Tear-film breakup: The role of membrane-associated mucin polymers

Mucin polymers in the tear film protect the corneal surface from pathogens and modulate the tear-film flow characteristics. Recent studies have suggested a relationship between the loss of membrane-associated mucins and premature rupture of the tear film in various eye diseases. This work aims to elucidate the hydrodynamic mechanisms by which loss of membrane-associated mucins causes premature tear-film rupture. We model the bulk of the tear film as a Newtonian fluid in a two-dimensional periodic domain, and the lipid layer at the air-tear interface as insoluble surfactants. Gradual loss of membrane-associated mucins produces growing areas of exposed cornea in direct contact with the tear fluid. We represent the hydrodynamic consequences of this morphological change through two mechanisms: an increased van der Waals attraction due to loss of wettability on the exposed area, and a change of boundary condition from an effective negative slip on the mucin-covered areas to the no-slip condition on exposed cornea. Finite-element computations, with an arbitrary Lagrangian-Eulerian scheme to handle the moving interface, demonstrate a strong effect of the elevated van der Waals attraction on precipitating tear-film breakup. The change in boundary condition on the cornea has a relatively minor role. Using realistic parameters, our heterogeneous mucin model is able to predict quantitatively the shortening of tear-film breakup time observed in diseased eyes.

Figure 1

Schematic representation of a tear film with a partially damaged mucus layer, with the membrane-associated mucins (MAMs) clipped from the central portion of the corneal epithelium and shed into the aqueous layer. The protrusions on the epithelium represent the corneal microplicae [2, 5].

Figure 2

Schematic of the computational domain. The MAM-depleted region, marked in red on the substrate, has length $l_2$ and sits in the middle of the domain between two MAM-intact regions of length $l_1$. The entire domain is periodic along the $x$ direction. The MAM-depleted region features a stronger van der Waals attraction with the interface above it (Hamaker constant ${\mathcal{A}}_2>{\mathcal{A}}_1$) and a no-slip condition on the substrate (Navier slip length ${\beta }_2=0$). The regions with intact MAMs has a negative slip (${\beta }_1<0$).

Figure 3

Determination of the negative slip length ${\beta }_1$ based on matching the tear-film rupture time in a healthy eye. The three curves are the temporal evolution of the minimum film thickness $h_{\text{min}}\left(t\right)$ predicted by the continuous viscosity model [12] and the current heterogeneous mucin model for two different values of ${\beta }_1$ over the entire substrate.

Figure 4

Validation of our numerical resolution for the capillary breakup of a Newtonian liquid filament against the numerical results of Ashgriz and Mashayek [54] at $\text{Re}=200$ and wavenumber $k=0.2/R$. $R$ is the mean radius of the filament, $r_s$ is the radius at the crest, ${\epsilon }_0$ is the initial amplitude of the sinusoidal perturbation, and $N$ is the number of axial elements. Our solutions at the two resolutions practically overlap.

Figure 5

Tear-film rupture with the central portion of the substrate between the vertical dotted lines being cleared of mucin ($f=1/3$, ${\mathcal{A}}_r=2$, ${\beta }_1=-0.011$, ${\beta }_2=0$, $d=0.01$). The gray scale contours and scale bar show the magnitude of velocity with streamlines superimposed, while the vectors represent velocities with logarithmic magnitude as its length. The frames are at times (a) $t=0.36$ s, (b) $t=2.57$ s, (c) $t=4.65$ s, and (d) $t=5.21$ s, shortly before the tear-film ruptures at $t=5.35$ s.

Figure 6

Effect of the transition width $d$. (a) The interfacial profile just before breakup. The three profiles for $d=0$, ${10}^{-3}$, and ${10}^{-2}$ practically overlap. (b) The rupture time increases moderately with the transition width.

Figure 7

Increasing the Hamaker constant ${\mathcal{A}}_2$ (or equivalently the ratio ${\mathcal{A}}_r$) hastens tear-film thinning and shortens the breakup time. We have fixed ${\mathcal{A}}_1$ at the baseline value, $f=1/3$ and ${\beta }_1={\beta }_2=-0.011$. (a) Time evolution of the film thickness $h_c$ at the center of the domain for different values of ${\mathcal{A}}_r$. (b) The rupture time decreases with increasing ${\mathcal{A}}_r$. We have also included a curve that corresponds to ${\beta }_2=0$, with no-slip condition on the MAM-depleted region.

Figure 8

Effect of expanding the MAM-depleted region on the tear-film rupture time. For the dashed curve, we have imposed the same Hamaker constant on the MAM-depleted region (${\mathcal{A}}_r=1$) but the no-slip boundary condition: ${\beta }_1=-0.011$, ${\beta }_2=0$. For the two solid curves, the MAM-depleted region features an elevated van der Waals attraction along with the no-slip boundary condition: ${\mathcal{A}}_r=2$, ${\beta }_2=0$. The two curves correspond to different choices of the domain length $\lambda$ as explained in the text.

Figure 9

Linear instability analysis of tear-film breakup on a homogeneous substrate with Hamaker constant $\mathcal{A}$ and slip coefficient $\beta =-0.011$. The other parameters are at baseline values. (a) Dispersion relation for three $\mathcal{A}$ values. (b) The wavelength ${\lambda }_m$ decreases monotonically with $\mathcal{A}$.