Tear Film Evolution

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Tear Film Evolution

• Katlyn Winter
• URCM
• George Mason University
• Professor Daniel Anderson

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Abstract

• After each blink, a thin film covers the
  surface of the eye to prevent irritation of the
  eyelid rubbing on the eye. A condition called dry
  eye exists in which holes form in the film
  between blinks, resulting in irritation of the
  eye. Based off of research done by R. J. Braun
  and A. D. Fitt, the film is modeled using a
  series of partial differential equations. To
  begin, no outside effects such as evaporation or
  gravity were taken into account. A new
  evaporation model, based on the work of Ajaev and
  Homsy, will be incorporated into the thin film
  equation and examined in order to take these
  effects into account and to try to more closely
  approximate the films behavior. We shall make
  comparisons of the present model to the
  Braun-Fitt research.

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Structure of Tear Film

• Mucus layer Closest to the surface of the eye
  and helps the film stay on the eye
• Aqueous layer Thickest layer washes away
  irritants (Layer we are modeling)
• Lipid layer Furthest from surface of they eye.
  Slows evaporation of aqueous layer

Information and picture from Schepens Eye
Research Inistitute http//www.schepens.harvard.ed
u/dry_eye_fact_sheet.htm
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Basic Problem

• Immediately after a blink, tear film begins to
  evolve due to effects such as gravity and
  evaporation
• Sometimes these effects cause holes in the film
  to form resulting in a condition known as dry eye
  syndrome
• Computer simulation of the evolution may help
  researchers better understand which effects are
  more influential on the evolution of the tear film

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Derivation(using a 2D thin film model)

• u is the velocity in the horizontal direction of
  the film, v is the velocity in the vertical
  direction, and u is the total velocity (u,v)
• Boundary conditions at y0 are u0 (no slip along
  surface) and v0 (no penetration of surface)
• Equation 1 maintains the conservation of mass
• Equation 2 is a form of Newtons second law ma
  F (the way it is shown here)
• Equation 3 takes into heat equation

Equations from Dr. Andersons notes on tear
film evaporation
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Set up

• L is the 1/2 length of the eye lid, in this case
  arbitrarily set to 14, a scaled distance
• h (x,t) is the thickness of the film, a function
  of space and time
• Notice the surface of the eye is flat. This is
  due to assumptions made in lubrication theory.
  Since the film is so thin in comparison to the
  length of the eye surface we can assume the film
  acts as if the surface is flat

Image from Braun and Fitt paper Modeling
drainage of precornial tear film after a blink
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Equation to Solve

• Boundary conditions fix h at L and hxx at L
• E/(Kh) is the evaporation term
• d is a new term added since the Braun Fitt
  equation
• On the right hand side of the equation, the G is
  the gravity term
• The A terms are related to Van der Waals
  attraction
• We are solving dh/dt using numerical
  approximation of a system of ODEs using Matlabs
  solver ODE23s

Equation from Dr. Andersons notes on tear film
evaporation
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Results

• The code just started working this week
• Assumptions I turned off all the extra
  parameters, delta, A, etc., to ensure the code
  was working
• Comparing to data from Braun and Fitt, it seems
  to have the correct basic shape
• The colors are different time values
• Red t1
• Yellow t10
• Green t25
• Cyan t100

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Side by Side Comparison
Similar, but one difference being that for Braun
Fitt data, 1,000s of points were used where as I
used 200
Graph on left obtained from data sent by Braun
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Problems/Difficulties

• One problem that came up was how to use a
  centered difference formula for the 3rd and 4th
  order derivative approximations. The solution
  that we used was based on the boundary condition
  that hxx was fixed on the end points, allowing us
  to create values for phantom points hN2 and
  h0. With these values, we then used a centered
  difference formula for all orders of the
  derivatives.

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Next steps

• Turn on and off different parameters to compare
  the effects
• Read Ajaev and Homsy, Journal of Colloid and
  Interface Science and Ajaev, Journal of Fluid
  Mechanics as suggested by Dr. Anderson to
  understand better the new parameters that we
  introduced
• Look into solving the porous surface model

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Thank you!

• Dr. Anderson
• URCM
• NSF