Parameter Estimation for Mixed-Mechanism Tear Film Thinning.

Etiologies of tear breakup include evaporation-driven, divergent flow-driven, and a combination of these two. A mathematical model incorporating evaporation and lipid-driven tangential flow is fit to fluorescence imaging data. The lipid-driven motion is hypothesized to be caused by localized excess lipid, or "globs." Tear breakup quantities such as evaporation rates and tangential flow rates cannot currently be directly measured during breakup. We determine such variables by fitting mathematical models for tear breakup and the computed fluorescent intensity to experimental intensity data gathered in vivo. Parameter estimation is conducted via least squares minimization of the difference between experimental data and computed answers using either the trust-region-reflective or Levenberg-Marquardt algorithm. Best-fit determination of tear breakup parameters supports the notion that evaporation and divergent tangential flow can cooperate to drive breakup. The resulting tear breakup is typically faster than purely evaporative cases. Many instances of tear breakup may have similar causes, which suggests that interpretation of experimental results may benefit from considering multiple mechanisms.

Parameter Estimation for Evaporation-Driven Tear Film Thinning.

Many parameters affect tear film thickness and fluorescent intensity distributions over time; exact values or ranges for some are not well known. We conduct parameter estimation by fitting to fluorescent intensity data recorded from normal subjects' tear films. The fitting is done with thin film fluid dynamics models that are nonlinear partial differential equation models for the thickness, osmolarity and fluorescein concentration of the tear film for circular (spot) or linear (streak) tear film breakup. The corresponding fluorescent intensity is computed from the tear film thickness and fluorescein concentration. The least squares error between computed and experimental fluorescent intensity determines the parameters. The results vary across subjects and trials. The optimal values for variables that cannot be measured in vivo within tear film breakup often fall within accepted experimental ranges for related tear film dynamics; however, some instances suggest that a wider range of parameter values may be acceptable.

High Resolution Images of Human Meibum Spread on Saline.

Purpose: Understanding of the role of the tear film lipid layer (TFLL) in evaporative dry eye requires knowledge of its structure. X-ray studies show 11.1-nm thick lamellae in meibum at tear film temperature (approximately 35°C), whereas below 30°C, 4.88-nm thick lamellae predominate. Here, high resolution microscopy of meibum spread on saline is studied as a function of temperature, to compare with x-ray results. Methods: A purpose-built high resolution color microscope, previously used to study the TFLL, was used to study meibum from 10 subjects. It was spread on buffered saline at near 40°C, and allowed to cool to room temperature. Analytical methods from previous studies were applied to measure meibum and lamellar thickness. Results: Initially, an irregular "island" was formed, surrounded by a "background layer" of 7.8 ± 0.3 nm thickness. Dewetting of the meibum layer always occurred, leading to the formation of lens-shaped droplets. Below 30°C, the lenses start to emit "tails" having a multilamellar structure containing up to about 49 lamellae superimposed on the background layer, each lamella being 4.82 ± 0.13 nm thick. Conclusions: Below 30°C, meibum spread on saline shows a multilamellar structure like the 4.88 nm thickness in x-ray studies, demonstrating the ability to observe and measure tightly stacked lamellae. In contrast, above 30°C, the 11.1 nm lamellae were not observed as in x-ray studies, indicating that these lamellae were not tightly stacked but may be separated by disordered lipid. The role of these findings in evaporative dry eye is discussed.

A Perspective on the Use of Fluorescent Imaging to Reveal Mechanisms of Breakup.

