Models for Dynamics of the Human Tear Film R.J. Braun 1 , K.L Maki 2 - - PowerPoint PPT Presentation

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Models for Dynamics of the Human Tear Film

R.J. Braun1, K.L ˙ Maki2, A. Heryudono3, T.A. Driscoll1, L.P . Cook1, P . Ucciferro1, P .E. King-Smith4, W.D. Henshaw5, G.B. McFadden6,

• R. Usha7, D.M. Anderson8 and K.N. Winter8

1 Mathematical Sciences, U of Delaware; 2 IMA, University of Minnesota; 3 Math, UMass Dartmouth; 4 College of Optometry, The Ohio State University; 5 CASC, LLNL; 6 MCSD, NIST; 7 IIT Madras, India; 8 Mathematical Sciences, George Mason U

Supported by the NSF

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Outline

Motivation Is tear film a complex fluid? (Modeling) Deciding from 1D models 1D results

Corneal substrate Blinks and partial blinks Reflex tearing Evaporation and conjoining pressure

2D computations – none today Summary and Future Directions

Goal: Quantify tear film dynamics!

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

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 sensation Foreign body or “gritty" sensation; Tearing Impact: Negative impact on, e.g., reading and driving from dry eye syndrome**.

*Stein et al. 1997; DEWS Report, 2007. **Miljanovic et al. 2007.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

What is Human Tear Film?

Tear film A multilayer structure playing a vital role in health and function of the eye.

Typical thickness of each layer in microns.

(M): A possible mucus film, if separate from aqueous debatable. A: Aqueous layer, primarily water (est. up to 98%). L: Lipid layer, polar surfactants at the A/L interface.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

What is Human Tear Film?

Tear film What’s in there? Properties? Mucins: 16 known mucins, most in aqueous layer 3 transmembrane at corneal surface Aqueous: Salts more than proteins mucins Lipid layer: insoluble in aqueous polar: sphingolipids, phospholipids nonpolar: waxes, cholesterol esters Shear thinning: Tiffany (1991), Pandit et al (1999) Not elastic: Tiffany (1994) gave inconclusive evidence for it Surface tension: 45mN/m, Tiffany and coworkers (1999); need proteins and lipids to get this value

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

What is Human Tear Film?

Tear film What’s in there? Properties? Mucins: 16 known mucins, most in aqueous layer 3 transmembrane at corneal surface Aqueous: Salts more than proteins mucins Lipid layer: insoluble in aqueous polar: sphingolipids, phospholipids nonpolar: waxes, cholesterol esters Shear thinning: Tiffany (1991), Pandit et al (1999) Not elastic: Tiffany (1994) gave inconclusive evidence for it Surface tension: 45mN/m, Tiffany and coworkers (1999); need proteins and lipids to get this value

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

What is Human Tear Film?

Tear film What’s in there? Properties? Mucins: 16 known mucins, most in aqueous layer 3 transmembrane at corneal surface Aqueous: Salts more than proteins mucins Lipid layer: insoluble in aqueous polar: sphingolipids, phospholipids nonpolar: waxes, cholesterol esters Shear thinning: Tiffany (1991), Pandit et al (1999) Not elastic: Tiffany (1994) gave inconclusive evidence for it Surface tension: 45mN/m, Tiffany and coworkers (1999); need proteins and lipids to get this value

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

What is Human Tear Film?

Tear film What’s in there? Properties? Mucins: 16 known mucins, most in aqueous layer 3 transmembrane at corneal surface Aqueous: Salts more than proteins mucins Lipid layer: insoluble in aqueous polar: sphingolipids, phospholipids nonpolar: waxes, cholesterol esters Shear thinning: Tiffany (1991), Pandit et al (1999) Not elastic: Tiffany (1994) gave inconclusive evidence for it Surface tension: 45mN/m, Tiffany and coworkers (1999); need proteins and lipids to get this value

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

What is Human Tear Film?

