## 3 Results
### Eigenfrequency of Approximated Vocal Fold  
Using numerical methodology discussed in Section 2.2, subject-specific vocal fold approximation was modeled for different laryngeal conditions. During in vivo experiments, vibration frequency in the HP condition ranged from 500 Hz to 900 Hz, consistent with the range observed in previous studies of rabbit phonation [30; 37]. For computational modeling, we assumed constant material properties for the soft tissue including Young's moduli, Poisson's ratio, and density as listed in Table 2. The Young's modulus of the vocal fold cover used here falls in the range of previous experimental measurements [57; 58]. The cartilages had negligible deformation and were thus assigned with much higher stiffness. These parameters would lead to a dominant eigenfrequency of around 600 Hz for the vocal fold in the HP condition. This frequency is consistent with the average frequency across the rabbit larynx samples in two previous studies from our group [29; 30]. The vibration \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & **Right (Hz)** & **Left (Hz)** & **Difference** \\ \hline \multirow{2}{*}{Subject 1} & HP & 875 & 875 & \textless{1}\% \\  & sUVFP & 815 & 908 & 11\% \\ \multirow{2}{*}{Subject 2} & HP & 660 & 660 & \textless{1}\% \\  & sUVFP & 605 & 670 & 10\% \\ \multirow{2}{*}{Subject 3} & HP & 763 & 763 & \textless{1}\% \\  & sUVFP & 650 & 722 & 11\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Right and left vocal fold vibration frequencies obtained through HSV and digital kymography. Figure 6: HSV analysis of Subject 1’s vocal folds in a unilateral suture condition. (**a**) An isolated frame from HSV; the red line depicts the plane from which the kymograph was generated; (**b**) kymograph of the HP condition, same periodicity observed for left and right vocal folds; (**c**) kymograph of the sUVFP condition, the unsutured vocal fold (right) has fewer periods than the sutured vocal fold (left). Videos are included in the supplemental material. frequency in the present study had more variations among individuals. However, the differences between the experiment and computation did not impact analysis, as we sought to compare left-right vibratory frequency as opposed to making one-on-one comparisons between the two models. A two-step process was used to compute the eigenfrequency of the approximated vocal fold. First, a static deformation was simulated for vocal fold adduction by the prescribed thyroid displacement according to the HP or sUVFP condition. As a result of this deformation, the vocal fold was pre-stressed. Then, the eigenfrequency and the eigenmode (or the vibration mode) were computed at the pre-stressed condition. Figure 7 shows vocal fold approximation under different conditions for Sample 1. Following reconstruction of the NA condition (Figure 7a), thyroid cartilage displacement for HP or sUVFP conditions was applied (Figure 7b or Figure 7c respectively). Using this subject-specific approach, we were able to model symmetrical approximation of the vocal folds for HP and asymmetrical approximation in sUVFP. The eigenfrequency and eigenmode calculations were done using the default algorithm in COMSOL Multiphysics. Multiple modes were obtained from this simulation. The vocal vibration is mainly related to the first two modes for each side of the vocal fold [59]. The higher-frequency modes involve out-of-phase vibration patterns on the medial surface of the vocal fold and were not found from the HSV results (see supplementary material for those mode shapes and frequencies). The first mode has the maximal displacement in the axial direction (\(Y\)-axis in Figure 8), while the second mode has the maximal displacement in the direction transverse to the midline (X-axis in Figure 8). The second mode was selected to represent the vibration pattern of the vocal fold under phonation by comparing it with the HSV results. This mode is shown in Figure 8 for sample 1.