On modeling tear breakup dynamics with a nematic lipid layer

Figure 1 shows a schematic of part of the TF, which lies on the ocular surface. At the anterior interface of the TF with air is an oily LL that retards evaporation [11, 12], and helps to retain a smooth, well-functioning tear film [13]. The LL also contains polar lipids that are surface active and lower the surface tension between the LL and the aqueous layer (AL), which is posterior to the LL [14, 15]. The AL consists mostly of water [16], but it also contains various ions and proteins [11]. At the corneal surface, which is posterior to the AL, there is a region with transmembrane mucins protruding from the epithelial cells of the cornea or conjunctiva. This forest of glycosolated mucins is called the glycocalyx. It is generally agreed that the presence of the healthy hydrophilic glycocalyx prevents the tear film from dewetting the ocular surface [17‐19].

The overall thickness of the TF is a few microns [20], while the average thickness of the LL is on the order of tens to 100 nanometers when the eye is open [10, 13, 21‐23]; the thickness of the glycocalyx is a few tenths of a micron [19, 24]. The overall structure is reformed on the order of a second after each blink in a healthy tear film [10, 14, 25, 26].

TBU occurs when a thinned region forms in the TF. Clinically, this is defined as the first dark area that is observed in the fluorescent TF following instillation of fluorescein dye [27]. Various mechanisms are thought to cause different types of TBU, or to combine in various ways: evaporation of water from the AL [5, 7, 28], Marangoni-driven tangential flow (from non-uniform polar lipid surface concentrations) [29], or dewetting-driven tangential flow (from uneven ocular surface conditions) [30‐32]. Here, we use the phrase tangential flow to mean flow along the eye. The evaporation of water from the AL causes relatively slow thinning [33], while one may expect rapid thinning to be explained by Marangoni-driven tangential flow [29] or dewetting-driven tangential flow. Concentration gradients from surface active polar lipids in the LL induce shear stress at the LL/AL interface and drive outward tangential flow via the Marangoni effect [14, 15]. Dewetting from irregularities in a corneal surface region has been hypothesized to generate outward tangential flow from internal pressure gradients inside the AL due to van der Waals type forces [30, 34, 35]. A related term is full-thickness tear film breakup (full thickness-TBU), which is when the AL has thinned to the point where the LL and glycocalyx touch [28, 36]. It is our belief that the strongest evidence to date regarding mechanisms for TBU are evaporative and Marangoni-driven thinning and their interaction [37, 38]; unambiguous evidence of van der Waals-driven dewetting is not yet in hand in our opinion [28]. We now discuss the studies of these mechanisms in more detail.

The aqueous part of tear fluid is primarily supplied from the lacrimal gland near the temporal canthus (corner) and the excess is drained through the puncta near the nasal canthus [39]. Water lost from the tear film due to evaporation into air is an important process [11, 40, 41]. We believe that this is the primary mechanism by which the osmolarity (essentially, the total concentration of ions on a molar basis) is increased in the tear film [10]. Some water is supplied from the ocular epithelia via osmosis when the osmolarity increases above the isotonic level (the so-called hyperosmolar state) [10, 40, 42‐44]. According to the Dry Eye WorkShop (DEWS) reports, the osmolarity is important because it plays an important etiological role in the development of dry eye disease [5]. An accepted view in the DEWS reports is that repeated exposure to sufficient hyperosmolarity in tears, as may occur in TBU, can result in inflammation and ocular surface damage in a vicious cycle that may lead to DED. There is still considerable ongoing research to more fully understand the causes and etiology of DED; some mathematical modeling results discussed in this paper help to quantify the relationship between tear breakup and hyperosmolarity.

Both TF instability and hyperosmolarity are important to study because they are proposed as etiological causes of dry eye disease [6, 7]. An osmolarity difference between the corneal epithelium and AL induces osmotic flow from the cornea to the TF [10, 45]. TF osmolarity may be measured in the inferior meniscus clinically [46]; the healthy range is 296–302 mOsM [46‐48]. Dry eye measurements in the same location can reach 316–360 mOsM [47, 49‐51] but estimates for the TF over the cornea reach 900 mOsM or higher [10, 45, 52, 53]. High levels of TF osmolarity are associated with pain, inflammation and cellular changes [52, 54, 55]. In support of these potentially high levels of TF osmolarity over the cornea, mathematical models without spatial variation have estimated peak osmolarities up to ten times the isotonic concentration [10, 42]. The modeling work of Peng et al. [45] found that evaporation elevates osmolarity in breakup regions.