Tear film What’s in there? Properties? Mucins: 16 known mucins, most in aqueous layer 3 transmembrane at corneal surface Aqueous: Salts more than proteins mucins Lipid layer: insoluble in aqueous polar: sphingolipids, phospholipids nonpolar: waxes, cholesterol esters Shear thinning: Tiffany (1991), Pandit et al (1999) Not elastic: Tiffany (1994) gave inconclusive evidence for it Surface tension: 45mN/m, Tiffany and coworkers (1999); need proteins and lipids to get this value

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

What is Human Tear Film?

Tear film What’s in there? Properties? Mucins: 16 known mucins, most in aqueous layer 3 transmembrane at corneal surface Aqueous: Salts more than proteins mucins Lipid layer: insoluble in aqueous polar: sphingolipids, phospholipids nonpolar: waxes, cholesterol esters Shear thinning: Tiffany (1991), Pandit et al (1999) Not elastic: Tiffany (1994) gave inconclusive evidence for it Surface tension: 45mN/m, Tiffany and coworkers (1999); need proteins and lipids to get this value

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Overview of the Dynamics

Tear film supply and drainage

Image from Wikipedia.

Lacrimal gland: The lacrimal gland supplies new tear fluid during a blink cycle. Meibomian glands: The meibomian gland supplies lipids from lid edges. Punctal drainage: Removes excess fluid starting at the halfway open position 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.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Characteristics of the Human Tear Film

From Wang et al. 2006.

Characteristic tear film thickness In the middle of the cornea, 3 − 5µm. Upper and lower menisci Volume: 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.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film dynamics

Upward post-blink motion: Marangoni effect Berger and Corrsin (74), Owens and Philips (01), King-Smith et al (04,05) 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, substrate (non-Newtonian) Gorla and Gorla (04), Braun et al (09) Tear film formation and relaxation Wong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt) Blink Cycles Braun and King-Smith (07), Heryudono et al (07) (today) Reflex tearing: Maki et al (08) (today) 2D numerics with Pressure bcs or flux bcs Maki et al (09)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film dynamics

Upward post-blink motion: Marangoni effect Berger and Corrsin (74), Owens and Philips (01), King-Smith et al (04,05) 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, substrate (non-Newtonian) Gorla and Gorla (04), Braun et al (09) Tear film formation and relaxation Wong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt) Blink Cycles Braun and King-Smith (07), Heryudono et al (07) (today) Reflex tearing: Maki et al (08) (today) 2D numerics with Pressure bcs or flux bcs Maki et al (09)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film dynamics

Upward post-blink motion: Marangoni effect Berger and Corrsin (74), Owens and Philips (01), King-Smith et al (04,05) 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, substrate (non-Newtonian) Gorla and Gorla (04), Braun et al (09) Tear film formation and relaxation Wong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt) Blink Cycles Braun and King-Smith (07), Heryudono et al (07) (today) Reflex tearing: Maki et al (08) (today) 2D numerics with Pressure bcs or flux bcs Maki et al (09)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film dynamics

Upward post-blink motion: Marangoni effect Berger and Corrsin (74), Owens and Philips (01), King-Smith et al (04,05) 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, substrate (non-Newtonian) Gorla and Gorla (04), Braun et al (09) Tear film formation and relaxation Wong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt) Blink Cycles Braun and King-Smith (07), Heryudono et al (07) (today) Reflex tearing: Maki et al (08) (today) 2D numerics with Pressure bcs or flux bcs Maki et al (09)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film dynamics

Upward post-blink motion: Marangoni effect Berger and Corrsin (74), Owens and Philips (01), King-Smith et al (04,05) 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, substrate (non-Newtonian) Gorla and Gorla (04), Braun et al (09) Tear film formation and relaxation Wong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt) Blink Cycles Braun and King-Smith (07), Heryudono et al (07) (today) Reflex tearing: Maki et al (08) (today) 2D numerics with Pressure bcs or flux bcs Maki et al (09)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film dynamics

Upward post-blink motion: Marangoni effect Berger and Corrsin (74), Owens and Philips (01), King-Smith et al (04,05) 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, substrate (non-Newtonian) Gorla and Gorla (04), Braun et al (09) Tear film formation and relaxation Wong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt) Blink Cycles Braun and King-Smith (07), Heryudono et al (07) (today) Reflex tearing: Maki et al (08) (today) 2D numerics with Pressure bcs or flux bcs Maki et al (09)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film dynamics

