Models for the Human Tear Film

Tear Film Group

This was the group in Summer 2016.
L to r, back row: Rich, Spencer, Kevin.
L to r, front row: Chris, Jerome, Amy, Mike.
Not pictured: Toby Driscoll, Christiaan Ketelaar, Javed Siddique, Tony Mastroberardino.

U of Delaware Members

Richard Braun, Faculty
Tobin Driscoll, Faculty
Michael Stapf, Graduate student for summer 2012 Unidel program and current PhD student.
Lan Zhong, Graduate student in AY 2014-15, through the present.
Kevin Aiton, Graduate student beginning in summer 2016.
Rayanne Luke, Graduate student for summer 2017.


Carolyn Begley, Indiana University
P. Ewen King-Smith, The Ohio State University
William Henshaw, Rensselaer Polytechnic Institute
Kara Maki, Rochester Institute of Technology
Javed Siddique, Pennsylvania State University at York
Tony Mastroberardino, Pennsylvania State University at Erie
Ranganathan Usha, IIT Madras, India (visited in 07-08 at UD)
Christiaan Ketelaar, Universidad Francisco Marroquin, Guatemala
Geoffrey McFadden, NIST
Daniel Anderson, George Mason University

Recent Progress

Intro videos

Inspired by ARVO's Thesis in Three competition in 2015, our students have produced short videos about their projects for the non-expert.

New! Journal of Modeling in Ophthalmology!

A new journal dedicated to the intersection of ophthalmology and mathematical modeling has published its first issue! The founding editors are Drs. Alon Harris (Ophthalmology) and Giovanna Guidoboni (Mathematics), both of IUPUI; Dr. Braun is on the editorial board.

Blue Rocks Game 2015

Posing with Rocky Bluewinkle, the Blue Rocks mascot.

More Articles on the tear film

Besides the Progress in Retinal and Eye Research paper (see below), some more papers have appeared this year.

  • "Tear Film Dynamics with Evaporation, Osmolarity and Surfactant Transport," J. Siddique and R.J. Braun, Appl. Math. Modeling, 39:255-269, 2015; doi.

  • The Effects of Increasing Ocular Surface Stimulation on Blinking and Tear Secretion," Ziwei Wu, C.G. Begley, N. Port, A. Bradley, R.J. Braun and P.E. King-Smith, Invest. Ophthalmol. Vis. Sci, 56:4211-4220, 2015; doi.

  • Computed Tear Film and Osmolarity Dynamics on an Eye-Shaped Domain," Longfei Li, R.J. Braun, T.A. Driscoll, W.D. Henshaw, J.W. Banks and P.E. King-Smith, Math. Med. Biol., 2015; doi.

Ongoing collaborations

Our group has been collaborating with Carolyn Begley's group at the Indiana University School of Optometry. They use several methods to image the tear film and are willing to share wonderful data with us too. The review article below is our biggest joint effort to date. This collaboration is funded by Carolyn's NIH grant from the National Eye Institute.

Review article on the tear film -- ocular surface oriented

A review article on tear films that includes many new results has appeared in Progress in Retinal and Eye Research, an invitation only journal, in early 2015. "Dynamics and function of the tear film in relation to the blink cycle" is currently available online at at the journal's site.

Some articles on tear break up and imaging

  • "Tear film dynamics with evaporation, wetting, and time-dependent flux boundary condition on an eye-shaped domain," Longfei Li, RJ Braun, KL Maki, WD Henshaw, PE King-Smith, Physics of Fluids 26, 052101 (2014); doi
  • "A Model for Tear Film Thinning With Osmolarity and Fluorescein," RJ Braun, NR Gewecke, CG Begley, PE King-Smith and JI Siddique, Invest. Ophthalmol. Vis. Sci. 55, 1133-1142, 201; doi
  • "Evaporation-driven instability of the precorneal tear film," C-C Peng, C Cerretani, RJ Braun and CJ Radke, Advances in Colloid and Interface Science 206, 250-264, 2014; doi
  • "Tear Film Breakup and Structure Studied by Simultaneous Video Recording of Fluorescence and Tear Film Lipid Layer Images," PE King-Smith, KS Reuter, RJ Braun, JJ Nichols and KK Nichols, Invest. Ophthalmol. Vis. Sci. 54, 4900-09, 2013; doi
  • "Tear Film Images and Breakup Analyzed Using Fluorescent Quenching," PE King-Smith, P. Ramamoorthy, RJ Braun and JJ Nichols, Invest. Ophthalmol. Vis. Sci. 54, 6003-11, 2013; doi

Some articles on blinking and cooling

The following articles involving blinking appeared recently.

