Tear Film Evolution

1
Tear Film Evolution

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

2
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, is 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. Comparisons of
  the present model using exaggeratedly high
  parameter values are made to the Braun-Fitt
  research. Findings show that including Van der
  Waals forces slows the evaporation process while
  there is very little effects noticed when
  introducing the curvature term.

3
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
4
Basic Problem

• Immediately after a blink, tear film begins to
  evolve due to effects such as gravity, curvature,
  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

5
Derivation(using a 2D thin film model)
Equations from Dr. Andersons notes on tear
film evaporation

• Equation 1 maintains the conservation of mass
• Equation 2 is a form of Newtons second law ma
  F (conservation of momentum)
• Eqns 1 2 are known as the Navier-Stokes
  Equations
• Equation 3 is the heat equation (conservation of
  energy)
• u is the total velocity (u,v), u is the velocity
  in the horizontal direction of the film, and v is
  the velocity in the vertical direction
• Boundary conditions at y0 are u0 (no slip along
  surface) and v0 (no penetration of surface)

6
Set up

• L is the 1/2 length of the eye lid, in this case
  set to 14, a scaled distance used by Braun-Fitt
• 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
7
Equation to Solve

• Boundary conditions fix h at L and hxx at L
• E/(Kh) evaporation term used in Braun-Fitt
• On the right hand side, the G gravity and
  underlined in purple is the surface tension term
  same as in Braun-Fitt
• The new A terms Van der Waals attraction
• ? is a new combined pressure effect on
  evaporation that is an effect of curvature
• We are solving ?h/ ?t using numerical
  approximation of a system of ODEs using Matlabs
  solver ODE23s
• Red underlined part will later be referenced as Q

Equation from Dr. Andersons notes on tear film
evaporation
8
Basic Cases

• Case 1 E 0, ? 0, A 0 (Braun-Fitt basic
  model)
• Case 2 E 14.1, ? 0, A 0 (Braun-Fitt basic
  evaporation)
• Case 3 E 0, ? 0 , A ? 0 (Just Van der
  Waals)
• Case 4 E 14.1, ? ? 0, A 0 (Evaporation and
  curvature)
• Case 5 E 14.1, ? 0, A ? 0 (Evaporation and
  Van der Waals)
• Case 6 E 14.1, ? ? 0, A ? 0 (All variables)

9
Thin Film Model

• Basic Model A 0, ? 0, E 0 (For current
  cases G 0)
• Interested in minimum or thinnest values for all
  time steps

10
Other film Plots
Case 6
N 2,000, A delta 10-3 E 14.1
Case 4
N 2,000 , E 14.1, delta 10-3, and A 0
11
Log-Log Plots

• Use log-log plot to compare minimum values of all
  cases at the same time

12
Q-Value

• Set Q 1 - ? (d2h/dx2 A/h3) to see the effect
  of evaporation
• Only important for 2 cases (4 6) since delta or
  E is zero in all other cases
• For case 6, all parameters turned on, you can
  approximate the film thickness where the
  evaporation will turn off since d2h/dx2 is
  small. ?A/h3 1 --gt (?A)1/3 h

13
Case 4 Q value plots
14
Case 6 Q-Value plots
15
Changing parameters --gt Break up time (BUT)

• Varied just ? and A separately and had general
  trend the lower the value the lower the BUT
• Same for this graph where both A and ? are varied

16
Conclusions

• The curvature effects had minimal impact on the
  evolution of the tear film
• Van der Waals effects had a slowing effect on
  evaporation such that the film took much longer
  to rupture or hypothetically did not rupture at
  all reaching an equilibrium value
• Using realistic values, film still ruptured
  early. No clear explanation for how to relate it
  back to dry eye

17
Thank you!

• Dr. Anderson
• URCM
• NSF