interaction of the tear film with the ocular surface · interaction of the tear film with the...

Intro During a blink TBU Overall osmolarity TF and Epithelium Stratified Epithelium The End

Interaction of the Tear Film with the OcularSurface

R.J. BraunDepartment of Mathematical Sciences, U of Delaware

Department of Mathematical SciencesDepartment of Biomedical Engineering

CAMM (www.mathandmedicine.org)

IMA Workshop on Electrohydrodynamics and Electrodiffusion in Material Sciences and BiologyMarch 14, 2018

Supported by the NSF and the NIH


Why Study Human Tear Film?

Normal tear film dynamics: Many aspects to understand!Dry eye syndrome: A common abnormality associated with the

disruption of the tear film. It is caused by an insufficientor malfunctioning tear film.

Prevalence: It is estimated that 10% - 15% of Americans over theage of 65 have one or more symptoms of dry eyesyndrome*.

Symptoms: Burning/stinging; Blurred visionIrritation/redness; Dry sensationForeign body or “gritty” sensation; Tearing

Impact: Tasks of daily life such as reading and driving arenegatively impacted by dry eye syndrome**.

*Stein et al. 1997.**Miljanovic et al. 2007.


What is the Tear Film?

Lipid layer floating fatty/oil slick at interface with air(≈ 50− 100nm)

Aqueous mostly water between lipid and ocular surface(≈ 3− 5µm)

Ocular surface Transmembrane mucins+microplicae (≤ 0.5µm)

Gipson 04 via Nichols 85(guinea pig) Govindarajan & Gipson

10


Overview of Dynamics

Tear film supply and drainage

Image from Wikipedia.

Lacrimal gland: The lacrimal glandsupplies new tear fluidduring a blink cycle.

Meibomian gland: The meibomiangland supplies lipids fromlid edges.

Punctal drainage: Removes excessfluid starting at the halfwayopen 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 During a Blink

Imaging the lipidlayer during ablink: stroboscopiclight sourceeliminates blurMore colorful isthickerMoving lid plowsthe lipid layer intothicker stateNot uncommonly,very similarstructurereappearsSource: RJB et. al(2015), Prog. Ret. EyeRes.



Imaging the lipidlayer during ablink: more colorfulis thickerMoving lid plowsthe lipid layer intothicker stateRapid TBU visible(Lan Zhong et alMMB 17)Source: RJB et. al(2015), PRER


Tear Breakup (TBU), Begley et al (Indiana)

Sodium fluorescein solution instilled as drop; quenching regime(Nichols et al 12)Glows green under blue lightHigh concentration quenches: Darker if thinned by evaporationSignificant regions of breakup as time increases (movie)

Saltiness in TBU not directly measurable, estimated by math models(Peng et al 2014; RJB et al, 2015, 2017).


Tear Breakup (TBU), King-Smith et al (OSU)

Interferometry aimed at lipid layer reveals moreLight and dark contours show relative change in LL thicknessRoughness appears in TBU regionsSignificant regions of breakup as time increases (movie)


Tear Film Dynamics: Simultaneous imaging

Lipid+aqueous: FL+Lipid IF images by King-Smith et al (IOVS 13)

Top: 15s post-blink;bottom: 20 pbLipid holes preceded dimFL areas.Elevated evaporationthrough holesHigh osmolarity in TBUestimated with mathmodels (Peng et al ’14,RJB et al ’15, ’17)


Part I: Blinking and Rippling(RJB, Nick Gewecke, et al, Prog. Ret. Eye Rsch., 2015)



Imaging the lipidlayer during ablink: more colorfulis thickerMoving lid causesrough surface,echoing thecorneaRoughness of TFdue to flow.Source: RJB et. al(2015), Prog. Ret. EyeRes.


Modeling of a blink

Ends of film move, thinlayer under movingendsEvaporation doesn’tmatter in short time ofblinkTear/air interface is”uniformly stretching”:limit of strongsurfactantJones et al, 05RJB and King-Smith, 07

A simplistic ”rough”cornea is assumed:similar to observationsGipson 2004King-Smith and Nichols 2014


Model for blinking domain

On − X (t) < x < X (t),

∂h∂t

+1− x

1− X (t)h2

dXdt

− 112

∂

∂x

(h3 ∂p∂x

)= 0,

p = −∂2h∂x2 −

∂2zc

∂x2 .

First eqn: PDE for TFthickness hMotion of whole surfaceinduced via dX/dt term:uniform stretchingSecond eqn: definition ofpressurePressure is leading order partof curvature of TF surface


Model for blinking domain

At the ends,

h(±X , t) = h0 − zc(±X )

−h3

0

12∂p(X , t)∂x

= ±−[h0 − zc(±Z )]

+[he − zc(±X )]/2 dXdt.

