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.

The influence of non-polar lipids on tear film dynamics

“…Evaporation model We assume that the evaporative transfer rate, E, depends only on the thickness of the non-polar lipid layer. In practice, E will also depend on the thickness of the aqueous layer, in that, as the aqueous layer becomes very thin, the evaporation rate will decrease due to the presence of the mucous layer (Zubkov et al 2012). Our model for the evaporation is the simplest possible form that is consistent with the data described earlier; the evaporation rate increases as the amount of lipid decreases, namely…”

“…This will combine the work presented in this paper with that described in Zubkov et al (2012). Such a study should enable us, for the first time, to link increases in tear film osmolarity with deficiencies in the non-polar lipid layer, and thus give valuable insight into evaporative dry eye.…”

“…We see that evaporation has a significant effect on the local film thickness near the upper black line by the end on the interblink, but there is no observable change in the menisci. As discussed in Zubkov et al (2012), evaporative reduction of the aqueous layer can result in significant increases in osmolarity (see Liu et al 2009), and thus discomfort to the patient. We note that our simulations show that the film thins to a thickness where intermolecular forces might become important; we do not examine this further in this paper.…”

“…The paper contains an asymptotic analysis of the model to explain the early time behaviour of the system. This model is used in Zubkov, Breward & Gaffney (2012) as the basis for examining tear film osmolarity. In this paper, the salt concentration is tracked during blinking and the interblink, as well as during a saccade.…”

A Model for the Tear Film and Ocular Surface Temperature for Partial Blinks

“…The tear film is commonly assumed to be Newtonian, and the underlying substrate (the cornea) is assumed to be flat (Berger and Corrsin, 1974; Braun et al, 2012). Mathematical studies have incorporated a variety of important effects: surface tension (McDonald and Brubaker, 1971;Wong et al, 1996; Sharma et al, 1998; Miller et al, 2002); polar lipid surface concentration gradients causing the Marangoni effect (Berger and Corrsin, 1974; Jones et al, 2005, 2006; Aydemir et al, 2010); evaporation (Braun and Fitt, 2003; Winter et al, 2010; King-Smith et al, 2009); wettability of the corneal surface via van der Waals terms (Zhang et al, 2003, 2004; Winter et al, 2010); motion of the eyelids in one space dimension (Jones et al, 2005, 2006; Aydemir et al, 2010; Braun and King-Smith, 2007; Heryudono et al, 2007; Maki et al, 2008; Jossic et al, 2009; Zubkov et al, 2012); heat transfer posterior to the tear film (Li and Braun, 2012); and the shape of the eye opening (Maki et al, 2009a, b). These effects may all contribute in different regions of the eye and at different times in the blink cycle.…”