Duplex Tear Film Evaporation Analysis

Abstract

Tear film thinning, hyperosmolarity, and breakup can cause irritation and damage to the human eye, and these form an area of active investigation for dry eye syndrome research. Recent research demonstrates that deficiencies in the lipid layer may cause locally increased evaporation, inducing conditions for breakup. In this paper, we explore the conditions for tear film breakup by considering a model for tear film dynamics with two mobile fluid layers, the aqueous and lipid layers. In addition, we include the effects of osmosis, evaporation as modified by the lipid, and the polar portion of the lipid layer. We solve the system numerically for reasonable parameter values and initial conditions and analyze how shifts in these cause changes to the system’s dynamics.

Appendix A

1.1 A.1 Fluid Model Derivation

The derivation for this fluid model follows closely to the work of Bruna and Breward (2014), while making use of lubrication theory techniques covered in a different context in Oron et al. (1997) and Craster and Matar (2009). We let numerical subscripts indicate the domain we are working in, or the surface we are considering, with $j=1$ corresponding to the aqueous domain, and $j=2$ corresponding to the lipid domain. We let $\mathbf {s}_{j}$ be the position on the upper interface of domain j, with corresponding unit normal and tangential vectors $\hat{n}_{j}$ and $\hat{t}_j$. We use letter subscripts x and t to indicate derivatives with respect to those variables.

At the aqueous–corneal interface, mass conservation and velocity continuity require the velocity is equal to the flux of water through the boundary from osmosis.

$$\begin{aligned} \text {Velocity Continuity}&\mathbf {u}_1 = \mathbf {J}_o&\text { at } y = 0. \end{aligned}$$

(12)

Inside the aqueous domain, we conserve mass and momentum, and so we use the incompressible Navier–Stokes equation.

$$\begin{aligned}&\text {Conserve Momentum}\quad \rho _1 \left( \mathbf {u}_{1 t} + \mathbf {u}_1 \cdot \nabla \mathbf {u}_1 \right) = -\nabla p_1 + \mu _1 \Delta \mathbf {u}_1 \quad \text { in } \varOmega _1. \end{aligned}$$

(13)

$$\begin{aligned}&\text {Conserve Mass} \qquad \qquad \nabla \cdot \mathbf {u}_1 = 0 \end{aligned}$$

(14)

At the aqueous–lipid boundary, velocity continuity requires that the horizontal aqueous and lipid velocities be equal. Meanwhile, the difference between a fluid’s velocity, $\mathbf {u}_{j}$, and the velocity of the film’s surface, $\mathbf {s}_{jt}(x)$, determines the mass flux through that surface. The position of the surface is governed by its movement due to surface forces, requiring a balance between the aqueous and lipid stresses with the effects of surface tension. Note that the surface tension at the aqueous–lipid boundary may vary.

$$\begin{aligned}&\text {Velocity Continuity}&\left( \mathbf {u}_1 - \mathbf {u}_2 \right) \cdot \hat{t}_1 = 0 \qquad \qquad \qquad \text { at } y = h_1. \end{aligned}$$

(15)

$$\begin{aligned}&\text {Aqueous Mass Conservation }&\rho _1 \left( \mathbf {u}_1 - \mathbf {s}_{1t} \right) = \mathbf {J}_e \qquad \qquad \qquad \qquad \qquad \quad \end{aligned}$$

(16)

$$\begin{aligned}&\text {Lipid Mass Conservation}&\rho _2 \left( \mathbf {s}_{1t} - \mathbf {u}_2 \right) = 0 \qquad \qquad \qquad \qquad \qquad \qquad \end{aligned}$$

(17)

$$\begin{aligned}&\text {Stress Balance}&\left( \mathbf {\mathbf {\tau }}_1 - \mathbf {\mathbf {\tau }}_2 \right) \cdot \hat{n}_1 = - \gamma _{s_1} \hat{n}_1 \nabla \cdot \hat{n}_1 + \nabla _{s_1} \gamma _{s_1} \end{aligned}$$

(18)

As with the aqueous domain, we use the incompressible Navier–Stokes equation to govern the lipid domain.