Upward post-blink motion: Marangoni effect Berger and Corrsin (74), Owens and Philips (01), King-Smith et al (04,05) 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, substrate (non-Newtonian) Gorla and Gorla (04), Braun et al (09) Tear film formation and relaxation Wong et al (96), Jones et al (05,06), Jossic et al (09, non-Newt) Blink Cycles Braun and King-Smith (07), Heryudono et al (07) (today) Reflex tearing: Maki et al (08) (today) 2D numerics with Pressure bcs or flux bcs Maki et al (09)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Modeling Choices

Idealized domain (I thru IV)

*Braun et al. 2003.

Modeling assumptions The aqueous fluid is Newtonian with properties of water. The mucus and lipid layers included with appropriate boundary conditions. Rate of evaporation is uniform in space and 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 scales ǫ = d′/L′ ≈ 10−3 ⇒ lubrication.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Part I: Corneal shape, with and without shear thinning

Braun, Usha, McFadden, Driscoll, Cook and Braun (submitted, 2009)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Idealized domain

Cornea → ellipsoidal surface Axisymmetric assumed First model geometry: 1/2 of prolate ellipsoid, sharp half

• f rugby ball

Second model geometry: up to 1/2 of ellipsoid with a meniscus µ = h′(ν′, t′) is film surface

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Tear film thinning and corneal shape

Post-blink thinning: King-Smith et al (OVS ’08)

Central corneal data: average 4µm/min not uncommon (fast) Capillary, Marangoni effects, flow into eye surface not it Substrate driving flow? Most important thought to be evaporation

Relaxation due to corneal shape?

Ellipsoidal substrate? shear thinning? meniscus? Roy et al (JFM ’O2): multiple orders, general curved substrate Howell (JEM ’03): general curved and/or moving substrate

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Tear film thinning and corneal shape

Post-blink thinning: King-Smith et al (OVS ’08)

Central corneal data: average 4µm/min not uncommon (fast) Capillary, Marangoni effects, flow into eye surface not it Substrate driving flow? Most important thought to be evaporation

Relaxation due to corneal shape?

Ellipsoidal substrate? shear thinning? meniscus? Roy et al (JFM ’O2): multiple orders, general curved substrate Howell (JEM ’03): general curved and/or moving substrate

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Domain and nondimensionalization

Wikipedia Commons, by WillowW

Film thickness d/L′ ≈ 10−3 PDE for film thickness on ellipsoidal surface Prolate spheroidal coordinates

µ is like radius; constant µ is ellipsoid with foci on z axis ν is polar angle, 0 ≤ ν ≤ π/2 φ is azimuthal angle: axisymmetric, won’t need

Film inside 0 ≤ ν ≤ π/2 and µ0 ≤ µ ≤ µ0 + ǫh(ν, t)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Corneal (substrate) geometry

Surface quantities we need:

N = [sinh2(µ0) + sin2(ν)]1/2 Principle curvatures: kν, kφ Twice mean curvature: k1 = kν + kφ k2 = k 2

ν + k 2 φ; k3 = kνkφ

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Corneal (substrate) geometry

Surface quantities we need:

N = [sinh2(µ0) + sin2(ν)]1/2 Principle curvatures: kν, kφ Twice mean curvature: k1 = kν + kφ k2 = k 2

ν + k 2 φ; k3 = kνkφ

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Governing equations

Shear thinning fluid Leading order: [η(u,µ)u,µ],µ − p,ν N = 0 On µ = µ0, u = v = 0 On free surface µ = µ0 + h(ν, t), kinematic condition and

Normal Stress: p = −˜ k, ˜ k = k1+ǫk2h + ǫ∇2

sh

Tangential Stress: u,µ = 0 or u = 0

Roy et al keep blue terms and more elsewhere

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Governing equations

Ellis constitutive relation: η =

1 1+βτ α−1 , τ = ηu,µ

α = 3 for -2/3 power decay in shear rate (Tiffany et al 91) Reasonable fit for normal tears is ˙ γ1/2 = 20s−1, η0 = 5mPa·s. Shear thinning: β =

• ηU/d

τ1/2

α−1 ≈ 10−7 for eyes; up to 102

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

PDE for h(ν, t)

Large substrate curvature limit of Howell (03) Stress-free bc, Ellis fluid (new) ηu,µ=0 on µ = µ0 + h ⇒ h,t = − 1 sin(ν)N sin ν N h3 3 (k1,ν) +sin ν Nα hα+2 α + 2β|k1,ν|α−1k1,ν

• ,ν

h(ν, 0) = 1, no bcs but symmetry inherited from substrate Newtonian: β → 0. (Howell 03) Newtonian, tangentially immobile: h3/3 → h3/12

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

PDE for h(ν, t)

Large substrate curvature limit of Howell (03) Stress-free bc, Ellis fluid (new) ηu,µ=0 on µ = µ0 + h ⇒ h,t = − 1 sin(ν)N sin ν N h3 3 (k1,ν) +sin ν Nα hα+2 α + 2β|k1,ν|α−1k1,ν

• ,ν

h(ν, 0) = 1, no bcs but symmetry inherited from substrate Newtonian: β → 0. (Howell 03) Newtonian, tangentially immobile: h3/3 → h3/12

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Methods

Two modes for getting answers Thinning rate at center of cornea. Numerical computation: method of lines.

Chebyshev spectral discretization in space (ν) Doubled domain, even number of modes for origin, symmetry 96 to 128 modes in 0 ≤ ν ≤ π/2 worked well

• de15s in Matlab in time

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Center of cornea, Newtonian

Consider Newtonian, tangentially immobile Large curvature: substrate drives flow ht = − 1 sin νN h3 12 sin ν N k1,ν

• ,ν

h(ν, 0) = 1, substrate curvature derivatives at ν = 0: ∂h′ ∂t′ (ν = 0, t = 0) = 2σ 3η∞ Q[h′(0, 0)]3 R3 . R = 0.0078m, Q = −0.19 (Read et al, IOVS ’06); σ = 0.045N/m; η∞ = 10−3Pa·s ∂h′/∂t′(0, 0) = −0.09µm/min. Expt: −4µm/min. K-S et al (OVS 08) Exact soln at ν = 0: hmid(t) =

• 1 +

2 N2(0) d2k1(0) dν2 t

−1/2

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Center of cornea, Newtonian

Consider Newtonian, tangentially immobile Large curvature: substrate drives flow ht = − 1 sin νN h3 12 sin ν N k1,ν

• ,ν

h(ν, 0) = 1, substrate curvature derivatives at ν = 0: ∂h′ ∂t′ (ν = 0, t = 0) = 2σ 3η∞ Q[h′(0, 0)]3 R3 . R = 0.0078m, Q = −0.19 (Read et al, IOVS ’06); σ = 0.045N/m; η∞ = 10−3Pa·s ∂h′/∂t′(0, 0) = −0.09µm/min. Expt: −4µm/min. K-S et al (OVS 08) Exact soln at ν = 0: hmid(t) =

• 1 +

2 N2(0) d2k1(0) dν2 t

−1/2

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Center of cornea, Newtonian

Consider Newtonian, tangentially immobile Large curvature: substrate drives flow ht = − 1 sin νN h3 12 sin ν N k1,ν

• ,ν

h(ν, 0) = 1, substrate curvature derivatives at ν = 0: ∂h′ ∂t′ (ν = 0, t = 0) = 2σ 3η∞ Q[h′(0, 0)]3 R3 . R = 0.0078m, Q = −0.19 (Read et al, IOVS ’06); σ = 0.045N/m; η∞ = 10−3Pa·s ∂h′/∂t′(0, 0) = −0.09µm/min. Expt: −4µm/min. K-S et al (OVS 08) Exact soln at ν = 0: hmid(t) =

• 1 +

2 N2(0) d2k1(0) dν2 t

−1/2

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Center of cornea, Newtonian

Consider Newtonian, tangentially immobile Large curvature: substrate drives flow ht = − 1 sin νN h3 12 sin ν N k1,ν

• ,ν

h(ν, 0) = 1, substrate curvature derivatives at ν = 0: ∂h′ ∂t′ (ν = 0, t = 0) = 2σ 3η∞ Q[h′(0, 0)]3 R3 . R = 0.0078m, Q = −0.19 (Read et al, IOVS ’06); σ = 0.045N/m; η∞ = 10−3Pa·s ∂h′/∂t′(0, 0) = −0.09µm/min. Expt: −4µm/min. K-S et al (OVS 08) Exact soln at ν = 0: hmid(t) =

• 1 +

2 N2(0) d2k1(0) dν2 t

−1/2

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Computation, large curvature, Newtonian

t−1/2 thinning at left, finite time singularity at right, to t = 24 Both predicted by Howell (JEM 03)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Computation, large curvature, Newtonian vs Ellis

Weak shear thinning on right, Newtonian left; to t = 15 Perhaps slightly faster blowup on right

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Computation, large curvature, Newtonian vs Ellis

Strong shear thinning on right; β = 100 Blowup no longer at ν = π/2 as for Newtonian

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Roy et al type PDE for h(ν, t)

Newtonian, p = −k1 + ǫhk2 + ǫ∇2

sh + more

Stress-free bc ηu,µ=0 on h ⇒ ζ,t = − 1 sin(ν)N h2ζ 3 sin ν N ˜ k,ν

• ,ν

ζ = h − ǫk1 2 h2 + ǫ2 k3 3 h3. h(ν, 0) = 1, symmetry at ν = 0 and ν = π/2 Increased order in space allows more bcs, meniscus

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Newtonian fluid, Howell vs Roy models

Left: large curvature; right: more terms; last time t = 24 Finite time blowup stabilized (Howell 03, Roy et al 02) Qualitatively different film shapes from flat substrate

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Meniscus away from center

Now have 0 ≤ ν ≤ π/2 (left) or π/4 (right) Polynomial ic with h(0, 0) = 1 and bcs h(π/2, t) = 13, no flux Thinning rate increased at ν = 0 but still not enough for expt

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Summary of corneal shape

Single layer, 1 PDE models on ellipsoidal cornea Stress-free surface allowed Newtonian vs Shear-thinning Ellis fluid Ellis fluid was bounding case with η → 0 as τ → ∞ Thinning from film and substrate curvature too slow to be significant in central cornea; expt by King-Smith and co-workers much faster

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Part II: Blink cycles

Heryudono et al, Math Med Biol (2007) 24:347-377

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

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 conditions Eye surface: Slip condition and impermeability: u = βuy, v = 0; β = 0.01. Free surface: Kinematic condition: ht + uhx = v − E, with E =

J′ Umǫρ ≈ 3 × 10−4.

Normal stress condition: p = −Shxx, where S = ǫ3σ

µU′

m ≈ 5 × 10−7.

Insoluble surfactant with strong Manangoni effect: Uniform stretching limit, u(x, h(x, t), t) = Xt 1−x

1−X ∗.

*Jones et al. 2005 and Heryudono et al. 2007.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

PDE for h(x, t)

Lubrication theory → separation of scales → pde for h(x, t) ht + E + Qx = 0, Q = h(x,t) u(x, y, t) dy Q(x, t) = h3 12

• 1 +

3β h + β

• (Shxxx + G) + Xt

1 − x 1 − X h 2

• 1 +

β h + β

• Blink cycle: E = 0, G = 0, β = 0.01

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

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 constant Burke 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 ≤ 1 Polynomial initial condition 1 + ξ2m Numerics: MOL: Chebyshev collocation in space (with bells and whistles)

• de15s in Matlab for ode at Chebyshev points

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

FPLM BCs

he = film thickness under lid; h0 at lid edge Want to control fluxes when end moves: input exposed fluid from he, subtract off that from h0

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Full blink cycle, FPLM BCs

blink separated by 5s of open Fluxes Proportional to Lid Motion (FPLM)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

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

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Full blink cycle, FPLM BCs, E = G = 0

