Eye drops are commonly instilled to treat a variety of ocular problems, such as dry eyes, glaucoma, infections, allergies, etc. The fluid instillation results in an increase in tear volume, and it slowly returns to its steady value due to tear drainage through canaliculi, and also fluid loss through other means such as evaporation or transport across the ocular epithelia.^{1} In fact, if the instilled fluid has a viscosity similar to that of tears, which is about 1.5 mPa · s, the instilled fluids or solutes are eliminated from the tears in a few minutes.^{2–4} As a result, the fluids or solutes have a short contact time with the eye surface, which results in reduced effects for artificial tears or low bioavailability for ophthalmic drugs. To increase the duration of comfort after drop instillation and to increase the bioavailability of the drugs delivered via eye drops, it is desirable to prolong the residence time for the instilled fluid. It has been suggested and also shown in a number of clinical and animal studies that increasing the viscosity of the instilled fluid leads to an increase in the retention time. Zaki et al.^{5} studied the clearance of solutions with viscosities from 10 to 100 mPa · s from the precorneal surface. These experiments showed a rather interesting effect of viscosity: the retention began to increase only after the fluid viscosity exceeded a critical value of about 10 mPa · s and also the relative increase in retention became smaller at very high viscosities. Although increasing fluid viscosity increases the residence time, it may also cause discomfort and damage to ocular epithelia due to an increase in the shear stresses during blinking. Shear thinning fluids such as sodium hyaluronate (NaHA) solutions can be used to obtain the beneficial effect of an increase in retention and yet avoid excessive stresses during blinking.^{2} The likely reason is that the shear rates during blinking are very high and at such high shear rates these shear-thinning fluids exhibit low viscosity but during the interblink which is the period during which tear drainage occurs, these fluids act as high viscosity fluids leading to reduced drainage rates and a concurrent increase in residence time.

Although the mechanisms of the impact of viscosity on residence time are qualitatively understood for both Newtonian and non-Newtonian fluids, no quantitative model has been yet proposed that can explain the detailed physics and predict the effect of viscosity on drainage rates and on retention time of eye drops. Such a model is likely to lead to an improved quantitative understanding of the effect of viscosity on tear dynamics, and also aid as a tool in development of better dry eye treatments and drug delivery vehicles. The goal of this study is to develop a mathematical model to predict the effect of viscosity on drainage rates and the residence time for both Newtonian and non-Newtonian fluids. The current study is an extension of our previous study that focused on modeling drainage of tears, which were considered to be Newtonian fluids with a viscosity of 1.5 mPa · s. Both the previous and the current models are based on the physiological description of canalicular tear drainage proposed by Doane,^{6} which is described in detail elsewhere,^{7} and are briefly presented below. According to Doane, the drainage of tears through lacrimal canaliculi is driven by the cyclic action of blinking. The entire blink cycle is divided into two phases, the blink phase and the interblink phase. During the blink phase, the eyelids move towards each other and meet, the puncta are closed and lacrimal canaliculi are squeezed by the surrounding muscles. The squeezing of canaliculi, along with puncta closure causes tear flow towards the nose. During the interblink phase, the eyelids separate leading to opening of puncta and a valve-like mechanism prevents any flow at the nasal end of canaliculi. Additionally, the muscles do not squeeze the canaliculi and this leads to a vacuum inside the canaliculi that sucks fluids from the ocular tear film. As a result of this cyclic process, tears are drained from the ocular tear film into the nose.

The tear drainage model developed by Zhu and Chauhan^{7} showed that for tears with a viscosity of 1.5 mPa · s, the canaliculus radius will reach steady states during both the blink and the interblink phase. The canaliculus reaches a steady state in the blink phase when the stresses generated by the deformation of canaliculi balance the pressure applied by the muscles. The steady state is reached in the interblink when canaliculi have relaxed to an extent at which the pressure in the canaliculi equals that in the tear film. Achieving steady state both in the blink and the interblink implies that if the duration of the interblink and the blink are further increased, there will be no changes in total tear drainage per blink. However, the drainage rates will decrease due to the reduction in the number of blinks per unit time. The canaliculus radius was shown to reach a steady state in a time