  • "Heat transfer and tear film dynamics over multiple blink cycles," Quan Deng, RJ Braun and TA Driscoll, Phys. Fluids 26, 071901 (2014); doi
  • "A model for the tear film and ocular surface temperature for partial blinks," Quan Deng, RJ Braun, TA Driscoll and PE King-Smith, Interfacial Phenomena and Heat Transfer 1, 357-381, 2013; doi
  • "A model for the human tear film with heating from within the eye," Longfei Li and RJ Braun, Phys. Fluids 24, 062103 (2012); doi

Progress from the Second NSF grant

Review article on the Tear Film -- fluid mechanics oriented

A review article on tear films in the Annual Review of Fluid Mechanics appeared in print in January 2012. "Dynamics of the Tear Film" is currently available online in volume 44 of the journal.

A recent talk on our work

At a BIRS workshop in December 2012, this talk summarized some of our recent work.

Elements of Thermal Models for the Tear Film

Longfei Li published a paper on mathematical modeling for the tear film that captures observed cooling of the ocular surface during the interblink period. He used models with a thick, thin or no substrate. The thick substrate is needed to get cooling of the ocular surface. Longfei was able to identify optimal thermal properties to match the observed cooling in eyes. The paper on the subject appeared in Physics of Fluids (A model for the human tear film with heating from within the eye, Phys. Fluids 24, 062103 (2012);

Processing of Lipid Microscope Images

Kaijing Wang(Math major, NSF REU supported) and Christian Paul (Civil engrg major) worked with Drs Braun and Driscoll to automate classification of images from the lipid microscope at the Ohio State University's College of Optometry. Three optometrists (Drs King-Smith, Nichols and Nichols) and their colleagues have made thousands of images of the lipid layer on the front of the eye. Kaijing and Christian worked on using clustering methods on a relatively small set of images used in a submitted paper from the Ohio State group. The resutls are very encouraging and work will continue on the project. This project was supported in part by an REU supplement to our current grant (1022706) and by the HHMI grant here at UD.

Models for Tear Film and Ocular Surface Interaction

Jen Bruhns (QBio major, HHMI supported) and Doug Freeman (Math major, NSF REU supported) worked with Dr Braun on models the connect the tear film and the ocular surface through osmosis within the cornea. The model connected the tear film with the corneal epithelium that was originally developed for mouse eyes. Work is ongoing after a great start during the summer. This project was supported in part by an REU supplement to our current grant (1022706).

Progress from the First NSF grant

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

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 dynamics 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 will appear in the Journal of Engineering Mathematics.

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.


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.


