## 2 Materials and Methods
### Development of Rules for Muscle Control  
In reality, the value of some parameters will vary with the movement of the laryngeal muscles during phonation, which is called the effective value of vocal fold vibration [21]. For a more realistic simulation, we improved the rules for controlling geometric and elastic parameters with muscle activation proposed by [PERSON] [19] to fit our model. These rules take the place of the physics of deformable bodies for vocal fold posturing. To reduce the number of control parameters, only the cricothyroid muscle activity \(a_{\rm CT}\) (ranging from 0.05 to 0.9), the throarytenoid muscle activity \(a_{\rm TA}\) (same range), and the lateral cricoarytenoid muscle activity \(a_{\rm LC}\) (same range) were considered. The reason for setting the different muscle activities in this range is that the muscle cannot actually relax and contract completely. The length of the vocal fold is the primary geometric variable for adjustment of \(F_{0}\), and it changes during phonation. It is therefore important to identify what is meant by effective  
Figure 1: Schematic of vocal fold model with a polyp. Vertical plane represents midline. vocal fold length. [PERSON] [22] showed that the fractional length change (vocal fold strain) due to rotation can be written as  
\[\varepsilon=G(Ra_{CT}-a_{TA})-Ha_{LC}\, \tag{1}\]  
where \(\varepsilon\) is the longitudinal vocal fold strain, \(G\) is a gain factor, \(R\) is a torque ratio, and \(H\) is the adductory strain factor. The range of \(\varepsilon\) for human must be higher to achieve a larger pitch range than canines. To obtain a larger maximum elongation, we needed to increase the values of the above parameters (\(G\), \(R\), \(H\)) compared to the canine model. For this study, we set \(G=0.2\), \(R=3.0\), and \(H=0.2\), as in a previous study [22]. When a polyp develops on one vocal fold, it can be approximated that the length and tension of the vocal folds remain constant, while the vocal fold linear density \(\rho\) increases slightly in the local area around the polyp [21]. In the present model, we treated the entire vocal fold cover as one with the polyp but still follow this formula. Using the above rule, the vocal fold length can be expressed in the form  
\[l=l_{0}(1+\varepsilon)\] \[=l_{0}[1+G(Ra_{CT}-a_{TA})-Ha_{LC}]\, \tag{2}\]  
where \(l_{0}\) is the resting length, for which the average value in humans is 1.4 cm. Vocal fold depth increases with vocal fold shortening during vocal fold vibration. We developed the following rules for thickness:  
\[D=\frac{D_{muc}+0.5D_{lig}}{1+0.2\varepsilon} \tag{3}\]  
where \(D_{muc}\), standing for depth of mucosa is 0.2 cm, and \(D_{lig}\), standing for depth of ligament, is 0.2 cm. Thickness is described as  
\[d=\frac{d_{0}}{1+0.8\varepsilon} \tag{4}\]  
where \(d_{0}\) is the vibrating thickness at resting length. We defined the ratio of the effective vibration parameter to the original parameter as the effective vibration ratio \(\omega\). The effective vibration ratio of length \(\omega_{l}\), depth \(\omega_{D}\) and thickness \(\omega_{d}\) can be written as  
\[\omega_{l}=l/l_{0}\ \omega_{D}=D/D_{0}\ \omega_{d}=d/d_{0} \tag{5}\]  
In addition to the above parameters, the stresses acting on the vocal fold also change with muscle activation during vibration. The cover layer has only a passive stress and can be expressed as  
\[\sigma_{c}=\left[\sigma_{muc}D_{muc}+0.5\sigma_{lig}D_{lig}\right]/D \tag{6}\]  
where \(\sigma_{muc}\) is the mucosa stress, and \(\sigma_{lig}\) is the ligament stress. According to previous work [23; 24], the passive stress of mucosa and ligament has been modeled with a combination of a linear and an exponential function and matched to vocal fold stress-strain curves. The form of this function can be expressed in the form  
\[\sigma_{p}=