Models for the Human Tear Film During Blink Cycles

Project Personnel

Richard J. Braun, U of Delaware

Tobin A. Driscoll, U of Delaware

L. Pamela Cook, U of Delaware

P. Ewen King-Smith, The Ohio State U

Kara Maki, U of Delaware

Alfa Heryudono, U of Delaware

William Henshaw, LLNL

Collaborators

Geoffrey B. McFadden, NIST

Ranganathan Usha, IIT Madras, India (currently visiting UD)

Shailesh Naire, U of Keele, UK

Alumni

Xiaolin Yang, Summer student, MS in CIS, now at Duquesne U

Peter Ucciferro, Summer student, REU supplement

Paul Parsons, Summer student, REU supplement

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 Bio, to appear).

An overset grid method has been developed, incorporating more physical effects, by Kara Maki. The effects include improved flux boundary conditions for tear supply and reflex tearing, gravity and evaporation. The write up of this work is nearing completion. Maki is also developing two dimensional models of the tear film using the Overture framework; Bill Henshaw (LLNL) is helping considerably with this effort.

Heryudono and Driscoll are currently developing 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.

Capturing Eyelid Motion

Ms. Xiaolin Yang, an MS student in the Department of Computer and Information Sciences, who worked with 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.

Opportunities

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

Last modified 8/27/07 by RJ Braun