Skip to main content

Advertisement

Log in

Parameter Estimation for Mixed-Mechanism Tear Film Thinning

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

Etiologies of tear breakup include evaporation-driven, divergent flow-driven, and a combination of these two. A mathematical model incorporating evaporation and lipid-driven tangential flow is fit to fluorescence imaging data. The lipid-driven motion is hypothesized to be caused by localized excess lipid, or “globs.” Tear breakup quantities such as evaporation rates and tangential flow rates cannot currently be directly measured during breakup. We determine such variables by fitting mathematical models for tear breakup and the computed fluorescent intensity to experimental intensity data gathered in vivo. Parameter estimation is conducted via least squares minimization of the difference between experimental data and computed answers using either the trust-region-reflective or Levenberg–Marquardt algorithm. Best-fit determination of tear breakup parameters supports the notion that evaporation and divergent tangential flow can cooperate to drive breakup. The resulting tear breakup is typically faster than purely evaporative cases. Many instances of tear breakup may have similar causes, which suggests that interpretation of experimental results may benefit from considering multiple mechanisms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  • Ajaev VS, Homsy GM (2001) Steady vapor bubbles in rectangular microchannels. J Coll Interface Sci 240(1):259–271

    Article  Google Scholar 

  • Argüeso P, Gipson IK (2001) Epithelial mucins of the ocular surface: structure, biosynthesis and function. Exp Eye Res 73(3):281–289

    Article  Google Scholar 

  • Arnold S, Walter A, Eppig T, Bruenner H, Langenbucher A (2010) Simultaneous examination of tear film break-up and the lipid layer of the human eye: a novel sensor design (part 1). Z Med Phys 20(4):309–315

    Article  Google Scholar 

  • Awisi-Gyau D (2020) Characterization of tear breakup and its sensory effects. PhD Thesis, Indiana University

  • Awisi-Gyau D, Begley CG, Braun RJ, Luke RA, Tichenor A, King-Smith P (2020) Characterization of spatial and temporal properties of tear breakup patterns (in preparation)

  • Aydemir E, Breward CJW, Witelski TP (2010) The effect of polar lipids on tear film dynamics. Bull Math Biol 73:1171–1201

    Article  MathSciNet  MATH  Google Scholar 

  • 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 54:2645–2653

    Article  Google Scholar 

  • Benedetto DA, Clinch TE, Laibson PR (1986) In vivo observations of tear dynamics using fluorophotometry. Arch Ophthalmol 102:410–412

    Article  Google Scholar 

  • Borchman D, Ramasubramanian A, Foulks GN (2019) Human meibum cholesteryl and wax ester variability with age, sex, and meibomian gland dysfunction. Invest Ophthalmol Vis Sci 60(6):2286–2293

    Article  Google Scholar 

  • Braun RJ (2012) Dynamics of the tear film. Annu Rev Fluid Mech 44:267–297

    Article  MathSciNet  MATH  Google Scholar 

  • 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):1133–1142

    Article  Google Scholar 

  • 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 Retin Eye Res 45:132–164

    Article  Google Scholar 

  • Braun RJ, Driscoll TA, Begley CG, King-Smith PE, Siddique JI (2018) On tear film breakup (TBU): dynamics and imaging. Math Med Biol 35(2):145–180

    Article  MATH  Google Scholar 

  • Bron A, Argüeso P, Irkec M, Bright F (2015) Clinical staining of the ocular surface: mechanisms and interpretations. Prog Ret Eye Res 44:36–61

    Article  Google Scholar 

  • Bruna M, Breward CJW (2014) The influence of nonpolar lipids on tear film dynamics. J Fluid Mech 746:565–605

    Article  MathSciNet  MATH  Google Scholar 

  • Butovich IA (2013) Tear film lipids. Exp Eye Res 117:4–27

    Article  Google Scholar 

  • Canuto C, Hussaini MY, Quarteroni A, Thomas A Jr et al (2012) Spectral methods in fluid dynamics. Springer, Berlin

    MATH  Google Scholar 

  • Carlson NB, Kurtz D, Hines C (2004) Clinical procedures for ocular examination, vol 3. McGraw-Hill, New York

    Google Scholar 

  • Casalini T, Salvalaglio M, Perale G, Masi M, Cavallotti C (2011) Diffusion and aggregation of sodium fluorescein in aqueous solutions. J Phys Chem B 115(44):12896–12904

    Article  Google Scholar 

  • Cerretani CF, Radke C (2014) Tear dynamics in healthy and dry eyes. Curr Eye Res 39(6):580–595

    Article  Google Scholar 

  • Cho P, Brown B, Chan I, Conway R, Yap M (1992) Reliability of the tear break-up time technique of assessing tear stability and the locations of the tear break-up in Hong Kong Chinese. Optom Vis Sci 69(11):879–885

