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Dynamics of Fluorescent Imaging for Rapid Tear Thinning

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Abstract

A previous mathematical model has successfully simulated the rapid tear thinning caused by glob (thicker lipid) in the lipid layer. It captured a fast spreading of polar lipid and a corresponding strong tangential flow in the aqueous layer. With the simulated strong tangential flow, we now extend the model by adding equations for conservation of solutes, for osmolarity and fluorescein, in order to study their dynamics. We then compare our computed results for the resulting intensity distribution with fluorescence experiments on the tear film. We conclude that in rapid thinning, the fluorescent intensity can linearly approximate the tear film thickness well, when the initial fluorescein concentration is small. Thus, a dilute fluorescein is recommended for visualizing the rapid tear thinning during fluorescent imaging.

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Acknowledgements

This work is funded by NSF Grant 1412085 (Braun), NIH Grant 1R01EY021794 (Begley), and NEI Grant R01EY017951 (King-Smith).

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Correspondence to L. Zhong.

Appendices

Appendix

Flow and Solute Transport Dimensional Model

This section gives a detailed derivation in Cartesian coordinates which is applicable to streak TBU.

The following governing equations enforce the conservation of mass of water, momentum and mass of solutes (osmolarity c and concentration of fluorescein f) within the aqueous layer. They apply on \(0< x'< X_L'\) and \(0<z'<h'(x',t')\).

$$\begin{aligned} \nabla ' \cdot \mathbf{u }'= & {} 0, \end{aligned}$$
(20)
$$\begin{aligned} \rho \left( \partial _{t'} \mathbf{u }' + \mathbf{u }'\cdot \nabla ' \mathbf{u }' \right)= & {} -\nabla ' p ' + \mu \nabla ^{'2} \mathbf{u }', \end{aligned}$$
(21)
$$\begin{aligned} \partial _{t'} c' + \nabla '\cdot (\mathbf{u }' c')= & {} D_{o} \nabla '^2 c', \quad \text{ and } \quad \partial _{t'} f' + \nabla '\cdot (\mathbf{u }' f') = D_{f} \nabla '^2 f'. \end{aligned}$$
(22)

The aqueous/cornea interface is located at \(z' = 0\) and \(0< x'< X_L'\). This is assumed to be a no-slip surface; we assume that there is osmotic flow across a semipermeable membrane and it is proportional to the concentration difference across the boundary:

$$\begin{aligned} u'_{x}= 0, \quad \text{ and } \quad u'_{z}= P_0 V_W(c' - c_o). \end{aligned}$$
(23)

The permeability of the membrane is \(\text {P}_o\), the molar volume is \(V_W\), and the isotonic concentration is \(c_0\). For osmolarity and fluorescein, we assume that the advective fluxes balance the diffusive fluxes at the corneal surface.

$$\begin{aligned} D_o \partial _{z'} c' - u'_{z} c' = 0, \quad \text{ and } \quad D_f \partial _{z'} f' - u'_{z} f' = 0 \end{aligned}$$
(24)

On the free surface of the tear film, \(z' = h'\) for \(0< x' < X_L'\). The kinematic boundary condition implies that evaporative flux is the only flux that thins the tear film,

$$\begin{aligned} J ' = \rho ( \varvec{u}'(x',h') - \varvec{u}_I')\cdot \varvec{n}. \end{aligned}$$
(25)

Here, \(\varvec{u}'(x',h')\) is the fluid velocity evaluated at the free surface, \(\mathbf{u }_I'\) is the interface velocity, and \(\mathbf{n } = \frac{(-\partial _{x'}h',1)}{(1 + (\partial _{x'}h')^2)^{1/2}}\) is the unit normal vector to the interface. The stress balance in the normal direction to the interface is as follows:

$$\begin{aligned} -p_v' - \varvec{n} \cdot \varvec{T} \cdot \varvec{n} = \sigma _0 \nabla ' \cdot \varvec{n} - \frac{A'}{h^{'3}}. \end{aligned}$$
(26)

The Newtonian stress tensor is \( \varvec{T} = -\,p' \varvec{I} + \mu (\nabla ' \varvec{u'} + \nabla ' \varvec{u'}^T)\). We enforce no flux of solutes across the tear film surface (\(0< x' < X'_L\) and \(z' = h'\)) via

$$\begin{aligned} D_o \varvec{n} \cdot \nabla ' c' - \varvec{n} \cdot (\varvec{u}' - \varvec{u}'_I) c' = 0. \end{aligned}$$
(27)

Here, \(D_o\) is the diffusivity of osmolarity in water, which is assumed to be that of salt ions in water (Riquelme et al. 2007). From Eq. (25), the equation above may be written

$$\begin{aligned} D_o \varvec{n} \cdot \nabla ' c' - \frac{J'}{\rho } c' = 0. \end{aligned}$$
(28)

Similar to osmolarity, no flux of fluorescein is enforced by

$$\begin{aligned} D_f \varvec{n} \cdot \nabla ' f' - \frac{J'}{\rho } f' = 0. \end{aligned}$$
(29)

The model includes a glob on the tear film surface \(z'=h'(x',t')\) as a tangentially immobile aqueous/glob interface with a higher concentration of lipid on \(0<x'<X'_I\); the glob is assumed to have a fixed size and a fixed lipid concentration. The aqueous/air interface on \(X'_I \le x' \le X'_L\) is mobile, and conservation of mass for the insoluble polar lipid is governed by the following transport equation there.

$$\begin{aligned} \varGamma ' = \varGamma _0 \quad \text{ in } \quad [0,X'_I], \quad \text{ and }\quad \partial _{t'} \varGamma ' + \nabla _{s}' \cdot (\varGamma ' \mathbf {u}') = D_s \nabla _{s}^{'2} \varGamma ' \quad \text{ in } \quad [X'_I, X'_L]. \end{aligned}$$
(30)

The surface Laplacian is defined as \(\nabla '_s = (I - \varvec{n}\cdot \varvec{n})\nabla '\). The corresponding definition in tensor notation can be found in (Slattery et al. 2007). While surfactant mass is conserved individually within the glob and in the mobile region outside the glob, when we put them together on the tear film, the glob will act as a source of surfactant for the mobile region; we believe that this caricature is value for understanding the tear film dynamics driven by globs.

The tangential stress boundary condition under the glob and outside the glob is different because of our assumptions about the glob. Under the glob, \(z' = h'\) and on \((0, X_I')\), the no-slip boundary condition is assumed to hold (which replaces the tangential stress boundary condition), while outside of the glob \(z' = h'\), and on \( (X_I', X_L')\), tangential stress balance is enforced:

$$\begin{aligned} u'_{x'} = 0 \quad \text{ in } \quad (0,X'_I), \end{aligned}$$
(31)

and

$$\begin{aligned} \quad \varvec{t} \cdot \varvec{T}\cdot \varvec{n} = \partial _S \sigma = \varvec{t} \cdot \nabla ' \sigma = \varvec{t} \cdot \nabla ' \left[ \sigma _0 - \left( \partial _{\varGamma } \sigma \right) _0 (\varGamma ' - \varGamma _0) \right] \quad \text{ in } \quad (X'_I,X'_L). \end{aligned}$$
(32)

The unit tangent vector to the interface is \(\varvec{t} = \frac{(1, \partial _{x'}h')}{(1 + (\partial _{x'}h')^2)^{1/2}}\).

Dimensionless Models and Model Reduction

The model in Cartesian coordinates and one independent space variable represents the glob in a streak shape. In the independent variables \((x',z')\), the corresponding fluid velocity is \((u'_{x'},u'_{z'})\). Similarly, for the axisymmetric spot model, in the cylindrical coordinates \((r', z')\), the fluid velocity is \((u'_{r'}, u'_{z'})\).

