Models for Dynamics of the Human Tear Film
R.J. Braun1, K.L. Maki5, A. Heryudono4, T.A. Driscoll1, L.P. Cook1,P. Ucciferro1, W.D. Henshaw2, and P.E. King-Smith3
1 Department of Mathematical Sciences, University of Delaware2 Center for Applied Scientific Computing, Lawrence Livermore National Laboratory
3 College of Optometry, The Ohio State University4 Department of Mathematics, University of Massachusetts Dartmouth
5 IMA, University of Minnesota
Supported by the NSF
Outline
MotivationIs tear film a complex fluid?(Modeling) Decisions, decisions...1D results
Blinks and partial blinksReflex tearing
2D computationsspecified pressure BCsspecified flux BCs
SummaryWhere to next...
Goal: Quantify tear filmdynamics!
Why Study Human Tear Film?
Normal tear film dynamics: Much to understand and quantify!Dry eye syndrome: Common abnormality from insufficient or
malfunctioning tear film causing disruption.Prevalence: An estimated 10% - 15% of Americans over the age of
65 have one or more symptoms of dry eye syndrome∗.Symptoms: Burning/stinging; Blurred vision
Irritation/redness; Dry sensationForeign body or “gritty" sensation; Tearing
Impact: Negative impact on, e.g., reading and driving from dryeye syndrome**.
*Stein et al. 1997; DEWS Report, 2007.**Miljanovic et al. 2007.
What is Human Tear Film?
Tear filmWhat’s in there? Properties?
Mucins:16 known mucins, most in aqueous layer3 transmembrane at corneal surfaceAqueous:Saltsmore than proteinsmucinsLipid layer: insoluble in aqueouspolar: sphingolipids, phospholipidsnonpolar: waxes, cholesterol esters
Shear thinning: Tiffany(1991), Pandit et al (1999)Not elastic: Tiffany (1994)gave inconclusive evidencefor itSurface tension: 45mN/m,Tiffany and coworkers(1999); need proteins andlipids to get this value
What is Human Tear Film?
Tear filmWhat’s in there? Properties?
Mucins:16 known mucins, most in aqueous layer3 transmembrane at corneal surfaceAqueous:Saltsmore than proteinsmucinsLipid layer: insoluble in aqueouspolar: sphingolipids, phospholipidsnonpolar: waxes, cholesterol esters
Shear thinning: Tiffany(1991), Pandit et al (1999)Not elastic: Tiffany (1994)gave inconclusive evidencefor itSurface tension: 45mN/m,Tiffany and coworkers(1999); need proteins andlipids to get this value
What is Human Tear Film?
Tear filmWhat’s in there? Properties?
Mucins:16 known mucins, most in aqueous layer3 transmembrane at corneal surfaceAqueous:Saltsmore than proteinsmucinsLipid layer: insoluble in aqueouspolar: sphingolipids, phospholipidsnonpolar: waxes, cholesterol esters
Shear thinning: Tiffany(1991), Pandit et al (1999)Not elastic: Tiffany (1994)gave inconclusive evidencefor itSurface tension: 45mN/m,Tiffany and coworkers(1999); need proteins andlipids to get this value
What is Human Tear Film?
Tear filmWhat’s in there? Properties?
Mucins:16 known mucins, most in aqueous layer3 transmembrane at corneal surfaceAqueous:Saltsmore than proteinsmucinsLipid layer: insoluble in aqueouspolar: sphingolipids, phospholipidsnonpolar: waxes, cholesterol esters
Shear thinning: Tiffany(1991), Pandit et al (1999)Not elastic: Tiffany (1994)gave inconclusive evidencefor itSurface tension: 45mN/m,Tiffany and coworkers(1999); need proteins andlipids to get this value
What is Human Tear Film?
Tear filmWhat’s in there? Properties?
Mucins:16 known mucins, most in aqueous layer3 transmembrane at corneal surfaceAqueous:Saltsmore than proteinsmucinsLipid layer: insoluble in aqueouspolar: sphingolipids, phospholipidsnonpolar: waxes, cholesterol esters
Shear thinning: Tiffany(1991), Pandit et al (1999)Not elastic: Tiffany (1994)gave inconclusive evidencefor itSurface tension: 45mN/m,Tiffany and coworkers(1999); need proteins andlipids to get this value
What is Human Tear Film?