    Article  Google Scholar 

  • Craig JP, Nichols KK, Nichols JJ, Caffery B, Dua HS, Akpek EK, Tsubota K, Joo CK, Liu Z, Nelson JD, Stapleton F (2017) The TFOS DEWS II Definition and classification Report. Ocul Surf 15:276–283

    Article  Google Scholar 

  • Craster RV, Matar OK (2009) Dynamics and stability of thin liquid films. Rev Mod Phys 81(3):1131

    Article  Google Scholar 

  • Dartt D (2009) Neural regulation of lacrimal gland secretory processes: relevance in dry eye diseases. Prog Retin Eye Res 28:155–177

    Article  Google Scholar 

  • Dartt D, Willcox M (2013) Complexity of the tear film: importance in homeostasis and dysfunction during disease. Exp Eye Res 117:1–3

    Article  Google Scholar 

  • Doane MG (1981) Blinking and the mechanics of the lacrimal drainage system. Ophthalmology 88:844–51

    Article  Google Scholar 

  • Dursch TJ, Li W, Taraz B, Lin MC, Radke CJ (2018) Tear-film evaporation rate from simultaneous ocular-surface temperature and tear-breakup area. Optom Vis Sci 95(1):5–12

    Article  Google Scholar 

  • Georgiev GA, Eftimov P, Yokoi N (2017) Structure-function relationship of tear film lipid layer: A contemporary perspective. Exp Eye Res 163:17–28

    Article  Google Scholar 

  • Gilbard JP, Farris RL, Santamaria J (1978) Osmolarity of tear microvolumes in keratoconjunctivitis sicca. Arch Ophthalmol 96(4):677–681

    Article  Google Scholar 

  • Gipson IK (2004) Distribution of mucins at the ocular surface. Exp Eye Res 78(3):379–388

    Article  Google Scholar 

  • Hamano H, Hori M, Mitsunaga S (1981) Measurement of evaporation rate of water from the precorneal tear film and contact lenses. Contacto 25(2):7–15

    Google Scholar 

  • Himebaugh N, Nam J, Bradley A, Liu H, Thibos LN, Begley CG (2012) Scale and spatial distribution of aberrations associated with tear breakup. Optom Vis Sci 89(11):1590–1600

    Article  Google Scholar 

  • Huang J, Hindman HB, Rolland JP (2016) In vivo thickness dynamics measurement of tear film lipid and aqueous layers with optical coherence tomography and maximum-likelihood estimation. Opt Lett 41(9):1981–1984

    Article  Google Scholar 

  • Jensen OE, Grotberg JB (1993) The spreading of heat or soluble surfactant along a thin liquid film. Phys Fluids A 75:58–68

    Article  MATH  Google Scholar 

  • Johnson ME, Murphy PJ (2004) Changes in the tear film and ocular surface from dry eye syndrome. Prog Ret Eye Res 23(4):449–474

    Article  Google Scholar 

  • Kimball SH, King-Smith PE, Nichols JJ (2010) Evidence for the major contribution of evaporation to tear film thinning between blinks. Invest Ophthalmol Vis Sci 51(12):6294–6297

    Article  Google Scholar 

  • King-Smith PE, Fink B, Hill R, Koelling K, Tiffany J (2004) The thickness of the tear film. Curr Eye Res 29(4–5):357–368

    Article  Google Scholar 

  • 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(6):2747–2756

    Article  Google Scholar 

  • 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):2418–2423

    Article  Google Scholar 

  • King-Smith PE, Nichols JJ, Braun RJ, Nichols KK (2011) High resolution microscopy of the lipid layer of the tear film. Ocul Surf 9(4):197–211

    Article  Google Scholar 

  • King-Smith PE, Ramamoorthy P, Braun RJ, Nichols JJ (2013a) Tear film images and breakup analyzed using fluorescent quenching. Invest Ophthalmol Vis Sci 54:6003–6011

    Article  Google Scholar 

  • King-Smith PE, Reuter KS, Braun RJ, Nichols JJ, Nichols KK (2013b) Tear film breakup and structure studied by simultaneous video recording of fluorescence and tear film lipid layer images. Invest Ophthalmol Vis Sci 54(7):4900–4909

    Article  Google Scholar 

  • King-Smith PE, Begley CG, Braun RJ (2018) Mechanisms, imaging and structure of tear film breakup. Ocul Surf 16:4–30

    Article  Google Scholar 

  • 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):792–798

    Article  Google Scholar 

  • Lemp MA et al (2007) The definition and classification of dry eye disease: Report of the Definition and Classification Subcommittee of the International Dry Eye WorkShop. Ocul Surf 5:75–92