$$\begin{aligned}&\text {Conserve Momentum} \quad \rho _2 \left( \mathbf {u}_{2 t} + \mathbf {u}_2 \cdot \nabla \mathbf {u}_2 \right) = -\nabla p_2 + \mu _2 \Delta \mathbf {u}_2 \quad \text { in } \varOmega _2.\qquad \end{aligned}$$

(19)

$$\begin{aligned}&\text {Conserve Mass} \qquad \qquad \nabla \cdot \mathbf {u}_2 = 0 \end{aligned}$$

(20)

At the lipid–air interface, we have no mass flux, and so the fluid velocity must equal the surface velocity. For surface forces, surface stress must balance with the effects of surface tension, but unlike the aqueous–lipid boundary, we will take this surface tension to be a constant, $\gamma _2$.

$$\begin{aligned}&\text {Lipid Mass Conservation} \quad \rho _2 \left( \mathbf {u}_2 - \mathbf {s}_{2t} \right) = 0 \qquad \qquad \text { at } h_2. \end{aligned}$$

(21)

$$\begin{aligned}&\text {Stress Balance} \quad \quad \quad \quad \quad \quad \,\, \mathbf {\mathbf {\tau }}_2 \cdot \hat{n}_1 = - \gamma _2 \hat{n}_2 \nabla \cdot \hat{n}_2 \end{aligned}$$

(22)

In addition to the usual fluid dynamics, we will require several additional equations governing the transport and effects of surfactants in the polar portion of the lipid, the transport of salts in the aqueous domain, evaporation, and osmosis. The polar portion of the lipid acts as a surfactant with a linear reduction from the base surface tension, and we take the concentration of these surfactants to be governed by advection–diffusion along the surface of the aqueous–lipid interface, where $\partial _{s_1}$ in Equation (24) indicates a derivative taken along the $\mathbf {s}_1$ surface.

$$\begin{aligned}&\text {Surface Tension} \qquad \quad \gamma _{s_1} = \gamma _1 - RT \varGamma \qquad \qquad \text { at } h_1. \end{aligned}$$

(23)

$$\begin{aligned}&\text {Surfactant Transport} \quad \varGamma _t + \partial _{s_1} \left( \varGamma \mathbf {u}_{s_1} \right) + \kappa \mathbf {u}_{n} \varGamma = D \partial _{s_1}^2 \varGamma . \end{aligned}$$

(24)

Salts are transported in the aqueous layer according to the advection diffusion equation. For no salt flux conditions at the top and aqueous–lipid and corneal interfaces, we require the salts transported away from the boundary by diffusion to equal those incident by advection.

$$\begin{aligned}&\text {Aqueous--Lipid No Flux} \quad&c \left( \mathbf {u}_1 - \mathbf {s}_{1t} \right) - D_1 \nabla c = 0 \qquad \,\, \text { at } y = h. \qquad \qquad \end{aligned}$$

(25)

$$\begin{aligned}&\text {Aqueous Salt Transport} \quad&c_t + \mathbf {u}_1 \cdot \nabla c - D_1 \Delta c = 0 \qquad \text { in } \varOmega _1. \end{aligned}$$

(26)

$$\begin{aligned}&\text {Aqueous--Cornea No Flux} \quad&c \left( \mathbf {u}_1 - \mathbf {s}_{0t} \right) - D_1 \nabla c = 0 \qquad \,\, \text { at } y = 0. \end{aligned}$$

(27)

Derivation for the evaporation model is given extensively in Appendix A.3. Lastly, osmotic flux is determined by the concentration difference between the aqueous layer and the cornea.

$$\begin{aligned} \text {Osmosis}&\mathbf {J}_o \cdot \hat{n}_0 = P_c \left( c - c_0\right)&\text { at } y=0. \end{aligned}$$

(28)

1.2 A.2 Reduction by Lubrication Theory

In order to drastically simplify this system, we will be applying the methods of lubrication theory, reducing this system as much as possible to leading-order terms in this thin film parameter $\displaystyle ~\epsilon ~=~\frac{H_1}{L}~\ll ~1$. We will dimensionally scale this system, define all relevant parameters, expand the velocities in orders of $\epsilon ^2$,

$$\begin{aligned} u_j = \epsilon ^0 u_j + \epsilon ^2 u_j^{(2)} + \epsilon ^4 u_j^{(4)} + \cdots \end{aligned}$$

(29)

and finally group in orders of $\epsilon ^2$. The dimensional scalings are given below. We assume the lipid viscosity is much larger than the aqueous viscosity (Rosenfeld et al. 2013). This requires a different pressure scales in the two layers, and establishes a shear balance in the aqueous layer and an extensional balance in the lipid layer at leading order.