Open phase; relaxation only, thin regions develop at ends: “black lines" Downstroke; flux out under moving lid

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Half blink comparison: King-Smith interferometry

8mm diam image, pre-lens tear film Experimental thicknesses from heavy line

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Half blink comparison, FPLM BCs, E = G = 0

V0 = 1.576, he = 0.35: valley depth not too good, shape of one side ok

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Punctal and lacrimal contribs: FPLM+ BCs

Temporary lacrimal gland supply Fluid to puncta via menisci near lids Visualized: Maurice (1973), Doane (unpub) Convert to 1d

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Half blink comp, FPLM+ BCs, E = G = 0

V0 = 1.576, he = 0.35: valley depth ok, shape good min value ok; add 0.25 of lac gland influx each partial blink

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Transition from periodicity: Realistic motion

Compute ∆ = ||h (x, ∆tbc + ∆tco) − h(x, 2∆tbc + ∆tco)||∞ Left, FPLM BCs; right, FPLM+ BCs Transition not too sensitive to he, flux in ends Effectively a full blink around 1/6 open

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Dynamics: wetting and evaporation

Need more than FPLM alone for good tear film distribution (1D) Fairly good agreement with partial blink valley Periodic solution, or full blink, if about 1/8 to 1/10 still open

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Part III: Study of Reflex Tearing

(Maki PhD thesis; Maki et al, Math Med Biol (2008) 25:187–214

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior Work: Measurements

Two Highlights Role of black lines:

Image taken by King-Smith.

The black line is a localized thin regions near the lid margin that is often described as a barrier to transfer of tear fluid between the film and meniscus. Reflex Tearing:

Taken from King-Smith et al. 2000.

Reflex tearing is the onset of tearing triggered by irritation. Figure: central corneal tear film thickness measurement.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Tear Film Evolution Model

The evolution of the free surface is given by ht + h3 12 (Shxxx + G) + Xt 1 − x 1 − X h 2

• x

+ E = 0, with E = 3 × 10−4, G = 2.5 × 10−3, β = 0 and upper lid motion X(t)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

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.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

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.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Reflex Tearing: Reduced Reflex Tearing continues

Evolution of the tear film thickness with reduced continuous reflex flux

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

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 pressure and 2D effects.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Reflex tearing summary

Control flux and lid motion Good qualitative agreement with one measurement Need residual flux to establish constant thickness after pulse

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Part IV: Evaporation and conjoining pressure

Winter, Anderson (GMU) and Braun, Math Med Biol (to appear, 09) (Ucciferro (ugrad), Tang (grad), Braun, Cook, summer and fall 09)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Motivation

A different choice for tear film break up The cornea is wettable (Tiffany 1986)

Not always thought so Many papers: rupture via disjoining pressure Sharma and coworkers, Matar and coworkers

Evaporation is more important that previously thought Models with precursor film (wetting surface) in microchannels, e.g. Morris; Ajaev; Ajaev and Homsy How to put in eye setting?

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Motivation

A different choice for tear film break up The cornea is wettable (Tiffany 1986)

Not always thought so Many papers: rupture via disjoining pressure Sharma and coworkers, Matar and coworkers

Evaporation is more important that previously thought Models with precursor film (wetting surface) in microchannels, e.g. Morris; Ajaev; Ajaev and Homsy How to put in eye setting?

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Motivation

A different choice for tear film break up The cornea is wettable (Tiffany 1986)

Not always thought so Many papers: rupture via disjoining pressure Sharma and coworkers, Matar and coworkers

Evaporation is more important that previously thought Models with precursor film (wetting surface) in microchannels, e.g. Morris; Ajaev; Ajaev and Homsy How to put in eye setting?

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Motivation

A different choice for tear film break up The cornea is wettable (Tiffany 1986)

Not always thought so Many papers: rupture via disjoining pressure Sharma and coworkers, Matar and coworkers

Evaporation is more important that previously thought Models with precursor film (wetting surface) in microchannels, e.g. Morris; Ajaev; Ajaev and Homsy How to put in eye setting?