    Article  Google Scholar 

  • LeVeque RJ (2007) Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Li L, Braun R, Maki K, Henshaw W, 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):052101

    Article  MATH  Google Scholar 

  • 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(2):123–157

    Article  MathSciNet  MATH  Google Scholar 

  • Liu H, Begley CG, Chalmers R, Wilson G, Srinivas SP, Wilkinson JA (2006) Temporal progression and spatial repeatability of tear breakup. Optom Vis Sci 83:723–730

    Article  Google Scholar 

  • 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 50:3671–79

    Article  Google Scholar 

  • Lu H, Wang MR, Wang J, Shen M (2014) Tear film measurement by optical reflectometry technique. J Biomed Opt 19(2):027001

    Article  Google Scholar 

  • Luke R, Braun R, Driscoll T, Begley C, Awisi-Gyau D (2020) Parameter estimation for evaporation-driven tear film thinning. Bull Math Biol 82(6):1–41. https://doi.org/10.1007/s11538-020-00745-8

    Article  MathSciNet  MATH  Google Scholar 

  • McCulley JP, Shine W (1997) A compositional based model for the tear film lipid layer. Trans Am Ophthalmol Soc 95:79–93

    Google Scholar 

  • Mertzanis P, Abetz L, Rajagopalan K, Espindle D, Chalmers R, Snyder C, Caffery B, Edrington T, Simpson T, Nelson JD et al (2005) The relative burden of dry eye in patients’ lives: comparisons to a US normative sample. Invest Ophthalmol Vis Sci 46(1):46–50

    Article  Google Scholar 

  • Miljanović B, Dana R, Sullivan DA, Schaumberg DA (2007) Impact of dry eye syndrome on vision-related quality of life. Am J Ophthalmol 143(3):409–415

    Article  Google Scholar 

  • Mishima S, Maurice D (1961) The oily layer of the tear film and evaporation. Exp Eye Res 1:39–45

    Article  Google Scholar 

  • Mota M, Carvalho P, Ramalho J, Leite E (1991) Spectrophotometric analysis of sodium fluorescein aqueous solutions, determination of molar absorption coefficient. Int Ophthalmol 15(5):321–326

    Article  Google Scholar 

  • Nagyová B, Tiffany J (1999) Components responsible for the surface tension of human tears. Curr Eye Res 19(1):4–11

    Article  Google Scholar 

  • Nelson JD, Craig JP, Akpek EK, Azar DT, Belmonte C, Bron AJ, Clayton JA, Dogru M, Dua HS, Foulks GN et al (2017) TFOS DEWS II Introduction. Ocul Surf 15(3):269–275

    Article  Google Scholar 

  • Nichols JJ, Mitchell GL, King-Smith PE (2005) Thinning rate of the precorneal and prelens tear films. Invest Ophthalmol Vis Sci 46(7):2353–2361

    Article  Google Scholar 

  • 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(9):5426–5432

    Article  Google Scholar 

  • Nocedal J, Wright S (2006) Numerical optimization. Springer, Berlin

    MATH  Google Scholar 

  • Nong K, Anderson DM (2010) Thin tilm evolution over a thin porous layer: modeling a tear film over a contact lens. SIAM J Appl Math 70:2771–2795

    Article  MathSciNet  MATH  Google Scholar 

  • Norn M (1969) Desiccation of the precorneal film: I. Corneal wetting-time. Acta Ophthalmol 47(4):865–880

    Article  Google Scholar 

  • Norn MS (1970) Micropunctate fluorescein vital staining of the cornea. Acta Ophthalmol 48:108–118

    Article  Google Scholar 

  • Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69(3):931

    Article  Google Scholar 

  • Paananen RO, Javanainen M, Holopainen JM, Vattulainen I (2019) Crystalline wax esters regulate the evaporation resistance of tear film lipid layers associated with dry eye syndrome. J Phys Chem Lett 10(14):3893–3898

    Article  Google Scholar 

  • Paananen RO, Viitaja T, Olżyńska A, Ekholm FS, Moilanen J, Cwiklik L (2020) Interactions of polar lipids with cholesteryl ester multilayers elucidate tear film lipid layer structure. Ocul Surf 18(4):545–553

    Article  Google Scholar 

  • Peng CC, Cerretani C, Braun RJ, Radke CJ (2014a) Evaporation-driven instability of the precorneal tear film. Adv Coll Interface Sci 206:250–264

    Article  Google Scholar 

  • Peng CC, Cerretani C, Li Y, Bowers S, Shahsavarani S, Lin M, Radke C (2014b) Flow evaporimeter to assess evaporative resistance of human tear-film lipid layer. Ind Eng Chem Res 53(47):18130–18139