1.1 Scalings

We simplify the streak glob model and spot glob model using the following scalings. The value of dimensional parameters used for scaling can be found in Table 2.

$$\begin{aligned} x'= & {} \ell x, \quad z' = d' z, \quad t' =\frac{\ell }{U}t, \quad h' = d' h, \quad u'_{x'} = U u_x, \quad u'_{z'} = \epsilon U u_z, \end{aligned}$$
(33)
$$\begin{aligned} p'= & {} \frac{\mu U}{\ell \epsilon ^2 } p, \quad J' = \epsilon \rho U J, \quad \varGamma ' = \varGamma _0 \varGamma , \quad c' = c_o c, \quad f' = f_\mathrm{cr}f. \end{aligned}$$
(34)
Table 2 Dimensional parameters used in this paper

1.2 Reduced Dimensionless Streak Model

We use the scalings in Sect. B.1 for the non-dimensionalization of our systems (in Sect. A). Dropping all the terms in the dimensionless equations, except the leading-order terms, we reduce our systems to be a simpler problem. The following are the reduced dimensionless equations for the streak TBU model.

The reduced governing equations inside the aqueous layer (\(0< z< h, 0< x < X_L\)), which enforce the conservation of mass of water, momentum and mass of solutes are:

$$\begin{aligned} \partial _{x}u_x + \partial _{z} u_z= & {} 0, \end{aligned}$$
(35)
$$\begin{aligned} \partial _{zz}u_x-\partial _{x}p= & {} 0, \quad \partial _{z}p =0, \end{aligned}$$
(36)
$$\begin{aligned} \partial _t c + u_x\partial _{x}c + u_z\partial _{z}c= & {} \text {Pe}_{c}^{-1} (\partial _{xx} c + \epsilon ^{-2}\partial _{zz}c), \quad \text{ and } \end{aligned}$$
(37)
$$\begin{aligned} \partial _t f + u_x \partial _{x}f + u_z \partial _{z}f= & {} \text {Pe}_{f}^{-1} (\partial _{xx} f + \epsilon ^{-2}\partial _{zz}f). \end{aligned}$$
(38)

At the aqueous/cornea interface \(z = 0\) and \(0< x < X_L\), we have no-slip and water permeability boundary conditions,

$$\begin{aligned} u_x = 0, \quad \quad u_z = \text {P}_c (c-1). \end{aligned}$$
(39)

We also assume there is no fluxes of solutes:

$$\begin{aligned} \epsilon ^{-2}\text {Pe}_c^{-1}\partial _z c = u_z c \quad \text{ and } \quad \epsilon ^{-2}\text {Pe}_f^{-1}\partial _z f = u_z f. \end{aligned}$$
(40)

Across the domain \([0,X_L]\) at tear film surface \(z = h(x,t)\), our kinematic boundary condition, tangential stress boundary condition and conservation of mass solutes result in the following dimensionless form,

$$\begin{aligned}&\partial _{t}h + u_x \partial _{x}h -u_z = -\,J(x), \end{aligned}$$
(41)
$$\begin{aligned}&p = -\,S\partial _{xx}h - Ah^{-3}, \end{aligned}$$
(42)
$$\begin{aligned}&\partial _z c = \epsilon ^2 (\text {Pe}_c Jc + \partial _x h \partial _x c) \quad \text{ and } \quad \partial _z f = \epsilon ^2 (\text {Pe}_f Jf + \partial _xh\partial _x f). \end{aligned}$$
(43)

Boundary conditions for the tangential immobility and fixed lipid concentration at the aqueous/glob interface \(0 \le x \le X_I\) become

$$\begin{aligned} u_x =0 \quad \text{ and } \quad \varGamma = 1. \end{aligned}$$
(44)

At the aqueous/air interface \([X_I,X_L]\), conservation of surfactant is now

$$\begin{aligned} \partial _{z}u_x = -\,M\partial _{x} \varGamma \quad \text{ and } \quad \partial _t \varGamma + \partial _{x} (u_s \varGamma ) = \text {Pe}_s^{-1}\partial _{x}^2 \varGamma . \end{aligned}$$
(45)

Here, \(\varvec{u}'_s\) is the surface velocity and is defined as \(\varvec{u}'_s = \nabla '_s \varvec{u}\). The dimensionless surface velocity is

$$\begin{aligned} \varvec{u}_s = \frac{1}{1 + \epsilon ^2 (\partial _x h)^2}(u + \epsilon ^2 u_z(\partial _x h)^2, \epsilon ^3 u_z (\partial _x h)^2 + \epsilon u \partial _x h) \end{aligned}$$
(46)

If only keep the leading order, \(\varvec{u}_s = (u, 0)\). To make a difference between \(\varvec{u_s}\) and u, we denote the surface velocity in the x direction using \(u_s\).