, where L, b, and R_{0} are the length, thickness, and the undeformed radius of canaliculi, E is the elastic modulus of canaliculi, and μ is the viscosity of the instilled Newtonian fluid. As the viscosity of the fluid increases, the time to achieve steady state increases, but as long as the canaliculus reaches a steady state in both the blink and the interblink, there is no change in the total amount of fluid drained in a blink, and so there is also no change in the drainage rates. This explains the observation of Zaki et al.^{5} that below a critical viscosity, increasing viscosity does not lead to enhanced retention. However, as the viscosity increases to a critical value at which the time to achieve steady state (τ_{s}) becomes larger than the duration of the blink phase, the canaliculus does not deform to the fullest extend possible, and so the amount of tears that drain into the nose during the interblink decreases. In the current study, the drainage model will be modified to calculate the drainage rate of instilled Newtonian fluids with viscosities that are larger than the critical viscosity, and so the system does not reach steady state in the blink phase. Additionally, the drainage rates will be calculated for power-law fluid, which is a reasonable approximation under physiological shear rates for typical non-Newtonian fluids used for ocular instillation. Finally, the modified tear drainage model will be incorporated into a tear balance model to predict the effect of viscosity on residence time of eye drops.

#### METHODS

Below we will first develop the models for the drainage of Newtonian and non-Newtonian fluids, respectively. The models are used to predict the drainage rates, residence time and bioavailability, and these predictions are compared with the experimental measurements^{2,8–10} reported in literature. In our mathematical models, the canaliculus is simplified as a straight pipe of length L with an undeformed radius R_{0}, wall thickness b, and modulus E. The canaliculus is considered to be a thin shell, i.e., its thickness is much smaller than the radius and thus axial deformation is neglected and the length of canaliculi L is assumed to be constant. Each of these assumptions is an approximation, and the impact of each of these on the model predictions are discussed elsewhere.^{7}

##### Drainage of a Newtonian Fluid

It has been shown by Zhu and Chauhan^{7} that for a Newtonian fluid, the time and position dependent radius of the canaliculus can be predicted by solving the following equation:

where R is the radius of canaliculi that depends on axial position and time, x is the position along the canaliculus, with x = 0 for the puncta side and x = L (length of canaliculi) for the nasal side. The details of the derivation of Eq. 1 are described elsewhere.^{7} The boundary conditions and the initial conditions for Eq. 1 are:

For the blink phase (0 < t < t_{b})

For the interblink phase (t_{b} < t < t_{c})

where t_{b} is the duration of the blink phase, t_{c} is the duration of one blink-interblink cycle, q is the flow rate of instilled fluids or tears through the canaliculus, L is the length of the canaliculus, p is the pressure inside the canaliculus, ς is the surface tension of the instilled fluids or tears, R_{m} is the radius of curvature of the tear meniscus, and R_{b} and R_{ib} are the steady state canaliculus radii in the blink and the interblink, respectively, and these are given by the following expressions:

where p_{0} and p_{sac} are the pressure applied by the surrounding muscles to the canaliculus during blinking and the pressure in the lacrimal sac. The pressure p in Eq. 3 can be written as a function of the canaliculus radius R through the following equation:

Equation 4 Image Tools |
Equation (Uncited) Image Tools |

The details of the above model development are described earlier by Zhu and Chauhan.^{7} The radius of the canaliculus can be solved analytically as a function of axial position and time from Eq. 1, 2, and 3. The volume of fluid contained in the canaliculus at any instant in time can be computed by using the following equation:

The volume of fluid drained in one blink-interblink cycle can then be computed as the difference between the volume at the end of an interblink (V_{interblink}) and that at the end of the blink (V_{blink}), and then the drainage rate through the canaliculus can be computed as

The above procedure can be used to calculate the effect of viscosity on tear drainage. To determine the effect of tear viscosity on the residence time of eye drops, the tear drainage rates are incorporated in a tear mass balance.