    Article  Google Scholar 

  • 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):2769

    Article  Google Scholar 

  • Stahl U, Willcox M, Stapleton F (2012) Osmolality and tear film dynamics. Clin Exp Optom 95(1):3–11

    Article  Google Scholar 

  • Stapf MR, Braun RJ, King-Smith PE (2017) Duplex tear film evaporation analysis. Bull Math Biol 79(12):2814–2846

    Article  MathSciNet  MATH  Google Scholar 

  • Sullivan BD, Whitmer D, Nichols KK, Tomlinson A, Foulks GN, Geerling G, Pepose JS, Kosheleff V, Porreco A, Lemp MA (2010) An objective approach to dry eye disease severity. Invest Ophthalmol Vis Sci 51(12):6125–6130

    Article  Google Scholar 

  • Tietz NW (1995) Clinical guide to laboratory tests. W.B. Saunders, Waltham

    Google Scholar 

  • Tiffany JM (1990a) Measurement of wettability of the corneal epithelium: I. Particle attachment method. Acta Ophthalmol 68(2):175–181

    Article  Google Scholar 

  • Tiffany JM (1990b) Measurement of wettability of the corneal epithelium: II. Contact angle method. Acta Ophthalmol 68(2):182–187

    Article  Google Scholar 

  • Tiffany JM (1991) The viscosity of human tears. Int Ophthalmol 15(6):371–376

    Article  Google Scholar 

  • 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):4309–4315

    Article  Google Scholar 

  • Tomlinson A, Doane M, McFadyen A (2009) Inputs and outputs of the lacrimal system: Review of production and evaporative loss. Ocul Surf 7(4):186–198

    Article  Google Scholar 

  • Trefethen LN (2000) Spectral methods in MATLAB. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Versura P, Profazio V, Campos E (2010) Performance of tear osmolarity compared to previous diagnostic tests for dry eye diseases. Curr Eye Res 35(7):553–564

    Article  Google Scholar 

  • Wang J, Fonn D, Simpson TL, Jones L (2003) Precorneal and pre-and postlens tear film thickness measured indirectly with optical coherence tomography. Invest Ophthalmol Vis Sci 44(6):2524–2528

    Article  Google Scholar 

  • Webber WRS, Jones DP (1986) Continuous fluorophotometric method measuring tear turnover rate in humans and analysis of factors affecting accuracy. Med Biol Eng Comput 24:386–392

    Article  Google Scholar 

  • Willcox MDP, Argüeso P, Georgiev GA, Holopainen JM, Laurie GW, Millar TJ, Papas EB, Rolland JP, Schmidt TA, Stahl U, Suarez T, Subbaraman LN, Ucakhan OO, Jones LW (2017) The TFOS DEWS II Tear Film Report. Ocul Surf 15:369–406

    Article  Google Scholar 

  • Wong S, Murphy PJ, Jones L (2018) Tear evaporation rates: what does the literature tell us? Cont Lens Anterior Eye 41(3):297–306

    Article  Google Scholar 

  • Wu Z, Begley CG, Port N, Bradley A, Braun R, King-Smith E (2015) The effects of increasing ocular surface stimulation on blinking and tear secretion. Invest Ophthalmol Vis Sci 56(8):4211–4220

    Article  Google Scholar 

  • Yokoi N, Georgiev GA (2013) Tear-film-oriented diagnosis and therapy for dry eye. In: Yokoi N (ed) Dry eye syndrome: basic and clinical perspectives. Future Medicine, London, pp 96–108

    Chapter  Google Scholar 

  • Yokoi N, Georgiev GA (2019) Tear-film-oriented diagnosis for dry eye. Jpn J Ophthalmol 63:127–136

    Article  Google Scholar 

  • Zhong L, Ketelaar CF, Braun RJ, Begley CG, King-Smith PE (2018) Mathematical modelling of glob-driven tear film breakup. Math Med Biol 36(1):55–91

    Article  MathSciNet  MATH  Google Scholar 

  • Zhong L, Braun RJ, Begley CG, King-Smith PE (2019) Dynamics of fluorescent imaging for rapid tear thinning. Bull Math Biol 81(1):39–80

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This work was supported by National Science Foundation Grant DMS 1909846 and National Institutes of Health Grant NEI R01EY021794. The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding sources.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Rayanne A. Luke.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 2210 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luke, R.A., Braun, R.J., Driscoll, T.A. et al. Parameter Estimation for Mixed-Mechanism Tear Film Thinning. Bull Math Biol 83, 56 (2021). https://doi.org/10.1007/s11538-021-00871-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11538-021-00871-x

Keywords

Navigation