1.3 Reduction of the PDE System

The model in preceding section can be reduced to the lubrication approximation in the following ways. The process are similar for streak model and spot model. We show example in the streak case.

Equation (36) shows that \(p = p(x,t)\), which results in

$$\begin{aligned} u_x = \frac{\partial _x p}{2}z^2 + C(x,t) z +D(x,t). \end{aligned}$$
(47)

The boundary conditions on corneal surface and tear film surface for u are:

$$\begin{aligned} u_x = 0 \quad \text{ at } \ z = 0, \qquad u (1-B(x)) + (\partial _z u_x + M \partial _x \varGamma ) B(x) = 0 \ \text{ at } \ z = h, \end{aligned}$$
(48)

which implies that

$$\begin{aligned} u_x = \frac{\partial _x p}{2} z^2 + C(x,t)z, \quad C(x,t) = \frac{-\frac{\partial _x p}{2} h [h(1-B) + 2B]-M B \partial _x \varGamma }{h(1-B) + B}. \end{aligned}$$
(49)

Integrating Eq. (35) in z and combined with Leibnitz rule, we have the following:

$$\begin{aligned} \partial _x h u_x(x,h,t) - u_z(x,h,t) = \partial _x (h {\bar{u}}) - \text {P}_c (c(x,0,t)-1), \end{aligned}$$
(50)

where \({\bar{u}} (x,t) = \frac{1}{h}\int _0^h u_x(x,z,t) dz\) is the depth averaged velocity along the film. Substituting it into Eq. (41), we arrives at

$$\begin{aligned} \partial _t h = -\,\partial _x \left( h{\bar{u}} \right) + \text {P}_c (c(x,0,t)-1) -J(x). \end{aligned}$$
(51)

Solving for c and f are similar (Jensen and Grotberg 1993); here we only illustrate the process for c. Assume c has the following expansion,

$$\begin{aligned} c(x,z,t) = c_0 (x,z,t) + \epsilon ^2 c_1(x,z,t)+ \cdots \end{aligned}$$
(52)

Substitution into Eq. (38) and collecting terms, we obtain the leading-order equation

$$\begin{aligned} \partial _{zz} c_0 = 0. \end{aligned}$$
(53)

The leading-order term from either boundary is \(\partial _z c_0=0\); using the boundary conditions results in

$$\begin{aligned} c_0 = c_0 (x,t). \end{aligned}$$
(54)

We then further collect terms corresponding to \(O(\epsilon ^2)\) in (38) to obtain

$$\begin{aligned} \text {Pe}_{c}(\partial _t c_0 + u_x \partial _x c_0) - \partial _{xx}c_0 = \partial _{zz}c_1. \end{aligned}$$
(55)

Integrating Eq. (55) with respect to z over (0, h), we obtain

$$\begin{aligned} \text {Pe}_{c}h(\partial _t c_0 + {\bar{u}}\partial _x c_0) -h\partial _{xx}c_0 = \partial _{z}c_1(x,h,t) - \partial _{z}c_1(x,0,t). \end{aligned}$$
(56)

The boundary condition at \(z = 0\) from Eq. (40) and \(z = h\) from Eq. (43) becomes, at the first nonzero order,

$$\begin{aligned} \partial _z c_1(x, 0,t) = \text {Pe}_c u_z(x,0,t) c_0, \quad \text{ and } \quad \partial _z c_1(x,h,t)= \text {Pe}_c J c_0 + \partial _x h \partial _x c_0. \end{aligned}$$
(57)

Using these conditions to eliminate \(c_1\) in Eq. (56) results in, after dropping the subscript,

$$\begin{aligned} h(\partial _t c + {\bar{u}}\partial _x c) = \text {Pe}_c^{-1}\partial _{xx} c + Jc + \partial _xh \partial _xc-\text {P}_c (c-1)c. \end{aligned}$$
(58)

Proceeding in a similar way with fluorescein, we obtain

$$\begin{aligned} h(\partial _t f + {\bar{u}}\partial _x f) = \text {Pe}_f^{-1}\partial _{xx} f + Jf + \partial _xh \partial _xf-\text {P}_c (c-1)f. \end{aligned}$$
(59)