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Model with wetting cornea and evaporation

Conjoining pressure to wet corneal surface Π′ = A′ 6π(h′)3 Flux rate from free surface (p term new) KJ = α(p − pv) + T − Ts The film is still thin, lubrication theory...

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Model with wetting cornea and evaporation

The evolution of the free surface is given by ∂h ∂t + E ¯ K + h

• 1 − δ
• S ∂2h

∂x2 + Π

• =

− ∂ ∂x h3 12

• S ∂3h

∂x3 + ∂Π ∂x + G

• ,

Π = Ah−3. E, ¯ K spec’d evap from before S and G as before A is nondimensional Hammaker constant δ ∝ α is pressure diff contribution to evap

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Model with wetting cornea and evaporation

The evolution of the free surface is given by ∂h ∂t + E ¯ K + h

• 1 − δ
• S ∂2h

∂x2 + Π

• =

− ∂ ∂x h3 12

• S ∂3h

∂x3 + ∂Π ∂x + G

• ,

Π = Ah−3. Stationary ends BCs: set h and pressure IC: pcwise poly or poly Uniform finite difference in space; ode23s in time

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Model with flat surface

If all free surface derivatives are zero then ∂h ∂t + E ¯ K + h

• 1 − δ
• Ah−3

= 0 When does ∂h/∂t = 0? When [] = 0, or heq = (Aδ)1/3. Choose heq ≈ 300nm, on scale of microplicae and trans-membrane mucins. Don’t have A and δ separately except by guessing right now.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Model with flat surface

If all free surface derivatives are zero then ∂h ∂t + E ¯ K + h

• 1 − δ
• Ah−3

= 0 When does ∂h/∂t = 0? When [] = 0, or heq = (Aδ)1/3. Choose heq ≈ 300nm, on scale of microplicae and trans-membrane mucins. Don’t have A and δ separately except by guessing right now.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Model with flat surface

If all free surface derivatives are zero then ∂h ∂t + E ¯ K + h

• 1 − δ
• Ah−3

= 0 When does ∂h/∂t = 0? When [] = 0, or heq = (Aδ)1/3. Choose heq ≈ 300nm, on scale of microplicae and trans-membrane mucins. Don’t have A and δ separately except by guessing right now.

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Open phase, compare vdW and w/o

On left, A = δ = G = 0: relaxation until hmin(82) = 0 On right, A = δ = 10−3: reach heq, and continue until across interior More robust than Braun and Fitt 03; need A and δ nonzero

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Open phase, capturing heq

A = δ = 10−3: reach heq = 10−2 and stay Other values consistent with heq = (Aδ)−1/3

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Long times approaching heq

Reach heq at black line first, middle next Eventually heq all across middle Evaporation rate?

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Long times approaching heq

Whenever h → heq, Qevap → 0 First at black line, then middle bit, then all of inside

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Tear film breakup

“Contact line" location: h = 2heq How does speed of opening compare with observations in vivo from King-Smith lab? Experimental estimate: 10 to 30µm/s; starts faster, slows down Theory: A = 10−2, δ = 10−3, d = 5µmheq = 0.11µm, about 10µm/s, constant rate

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Tear film breakup

“Contact line" location: h = 2heq How does speed of opening compare with observations in vivo from King-Smith lab? Experimental estimate: 10 to 30µm/s; starts faster, slows down Theory: A = 10−2, δ = 10−3, d = 5µmheq = 0.11µm, about 10µm/s, constant rate

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Tear film breakup

“Contact line" location: h = 2heq How does speed of opening compare with observations in vivo from King-Smith lab? Experimental estimate: 10 to 30µm/s; starts faster, slows down Theory: A = 10−2, δ = 10−3, d = 5µmheq = 0.11µm, about 10µm/s, constant rate

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Wetting, evaporation and gravity, G = 0.0378

More fluid brought into domain!

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Wetting, evaporation and gravity, G = 0.016

Not really any effect!