Tear filmWhat’s in there? Properties?
Mucins:16 known mucins, most in aqueous layer3 transmembrane at corneal surfaceAqueous:Saltsmore than proteinsmucinsLipid layer: insoluble in aqueouspolar: sphingolipids, phospholipidsnonpolar: waxes, cholesterol esters
Shear thinning: Tiffany(1991), Pandit et al (1999)Not elastic: Tiffany (1994)gave inconclusive evidencefor itSurface tension: 45mN/m,Tiffany and coworkers(1999); need proteins andlipids to get this value
What is Human Tear Film?
Tear filmA multilayer structure playing a vital role in health and
function of the eye.
Typical thickness of each layer inmicrons.
(M): A possible mucus film, ifseparate from aqueousdebatable.
A: Aqueous layer, primarilywater (est. up to 98%).
L: Lipid layer, polarsurfactants at the A/Linterface.
Overview of the Dynamics
Tear film supply and drainage
Image from Wikipedia.
Lacrimal gland: The lacrimal glandsupplies new tear fluidduring a blink cycle.
Meibomian glands: The meibomian glandsupplies lipids from lidedges.
Punctal drainage: Removes excess fluidstarting at the halfway openposition of the lids.
Typical blink cycle
Upstroke/Formation: Opening of lids, 0.1758s.Interblink/Relaxation: Lids remain open, 5s (wide variation).Downstroke: Closing of lids, 0.0821s.
Characteristics of the Human Tear Film
From Wang et al. 2006.
Characteristic tear film thicknessIn the middle of the cornea, 3− 5µm.
Upper and lower menisciVolume: Contain an estimated 73% of exposed tear
film volume (experimental range 2.45− 4.0µl).Tear meniscus height (TMH): 181− 336µm in expt.Tear meniscus width (TMW): 48− 66µm in expt.Tear meniscus radius of curvature (TMC):
127− 351µm in expt.
Image from Jones et al. 2005.
Prior work: tear film evolution
Post-blink relaxation (Newtonian)Wong et al (96), Sharma et al (98), Miller et al (02)Braun and Fitt (03), Winter et al (09): evaporation
Post-blink relaxation (non-Newtonian)Gorla and Gorla (04), RJB et al (09, nearly done)Tear film formation and relaxationWong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt)Blink CyclesBraun and King-Smith (07), Heryudono et al (07) (today)Reflex tearing (today)Maki et al (08)2D numerics with Pressure bcs or flux bcs(today)Maki et al (09)
Prior work: tear film evolution
Post-blink relaxation (Newtonian)Wong et al (96), Sharma et al (98), Miller et al (02)Braun and Fitt (03), Winter et al (09): evaporation
Post-blink relaxation (non-Newtonian)Gorla and Gorla (04), RJB et al (09, nearly done)Tear film formation and relaxationWong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt)Blink CyclesBraun and King-Smith (07), Heryudono et al (07) (today)Reflex tearing (today)Maki et al (08)2D numerics with Pressure bcs or flux bcs(today)Maki et al (09)
Prior work: tear film evolution
Post-blink relaxation (Newtonian)Wong et al (96), Sharma et al (98), Miller et al (02)Braun and Fitt (03), Winter et al (09): evaporation
Post-blink relaxation (non-Newtonian)Gorla and Gorla (04), RJB et al (09, nearly done)Tear film formation and relaxationWong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt)Blink CyclesBraun and King-Smith (07), Heryudono et al (07) (today)Reflex tearing (today)Maki et al (08)2D numerics with Pressure bcs or flux bcs(today)Maki et al (09)
Prior work: tear film evolution
Post-blink relaxation (Newtonian)Wong et al (96), Sharma et al (98), Miller et al (02)Braun and Fitt (03), Winter et al (09): evaporation
Post-blink relaxation (non-Newtonian)Gorla and Gorla (04), RJB et al (09, nearly done)Tear film formation and relaxationWong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt)Blink CyclesBraun and King-Smith (07), Heryudono et al (07) (today)Reflex tearing (today)Maki et al (08)2D numerics with Pressure bcs or flux bcs(today)Maki et al (09)
Prior work: tear film evolution
Post-blink relaxation (Newtonian)Wong et al (96), Sharma et al (98), Miller et al (02)Braun and Fitt (03), Winter et al (09): evaporation