Now, in summary, we have the following five PDEs.

$$\begin{aligned} \partial _t h= & {} -\,\partial _x (h{\bar{u}}) + \text {P}_c (c-1) -J , \end{aligned}$$
(60)
$$\begin{aligned} p= & {} -\,\partial _{xx}h - Ah^{-3}, \end{aligned}$$
(61)
$$\begin{aligned} \partial _t \varGamma= & {} [\text{ Pe }_s^{-1}\partial _{x}^2 \varGamma -\partial _{x} (u_s \varGamma )]B(x), \end{aligned}$$
(62)
$$\begin{aligned} h(\partial _t c + {\bar{u}}\partial _x c)= & {} \text {Pe}_c^{-1}\partial _{xx} c + Jc + \partial _xh \partial _xc-\text {P}_c (c-1)c, \end{aligned}$$
(63)
$$\begin{aligned} h(\partial _t f + {\bar{u}}\partial _x f)= & {} \text {Pe}_f^{-1}\partial _{xx} f + Jf + \partial _xh \partial _xf-\text {P}_c (c-1)f. \end{aligned}$$
(64)

Here,

$$\begin{aligned} {\bar{u}}= & {} \frac{-\frac{1}{3}\partial _{x}p h^2 [ \frac{1}{4}(1-B) h+B]-\frac{1}{2} \partial _{x}\varGamma B h}{(1-B)(\beta + h) + B}, \end{aligned}$$
(65)
$$\begin{aligned} u_s= & {} \frac{ - B \left[ (h^2/2) \partial _{x}p + h \partial _{x} \varGamma \right] }{(1-B) h + B}. \end{aligned}$$
(66)

1.4 Streak Boundary and Initial Conditions

As in the spot problem, we apply symmetry at the origin and no-flux conditions at the other end of the domain. At the center of glob \(x = 0\), the boundary conditions enforce symmetry and thus no flux via

$$\begin{aligned} \partial _{x}h = \partial _{x}p = \partial _x \varGamma = \partial _{x}c = \partial _{x}f = 0. \end{aligned}$$
(67)

We enforce no-flux conditions on h, p, \(\varGamma \), c and f at \(x = X_L\), and these end up being Eq. (67) applied at the other end.

The initial conditions are again spatially uniform (except for \(\varGamma \)); on \(0\le x \le X_L\),

$$\begin{aligned} h(x,0) = c(x,0) = f(x,0) = 1, \quad \text{ and } \quad p(x,0)= -\,A. \end{aligned}$$
(68)

We assume that the glob has high surfactant concentration and the mobile tear/air interface outside of it has a lower concentration initially. From our scalings, \(\varGamma =1\) under the glob and \(\varGamma = 0.1\) outside the glob. Using the transition function B(x), the initial condition for \(\varGamma \) is then

$$\begin{aligned} \varGamma (r, 0) = 1 \cdot [1-B(x)] + 0.1 \cdot B(x). \end{aligned}$$
(69)

1.5 Reduced Dimensionless Spot Model

The derivation for spot model in axisymmetric cylindrical coordinate is similar to the streak model in one dimension. In this subsection, we will just list the dimensionless model with only leading-order terms.

Inside the aqueous layer \(0<z<h, 0<r<R_L\), the conservation laws of mass of water, momentum and mass of solutes are

$$\begin{aligned} \frac{1}{r} \frac{\partial }{\partial r} (r u_r) + \frac{\partial u_z}{\partial z}= & {} 0, \end{aligned}$$
(70)
$$ \begin{aligned} \partial _r p =-\,\partial _z^2 u_r \quad \& \quad \partial _z p= & {} 0, \end{aligned}$$
(71)
$$\begin{aligned} \partial _t c + u_r\partial _r c + u_z\partial _z c= & {} \text {Pe}_c^{-1} \left[ \frac{1}{r}\partial _r (r\partial _r c) + \epsilon ^{-2}\partial ^2_z c\right] , \end{aligned}$$
(72)
$$\begin{aligned} \partial _t f + u_r\partial _r f + u_z\partial _z f= & {} \text {Pe}_f^{-1} \left[ \frac{1}{r}\partial _r (r\partial _r f) + \epsilon ^{-2}\partial ^2_z f\right] . \end{aligned}$$
(73)