TF thinning rates have been measured experimentally or estimated in many studies. A few experimental methods include spectral interferometry [33, 41, 56], an open chamber [57], an evaporimeter [58], and averaging pure water and functioning LL rates over the cornea obtained by heat transfer analysis and thermal imaging [59]. Parameter estimation schemes were developed in Luke et al. [38, 53] for fitting partial differential equation (PDE) models to experimental fluorescent intensity distributions [60]. They found evaporative thinning rates ranging from 36.9 to \(-\,4.91\) \(\upmu \)m/min (negative indicates thickening) and overall TF thinning rates ranging from 23.5 to 1.85 \(\mu \)m/min. These estimates were quite similar to previous experimental spot-thinning rates that did not specifically seek TBU instances [56].

Tiffany and Dart [61] measured values of the viscosity of human meibum (the material expressed from the meibomian glands inside the lids) using a capillary tube and they estimated values from 9.7 to 19.5 Pa s at 308 K. This is a representative value for the temperature on the surface of the tear film [62]. In a series of papers by Tiffany et al., the human tear film has been shown to have non-Newtonian properties [63‐65]. Tiffany [65] found that whole tears, that is, tear sampled directly from the eye, are shear thinning. The shear thinning has been fit with empirical models, namely Cross [65] and Ellis models [66]. Tiffany [63] also found weak elastic effects; Tiffany and Nagyová [67] found that removing all lipids from tears caused tear fluid to become Newtonian. The rheology of the LL depends on temperature [68]; in vitro observations showed temperature dependence of meibum on saline for monolayers [69, 70] and for bulk samples [71]. The elastic contributions to viscous response were small above 305 K for these measurements [69]. King-Smith et al. [23] used high-resolution microscopy to observe LL dynamics, and they saw fine structure of the LL during the interblink period that included islands of lipids.

The surface tension of the tear/air interface was measured to be 46.2 mN/m by Miller [72] in vivo using a special contact lens; this is about 2/3 that of an air/pure water interface. Pandit et al. [64] measured the surface tension of whole tears with a capillary tube and obtained 43 mN/m for unstimulated tears and 46 mN/m for stimulated tears; Miller [72] likely measured stimulated tears. A value of 45 mN/m is often used in theoretical studies. The components responsible for lowering the surface tension from that of pure water has been the subject of a number of papers [67, 73‐75]. A challenge for this measurement is the dependence of surface tension on surface active components like phospholipids and other lipids [76‐78] and still other species [79]. Detailed equations of state for studying the Marangoni effect from these species are not available for the tear film in vivo to our knowledge, and this is an ongoing area of research.

A variety of mathematical models have incorporated various important effects of tear film dynamics as has been recently reviewed [10, 42, 80]. The most common assumptions for these models are a Newtonian tear fluid and a flat cornea [14, 81]. One category of models treats the flow over an open eye-shaped domain; these models have used lubrication theory with two space dimensions and time as independent variables [82‐86]. These papers were on a stationary domain, and emphasized the effects of supply and drainage, osmolarity in various locations in the tear film, and the overall appearance of fluorescein imaging of the tear film. Some initial studies have examined flow on a blinking domain [80, 87, 88], but they have yet to incorporate the same level of effects as in the static boundary papers. There have also been useful compartment models developed for tear film dynamics [43, 89]. These models use compartments of the tear film as dependent variables in systems of ordinary differential equations (ODE) and thus avoid the computational difficulties of solving PDEs on 2D domains. The models can be solved over many blinks and give insight into the osmolarity and other variables at various operating conditions.

There are quite a few models that have been developed using a single variable in space and time for independent variables. There are two main categories here: (1) a spatial domain that extends across the eye opening vertically, with or without a moving end, and (2) a shorter domain that is a local model for TF dynamics. We first discuss category (1). First, a number of papers considered a domain that extended across the open eye with fixed ends which typically called post-blink drainage models [90‐93]. These models typically imposed large menisci at the domain ends that drained fluid from the relatively flat interior of the TF, and that drove the dynamics. Notable among these papers is that evaporation was included to find that it could drive thinning to TBU [93], and that a moving end, representing the upstroke of the blink, was treated as a coating flow problem [91]. Both of those aspects would find many applications in later papers. Heat transfer and evaporation were considered in Li and Braun [94], and later treated together with thermal imaging in Dursch et al. [59]. A counterpoint to these papers Berger and Corrsin’s treatment of the Marangoni effect in post-blink spreading [14], which provided a basis for using this effect in many later papers.