$x^* \rightarrow L x$	$h_1^* \rightarrow \epsilon L h_1$	$h_2^* \rightarrow \delta \epsilon L h_2$
$y^* \rightarrow \epsilon L y$	$u_1^* \rightarrow U u_1$	$u_2^* \rightarrow U u_2$
$\displaystyle t^* \rightarrow \frac{L}{U} t$	$v_1^* \rightarrow \epsilon U v_1$	$v_2^* \rightarrow \epsilon U v_2$
$J^* \rightarrow \rho _1 \epsilon U J$	$\displaystyle p_1^* \rightarrow \frac{\mu _1 U}{\epsilon ^2 L} p_1$	$\displaystyle p_2^* \rightarrow \frac{\mu _2 U}{L} p_2$
$\varGamma ^* \rightarrow \varGamma _0 \varGamma$

The parameters chosen are based on commonly used fluids parameters. We require our nondimensional parameters be on the order of 1 in comparison to $\epsilon$, and so we include factors of $\epsilon$ to keep the parameters near unit sized. The parameters are defined below.

$\displaystyle \epsilon = \frac{H_1}{L}$	$\displaystyle \mathcal {C}_1 = \epsilon ^3 \frac{\gamma _1}{\mu _1 U}$	$\displaystyle \mathrm{Pe}_1 = \frac{UL}{D_1}$
$\displaystyle \delta = \frac{H_2}{H_1}$	$\displaystyle \mathcal {C}_2 = \epsilon \frac{ \gamma _2}{\mu _2 U}$	$\displaystyle \mathrm{Pe}_2 = \frac{UL}{D_2}$
$\displaystyle \varUpsilon = \epsilon ^2 \frac{\mu _2}{\mu _1}$	$\displaystyle \mathrm {Re}_1 = \frac{\rho _1 U L}{\mu _1}$	$\displaystyle \mathcal {E} = \frac{k_m}{\epsilon \rho _1 U} \frac{M_1}{R} \left( \frac{p_0}{T_0} - \frac{ R_h p_\infty }{T_\infty } \right)$
$\displaystyle \mathcal {M} = \epsilon \frac{RT \varGamma _0}{\mu _1 U}$	$\displaystyle \mathcal {R}_2 = \epsilon ^{-2} \frac{\rho _2 U L}{\mu _2}$	$\displaystyle \mathcal {R} = \frac{k_m H_2}{Dk}$
		$\displaystyle \mathcal {P} = \frac{C P_c}{\epsilon U}$.

After performing the above substitutions, expanding via perturbation theory, and grouping at leading orders, the number of terms in each of these equations is greatly reduced. We may now summarize these more simplified equations by region.

At the corneal–aqueous interface, we have

$$\begin{aligned} \text {Velocity Continuity} \quad \mathbf {u}_1= & {} 0 \quad \text { at } y = 0. \end{aligned}$$

(30)

$$\begin{aligned} v_1= & {} J_o. \end{aligned}$$

(31)

Inside the aqueous layer,

$$\begin{aligned}&\text {Incompressibility}&v_{1y} = - u_{1x} \quad \text { in } \varOmega _1. \end{aligned}$$

(32)

$$\begin{aligned}&\text {Horizontal Momentum}&u_{1yy} = p_{1x}. \end{aligned}$$

(33)

$$\begin{aligned}&\text {Vertical Momentum}&p_{1y} = 0. \end{aligned}$$

(34)

At the aqueous–lipid interface,

$$\begin{aligned}&\text {Aqueous Mass} \quad&h_{1t} + h_{1x} u_1 - v_{1} + J_e = 0 \quad \text { at } y=h_1. \end{aligned}$$

