Fitting ODE models of tear lm breakup Tobin A. Driscoll1 Richard J. Braun1 Rayanne A. Luke2 Dominick Sinopoli1 Aashish Phatak1 Julianna Dorsch1 Carolyn G. Begley3 and Deborah Awisi-Gyau4

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Fitting ODE models of tear ﬁlm breakup

Tobin A. Driscoll1, Richard J. Braun1, Rayanne A. Luke2, Dominick Sinopoli1, Aashish

Phatak1, Julianna Dorsch1, Carolyn G. Begley3, and Deborah Awisi-Gyau4

1Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

2Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD 21218

USA

3School of Optometry, Indiana University, Bloomington, IN 47405, USA

4Alcon Research LLC, 6201 South Freeway, Fort Worth, TX 76134, USA

Abstract

Purpose. Several elements are developed to quantitatively determine the contribution of

diﬀerent physical and chemical eﬀects to tear breakup (TBU) in subjects with no self-reported

history of dry eye or other ocular surface disease. Fluorescence (FL) imaging is employed to

visualize the tear ﬁlm and to determine tear ﬁlm (TF) thinning and potential TBU.

Methods. An automated system using a convolutional neural network is deployed that was

trained and tested on more than 50,000 images from FL imaging experiments. The trained

system could identify multiple TBU instances in each trial. Once identiﬁed, extracted FL

intensity data was ﬁt by mathematical models that included tangential ﬂow along the eye,

evaporation, osmosis and FL intensity of emission from the tear ﬁlm. The mathematical models

consisted of systems of ordinary diﬀerential equations for the aqueous layer thickness, osmolarity,

and the FL concentration; they are a local approximation to TF thinning and/or TBU dynamics.

FL intensity was computed using the resulting thickness and FL concentration. Optimizing the

ﬁt of the models to the FL intensity data determined the mechanism(s) driving each instance

of TBU and produced an estimate of the osmolarity within TBU.

Results. Initial estimates for FL concentration and initial TF thickness agree well with prior

results. Fits were produced for N= 467 instances of potential TBU from 15 non-DED subjects.

The results showed a distribution of causes of TBU in these healthy subjects, as reﬂected by

estimated ﬂow and evaporation rates, which appear to agree well with previously published

data. Final osmolarity depended strongly on the TBU mechanism, generally increasing with

evaporation rate but complicated by the dependence on ﬂow.

Conclusion. The method has the potential to classify TBU instances based on the mechanism

and dynamics and to estimate the ﬁnal osmolarity at the TBU locus. The results suggest that

it might be possible to classify individual subjects and provide a baseline for comparison and

potential classiﬁcation of dry eye disease subjects.

1 Introduction

In this paper, we generate quantitative estimates of important parameters for the tear ﬁlm on the

surface of the eye in healthy subjects. We do this with what we believe, at the time of writing, to

arXiv:2210.03593v2 [math.NA] 21 Feb 2023

Fitting ODE models of TBU 2

be unprecedented precision and quantity. The dataset creates a preliminary baseline for a small

population of subjects without dry eye disease (DED). The importance of this baseline is that it may

be used to contrast what is found for a population with DED, thus leading to better understanding

of the mechanisms at work in this disease that aﬀects millions of people [86,87,95,96]. Though this

work does not give a complete baseline for non-DED eyes, or a contrast with data for DED eyes,

we develop the method in detail and explain how it can reveal the mechanisms behind individual

instances of thinning and tear breakup (TBU) in the tear ﬁlm (TF).

The introduction is structured as follows. Firstly, we give some background on the tear ﬁlm,

ocular surface and DED. Secondly, we brieﬂy discuss some related methods for imaging the tear

ﬁlm. Thirdly, we discuss methods to extract data about tear ﬁlm dynamics. Finally, we discuss

mathematical models for tear ﬁlm dynamics, and best ﬁts of those models to data extracted from

the tear ﬁlm.

