1040-5488/08/8508-0715/0 VOL. 85, NO. 8, PP. E715–E725
`OPTOMETRY AND VISION SCIENCE
`Copyright © 2008 American Academy of Optometry
`
`ORIGINAL ARTICLE
`
`Effect of Viscosity on Tear Drainage and Ocular
`Residence Time
`
`HENG ZHU, PhD, and ANUJ CHAUHAN, PhD
`University of Florida, Chemical Engineering Department, Gainesville, Florida
`
`ABSTRACT
`Purpose. An increase in residence time of dry eye medications including artificial tears will likely enhance therapeutic
`benefits. The drainage rates and the residence time of eye drops depend on the viscosity of the instilled fluids. However,
`a quantitative understanding of the dependence of drainage rates and the residence time on viscosity is lacking. The
`current study aims to develop a mathematical model for the drainage of Newtonian fluids and also for power-law
`non-Newtonian fluids of different viscosities.
`Methods. This study is an extension of our previous study on the mathematical model of tear drainage. The tear drainage
`model is modified to describe the drainage of Newtonian fluids with viscosities higher than the tear viscosity and
`power-law non-Newtonian fluids with rheological parameters obtained from fitting experimental data in literature. The
`drainage rate through canaliculi was derived from the modified drainage model and was incorporated into a tear mass
`balance to calculate the transients of total solute quantity in ocular fluids and the bioavailability of instilled drugs.
`Results. For Newtonian fluids, increasing the viscosity does not affect the drainage rate unless the viscosity exceeds a
`critical value of about 4.4 cp. The viscosity has a maximum impact on drainage rate around a value of about 100 cp. The
`trends are similar for shear thinning power law fluids. The transients of total solute quantity, and the residence time agrees
`at least qualitatively with experimental studies.
`Conclusions. A mathematical model has been developed for the drainage of Newtonian fluids and power-law fluids
`through canaliculi. The model can quantitatively explain different experimental observations on the effect of viscosity on
`the residence of instilled fluids on the ocular surface. The current study is helpful for understanding the mechanism of
`fluid drainage from the ocular surface and for improving the design of dry eye treatments.
`(Optom Vis Sci 2008;85:E715–E725)
`
`Key Words: canaliculi, model, tear balance, tear drainage, viscosity, shear-thinning
`
`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 vol-
`ume, 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 䡠 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 䡠 s from
`the precorneal surface. These experiments showed a rather inter-
`esting effect of viscosity: the retention began to increase only after
`the fluid viscosity exceeded a critical value of about 10 mPa 䡠 s and
`also the relative increase in retention became smaller at very high
`viscosities. Although increasing fluid viscosity increases the resi-
`dence 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) solu-
`tions 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
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`which the time to achieve steady state (␶
`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 New-
`tonian and non-Newtonian fluids, respectively. The models are
`used to predict the drainage rates, residence time and bioavailabil-
`ity, and these predictions are compared with the experimental
`measurements2,8 –10 reported in literature. In our mathematical
`models, the canaliculus is simplified as a straight pipe of length L
`with an undeformed radius R0, 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 ne-
`glected 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 Chauhan7 that for a Newtonian
`fluid, the time and position dependent radius of the canaliculus
`can be predicted by solving the following equation:
`⳵2R
`⳵R
`⳵x2
`⳵t
`
`bER0
`16␮
`
`⫽
`
`(1)
`
`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 con-
`ditions and the initial conditions for Eq. 1 are:
`For the blink phase (0 ⬍ t ⬍ tb)
`q ⫽ 0 f ⳵R
`(x ⫽ 0, t) ⫽ 0
`⳵x
`p (x⫽ L, t) ⫽ 0
`R (x, t ⫽ 0) ⫽ Rib
`⬍ t ⬍ tc)
`For the interblink phase (tb
`
`(2)
`
`␴ R
`
`m
`
`p (x⫽ 0, t) ⫽ ⫺
`q (x⫽ L, t) ⫽ 0
`R (x, t ⫽ tb) ⫽ Rb
`where tb is the duration of the blink phase, tc 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
`
`(3)
`
`E716 Viscosity and Tear Drainage—Zhu and Chauhan
`
`leading to reduced drainage rates and a concurrent increase in
`residence time.
`Although the mechanisms of the impact of viscosity on resi-
`dence time are qualitatively understood for both Newtonian and
`non-Newtonian fluids, no quantitative model has been yet pro-
`posed 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 vehi-
`cles. 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 mod-
`eling drainage of tears, which were considered to be Newtonian
`fluids with a viscosity of 1.5 mPa 䡠 s. Both the previous and the
`current models are based on the physiological description of can-
`alicular 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 sur-
`rounding 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 canal-
`iculi. 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 Chauhan7
`showed that for tears with a viscosity of 1.5 mPa 䡠 s, the canaliculus
`radius will reach steady states during both the blink and the inter-
`blink 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
`16␮L2
`shown to reach a steady state in a time ␶ ⫽
`, where L, b,
`␲2bER0
`and R0 are the length, thickness, and the undeformed radius of
`canaliculi, E is the elastic modulus of canaliculi, and ␮ 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
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`Viscosity and Tear Drainage—Zhu and Chauhan E717
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`smaller than that of conjunctiva. The effects of such approxima-
`tions 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
`
`d(cV)
`dt
`
`⫽ ⫺cqdrainage
`
`(10)
`
`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 qdrainage depends on viscosity, and this dependence
`leads to the dependence of dynamic concentration profiles on vis-
`cosity. The normal viscosity of tears is about 1.5 mPa 䡠 s and after
`instillation, it increases to a value ␮
`i that is close to the viscosity of
`the eye drops ␮
`drop. Immediately after instillation, due to the di-
`i will be smaller than ␮
`lution by the tears ␮
`drop, but here for
`
`i is assumed to be equal to ␮drop. Subsequently, the
`simplicity ␮
`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.,
`
`(11)
`
`(␮i ⫺ ␮tears)
`
`c c
`
`0
`
`␮ ⫽ ␮tears ⫹
`
`where c0 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
`
`qproduction ⫺ qevaporation
`V
`
`冊dt
`
`(12)
`
`exp冕
`
`t冉⫺
`
`0
`
`V V
`
`0
`
`⫽
`
`I I
`
`0
`
`where I (⫽cV) is the total quantity of solutes, and I0 and V0 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 vol-
`ume can be related to the meniscus curvature by the following
`equation11:
`
`(13)
`
`␲)Rm
`2 Llid
`
`1 4
`
`V(Rm) ⫽ Vfilm ⫹ (1 ⫺
`
`pressure inside the canaliculus, ␴ is the surface tension of the
`instilled fluids or tears, Rm is the radius of curvature of the tear
`meniscus, and Rb and Rib are the steady state canaliculus radii in
`the blink and the interblink, respectively, and these are given by the
`following expressions:
`
`(4)
`
`(5)
`
`Rb ⫽
`
`R0
`(p0 ⫺ psac)R0
`bE
`
`1 ⫹
`
`R0
`
`␴ R
`
`R0
`
`m
`bE
`
`Rib ⫽
`
`1 ⫹
`
`where p0 and psac 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:
`R ⫺ R0
`R
`
`p ⫽ p0 ⫹
`
`bE
`R0
`
`(6)
`
`The details of the above model development are described ear-
`lier 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:
`
`Vcanaliculus ⫽冕
`
`L
`
`␲R2(x)dx
`
`(7)
`
`0
`
`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 (Vinterblink) and that at the end of the blink (Vblink),
`and then the drainage rate through the canaliculus can be com-
`puted as
`
`qdrainage ⫽
`
`Vinterblink ