Bulletin of Mathematical Biology

, Volume 79, Issue 12, pp 2814–2846 | Cite as

Duplex Tear Film Evaporation Analysis

  • M. R. Stapf
  • R. J. Braun
  • P. E. King-Smith
Original Article


Tear film thinning, hyperosmolarity, and breakup can cause irritation and damage to the human eye, and these form an area of active investigation for dry eye syndrome research. Recent research demonstrates that deficiencies in the lipid layer may cause locally increased evaporation, inducing conditions for breakup. In this paper, we explore the conditions for tear film breakup by considering a model for tear film dynamics with two mobile fluid layers, the aqueous and lipid layers. In addition, we include the effects of osmosis, evaporation as modified by the lipid, and the polar portion of the lipid layer. We solve the system numerically for reasonable parameter values and initial conditions and analyze how shifts in these cause changes to the system’s dynamics.


Tear film Lipid layer Evaporation Tear osmolarity 



This work was supported by National Science Foundation Grant DMS 1412085 (MS, RJB, PEKS). The content is solely the responsibility of the authors and does not necessarily represent the official views of the NSF.


  1. Anonymous (2007) The definition and classification of dry eye disease. Ocul Surf 5(2):75–92Google Scholar
  2. Aydemir E, Breward CJW, Witelski TP (2011) The effect of polar lipids on tear film dynamics. Bull Math Biol 73(6):1171–1201MathSciNetCrossRefzbMATHGoogle Scholar
  3. Begley CG, Simpson T, Liu H, Salvo E, Wu Z, Bradley A, Situ P (2013) Quantative analysis of tear film fluorescence and discomfort during tear film instability and thinning. Invest Ophthalmol Vis Sci 54:26452653CrossRefGoogle Scholar
  4. Benedetto DA, Clinch TE, Laibson PR (1986) In vivo observations of tear dynamics using fluorophotometry. Arch Ophthalmol 102:410–412CrossRefGoogle Scholar
  5. Berger RE, Corrsin S (1974) A surface tension gradient mechanism for driving the pre-corneal tear film after a blink. J Biomech 7:225–238CrossRefGoogle Scholar
  6. Bitton E, Lovasik JV (1998) Longitudinal analysis of precorneal tear film rupture patterns. In: Sullivan DA, Dartt DA, Meneray MA (eds) Advances in experimental medicine and biology, vol 438. Lacrimal gland, tear film, and dry eye syndromes 2. Springer, Berlin, pp 381–389Google Scholar
  7. Braun RJ (2012) Dynamics of the tear film. Annu Rev Fluid Mech 44:267–297MathSciNetCrossRefzbMATHGoogle Scholar
  8. Braun RJ, Gewecke NR, Begley CG, King-Smith PE, Siddique JI (2014) A model for tear film thinning with osmolarity and fluorescein. Invest Ophthalmol Vis Sci 55(2):1133CrossRefGoogle Scholar
  9. Braun RJ, King-Smith PE, Begley CG, Li L, Gewecke NR (2015) Dynamics and function of the tear film in relation to the blink cycle. Prog Ret Eye Res 45:132–164CrossRefGoogle Scholar
  10. Braun RJ, Driscoll TA, Begley CG, King-Smith PE, Siddique JI (2017) On tear film breakup (TBU): Dynamics and imaging. Math Med Biol. doi: 10.1093/imammb/dqw023
  11. Bron AJ, Tiffany JM, Gouveia SM, Yokoi N, Voon LW (2004) Functional aspects of the tear film lipid layer. Exp Eye Res 78(3):347–360CrossRefGoogle Scholar
  12. Bruna M, Breward CJW (2014) The influence of non-polar lipids on tear film dynamics. J Fluid Mech 746:565–605CrossRefzbMATHGoogle Scholar
