The Role of Glottal Surface Adhesion on Vocal Folds Biomechanics

Abstract

The airway surface liquid (ASL) is a very thin mucus layer and covers the vocal fold (VF) surface. Adhesion mediated by the ASL occurs during phonation as the VFs separate after collision. Such adhesion is hypothesized to determine voice quality and health. However, biomechanical insights into the adhesive processes during VF oscillation are lacking. Here, a computational study is reported on self-sustained VF vibration involving contact and adhesion. The VF structural model and the glottal airflow are considered fully three-dimensional. The mechanical behavior of the ASL is described through a constitutive traction–separation law where mucosal cohesive strength, cohesive energy and rupture length enter. Cohesive energy values considered are bound below by the cohesive energy of water at standard temperature and pressure. Cohesive strength values considered are bound above by prior reported data on the adhesive strength of mucosal surface of rat small intestine. This model introduces a mechanical length scale into the analysis. The sensitivity of various aspects of VF dynamics such as flow-declination rate, VF separation under adhesive condition and formation of multiple local fluid bridges is determined in relation to specific ASL adhesive properties. It is found that for the ASL considered here, the characteristics of the VF separation process are of debond type. Instabilities lead to the breakup of the bond area into several smaller bond patches. Such finding is consistent with in-vivo observations.

1 Introduction

The superficial mucus layer present on the vocal fold (VF) surfaces, known as the airway surface liquid (ASL), is postulated to induce an adhesive behavior at the glottal surface (Nakagawa et al 1998; Ayache et al 2004). VF surface adhesion in turn can influence phonation (Chodara et al 2012) as well as voice health (Leydon et al 2009). Electrolytic composition of ASL (Kutta et al 2002) and characteristics of its spatial agglomeration (Bonilha et al 2008) have been studied but its mechanical characteristics remain difficult to measure.

The objective of the present study is to establish a computational framework for the analysis of VF vibration under the consideration of VF collision and subsequent ASL mediated adhesion. Our specific aims are

1. to understand what changes are induced into VF vibration by VF adhesion,

2. to investigate how characteristic parameters of the constitutive model describing adhesion affect the VF vibration process, and

3. to characterize the VF separation process in view of the multiple (biomechanical and geometrical) length scales of the problem.

Computational investigation of VF surface adhesion pose substantial challenges since the adhesive nature of the ASL is relevant only in the presence of VF collision, and because simulation of VF vibration including collision effects already poses many complexities. Key challenges in modeling VF collision arise from the need for the use of a three-dimensional (3D) VF geometry, the complex behavior of airflow associated with the substantial changes in Reynolds number during open and closed phases, and the requirement to both resolve local stresses in the contact well while also accounting for the large deformation of the VFs at high-amplitude vibration.

Numerical studies available in literature that use a Navier–Stokes model for the air flow and a continuum VF model to determine 3D VF collision due to high-amplitude self-sustained oscillation are few (Zheng et al 2009; Bhattacharya and Siegmund 2014). Problems involving FSI can be solved either by employing a monolithic strategy where the governing equations of the coupled system are solved by a single solver (Zheng et al 2009), or by following a segregated approach where bi-directional coupling is used to communicate between distinct solvers for the fluid and solid domains (Bhattacharya and Siegmund 2014). This study follows recent work by (Bhattacharya and Siegmund 2014) and the solution of the coupled fluid–structure interaction (FSI) model use an arbitrary Lagrangian–Eulerian approach. On the other hand, currently the only known quantitative study of glottal surface adhesion on VF dynamics was performed by Decker (2006), where lubrication theory based continuum model was used to determine tractions on the ASL–VF interface as a function of the separation velocity between the VFs. A 2D geometry of the VF was used and the flow model was 1D.

The present study substantially expands the modeling of contact and adhesion interactions (referred to as the collisional interaction problem) and connects the collisional interaction problem into a full scale 3D FSI framework.

The ASL mediated adhesion effects are explicitly accounted for in the model simulation by employing constitutive relations describing the relationship between VF adhesive tractions and VF separation distance, and by embedding the collisional interaction model into a fully 3D transient dynamic FSI computational model. The 3D geometry of the VFs used is based on a canonical model known as M5 that was introduced in Scherer et al (2001). M5 geometry-based models are relevant as these have been employed in several studies on phonation (Thomson et al 2005; Spencer et al 2006; Mihaescu et al 2010; Pickup and Thomson 2011; Bhattacharya and Siegmund 2012). Constitutive properties corresponding to air and VF tissue (Zhang et al 2006) are employed without any scaling of these quantities. A Navier–Stokes model is used for air flow, and the VF tissue is considered to be isotropic linear elastic with viscoelastic behavior. The values of the constitutive parameters of the traction–separation law are bound by data on surface-tension of water and by data on intestinal mucus. To solve the governing equations of the overall system, a segregated solution approach is used, with coupling between the solvers enforced between consecutive time increments.

A detailed analysis is conducted based on parametric variation of traction–separation law parameters. Biomechanical properties of ASL are thus linked to aspects of VF dynamics. Adhesion characteristics are found to influence both local and global VF dynamical behavior. Local aspects relate to the actual process of VF separation, and global aspects to the overall flow rate and frequency. Fundamental insights into separation characteristics are obtained using concepts from the field of non-linear fracture mechanics.

2 Method

2.1 Computational model

The present model of self-sustained VF vibration in the presence of adhesion comprises separate continuum region definitions for the glottal airflow and the pair of VFs, a collisional interaction model (comprising both contact and adhesion) and an FSI model. The collisional interaction model describes the details of the interaction between the VFs. The FSI model describes the interaction between each VF and the glottal airflow. Details of the model and model results in the absence of adhesion are given by the authors in Bhattacharya and Siegmund (2014).

Figure 1a shows the geometry of the airflow domain which is based on the M5 description (Scherer et al 2001). Its geometric dimensions are provided in table 1; governing equations (continuity and Navier–Stokes) are given in table 2a and Newtonian fluid constitutive behavior is prescribed in table 2b along with (constant) density and dynamic viscosity of air in table 1; boundary conditions in table 2c prescribe a time-varying pressure at the inlet

Fig. 1.