##### Incorporation of Tear Drainage into Tear Balance

A mass balance for the fluids on the ocular surface yields

where V is the total volume of the fluids on the ocular surface, including tears and the instilled fluids, q_{production} is the combined tear production rate from the lacrimal gland and conjunctiva secretion, q_{evaporation} is the tear evaporation rate, and both of these are assumed to be constant. In the above equation, tear transport across the cornea is neglected because the area of cornea is much smaller than that of conjunctiva. The effects of such approximations are discussed elsewhere.^{7}

A mass balance for the solutes in the instilled fluid, such as radioactive tracers or fluorescent dye molecules, can be written as

where c is the concentration of the solutes and V is the total volume of ocular fluids. The above equation is only valid for solutes that do not permeate the ocular epithelia such as fluorescent dextran. The drainage rate q_{drainage} depends on viscosity, and this dependence leads to the dependence of dynamic concentration profiles on viscosity. The normal viscosity of tears is about 1.5 mPa · s and after instillation, it increases to a value μ_{i} that is close to the viscosity of the eye drops μ_{drop.} Immediately after instillation, due to the dilution by the tears μ_{i} will be smaller than μ_{drop}, but here for simplicity μ_{i} is assumed to be equal to μ_{drop}. Subsequently, the viscosity of the ocular fluid begins to decrease due to changes in the polymer concentration c. The viscosity of a polymer solution can be a complex function of concentration, and here for simplicity, the viscosity is assumed to be a linear function of concentration, i.e.,

where c_{0} is the solute concentration immediately after instillation. Since experimental studies on the residence time of instilled fluids often measure the transient of total quantity of radioactive tracers or dye molecules in the ocular fluids, we combine Eq. 9 and 10 to yield the following equation for the total quantity of solutes

where I (=cV) is the total quantity of solutes, and I_{0} and V_{0} are the values of I and V immediately after instillation. It is noted that the drainage rate calculations are coupled to the tear balance because the radius of curvature of the meniscus depends on the total tear volume, and the drainage rate is affected by the curvature through boundary condition (3). By geometric considerations, the tear volume can be related to the meniscus curvature by the following equation^{11}:

where the second term accounts for the fluid in the meniscus and V_{film} accounts for the remaining tear volume, i.e., fluid in the exposed and the unexposed tear film. In the above equation L_{lid} is the perimeter of the lid margin. The volume V_{film} depends on the tear film thickness (h), which in turn is related to viscosity through h = 2.12R_{m}(μU/ς)^{2/3}, where U is the velocity of the upper lid; and ς is the tear surface tension.^{12} The relationship yields unrealistically large value of film thickness for large viscosities and therefore is likely invalid. Therefore in this study V_{film} is assumed to be independent of viscosity and based on our earlier calculations^{11} and the measurements by other researchers,^{13} its value is fixed at 5.37 μl. This yields a total normal tear volume of 7 μl for a meniscus radius value of 0.365 mm.^{14} It is noted that the instillation of extra fluids may cause large variations of the radius of curvature, changes of the geometry of the tear menisci, and even overflow of the ocular fluids onto the cheeks, which may render Eq. 13 invalid. However, such factors may vary across different subjects and there is no quantitative expression to account for these factors. Therefore in this study, Eq. 13 will be used, while noting that it is only an approximation, and a more accurate expression can be used instead if more detailed physiological information is available. By solving Eq. 1, 7, 8, 9 and 12 simultaneously using finite difference method, the transient quantity of the solutes in the ocular fluids can be obtained as a function of time.

Equation 8 Image Tools |
Equation 9 Image Tools |
Equation 13 Image Tools |

Equation 14 Image Tools |

In Eq. 10 above, it is assumed that the tracers do not permeate into the ocular surface, which is a reasonable assumption for some commonly used tracers. However it is not a reasonable assumption for ocular drugs delivered via drops. For such solutes, the mass balance needs to be modified to include drug transport through the ocular tissue. The modified mass balance is

where the constants K_{cornea} and K_{conj} are the permeabilities of cornea and conjunctiva to timolol, respectively, and A_{cornea} and A_{conj} are the areas of cornea and conjunctiva, respectively. By combining the above mass balance with Eq. 9, bioavailability (β), i.e., the fraction of the instilled drug that permeates into cornea can be calculated as

where V_{0} is the sum of the tear volume and the volume of the instilled fluid immediately after the instillation. In Eq. 15 the transient total ocular fluid volume (V(t)) can be calculated using Eq. 1, 7, 8 and 9. It is assumed that all the drug that is absorbed into the conjunctiva or drained in the canaliculi goes to the systemic circulation. The derivation and the details of Eq. 15 are given elsewhere.^{11}