  13. Buck AL (1981) New equations for computing vapor pressure and enhancement factor. J Appl Meteorol 20(12):1527–1532CrossRefGoogle Scholar
  14. Cerretani CF, Radke CJ (2014) Tear dynamics in healthy and dry eyes. Curr Eye Res 39:580–595CrossRefGoogle Scholar
  15. Cerretani CF, Ho NH, Radke CJ (2013) Water-evaporation reduction by duplex films: application to the human tear film. Adv Colloid Interface Sci 197:33–57CrossRefGoogle Scholar
  16. Craster RV, Matar OK (2009) Dynamics and stability of thin liquid films. Rev Mod Phys 81(3):1131CrossRefGoogle Scholar
  17. Doane MG (1981) Blinking and the mechanics of the lacrimal drainage system. Ophthalmology 88:844–51CrossRefGoogle Scholar
  18. Doane MG (1989) An instrument for in vivo tear film interferometry. Optom Vis Sci 66:383–388CrossRefGoogle Scholar
  19. Gilbard JP, Farris RL II, Santamaria J (1978) Osmolarity of tear microvolumes in keratoconjunctivitis sicca. Arch Ophthalmol 96:677–681CrossRefGoogle Scholar
  20. Gipson IK (2004) Distribution of mucins at the ocular surface. Exp Eye Res 78(3):379–388CrossRefGoogle Scholar
  21. Goto E, Tseng SCG (2003) Kinetic analysis of tear interference images in aqueous tear deficiency dry eye before and after punctal occlusion. Invest Ophthalmol Vis Sci 44:1897–1905CrossRefGoogle Scholar
  22. Govindarajan B, Gipson IK (2010) Membrane-tethered mucins have multiple functions on the ocular surface. Exp Eye Res 90(6):655–663CrossRefGoogle Scholar
  23. Hogan MJ, Alvarado JA, Weddell JE (1971) Histology of the human eye. An atlas and textbook. W. B.Saunders, PhiladelphiaGoogle Scholar
  24. Jones MB, McElwain DLS, Fulford GR, Collins MJ, Roberts AP (2006) The effect of the lipid layer on tear film behavior. Bull Math Biol 68:1355–1381MathSciNetCrossRefzbMATHGoogle Scholar
  25. Kimball SH, King-Smith PE, Nichols JJ (2010) Evidence for the major contribution of evaporation to tear film thinning between blinks. Invest Opthalmol Vis Sci 51:6294–6297CrossRefGoogle Scholar
  26. King-Smith PE, Fink B, Hill R, Koelling K, Tiffany JM (2004) The thickness of the tear film. Curr Eye Res 29(4–5):357–368CrossRefGoogle Scholar
  27. King-Smith PE, Fink BA, Nichols JJ, Nichols KK, Braun RJ, McFadden GB (2009) The contribution of lipid layer movement to tear film thinning and breakup. Invest Ophthalmol Vis Sci 50:2747–2756CrossRefGoogle Scholar
  28. King-Smith PE, Hinel EA, Nichols JJ (2010) Application of a novel interferometric method to investigate the relation between lipid layer thickness and tear film thinning. Invest Ophthalmol Vis Sci 51(5):2418–2423CrossRefGoogle Scholar
  29. King-Smith PE, Nichols JJ, Nichols KK, Braun RJ (2011) A high resolution microscope for imaging the lipid layer of the tear film. Ocul Surf 9(4):197–211CrossRefGoogle Scholar
  30. King-Smith PE, Bailey MD, Braun RJ (2013a) Four characteristics and a model of an effective tear film lipid layer (TFLL). Ocul Surf 11(4):236–245CrossRefGoogle Scholar
  31. King-Smith PE, Ramamoorthy P, Braun RJ, Nichols JJ (2013b) Tear film images and breakup analyzed using fluorescent quenching. Invest Ophthalmol Vis Sci 54:6003–6011CrossRefGoogle Scholar
  32. King-Smith PE, Reuter KS, Braun RJ, Nichols JJ, Nichols KK (2013c) Tear film breakup and structure studied by simultaneous video recording of fluorescence and tear film lipid layer, TFLL, images. Invest Ophthalmol Vis Sci 54(7):4900–4909CrossRefGoogle Scholar
  33. Leiske DL, Leiske CI, Leiske DR, Toney MF, Senchyna M, Ketelson HA, Meadows DL, Fuller GG (2011) Temperature-induced transitions in the structure and interfacial rheology of human meibum. Biophys J 102:369–376CrossRefGoogle Scholar
  34. Leiske DL, Miller CE, Rosenfeld L, Cerretani C, Ayzner A, Lin B, Meron M, Senchyna M, Ketelson HA, Meadows D, Srinivasan S, Jones L, Radke CJ, Toney MF, Fuller GG (2012) Molecular structure of interfacial human meibum films. Langmuir 28:11858–11865CrossRefGoogle Scholar
  35. Lemp MA, Bron AJ, Baudouin C, del Castillo JMB, Geffen D, Tauber J, Foulks GN, Pepose JS, Sullivan BD (2011) Tear osmolarity in the diagnosis and management of dry eye disease. Am J Ophthalmol 151(5):792–798CrossRefGoogle Scholar
  36. Li L, Braun RJ, Maki KL, Henshaw WD, King-Smith PE (2014) Tear film dynamics with evaporation, wetting, and time-dependent flux boundary condition on an eye-shaped domain. Phys Fluids 26(5):052101CrossRefzbMATHGoogle Scholar
  37. Li L, Braun RJ, Driscoll TA, Henshaw WD, Banks JW, King-Smith PE (2016) Computed tear film and osmolarity dynamics on an eye-shaped domain. Math Med Biol 33:123–157MathSciNetCrossRefGoogle Scholar
  38. Liu H, Begley CG, Chalmers R, Wilson G, Srinivas SP, Wilkinson JA (2006) Temporal progression and spatial repeatability of tear breakup. Optom Vis Sci 83(10):723–730CrossRefGoogle Scholar
  39. Liu H, Begley C, Chen M, Bradley A, Bonanno J, McNamara NA, Nelson JD, Simpson T (2009) A link between tear instability and hyperosmolarity in dry eye. Invest Ophthalmol Vis Sci 50(8):3671–3679CrossRefGoogle Scholar
  40. Matar OK, Craster RV, Warner MRE (2002) Surfactant transport on highly viscous surface films. J Fluid Mech 466:85–111CrossRefzbMATHGoogle Scholar
  41. Mishima S, Maurice DM (1961) The oily layer of the tear film and evaporation from the corneal surface. Exp Eye Res 1:39–45CrossRefGoogle Scholar
  42. Montés-Micó R, Cervino A, Ferrer-Blasco T, García-Lázaro S, Madrid-Costa D (2010) The tear film and the optical quality of the eye. Ocul Surf 8(4):185–192CrossRefGoogle Scholar
  43. Nagyova B, Tiffany JM (1999) Components responsible for the surface tension of human tears. Curr Eye Res 19(1):4–11CrossRefGoogle Scholar
  44. Naire S, Braun RJ, Snow SA (2000) Limiting cases of gravitational drainage of a vertical free film for evaluating surfactants. SIAM J Appl Math 61:889913MathSciNetzbMATHGoogle Scholar
  45. Nichols B, Dawson CR, Togni B (1983) Surface features of the conjunctiva and cornea. Invest Ophthalmol Vis Sci 24(5):570–576Google Scholar
  46. Nichols JJ, King-Smith PE, Hinel EA, Thangavelu M, Nichols KK (2012) The use of fluorescent quenching in studying the contribution of evaporation to tear thinning. Invest Ophthalmol Vis Sci 53:54265432Google Scholar
  47. Norn MS (1969) Dessication of the precorneal film I. Corneal wetting time. Acta Ophthalmol 4:865–880Google Scholar
  48. Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69(3):931CrossRefGoogle Scholar
  49. Owens H, Phillips J (2001) Spreading of the tears after a blink: velocity and stabilization time in healthy eyes. Cornea 20:484–487CrossRefGoogle Scholar
  50. Peng C, Cerretani C, Braun RJ, Radke CJ (2014) Evaporation-driven instability of the precorneal tear film. Adv Colloid Interface Sci 206:250–264CrossRefGoogle Scholar