Purpose: Tear film instability, which can lead to rapid tear film breakup (TBU), is considered to be a major etiological factor in dry eye. However, experimental support for many of the proposed theories for TBU mechanisms is relatively scarce. The major aim of this perspective is to show that fluorescence studies of TBU can be used to provide experimental evidence for two proposed underlying mechanisms of TBU, evaporation and divergent flow.Methods: To understand the effects of TBU on tear film fluorescence, we show that local fluorescence is the product of three main factors: tear film thickness, fluorescein concentration and fluorescent efficiency. In divergent tear flow, tear film thickness is reduced without change in fluorescein concentration and fluorescent efficiency, thus leading to reduced fluorescence intensity. Evaporation causes decreased fluorescence mainly by self-quenching due to high fluorescein concentration. Fluorescent efficiency is reduced by quenching at high fluorescein concentration but is independent of concentration for very low fluorescein concentration; thus, comparison of high and very low concentrations of fluorescein can be used to discriminate between divergent flow and evaporation. Finally, it is shown how the fluorescent pattern can change greatly during the development of breakup.Conclusions: This analysis demonstrates that the study of tear film fluorescence in TBU and dry eye may be underutilized as additional information pertinent to clinical practice may be obtained. A better understanding of TBU mechanisms may lead to improved diagnosis and treatment of dry eye.

High resolution microscopy of the lipid layer of the tear film.

Tear film evaporation is controlled by the lipid layer and is an important factor in dry eye conditions. Because the barrier to evaporation depends on the structure of the lipid layer, a high resolution microscope has been constructed to study the lipid layer in dry and in normal eyes. The microscope incorporates the following features. First, a long working distance microscope objective is used with a high numerical aperture and resolution. Second, because such a high resolution objective has limited depth of focus, 2000 images are recorded with a video camera over a 20-sec period, with the expectation that some images will be in focus. Third, illumination is from a stroboscopic light source having a brief flash duration, to avoid blurring from movement of the lipid layer. Fourth, the image is in focus when the edge of the image is sharp - this feature is used to select images in good focus. Fifth, an aid is included to help align the cornea at normal incidence to the axis of the objective so that the whole lipid image can be in focus. High resolution microscopy has the potential to elucidate several characteristics of the normal and abnormal lipid layer, including different objects and backgrounds, changes in the blink cycle, stability and fluidity, dewetting, gel-like properties and possible relation to lipid domains. It is expected that high resolution microscopy of the lipid layer will provide information about the mechanisms of dry eye disorders. Illustrative results are presented, derived from over 10,000 images from 375 subjects.

Dynamics and mechanisms for tear breakup (TBU) on the ocular surface.

The human tear film is rapidly established after each blink, and is essential for clear vision and eye health. This paper reviews mathematical models and theories for the human tear film on the ocular surface, with an emphasis on localized flows where the tear film may fail. The models attempt to identify the important physical processes, and their parameters, governing the tear film in health and disease.

The influence of a lipid reservoir on the tear film formation.

We present a mathematical model to study the influence of a lipid reservoir, seen experimentally, at the lid margin on the formation and relaxation of the tear film during a partial blink. Applying the lubrication limit, we derive two coupled non-linear partial differential equations characterizing the evolution of the aqueous tear fluid and the covering insoluble lipid concentration. Departing from prior works, we explore a new set of boundary conditions (BCs) enforcing hypothesized lipid concentration dynamics at the lid margins. Using both numerical and analytical approaches, we find that the lipid-focused BCs strongly impact tear film formation and thinning rates. Specifically, during the upstroke of the eyelid, we find specifying the lipid concentration at the lid margin accelerates thinning. Parameter regimes that cause tear film formation success or failure are identified. More importantly, this work expands our understanding of the consequences of lipid dynamics near the lid margins for tear film formation.

Mechanisms, imaging and structure of tear film breakup.

Tear film breakup (BU) is an important aspect of dry eye disease, as a cause of ocular aberrations, irritation and ocular surface inflammation and disorder. Additionally, measurement of breakup time (BUT) is a common clinical test for dry eye. The current definition of BUT is subjective; here, a more objective concept of "touchdown" - the moment when the lipid layer touches down on the corneal surface - is proposed as an aid to understanding processes in early and late stages of BU development. Models of BU have generally been based on the assumption that a single mechanism is involved. In this review, it is emphasized that BU does not have a single explanation but it is the end result of multiple processes. A three-way classification of BU is proposed - "immediate," "lid-associated," and "evaporative." Five different types of imaging systems are described, which have been used to help elucidate the processes involved in BU and BUT; a new method, "high resolution chromaticity images," is presented. Three directions of tear flow - evaporation, osmotic flow out of the ocular surface, and "tangential flow" along the ocular surface - determine tear film thinning between blinks, leading to BU. Ten factors involved in BU and BUT, both before and after touchdown, are discussed. Future directions of research on BU are proposed.