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Wetting, evaporation and gravity, vary G

In narrow range, thicken film or not Evaporation removes fluid fast enough to arrest thickening if G not to large (other parameters fixed)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Dynamics: wetting and evaporation

Opening speeds of holes reasonable for obvious parameter choices With gravity, fluid dragged into domain if G big enough Seems to suggest controlling flux is more what eye does

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

More and Future directions

more King-Smith/OSU expts! 2D film models: Moving geometry for blinks! Blinking ellipse working (Maki) Uniform stretching equations on moving domain Other directions: wetting and evaporation 1D moving ends: J Tang Flux BCs insoluble surfactant models two layer models (separate lipid layer) Thank You!

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

More and Future directions

more King-Smith/OSU expts! 2D film models: Moving geometry for blinks! Blinking ellipse working (Maki) Uniform stretching equations on moving domain Other directions: wetting and evaporation 1D moving ends: J Tang Flux BCs insoluble surfactant models two layer models (separate lipid layer) Thank You!

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

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)

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film evolution

Upward post-blink motion: Marangoni effect

Berger and Corrsin (74) Owens and Philips (01) King-Smith et al (04,05)

Post-blink relaxation

Localized power law thinning near film ends Braun and Fitt (03): evaporation

Tear film formation and relaxation

Wong, Fatt and Radke (96): quasi-static dip coating and post blink models Jones et al (05): opening with coating and subsequent relaxation for one-equation models Jones et al (06): same except mobile surface and surfactant transport

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film evolution

Upward post-blink motion: Marangoni effect

Berger and Corrsin (74) Owens and Philips (01) King-Smith et al (04,05)

Post-blink relaxation

Localized power law thinning near film ends Braun and Fitt (03): evaporation

Tear film formation and relaxation

Wong, Fatt and Radke (96): quasi-static dip coating and post blink models Jones et al (05): opening with coating and subsequent relaxation for one-equation models Jones et al (06): same except mobile surface and surfactant transport

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Prior work: tear film evolution

Upward post-blink motion: Marangoni effect

Berger and Corrsin (74) Owens and Philips (01) King-Smith et al (04,05)

Post-blink relaxation

Localized power law thinning near film ends Braun and Fitt (03): evaporation

Tear film formation and relaxation

Wong, Fatt and Radke (96): quasi-static dip coating and post blink models Jones et al (05): opening with coating and subsequent relaxation for one-equation models Jones et al (06): same except mobile surface and surfactant transport

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Recent results

Blink cycle:

Sinusoidal motion and fluxes: Braun (06), with King-Smith (JFM 07) Quantitative comparison with experiment

Realistic lid motion and fluxes Heryudono et al (MMB 07)

Lid motion from filmed eye blinks: Doane (80), Berke and Müller (98), modified by us Fluxes 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 tears Maki et al (07)

Lid motion from filmed eye blinks: Doane (80), Berke and Müller (98), modified by us Fluxes of tear fluid estimated from tear film literature: Mishima and Maurice (65), Jones et al (05,06) Quantitative comparison with in vivo thickness measurements

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Recent results

Blink cycle:

Sinusoidal motion and fluxes: Braun (06), with King-Smith (JFM 07) Quantitative comparison with experiment

Realistic lid motion and fluxes Heryudono et al (MMB 07)

Lid motion from filmed eye blinks: Doane (80), Berke and Müller (98), modified by us Fluxes 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 tears Maki et al (07)

Lid motion from filmed eye blinks: Doane (80), Berke and Müller (98), modified by us Fluxes of tear fluid estimated from tear film literature: Mishima and Maurice (65), Jones et al (05,06) Quantitative comparison with in vivo thickness measurements

Tear Film Corneal shape Blink Cycles Reflex Tearing vdW wetting The End?

Recent results

Blink cycle:

Sinusoidal motion and fluxes: Braun (06), with King-Smith (JFM 07) Quantitative comparison with experiment

Realistic lid motion and fluxes Heryudono et al (MMB 07)

Lid motion from filmed eye blinks: Doane (80), Berke and Müller (98), modified by us Fluxes 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 tears Maki et al (07)

Lid motion from filmed eye blinks: Doane (80), Berke and Müller (98), modified by us Fluxes of tear fluid estimated from tear film literature: Mishima and Maurice (65), Jones et al (05,06) Quantitative comparison with in vivo thickness measurements