Using a moving end on a one-dimensional domain to simulate the upper eyelid during a blink has been employed in a number of papers. The first to do this was by Wong et al. [91]; they treated the problem as a coating flow analysis and developed a closed form equation for the thickness of the deposited film as a function of end (lid) speed. Later papers extended this analysis by computing the thickness and other TF variables numerically. These included a combined TF formation and drainage model [95], and then adding surfactant transport on the surface of the TF [96]. Additional papers using the eye opening and interblink together included gravity and reflex tearing [97]; using optimization varying viscosity to simulate and design eyedrops [66]; a detailed look at insoluble surfactant [98]; a (Newtonian) two-layer model of the tear film with insoluble surfactant between them [99]; the effect of (constant) globe curvature on tear film thickness [100]; and a shear-thinning (power law) model for TF thickness on a curved substrate [101]. Some papers also treated the moving end for complete blink cycles. Simple sinusoidal motion was treated in Braun and King-Smith [102], while more realistic motion was treated in Heryudono et al. [103], and insoluble surfactant was added in Zubkov et al. [104]. Heat transfer was later considered with either full [105] or partial [106] blink cycles.

We next discuss category (2) which are local one-dimensional models for TF dynamics. These models are along a short line segment and are designed to be local models of TF flow. The effects of evaporation and the Marangoni effect on TBU have been studied in a number of papers. Evaporation drove all the dynamics in the models in [10, 45, 60], though the treatments were different. In [45], a stationary LL with variable thickness caused increased localized thinning; also included were osmolarity transport in the AL and osmosis across the AL/cornea interface. An important result from that work is that diffusion of osmolarity out of the region of evaporation prevents osmosis from stopping thinning as it would in spatially uniform models [10, 42]. Simpler evaporation models were used in [10, 60], but they included transport of fluorescein so that they could explain what was seen in TF visualization experiments. Zhong et al. [29] developed a PDE model with one spatial variable that incorporated both mechanisms. Including fluorescein dye transport and fluorescence enabled fitting of models to in vivo fluorescence data within TBU regions to estimate parameters that are not possible to directly measure there at the time of writing [38, 53]. Recently, ODE models with no space dependence have been successfully fit to fluorescence data in small TBU spots and streaks [107]. Those models have been coupled to a neural network-based data extraction system to greatly expand the amount of TBU instances that may be studied [108].

Other effects have been included in local models of TBU [29‐31, 45, 60, 109‐113]. Some models incorporated the effect of soluble mucins in the AL necessitating departure from more standard lubrication models [111, 112]. Localized non-wettability of the ocular surface was modeled in [113]. Other models have provided fundamental analyses on evaporation [114, 115] and instability [116, 117].

The composition and structure of the LL, and their relation to function, are the subject of much ongoing research [76, 118, 119]. The complexity of the composition affects the determination of structure and function and has led to a variety of perspectives in this area [79, 120]. It is generally thought that the LL has an outermost nonpolar LL with a layer of polar lipids and other surface active components at the LL/AL interface [1, 121]. Measuring all but the most basic properties can be a challenge, however. Microscopy has shown some example forms that the LL may take [23]; the shapes observed there suggest non-Newtonian behavior of the LL [70, 71]. For a number of years, it was very difficult to reproduce significant resistance to evaporation using in vitro meibum or LL films [122, 123], even though resistance to evaporation has been thought to be an important property in vivo [11, 33]. However, recent progress on in vitro films with many components has been made [12, 124]. The barrier function does depend on thickness [33, 125], but clearly, there is an effect from internal structure as well [126, 127].