(a) Geometry of the glottal airflow domain: the inlet, outlet and glottal surfaces are shaded, the coordinate origin (at the intersection of the mid-coronal plane, the mid-saggital plane and the VF superior surface) is denoted by ⊗, and axes x_is, x_ml and x_ap are in the inferior–superior, medial–lateral and anterior–posterior directions respectively; (b) geometry of the left half of the solid VF model: surface C_L, line AB, and point X_MC are reference regions expected to participate in contact and adhesion; (c) mid-coronal section showing both pairs of VFs and rigid planes: coordinate axes are offset from the origin for clarity

Table 1.

Model constants

Geometric dimensions
Tentry	10.0 mm	Texit	20.0 mm
W	17.4 mm	T	10.7 mm
L	20.0 mm	D	8.40 mm
dg	0.600 mm	dp	0.200 mm
Constitutive properties of air and VF tissue
ρf	1.23 kg/m3	μ	1.79 · 10−5 kg/m·s
E	6.00 kPa	ν	0.450
ρs	1070 kg/m3	g1	0.100
k1	0.100	τ1	0.100 s

Table 2.

Governing equations, boundary conditions and constitutive relations

Airflow domain (a) $\{\begin{array}{l}{\oint }_{\partial (V^f)}(\mathbf{v}-{\mathbf{v}}_g)·\mathrm{d}\mathbf{S}=0\hfill \\ {\rho }_f\frac{\mathrm{d}}{\mathrm{d}t}{\int }_{V^f}\mathbf{v}\mathrm{d}V+{\rho }_f{\oint }_{\partial (V^f)}\mathbf{v}(\mathbf{v}-{\mathbf{v}}_g)·\mathrm{d}\mathbf{S}=-{\oint }_{\partial (V^f)}p\mathbf{I}·\mathrm{d}\mathbf{S}+{\oint }_{\partial (V^f)}{\mathit{\tau }}_f·\mathrm{d}\mathbf{S}\hfill \end{array}$ (b)τf= μ[▽v + (▽v)T] (c) $\{\begin{array}{l}p(x_{\text{is}}=-T_{\text{entry}}-T)=p_{\text{in}}(t)\hfill \\ p(x_{\text{is}}=T_{\text{exit}})=0,\hfill \\ \mathbf{v}(x_{\text{ap}}=\pm L/2)={\mathbf{v}}_g(x_{\text{ap}}=\pm L/2)=0,\hfill \\ \mathbf{v}(x_{\text{ml}}=\pm W/2)={\mathbf{v}}_{\mathrm{g}}(x_{\text{ml}}=\pm W/2)=0\hfill \end{array}$

Vocal fold domain (d) ${\int }_{V^s}\mathit{\sigma }:\delta {\mathit{D}}_v\mathrm{d}V-{\oint }_{\partial (V^s)}{\mathit{\tau }}_s·\delta {\mathbf{u}}_v\mathrm{d}S-{\int }_{V^s}{\rho }_s\ddot{\mathbf{u}}·\delta {\mathbf{u}}_v\mathrm{d}V$ (e) $\{\begin{array}{l}\mathit{\sigma }(t)={\int }_0^{t}2G(t-t^{\prime })\stackrel{.}{\mathit{e}}\mathrm{d}{t}^{\prime }+\mathit{I}{\int }_0^{t}K(t-t^{\prime })\stackrel{.}{\epsilon }\mathrm{d}{t}^{\prime }\hfill \\ G(t)=\frac{E}{2(1+\nu )}\left[1-g_1+g_1{\mathrm{e}}^{-t/{\tau }_1}\right]\hfill \\ K(t)=\frac{E}{3(1-2\nu )}\left[1-k_1+k_1{\mathrm{e}}^{-t/{\tau }_1}\right]\hfill \end{array}$ (f)u(xap= ± L/2) = 0, u(xml = ±W/2) = 0

$$p_{\text{in}}(t)\equiv p_{max}\{\begin{array}{ll}{(t/t_{\text{ramp}})}^2[3-2(t/t_{\text{ramp}})]\hfill & \forall t\in [0,t_{\text{ramp}}]\hfill \\ 1\hfill & \forall t\in [t_{\text{ramp}},\infty ]\hfill \end{array}$$ (1)

where p_max = 400 Pa and t_ramp = 0.150 s, zero pressure at the outlet and no-slip and no-penetration at all bounding surfaces except the inlet and outlet. The motion of the moving–deforming glottal surface, specifically the grid velocity v_g on the glottal surface, is determined by the FSI model (described later). The fluid volume is discretized using tetrahedral cells, with a minimum cell size of 0.050 mm near the glottis ensured throughout the computation. The fluid model is implemented in ANSYS/FLUENT 12.0 (ANSYS, Inc. Canonsburg, PA, USA). A fixed time increment of 50 μs is used throughout the solution process. The implicit PISO algorithm (with neighbor and skewness correction) was used to advance the solution in time.

The VF domain comprises identical and disjoint left and right solid parts. Considering only the left VF the geometry is shown in figure 1b; dimensions are given in table 1; the principle of virtual work (table 2d) governs the VF mechanics; homogeneous isotropic linear viscoelasticity defines the VF constitutive behavior (table 2e); constitutive property values for the tissue domain are given in table 1; boundary conditions listed in table 2f constrain all degrees of freedom on the lateral, anterior and posterior surfaces. Displacement and traction boundary conditions on the glottal surface are determined as a result of collisional interaction and FSI models (described later). The VF volumes are discretized using first-order hexahedral elements with minimum edge length 0.110 mm near the medial surface where contact and adhesion processes are relevant. A line AB oriented in the anterior–posterior direction and lying on the left VF surface and a nodal location X_MC corresponding to the mid-point of AB are defined for further reference (figure 1b).

Surface definitions are indicated in figure 1c. The FSI model defines the interaction between the VF surfaces (left: S_L, right: S_R) and the glottal surface boundary of the airflow domain. Auxiliary static and rigid planes (left: P_L, right P_R) are positioned as mutually parallel (separation d_p, table 1) and symmetrically between the VFs (figure 1c). A collisional interaction is defined between the contact-prone regions of the two VF surfaces (left: C_L, right: C_R) and the corresponding rigid planes (C_L with left face of P_L and C_R with right face of P_R). The rigid planes thus restrict the topology of the deformable airflow domain to its initial undeformed topology (genus 0) and thereby satisfy a limiting condition in the flow solver implementation. As a result of the above topology restriction a leakage flow is present between the surfaces of the opposing VFs that are in active collisional interaction with the rigid planes.