  51. Riquelme R, Lira I, Pérez-López C, Rayas JA, Rodríguez-Vera R (2007) Interferometric measurement of a diffusion coefficient: comparison of two methods and uncertainty analysis. J Phys D Appl Phys 40(9):2769CrossRefGoogle Scholar
  52. Rosenfeld L, Cerretani C, Leiske DL, Toney MF, Radke CJ, Fuller GG (2013) Structural and rheological properties of meibomian lipid. Invest Ophthalmol Vis Sci 54:2720–2732CrossRefGoogle Scholar
  53. Sharma A (1998) Surface-chemical pathways of the tear film breakup. In: Sullivan DA, Dartt DA, Meneray MA (eds) Advances in experimental medicine and biology, vol 438. Lacrimal Gland, Tear Film, and Dry Eye Syndromes 2. Springer, Berlin, pp 361–370Google Scholar
  54. Sharma A, Ruckenstein E (1985) Mechanism of tear film rupture and formation of dry spots on cornea. J Colloid Interface Sci 106(1):12–27CrossRefGoogle Scholar
  55. Siddique JI, Braun RJ (2015) Tear film dynamics with evaporation, osmolarity and surfactant transport. Appl Math Model 39:255–269MathSciNetCrossRefGoogle Scholar
  56. Stebe KJ, Maldarelli C (1994) Remobilizing surfactant retarded fluid particle interfaces: II. Controlling the surface mobility at interface of solutions containing surface active components. J Colloid Interface Sci 163:177–189CrossRefGoogle Scholar
  57. Tietz NW (1995) Clinical guide to laboratory tests. W. B. Saunders, PhiladelphiaGoogle Scholar
  58. Tiffany JM (1987) The lipid secretion of the meibomian glands. Adv Lipid Res 22(1):1–62Google Scholar
  59. Tiffany JM (1991) The viscosity of human tears. Int Ophthalmol 15(6):371–376CrossRefGoogle Scholar
  60. Tomlinson A, Khanal S, Ramaesh K, Diaper C, McFadyen A (2006) Tear film osmolarity: determination of a referent for dry eye diagnosis. Invest Ophthalmol Vis Sci 47(10):4309–4315CrossRefGoogle Scholar
  61. Tomlinson A, Doane MG, McFayden A (2009) Inputs and outputs of the lacrimal system: review of production and evaporative loss. Ocul Surf 7:17–29CrossRefGoogle Scholar
  62. Trefethen LN (2000) Spectral methods in MATLAB. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  63. Weast RC (ed) (1977) CRC handbook of chemistry and physics, 58th edn. Chemical Rubber Company, West Palm BeachGoogle Scholar
  64. Yanez-Soto B, Mannis MJ, Schwab IR, Li JY, Leonard BC, Abbott NL, Murphy CJ (2014) Interfacial phenomena and the ocular surface. Ocul Surf 12:178–201CrossRefGoogle Scholar
  65. Yokoi N, Georgiev GA (2013a) Tear dynamics and dry eye disease (Chapter 7). In: Benitez del Castilo JM, Lemp MA (eds) Ocular surface disorders. JP Medical Ltd., London, pp 47–54Google Scholar
  66. Yokoi N, Georgiev GA (2013b) Tear-film-oriented diagnosis and therapy for dry eye. In: Dry eye syndrome: basic and clinical perspectives, pp 96–108Google Scholar
  67. Yokoi N, Takehisa Y, Kinoshita S (1996) Correlation of tear lipid layer interference patterns with the diagnosis and severity of dry eye. Am J Ophthalmol 122:818824Google Scholar
  68. Zhong L, Ketelaar C, Braun RJ, King-Smith PE, Begley CG (2017) A mathematical model for glob-driven tear break up (TBU) (submitted)Google Scholar
  69. Zubkov VS, Breward CJ, Gaffney EA (2012) Coupling fluid and solute dynamics within the ocular surface tear film: a modelling study of black line osmolarity. Bull Math Biol 74:2062–2093MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.College of OptometryThe Ohio State UniversityColumbusUSA

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