Abstract
Extensive studies have explored the dynamics of the ocular surface fluid, though theoretical investigations are typically limited to the use of the lubrication approximation, which is not guaranteed to be uniformly valid a-priori throughout the tear meniscus. However, resolving tear film behaviour within the meniscus and especially its apices is required to characterise the flow dynamics where the tear film is especially thin, and thus most susceptible to evaporatively induced hyperosmolarity and subsequent epithelial damage. Hence, we have explored the accuracy of the standard lubrication approximation for the tear film by explicit comparisons with the 2D Navier–Stokes model, considering both stationary and moving eyelids. Our results demonstrate that the lubrication model is qualitatively accurate except in the vicinity of the eyelids. In particular, and in contrast to lubrication theory, the solution of the full Navier–Stokes equations predict a distinct absence of fluid flow, and thus convective mixing in the region adjacent to the tear film contact line. These observations not only support emergent hypotheses concerning the formation of Marx’s line, a region of epithelial cell staining adjacent to the contact line on the eyelid, but also enhance our understanding of the pathophysiological consequences of the flow profile near the tear film contact line.
Similar content being viewed by others
Notes
We note at this stage that the value of x f may be different in the lubrication approximation and the full model.
h m and R are dimensional in this expression.
References
Aydemir, E., Breward, C. J. W., & Witelski, T. P. (2011). The effect of polar lipids on tear film dynamics. Bull Math Bio, 73(6), 1171–1201.
Baudouin, C. (2007). The vicious circle in dry eye syndrome: a mechanistic approach. J. Fr. Ophthalmol., 30, 239–246.
Benilov, E., & Zubkov, V. (2008). On the drag-out problem in liquid film theory. J. Fluid Mech., 617, 283–299.
Berger, R. E., & Corrsin, S. (1974). A surface tension gradient mechanism for driving the pre-corneal tear film after a blink. J. Biomech., 7, 225–238.
Braun, R. J. (2012). Dynamics of the tear film. Annu. Rev. Fluid Mech., 44, 267–297.
Breward, C. J. W., Bruna, M., Gaffney, E. A., & Zubkov, V. S. (2012, in preparation). The influence of nonpolar lipids on tear film dynamics.
Bron, A., Tiffany, J., Gouveia, S., Yokoi, N., & Voon, L. (2004). Functional aspects of the tear film lipid layer. Exp. Eye Res., 78, 347–360.
Bron, A. J., Yokoi, N., Gaffney, E. A., & Tiffany, J. M. (2011a). A solute gradient in the tear meniscus I. An hypothesis to explain Marx’s line. Ocul. Surf., 7, 92–97.
Bron, A. J., Yokoi, N., Gaffney, E. A., & Tiffany, J. M. (2011b). A solute gradient in the tear meniscus II. Implications for lid margin disease, including Meibomian gland dysfunction. Ocul. Surf., 9, 70–91.
DEWS (2007). The epidemiology of dry eye disease: report of the epidemiology subcommittee of the international dry eye workshop. Ocul. Surf., 5, 93–107.
Gaffney, E. A., Tiffany, J. M., Yokoi, N., & Bron, A. J. (2010). A mass and solute balance model for tear volume and osmolarity in the normal and the dry eye. Prog. Retin. Eye Res., 29, 59–78.
Gilbard, J. P., Carter, J. B., Sang, D. N., Refojo, M. F., Hanninen, L. A., & Kenyon, K. R. (1984). Morphologic effect of hyperosmolarity on rabbit corneal epithelium. Ophthalmology, 91, 1205–1212.
Gilbard, J. P., Rossi, S. R., & Heyda, K. G. (1989). Tear film and ocular surface changes after closure of the meibomian gland orifices in the rabbit. Ophthalmology, 96, 1180–1186.
Harwood, M. R., Mezey, L. E., & Harris, C. M. (1999). The spectral main sequence of human saccades. J. Neurosci., 19, 9098–9106.
Huang, A. J. W., Belldegrun, R., Hanninen, L., Kenyon, K. R., Tseng, S. C. G., & Refojo, M. F. (1989). Effect of hypertonic solutions on conjunctival epithelium and mucin like glycoprotein discharge. Cornea, 8, 15–20.
Johnson, M. E., & Murphy, P. J. (2005). The agreement and repeatability of tear meniscus height measurement methods. Optom. Vis. Sci., 82, 1030–1037.
Jones, M. B., Please, C. P., McElwain, D. L. S., Fulford, G. R., & Robert, A. P. (2005). Dynamics of tear film deposition and draining. Math. Med. Biol., 22, 265–288.