At corneal surface \(z = 0, 0<r<R_L\), the no-slip boundary conditions and osmotic flow through semipermeable membrane are defined as follows:

$$\begin{aligned} u_r(r,0,t) =0 \quad \text{ and } \quad u_z(r,0,t) = \text {P}_c (c-1). \end{aligned}$$
(74)

In addition, there are two more conditions for non-flux of solutes,

$$\begin{aligned} \epsilon ^{-2}\text {Pe}_c^{-1}\partial _z c = u_zc \quad \text{ and } \quad \epsilon ^{-2}\text {Pe}_f^{-1}\partial _z f = u_zf. \end{aligned}$$
(75)

At the tear film surface \((z = h, \; 0<r<R_L)\), the kinematic boundary condition and normal stress balance are simplified to be

$$\begin{aligned} \partial _t h + u_r \partial _r h - u_z = -\,J, \quad \text{ and } \quad p - p_{v} = -\,S\frac{1}{r} \partial _r (r \partial _r h) - Ah^{-3}. \end{aligned}$$
(76)

The equations of no fluxes of solutes are

$$\begin{aligned} \partial _z c= \epsilon ^2 [\text {Pe}_c Jc + \partial _rh \partial _r c], \quad \text{ and }\quad \partial _z f = \epsilon ^2[ \text {Pe}_f Jf + \partial _rh \partial _r f]. \end{aligned}$$
(77)

At the aqueous/glob interface, \(z = h\) and \(0<r< R_I\), the no-slip boundary condition and a fixed surfactant concentration are as follows:

$$\begin{aligned} u_r = 0, \quad \varGamma = 1. \end{aligned}$$
(78)

At the aqueous/air interface, \(z = h\) and \(R_I<r<R_L\), the boundary condition for \(\varGamma \) reduced to be

$$\begin{aligned} \partial _z u_r = -\,M\partial _r \varGamma , \quad \text{ and } \quad \partial _t \varGamma = \hbox {Pe}_s^{-1}\left[ \frac{1}{r}\partial _r (r \partial _r \varGamma )\right] - \frac{1}{r}\partial _r (r \varGamma u_r). \end{aligned}$$
(79)

The boundary and initial conditions are given in the text.

Other Evaporation Fluxes

Besides the evaporation fluxes in Eq. (10), three other fluxes also have been used here.

Case (a): If lipid with higher concentration provides a better barrier to evaporation, we may assume that there is no evaporation over the glob. The edge of the glob corresponds to the highest evaporation rate \(v_{\max }\) due to the abnormal decreasing of concentration of lipid. The aqueous/air interface corresponds to the uniform evaporation rate \(v_{\min }\) for the uniform distributed lipid concentration.

$$\begin{aligned} J (r)= a(1-v_{\min })(r-R_I)e^{-\frac{(r-R_I)^2}{2R_W^2 }}+v_{\min }\tanh \left( \frac{r-R_I}{R_W}\right) , \end{aligned}$$
(80)

where the constant a is given by

$$\begin{aligned} a = \frac{e^{1/2}[v_{\max }-v_{\min }\tanh (1)]}{(1-v_{\min })R_W}. \end{aligned}$$
(81)

Case (b): If the higher concentration of lipid results in a poor barrier to evaporation, we let the evaporation rate be low on the glob and high at the aqueous/air interface:

$$\begin{aligned} J(r) = v_{\min }\left[ 1- B(r) \right] +v_{\max }B(r). \end{aligned}$$
(82)

Case (c): The different compositions of glob are assumed to have no effect on evaporation; in that case, the evaporation rate is uniform across the entire domain,

$$\begin{aligned} J(r) = v_{\max }. \end{aligned}$$
(83)

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Zhong, L., Braun, R.J., Begley, C.G. et al. Dynamics of Fluorescent Imaging for Rapid Tear Thinning. Bull Math Biol 81, 39–80 (2019). https://doi.org/10.1007/s11538-018-0517-0

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