Tear Film The TF plays an important role in vision and ocular surface health [75]. The TF

is established during a blink, and lubricates the cornea and the conjunctival surfaces lining the

gap between the lids and the globe [83]. The air/tear ﬁlm interface causes the tear ﬁlm to have

the most powerful refractive surface in the eye; thus, keeping that surface smooth and regular is

essential to clear vision [99]. When the TF fails to uniformly coat the ocular surface, it is said that

tear breakup has occurred [24,78]. TBU may cause the ocular surface to be exposed to cooling [8,

36,66] and evaporation [35,76], and evaporation may lead to tear hyperosmolarity [15,27,53,58]

and mechanical stimulus to the surface [3]. The exposure of the ocular surface to hyperosmolarity

from TBU is thought to play a central role in the etiology of DED [27,53] which aﬀects millions

of people [95]. As a result of this signiﬁcance, TBU dynamics have been studied for more than 50

years using a variety of methods [78,107]. Clinically, the instability of the tear ﬁlm is measured

by the technique of tear breakup time (TBUT), in which the time to the ﬁrst break or irregularity

of the tear ﬁlm is measured.

Imaging methods The imaging methods for TBU dynamics are numerous. Here we list a few of

them: visualization with dyes such as ﬂuorescein (FL) [23,78]; reﬂection of a pattern using a grid

[69] or placido disc images [59]; interferometry and spectrometry [29,33,40,51,88]; simultaneous

imaging with ﬂuorescence (FL) imaging and retroillumination [15]; and simultaneous FL imaging

with interferometry [50].

These and other approaches have quantiﬁed various aspects of TF parameters such as thick-

nesses, thinning rates, TBUTs and more. In this work, we focus on ﬂuorescence imaging as an

experimental method to collect data on aqueous layer (AL) dynamics. This method is chosen due

to the relatively low cost, ease of use and widespread use in the clinic. Clinically, short TBUTs

indicate an unstable TF and the possible presence of DED [107]. Despite the utility of the method,

repeatability from one clinician or researcher to the next and one clinic to the next can be a chal-

lenge [79], though some maintain that TBUT measurements can be generally repeatable under

some circumstances [23]. In this work, we aim to use automated detection of FL imaging to (i)

repeatably extract FL imaging data of TF thinning and TBU, and subsequently to (ii) optimize

the ﬁt of mathematical models to that data to identify mechanism and (iii) estimate important

parameters within TBU.

Eﬀorts to automate TBU and DED measurements were recently reviewed by Vyas and Mehta

Fitting ODE models of TBU 3

[103]. Early eﬀorts generally aimed at quantifying TF breakup time measurement and related

quantities[84,98]. Vyas and Mehta [103] surveyed various methods for automating measurements

and diagnoses, including: tear meniscus evaluation using optical coherence tomography [6]; thermal

imaging to attempt to diagnose DED [1]; and ﬂuorescence imaging of the TF for tear breakup time

detection [97] and DED diagnosis [85].

Extraction of data Our method in this paper is adapted from that of Su et al [97]. In their

system, a convolutional neural network (CNN) is implemented that determines a region of interest

where TBU is most likely to occur. Then, the region of interest is followed in time and the ﬁrst

frame where TBU is found determines the TBUT. Their method is trained on TBU and TBUT data

from experienced clinical researchers, and is therefore designed to imitate the clinical determination

of TBUT for the purpose of DED diagnosis. While we retained the CNN design from their work,

we introduced several changes to the approach of Su et al [97]. The method is adapted to identify

multiple regions of TBU in every trial. We extracted a time series of FL thinning data from each

TBU region. We used that FL imaging time series to determine TBUT (if appropriate) as well

as optimal parameters for mathematical models to determine important quantities of interest with

thinning and TBU areas. The optimal parameters allow us to identify the mechanism(s) driving

each instance of TBU.

Mathematical models A variety of mathematical modeling approaches for the TF have been

developed. For overall ﬂows and concentrations of interest in the TF, there have been compart-

ment models, systems of ordinary diﬀerential equations (ODEs), or diﬀerential algebraic equations

(DAEs) that have included the eﬀect of blinks [21,37] and contact lenses [38,49]. TBU and TF

dynamics with contact lenses are beyond the scope of this paper.

