## 2 Contact Pressure Estimation from Laryngeal High-Speed Video
### Contact Pressure Estimation  
Hertzian contact theory [26; 27] describes the stresses occurring in elastic bodies in contact when the local deformation is small in comparison to the characteristic scale of the body. As such, the contact mechanics can be described in terms of the effective contact surface and apparent penetration, that is, the degree to which a body with the same geometry would \"penetrate\" the contacting body under the same displacement condition [25]. As a first-order approximation, we herein assume that the contact area between opposing vocal folds begin as a line in the inferior-superior direction, with the contact area increasing as the folds progress through the collision event (refer back to Figure 1 for the collision geometry). Noting that the vertex of a quadratic (\(p=2\) from the previous sections) is locally circular,  
Figure 4: KF Edge tracking result. Three representative instants of the closure and a midline kymogram are shown. An overlap between VF is now visible during contact phases despite the lost of detected points in the detection stage. Figure 3: Coefficient tracking using the Kalman Filter(KF) during closure. Quadratic curves are employed (\(p=2\)). The estimates of \(\Theta_{k}\) are converted to standard quadratic coefficients as \((a,b,c)_{k}=(1,-(y_{a}+yb),x_{a}+y_{a}y_{b})_{k}\theta_{0,k}\) and are plotted across time. The polynomial coefficients of each fold are shown before and after the KF edge tracking module, with the ratio \(\lambda_{k}\) and uncertainty factor \(\rho_{k}\) (only the uncertainty of the right side is shown for simplicity). The VF description fails when collision occurs, making registered coefficients not valid at certain times. The KF completes the temporal sequence of the fitted model by making predictions of its location during the collision phases (high \(\rho_{k}\) values). we model the collision event as a Hertzian contact between two cylinders with parallel axes, assuming linear elasticity. Following [PERSON] et al. [34], the maximum contact pressure \(P_{c}\) experienced by colliding cylinders is given by  
\[\frac{P_{c}}{E^{*}}=4\,\alpha\,\frac{\delta_{c}}{L_{c}}\quad, \tag{20}\]  
where \(\delta_{c}\) is the penetration depth and \(L_{c}\) is the contact line length observed from the HSV; refer to Figure 1 for a schematic. The parameter \(\alpha\) is a correction factor to account for VF inferior-superior contact bias; we note that this is not included in standard derivations of Equation (20). This factor is initially assumed to be unity, meaning that there is no bias due to the contact geometry assumption; however, it will be later calibrated as part of the numerical validation. The parameter \(E^{*}\) in Equation (20) is the effective [PERSON]'s modulus, defined as  
\[E^{*}=\frac{1}{\frac{(1-\
u_{1}^{2})}{E_{1}}+\frac{(1-\
u_{2}^{2})}{E_{2}}}= \frac{E}{2(1-\
u^{2})}\quad, \tag{21}\]  
where \(E\) is [PERSON]'s modulus, \(\
u\) is [PERSON]'s ration, and the subscripts \(1\) and \(2\) indicate the two VFs, thus allowing consideration of cases wherein the two VFs have different material properties. For the remainder of this study, however, the two VFs are assumed symmetric with regards to tissue properties, that is, \(E_{1}=E_{2}=E\) and \(\
u_{1}=\
u_{2}=\
u\), and \(E^{*}\) is given by the simplified right hand side of the equation. We note that the normalized maximum contact pressure is purely a function of measured quantities. Alternative geometric models for the vocal folds, such as spherical contacting bodies, result in similar expressions for \(P_{c}/E^{*}\), though they generally include additional geometric factors, such as the thickness of the VFs, which are not readily measured with HSV. It is also important to note that the result is independent of the cylinder radius in contrast to other shapes (e.g., colliding spheres). Equation (20) yields the dimensionless maximum contact pressure.