Models for the Human Tear Film

Tear Film Group

This was the group in Spring 2008.
Back row, l to r: Toby, Rich and Pam.
Front row, l to r: Alfa, Kara and Usha.
Alfa had successfully defended his thesis that day.

Members

Richard Braun, U of Delaware

Tobin Driscoll, U of Delaware

Pamela Cook, U of Delaware

Jiahua Tang, U of Delaware, Graduate summer student 2009, GEMS program

Longfei Li, U of Delaware, Graduate summer student 2010, GEMS program

Collaborators

P. Ewen King-Smith, The Ohio State U

William Henshaw, LLNL

Ranganathan Usha, IIT Madras, India (visited in 07-08 at UD)

Geoffrey McFadden, NIST

Daniel Anderson, George Mason University

Katlyn Winter, now a grad student at George Mason University

Shailesh Naire, U of Keele, UK

Alumni

Kara Maki, U of Delaware, PhD in Applied Math 2009, now an IMA Postdoc

Alfa Heryudono, U of Delaware, PhD in Applied Math 2008 , now at U Mass Dartmouth

Xiaolin Yang, U of Delaware summer student, MS in CIS, now at Duquesne U

Paul Parsons, Undergraduate summer student 2007 (REU support)

Pete Ucciferro, Undergraduate summer student 2007-2009 (REU support, 3 times), Fall 09 independent study

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Multiple Blink Cycles and the Tear Film

We are using lubrication theory to develop nonlinear partial differential equation(s) that govern the free surface of the human tear film during the complete blink cycle. In the simplest case, the surface of the film is assumed to be stress free (SF), as if the tear fluid were pure water; another simplifying limit is that of a very strong insoluble surfactant, where the film surace stretches uniformly (the uniform stretching limit, or USL). In either limit, a single pde governs the shape of the free surface; we have extended the work of Jones et al (Math Med Bio, 2005) by addition additional effects and computing for the whole blink cycle.

We have computed multiple blink cycles for these limiting cases with the simplification of the sinusoidal motion of the moving lid. In this instance, an MOL method based on uniform finite differences in space and BDF methods for the resulting ODEs (via DASPK) was used. While this seems like a radical simplification, it still gets a number of things right. For example, there is a transition between periodic and non-periodic tear film evolution for incomplete blinks; that it, the tear film behaves as if there was a full blink even if the lids d t fully close. Also, after a half-blink, there is a valley in the tear fluid corresponding to where the lid was; we have quantitative film thickness measurements from in vivo interference patterns to use for comparison. The sinusoidal lid motion captures the existence of this valley qualitatively. This work has appeared in JFM (Braun and King-Smith, JFM 586 (2007) 465-490).

We have solved the pdes using a modified spectral method and with realistic lid motion from blinks. The MOL method maps Chebyshev points in space to minimize round off error in the higher derivatives and uses exact flux values from the boundary conditions when evaluating the ODEs at grid points; the ODEs are solved using ode15s in Matlab and the code was developed by Alfa Heryudono. The approximation conserves volume very well, typically below 0.0001 or better relative error over multiple blink cycles; this is a significant improvement over the previous method based on the uniform finite difference grid. Using realistic lid motion, we get better agreement with the in vivo film thickness measurements from the half blink, and modified results for the transition from periodic to non-periodic solutions for the film. This work is accepted for publication in "Single-Equation models for the Tear Film in a Blink Cycle: Realistic Lid Motion," Mathematical Medicine and Biology (Heryudono, Braun, Driscoll, Maki, Cook and King-Smith, Math Med Biol 24, (2007) 347-377). The presentation Alfa gave at the 2007 APS DFD meeting is available.

Heryudono and Driscoll have developed radial basis function methods for the these kinds of problems. They published results for an adaptive rbf method for problems that are second order in space, which has been published (Driscoll and Heryudono, "Adaptive residual subsampling methods for radial basis function interpolation and collocation problems," Comp. Appl. Math. 53 (2007) 927). They have also published the rbf algorithms on www.matlabcentral.com.

Overset Grids in 1D and 2D

Reflex Tearing in 1D

An overset grid method has been developed, for a tear film problem incorporating opening and relaxation while open, as well as more physiological effects, by Kara Maki. The effects include improved flux boundary conditions for tear supply and reflex tearing, gravity and evaporation. Kara gave this talk at the APS DFD meeting in 2007 (and the talk uses this 37 MB movie. This manscript has appeared in "An Overset Grid Method for the Study of Reflex Tearing" in Mathematical Medicine and Biology (K.L. Maki, R.J. Braun, T.A. Driscoll, and P.E. King-Smith, Math Med Biol 25, (2008) 187-214.)

