The effect of swelling on vocal fold kinematics and dynamics

Abstract

Swelling in the vocal folds is caused by the local accumulation of fluid, and has been implicated as a phase in the development of phonotraumatic vocal hyperfunction and related structural pathologies, such as vocal fold nodules. It has been posited that small degrees of swelling may be protective, but large amounts may lead to a vicious cycle wherein the engorged folds lead to conditions that promote further swelling, leading to pathologies. As a first effort to explore the mechanics of vocal fold swelling and its potential role in the etiology of voice disorders, this study employs a finite-element model with swelling confined to the superficial lamina propria, which changes the volume, mass, and stiffness of the cover layer. The impacts of swelling on a number of vocal fold kinematic and damage measures, including von Mises stress, internal viscous dissipation, and collision pressure, are presented. Swelling has small but consistent effects on voice outputs, including a reduction in fundamental frequency with increasing swelling (10 Hz at 30 % swelling). Average von Mises stress decreases slightly for small degrees of swelling but increases at large magnitudes, consistent with expectations for a vicious cycle. Both viscous dissipation and collision pressure consistently increase with the magnitude of swelling. This first effort at modeling the impact of swelling on vocal fold kinematics, kinetics, and damage measures highlights the complexity with which phonotrauma can influence performance metrics. Further identification and exploration of salient candidate measures of damage and refined studies coupling swelling with local phonotrauma are expected to shed further light on the etiological pathways of phonotraumatic vocal hyperfunction.

Appendices

Appendix A Swelling constitutive equation

Let the deformation from a reference configuration, $\varvec{X}$, to spatial coordinates, $\varvec{x}$, be given by $\varvec{x}=\varvec{X}+ \varvec{u}(\varvec{X})$. Then, $\varvec{F}=\partial {\varvec{u}}/\partial {\varvec{X}}$ is the deformation gradient and $\varvec{E}={1}/{2}(\varvec{F}^\top \varvec{F}-\varvec{I})$ is the Green strain tensor. To incorporate the effects of swelling, Gou and Pence (Tsai et al. 2004; Pence and Tsai 2005a) proposed an extension of a hyperelastic strain energy $\psi$ to the form (the strain energy is formulated here with dependence on $\varvec{E}$ instead of $\varvec{F}$ where the relation between the two strain energies is $\psi (\varvec{E})=\psi (1/2(\varvec{F}^\top \varvec{F}-\varvec{I}))=\psi (\varvec{F})$)

$$\begin{aligned} \bar{\psi }(\varvec{E}; v) = m(v) \psi (\bar{\varvec{E}}(\varvec{E};v)) \end{aligned}$$

(A1)

where $\bar{\psi }$ is the swelling-generalized strain energy, $\psi$ is the original strain energy, $v$ is the swelling, and $m(v)$ is a scalar valued monotonic function that satisfies $m(1)=1$. The swelling-modified deformation gradient is given by

$$\begin{aligned} \bar{\varvec{F}}(\varvec{F};v)=v^{-1/3}\varvec{F}; \end{aligned}$$

and the swelling-modified Green strain by

$$\begin{aligned} \bar{\varvec{E}}(\varvec{E};v) = \frac{1}{2}(\bar{\varvec{F}}^T\bar{\varvec{F}}-\varvec{I}) = v^{-2/3} \varvec{E}+ \frac{1}{2}(v^{-2/3}-1)\varvec{I}. \end{aligned}$$

The second Piola–Kirchhoff stress for the swelling-modified strain energy is

$$\begin{aligned} \begin{aligned} \bar{\varvec{S}}&= \frac{\partial {\bar{\psi }(\varvec{E}; v)}}{\partial {\varvec{E}}} \\&= m(v)\frac{\partial \psi (\bar{\varvec{E}})}{\partial \bar{\varvec{E}}}\frac{\partial \bar{\varvec{E}}}{\partial \varvec{E}} \\&= m(v)v^{-2/3} \frac{\partial \psi (\bar{\varvec{E}})}{\partial \bar{\varvec{E}}} \\&= m(v)v^{-2/3} \varvec{S}\vert _{\bar{\varvec{E}}(\varvec{E}; v)} \end{aligned} \end{aligned}$$

(A2)

For a Saint Venant–Kirchhoff material $\psi (\varvec{E})=\lambda ({{\,\textrm{Tr}\,}}{\varvec{E}})^2 +\mu {{\,\textrm{Tr}\,}}{\varvec{E}^2}$ so that the second Piola–Kirchhoff stress is

$$\begin{aligned} \varvec{K}_{ijkl}&= \lambda \delta _{ij}\delta _{kl} + \mu (\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk}), \end{aligned}$$