##### Non-Newtonian Fluid

The residence time of non-Newtonian fluids can also be calculated by following the same approach as outlined above for Newtonian fluid except that Eq. 1 needs to be modified. Unlike Newtonian fluids, which have a linear relationship between the shear stress and the shear rate, non-Newtonian fluids have more complicated relation between the shear stress and the shear rate. One of the most common non-Newtonian fluids for dry eye treatment is sodium hyaluronate solution. Rheological measurements have shown that at the concentration used for ocular instillation, it can be approximated as power-law (shear-thinning) fluid, i.e., the relation between the shear stress τ and the shear rate γ can be written as

where Κ and n are rheological parameters that can be obtained by fitting the viscosity vs. shear rate data. Here, the constant Κ is assumed to be related linearly to the instantaneous polymer concentration in the tear film by using a linear relationship as given by Eq. 11. Using Eq. 16 the equation for the deformation of canaliculi as a result of blinking can be derived as

Equation 12 Image Tools |
Equation 17 Image Tools |

where a is a constant that is defined as

The derivation of Eq. 17 is described in the appendix (available online at www.optvissci.com.). It is noted that Newtonian fluid is a special case of a power-law fluid with n = 1 and Eq. 17 correctly reduces to Eq. 1 for this case. By solving Eq. 17, 7, 8, 9, and 12 simultaneously using finite difference method, the transient quantity of the solutes in the ocular fluids can be obtained as a function of time. Similar to the Newtonian fluid case, the bioavailability can be also calculated using Eq. 15.

##### Model Parameters

Most of the parameters needed in the model are available in literature and these are listed in Table 1.^{6,7,15–23}

The rheological parameters Κ and n were obtained by fitting the viscosity vs. shear rate data, and these are listed in Table 2 for a variety of fluids that are commonly used in ocular studies. The non-linear least square fitting was applied to the original data points shown on the respective plots in the references, and was conducted using the curvefitting toolbox in Matlab and the non-linear equation given in the caption of Table 2. In Table 2, “R_{fitting}” represents the correlation coefficient.

#### RESULTS

##### Effect of Viscosity on Tear Drainage

The effect of viscosity on tear drainage rate q_{drainage} immediately after instilling 25 μl of fluids is shown in Fig. 1 for a Newtonian fluid. The drainage rates for shear-thinning fluids depend on Κ and n, and the results for shear thinning fluids are shown in Fig. 2. The drainage rates depend on the tear volume, and the results reported in Figs. 1, 2 correspond to a tear volume immediately after instillation, which is taken to be 32 μl.

Figure 1 Image Tools |
Figure 2 Image Tools |

##### Effect of Viscosity on Residence Time of Instilled Fluids

The effect of viscosity on residence time in eyes is typically measured by instilling the high viscosity fluid laden with tracers such as radioactive or fluorescent compounds, and then following the total amount of tracer present in the tear volume by measuring the radioactivity or fluorescence. The transients of the total signal from the tracer I(t), which is a measure of the total solute quantity are plotted in Fig. 3 for Newtonian fluids for a range of viscosities. Similar data is compared with experiments in Fig. 4, 5. It is noted that for fluids with viscosities lower than 4.4 mPa · s, the transients of I(t) overlap. In these and all other calculations reported below, the volume of all the instilled drops is set to be 25 μl.

Figure 3 Image Tools |
Figure 4 Image Tools |
Figure 5 Image Tools |

For non-Newtonian fluids, the transients of I are calculated for sodium hyaluronate acid of 0.2 and 0.3% w/v concentrations, which are commonly used for ocular instillation. The initial values of Κ and n for these and all other shear-thinning fluids that are discussed in this paper are listed in Table 2. The solute quantity transients I(t) are plotted in Fig. 6a to c for these two fluids in solid lines after rescaling. The details of the rescaling will be given in the discussion section.

##### Bioavailability of Instilled Drugs

The bioavailability of timolol instilled as an eye drop is calculated for a variety of drop compositions including both Newtonian and non-Newtonian fluids and the bioavailability values are listed in Table 3.