Since C_L ⊂ S_L and C_R ⊂ S_R, a determination needs to be made during the simulation regarding whether the collisional interaction model or the FSI model is to be used to compute surface tractions on C_L and C_R. This choice is based on the VF opening distance Δ ≡ |x_ml| − d_p/2. Specifically, the variable

$$\chi (t)=\{\begin{array}{ll}1,\hfill & \text{for}\mathrm{\Delta }=0\hfill \\ 0,\hfill & \text{for}\mathrm{\Delta }>{\delta }_f\hfill \\ \chi (t-\mathrm{\Delta }t),\hfill & \text{for}\mathrm{\Delta }\in (0,{\delta }_f]\hfill \end{array}$$ (2)

with initial condition χ(0) = 0 is defined in order to track the state of collisional interaction (χ= 1: active and χ= 0: inactive) at every location on C_L and C_R. Above Δt = 50 μs is the fixed solution time increment in the solid domain solver. Tractions on C_L and C_R are computed from the contact-adhesion model when χ= 1 and from the FSI model when χ= 0. A continuous line segment within AB (figure 1b) is referred to as a collisional interaction line segment (CILS) if χ= 1 for each nodal location on the segment. There may exist multiple CILSs within AB that are mutually disjoint, and any CILS may grow or shrink with time, as well as merge with adjacent segments. The total length of all CILSs on AB at a given time is henceforth denoted as length l_c which is bound above by the VF length L.

Collisional interaction comprises both contact and adhesion. In contact a location on the VF surface is coincident with the corresponding rigid plane. Compressive tractions are computed such that the location does not penetrate the rigid plane (hard contact). During VF separation, adhesion of the VFs leads to tensile tractions. When the VF opening distance Δ is in the range (0, δ_f] the normal traction on the VF surface is given by a traction–separation law (Figure 2)

Fig. 2.

The VF adhesion model: cohesive strength σ₀, the elastic separation limit δ₀ and rupture length δ_f

$${\mathit{\tau }}_s·\widehat{n}=\{\begin{array}{ll}{\sigma }_0\frac{\mathrm{\Delta }}{{\delta }_0},\hfill & \mathrm{\Delta }\in (0,{\delta }_0]\hfill \\ {\sigma }_0\frac{{\delta }_f-\mathrm{\Delta }}{{\delta }_f-{\delta }_0},\hfill & \mathrm{\Delta }\in [{\delta }_0,{\delta }_f],\hfill \end{array}$$ (3)

where the parameters are the cohesive strength σ₀, the elastic separation limit δ₀ and rupture length δ_f. The cohesive energy ϕ = σ₀δ_f/2 is the area under the traction–separation curve and represents the energy dissipated or work done against surface adhesion per unit surface area. Tangential tractions (τ_s·r̂) are assumed negligible because the relative tangential motion between the symmetrically deforming opposite VFs is expected to be negligible. The solid domain model including the collisional interaction model is implemented in Abaqus/Standard v6.11 (Dassault Systèmes Simulia Corp., Providence, RI, USA). The solution is integrated implicitly in time using the Hilber-Hughes-Taylor algorithm.

At locations within C_L and C_R where χ(t) = 0 (as determined from (2) at time t) the FSI model applies the traction boundary condition (also called the dynamic boundary condition)

$${\mathit{\tau }}_s=(-p\mathit{I}+{\mathit{\tau }}_f)·\widehat{n}.$$ (4)

This condition ensures that in both models the tractions acting on the glottal surfaces domains are equal and opposite to each other. Equation (4) is also applied unconditionally on S_L \ C_L and S_R \ C_R at all times. In (4) terms on the left and right sides of the equation are evaluated by interpolating between neighbouring nodes taken from the VF and airflow models respectively. The FSI model is used to determine the grid velocity of the glottal surfaces in the airflow model from the kinematic boundary condition v_g = u̇ where the right hand side is evaluated on S_L and S_R. Thus the deformed glottal surface geometries in the airflow and VF models always remain coincident. Both the dynamic and the kinematic boundary conditions are applied at intervals of 50 μs, i.e. after every solution increment in the flow and solid domain solvers. The FSI model is implemented in MpCCI v4.1 (Fraunhofer SCAI, Sankt Augustin, Germany).

2.2 Glottal surface adhesion properties

Several experimental studies have focused on characterising mucosal surfaces in animals, and have been reviewed in Davidovich-Pinhas and Bianco-Peled (2010); das Neves et al (2011). Although it is well known that indices such as temperature, pH and underlying biochemistry can vary significantly across anatomical sites (e.g. nasal mucosa, intestinal mucosa, corneal mucosa), the role of these indices on mechanical characteristics is poorly understood. Adhesion on the rat small intestinal mucosal surface was studied in Mortazavi and Smart (1994) to suggest that the cohesive strength of mucosal surface is in the order of σ̂₀ = 3 kPa (specifically an applied load of 10 g over a surface area of 30.2 mm² could be sustained). With σ₀/σ̂₀ ∈ [0.18, 0.72] in the present study (table 3) the value reported in Mortazavi and Smart (1994) is considered as an upper bound for cohesive strength values. On the other hand liquid mediated VF adhesion is understood to have its primary source in surface tension (Ayache et al 2004). Since cohesive energy ϕ is identical to surface tension, at standard temperature and pressure the cohesive energy for a purely aqueous ASL is equal to its surface tension i.e. ϕ_aq = 0.072 J/m² (Dean 1999). However in interpreting this value some care is needed because the model geometry detailed above allows for ASL bridges to form only between the VF surfaces and the rigid planes. Cohesive energy, being the work of adhesion done per unit area by a continuous ASL between the VFs, can be equally divided into works of adhesion done per unit area (or cohesive energies) by two separate ASLs forming between each of the two VFs and the corresponding rigid planes. An equivalent one-sided ASL possesses a cohesive strength identical to the original, but half the rupture length of the original ASL. The cohesive energy of an equivalent purely aqueous one-sided ASL is hence ${\varphi }_{\text{aq}}^{\ast }=0.036\mathrm{J}/{\mathrm{m}}^2$. This value denotes the lower bound of the ϕ values used in the present study (table 3). Rupture length values are determined as δ_f = 2ϕ/σ₀. The parameter δ₀ is held fixed for all models and its variation is not considered within the scope of the present study.

