification in our full arc calcification computations. Our study, however, was limited to static in vitro loading, to validate it against experiments in the literature. A follow-up study should be performed with dynamic and cyclic loading using an in vivo configuration of the AV leaflet. Furthermore, the extension study should feature stochastic representations of the calcification topology, perhaps via strain-driven growth models [3; 18].  
Conceptualization, methodology, A.A.B., S.G., M.R.K.M.; software, A.A.B., S.G.; writing--original draft preparation, A.A.B.; writing--review and editing, S.G., M.R.K.M.; supervision, S.G., M.R.K.M.; funding acquisition, S.G., M.R.K.M. All authors have read and agreed to the published version of the manuscript.  
Financial support by the National Science Foundation through grant CMMI-1538707 is gratefully acknowledged. [PERSON]'s and [PERSON]'s participation in this work was supported in part by the [PERSON], [PERSON], and [PERSON] Chair in Engineering.  
This research used the SAVIO computational cluster resource provided by the Berkeley Research Computing program at the University of California, Berkeley (supported by the UC Berkeley Chancellor, Vice Chancellor for Research, and Chief Information Officer). [PERSON] and [PERSON] thank Professor [PERSON] for their assistance with the finite element code FEAP.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
2D two dimension(s)(al)  
3D three dimension(s)(al)  
AV aortic valve  
CAS calcified aortic stenosis  
CRT circumferential, radial, and transmural  
CT computed tomography  
ECM extracellular matrix  
FE finite element  
FE2 finite element method squared, computational homogenization  
OOP out-of-plane  
RVE representative volume element  
VIC valvular interstitial cells  
VICAR valvular interstitial cell aspect ratio  
VR volume ratio