(A3)

$$\begin{aligned} \varvec{S}&= \varvec{K}\varvec{E}= \lambda {{\,\textrm{Tr}\,}}{\varvec{E}}\varvec{I}+ 2\mu \varvec{E}, \end{aligned}$$

(A4)

where $\mu$ and $\lambda$ are Lame’s parameters. Substituting the above [Eq. (A4)] into $\bar{\varvec{S}}$ [Eq. (A2)] results in

$$\begin{aligned} \begin{aligned} \bar{\varvec{S}}&= \frac{m(v)}{v} v^{1/3} \left( \lambda {{\,\textrm{Tr}\,}}{\bar{\varvec{E}}}\varvec{I}+ 2\mu \bar{\varvec{E}}, \right) , \end{aligned} \end{aligned}$$

(A5)

where $\varvec{K}$ is the constant elasticity tensor for a Saint Venant–Kirchhoff material.

To determine how $m(v)$ changes the modulus with swelling, consider the reference configuration coordinate, $\varvec{X}^\star$, corresponding to equiaxial expansion by the prescribed swelling such that $\partial {\varvec{X}^\star }/\partial {\varvec{X}} = v^{1/3}\varvec{I}$. The deformation gradient and Green strain measured with respect to the unswollen reference configuration, $\varvec{X}$, and with respect to the swollen reference configuration, $\varvec{X}^\star$, are then related by

$$\begin{aligned} \varvec{F}^\star= & {} v^{-1/3}\varvec{F}\\ \varvec{E}^\star= & {} \frac{1}{2}(\varvec{F}^{\star \top }\varvec{F}^\star -\varvec{I}) = v^{-2/3} \varvec{E}+ \frac{1}{2}(v^{-2/3}-1)\varvec{I}, \end{aligned}$$

identical to the relation between $\bar{\varvec{E}}$ and $\varvec{E}$ described above. The strain energy of the material with respect to the swollen reference configuration is

$$\begin{aligned} \psi ^\star (\varvec{E}^\star ; v) = m(v)\frac{\psi (\bar{\varvec{E}}(\varvec{E},v))}{v} = m(v)\frac{\psi (\varvec{E}^\star )}{v} \end{aligned}$$

where the factor $1/v$ is due to the volume increase. The tangent modulus with respect to the swollen configuration is then given by

$$\begin{aligned} \frac{\partial ^{2}\psi ^\star (\varvec{E}^\star ; v)}{{\partial (\varvec{E}^\star )}^{2}} = \frac{m(v)}{v}\frac{\partial ^{2}{\psi (\varvec{E}^\star )}}{{\partial (\varvec{E}^\star )}^{2}} = \frac{m(v)}{v}\varvec{K}, \end{aligned}$$

where $\varvec{K}$ is the elasticity tensor of the swollen material. This shows that the change in modulus is controlled by ${m(v)}/{v}$,

To simplify investigating different functional forms of $m(v)$, we approximate the effect of ${m(v)}/{v}$ with a linear approximation

$$\begin{aligned} \begin{aligned} \bar{\varvec{S}}&\approx (\bar{m}'\vert _{v=1}(v-1)+1) v^{-1/3} \left( \lambda {{\,\textrm{Tr}\,}}{\bar{\varvec{E}}}\varvec{I}+ 2\mu \bar{\varvec{E}}, \right) \end{aligned} \end{aligned}$$

(A6)

where $\bar{m}= m(v)/v$.

Appendix B Independence study

Fig. 12

Mesh and time step independence study results in terms of four derived quantities for the case $(v,\bar{m}')=({1.3}, -1.6)$. Solid lines show absolute values while dotted lines show errors relative to the finest mesh and time step case (mesh refinement of 2 and time step refinement of 16)

The mesh density in Fig. 1 and time step $\Delta t$ were chosen based on a mesh and time step independence study shown in Fig. 12. The mean (over time and the cover region) for von Mises stress and viscous dissipation rate both converge to within 1 % of the finest discretization case (mesh size scale of 0.5 and $\Delta t$ refinement factor of 16) by a $\Delta t$ factor of 8 for the mesh refinement factor of 1. Errors in mean (over time) for contact force and area similarly converge by the same refinement condition. Therefore, a time step of $\Delta t=1.25 \times 10^{-5}\,\hbox {s}$ and the mesh refinement factor of 1 (corresponding to the mesh shown in Fig. 1) were chosen.