On tear film breakup (TBU): dynamics and imaging.

We report the results of some recent experiments to visualize tear film dynamics. We then study a mathematical model for tear film thinning and tear film breakup (TBU), a term from the ocular surface literature. The thinning is driven by an imposed tear film thinning rate which is input from in vivo measurements. Solutes representing osmolarity and fluorescein are included in the model. Osmolarity causes osmosis from the model ocular surface, and the fluorescein is used to compute the intensity corresponding closely to in vivo observations. The imposed thinning can be either one-dimensional or axisymmetric, leading to streaks or spots of TBU, respectively. For a spatially-uniform (flat) film, osmosis would cease thinning and balance mass lost due to evaporation; for these space-dependent evaporation profiles TBU does occur because osmolarity diffuses out of the TBU into the surrounding tear film, in agreement with previous results. The intensity pattern predicted based on the fluorescein concentration is compared with the computed thickness profiles; this comparison is important for interpreting in vivo observations. The non-dimensionalization introduced leads to insight about the relative importance of the competing processes; it leads to a classification of large vs small TBU regions in which different physical effects are dominant. Many regions of TBU may be considered small, revealing that the flow inside the film has an appreciable influence on fluorescence imaging of the tear film.

Applications of small-world network theory in alcohol epidemiology.

OBJECTIVE: This study develops a mathematical model of alcohol abuse in structured populations, such as communities and college campuses. The study employs a network model that has the capacity to incorporate a variety of forms of connectivity membership besides personal acquaintance, such as geographic proximity and common organizations. The model also incorporates a resilience dimension that indicates the susceptibility of each individual in a network to alcohol abuse. The model has the capacity to simulate the effect of moving alcohol abusers into networks of nonabusers, either as the result of treatment or membership in self-help organizations. METHOD: The study employs a small-world model. A cubic equation for each person (vertex on a graph) governs the evolution of an individual's state between 0 and 1 with regard to alcohol dependence, with 1 indicating absolute certainty of alcohol dependence. The simulations are dependent on initial conditions, the structure of the network, and the resilience distribution of the network. The simulations incorporate multiple realizations of social networks, showing the effect of different network structures. RESULTS: The model suggests that the prevalence of alcohol abuse can be minimized by treating a relatively small percentage of the study population. In the small populations that we studied, the critical point was 10% or less of the study population, but we emphasize that this is within the limitations and assumptions of this model. CONCLUSIONS: The use of a simple model that incorporates the influence of the social network neighbors in structured populations shows promise for helping to inform treatment and prevention policy.

Computed tear film and osmolarity dynamics on an eye-shaped domain.

The concentration of ions, or osmolarity, in the tear film is a key variable in understanding dry eye symptoms and disease. In this manuscript, we derive a mathematical model that couples osmolarity (treated as a single solute) and fluid dynamics within the tear film on a 2D eye-shaped domain. The model includes the physical effects of evaporation, surface tension, viscosity, ocular surface wettability, osmolarity, osmosis and tear fluid supply and drainage. The governing system of coupled non-linear partial differential equations is solved using the Overture computational framework, together with a hybrid time-stepping scheme, using a variable step backward differentiation formula and a Runge-Kutta-Chebyshev method that were added to the framework. The results of our numerical simulations provide new insight into the osmolarity distribution over the ocular surface during the interblink.

Hydrodynamics of a bounded vertical film with nonlinear surface properties.

The drainage of a thin liquid film with an insoluble monolayer down a vertical wall is studied. Lubrication theory is used to develop a model where the film is pinned at the top with a given thickness and the film drains into a bath at the bottom. A nonlinear equation of state is used for the surface tension and the surface viscosity is a nonlinear function of the surfactant concentration; these are appropriate for some aqueous systems. The three partial differential equations are solved via discretization in space and then the resulting differential algebraic system is solved. Results are described for a wide range of parameters, and the conditions under which the free surface is immobilized are discussed.

Evaporation-driven instability of the precorneal tear film.

Tear-film instability is widely believed to be a signature of eye health. When an interblink is prolonged, randomly distributed ruptures occur in the tear film. "Black spots" and/or "black streaks" appear in 15 to 40 s for normal individuals. For people who suffer from dry eye, tear-film breakup time (BUT) is typically less than a few seconds. To date, however, there is no satisfactory quantitative explanation for the origin of tear rupture. Recently, it was proposed that tear-film breakup is related to locally high evaporative thinning. A spatial variation in the thickness of the tear-film lipid layer (TFLL) may lead to locally elevated evaporation and subsequent tear-film breakup. We examine the local-evaporation-driven tear-film-rupture hypothesis in a one-dimensional (1-D) model for the evolution of a thin aqueous tear film overriding the cornea subject to locally elevated evaporation at its anterior surface and osmotic water influx at its posterior surface. Evaporation rate depends on mass transfer both through the coating lipid layer and through ambient air. We establish that evaporation-driven tear-film breakup can occur under normal conditions but only for higher aqueous evaporation rates. Predicted roles of environmental conditions, such as wind speed and relative humidity, on tear-film stability agree with clinical observations. More importantly, locally elevated evaporation leads to hyperosmolar spots in the tear film and, hence, vulnerability to epithelial irritation. In addition to evaporation rate, tear-film instability depends on the strength of healing flow from the neighboring region outside the breakup region, which is determined by the surface tension at the tear-film surface and by the repulsive thin-film disjoining pressure. This study provides a physically consistent and quantitative explanation for the formation of black streaks and spots in the human tear film during an interblink.

A model for tear film thinning with osmolarity and fluorescein.

PURPOSE: We developed a mathematical model predicting dynamic changes in fluorescent intensity during tear film thinning in either dilute or quenching regimes and we model concomitant changes in tear film osmolarity. METHODS: We solved a mathematical model for the thickness, osmolarity, fluorescein concentration, and fluorescent intensity as a function of time, assuming a flat and spatially uniform tear film. RESULTS: The tear film thins to a steady-state value that depends on the relative importance of the rates of evaporation and osmotic supply, and the resulting increase of osmolarity and fluorescein concentrations are calculated. Depending on the initial thickness, the rate of osmotic supply and the tear film thinning rate, the osmolarity increase may be modest or it may increase by as much as a factor of eight or more from isosmotic levels. Regarding fluorescent intensity, the quenching regime occurs for initial concentrations at or above the critical fluorescein concentration where efficiency dominates, while lower concentrations show little change in fluorescence with tear film thinning. CONCLUSIONS: Our model underscores the importance of using fluorescein concentrations at or near the critical concentration clinically so that quenching reflects tear film thinning and breakup. In addition, the model predicts that, depending on tear film and osmotic factors, the osmolarity within the corneal compartment of the tear film may increase markedly during tear film thinning, well above levels that cause marked discomfort.

Four characteristics and a model of an effective tear film lipid layer (TFLL).

It is proposed that a normal, effective tear film lipid layer (TFLL) should have the following four characteristics: 1) high evaporation resistance to prevent water loss and consequent hyperosmolarity; 2) respreadability, so it will return to its original state after the compression-expansion cycle of the blink; 3) fluidity sufficient to avoid blocking secretion from meibomian glands; 4) gel-like and incompressible structure that can resist forces that may tend to disrupt it. These characteristics tend to be incompatible; for example, lipids that form good evaporation barriers tend to be disrupted by compression-expansion cycles. It is noted that clues about the function and organization of the TFLL can be obtained by comparison with other biological lipid layers, such as lung surfactant and the lipid evaporation barrier of the skin. In an attempt to satisfy the conflicting characteristics, a "multilamellar sandwich model" of the TFLL is proposed, having features in common with the skin evaporation barrier.

Tear film images and breakup analyzed using fluorescent quenching.

PURPOSE: Tear evaporation should increase fluorescein concentration, causing fluorescence dimming from self-quenching for high but not low fluorescein concentration. This prediction was tested and compared to the predicted effect of "tangential flow" that fluorescence dimming should be similar for high and low concentrations. METHODS: A custom optical system was used for video recording of tear film fluorescence in 30 subjects. The subjects were asked to blink at the start of the recording and try to keep their eyes open for the rest of the 60-second recording. An initial recording was made after instillation of 1 µL 0.1% fluorescein followed by further recordings at 5-minute intervals using 0.5% and 5% fluorescein. RESULTS: Decay of fluorescence was considerably greater for the high (5%) concentration condition than for the low (0.1%) concentration. This is shown by "ratio images" (ratio of the intensity of a fluorescence image at a later time divided by that of an earlier image), fluorescence decay curves, fluorescence decay rates, and histograms of estimated tear thickness decrease. For example, for the high concentration condition, decay rates were higher than for the low concentration for all 30 subjects (P < 0.0001, binomial test). Additionally, breakup time was significantly reduced for the high compared to the low concentration condition. CONCLUSIONS: The greater fluorescence decay and more rapid breakup for the high concentration condition are the results expected if thinning and breakup are mainly due to evaporation, hence causing self-quenching. Fluorescence decay rate for the low concentration condition was not significantly greater than zero.

Tear film breakup and structure studied by simultaneous video recording of fluorescence and tear film lipid layer images.

PURPOSE: The thinning of the precorneal tear film between blinks and tear film breakup can be logically analyzed into contributions from three components: evaporation, flow into the cornea, and tangential flow along the corneal surface. Whereas divergent tangential flow contributes to certain types of breakup, it has been argued that evaporation is the main cause of tear thinning and breakup. Because evaporation is controlled by the tear film lipid layer (TFLL) it should therefore be expected that patterns of breakup should match patterns in the TFLL, and this hypothesis is tested in this study. METHODS: An optical system is described for simultaneous video imaging of fluorescein tear film breakup and the TFLL. Recordings were made from 85 subjects, including both with healthy and dry eyes. After instillation of 5 µL2% fluorescein, subjects were asked to blink 1 second after the start of the recording and try to maintain their eyes open for the recording length of 30 or 60 seconds. RESULTS: Areas of tear film thinning and breakup usually matched corresponding features in the TFLL. Whereas thinning and breakup were often matched to thin lipid, surprisingly, the corresponding lipid region was not always thinner than the surrounding lipid. Occasionally, a thin lipid region caused a corresponding region of greater fluorescence (thicker aqueous layer), due to convergent tangential flow. CONCLUSIONS: Areas of tear thinning and breakup can generally be matched to corresponding regions of the TFLL as would be expected if breakup is largely due to evaporation. Surprisingly, in some examples, the corresponding lipid area was not thinner and possibly thicker than the surrounding lipid. This indicates that the lipid was a poor barrier to evaporation, perhaps because of deficiency in composition and/or structure. For example, bacterial lipases may have broken down esters into component acids and alcohols, causing a defective TFLL structure with increased evaporation.

A model for wetting and evaporation of a post-blink precorneal tear film.

We examine a fluid dynamic model for the evolution of a precorneal tear film that includes evaporation of the aqueous layer and a wetting corneal surface. Our model extends previous work on the break-up time for a post-blink tear film to include a more realistic model for evaporation. The evaporation model includes the effects of conjoining pressure and predicts the existence of an equilibrium adsorbed fluid layer that serves as a model for a wetting corneal surface/mucin layer. The model allows the prediction of dewetting rates that are compared with experimental measurements. By choosing an expected thickness where evaporation and conjoining pressure balance, we obtain qualitative agreement for the opening rate with in vivo observations.

Tear film dynamics on an eye-shaped domain I: pressure boundary conditions.

We study the relaxation of a model for the human tear film after a blink on a stationary eye-shaped domain corresponding to a fully open eye using lubrication theory and explore the effects of viscosity, surface tension, gravity and boundary conditions that specify the pressure. The governing non-linear partial differential equation is solved on an overset grid by a method of lines using a finite-difference discretization in space and an adaptive second-order backward-difference formula solver in time. Our 2D simulations are calculated in the Overture computational framework. The computed flows show sensitivity to both our choices between two different pressure boundary conditions and the presence of gravity; this is particularly true around the boundary. The simulations recover features seen in 1D simulations and capture some experimental observations including hydraulic connectivity around the lid margins.