First eqn: Pin theTF thickness atconstant valueSecond eqn, rtside: flux into or outof TF via motion ofendSecond eqn, leftside: response offilm at end


Nonlinear model results

Ends of film move and drag fluidMoving film echoes corneal surfaceLike ripples on a shallow stream



Downstrokeandupstrokeshift peaksrelative tocornealsurfaceSeen inexpt!



A set of dotsdid not movewith lids ortear filmBright anddark sidesflipped withdirection oflidsConsistentwith phaseshiftSeen intheory!


Part II: TBU and Rippling(Amy Janett et al, in prep, 2017)


Roughness in TBU

P.E. King-Smith et al, OVS 2008

Left: TFLLinterferometryBelow:Retroillumination(Begley lab, IU)

Braun et al, PRER 2015

How?


Modeling

In interblink periodnowEvaporation drivesthinning hereCapillarity pushesflow into thin regionbut not fast enoughif TBUCorneal surfaceassumed to bewettable, withassociatedequilibriumthickness ofglycocalyx


Scalings and nondimensionalization

Scalings based on tear break up (TBU):v0 is measured peak thinning rated is characteristic thicknessd/v0 is time scale: TBUT for flat filmsurface tension γ, viscosity µbalance surface tension and viscosity: S = γε4/(µv0) = 1

so that ` =(

γµv0

)1/4d

For v0 = 20µm/min, d = 3.5µm, ` ≈ 0.35mm→ ε = d/` = 0.01→Lubrication theory


Model for TBU, rough corneaFor the TF thickness h:

∂h∂t− 1

12∂

∂x

(h3 ∂p∂x

)= −J,

p = −∂2h∂x2 −

∂2zc

∂x2 − Π(h).

For the Gaussian evaporation function J(x):, and wetting forces3

given by

J(x) =v1

v0+

(1− v1

v0

)· exp

(−(x − xL

2

)2

2x2w

).

Ocular surface function zc(x):

zc(x) = za [1 + sin(2πkx)] ,

Wetting forces Π(h):Π(h) = Ah−3.

Flat IC’s:

h(x ,0) = 1− zc(x)− za, p(x ,0) = Π(h(x ,0)).


Model for TBU, rough cornea

Typical parameter values are:v1/v0 = 1/20, slow evaporation around peak value (King-Smithet al 2005)xL = 8, wide domainxw = 1.2, evaporation width about natural scaleza = 1/14, amplitude about same as glycocalys, surfaceroughnessk = 10, based on average corneal cell size of 36 µmA = 9.9× 10−3, stops thinning at 0.25 µm (glycocalyx size).

Numerical solution method similar except Fourier spectral in space


Numerical Solutions

Film thinning fastest at peak evaporationSurface roughness visible at TBU


Numerical Solutions

5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

k

r

Relative amplitude of TF during TBU

Approximate ha computed at TBUApproximate relative amplitude r = ha/za decreases withincreasing k


Summary for rippling

Lipid imaging system caught rippling during blink

Can explain observations with flow over rough substrate

Shifting of image, scalesNonlinear (shown) and linear theory (not shown)

Simplified TBU model sees rippling quite close to TBU: good forexperitmenters!

Generalizations: better model for glycocalyx (first effort started withMastroberardino, Siddique, Anderson)


Part III: Osmolarity increase(RJB, Nick Gewecke, et al, Prog Ret Eye Rsch, 2015;

Peng et al, Adv Coll Interface Sci 2014;RJB et al Math Med Biol 2017;

Longfei Li et al, Math Med Biol 2016)


Observations

Overall dynamics

King-Smith imaging (09)Sodium fluorescein solution instilled as drop; quenching regime(Nichols et al 12)Glows green under blue light; lighter is less concentrated (movie)


Osmolarity and Dry Eye Syndrome (DES)Report of the International Dry Eye WorkShop (DEWS 2007):

Osmolarity: concentration ofions (primarily salts andproteins) in aqueous.

Osmolarity thought to beimportant in DES (DEWSReport, 2007)

Tear film instability:evaporation, breakup (TBU)

Elevated osmolarity: pain,inflammation, damage

Water permeability of cornealess than conjunctiva

Measurements only intemporal canthus; elsewhere?

Estimate osmolarity in TBUfrom imaging or models?

