## 4 Results  
### Numerical Validation  
Examples of predicted VF trajectories for both synthetic tests are shown in Figure 7. The CPA method detects the minimum glottal distance and is capable of reproducing the apparent penetration between the tissues with reasonable accuracy. We note that, due to the phase lag between the upper and lower masses, inferior mass impacts are tracked. The MBCM shows a relatively low penetration of the upper VF mass, and thus, it contributes less to the total contact force in these simulations. CPA is found to be sensitive to different penetration depth values, predicting correctly more overlap displacement in complete closure phonation (TEST1) than incomplete closure phonation (TEST2) at equal subglottal pressures.  
Results from synthetic tests are shown in Figure 8. The apparent penetration and normalized pressure are compared with simulated data. Contact pressure was calculated by taking the average of all peak values. Though not shown for brevity, the effective contact area in both tests was  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Parameter** & **Symbol** & **Synthetic** & **In Vivo HSV** & **Unit** \\ \hline Sampling rate & \(f_{s}\) & 8750 & 8000 & (fps) \\ Young modulus & \(E\) & 24.75 & 24.75 & (kPa) \\ Poisson ratio & \(v\) & 0.5 & 0.5 & - \\ Polynomial order & \(p\) & 2 & 2 & - \\ Gradient threshold & \(t_{h}\) & 0.5 & 0.25 & - \\ Beta & \(\beta\) & 20 & 20 & - \\ Gamma & \(\gamma\) & 0.5 & 0.5 & - \\ \hline \end{tabular}
\end{table}
Table 2: Contact Pressure Analysis (CPA) parameters for synthetic and in vivo high-speed videoundoscopy (HSV) material.  
Figure 7: Right side kymograms with upper and lower mass displacements from Modified Body Cover Model (MBCM) and CPA trajectory predictions. Two particular simulations from (**a**) TEST1 and (**b**) TEST2 at \(Ps=1500\) Pa are shown. CPA-detected and -tracked trajectories were obtained only by processing the videos. CPA-detected points were used to reproduce the lower mass penetration in both cases, considering their kinematics.  
overestimated, as expected, since the CPA method considers an area with full contact along the VF thickness (i.e., there is no inferior-superior variation in contact).  
Included in the contact pressures plotted in Figure 8 is a line labeled \"HERTZ\". This line was obtained by using the actual inferior and superior mass overlap and contact area from the MBCM model, rather than that estimated through the CPA, in Equation (20). Due to the consistent positive bias observed in the idealized HERTZ pressure values, it is reasonable to conjecture that bias errors arise from a missing term or formulation in the Hertz equations for obliquely contacting bodies. Thus, we calibrated the correction factor \(\alpha\) in Equation (20) to compensate for this effect through an optimization scheme for all simulations, yielding \(\alpha=1.679\). This factor is used in all subsequent results and discussions to compensate for inferior-superior variations in contact area geometry.  
The results from TEST1 shown in Figure 8 suggest that the corrected Hertzian model is an accurate contact estimator for a complete closure phonation scenario. Here, penetration and contact pressure predictions have similar values to the MBCM. MBCM pressure is well-predicted by HERTZ, and the CPA estimation errors for this test remain bounded when compared with the numerical simulations. The ideal prediction by HERTZ of TEST2 in Figure 8 shows some error compared to the MBCM results, which suggest that the Hertz model is less accurate in estimating the contact pressure for incomplete glottal closure configurations.