# A Coupled Multiscale Approach to Modeling Aortic Valve Mechanics in Health and Disease
## 2 Materials and Methods
### Multiscale Modeling Framework  
We modeled the AV tissue within a classical continuum mechanics framework (see e.g., [29] or [30]), wherein we seek to solve the governing equations of motion for a body, i.e., the manifold with boundaries, subject to boundary conditions, specifying tractions and displacements. We define a one-parameter (time \(t\)) family of finite deformation maps \(\mathbf{\varphi}_{t}:\mathbb{R}^{3}\to\mathbb{R}^{3}\) of a hyperelastic body from a reference configuration (\(\mathcal{B}_{0}\)) to a current (deformed) configuration (\(\mathcal{B}_{t}\)) (see [PERSON] [31] or [PERSON] [29]); viz., \(\mathcal{B}_{t}=\mathbf{\varphi}_{t}(\mathcal{B}_{0})\). We define the deformation gradient \(\mathbf{F}=\
abla\mathbf{\varphi}\). Note that \(\mathbf{F}\) is a 2-point tensor mapping vectors from the reference manifold to the deformed manifold. We define the left and right Cauchy-Green deformation tensors by \(\mathbf{C}=\mathbf{F}^{T}\mathbf{F}\) and \(\mathbf{b}=\mathbf{F}\mathbf{F}^{T}\), respectively. We further define the associated strain tensors: the Green-Lagrange strain tensor \(\mathbf{E}=\frac{1}{2}(\mathbf{C}-\mathbf{I})\) for the reference configuration and the Almansi strain tensor \(\mathbf{e}=\frac{1}{2}(\mathbf{I}-\mathbf{b}^{-1})\) for the current configuration. \(\mathbf{I}\) is the identity tensor for vectors in \(\mathbb{R}^{3}\). The equilibrium deformation map at time \(t\) is the one that minimizes the potential energy (\(\Pi\)) of the elastic system subject to conservative traction loading \(\mathbf{\dot{t}}_{t}\):  
\[\mathbf{\varphi}_{t}^{eq}=\arg\inf_{\mathbf{\dot{\varphi}}_{t}}\Pi(\mathbf{\varphi}_{t}; \mathbf{\ddot{t}}_{t}). \tag{1}\]  
Under the assumption of hyperelasticity, the 1 st Piola-Kirchhoff stress of the system, \(\mathbf{P}\), is obtained from the Helmholtz free energy, \(\dot{\psi}\), of the material:  
\[\mathbf{P}=\frac{\partial\dot{\psi}}{\partial\mathbf{F}}. \tag{2}\]  
Although the solution to (1) is in general not unique, the polyconvexity (in the sense of Ball [32] along with suitable growth conditions) of the energy function guarantees the existence of a solution. We solve the problem with a standard finite element (FE) numerical procedure (Section 2.1). Our challenge is to specify \(\dot{\psi}\) such that the FE model is consistent with the observed experimental response. We can further define two additional stress tensors: the 2 nd Piola-Kirchhoff stress tensor \(\mathbf{S}\) and the Cauchy stress tensor \(\mathbf{T}\). The relationship between the stress tensors is:  
\[\mathbf{T}=J^{-1}\mathbf{F}\mathbf{P}=J^{-1}\mathbf{F}\mathbf{S}\mathbf{F}^{T}, \tag{3}\]  
where \(J=\det(\mathbf{F})\) is the Jacobian (of the deformation gradient). To use an FE approach to solve (1), let \(\partial\mathcal{B}_{u}\) and \(\partial\mathcal{B}_{t}\) denote the partitions of the boundary (\(\partial\mathcal{B}_{0}\)) of the body, \(\mathcal{B}_{0}\), where deformation and tractions are imposed, respectively, with \(\partial\mathcal{B}_{u}\cap\partial\mathcal{B}_{t}=\emptyset\), \(\overline{\partial\mathcal{B}_{u}\cup\partial\mathcal{B}_{t}}=\partial \mathcal{B}_{0}\).