Post-blink relaxation (non-Newtonian)Gorla and Gorla (04), RJB et al (09, nearly done)Tear film formation and relaxationWong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt)Blink CyclesBraun and King-Smith (07), Heryudono et al (07) (today)Reflex tearing (today)Maki et al (08)2D numerics with Pressure bcs or flux bcs(today)Maki et al (09)
Prior work: tear film evolution
Post-blink relaxation (Newtonian)Wong et al (96), Sharma et al (98), Miller et al (02)Braun and Fitt (03), Winter et al (09): evaporation
Post-blink relaxation (non-Newtonian)Gorla and Gorla (04), RJB et al (09, nearly done)Tear film formation and relaxationWong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt)Blink CyclesBraun and King-Smith (07), Heryudono et al (07) (today)Reflex tearing (today)Maki et al (08)2D numerics with Pressure bcs or flux bcs(today)Maki et al (09)
Modeling Choices
Idealized domain (I & II)
*Braun et al. 2003.
Modeling assumptions
The aqueous fluid is Newtonian withproperties of water.The mucus and lipid layers included withappropriate boundary conditions.Rate of evaporation is uniform in spaceand constant in time*.
Characteristic length scales
For x ′ direction:L′ = 5mm, half width of cornea.
For y ′ direction:d ′ = 5µm, thickness of film.
The ratio of length scalesε = d ′/L′ ≈ 10−3 ⇒ lubrication.
Part I: Blink cycles(Heryudono PhD thesis)
Formulation of Model
At leading order, on 0 ≤ y ≤ h(x , t) and X (t) ≤ x ≤ 1 we have
ux + vy = 0, uyy − px + G = 0 and py = 0,
where G = ρgd ′2
µUm≈ 2.5× 10−3.
Boundary conditionsEye surface:
Slip condition and impermeability: u = βuy , v = 0; β = 0.01.Free surface:
Kinematic condition: ht + uhx = v −E , with E = J′
Umερ ≈ 3× 10−4.
Normal stress condition: p = −Shxx , where S = ε3σµU′
m≈ 5× 10−7.
Insoluble surfactant with strong Manangoni effect:Uniform stretching limit, u(x , h(x , t), t) = Xt
1−x1−X
∗.
*Jones et al. 2005 and Heryudono et al. 2007.
PDE for h(x , t)
Lubrication theory → separation of scales → pde for h(x , t)
ht + E + Qx = 0, Q =
∫ h(x,t)
0u(x , y , t) dy
Q(x , t) =h3
12
(1 +
3β
h + β
)(Shxxx +G)+Xt
1− x1− X
h2
(1 +
β
h + β
)Blink cycle: E = 0, G = 0, β = 0.01
End motion and fluxes
At ends, h(±1, t) = h0, Q(1, t) = Qbot(t), Q((X (t), t) = Qtop(t)(hxxx(±1, t) spec’d)Realistic lid motion and fluxes
Lid motion like t2e−t or constantBurke and Mueller (98), Doane (80)Fluxes Proportional to Lid Motion (FPLM) Jones et al (05)FPLM and punctal drainage/lacrimal gland supply:FPLM+ BCs (Heryudono et al (07))
Map to fixed domain −1 ≤ ξ ≤ 1Polynomial initial condition 1 + ξ2m
Numerics: MOL: Chebyshev collocation in space (with bells andwhistles)ode15s in Matlab for ode at Chebyshev points
FPLM BCs
he = film thickness under lid; h0 at lid edgeWant to control fluxes when end moves: input exposed fluid fromhe, subtract off that from h0
Full blink cycle, FPLM BCs
blink separated by 5s of openFluxes Proportional to Lid Motion (FPLM)
Full blink cycle, FPLM BCs, E = G = 0
Opening; Q(X (t), t) = −Xthe (FPLM), Jones et al(05)exposes film under lid; better uniformity than no flux
Full blink cycle, FPLM BCs, E = G = 0
Open phase; relaxation only, thin regions develop at ends: “blacklines"Downstroke; flux out under moving lid
Half blink comparison: King-Smith interferometry8mm diam image, pre-lens tear film
Experimental thicknesses from heavy line
Half blink comparison, FPLM BCs, E = G = 0
V0 = 1.576, he = 0.35: valley depth not too good,shape of one side ok
Punctal and lacrimal contribs: FPLM+ BCs
Temporary lacrimalgland supplyFluid to puncta viamenisci near lidsVisualized: Maurice(1973), Doane(unpub)Convert to 1d
Half blink comp, FPLM+ BCs, E = G = 0
V0 = 1.576, he = 0.35: valley depth ok, shape goodmin value ok; add 0.25 of lac gland influx each partial blink
Part II: Study of Reflex Tearing(Maki PhD thesis here after;
Maki et al, Math Med Biol (2008) 25:187–214)
Prior Work: Measurements
Two Highlights
Role of black lines:
Image taken by King-Smith.
The black line is a localized thinregions near the lid margin thatis often described as a barrier totransfer of tear fluid between thefilm and meniscus.
Reflex Tearing:
Taken from King-Smith et al. 2000.
Reflex tearing is the onset oftearing triggered by irritation.Figure: central corneal tear filmthickness measurement.
Tear Film Evolution Model
The evolution of the free surface is given by
ht +
[h3
12(Shxxx + G) + Xt
1− x1− X
h2
]x
+ E = 0,
with E = 3× 10−4, G = 2.5× 10−3, β = 0and upper lid motion X(t)
Boundary Conditions
Fix TMW:h(±1, t) = h0, where h0 = 13 (note, h′0 = 65µm).
Specify Flux:
Modified from Jones et al. 2005 and Heryudono et al. 2007.
Numerical Method
Map moving domain X (t) ≤ x ≤ 1 into the fixed domain −1 ≤ ξ ≤ 1
ξ = 1− 2(1− x)/(1− X (t)).
Overset grid:
Method of lines
Spatial discretization: Second-order finite differences.Time discretization: ode15s in MATLAB.
Reflex Tearing: Reduced Reflex Tearing continues
Evolution of the tear film thickness with reduced continuous reflex flux
Comparison with in vivo Measurements
The small flux causes the tendency to a constant thickness.Omitting the constant flux lets film thin after pulse.Possible improvements: Include van der Waals conjoining pressureand 2D effects.
Part III: Eye-shaped Domain with Pressure BCs(Maki et al, Math Med Biol (2009), to appear)
Prior Work: Measurements
Evidence of hydraulic connectivity
Observation 1:Suspension of dark particles is introduced into the eye. The particlesmove with the tear fluid. Maurice observed communication betweenthe menisci with a slit lamp.Observation 2:
Image taken by Harrison et al.
Stationary Domain Ω
Boundary ∂Ω
Upper/Lower Lid:
Polynomials fit to measure lid data. (help: Xaolin W, Pete U)Temporal/Nasal Corner:
Polynomials constructed to create a smooth boundary.
Tear Film Evolution Model
The evolution of the free surface is given by
ht +∇ ·[− h3
12∇ (p + Gy)
]= 0, p + S∆h = 0.
Boundary conditions
Fix TMW:h|∂Ω = h0, where h0 = 13.
Specify pressure or TMC:
Previous works assume con-stant TMC. For example, Wonget al. 1996 derived the formula
h′ = 2.1234R′(
µU ′m
σ
)2/3
.
Numerical Method
Overset grid: Generated in Overture. (Henshaw and coworkers,LLNL)
Temporal corner.
Boundary curves:
Defined by NURBS mapping.Grid for upper/lower lid:
Defined by a normal mapping.Grid for temporal/nasal corner:
Defined by a normal mapping.
Method of lines
Spatial discretization: Second-order curvilinear finite differencesgenerated in Overture.
Time discretization: Variable stepsize (fixed leading coefficient)second-order backward differentiation formula.