Diagnose dry eye with TearLab


Model Formulation

PDEs for thickness h andosmolarity c:

∂th +∇ ·Q = −EJ + Pc(c − 1)

J =1− δ

(S∆h + Ah−3

)− Bi(1−T∞)h

1+Bih

K + h1+Bih

Q =h3

12∇(S∆h + Ah−3 −Gy

)h∂tc +∇c ·Q =

1Pec∇ · (h∇c)

+ EJc − Pc(c − 1)c

ICs and BCs:

h(x,0) = ”flat- c(x,0) = 1bottom bowl” c|∂Ω = 1

Solute Parameters:Pec ≈ 9.6× 103 is the PecletnumberPc ≈ 1.305× 10−2 is the thewater permeability of cornea

c equation based on Jensen andGrotberg , 1993.Steady BC with E = A = 0, Makiet al, 2010Unsteady BC with all but c, Li etal, 2013


Permeability Distribution

We consider different water permeability for cornea (12.0µm/s, 0.013nondimensionally) and conjunctiva (55.4µm/s, 0.06nondimensionally).

Water permeability of the ocular surface


Thickness & Osmolarity (Thinning Rate: 4 µm/min)

Thickness left, osmolarity right. Black line forms from capillarity

as a dark blue band. Tear film thinner in interior from

evaporation; worst on cornea. Lower water permeability of

cornea⇒ thinner there. Highest osmolarity over cornea.

max(c)≈ 1.5 is 450mOsM:threshold of feeling (Liu et al 09)

Faster evaporation: max(c)≈ 6 or1800 mOsM possible with 20µm/min! Similar to TBU (Peng etal 14; Braun et al 15)

Estimates of 800-900mOsM inTBU (in vivo, Liu et al IOVS ’09)


Part IV: Osmolarity and the epithelium(Jen Bruhns et al, in prep;

Spencer Walker et al, in prep)


What are Corneal Epithelia?

Corneal Epithelium: The outermost layer of the cornea, which iscomposed of a single layer of basal cells, and multiplelayers of stratified squamous epithelial cells.

Figure: Hogan, Alvarado and Weddell, 1971.


What are Corneal Epithelia?

Corneal Epithelium: The outermost layer of the cornea, which iscomposed of a single layer of basal cells, and multiplelayers of stratified squamous epithelial cells.

Figure: Hogan 1971.

How does evporation of the TF affect this epithelium?


Osmolarity in Tear Film Models

How do tear film models deal with these cells?No Flux: Some tear film models assume that there is no flux of

ions or water at the corneal surface (Zubkov et al, JFM2012).

Semipermeable Membrane: Other models treat the corneal surfaceas a semipermeable membrane, where as theosmolarity of the tear film becomes large an osmoticflux of water crosses the cornea (e.g., Li et al. MMB2016).


Modeling the Cornea as a Stack of Cells

Water Transport Models: Levin et al.(2004) modeled water transport in a stackof these epithelial cells.

Strengths: Keeps track of water in theentire stack.

Weaknesses: Ignores ion (salt) transport.

Figure: Levin et al. 2004. (mice)


Modeling the Cornea With Ion Transport

Ion Transport Models: Levin et al.2006 modeled ion transport throughoutthe epithelium.

The multilayered epithelium was modeledas a single layer.

Strengths: This model is able to keeptrack of ions throughout the compart-ments.

Weaknesses: Simplified water transportwith osmolarity throughout the system(320 mOsM).

The stack of epithelium was modeled assingle layer.

Figure: Levin et al. 2006.(mice)


The Model

Model Assumptions:- The basolateral solutions has fixed composition.- The evaporation rate is constant.- There is no water transport through the paracellular

space (tight junctions).


Mass Conservation

Water Transport: Governed by osmosis and evaporation.

JWi = Vw pw

i (ψout − ψin).

JWe = 20× 10−6cm/s


Mass Conservation (continued)

Passive Transport (GHK): Governed by electrodiffusion.

Jxi = px

i zxVi FRT

Cxin−Cx

out exp(−zx

Vi FRT

)1−exp

(−zx

Vi FRT

) .1

1Hille 1992



Active Transport: Pumps 3 sodium ions out of the cell and 2 potassiumions into the cell.

Jact = pact

[CNa

inCNa

in +K Naact

]3 [ CKb

CKb +K K

act

]2.2

2Levin et al. 2006



Co-Transport: Pumps 2 chloride ions, 1 potassium ion, and 1 sodiumion per cycle.

Jco = pcoCNa

in CKin(CCl

in )2−CNab CK

b (CClb )2[

1+CNa

bK Na

co

][1+

CKb

K Kco

][1+

CClb

K Clco

]2 .3

3Levin et al. 2006


Model System

From the given mass fluxes a system can be set up via massconservation and Kirchoff’s laws (solved using ODE15s in MATLAB).Concentration (differential):

dCxi

dt=[ax

i Jx,Neti − Cx

i

] 1hi.

Height (differential): Ions do not contribute to the volume.

dhcell

dt= −JW

A − JwB .

dhtf

dt= JW

A − JWe .

Kirchhoff’s Laws (algebraic): Prevents charge buildup.IA + IP = 0 and IB + IP = 0.4

4VA, VB , and VP are the algebraic variables.