Table 3.

Adhesion parameters: adhesion is absent in model 1; model 3 is considered as baseline

Model	σ0 [kPa]	δ0 [mm]	δf [mm]	$\varphi /{\varphi }_{\text{aq}}^{\ast }$
1	0.00	0.00	0.00	0.00
2	0.540	0.100	0.267	2.00
3	1.08	0.100	0.267	4.00
4	1.08	0.100	0.400	6.00
5	2.16	0.100	0.267	8.00

To increase the numerical stability of the computation, viscous contributions to surface tractions are considered as proportional to the approach velocity of the surfaces. Such contributions can also be interpreted to arise from the viscous contribution to liquid mediated adhesion. Here, the magnitudes of viscous tractions are negligible compared to tractions caused by contact and surface-tension based adhesion.

In computations with cohesive constitutive models, mesh convergence depends on the number of elements present in the zone of adhesive break-down i.e. the process zone l_p. Process zone length depends on the traction–separation law parameters and the bulk constitutive properties. One fundamental estimate defines l_p as $l_p\approx [E/(1-{\nu }^2)]\varphi /{\sigma }_0^2$. For the model considered herein the minimum l_p is 0.815 mm (model 5). A comparison of element size in the contact-prone region (0.110 mm) and the minimum l_p required indicates that the mesh used in the solid domain is sufficiently refined to obtain convergence since the process zone is spatially resolved by several elements.

3 Results

Results of the FSI computations are initially presented in terms of the temporal development of the glottal opening at the location X_MC (figure 1b). Figure 3 depicts such data in already well-established and self-sustained VF vibration cycles for (a) a case without VF adhesion (model 1) and for (b) a case with VF adhesion (model 3). Henceforth, the time origin (t = 0) is reset to the start of the cycle. The instant at which the cycle ends relative to the time origin is referred to as t_cycle and equals the reciprocal of the vibration frequency f.

Fig. 3.

VF opening distance at X_MC in dependence of time in the (a) absence of adhesion (model 1) and (b) presence of adhesion (model 3)

Several characteristic time-instants within the cycle are identified below. In the absence of adhesion two time instants characterize VF contact. Instant t_c(=3.3 ms) denotes the onset of VF contact at X_MC. From t_c to a subsequent instant t_o(= 4.6 ms) the VF is closed. During the interval [t_c, t_o] compressive tractions develop on the VF surface due to VF contact. Starting at t_o VF opening starts at X_MC, and VF tractions are due to the airflow until contact sets in again in the next cycle. In the presence of VF adhesion the process is more complex. Contact is initiated at t_c(= 3.7 ms), and opening begins at t_o(= 4.5 ms). However, the development of the glottal opening is slowed by the tensile tractions on the VF surface. As the VF opening distance increases the tensile tractions increase correspondingly. The initial elastic response of the ASL at X_MC lasts until t_e(= 5.9 ms). For larger values of VF opening distance the traction at X_MC decreases but remains positive (tensile). This softening behavior is reflected in the time rate of change of the VF opening distance at instant t_e. At a subsequent instant t_f (= 6.3 ms) the collisional (and specifically adhesive) interaction ceases, and for the remainder of the cycle the VF surfaces are free of adhesive tractions, but subjected to tractions from the air flow until contact is initiated in the next cycle.

For the adhesive case (model 3), normal tractions at X_MC in the interval [t_c, t_f] are plotted in figure 4. The maximum compressive tractions (i.e. the impact stress) is found to be 574 Pa. This is within the range of values predicted or measured previously (Gunter et al 2005; Jiang and Titze 1994; Spencer et al 2006; Verdolini et al 1999). The maximum tensile traction is identical to the prescribed cohesive strength σ₀ of the ASL. Even though the elastic characteristic length δ₀ of the traction–separation law is smaller than the softening length (δ_f − δ₀), the duration in which airflow pressure deforms the VF against the build up of adhesive tractions (t_e − t_o) is 3.5 times longer than the duration in which the VF is deformed against the softening ASL (t_f − t_e). The subintervals [t_c, t_o], [t_o, t_e] and [t_e, t_f] correspond to compressive, tensile but increasing, and tensile but diminishing tractions at X_MC, respectively. These subintervals are referred to as compression, tension build-up, and tension break-down, respectively.

Fig. 4.

Tractions due to VF contact and adhesion at X_MC in dependence of time over one collision cycle each from model 1 (no-adhesion) and model 3 (baseline)

Next, tractions along line AB are considered. The distribution along AB of normal traction only due to contact at instants 3.70 ms through 6.35 ms is shown in figure 5. In the compression phase (figure 5a), which starts when at least one location on AB comes into contact, a single CILS grows until it reaches a maximum length. This maximum length is henceforth referred to as l_c,max. The highest compressive traction recorded along AB is 686 Pa. This value is found to occur slightly offset from the mid-coronal plane (x_ap = 0). The subsequent tension build-up phase (figure 5b) begins with tractions on the CILS gradually changing from compressive to tensile. This change is not uniform along AB with tensile and compressive tractions present simultaneously, especially in the beginning of this phase. Towards the end of this phase the distribution of traction becomes more uniform compared to that during compression phase. Yet, non-uniformity in normal traction exists and is highest at the edges of the CILS. The length of the CILS remains constant in the tension build-up phase.

Fig. 5.

Variation of normal traction along AB at different instants: labels identify corresponding times instants (in milliseconds) with respect to cycle start time

The final tension break-down phase (figure 5b) begins once the glottal opening reaches a critical value δ₀ at at least one point on the CILS. The softening behavior of the adhesive interaction initiates predominantly at the ends of the CILS (figure 5c). Furthermore, due to off-center contact the degradation development has an anterior–posterior asymmetry. At 6.25 ms the CILS disintegrates into three disjoint CILSs; each CILS subsequently shrinks to zero.

Table 4 summarizes data for models 1–5 with respect to various energy contributions. One representative vibration cycle from each model is selected such these cycles possess motion characteristics similar to the model 3 cycle as considered above. Note that for the selected cycles the vibration frequency f varies in the range 108–109 Hz. For the cycle corresponding to model 1 table 4 reports the following energy contributions averaged over the whole cycle: strain energy, kinetic energy and viscous damping. For cycles corresponding to models 2–5 energy contributions are reported at the instant at which the energy dissipated to surface adhesion peaks. The energy contributions reported are strain energy, kinetic energy, viscous damping and the energy dissipated to surface adhesion. These energy contributions along with the energy lost to numerical contact damping arise from the (external) work done by the flow pressure on the VF glottal surfaces. In all cases the majority (60 – 75%) of external work is converted to strain energy, approximately 20–30% is converted to kinetic energy and approximately 3–4% is dissipated in viscoelastic damping in the VF. The amount of energy dissipated in surface adhesion in models 2–5 is never more that 2.2% of the total external work. Energy lost to numerical contact damping is always < 0.001% for all models.

Table 4.

Magnitudes of different forms of energy during a representative vibration cycle: for model 1 the cycle average is reported; for models 2–5 values are reported at the instant at which energy dissipated to surface adhesion peaks

Model	Strain energy [μJ]	Kinetic energy [μJ]	Viscous dissipation [μJ]	Surface adhesion [μJ]
1	75.1	20.1	4.83	-
2	35.5	13.6	2.14	0.162
3	33.2	14.5	1.80	0.450
4	34.5	16.5	2.25	1.22
5	32.5	11.6	1.71	0.810

Table 5 summarizes cycle characteristic time variables for the five model conditions considered in terms of the computed quantities t_o, t_c, t_e, t_f and maximum compressive traction at X_MC and the maximum values recorded for l_c for the left VF. For the VF without adhesion (model 1) the time interval during which collisional interaction is active (t_o − t_c) is found to be 1.6 ms. In the presence of adhesion (models 2–5) the active collisional interaction interval (t_f − t_o) substantially increases (from 2.3 ms in model 2 to 3.6 ms in model 5) and this increase closely follows the increase in σ₀ across these models. This extended interval of collisional interaction causes the glottal opening to be inhibited and glottal air flow to be restricted. To compare across models the glottal airflow rate in the cycles considered above is normalized with respect to the airflow rate at cycle start and the minimum airflow rate in the cycle. In figure 6 significant differences between models are evident in derivative of the normalized flow rate immediately following the contact event. Specifically, in the duration 4–7 ms (with respect to start of the cycle) the slope of the airflow rate decreases by nearly 50% going from the no-adhesion model 1 to the high cohesive energy models 4 and 5.

Table 5.

Cycle characteristics for different models of ASL cohesive properties: quantities t_o, t_c, t_e, t_f and maximum compressive traction determined at X_MC, whereas remaining quantities are global measures

Model	tc	to − tc	te − to	tf − te	max. impact stress	max lc	vc/cR	η
	[s]	[s]	[s]	[s]	[kPa]	[mm]	-	-
1	0.0033	0.0016	-	-	1.03	3.04	-	-
2	0.0037	0.0011	0.0010	0.0002	0.679	1.77	10.8	0.523
3	0.0037	0.0008	0.0014	0.0004	0.574	1.32	15.6	0.289
4	0.0032	0.0017	0.0012	0.0006	1.11	3.62	7.58	0.318
5	0.0035	0.0015	0.0020	0.0001	0.949	3.26	4.59	0.102

Fig. 6.

Variation of normalized flow rate during corresponding collision cycles in models 1–5: time origin is set to cycle start time (maximum open state); flow rate is normalized with respect to the initial and minimum values

The first instant with respect to the cycle start time at which any location on AB first undergoes contact is referred to as t_c,min. For the same VF vibration cycles analysed, consider the interval [t_c,min, t_cycle]. Figure 7a shows the combined length l_c, of the CILSs that appear on AB, in dependence of time for the models 1–5. Time and collisional interaction line length are normalized as

Fig. 7.

(a) Variation of normalized collisional interaction line length l_c/l_c,max in dependence of normalized time (t − t_c,min)/(t_cycle − t_c,min); (b) number of disjoint collisional interaction line segments in dependence of time

$$\widehat{t}\equiv (t-t_{c,min})/(t_{\text{cycle}}-t_{c,min}),t\in [t_{c,min},t_{\text{cycle}}]$$ (5)

$$\lambda \equiv l_c/l_{c,max},\text{where}{l}_{c,max}=\underset{\widehat{t}\in [0,1]}{max}{l}_c,$$ (6)

in order to remove the variations due to differences in severity of collision between models. The dependence of λ on t̂ is then expected to vary mostly due to the variation in adhesion. Here a clear order of adhesive strength on the development of l_c emerges. The stronger the adhesion the further extended l_c=1 becomes in time.

During the process of tension break-down the CILS on AB may disintegrate into multiple pieces. In contrast with l_c which refers to the combined length of all segments figure 7b shows the number of disjoint CILSs that are contained in AB as a function of time in the interval [t_c,min, t_cycle].

4 Discussion

Before discussing the influence of ASL adhesive properties on the mechanics of VF vibration, some remarks are made about the present model. Stroboscopic images of oscillating VFs (Hsiung 2004) indicate that in severe cases of mucus aggregation, the ASL on the opposite VFs can form a connected fluidic bridge during phonation. For all the models studied here, VF vibration is found to be sufficiently high to cause total failure of the ASL in each cycle. Across all models medial–lateral displacement u_ml at X_MC is found to be ~ −0.180 mm when averaged over the collision cycles. For this displacement level a continuous ASL connecting the two VFs is estimated to rupture at δ_f = 2 |x_ml(X_MC)| = 2 |−d_g/2 + 〈u_ml〉 (X_MC)| = O(1 mm) where 〈·〉 denotes an average taken over the collision cycle. If such an ASL is purely aqueous in composition then it follows from (3) that it cohesive strength is σ₀ = 144 Pa. The values of σ₀ considered in this study (table 3) are then up to an order of magnitude higher than that in the purely aqueous ASL. The evidence that adhesive nature of the ASL varies significantly within subjects (Hsiung 2004) supports the range considered herein.