#### DISCUSSION

##### Drainage Rates

There are three important time scales relevant to the drainage process; the duration of the blink phase, which is the duration of canaliculus compression t_{b} (∼0.04 s), the time of the interblink period t_{ib} (∼6 s) and the time scale for achieving steady canaliculus deformation τ. Based on the relative magnitude of τ with respect to t_{b} and t_{ib}, there are three different scenarios for the tear drainage. When the viscosity is <4.4 mPa · s, τ_{s} < t_{b} and τ_{s} < t_{ib} and so the canaliculus radius can reach steady state during both the blink phase and the interblink phase. In this case, the canaliculus radius is uniform at the end of both the blink and the interblink phase, and V_{interblink} and V_{blink} can be calculated from the steady state radii, and changing viscosity within this range will not change the drainage rate q_{drainage}. When the viscosity is larger than 4.4 mPa · s but smaller than 654 mPa · s, t_{b} < τ_{s} < t_{ib}, and so the canaliculus radius cannot reach steady state during the blink phase but still can reach steady state during the interblink phase. In this case the canaliculus radius can still reach the same steady state and be uniform at the end of the interblink phase, but the canaliculus radius is not uniform at the end of the blink phase and V_{blink} need to be calculated by determining the position dependent radius at the end of the blink and then using Eq. 7. Increasing viscosity within this range will decrease the drainage rate, and this range is applicable to most of the high viscosity Newtonian fluids that are used for ocular instillation. When the viscosity is larger than 645 mPa · s, τ_{s} > t_{ib} > t_{b}, and the canaliculus radius cannot reach steady state during both in the blink phase and the interblink phase. However at this high viscosity, the shearing to the ocular surface is likely to be high and may cause irritation. Therefore Newtonian fluids with viscosities higher than 645 mPa · s are not likely to be used for ocular instillation and are not considered in this study.

Since drainage rates through canaliculi are not typically measured, it is not possible to compare the predictions with experiments. It is noted that the data shown in Fig. 1 has three distinct regions. In the first region (μ < 4.4 mPa · s), there is no effect of viscosity on drainage rates. In the second region (4.4 < μ < 100 mPa · s), the viscosity has the maximum impact on the drainage. In the last region (μ > 100 mPa · s), the effect of viscosity on drainage rates becomes small. These trends qualitatively agree with observations noted in literature. Also, the data in Fig. 1 suggests that it may be best to use eye drops with a viscosity of about 100 mPa · s to increase the retention time, and yet not cause damage to ocular epithelia due to excessive shear during blinking.

The trends shown in Fig. 2 for shear-thinning non-Newtonian fluids are similar to those for Newtonian fluids. The trends for the three typical n values are similar, with larger n yielding smaller drainage rates at the same Κ because of the smaller viscosities. The data in Fig. 2 suggests that the best range of Κ for non-Newtonian eye drops should be below 400 mPa · s^{n}. It has been reported that some polymers in such non-Newtonian eye drops have the ability to bind to the mucous ocular surface^{23} and thus increase the residence time of such eye drops. The binding effect is not included in the current study due to the insufficient information about the binding isotherms of such polymers, and the actual drainage rates for polymers that bind to the ocular surface should be lower than that predicted by this model.

All the parameters required for calculating the drainage rates have been measured and are available in literature except bE, which is the product of the canaliculus thickness and the elasticity. The mechanical properties of porcine canaliculus have been measured, and these results show that the canaliculus is a viscoelastic material, with frequency dependent moduli.^{24} The model presented here still assumes the canaliculus to be elastic with a value of 2.57 Pa · m for bE, which is based on measurements of the zero frequency mechanical properties of porcine canaliculus.^{24} The properties of human canaliculus may be different and so in the calculations for concentration transients shown above, three different values of bE, 0.64, 1.29 and 2.57 Pa · m, are used.

##### Comparison of Residence Time

The transients of solute quantity predicted by our model can be compared with the experimental studies in which a tracer-laden fluid is instilled and the transients of the tracer amount are measured. In these experiments, the tracer laden fluid is typically administered in the exposed tear film, and the intensity transients are measured. The transients generally show a fast decrease in tracer quantity, followed by a slower decay. The fast decrease in tracer quantity may be due to mixing of the instilled fluids on the ocular surface, into the upper sac and into the canthus regions, which takes only a few blinks for low viscosity fluids, and also due to transport of tracer into the fluid in the lower sac, which is a slower process due to smaller motion of the lower eyelid. The model presented here assumes an instantaneous mixing, and so it cannot capture the initial rapid decay in intensity transients. Additionally, in the model presented above, we have neglected transport of tracers through the ocular epithelia and also binding to the ocular surface. Such binding could lead to residual intensity, which is also commonly observed in experiments. To account for the issues of mixing and binding, the tracer quantity transients predicted by the above model are rescaled in the following manner. If the tracer quantity I decreases from I_{0} to I_{0}^{’} immediately due to the initial mixing, and the amount of residual quantity is I_{r}, the predicted tracer I(t) quantity can be rescaled using the following equation.

