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.
There are three important time scales relevant to the drainage process; the duration of the blink phase, which is the duration of canaliculus compression tb (∼0.04 s), the time of the interblink period tib (∼6 s) and the time scale for achieving steady canaliculus deformation τ. Based on the relative magnitude of τ with respect to tb and tib, there are three different scenarios for the tear drainage. When the viscosity is <4.4 mPa · s, τs < tb and τs < tib 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 Vinterblink and Vblink can be calculated from the steady state radii, and changing viscosity within this range will not change the drainage rate qdrainage. When the viscosity is larger than 4.4 mPa · s but smaller than 654 mPa · s, tb < τs < tib, 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 Vblink 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, τs > tib > tb, 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 · sn. It has been reported that some polymers in such non-Newtonian eye drops have the ability to bind to the mucous ocular surface23 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 I0 to I0 ’ immediately due to the initial mixing, and the amount of residual quantity is Ir, 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 I0 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 (99Tcm), 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)/I0 for the two fluids almost overlap (dashed curve) and these are plotted in Fig. 4 along with the scaled theoretical profiles I′(t)/I0 with I0′ = 0.6 I0 and Ir = 0.13 I0 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 I0′ = 0.4 I0 and Ir = 0.122 I0 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 I0′ = 0.80 I0 and Ir = 0.30 I0, 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 model7 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 · sn, 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 τabsorption = V/(KconjAconj) 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.
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.
This research was partly supported by the National Science Foundation (CTS 0426327).
University of Florida, Chemical Engineering Department
Room 237 CHE PO Box 116005 Gainesville, Florida 32611-6005
The appendix is available online at www.optvissci.com.
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 R0, 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 R0 are always much smaller than R0. To validate this assumption, the values of Rb and Rib 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 Rm. The calculated values for Rb and Rib are 0.96R0 and 0.99R0. 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 R0 are small and so the solution for the radius can be expanded around the base state of constant radius of R0.
The appendix is available online at www.optvissci.com.
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Keywords:© 2008 American Academy of Optometry
canaliculi; model; tear balance; tear drainage; viscosity; shear-thinning