(35)

$$\begin{aligned}&\text {Lipid Mass} \quad&h_{1t} + h_{1x} u_2 - v_{2} = 0. \end{aligned}$$

(36)

$$\begin{aligned}&\text {Velocity Continuity} \quad&u_1 = u_2. \end{aligned}$$

(37)

$$\begin{aligned}&\text {Normal Stress} \quad&p_1 = \varUpsilon ( p_2 - 2v_{2y} ) - \mathcal {C}_1 h_{1xx}. \end{aligned}$$

(38)

$$\begin{aligned}&\text {Shear Stress} \quad&u_{2y} = 0. \end{aligned}$$

(39)

Inside the lipid layer,

$$\begin{aligned}&\text {Incompressibility} \quad v_{2y} = - u_{2x} \quad \text { in } \varOmega _2. \end{aligned}$$

(40)

$$\begin{aligned}&\text {Horizontal Momentum} \quad u_{2yy} = 0 \end{aligned}$$

(41)

$$\begin{aligned}&\text {Vertical Momentum} \quad v_{2yy} = p_{2y}. \end{aligned}$$

(42)

At the lipid–air interface,

$$\begin{aligned}&\text {Lipid Mass} \quad \qquad h_{t} + h_{x} u_2 - v_{2} = 0 \quad \text { at } y=h. \end{aligned}$$

(43)

$$\begin{aligned}&\text {Normal Stress} \quad \,\, p_2 = 2 v_{2y} - \mathcal {C}_2 h_{xx}. \end{aligned}$$

(44)

$$\begin{aligned}&\text {Shear Stress} \quad \,\,\,\,\,\, u_{2y} = 0 \quad \qquad \text { at } y=h_2. \end{aligned}$$

(45)

The leading-order equations governing salt transport are given by

$$\begin{aligned}&\text {Salt Transport} \quad c_{yy} = 0 \quad \,\,\,\,\,\,\text { in } \varOmega _1. \end{aligned}$$

(46)

$$\begin{aligned}&\text {No Flux} \qquad \qquad c_{y} = 0 \qquad \quad \text { at }\quad y = 0 \text { and } y = h_1. \end{aligned}$$

(47)

And for auxiliary equations,

$$\begin{aligned}&\text {Surfactants} \quad \varGamma _t + \partial _x \left( u_2 \varGamma \right) = \mathrm{Pe}^{-1} \varGamma _{xx} \quad \text { at } y=h_1 \end{aligned}$$

(48)

$$\begin{aligned}&\text {Evaporation} \quad J_e = \frac{\mathcal {E}}{1 + \mathcal {R} h_2}. \end{aligned}$$

(49)

$$\begin{aligned}&\text {Osmosis} \quad \qquad J_o = P_c \left( c - c_0 \right) \qquad \qquad \text { at } y=0. \end{aligned}$$

(50)

Lastly, we require second-order terms to determine the velocity of the lipid layer extensional flow as well as the salt transport equations. The second-order lipid layer equations, with corresponding variable $u_2^{(2)}$, are given by

$$\begin{aligned}&\text {Aqueous-Lipid Shear} \quad u_{2y}^{(2)} = \varUpsilon ^{-1} \left( u_{1y} + \mathcal {M} \varGamma _x \right) - v_{2x} - 2 h_{1x} v_{2y} + 2 h_{1x} u_{2x} \quad \text { at } y = h_1. \end{aligned}$$

(51)

$$\begin{aligned}&\text {Lipid Momentum} \qquad \qquad u_{2yy}^{(2)} = p_{2x} - u_{2xx} \qquad \qquad \qquad \qquad \qquad \quad \qquad \text { in } \varOmega _2. \end{aligned}$$

(52)

$$\begin{aligned}&\text {Lipid-Air Shear} \quad \qquad \qquad u_{2y}^{(2)} = - v_{2x} - 2h_x v_{2y} + 2 h_{x} u_{2x} \qquad \qquad \qquad \text { at } y=h. \end{aligned}$$

(53)