Results

Ions: In this model ion concentrations remain in similarranges to the ion transport model by Levin et al. 2006.

- Even with the fixed osmolarity assumption removed,the various osmolarities throughout remain at high (dueto evaporation), but physiological levels.

Figure: Osmolarity for short time (left), and long time (right).


Results (continued)

Water: It is clear that there are two time scales for watertransport.

- The tear film equilibrates more slowly than the cell forshort time, but continues changing long after the cellreaches an equilibrium.

Figure: Height for short time (left), and long time (right).


Background

Dealing With a Stack of Epithelium:Prior models deal with ion transport in asingle epithelium. We hope to model anentire stack.

Known:-Potential difference across individual lay-ers (rabbit).-Epithelial cells are polar.-The dynamics of the stack as one layer.

Unknown:-Transporters on each individualepithelium?-Should the system even reach anequilibrium?

Figure: Klyce 1972 (rabbit)


Conclusion

We have some partial knowledge of the distribution of some waterchannels (aquaporins).

Aquaporins:- More AQP1 in endothelium and stroma.- More AQP3 at back of epithelium.- More AQP5 and CFTR at apical side.

Figure: Levin et al. 2006.


The Model

Assumptions:- since the transporters are largely unknown it is simplest to

first model two identical cells on top of one another.

- The paracellular space has a fixed height (h1(t = 0)/1000).


Model System

This system is larger but is nearly identical the case for one cell. Newstate equations must be added to deal with the paracellular solution.Concentration (differential):

dCxi

dt=[ax

i Jx,Neti − Cx

i

] 1hi.

Height (differential):

dhadt = JW

A,1 − JWe . dh1

dt = −JWA,1 − JW

B,1.

dh2dt = −JW

A,2 − JWB,2. JW

B,1 = −JWA,2.

Kirchhoff’s Laws (algebraic):IA,1 + IP,1 = 0. IB,1 + IP,1 = 0.

IA,2 + IP,2 = 0. IB,2 + IP,2 = 0.


Results

Steady State: When the evaporationrate is set to zero the entire system shouldbe at an equilibrium.

- Ions are pumped into the paracellu-lar space from active transport in thetop cell which drives up the somolarityand causes the system to drift away fromphysiological conditions at very large time.

Potentials: In mice the potential differ-ence across the cornea should be -23 mV.In this model with two stacked the poten-tial difference is about -49 mV.


Three cell layers

We attempted to adjust the parameters for a stack of cells with threelayers using fminsearch. We found:

Trying to optimize for keeping equilibria from one layer producedparacellular space much too mobileUsing fminsearch as a heuristic to keep roughly equilibriumvalues gave reasonable valuesLower values for ion transport at interior apical sidesMore permeability in basolateral paracellular spacesSimilar apical and basolateral end valuesStays very close to equilibrium if zero evaporationReasonable dynamics with evaporation: voltage drop -85mV for1 minute at 12µm/min thinningNeed to refine with 4 cell layers for rabbit or mouse.


Conclusion for cell models

Understanding how the tear film interacts with ocular epithelium willbe useful in gaining a better understanding of dry eye syndrome.

Future Work Hopes To:Find more info re distribution ofchannels in each cell later.

Better channel/membrane models?

Model the entire epithelium.Figure: Levin et al. 2006.


Summary

Brief survey of tear film imaging in blinking and interblink

During blinks, pattern consistent with theory of flow over rough surface

In TBU, theory confirms experiment assumption that rough pattern isTBU

Osmolarity in tear film with evaporation: elevated, especially in TBU

Effect of evaporating tear film on corneal epithelium: ion and watertransport

Increasing tear film osmolarity brings water out of and Na+ throughepithelium

Not discussed today:

Rapid TF breakup dynamics (Lan Zhong, finishing 2018)Two-layer TF breakup (Mike Stapf, finishing 2018)FL imaging models (several papers)Blinking eye, ”lens” shape: with J Brosch, TA DriscollBlinking eye, overlapping grids: with K Maki (RPI), TAD, et al

CAMM for more info: www.mathandmedicine.org


Thank you!

UD:Back, l to r: Rich, Spencer Walker,Kevin Aiton.

Front, l to r: Chris Cornwell, JeromeTroy, Amy Janett, Mike Stapf.

Not Pictured: Lan Zhong, TobyDriscoll.

Ohio State Optometry:Ewen King-Smith

Indiana Optometry:Carolyn Begley andAdam WinkelerRPI: Kara MakiRPI: Bill Henshaw, JeffBanks, Longfei LiPSU York: JavedSiddique


A question for you allHow to treat this layer of mucins and proteins that allow water topass, but not at least some ions? (Argueso et al, J Biol Chem 2009)

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