## 2 Models
### Geometric Models  
Figure 1a shows the geometric models of the vocal folds, SGS, supraglottal tract, and subglottal tract, as well as the dimension of the computational domain. The configuration of the supraglottal airway was generated based on a neutral area function reported in [PERSON] [25], which is the mean area function of 12 vowels of a male speaker. According to another study by [PERSON] [26], the first and second natural frequencies (F1 and F2) of the mean area function are similar to those from a uniform tube, but higher frequencies are more subject-specific. The overall length of the supraglottal airway centerline is 17.4 cm. The geometries of the ventricle, false vocal folds, and subglottic tract were superimposed to the tract based on CT scans of a male subject larynx [27].  
The geometry of the vocal fold (Figure 1b) was generated by using a mathematical model proposed by [PERSON] and [PERSON] [28] that matches physiologically realistic key parameters and features (e.g., shape of medial surface, anterior-posterior shape variation, and the subglottal angle). The model assumes a three-layer structure of the vocal fold including the cover, ligament, and body, as shown in Figure 1b. The thickness of the cover and ligament layer was set as 0.5 and 1.1 mm, respectively, based on the average values of human measurement [29]. The vocal fold tissues were modeled as viscoelastic transversely isotropic materials. The material properties of each layer are listed in Table 1. These valueswere adopted from previous studies which showed how to reproduce typical vocal fold dynamics [24; 30; 31].  
The subglottic tract was extended downward by 6.6 cm to incorporate the SGS. An idealized geometry of the SGS was created by using a cosine function (Figure 2a). Specifically, for a given surface point \((x_{1},\ y_{1},\ z_{1})\) at the SGS region where \(x_{1},y_{1},\) and \(z_{1}\) are the original coordinates of the point, the new coordinates \((x_{s},\ y_{s},\ z_{s})\) of the point after adding the SGS were determined by using the equation below:  
\[\begin{split} x_{s}&=x_{1}-x_{1}\times\frac{a}{2} \times\left(1-cos\big{(}2\pi\frac{y_{c}}{L}\big{)}\right)\\ z_{s}&=z_{1}-z_{1}\times\frac{a}{2}\times\left(1- cos\big{(}2\pi\frac{y_{c}}{L}\big{)}\right)\\ y_{s}&=y_{1}\end{split} \tag{5}\]  
where \(L\) is the SGS length and was set as 1.0 cm, which is the most common SGS length [32]; \(y_{c}\) is the distance of the lower bound of the SGS to the \(y\) component of the point (\(y_{1}\)); and \(a\) is the percentage reduction of the hydraulic diameter of the airway cross-section due to the SGS.  
In this study, the severity and location of SGS were both varied. The SGS severity was quantified by the percentage of area reduction due to SGS. Seven values were used, including 0% (baseline case), 25%, 50%, 75%, 90%, 96%, and 99%, by varying \"\(a\)\" in Equation (5) from 0 to 0.9 (Figure 2b). The SGS location was defined as the distance between the narrowest section of the SGS and the superior surface of the vocal folds  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \(E_{p}\) (kPa) & \(E_{p2}\) (kPa) & \(G_{p2}\) (kPa) & \(\eta\) (Pa\(\cdot\)s) \\ \hline Cover & 2.014 & 40 & 10 & 0.15 \\ Ligament & 3.306 & 66 & 40 & 0.225 \\ Body & 3.99 & 80 & 20 & 0.375 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Material properties of the inner layers of the vocal fold. \(E_{p}\) is the transversal Young’s modulus, \(E_{p2}\) is the longitudinal Young’s modulus, \(G_{p2}\) is the longitudinal shear modulus, and \(\eta\) is the damping ratio.  
Figure 1: (**a**) The computational domain and geometry of the larynx, subglottic stenosis, vocal fold, and vocal tract. (**b**) The inner layers’ structure of the vocal fold.  
(Figure 2c). Two values were used, 2.1 and 3.13 cm, which respectively correspond to the minimum and average values of 92 SGS patients reported in the past studies [32; 33].