Tear film evolution in 2D

Kara has also developed 2D models of the post-blink tear film using the Overture framework; Bill Henshaw (LLNL) is helping considerably with this effort. In the first case, we used lubrication equations for the tear film and the boundary conditions that specify the film thickness and the pressure at the boundary. This work is has appeared elctronically in Mathematical Medicine and Biology (doi:10.1093/imammb/dqp023).

We have also implemented flux boundary conditions; some preliminary work was presented in this talk at the 2008 APS DFD meeting and elsewhere. (These three movies were used for this talk: thickness difference (27MB), thickness difference (50MB), and thickness difference (28MB).) The image at left above is the flux direction vectors superimposed over magnitude of the flux; darker indicates slower flow. Tear fluid is supplied from superior temporal location of the lacrimal gland, and is extracted at the locations of the puncta which are near the nasal canthus (corner) at the left end of the domain. The flux conditions are independent of time, which is an initial model. (In a real blink, the supply and drainage of tear fluid are time dependent and are closely related to lid motion.) Most of the flow is around the lid margins.

The image at right is the thickness distribution at time 10 when gravity is included in the simulation. Maroon indicates greater than or equal to 3 microns; the dark blue is the minimum thickness. The drooping area of maroon indicates a bulge in the meniscus outward from the lid margin. For the flux boundary conditions we used, the fluid in the upper meniscus can break through the black line if enough time is allowed between blinks. This work with flux BCs specified is accepted for publication in Journal of Fluid Mechanics and will be out in March 2010.

Wetting and evaporation on the cornea in 1D

Dan Anderson and Kat Winter (then a CSUMS undergrad at George Mason) worked with Dr Braun on a model of the tear film that has a wetting cornea and evaporation from the tear film. The model was compared with some in vivo observations from King-Smith at Ohio State, and choosing two parameters to fit the "dry" film thickness and the opening speed of the thin region seemed to give reasonable agreement between this most basic theory and the experiment. This work has appeared electronically in Mathematical Medicine and Biology (doi:10.1093/imammb/dqp019). A talk including some of these results is given in the next section. The talk also includes some extensions from Pete Ucciferro, an REU student in Summer 09 and an independent study student in Fall 09; the work was done as part of a GEMS team that included graduate studente Jiahua Tang, Pam, Chris Raymond and me.

Effect of Corneal Shape

The effect of the shape of the cornea was studied by R Usha of IIT Madras, Jeff McFadden of NIST, Ewen King-Smith of OSU as well as Pam, Toby and me. The substrate is nearly always assumed to be flat, and Berger's 1973 thesis on the tear film does not explicitly justify the assumption. The upshot is that we don't think the substrate has an important effect on tear film dynamics. Along the way, however, we tried out an Ellis model for the tear fluid and it gave some interesting results for shock formation on the prolate spheroidal substrate. I gave a talk that included these results as well as the competition of a conjoining pressure, evaporation and gravity at the U of Minnesota in their Chemical Engineering and Materials Science Department. A paper on this work is submitted.

Capturing Eyelid Motion

Ms. Xiaolin Yang, an MS student in the Department of Computer and Information Sciences, who worked with Dr Braun on capturing the lid motion during a blink automatically from high speed digital movies of blinks. The movies were made in the MEC Lab in the UD Department of Mathematical Sciences with the help of Dr. John Pelesko. She developed a code that could extract least squares polynomial fits from the blink movies using Sobel edge detection with some pre- and post-processing. That mathematical approximation that she generated of a blink has already been incorporated into our efforts to compute the tear film evolution in two dimensions. She worked with Braun in summer 2006 and January 2007. The code was developed in Matlab, and the code with some results will appear on this website in the near future.

Mr. Peter Ucciferro (a Quantitative Biology major) and Mr. Paul Parsons (a Physics major) were supported by an REU supplement for this NSF grant for the summer of 2007. They worked on evaluating and extending Xiaolin's algorithms for capturing and mathematically representing eyelid motion during a blink. Ucciferro and Parsons recorded some observations of blinks and extended the software for this aspect of the project. One aspect of their work was to use roifill to blur regions around the eye so that more blink movies could be used. They presented their results in an excellent joint presentation during the Departmental symposium on summer undergraduate research in August 2007.

Publicity

From the APS DFD 2008 virtual press room, LiveScience did an article on our work that was presented at the meeting. A seemlingly related article appeared in SoftPedia at close to the same time.

Our work was mentioned in a short summary of highlights from the 2007 meeting of the Tear Film and Ocular Surface Society.

Opportunities

Many aspects of this project remain open to investigation; join the team! If you're interested, please contact Dr. Braun.

Last modified 8/24/09 by RJ Braun