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


Nick Gewecke, UD postdoc, 2011-14, now in a tenure track position at Dalton State in Georgia.
Graduate students
Amy Janett, U of Delaware, MS in Applied Math in Summer 2016. Amy is starting at Mitre Corporation in Fall 2016.
Longfei Li, PhD in Applied Math 2014. Longfei accepted the Margaret A. Darrin postdoctoral fellowship at RPI with Bill Henshaw.
Quan Deng, PhD in Applied Math 2013. Quan joined Amazon in a research position.
Kara Maki , U of Delaware, PhD in Applied Math 2009. After an IMA Postdoc, Kara joined the School of Mathematical Sciences at Rochester Institute of Technology
Alfa Heryudono, U of Delaware, PhD in Applied Math 2008, now on the faculty in the Department of Mathematics at U Mass Dartmouth.
Ryan Evans, U of Delaware graduate student, summer 2013 and AY 2013-14.
Kevin Buckman, U of Delaware graduate student 2013 (Unidel program).
Tianyu Qiu, Graduate student for summer 2012 GEMS program.
Jiahua Tang, U of Delaware graduate student 2010 (GEMS support)
Xiaolin Yang, U of Delaware volunteer summer student 2007, MS in CIS at UD, then to Duquesne U.
Undergraduate students
Pete Ucciferro, Undergraduate summer student 2007-2009 (NSF REU support, 3 times), Fall 2009 independent study; went on to Philadelphia College of Osteopathic Medicine
Paul Parsons, Undergraduate summer student 2007 (NSF REU support)
Stacey Watro, Undergraduate summer student 2010 (NSF REU support)
Douglas Freeman, Undergraduate summer student 2011.
Wai-Kit Ricky Shum, Undergraduate summer student 2011
R. Christian Paul, Undergraduate summer student 2011.
Kaijing Kevin Wang, Undergraduate student summer and fall 2011 (NSF REU support), currently at Ernst and Young.
Jennifer Bruhns, Undergraduate HHMI scholar (Summer 2011, 2012), Independent study 12S and UG thesis 12-13; starting New England College of Optometry in 2013. Jen won the departmental undergraduate research award in Spring 2013.
Vikramjit Singh, Undergraduate summer scholar 2012 and UG thesis 12-13.
Dylan Chapp, Undergraduate summer scholar 2012 and independent study 2012-13. BS in Mathematics 2014. Dylan went on to an RA in the UD CIS department; he won the departmental undergraduate research award in Spring 2014.
Matthew Moye, Summer 2013 NSF REU student and AY 2013-14. Matt went on to graduate study in mathematics at NJIT.
Justin Grier, Summer 2013 NSF REU student. Justin went on to attend Nova Southeastern College of Optometry in Fall 2015.
Laura Cahill, Summer 2015 NSF REU student. Laura went on to study in public health.
Joe Brosch, Undergraduate student, Spring and Summer 2013, through 2016. Joe won two awards for poster presentations at Joint Math Meetings, won the departmental undergraduate research award in Spring 2016, and started a job at Arotech in Michigan in Fall 2016.
Spencer Walker, Undergraduate student, Summer 2015, 2016 (NSF REU).
Jerome Troy, Undergraduate student, Summer 2016 (UDMS).
Chris Cornwell, Undergraduate student, Summer 2016 NSF REU student.

Group Photos

This was the group in Summer 2015.
L to r, front row: Rich, Mike, Spencer, Amy.
L to r, back row: Christiaan, Laura, Joe, Toby, Lan.
Not pictured: Javed Siddique, Tony Mastroberardino.

This was the group in Summer 2014.
L to r: Rich, Mike, Joe, Longfei, Toby.
Not pictured: Javed Siddique, Ryan Evans.

This was the group in Summer 2013.
Back row, l to r: Quan, Joe, Longfei, Dylan, Kevin, Ryan.
Front row, l to r: Toby, Nick, Rich, Matt, Justin.
Not pictured: Christian.
We celebrated a productive summer at lunch that day.

This was the group in Summer 2012.
Back row, l to r: Nick, Jen, Vikram, Longfei, Quan and Tianyu.
Front row, l to r: Rich, Toby, Javed, Matt, Dylan and Michael.
We had a mid-summer review that day.

This was the group in Summer 2011.
Back row, l to r: Longfei, Jen, Ricky and Quan.
Front row, l to r: Javed, Rich, Toby, Doug and Kaijing.
Not pictured: Christian.
We celebrated a productive summer at lunch that day.

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.


This material is based upon work supported by the National Science Foundation under Grant Nos. 1412085, 1022706 and 0616483, and by the National Insitutes of Health (NEI) under Grant No. 1R01EY021794. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or the NIH.

The group is also grateful for support from the Howard Hughes Medical Institute for undergraduates, and from the University of Delaware via its Office of Undergraduate Research and Experiential Learning as well as the Department of Mathematical Sciences.


View the June 2019 version from Romanian translation courtesy of EWS Translate.

View the January 2015 version in Russian courtesy of Oleg Segal at Eskipaper.

View the August 2015 version in Slovak courtesy of Juraj Rehorovsky at Coupofy.

View the August 2015 version in Belarussian courtesy of Natalie Harmann.

View the December 2015 version in French courtesy of Patricia Motosan.

View the April 2016 version in Latvian courtesy of Evelina Koprziwova.

View the May 2017 version in Swedish courtesy of Catherine Desroches.

View the May 2018 version in Hindi courtesy of Nathan Zed.

View the October 2018 version in Dutch courtesy of Anna Romanova.

Last modified 06/07/2017 by RJ Braun