The second-order equations for salt transport, with corresponding variable $c^{(2)}$, are given by

$$\begin{aligned}&\text {Aqueous--Cornea No Flux} \quad c_y^{(2)} = \mathrm{Pe}_1 c J_o \qquad \qquad \quad \quad \quad \quad \quad \text { at } y = 0. \end{aligned}$$

(54)

$$\begin{aligned}&\text {Aqueous Salt Transport} \,\,\qquad c_{yy}^{(2)} = \mathrm{Pe}_1 \left( c_t + + u c_x \right) - c_{xx} \quad \text { in } \varOmega _1. \end{aligned}$$

(55)

$$\begin{aligned}&\text {Aqueous-Lipid No Flux} \quad \,\quad c_y^{(2)} = \mathrm{Pe}_1 c J_e + h_{1x} c_x \qquad \quad \quad \,\, \text { at } y=h_1. \end{aligned}$$

(56)

Next, we apply the methods of lubrication theory, with a shear balance in the aqueous layer and an extensional balance in the lipid, in order to eliminate the vertical dimension in this problem, giving us this system to be modeled. After significant algebraic manipulation, this system simplifies to an evolution equation for aqueous thickness, an evolution equation for lipid thickness, a transport equation for salt concentration, a transport equation for surfactants, and an extensional equation that determines the horizontal lipid velocity. This system is given by

$$\begin{aligned}&h_{1t} + \partial _x \left( h_1 \overline{u}_1 \right) = J_o - J_e \end{aligned}$$

(57)

$$\begin{aligned}&h_{2t} + \partial _x \left( h_2 u_2 \right) = 0 \end{aligned}$$

(58)

$$\begin{aligned}&\left( h_1 c\right) _t = \mathrm{Pe}_1^{-1} \partial _x ( h_1 c_x ) - \partial _x( h_1 c \overline{u}_1 ) \end{aligned}$$

(59)

$$\begin{aligned}&\varGamma _t + \partial _x ( u_2 \varGamma ) = \mathrm{Pe}_2^{-1} \varGamma _{xx} \end{aligned}$$

(60)

$$\begin{aligned}&4 \delta \varUpsilon \partial _x ( h_{2} u_{2x} ) - \frac{u_2}{2} = \frac{h_1}{2} p_{1x} - \delta \varUpsilon \mathcal {C}_2 h_2 h_{xxx} + \mathcal {M} \varGamma _x. \end{aligned}$$

(61)

The above system contains a few direct equations which have been left to improve the system’s readability.

$$\begin{aligned}&\overline{u}_1 = - \frac{h_1^2}{12} p_{1x} + \frac{u_2}{2} \end{aligned}$$

(62)

$$\begin{aligned}&p_1 = -\mathcal {C}_1 h_{1xx} - \varUpsilon \mathcal {C}_2 h_{xx} \end{aligned}$$

(63)

$$\begin{aligned}&p_2 = - 2 u_{2x} - \mathcal {C}_2 h_{xx} \end{aligned}$$

(64)

$$\begin{aligned}&J_e = \frac{\mathcal {E}}{1 + \mathcal {R} h_2} \end{aligned}$$

(65)

$$\begin{aligned}&J_o = \mathcal {P} \left( c-1 \right) . \end{aligned}$$

(66)

1.3 A.3 Evaporation Model Derivation

For evaporation, we use a modified boundary layer model where mass transfer is driven by the chemical potential difference across the liquid gas interface, as expressed by a concentration difference.

$$\begin{aligned} J = -k(c_l - c_g). \end{aligned}$$

(67)

Here, k is a kinetic mass transfer coefficient. We may then use the molar mass M to convert to molar concentration to partial pressure via the ideal gas law. This results in the mass flux equation

$$\begin{aligned} J = k \left( \frac{M p_0}{RT_0} - \frac{M p_\infty }{RT_\infty } \right) \end{aligned}$$

(68)

where subscripts 0 and $\infty$ correspond to liquid and gas, respectively. Next, we then express $p_0$ and $p_\infty$ more naturally in terms of saturation vapor pressure. The liquid water layer will be at saturation vapor pressure, while the water vapor pressure $p_0$ is expressed by the saturation vapor pressure and relative humidity $R_h$. We use the saturation vapor pressure model given in Buck (1981) to express saturation vapor pressure p(T).