Molecular dynamics simulations, together with Langmuir trough experiments, have also given insight into the structure and function of the LL or model systems for it. Coarse grained models were initially employed [128, 129] and those models suggest that the LL may not be so as well organized as layers of nonpolar lipids atop a polar lipid monolayer. Later, all-atom models have suggested that the specific polar lipids [130, 131] and wax esters [132] may form ordered states to slow evaporation of water from the AL. Molecular dynamics simulations are likely to continue to give important insights into LL structure and function, but they currently can only be solved at small space and time scales. A link to bigger scales has been made via Langmuir trough observations with Brewster angle microscopy, e.g., [130, 131] for equilibrium states. Those insights need to be converted to larger scale dynamic models to address other situations such as TBU. In this paper, we develop a two-layer continuum model that attempts to incorporate some internal structure of the LL that may be expected to produce evaporation resistance [28, 37] that depends on that structure. We use a nematic liquid crystal structure as a model for the LL, which can show birefringence as has been observed for meibum [133].

Adapting two-layer models of tear dynamics [45, 99] to dynamic models of local TBU regions [134, 135] has made use of extensional flow for the LL and shear-dominated flow for the AL. These models have a common basis from the approach of Matar et al. [136]. Of particular interest here is including extensional flow of a nematic liquid crystal [137, 138] for the LL with a Newtonian shear-dominated AL layer. That extensional flow modeling used the Ericksen–Leslie equations and multiple scale perturbation methods to reduce the equations to a small PDE system. A detailed derivation and preliminary results for the combined model were developed in Taranchuk and Braun [135]. With this new model, they found dependence of TF dynamics on the director angle \(\theta _B\) with all other parameters fixed, and we continue this work with a detailed parametric study. We find new ways the effects combine to affect TF dynamics and TBU.

The paper is organized as follows. In Sect. 2, we describe the problem formulation. In Sect. 3, we present our results. These include profiles of the film thicknesses, velocities, and pressures that lead to different outcomes depending on the internal orientation of the liquid crystal molecules. Finally, in Sect. 4, we discuss the results and outline our conclusions.

After nondimensionalizing, we asymptotically expand all of the dependent variables in powers of the small parameter \(\epsilon \). We then collect like powers of \(\epsilon \) and solve to find a closed system of leading order equations. In the process, we find that the director angle is constant at leading order, \(\theta =\theta _B\). We also determine the leading order AL pressure \(p_1\), and the AL depth averaged axial velocity \(\bar{u}_1\). This process is described in more detail in the Appendix A and derived thoroughly in Taranchuk and Braun [135]. We find the closed leading order system to bewhereandEquations (2) and (3) represent conservation of mass (of solvent) in the AL and LL, respectively. Equation (4) is the axial force balance in the LL. Equation (5) conserves mass of surfactant, and Eq. (6) conserves osmolarity in the AL.

The evaporation function in Eq. (9) depends on the evaporative resistance in the LL which is described by Eq. (10). By choosing to implement dependence in this way, smaller director angles provide a stronger barrier to evaporation, and larger director angles provide a weaker barrier to evaporation [37]. This is a representative choice based upon in vitro experiments using molecules with long hydrocarbon chains, similar to those found in the LL of the eye. These experiments show that evaporation resistance is increased when the hydrocarbon tails align in an upright configuration [130, 146, 147]; this corresponds to an angle of \(\theta _B=0\) in our model. Further details are given in the Appendix A.3 and in Taranchuk and Braun [135]. We vary the evaporative resistance function \({\mathcal {R}}(\theta _B)\) by changing both \({\mathcal {R}}_0\) and \(\theta _B\). Various director angles are considered throughout the paper, and Sects. 3.1.3 and 3.2.2 focus on the effect of \({\mathcal {R}}_0\) for two sets of initial conditions.

Before solving numerically, we make two change of variables. In order to reduce the order of the system, we define \(p_{1}\) as a dependent variable (Eq. (11)) and substitute into Eq. (4). We also use the substitution \(m=h_{1}c\) in Eq. (6), where m is the mass. This yields the following closed system which we wish to solvewhereWe then use the method of lines to solve the system numerically, subject to the initial and boundary conditions of Sect. 2.2. Spatial derivatives are approximated with second-order finite differences on a uniform grid of 512 points. This results in a system of differential algebraic equations that we solve forward in time in Matlab (MathWorks, Natick, MA, USA) using ode15s. We found that the solutions were converged using this number of grid points, and that the mass of water and osmolarity were conserved to the tolerances of the computation.