A few categories of 1D partial diﬀerential equation (PDE) models in space and time have been

developed; this includes TF drainage for the open eye during the interblink [71,91,108]. Those

models used a Newtonian ﬂuid close to water in viscosity and measured TF values. Boundary

conditions (BCs) at the end of the ﬁlm mimicked the TF and drove ﬂows to redistribute TF.

Eﬀects added to this type of model include Marangoni eﬀects [10], evaporation [13], van der Waals

wetting terms [106] and curvature of the ocular surface [17]. Local models for TF thinning and TBU

include those which have been studied for the following eﬀects: evaporation to air and osmosis from

corneal surface [82] and with ﬂuorescence [16]; Marangoni eﬀects [110]; a non-polar lipid layer (LL)

[19,94]; dewetting of the ocular surface from long-range van der Waals forces [89,90]; dewetting of

the ocular surface with mucin-dependent viscosity [31,32] and membrane-associated mucins [25].

Some models for TBU are discussed in more detail below.

Models for TF formation, which occurs during the opening phase of the blink cycle, have been

studied as well. A seminal work in this area is Wong et al [108], which treated the TF deposition

as a thin ﬁlm coating ﬂow model; this is a cornerstone of later papers although they modiﬁed the

approach. Later models have included the eﬀect of polar lipids via the Marangoni eﬀect [4,46,47,

63]; partial blinks [30,42]; a non-polar LL [19,112]; the curvature of the ocular surface [2] and

non-Newtonian eﬀects [48,67,68].

Models for ﬂow over the (2D) exposed ocular surface have been developed [12,18,54,56,64,

65]. The 2D models capture a number of aspects of the overall ﬂows, osmolarity and ﬂuorescence

imaging. Some 2D models may take into account the eﬀect of blinking via time-dependent ﬂow

Fitting ODE models of TBU 4

BCs with no lid motion [55], or via lid motion with model problems plus simple BCs [18], but there

is much room to develop blinking models.

Local models have been developed for ﬂow in TBU regions. Peng et al.[82] studied TBU

driven by tear evaporation through a LL distribution that was ﬁxed in space. In their model,

evaporation rate depended on the temperature of the ocular surface, as well as the temperature,

relative humidity and wind conditions of the surroundings. They found that evaporation could drive

the AL thickness to very small values and thus TBU. Simple ODE models of TF thinning with

osmosis could develop suﬃciently elevated osmolarity that could stop thinning and TBU [11,15];

however, Peng et al.[82] found that diﬀusion of osmolarity (salt ions) out of the high concentration

region within TBU prevented suﬃcient osmosis to stop TBU [82]. A dynamic LL was introduced

in Stapf et al.[94]. The model consisted of two Newtonian layers: a relatively thick and less viscous

shear layer topped by a relatively thin but more viscous extensional layer through which evaporation

occurred. Stapf et al.[94] found that TBU could occur, but the model could yield longer TBUTs

than would be observed in vivo. This also happened with models that incorporated mucin eﬀects

[25,31].

Braun et al. [16] simpliﬁed TF dynamics to a single layer for the AL with evaporation modeled

as a ﬁxed Gaussian, but they included ﬂuorescein concentration and ﬂuorescence in their models

of TBU. They found that the ﬂuorescence dynamics depended on initial FL concentration, evapo-

ration distribution width (related to TBU size) and ﬁlm thickness in a complicated way, but the

mechanisms at work in various instances were clariﬁed by the model. Subsequently, models were

proposed to include rapid thinning that could be induced by excess lipid acting as a surfactant [62,

110,111]. The models explained many aspects of TBU, but they tend to overestimate the size of

the TBU region [62].

In this work, we use local models for tear break up involving tangential ﬂow, evaporation,

osmosis and ﬂuorescence, but the models have been simpliﬁed to ODEs for the thickness, osmolarity,

ﬂuorescein and ﬂuorescent intensity[60]. We ﬁnd the optimal parameters for these models that make

them as close as possible to FL intensity data extracted from video recordings of in vivo TFs. With

those optimal parameters, we can infer which eﬀects were most important in each TBU instance.

