## 2 Materials and Methods
### Numerical Methodology
#### 2.2.1. Vocal Fold Reconstruction from MRI Scan  
Computational laryngeal reconstruction was performed using MRI images from three samples that were scanned ex vivo. Using sample 1 as an example (see Figure 3), we first scanned the vocal fold in the NA condition. Following this, the MRI scans were  
Figure 2: Cricothyroid suture approximation conditions. HSV was captured in three conditions in vivo. The NA condition was captured before suturing to observe the natural resting vocal fold position. The HP condition was captured following bilateral sutures between the cricoid and thyroid cartilages, shortening the distance between these structures and lengthening the vocal folds to achieve approximation. The sUVFP condition was captured by releasing the right-sided sutures, allowing lengthening of the distance between the cricoid and thyroid unilaterally, leading to the right vocal fold having less tension applied and appearing “floppy”. imported to ITK-SNAP to manually segment the images. We considered the following structural components for model reconstruction: thyroid cartilage, arytenoid cartilages, cricoid cartilage, and soft tissue (vocal fold). Two distinct layers were segmented within the vocal folds due to known differences in tissue properties [54]: the pliable vocal fold cover, composed of the epithelium and superficial lamina propria, and the much stiffer vocal fold body, composed of the deep lamina propria and throarytenoid muscles. The layers were segmented in MRI images as depicted in Figure 3.  
Following segmentation, we output surface meshes for each structural component from ITK-SNAP and processed them with Mesh Mixer (Autodesk, San Francisco, CA, USA) to avoid uneven surfaces from the manual segmentation. However, this procedure leads to a separate surface mesh for each segmented component, and the two components adjacent to each other, e.g., the thyroid cartilage and the vocal fold body do not share the same surface on their meshes, and there exists a thin gap in between. To handle this problem, we output a surface mesh for the larynx as an entire segment from ITK-SNAP, and this mesh was also smoothed with MESH Mixer. Next, all the smoothed surface meshes are imported into COMSOL Multiphysics (COMSOL Inc., Burlington, MA, USA) to generate solid bodies and create unstructured volume meshes sequentially. A tetrahedron mesh with 10 nodes on each element was applied. Different structural components need to be identified on the entire larynx mesh as this mesh will be used as the FEM model in the following simulations. To do so, we compared the nodes on the volume mesh of the entire larynx with the nodes on the volume mesh of individual structural components and set a minimum distance criterion to determine whether the nodes of the entire larynx's mesh belong to a specific structural component. Using this approach, all the nodes on the entire larynx's mesh were marked as individual structural components as shown in Figure 4a and can be assigned with the material properties related to that component. Since the vocal fold cover is thin but important for the vocal fold vibration, this region is refined with denser mesh as shown in Figure 4b. In the mesh-independence study, four different unstructured meshes with 76,181, 88,334, 125,351, and 135,748 elements respectively were used to compute the eigenfrequency of approximated vocal folds in the HP condition. The eigenfrequency of the left vocal fold is 578 Hz, 588 Hz, 597 Hz, and 598 Hz, respectively, using the aforementioned four meshes. The third mesh resolution was selected in all the cases, as further refinement did not significantly improve the result. In this mesh set, the thyroid cartilage, arytenoid cartilage, cricoid cartilage, vocal fold cover, and vocal fold body for Sample 1 have 16,510, 3232, 11,489, 13,021, and 81,099 elements respectively. Figure 3: Vocal fold segmentation for sample 1. Green = arytenoid cartilages; red = cricoid cartilage; cyan = thyroid cartilage; blue = vocal fold body; yellow = vocal fold cover.