Coupling Fluid and Solute Dynamics Within the Ocular Surface Tear Film: A Modelling Study of Black Line Osmolarity

Abstract

We present a mathematical model describing the spatial distribution of tear film osmolarity across the ocular surface of a human eye during one blink cycle, incorporating detailed fluid and solute dynamics. Based on the lubrication approximation, our model comprises three coupled equations tracking the depth of the aqueous layer of the tear film, the concentration of the polar lipid, and the concentration of physiological salts contained in the aqueous layer. Diffusive boundary layers in the salt concentration occur at the thinnest regions of the tear film, the black lines. Thus, despite large Peclet numbers, diffusion ameliorates osmolarity around the black lines, but nonetheless is insufficient to eliminate the build-up of solute in these regions. More generally, a heterogeneous distribution of solute concentration is predicted across the ocular surface, indicating that measurements of lower meniscus osmolarity are not globally representative, especially in the presence of dry eye.

Vertical saccadic eyelid motion can reduce osmolarity at the lower black line, raising the prospect that select eyeball motions more generally can assist in alleviating tear film hyperosmolarity. Finally, our results indicate that measured evaporative rates will induce excessive hyperosmolarity at the black lines, even for the healthy eye. This suggests that further evaporative retardation at the black lines, for instance due to the cellular glycocalyx at the ocular surface or increasing concentrations of mucus, will be important for controlling hyperosmolarity as the black line thins.

Appendix: Derivation of the Lubrication Model

We derive a model to describe the fluid motion, polar lipid concentration and salt concentration during a blink and during a saccade. In both cases, we neglect inertial terms. In addition, for saccadic motion we also neglect the non-inertial forces arising from the rotating frame, since it is straightforward to show that viscous effects dominate over rotational effects everywhere, even in the menisci. Hence, we proceed by substituting (29) and (30) into the model equations (3), (14), (17). Omitting the primes, we obtain the following system of equations:

(48)

(49)

(50)

(51)

(52)

where

$$ C=\frac{\gamma_{0}\varepsilon^{3}}{\mu U},\qquad B=\frac{\rho g{L_{\mathrm{op}}^{2}}}{\varepsilon \gamma_{0}},\qquad \mathit{Pe}_{c}= \frac{UL_{\mathrm{op}}}{D_{c}},\qquad \mathit{Pe}_{\varGamma }=\frac{UL_{\mathrm{op}}}{D_{\varGamma }}, $$

(53)

and we note that we choose to retain u _s as a primary variable. The boundary conditions (1), (2), (4)–(13), (15), (16), (18)–(24) become

(54)

(55)

(56)

(57)

where

$$ M=\frac{\varepsilon RT\varGamma^{\ast }}{\mu U}. $$

(58)

The evaporation term in (57) is given by

$$ \tilde{E}=\frac{\tilde{E}_{0}h}{\tilde{\delta}+h}\tilde{f}(x,\tilde{a}),\quad \tilde{f}(x,\tilde{a})=\left \{ \begin{array}{l@{\quad }l} 0.5-0.5\cos (\pi x/\tilde{a}), & {0\leqslant x<\tilde{a}} ,\\ 1, & {\tilde{a}\leqslant x<1-\tilde{a}} ,\\ 0.5+0.5\cos (\pi ( x-1+\tilde{a} ) /\tilde{a}), & {1-\tilde{a}\leqslant x\leqslant 1}.\end{array} \right . $$

(59)

The boundary conditions on the fluid problem are

(60)

(61)

where

$$ \tilde{L}_{\mathrm{up}}(t)=\left \{ \begin{array}{l@{\quad }l} \tilde{L}_{\mathrm{cl}}+\tilde{U}_{0}\tilde{\tau} ( -\frac{1}{2} [ \frac{t}{\tilde{\tau}} ]^{2}+\lambda [ \frac{\sqrt{\pi }}{2}\operatorname {erf}( \sqrt{\frac{t}{\tilde{\tau}}} ) -\sqrt{\frac{t}{\tilde{\tau}}}e^{-t/\tilde{\tau}} ] ) , & 0\leqslant t<\tilde{t}_{\mathrm{op}}, \\ 1, & \tilde{t}_{\mathrm{op}}\leqslant t\leqslant \tilde{t}_{\mathrm{ib}}, \\ 1- ( 1-\tilde{L}_{\mathrm{cl}} ) ( \frac{t-\tilde{t}_{\mathrm{ib}}}{\tilde{\tau}_{\mathrm{cl}}} )^{2}\exp ( 1- ( \frac{t-\tilde{t}_{\mathrm{ib}}}{\tilde{\tau}_{\mathrm{cl}}} )^{2} ) , & \tilde{t}_{\mathrm{ib}}<t\leqslant \tilde{t}_{\mathrm{ib}}+\tilde{\tau}_{\mathrm{cl}} \end{array} \right . $$