It was remarked earlier that due to computational modeling limitations a leakage flow occurs when the opposing VF surfaces are actively undergoing collisional interaction with the corresponding rigid planes. Since the state of collisional interaction is tracked by the variable χ, the instantaneous leakage flow is higher when the condition χ = 1 hold over a larger anterior–posterior extent of the VF surface. Hence an upper bound of the glottal area through which the leakage occurs can be estimated as (2δ_f + δ_p)l_c,max. The effect of the leakage flow is expected to confound determination of absolute flow rate values, but not the normalized flow rate presented in this paper. The reduction in the derivative of the airflow rate (or glottal flow derivative GFD as referred to in voice literature) corresponding to an increase in cohesive energy perhaps provides an interesting biomechanical insight. Peterson-Falzone et al (1981) found that the absence of ASL on vocal fold surfaces led to breathy voice in patients of ectodermal dysplasia, whereas it is well known that a smooth and sinusoidal GFD (such as in the absence of ASL in case 1) is correlated with increased breathiness in speech (Epstein 1999). Thus the present results indicate that increased ASL activity can decrease the breathiness in speech by directly decreasing GFD. It is also interesting to note that GFD has been studied extensively in the field of speaker identification (Plumpe et al 1999). Modulation in GFD through ASL adhesive properties further suggests the possibility of variability in speech characteristics of the same speaker.

Implicit algorithms used in the fluid and solid solvers ensure that the effect of varying the time-step size is limited to the accuracy of the solution while the stability of the solution remains unaffected. The influence of the time-step used herein was evaluated separately for the flow solver. A 2D model of flow past rigid VF was constructed to possess geometry, mesh refinement and boundary conditions similar to the present model (identical to Suh and Frankel 2007). This 2D model was analyzed with time-step and time-integration algorithm identical to the present model. The computed flow pressures on the VF surface were found to agree with experimental measurements (Scherer et al 2001) within 8 % accuracy. In the solid domain part of the FSI model, the most rapidly varying quantity is the tensile traction due to adhesive ASL. The variation of tensile traction occurs within a duration that is orders of magnitude smaller than characteristic durations of VF vibration and of VF viscoelastic stress-relaxation. Thus accuracy of the solid model is established by ensuring only the accuracy of the tensile tractions. Considering model 3 during the collisional cycle, all nodes on line AB that went into contact were found to attain peak tensile stresses that were within 95 % of the imposed cohesive strength value. Thus the fixed solution time increment of 50 μs is found to accurately capture all relevant details in the flow and solid domain solutions, and is not expected to influence the results presented here.

In comparing across models 1–5 it is firstly noted that according to table 4 the strain energy, kinetic energy and viscous damping contributions always account for > 97 % of the external work (i.e. by the airflow) on the VF. Adhesion accounts for only a minor part (< 3 %). Despite the small amount of energy dissipated in adhesion, ASL adhesive properties significantly influence the VF vibration characteristics as detailed in table 5. The discussion below attempts to elucidate the underlying mechanics leading to the predicted differences in vibration characteristics.

A variable that captures VF mechanics just prior to adhesive interaction and yet due to contact interaction is the computed maximum impact stress. Both table 5 and figure 4 show significant differences in maximum compressive stress achieved at X_MC between the models. The maximum impact stress is expected to be dependent strongly on the severity of collision. A measure of the severity of collision is the closed quotient (CQ) defined as the fraction of the vibration period during which the VF opening distance at X_MC is zero (i.e. contact is closed). Since the vibration frequency does not differ significantly across the different models, the compressive interval duration t_o − t_c in each model is proportional to its CQ. Therefore, the impact stress is expected to scale with t_o − t_c. Indeed, the maximum impact stress is found to increase with increase in the t_o − t_c (table 5).

Beyond the marginally open instant t_o, the surface normal stress increases from zero to σ₀ over the tensile interval [t_o, t_e] (table 5). The interval length t_e − t_o is expected to decrease as t_o − t_c increases, since for fixed f a smaller duration is available to return to the fully open state. Simultaneously, t_e − t_o is expected to increase as σ₀ increases. However, it is difficult to determine a quantitative relationship explaining the variation of t_e − t_o in dependence of t_o − t_c and σ₀. Qualitatively, the effect of t_o − t_c can be inferred by comparing models 3 and 4 (identical σ₀). The smaller t_e − t_o in model 4 compared to model 3 is explained by the larger t_o − t_c of model 4. The effect of σ₀ is inferred by comparing model 2 with model 3 or comparing model 4 with model 5. In each model pair t_o − t_c is of similar order. The increase in t_e − t_o from model 2 to model 3, and from models 4 and 5 is explained by corresponding increases in σ₀.

The length of the degrading interval [t_e, t_f] (table 5) expectedly increases from model 3 to model 4 because δ_f is relatively larger in the model 4. The significantly shorter degrading interval for model 5 compared to other models is attributed to σ₀ being the largest in model 5 whereas δ_f in model 5 is identical or smaller than in other models. The net result is that in model 5 the (restitutive) stress state at t_e is higher than in other models at corresponding t_e instants. When the ASL degrades entirely, the higher stress-state produces a higher restitutive acceleration.

In the same manner as the ASL adhesive properties influence the vibration characteristics of point X_MC, so also do ASL adhesive properties determine vibration characteristics of line AB as a whole. In this respect consider the normalized forms of time and total CILS length i.e. t̂ and λ. The variable t̂_o defines the normalized time instant t̂ when λ increases to 1. It is expected that for t̂ ≤ t̂_o, most points in AB are in compression phase (figure 5a) and the dynamics is not influenced by the ASL. This explains t̂_o ~ 0.1 for all models in figure 7a.

For t̂ > t̂_o the VF without ASL (model 1) loses contact on AB such that for t̂ ≥ 0.3, λ = 0 up to the end of the vibration cycle. For the models with ASL, the behavior after t̂_c is significantly different from model 1. As the VF begins to move laterally, various locations on the collisional interaction line are in the compression phase, in the tensile phase, and in the degrading phase. The variable t̂_d refers to earliest time instant when at least one point on the CILS is separated from P_L by δ_f. Thus λ = 1 for all t̂ ∈ [t̂_o, t̂_d]. The duration t̂_d − t̂_o is a complex interplay between the airflow forces on the non-contacting surface of the VF and the cohesive tractions. Compared to model 3, the lower σ₀ in model 2 slightly decreases t̂_d − t̂_o whereas the larger δ_f in model 4 causes t̂_d − t̂_o to increase. Compared to model 4, model 5 has larger σ₀ but smaller δ_f, and these changes produce opposite effects. However, it can be remarked that between models 4 and 5 the effect of σ₀ dominates the effect of δ_f resulting in a net increase of t̂_d − t̂_c in model 5.

