## 4 Results
### In Vivo Example and Contrast with Prior Studies  
The glottal area from HSV at different loudness levels is presented in Figure 9 along with predictions of the penetration depth and normalized contact pressure from the CPA procedure. We utilize the material properties listed in Table 2 to obtain contact pressure in physical units. Note that the \(E=24.75\) kPa, which implies that \(E^{*}=16.5\) kPa, and if we use the same correction factor as in the silicone models, we would have \(E^{*}_{cover}=1.80\) kPa, which is in the range of experimental data [45]. From the glottal area waveform, it is evident that the phonation frequency increases with increasing loudness and that the peak glottal area and rate of closure is highest for the loud gesture. The normal and soft gestures exhibit approximately the same peak glottal area and similar closure rates, thus offering a diverse set of cases to study the robustness of CPA. The penetration and normalized contact pressure estimations exhibit increasing peak values with increasing loudness, as intuitively expected. Phase differences in the VF positions during contact were observed, exhibiting more displacement and delay in the right VF of the subject. The fictitious overlap observed is not symmetric and depends on the kinematic behavior of each VF border. CPA was sensitive to different VF displacement magnitudes, depending on contact time and impact velocity, which is noted in the glottal area amplitude and rate of change. Abrupt closure develops more deflection and, therefore, presents higher penetration values. CPA estimations are further quantified in Table 5. The mean peak values of contact pressure for absolute and relative values to the normal gesture (in brackets) are shown. Comparing the contact pressure with the normal loudness scenario, the louder case is at least two times higher and the softer case is roughly a half of it. This implies that doubling the contact pressure results in roughly +10 dB in the SPL in this scenario. Some previous evidence of this variations was documented by [PERSON] et al. [8] and [PERSON] et al. [7], where comparable values were reported across similar loudness intensities. Figure 9: Hertzian pressure estimations for in vivo HSV recordings. Glottal area, penetration, and contact area estimates for soft (solid line), normal (dash line), and loud (dash-dot line) gestures are shown. Increasing impact slopes of glottal area are observed with increasing loudness. Increasing penetration depth and estimated normalized contact pressure are observed with louder gestures. A further quantitative comparison is obtained by contrasting CPA-predicted contact pressure estimations with the published data of time series of contact pressure from hemilarynx experiments [16], finite element models of vocal fold collision [20; 21], and an analytical viscous contact model [22]. As an example, we use the normal loudness condition to better resemble these prior studies. Figure 10 shows the contrast of the proposed Hertz contact model (red line) against prior studies (blue lines) [16; 20; 21; 22]. Given that the duration of the contact pulse is a function of the fundamental frequency, contact pulses can vary between 0.8 and 1.8 ms in these cases. For ease of comparison, the time axis was normalized in Figure 10 to maintain a contact duration of 1 ms. It can be seen that the resulting pulse yields comparable magnitude, although it does not capture the waveform asymmetry that some of prior studies report. This limitation, however, does not hamper the correct estimation of the maximum peak in the contact pressure, which is considered more clinically relevant. In addition, note that this pulse asymmetry is not observed in one of the prior studies using finite element models [21]. Note that the resulting contact pressure waveform is symmetric due to the underlying lumped mass model in the Kalman filter, which uses a linear spring as a first approximation to describe the overlap penetration.