Capillarity OnlyFrozen Pressure BC
Relaxation: Thickness Contours
Evolution of the contours of the tear film thickness
Relaxation: Pressure
Evolution of the pressure of the tear film thickness
Tear Film Movement
Flux vector field at 5 seconds:
Does not capture hydraulic connectivity. Tear fluid exits domain inupper and lower menisci.
Capillarity and GravityFrozen Pressure BC
Relaxation: Thickness Contours
Evolution of the contours of the tear film thickness
Tear Flux: frozen pressure on bndry
Flux vector field at 5 seconds, G 6= 0:
Captures weak hydraulic connectivity.
Dynamics: frozen pressure bc
Thickness dynamics give black lines, bulge in middleCanthi’s large curvature extracts fluidWith gravity, fluid dragged down slowlyWith gravity, weak hydraulic connectivityEnforced low pressure at boundary promotes pressure steeping;2D version of Bertozzi et al (94)
Part IV: Eye-shaped Domain with Flux BCs (Makiet al, (2009), submitted)
Capillarity OnlyNo Flux BC
Tear Flux: no flux bc
Flux vector field at 1 seconds:
No flux out boundary now.No hydraulic connectivity.
Tear Flux: no flux bc
Flux vector field at 10 seconds:
No flux out boundary now.No hydraulic connectivity.
No flux BCs: Pressure
Evolution of the pressure of the tear film thickness
No pressure gradient normal to bndry now.Small pressure gradient around bndry.
No flux BCs: Pressure
Evolution of the pressure of the tear film thickness
Small pressure gradient around bndry.Much like time-varying pressure case.
Dynamics: no flux bc
Thickness dynamics similar to previous caseSmall flux out boundary eliminated.Rapid decrease to small or pressure variations around bndryRapid change in p from interior to meniscus weakens with timePressure peaks smoothed with time
Capillarity and GravityNonzero Flux BC
Tear Film Evolution Model
The evolution of the free surface is given by
ht +∇ ·[− h3
12∇ (p + Gy)
]= 0, p + S∆h = 0.
Boundary conditions
Fix TMW:h|∂Ω = h0, where h0 = 13.
Specify flux: (with P. Ucciferro)
Tear Flux: nonzero flux bc (G = 0)
Tear film thickness at 10 seconds:
Flux from upper lid splits.Some hydraulic connectivity.
Tear Flux: nonzero flux bc (G = 0)
Flux vector field at 1 seconds:
Flux from upper lid pushes in black line.Black line not as easy to penetrate.
Tear Flux: nonzero flux bc (G = 0)
Flux vector field at 10 seconds:
Black line being pushed out of way.Some hydraulic connectivity.
Tear Flux: nonzero flux bc (G = 0)
Pressure field at 10 seconds:
Dramatic steepening near puncta limits calculation.
Tear Flux: nonzero flux bc (G 6= 0)
Tear film thickness at 10 seconds:
Flux from upper lid pushes in black line.Black line not as easy to penetrate.
Tear Flux: nonzero flux bc (G 6= 0)
Flux vector field at 1 seconds:
Flux from lac gland pushing black line.Some hydraulic connectivity.
Tear Flux: nonzero flux bc (G 6= 0)
Flux vector field at 10 seconds:
Flux doesn’t split; all to canthus.Some hydraulic connectivity.
Dynamics: nonzero flux bc
Thickness dynamics similar to previous caseNo gravity, flux splits as in exptWith gravity flux doesn’t split for our choicesSteepening pressure in this model with some BCs limitscomputationHydraulic connectivity present at outer canthus
More and Future directions
more King-Smith/OSU expts!
2D film models:
Moving geometry for blinks! Blinking ellipse working...Uniform stretching equations on moving domainEllipsoidal cornea, with McFadden (NIST) and Usha (IIT Madras)(09)
Other directions: vdW conjoining pressure and evaporation
1D stationary ends: with Winter and Anderson (09); P Ucciferro1D moving ends: J Tang
two layer models (separate lipid layer)
Thank You!
More and Future directions
more King-Smith/OSU expts!
2D film models:
Moving geometry for blinks! Blinking ellipse working...Uniform stretching equations on moving domainEllipsoidal cornea, with McFadden (NIST) and Usha (IIT Madras)(09)
Other directions: vdW conjoining pressure and evaporation
1D stationary ends: with Winter and Anderson (09); P Ucciferro1D moving ends: J Tang
two layer models (separate lipid layer)
Thank You!