We use a CNN to extract data for many TBU instances in order to get a more complete picture of

TBU for the cohort of healthy subjects studied.

Paper structure This paper is structured as follows. The methods section will describe in some

detail the FL imaging used to generate data; the extraction method we used to obtain the detailed

thinning data; and mathematical methods and models used to ﬁt that data and determine TBU

parameters of interest. In the results section, we present the results of applying these methods.

In the discussion section we explain the context and signiﬁcance of the results. In the conclusion

section, we summarize our ﬁndings and discuss possible future directions.

Fitting ODE models of TBU 5

Table 1: Architecture of the neural network trained to classify 96 ×96 RGB image tiles.

Layer type Number Size Stride, Pad Output size Activation

Convolution 32 5 ×5 1,2 96 ×96 ×32 ReLU

Max pool 2 ×2 2,0 48 ×48 ×32

Convolution 32 5 ×5 1,2 48 ×48 ×32 ReLU

Average pool 3 ×3 2,1 24 ×24 ×32

Convolution 64 5 ×5 1,2 24 ×24 ×64 ReLU

Average pool 3 ×3 2,1 12 ×12 ×64

Convolution 64 5 ×5 1,0 8 ×8×64 ReLU

Average pool 3 ×3 2,1 4 ×4×64

Convolution 64 4 ×4 1,0 1 ×1×128 ReLU

Dropout, p= 0.4

Dense 5 softmax

2 Methods

2.1 Fluorescence imaging

The experimental data was collected at Indiana University and was approved by the Biomedical

Institutional Review Board of Indiana University. The principles of the Declaration of Helsinki

were followed during data collection, and informed consent was obtained from all subjects. Data

collection is described in a previous publication [3] and discussed in several papers [3,60–62,

110], but will be summarized brieﬂy here. Twenty-ﬁve subjects with no self-reported history of

DED, ocular surface or systemic disease, ocular surgery or medications aﬀecting ocular sensation

participated in the study. Subjects were seated behind a slit lamp biomicroscope and 2 µl of 2%

sodium ﬂuorescein solution was instilled in the subject’s eye. Subjects were asked to keep the

tested eye open as long as possible (STARE trial) while the tear ﬁlm was imaged with a cobalt blue

excitation ﬁlter over the illumination system and a Wratten #12 ﬁlter over the observation port.

With this illumination system, the aqueous layer of the TF ﬂuoresced green [20] with dark areas

appearing due to TBU.

A trial is the sequence of images of the subject’s eye following a few quick blinks. The trial

records the ﬂuorescence of the aqueous part of the TF. The trials typically start with an FL

concentration close to 0.2% (discussed more below), which is the so-called critical concentration

where peak ﬂuorescence occurs for thin TFs [105]. The critical FL concentration may also be

expressed as 0.0053 M [60].

2.2 TBU Detection

We implemented a deep CNN [41,43] similar to the one used by Su et al.[97] to classify small

square patches within an image as belonging to eyelids, eyelashes, sclera, TBU, and non-TBU. The

architecture of the CNN is described in Table 1and requires a total of 313,637 parameters for

training.

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FittingODEmodelsoftearlmbreakupTobinA.Driscoll1,RichardJ.Braun1,RayanneA.Luke2,DominickSinopoli1,AashishPhatak1,JuliannaDorsch1,CarolynG.Begley3,andDeborahAwisi-Gyau41DepartmentofMathematicalSciences,UniversityofDelaware,Newark,DE19716,USA2DepartmentofAppliedMathematicsandStatistics,TheJohnsHopkins...

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Fitting ODE models of tear lm breakup Tobin A. Driscoll1 Richard J. Braun1 Rayanne A. Luke2 Dominick Sinopoli1 Aashish Phatak1 Julianna Dorsch1 Carolyn G. Begley3 and Deborah Awisi-Gyau4

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