$$\begin{aligned} p^{\text {sat}}(T) = p_{\text {ref}} \exp \left[ \frac{\Delta H}{RT_{\text {ref}}} \left( 1 - \frac{T_{\text {ref}}}{T} \right) \right] . \end{aligned}$$

(69)

For calculating saturation vapor pressure, we use the values chosen in Buck (1981) for environmental conditions closest to our simulations.

$$\begin{aligned} p^{\text {sat}}(T) = 6.1142 \exp \left[ 17.368 \ \frac{T}{T + 238.88} \right] \end{aligned}$$

(70)

Based on the values given in Buck (1981), the input temperature is in C, while the output pressure is in Pa. Lastly, we modify this boundary layer model by altering the boundary layer resistance to include a second layer, representing the resistance to permeation applied by the lipid layer in series with the evaporative resistance. As is done in Bruna and Breward (2014), Cerretani et al. (2013), Peng et al. (2014), we assign the lipid layer a resistance based on its thickness and permeability (given by Dk),

$$\begin{aligned} J = \frac{1}{\frac{1}{k_m} + \frac{1}{Dk} h_2} \frac{M}{R} \left( \frac{p^{\text {sat}}(T_0)}{T_0} - R_h \frac{p^{\text {sat}}(T_\infty )}{T_\infty } \right) . \end{aligned}$$

(71)

Many models simplify this one step further, grouping $k_m$ with the concentration terms on the right. They may also express evaporation in terms of a thinning rate, rather than mass flux term. Performing this simplification reduces all environmental terms on the right to a single evaporative thinning rate constant, $E_0$.

$$\begin{aligned}&E_0 = \frac{k_m}{\rho } \frac{M}{R} \left( \frac{p^{\text {sat}} (T_0)}{T_0} - R_h \frac{p^{\text {sat}}(T_\infty )}{T_\infty } \right) \end{aligned}$$

(72)

$$\begin{aligned}&J_{\text {thinning}} = \frac{E_0}{1 + \frac{k_m}{Dk} h_2}. \end{aligned}$$

(73)

Table 5 Evaporation rate physical parameters

Table 6 Nonlinear least squares (NLS) fit physical parameters

Fig. 15

Comparison of the lipid resistance versus evaporation rate profiles with the experimental data from King-Smith et al. (2010). (A) Cerretani et al. (2013) experimental estimate. (B) Peng et al. (2014). (C) NLS fit to King-Smith et al. (2010). (D) Cerretani et al. (2013) estimate for in vivo results found in the literature. (E) Bruna and Breward (2014)

For the constants relating to concentration and saturation vapor pressure, we have used values given in Table 5. However, there is a significant variation in the values relating to boundary layer resistances, $k_m$ and Dk. To resolve this, we determine suitable values for $k_m$ and Dk using nonlinear least squares fitting to data from the simultaneous aqueous thinning rate and lipid thickness measurements found in King-Smith et al. (2010). Due to the constraints of the original measurement method, there is no data for lipid thicknesses near 0 $\upmu$m, and so we have augmented the data with one additional data point for peak observed evaporation rates in the human eye. We use a peak rate of 38 $\upmu$m/min, as used in Peng et al. (2014). We consider the thinning rate $J_{\text {thinning}}$ in $\upmu$m/min as a function of lipid thickness $h_2$ in nm,

$$\begin{aligned} J_{\text {thinning}}(h_2) = \frac{1}{\frac{1}{k_m} + \frac{1}{Dk} h_2} \frac{M}{\rho R} \left( \frac{p^{\text {sat}}(T_0)}{T_0} - R_h \frac{p^{\text {sat}}(T_\infty )}{T_\infty } \right) \end{aligned}$$

(74)

and perform the nonlinear least squares (NLS) fit using MATLAB’s lsqcurvefit command. This results in the values for $k_m$ and Dk given in Table 6. In addition, in Fig. 15, we provide a comparison to the raw data, as well as to the evaporation profiles from several other sources (Bruna and Breward 2014; Cerretani et al. 2013; Peng et al. 2014).