Finally, the instant when degradation of the ASL is complete at all points of the CILS, is indicated by t̂_f. Figure 7a indicates that the length of the interval [0, t̂_f] increases with increase in cohesive surface energy ϕ of the ASL. The reciprocal of (t̂_f − t̂_d) is a measure of the average speed with which the collisional interaction line recedes. For models 2–5, this dimensionless speed was found to be 14.1, 22.9, 8.76 and 5.02 respectively. The average speed of reduction in l_c is determined as

$$v_c=\frac{l_{c,max}}{({\widehat{t}}_f-{\widehat{t}}_d)(t_{\text{cycle}}-t_{c,min})}.$$ (7)

In figure 7a the significant changes in the slope of λ with respect to t̂ in model 5 suggests that the instantaneous speed of CILS reduction can deviate significantly from the average speed v_c. For all models table 5 shows the ratio of v_c to the Rayleigh wave speed in the VF tissue (Freund 1990)

$$c_R-\frac{0.862+1.14\nu }{1+\nu }\sqrt{\frac{E}{2{\rho }_s(1+\nu )}},$$ (8)

which is always found to be O(10¹). Note that v_c does not capture the propagation speed of an individual VF bond patch, and hence v_c > c_R does not imply a necessarily supersonic decohesion process. Specifically, l_c remains constant even as the separation between the line AB and rigid plane increases until the ASL at at least one location fails completely. Moreover, due to the three-dimensionality of the ASL decohesion, decrease in l_c is due to the propagation of multiple debonds. For e.g. figure 7b shows that the original CILS may disintegrate into several disjoint CILSs. The multiplicity of disjoint CILS suggests a fingering instability phenomenon.

In adhesive contact of soft elastic materials (similar to the ASL in tension) instabilities can occur in the debond process (Ghatak and Chaudhury 2003; Vilmin et al 2009). Such fingering instabilities are understood to occur with a characteristic wavelength that can be related to the constitutive properties of the ASL (i.e. elasticity and traction–separation law parameters). Given the present configuration, is expected that the range of ASL constitutive parameters considered lead to differences observed in CILS disintegration patterns between models in figure 7b.

For E/σ₀ ≫ 1, referred to as adhesive regime, the failure process can be modeled by an interface of infinitesimal thickness. In the present study, E/σ₀ was O(10¹) and hence the ASL has zero initial thickness. In Needleman (1990), failure of an adhesive interface under tension was analyzed considering E/σ₀ fixed at 167. The interface failure mechanism was studied in dependence of a parameter that corresponds to the ratio l_c,max/δ_f in the present study. Needleman (1990) found that for l_c,max/d_f ≪ 10³ the interface fails in a manner characteristic of a uniform separation process, as opposed to a progressive debond propagation process. The main feature of a uniform separation process is that cohesive tractions are distributed homogeneously along the interface length, and degradation proceeds uniformly. In figure 5b, c for model 3, and also for all models with ASL (models 2–5) considered in this study, the process of VF separation under adhesive condition demonstrated a uniform separation type behavior. This is consistent with the fact that l_c,max/δ_f was found to be O(10¹) for all the models. Table 5, column 8, shows that the ratio

$$\eta \equiv \frac{E/{\sigma }_0}{l_{c,max}/{\delta }_f}$$ (9)

is indeed O(1) for models 2–5. It is interesting to note that a higher η corresponds to an increased number of segments of the original collisional interaction line during the degradation process (figure 7b).

It is perhaps biomechanically relevant to note that immediately outside the CILS the normal tractions (due to airflow) and are typically compressive and thus opposite in sense to the normal tractions inside the CILS due to adhesion. Hence large gradients in normal traction can result at the CILS boundary and possibly lead to tissue damage in the interior.

The formation of multiple ASL bridges has been reported previously in clinical visualization studies (Hsiao et al 2002; Bonilha et al 2008, 2012). Qualitative characterization undertaken in these studies has aided in distinguishing between voice disorders (Hsiao et al 2002; Hsiung 2004; Bonilha et al 2012). A typical ASL characteristic evaluated is referred to as pooling, and is defined as the portion of VF length over which ASL bridges form (Bonilha et al 2012). Thus ASL pooling is expected to be related closely to the quantity l_c arising from the present definition of the CILS. This highlights the relevance of quantitative descriptors such as l_c, v_c and η detailed in this study.

In characterizing the surface interaction of the ASL, the present study used the measured properties of water in surface tension. It is expected that direct experimental characterization of the ASL will lead to a better understanding of the ASL mechanical behavior, and thus enable a more precise computation of its influence on VF dynamics. A major challenge in experimentally characterizing any surface interaction is to isolate the surface interaction from the background mechanical response of materials on either side of the interface. In the computational model the ASL is attached to the VF tissue (the mechanical response of which varies across samples) and on the other side the ASL interacts with a rigid surface. In Atomic Force Microscopy (AFM) a tip of a known shape (e.g. sphere) and mechanical properties is attached to a cantilever. Typically the tip material is significantly stiffer than the substrate (VF in this case) and hence the tip can be idealized as rigid. Using techniques developed for analyzing nano-scale contact in the presence of adhesion (Lin et al 2007a,b; Leite et al 2012) mechanical properties of the ASL and the underlying VF tissue can perhaps be better quantified.

5 Conclusion

The present study documents numerical simulations of VF vibration taking into account both collision as well as adhesion on the VF surface. Prior work on simulation on VF adhesion and phonation had been substantially more restrictive than the present study. The results presented highlight the important role ASL mediated adhesion can play in influencing both flow and tissue relevant characteristics, as well as collisional interaction on the VF surface. Specifically, it is found that an increase in cohesive energy of the ASL adhesion was found to lead to a reduction in GFD. It may be inferred that through its influence on GFD, the ASL influences characteristics of speech quality e.g. breathiness.

