models for dynamics of the human tear film · models for dynamics of the human tear film r.j....

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

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)

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

h + β

)(Shxxx +G)+Xt

1− x1− X

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

12(Shxxx + G) + Xt

1− x1− 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.

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

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

No flux out boundary now.No hydraulic connectivity.

Tear Flux: no flux bc

No flux out boundary now.No hydraulic connectivity.

No flux BCs: Pressure

No pressure gradient normal to bndry now.Small pressure gradient around bndry.

No flux BCs: Pressure

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

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.

Flux from upper lid pushes in black line.Black line not as easy to penetrate.

Black line being pushed out of way.Some hydraulic connectivity.

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.

Flux from lac gland pushing black line.Some hydraulic connectivity.

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)

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

models for dynamics of the human tear film · models for dynamics of the human tear film r.j....

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