The rescaled I′(t) can then be compared with the experiments. Eq. 19 is in principle only applicable when the instillation of fluids with lower viscosities so that the mixing time is negligible compared to the drainage through canaliculi. The transient profiles can also be used to determine the residence time, which is defined as the time for I to decrease to within 0.01 I_{0} to the end value. In the model calculation and the experimental studies mentioned below, the volume of the instilled fluids is 25 μl.

The results for I(t) for viscosities of 5 and 8 mPa · s (Fig. 4) can be compared with the experimental study by Snibson et al.^{8} using radioactive tracers (^{99}Tc^{m}), in which the viscosities of two kinds of Newtonian fluids, 1.4% PVA and 0.3% HPMC were measured to be about 5 and 8 mPa · s, respectively. The experimentally measured I(t)/I_{0} for the two fluids almost overlap (dashed curve) and these are plotted in Fig. 4 along with the scaled theoretical profiles I′(t)/I_{0} with I_{0}′ = 0.6 I_{0} and I_{r} = 0.13 I_{0} for the three different values of bE. The residence time decreases on reducing bE because smaller elasticity leads to a larger deformation of the canaliculus in each blink. The theoretical profiles best match the experimental data for bE = 1.29 Pa · s. Greaves et al.^{9} measured the precorneal residence of 0.3% HPMC, which is Newtonian and has a viscosity of 6.649 mPa · s. The solute quantity transients predicted by the current study after rescaling using Eq. 19 with I_{0}′ = 0.4 I_{0} and I_{r} = 0.122 I_{0} is plotted in Fig. 5 for the three values of bE (solid lines). The three dashed lines represent the experimental data by Greaves et al. (dashed line) with the middle dashed line representing the mean and the upper and the lower representing the standard deviation in the measurements. The residence time predicted by the model is about 1362, 1044, and 954 s, respectively for bE values of 2.57, 1.29 and 0.64 Pa · m, but the residence time from the experimental data is only about 550 s. The agreement between the experiments and the model is poor but it should be noted that the results of the experiments of Greaves et al. (Fig. 5) strongly disagree with those of Snibson et al. (Fig. 4) even though they both used 0.3% HPMC solution. It should be noted that Snibson et al. conducted measurements on the entire ocular surface, while Greaves et al. focuses only on the precorneal area. However, these differences cannot completely account for the significant differences between the two studies.

For non-Newtonian fluids, the model predictions can be compared to experiments for 0.2% NaHA, 0.3% NaHA conducted by Snibson et al.^{2} The predicted transient profiles for these two cases are plotted in solid lines Fig. 6 after rescaling using Eq. 19 with I_{0}′ = 0.80 I_{0} and I_{r} = 0.30 I_{0}, together with the experimental data shown in dash lines. It is noted that in this case since the viscosity is much higher than that of tears, the initial mixing of tracers is not likely to be immediate, as supported by the lack of sudden drop of tracer quantity in the experimental data. Also, from the experimental data presented for both NaHA solutions, it is difficult to judge whether the tracer quantity transient had reached steady state at the end of the measurement. Therefore the rescaling of intensity may introduce additional errors. The predicted residence times for the two solutions are 2214 and 3336 s. It is noted that in the experimental study lasted only 2000 s after the instillation and it is not possible to determine the residence time from the data because the profiles had not leveled off by 2000 s. Therefore it is not possible to compare the residence time predictions with experiments. The predictions best match the experiments for bE = 0.64 Pa · m. The residence time for non-Newtonian fluids can be compared with the experimental results for Carboxyl methyl cellulose solutions by Paugh et al.^{10} using FITC-dextran, which has a molecular weight of about 72 000 Da and is not likely to permeate into the ocular surface. According to the experiments two kinds of carboxyl methyl cellulose solutions, “L-100” and “H-100,” had residence time of 14.6 ± 9.7 min and 15.4 ± 5.4 min longer that of the control saline solution that has a viscosity similar to tear viscosity, respectively. The model predictions for the same variables using the fitted rheological parameters for “L-100” and “H-100” in Table 2 are 27.0 min and 8.7 min, respectively.