We integrate the system until either of the following happens: (i) a final time for 60 s is reached, or (ii) TBU occurs. We define TBU as the point at which the AL thickness reaches a minimum value of \(0.5\upmu \)m and use event detection to stop the simulation if this happens. When breakup occurs, the tear breakup time (TBUT) becomes the final time.

As an example of evaporation-driven thinning, we consider the initial conditions described in Sect. 2.2.1 with a director angle of \(\theta _B=\pi /2\). This choice maximizes the evaporation rate for our chosen evaporation function. Evaporative resistance in the LL is maximized for \({\mathcal {R}}(0)\); \({\mathcal {R}}(\pi /2)\) reduces that resistance by a factor of 10.

Solutions are shown in Fig. 3. The LL changes very little over time, maintaining its original shape with a valley in the center of the domain. The initial velocity in the LL, which ranges between \(\pm 50\,\upmu \)m, is moderate compared to the dynamics of other scenarios considered later. The thinning in the LL leads to increased evaporation of the AL, which quickly thins, reaching breakup conditions in the center of the domain by 15.9 s. AL pressure begins to drop in the center of the domain; in general, this would help to restore fluid to the area; however, TBU occurs because that capillarity-driven contribution is too weak.

The surface surfactant concentration \(\Gamma \), shown relative to its initial value, builds up slightly in the center of the domain. We explore Marangoni-driven flow in detail in Sect. 3.2; however, in this example, the increase in surfactant concentration is not large enough to contribute significantly to the flow.

These solutions are representative of evaporation-driven thinning that leads to breakup before the final time of 60 s. With a smaller director angle, evaporation resistance of the LL is increased, and the AL thins more slowly. When \(\theta _B\gtrsim 11\pi /25\), TBU is not reached within the simulation time of 60 s. Because evaporation is slower, the valley that develops in the AL is less pronounced, and the resulting peak in osmolarity is smaller; see Taranchuk and Braun [135] for more details.

We now turn to the initial conditions of Sect. 2.2.2, in which concentration gradients drive the tear film dynamics. The Marangoni effect results in rapid tangential flow; thus we consider solutions on a short timescale, shown in Fig. 10, and a longer timescale, shown in Fig. 11. Figure 10 shows the evolution of the variables over the first 0.06 s, when \(\theta _B=0\). These solutions are representative of the dynamics of tear film thinning due to surfactant concentration gradients regardless of the orientation of the liquid crystal molecules; the director angle \(\theta _B\) has little effect on the short timescale. The large surfactant gradients immediately cause rapid thinning in both the lipid and the aqueous layers. As the local excess of surfactant flattens, the Marangoni effect concludes. The LL thins dramatically, with the valley thickness roughly half of the initial value of 50 nm. Fluid is pulled away from the excess surfactant and builds up on the edges of the domain. The pattern of thinning is very similar in the AL, although to a lesser degree. During this initial redistribution of fluid, the LL fluid velocity is very large in magnitude (around \(3\times 10^5\,\upmu \)m/min), but quickly slows to the range of \(\pm 5\,\upmu \)m/min throughout the domain (not shown). Similarly, AL pressure (not shown) drops in magnitude from a range of \(\pm 3\) Pa during the first 1 s, to \(\pm 1\) Pa, and thinning continues over a longer timescale.

Figure 11 shows solutions at the final time for fives values of \(\theta _B\) ranging from 0 to \(\pi /2\). Breakup conditions are reached when \(\theta _B=\pi /2\) by 16.6 s; for all other values of \(\theta _B\), the final time of 60 s is reached. The surfactant concentration remains effectively uniform around 1.58 times the original background concentration, and over the longer timescale, it is the evaporative resistance in the LL that influences thinning dynamics. When \(\theta _B=0\) or \(\pi /4\), thinning becomes more localized in the center of the domain, while larger values of \(\theta _B\) maintain a wider valley of thinning through time. For smaller values of \(\theta _B\), we observe that negative pressure in the center of the AL domain pulls fluid in and helps to reduce thinning. This results in an overall thicker AL at the final time. Arrows in the velocity and pressure plots of Fig. 11 point to the difference in behavior when \(\theta _B=0\) or \(\pi /2\) as compared to smaller director angles.