FPLM BCs
No flux: he = 0, add flux of Xth0
For Fluxes Proportional to Lid Motion (FPLM), add another flux−Xthe (Jones et al, 2005)
Prior work: tear film evolution
Upward post-blink motion: Marangoni effectBerger and Corrsin (74)Owens and Philips (01)King-Smith et al (04,05)
Post-blink relaxationLocalized power law thinning near film endsBraun and Fitt (03): evaporation
Tear film formation and relaxationWong, Fatt and Radke (96): quasi-static dip coating and post blinkmodelsJones et al (05): opening with coating and subsequent relaxationfor one-equation modelsJones et al (06): same except mobile surface and surfactanttransport
Prior work: tear film evolution
Upward post-blink motion: Marangoni effectBerger and Corrsin (74)Owens and Philips (01)King-Smith et al (04,05)
Post-blink relaxationLocalized power law thinning near film endsBraun and Fitt (03): evaporation
Tear film formation and relaxationWong, Fatt and Radke (96): quasi-static dip coating and post blinkmodelsJones et al (05): opening with coating and subsequent relaxationfor one-equation modelsJones et al (06): same except mobile surface and surfactanttransport
Prior work: tear film evolution
Upward post-blink motion: Marangoni effectBerger and Corrsin (74)Owens and Philips (01)King-Smith et al (04,05)
Post-blink relaxationLocalized power law thinning near film endsBraun and Fitt (03): evaporation
Tear film formation and relaxationWong, Fatt and Radke (96): quasi-static dip coating and post blinkmodelsJones et al (05): opening with coating and subsequent relaxationfor one-equation modelsJones et al (06): same except mobile surface and surfactanttransport
Recent results
Blink cycle:Sinusoidal motion and fluxes:Braun (06), with King-Smith (JFM 07)Quantitative comparison with experiment
Realistic lid motion and fluxesHeryudono et al (MMB 07)
Lid motion from filmed eye blinks:Doane (80), Berke and Müller (98), modified by usFluxes of tear fluid estimated from tear film literature:Mishima and Maurice (65), Jones et al (05,06)Quantitative comparison with in vivo thickness measurements
Realistic lid motion and reflex tearsMaki et al (07)
Lid motion from filmed eye blinks:Doane (80), Berke and Müller (98), modified by usFluxes of tear fluid estimated from tear film literature:Mishima and Maurice (65), Jones et al (05,06)Quantitative comparison with in vivo thickness measurements
Recent results
Blink cycle:Sinusoidal motion and fluxes:Braun (06), with King-Smith (JFM 07)Quantitative comparison with experiment
Realistic lid motion and fluxesHeryudono et al (MMB 07)
Lid motion from filmed eye blinks:Doane (80), Berke and Müller (98), modified by usFluxes of tear fluid estimated from tear film literature:Mishima and Maurice (65), Jones et al (05,06)Quantitative comparison with in vivo thickness measurements
Realistic lid motion and reflex tearsMaki et al (07)
Lid motion from filmed eye blinks:Doane (80), Berke and Müller (98), modified by usFluxes of tear fluid estimated from tear film literature:Mishima and Maurice (65), Jones et al (05,06)Quantitative comparison with in vivo thickness measurements
Recent results
Blink cycle:Sinusoidal motion and fluxes:Braun (06), with King-Smith (JFM 07)Quantitative comparison with experiment
Realistic lid motion and fluxesHeryudono et al (MMB 07)
Lid motion from filmed eye blinks:Doane (80), Berke and Müller (98), modified by usFluxes of tear fluid estimated from tear film literature:Mishima and Maurice (65), Jones et al (05,06)Quantitative comparison with in vivo thickness measurements
Realistic lid motion and reflex tearsMaki et al (07)
Lid motion from filmed eye blinks:Doane (80), Berke and Müller (98), modified by usFluxes of tear fluid estimated from tear film literature:Mishima and Maurice (65), Jones et al (05,06)Quantitative comparison with in vivo thickness measurements