The model predictions are in qualitative agreement with most of the experiments for both Newtonian and non-Newtonian fluids but there are quantitative differences. There are several possible causes for the discrepancies between experiments and model predictions. First, the rescaling in Eq. 19 assumes the tracers are well-mixed instantaneously after instillation, while it is expected that such mixing is viscosity-dependent. The rescaling can be improved if quantitative information of the dependence of mixing on viscosity is available. Also, a more realistic description of tracer permeation and binding would require permeability measurements and binding isotherms for these tracers, which if available can be included in the current model. Another possible source of discrepancy between the experiments and the model is the assumption of the linear relationship between viscosity and polymer concentration in the tears. After the solutions are instilled onto the ocular surface, they are subject to dilution because of tear refreshing and such dilution is expected to decrease the viscosity of the solution, and change in viscosity is often non-linearly related to the polymer concentration. This issue could also be included more accurately in the model if the relationship between polymer concentration and solution viscosity is known for the solution of interest. It is also noted that this issue will cause only a small difference for solutions with starting viscosities of 10 mPa · s or less. It is also noted that the current model assumes that the surface tension of tears (∼43 × 10^{−3} N/m) is not changed after the instillation of extra fluids, which may not be good assumption particularly if the tracers are surface active. Finally, there are several assumptions involved in developing the tear drainage model^{7} including values of parameters needed for the model, and these could cause additional errors in model predictions.

Besides the experimental studies mentioned in the above comparison, there are also other studies on the effect of viscosity on the residence time of instilled fluids. However, they are not used to compare with the model predictions because some of them used rabbits as subjects instead of human, and the others lack either the clear definition of the area being measured or the viscosity data of the exact solution used for instillation. It is known that rabbits have a much lower blinking rate than human and this will likely result in smaller drainage rates. Although the effect of viscosity on the residence time showed a similar qualitative trend to that of human, the rabbit studies were not used in the comparison because the current model is based on the tear drainage physiology of human. Several other studies also cannot be compared directly with the model developed here because these studies did not clearly indicate the region of ocular surface which were used for measurements, or we could not find the rheological data for the fluids used in these studies.

##### Validity of the Power-Law Equation

Typical ocular fluids can be described by power law only in a limited range of shear rates. Since we use the power law assumption over the entire process of caniculus deformation, it is important to ensure that the shear rates experienced by the fluid during this step lie in the range of the validity of the power law equation. It is noted that for the range of viscosities considered in this paper the amount of fluid that drains from the canaliculi to the nose during the blink equals the amount that enters the canaliculi from the tear film during the interblink. To calculate the drainage rates, we only determine the volume of fluid drained from the canaliculi in the blink, and so we only need to validate the power law assumption during the blink. The shear rates during the blink periods decrease monotonically during the blink period, and also decrease monotonically as the total tear volume approaches the steady state value at which the concentration of the polymer in the tears approaches zero. It was verified that for all the non-Newtonian fluids in this study, the shear rates during the drainage are generally within the range measured in the experimental studies except the case when the K value in the power law equation is about 100 mPa · s^{n}, i.e., the case of least viscous power-law fluid considered here. Even for this case, the power law assumption is reasonable for all times except near steady state and so the results for drainage times and dynamic profiles are not unduly impacted by this assumption.

##### The Effect of Viscosity on Bioavailability

Podder et al.^{25} showed that by using eye drops with viscosity-enhancing polymers the ocular absorption could be increased two-fold for rabbits. The current model predicts that the timolol bioavailability increases with increasing viscosity of eye drops (Table 3), the predicted increase is much smaller. The increase in bioavailability due to increase in viscosity occurs due to the reduction in drainage rates. However for timolol, the time scale for drug absorption into the conjunctiva, which can be estimated by τ_{absorption} = V/(K_{conj}A_{conj}) is only 35 s. This time is much smaller than the time scale of the drainage through canaliculi (∼1000 s). This implies that the drug timolol exits the tear volume mainly by transport across the ocular epithelium (cornea and conjunctiva), and only a negligible amount drains into the nasal cavity. Thus any change in drainage rates as a result of changes in viscosity will make only minimal improvements in bioavailability. The increase in viscosity will begin to impact the bioavailability if the permeabilities of the cornea and the conjunctiva are sufficiently small so that the conjunctiva uptake time scale increases to about 1000 s. These results agree with the theoretical study by Keister et al.^{26} In fact, in an extreme case where punctal plugs are applied and the drainage through canaliculi is completely blocked, the bioavailability will be

and according to the parameter values in Table 1, β would be 1.34%, which is the maximal bioavailability achievable for this drug, and it is only marginally larger than the bioavailability under normal circumstances. It should be noted that some polymers that are used as viscosity enhancers can impact the bioavailability in an indirect manner. For instance they could preferentially bind to corneal surface or cause formation of a gel that traps the drug, leading to larger drug retention in the precorneal tear film. These effects are not included in the current model.

