## 2 Materials and Methods
### Fiber-Gel Finite Element Model
The stress \(\sigma_{y}\) is converted into an equivalent shear modulus \(\upmu\)', to which a gel shear modulus \(\upmu\) is added for transversally isotropic gel properties. In combination, the viscoelastic model becomes a fiber-gel model. The active properties of all intrinsic muscles are determined by a modified Kelvin model ([43], p. 76). The modified Kelvin model is governed by maximum isometric active stress, \(\sigma_{m}\). Table 1 shows the parameters of Equation (6) and \(\sigma_{m}\) used in this study. The full details of the viscoelastic properties obtained from measurements on multiple species' biological tissues are given in ([43], Chapter 2). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **CT** & **LCA** & **TA** & **IA** & **PCA** & **LIG** & **SLLP** \\ \hline \(\sigma_{0}\left(\text{dyn}/\text{cm}^{2}\right)\) & \(2.2\times 10^{4}\) & \(3\times 10^{4}\) & \(2\times 10^{4}\) & \(2\times 10^{4}\) & \(5\times 10^{4}\) & \(2\times 10^{4}\) & \(0.2\times 10^{4}\) \\ \hline \(\sigma_{2}\left(\text{dyn}/\text{cm}^{2}\right)\) & \(5\times 10^{4}\) & \(59\times 10^{4}\) & \(1.5\times 10^{4}\) & \(30\times 10^{4}\) & \(55\times 10^{4}\) & \(0.15\times 10^{4}\) & \(1.5\times 10^{4}\) \\ \hline B & 7 & 4 & 6.5 & 3.5 & 5.3 & 12.8 & 6.5 \\ \hline \(\varepsilon_{1}\) & \(-0.9\) & \(-0.9\) & \(-0.9\) & \(-0.9\) & \(-0.9\) & \(-0.9\) & \(-0.9\) \\ \hline \(\varepsilon_{2}\) & \(-0.06\) & \(0.05\) & \(-0.5\) & \(0\) & \(0.1\) & \(-0.5\) & \(-0.2\) \\ \hline \(\sigma_{m}\left(\text{dyn}/\text{cm}^{2}\right)\) & \(400\times 10^{4}\) & \(140\times 10^{4}\) & \(180\times 10^{4}\) & \(140\times 10^{4}\) & \(96\times 10^{4}\) & & \\ \hline \end{tabular} CT-circthyroid; LCA-lateral cricthyroid; TA-thyroid; IA-interarytenoid; PCA-posterior cricoarytenoid; SLLP-superficial layer of lamina propria.  
\end{table} 
Table 1: Parameter values used in fiber-gel finite element model for modeling the five intrinsic laryngeal muscles (CT, LCA, TA, IA, and PCA) along with vocal ligament (LIG) and SLLP layers. Figure 2: (**a**) Right vocal fold geometry in the Fiber-Gel Model, and (**b**) coronal section of the right vocal fold in finite element implementation. Purple is the superficial layer of the lamina propria (SLLP), yellow is the ligament, and red is the throarytenoid (TA) muscle. Each vocal fold consisted of three layers: SLLP, vocal ligament, and TA muscle. A two-dimensional finite-element mesh as seen in Figure 2b was constructed for each of \(M\) = 15 coronal slices in the posterior-anterior direction of each vocal fold, following [48]. The elements were triangular, allowing the shape functions (interpolations within the elements) to be computed analytically from the nodal points. This choice was based on speed of computation. There were two triangular elements per cell, with \(L\) = 10 columns of cells and \(N\) = 6 rows of cells in the coronal plane ([43], pp. 214-231). In solving the matrix equations for (L+1)*(N+1) nodal displacements, aerodynamic forces were applied at the open boundaries and string (fiber) forces were applied between the slices. The material in each slice was considered incompressible and transversally isotropic [48]. Thus, one elastic constant (the transverse shear modulus \(\mu\) = 0.5 kPa for all the layers) and a viscosity \(\eta\) (= 0.1 Pa-s for all the layers) defined the gel property. A second shear modulus \(\mu^{\prime}\) defined the fiber property (Equation (6) above). Boundaries were fixed at anterior, lateral, and posterior surfaces. Boundaries were free to move with pressure distributions on inferior, medial, and superior surfaces. The lateral boundary was curved as shown in Figure 2a. The vibrational displacement was superimposed on the postural displacement. A Bernoulli approach was used to calculate glottal pressures up to the minimum glottal diameter, from which point jet flow was assumed in the remainder of the glottis ([43], Chapter 7).