The effects of ASL adhesive properties on VF collisional interaction were high-lighted by focusing on an anterior-posterior oriented line AB situated on the medial plane. The following observations were found to hold in general

1. length of the tensile interval increases with increase in σ₀,

2. for fixed σ₀, the length of the degrading interval increases with increase in δ_f, and

3. for fixed δ_f, the length of the degrading interval decreases with increase in σ₀.

In this study a CILS was defined as a continuous line segment within line AB such that at every point on it is active in collision or adhesion. With respect to the anterior-posterior line AB the variables l_c, v_c were defined to represent, respectively, the cumulative length of CILSs on AB and the average speed with which this cumulative length recedes to zero. It was found that cohesive surface energy ϕ strongly influenced the variation of l_c with time. Specifically, a higher cohesive surface energy ϕ resulted in a delayed onset of degradation and a longer time spent in contact (figure 7a). For all the models v_c was found to be larger than the Rayleigh wave speed of the VF tissue. This high speed of ASL failure agrees with the finding that ASL failure is of a rather uniform separation type than a progressive debonding event. The determination that the VF separates uniformly rather than by growth from a debond tip was also inferred from the typical computed values of the length scale ratio η using concepts from the field of non-linear fracture mechanics. Lastly, the number of smaller disjoint CILSs formed on AB during breaking of ASL adhesion was also considered in dependence of ASL adhesive properties. For models in which η was higher, the number of smaller disjoint CILSs was found to decrease. ASL kinematics has been well visualized in a clinical setting. Until now the kinematics was characterized by qualitative parameters such as ASL pooling. From the expected link between ASL pooling and CILS quantitative variables (l_c, v_c and η), ASL pooling is inferred to be ultimately controlled by its adhesive properties.

The effect of ASL adhesion on VF tissue is a direct result of the altered VF vibration characteristics outlined above. Specifically, it was noted that ASL adhesion can cause sharp gradients in normal tractions at the boundary of the CILSs but also in general on the boundary of the collisional interaction zone. The magnitude of the gradients will depend on all ASL adhesive traction–separation law parameters considered here i.e. σ₀, ϕ and δ_f, since these parameters determine how long collisional interaction lasts and the nature of the collisional interaction (compressive, tensile or degrading) over time.

The present study advances the current knowledge of biomechanical aspects of VF dynamics under the influence of glottal surface adhesion. In this study, ASL adhesive behavior as parameterized by the cohesive strength, cohesive energy and rupture length was varied in a potentially physiologically representative range. The results of the study strongly suggest that ASL adhesive behavior might strongly influence VF tissue health and voice quality. Accurate experimental characterization of ASL adhesive behavior is thus imperative to assessing voice health, and further research in this direction is recommended.

Glossary

Roman symbols

AB

Line oriented in anterior–posterior direction and situated on medial plane

C_L,C_R

Collision-prone parts of S_L and S_R

c_R

Rayleigh wave-speed in VF tissue

d_g

Initial distance between VFs

d_p

Fixed separation between planes P_L and P_R

D

VF medial-lateral extent (depth)

δD_v

Virtual strain tensor associated with δu_v

ε

Volumetric strain

e

Deviatoric strain tensor

E

Elastic modulus of VF tissue

f

Vibration frequency

g₁

Viscoelastic shear-modulus relaxation factor

G

VF tissue shear modulus

I

Second-order identity tensor

k₁

Viscoelastic bulk-modulus relaxation factor

K

VF tissue bulk modulus

l_c

Total length of all interaction line segments

l_cmax

Maximum value of l_c over a cycle

L

VF anterior-posterior extent (length)

n̂

Unit vector directed normal to surface

p

Flow pressure

p_in

Time-dependent flow pressure at glottal air tract inlet

p_max

Maximum flow pressure at glottal air tract inlet

P_L,P_R

Rigid planes

r̂

Arbitrary unit vector parallel to surface

S_L, S_R

Glottal surfaces; flow–structure interaction surfaces

t

Time

Δt

Time increment

t_c,t_o, t_e, t_f

Dimensional times denoting local events related to ASL

t_c,_min

Time instant corresponding to initiation of contact along line AB

t_cycle

Time instant corresponding to end of collision cycle

t_ramp

Time duration for inlet pressure ramp

t̂,t̃

Non-dimensional time variables

t̂_o,t̂_d, t̂_f

Non-dimensional times denoting global events related to ASL

T

VF inferior-superior extent (thickness)

T_entry

Subglottal channel length

T_exit

Supraglottal channel length

δu_v

Arbitrary virtual displacement

u

VF displacement

v

Airflow velocity

v_g

Airflow domain grid velocity

v_c

Average speed at which interaction line segment recedes

V^f

Glottal air tract volume

V^s

Volume of space occupied by VF pair

∂(V^f)

Glottal air tract boundary

∂(V^s)

Bounding surfaces of VF pair

W

Glottal air tract width

x_is,x_ml, x_ap

Coordinate axes in inferior–superior, medial–lateral and anterior–posterior directions respectively

X_MC

Reference location on mid-coronal plane prone to collision

Operators

(^·)

Time derivative

d

Differential operator

∇

Spatial gradient

()^T

Transpose

〈·〉

Average taken over cycle time period

:

Double contraction between two tensors

Greek symbols

δ₀

Linear-elastic length limit of VF mucus layer

δ_f

Rupture length limit of VF mucus layer

δ_n

Length of mucus layer

ϕ

Cohesive energy or surface tension

ϕ_aq

Surface tension of water

${\varphi }_{\text{aq}}^{\ast }$

Cohesive energy of an equivalent one-sided aqueous ASL

η

Non-dimensional ratio of VF bulk and ASL adhesive properties

λ

Normalized interaction line segment length

μ

Dynamic viscosity of air

ν

Poisson’s ratio of VF tissue

ρ_f

Density of air

ρ_s

Density of VF tissue

σ

Cauchy stress tensor in VF volume

σ₀

Cohesive strength of mucus layer

σ_n

Cohesive traction due to mucus layer

τ₁

Viscoelastic relaxation rate

τ_f

Stress tensor within air flow domain

τ_s

Traction on VF surface

χ

Collision state variable