Jones, M. B., Please, C. P., McElwain, D. L. S., Fulford, G. R., & Robert, A. P. (2006). The effect of the lipid layer on tear film behaviour. Bull. Math. Biol., 86, 1355–1381.
King-Smith, P. E., Fink, B. A., Hill, R. M., Koelling, K. W., & Tiffany, J. M. (2004). The thickness of the tear film. Curr. Eye Res., 29, 357–368.
King-Smith, P. E., Nichols, J. J., Nichols, K. K., Fink, B. A., & Braun, R. J. (2008). Contributions of evaporation and other mechanisms to tear film thinning and break-up. Optom. Vis. Sci., 85, 623–630.
Knop, E., Knop, N., Zhivov, A., Kraak, R., Korb, D., Blackie, C., Greiner, J., & Guthoff, R. (2011). The lid wiper and muco-cutaneous junction anatomy of the human eyelid margins: an in vivo confocal and histological study. J. Anat., 218, 449–461.
Maki, K. L., Braun, R. J., Henshaw, W. D., & King-Smith, P. E. (2010a). Tear film dynamics on an eye-shaped domain i: pressure boundary conditions. Math. Med. Biol., 27, 227–254.
Maki, K. L., Braun, R. J., Henshaw, W. D., & King-Smith, P. E. (2010b). Tear film dynamics on an eye-shaped domain part 2. flux boundary conditions. J. Fluid Mech., 647, 361–390.
Miller, K. L., Polse, K. A., & Radke, C. J. (2002). Black-line formation and the “perched” human tear film. Curr. Eye Res., 25, 155–162.
Moffatt, H. K. (1963). Viscous and resistive eddies near a sharp corner. J. Fluid Mech., 18(1), 1–18.
Owens, H., & Phillips, J. R. (2001). Spreading of the tears after a blink - velocity and stabilization time in healthy eyes. Cornea, 20(5), 484–487.
Sharma, A., Tiwari, S., Khanna, R., & Tiffany, J. M. (1998). Hydrodynamics of meniscus induced thinning of the tear fluid. In D. Sullivan, D. Dartt, & M. Meneray (Eds.), Lacrimal gland, tear film, and dry eye syndromes (p. 2). New York: Plenum.
Tiffany, J. M., Winter, N., & Bliss, G. (1989). Tear film stability and tear surface tension. Curr. Eye Res., 8, 507–515.
Tsubota, K., Hata, S., Okusawa, Y., Egami, F., Ohtsuk, T., & Nakamori, K. (1996). Quantitative videographic analysis of blinking in normal subjects and patients with dry eye. Arch. Ophthalmol., 114, 715–720.
Wilson, S. D. R. (1982). The drag-out problem in film coating theory. J. Eng. Math., 16, 209–221.
Winter, K. N., Anderson, D. M., & Braun, R. J. (2010). A model for wetting and evaporation of a post-blink precorneal tear film. Math. Med. Biol., 27, 211–225.
Wong, H., Fatt, I., & Radke, C. J. (1996). Deposition and thinning of the human tear film. J. Colloid Interface Sci., 184, 44–51.
Yarbus, A. L. (1967). Eye movements and vision. New York: Plenum.
Yokoi, N., Bron, A. J., Tiffany, J. M., Brown, N., Hsuan, J., & Fowler, C. (1999). Reflective meniscometry: A. non-invasive method to measure tear meniscus curvature. Br. J. Ophthalmol., 83, 92–97.
Yokoi, N., Yameda, H., Mizukusa, Y., Bron, A. J., Tiffany, J. M., Kato, T., & Kinoshita, S. (2008). Rheology of tear film lipid layer spread in normal and aqueous tear-deficient dry eyes. Investig. Ophthalmol. Vis. Sci., 49, 5319–5324.
Zubkov, V. S., Breward, C. J. W., & Gaffney, E. A. (2012). Coupling fluid and solute dynamics within the ocular surface tear film: a modelling study of black line osmolarity. Bull. Math. Biol. 74(9), 2062–2093. doi:10.1007/s11538-012-9746-9.
Acknowledgements
This paper is based on work supported by Award No. KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST). We are grateful to Professor Richard Braun, Professor Anthony Bron, Professor Colin Please, and Dr. John Tiffany for insightful discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zubkov, V.S., Breward, C.J.W. & Gaffney, E.A. Meniscal Tear Film Fluid Dynamics Near Marx’s Line. Bull Math Biol 75, 1524–1543 (2013). https://doi.org/10.1007/s11538-013-9858-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-013-9858-x