## 3 Results and Discussion  
### Qualitative Features of Pathological Vocal Fold  
The glottal volume flow velocity \(U\) of the normal and pathological vocal folds is shown in Figure 2. As can be seen from the figure, the mass block displacement in both normal and pathological vocal fold models vibrates periodically and produces a valley-cut sinusoidal shape for airflow. However, the amplitude of volume flow velocity \(U\) decreases when the polyp is present. In the closing phase of the whole cycle, the pathological vocal fold can not close entirely, and the duration of the closed phase is reduced. As we can see, vocal fold polyps have an impact on the quality and intensity of phonation. Hence, the study focused on simulating muscle activations that may reduce the adverse effect of polyps on phonation. Previous work has identified that for \(a_{\rm LC}>0.5\), the vocal processes are approximated, and for \(a_{\rm LC}<0.5\), the vocal processes are separated. Oscillation occurs approximately along the constant horizontal line of \(a_{\rm LC}=0.5\)[19]. Therefore, we fixed \(a_{\rm LC}\) at 0.5 to better understand the implication of muscle activation on adduction and abduction. When \(a_{\rm CT}=0.25\), \(a_{\rm TA}=0.25\), and \(a_{\rm LC}=0.5\), the longitudinal strain of the vocal fold \(\varepsilon\) will be 0, and all laryngeal muscles are activated. Hence, fixing \(a_{\rm CT}\) at 0.25 for the study of \(a_{\rm TA}\) was seen as the best choice. The same method used for fixing \(a_{\rm TA}\) was used for the study of \(a_{\rm CT}\). Figure 3 shows glottal volume flow velocity waveforms of different cricothyroid muscle activities (\(a_{\rm CT}\)). \(a_{\rm CT}\) now increases from 0.05 to 0.9 with \(a_{\rm TA}\) constant at 0.25 and \(a_{\rm LC}\) constant at 0.5. Note that the effect of increasing \(a_{\rm CT}\) is an initial increase in fundamental frequency. The average amplitude of the glottal airflow first increases and then decreases with increasing \(a_{\rm CT}\). It can be seen the vocal polyp cannot change the status of the glottal cycle on phonation. However, the figure shows that the vocal folds can maintain a more periodic vibration only when \(a_{\rm CT}\) is at a moderate value. When the level of CT muscle activity is too low, there is low stiffness of the vocal fold cover, causing the vocal fold to vibrate with little speed and amplitude. However, when \(a_{\rm CT}\) increases, the effect of the polyp on vibration becomes obvious, and there is abnormal glottal airflow. As the stiffness increases with \(a_{\rm CT}\) to the maximum of the range, the stabilized waveforms of glottal airflow tend to be a straight line, and vibration is severely impaired. Figure 4 shows glottal volume flow velocity waveforms for different levels of thyroartytenoid muscle activity (\(a_{\rm TA}\)). The waveform is nearly sinusoidal at small values of \(a_{\rm TA}\) but decreases in amplitude when \(a_{\rm TA}\) increases. The waveforms in Figures 3a and 4d are similar, showing the effect of low stiffness on vocal fold vibration. The fundamental frequency slightly decreases with increasing values of \(a_{\rm TA}\). This supports the function of the thyroartytenoid and cricothyroid muscles in controlling fundamental frequency [21]. Figure 3: Glottal volume flow velocity \(U\) for a unilateral polyp vocal fold model across differing levels of cricothyroid muscle activity, \(a_{\rm CT}\). In each case, \(a_{\rm TA}\) was fixed at 0.25 and \(a_{\rm LC}\) was fixed at 0.5. Figure 2: Glottal volume flow velocity rate. (**a**) The normal vocal fold. (**b**) The vocal fold with a unilateral polyp. Figure 5 shows glottal volume flow velocity waveforms for different levels of lateral cricoarytenoid muscle activity (\(a_{\text{LC}}\)). There is a decrease in glottal airflow amplitude with increasing \(a_{\text{LC}}\). Unlike for the thyroarytenoid muscle, the glottal airflow waveform is almost sinusoidal when \(a_{\text{LC}}\) is around 0.5. Irregular vocal fold vibration occurs with increasing or decreasing \(a_{\text{LC}}\).