On the other hand, the LL profiles are indistinguishable from each other at the final time and look very similar to the solution at 0.06 s shown in Fig. 10. Virtually all of the movement in the LL happen within the first second; this immobilization of the fluid surface due to the Marangoni effect has been studied in other contexts such as vertical drainage in Braun et al. [148] and remobilization of droplet surfaces in Stebe and Maldarelli [149].

Osmolarity in the AL increases for all values of \(\theta _B\) over the original value of 300 mOsM; however, it is most pronounced when \(\theta _B=\pi /2\). While the solutions in the AL are not dissimilar for \(\theta _B=4\pi /9\) and \(\theta _B=\pi /2\), the osmolarity is very different. When \(\theta _B=\pi /2\), peak osmolarity reaches nearly 1400 mOsM by 16.6 s, more than one and a half-times the peak osmolarity at 60 s when \(\theta _B=4\pi /9\). When \(\theta _B=\pi /2\), thinning and TBU happen so rapidly that the solutes do not have enough time to diffuse through the AL.

We now compare the two scenarios which combine perturbations in both the LL thickness and the surfactant concentration. The first combination is a bump in \(h_2\) with a local excess of \(\Gamma \), and the second is a valley in \(h_2\) with a local excess of \(\Gamma \). Figure 16 compares the solutions for AL and LL thickness, as well as osmolarity for these two cases with \(\theta _B=0\). While neither case reaches breakup for this angle of the director, thinning in the AL is more pronounced when there is a valley in the LL, with a minimum value of 0.94 \(\upmu \)m, as compared to 1.66 \(\upmu \)m when the LL contains a bump. The LL behavior is similar in both cases; it undergoes rapid redistribution of fluid initially and then remains tangentially immobile for the remainder of the time. When the LL contains a bump, rapid redistribution of fluid throughout the domain results in an area of thinning where the bump was previously located, with the minimum thickness of 28.46 nm. When the LL contains a valley, the Marangoni effect magnifies this profile resulting in a solution similar to those of Fig. 11, but with a deeper valley: the depth is now 14.15 nm versus 28.1 nm (when \(\theta _B=0\)).

Osmolarity in both cases reflects the respective changes in the AL. When the LL contains a valley, the AL thinning is more pronounced, and thus osmolarity reaches a peak of 650 mOsM, with little increase in concentration at the edges of the domain. When the LL contains a bump, the AL thins to a lesser degree, and the osmolarity steadily rises throughout the domain, with a smaller peak in the center. Peak osmolarity at the final time reaches 485 mOsM. In all scenarios that contain a local excess of surfactant initially, the Marangoni effect results in a rapid flattening of the surfactant concentration to a more or less uniform value of 1.58.

We can consider the influence of the director field angle, by observing the final time solutions for various angles as shown in Fig. 17. When there is a bump in the LL, we see more variation resulting from changes in the director angle. When \(\theta _B=\pi /2\), TBU occurs at 46.8 s, but no other angles of the director result in breakup. When there is a valley in the LL, the director angle has less of an impact because rapid thinning in the AL leads to TBU for \(\theta _B=5\pi /12,\,4\pi /9\), and \(\pi /12\). Breakup times are detailed further in Table 5 which is discussed more below. Osmolarity at the final time is strongly influenced by the director angle. When there is a bump in the LL, peak osmolarity is almost three times higher for \(\theta _B=\pi /2\) (1409 mOsM) than for \(\theta _B=0\) (485 mOsM). The difference is just over double when there is a valley in the LL (1442 versus 650 mOsM).

Thus far we have considered a valley in the LL with depth of 25 nm in a 50 nm LL. We now vary this profile keeping a valley of half-thickness. Table 5 compares the TBUT for three different LL background thicknesses: 30, 40, and 50 nm. We see that the dynamics are not sensitive to LL thickness when \(\theta _B=\pi /2\). This is because evaporative resistance is at its lowest for this angle of the director; for all of the scenarios, we have considered \(\theta _B=\pi /2\) results in TBU before 60 s. LL thickness has a stronger effect for intermediate angles of the director. For example, when \(\theta _B=4\pi /9\), decreasing the LL thickness by 20%, from 50 to 40 nm, results in TBU in half the time. When \(\theta _B\) is small enough, TBU does not occur; for all three LL thicknesses, TBU does not occur for \(\theta _B=0\) or \(\pi /4\),