(62)

and

(63)

(64)

The boundary conditions on the lipid problem are

$$ u_s\varGamma -\frac{1}{\mathit{Pe}_{\varGamma } (1+\varepsilon^2 h_x^2 )^{\frac{1}{2}}} \varGamma_x= \tilde{U}_{\mathrm{low}/\mathrm{up}}\varGamma \quad \mathrm{at}\ x= \tilde{L}_{\mathrm{low}/\mathrm{up}}, $$

(65)

while the boundary conditions on the salt problem are

(66)

(67)

(68)

(69)

The initial conditions are given by

$$ h=\tilde{h}_{0}(x),\qquad\varGamma =1,\qquad c=1\quad \mathrm{at}\ t=0, $$

(70)

where $\tilde{h}_{0}(x)$ satisfies

$$ \int_{0}^{\tilde{L_{\mathrm{cl}}} }\tilde{h}_{0}\, \mathrm{d}x=\tilde{V}_{0},\qquad \tilde{h}_{0}(x=0, \tilde{L}_{\mathrm{cl}})=\tilde{h}^{\ast },\qquad \biggl[\frac{\tilde{h}_{0xx}}{ ( 1+\varepsilon^{2}\tilde{h}_{0x}^{2} )^{3/2}} \biggr]_x=B. $$

(71)

Assuming that ε≪1, we can take advantage of the lubrication approximation. Following the usual routine of lubrication analysis, we solve the boundary-value problem for u, v and p expressing them as expansions in ε ², which yields (e.g.)

$$ u=-C [ h_{xxx}-B ] \biggl( \frac{y^{2}}{2}-hy \biggr) -M \varGamma_{x}y+O \bigl( \varepsilon^{2} \bigr) . $$

(72)

Substituting (72) into (51), (57), and omitting O(ε ²) terms, we obtain

$$ \begin{aligned} &h_{t}+(\bar{u}h)_{x}=-\tilde{E},\qquad \bar{u}=\frac{1}{3}Ch^{2} [ h_{xxx}-B ] - \frac{1}{2}Mh\varGamma_{x},\\ &\varGamma_{t}+ \biggl( u_{s}\varGamma -\frac{1}{\mathit{Pe}_{\varGamma }} \varGamma_{x} \biggr)_{x}=0,\qquad u_{s}= \frac{1}{2}Ch^{2} [ h_{xxx}-B ] -Mh \varGamma_{x}.\end{aligned} $$

(73)

A similar system was previously derived by Jones et al. (2006) and Aydemir et al. (2011).

Let us now consider the equation for the salt concentration. Substituting the salt concentration expansion

$$ c=c_{0}+\varepsilon^{2}c_{1}+O \bigl( \varepsilon^{4} \bigr) $$

(74)

into (52), (66), (67), and additionally assuming that Pe _c ∼O(1) and $\tilde{E}_{0}\sim O(1)$, our leading-order model is simply

(75)

(76)

Thus,

$$ c_{0}=c_{0}(x,t). $$

(77)

In order to find an equation for c ₀, we proceed to next order in the field equations and boundary conditions, and find that

(78)

(79)

(80)

Integrating (78) across the tear film and using the boundary conditions (79), (80), we obtain

$$ h ( c_{0t}+\bar{u}c_{0x} ) =\frac{1}{\mathit{Pe}_{c}} [ hc_{0x} ]_{x}+\tilde{E}c_{0} . $$

(81)

To obtain boundary conditions for Eqs. (73), (81), we use (65) and integrate (61), (68), (69) across the tear film, giving

(82)

Omitting O(ε ²) terms in Eqs. (71), we find that the initial tear film depth is given by

$$ \begin{aligned}[b] \tilde{h}_{0}(x,\tilde{V}_{0})&=\frac{B}{6}x^{3}- \biggl( \frac{6 ( \tilde{V}_{0}-\tilde{h}^{\ast }\tilde{L}_{\mathrm{cl}} ) }{\tilde{L}_{\mathrm{cl}}^{3}}+\frac{\tilde{L}_{\mathrm{cl}}B}{4} \biggr) x^{2}\\ &\quad + \biggl( \frac{6 ( \tilde{V}_{0}-\tilde{h}^{\ast }\tilde{L}_{\mathrm{cl}} ) }{L_{\mathrm{cl}}^{2}}+\frac{\tilde{L}_{\mathrm{cl}}^{2}B}{12}\biggr) x+\tilde{h}^{\ast } \quad \mathrm{at}\ t=0.\end{aligned} $$

(83)