##### Importance of Tear Rheology on Drainage

In our previous study of the tear drainage model, tears were assumed to be Newtonian and a constant viscosity of 1.5 mPa · s is used. However tears are in fact non-Newtonian (rheology parameters listed in Table 2) possibly due to the presence of mucin and other large molecules. The current study shows that any variance of viscosity below 10 mPa · s leads to indistinguishable changes in the drainage rate through canaliculi, as evident from Fig. 1. Therefore assuming tears to be Newtonian with constant viscosity is unlikely to have a major effect on tear drainage calculations.

#### CONCLUSIONS

The effect of the viscosity of instilled fluids on drainage rates and ocular residence time is studied using a mathematical model for both Newtonian and non-Newtonian fluids. This model can predict the transients of solute quantity, fluid volume, and the concentration of solutes in instilled solutions. It can also predict the residence time and the bioavailability of instilled drugs. It is shown that for Newtonian fluids, increasing the viscosity till about 10 mPa · s causes only indistinguishable difference in the residence time, which agrees with experimental observations. The model predictions are compared with the experimental studies of the fluid clearance from the ocular surface using human subjects, and they agree at least qualitatively with the experimental data of several studies using radioactive or fluorescent tracers.

The current study is the continuation of our previous effort of the physiology-based modeling for the tear drainage. Using this approach, the physics behind the drainage process can be quantitatively assessed and therefore it can help explain some phenomena that are difficult to understand by experiments alone. The model presented in the current study can serve as a guide in designing treatments for various eye ailments, such as high-viscosity artificial tears for dry eyes or ophthalmic drug delivery by high-viscosity eye drops with improved bioavailability. In addition, this study can also provide insights of the drainage of non-Newtonian fluids, which may be a closer approximation for many kinds of eye drops.

The model predictions are in qualitative agreement with experiments but there are quantitative differences. These differences are partly due to variability in model parameters and partly due to certain model assumptions that are not strictly valid. The model developed here still suffers from several deficiencies and in future we hope to eliminate some of these so that the model can make more realistic predictions. Although the model developed here is expected to be useful, it can in no way serve as a replacement for experiments. It is only intended to serve as a guide in selection of the viscosities of the eye drops.

#### ACKNOWLEDGMENTS

This research was partly supported by the National Science Foundation (CTS 0426327).

Anuj Chauhan

University of Florida, Chemical Engineering Department

Room 237 CHE PO Box 116005 Gainesville, Florida 32611-6005

e-mail: chauhan@che.ufl.edu

##### APPENDIX

The appendix is available online at www.optvissci.com.

##### APPENDIX

Poiseuille flow of non-Newtonian shear thinning fluid (Derivation of Eq. 17).

A mass balance for fluid in a small section from x to x + dx in a canaliculus yields

The flow rate q through a cylindrical pipe of power-law fluids that obeys Eq. 16 can be written as

Eq. 17 can be obtained by combing equations (A1) and (A2) and simplifying the equation by expanding the solution for radius R(x, t) around the base state of constant radius of R_{0}, and using the lubrication approximation in which x-derivatives of R are small and so any terms that involve products of derivatives are neglected. The derivation assumes that the differences between the time and position dependent radius R(x, t) and the base radius of R_{0} are always much smaller than R_{0}. To validate this assumption, the values of R_{b} and R_{ib} given by Eqs. 4 and 5 were calculated for the parameter values given in Table 1 and values of 2.57 Pa.m for bE and 0.365 mm for R_{m}. The calculated values for R_{b} and R_{ib} are 0.96R_{0} and 0.99R_{0}. The radius at any time must lie in between these two limits. This proves the validity of the assumption that the deviations between R and R_{0} are small and so the solution for the radius can be expanded around the base state of constant radius of R_{0}.

Equation 5 Image Tools |
Equation 6 Image Tools |
Equation 22 Image Tools |

Equation 23 Image Tools |

The appendix is available online at www.optvissci.com. Cited Here...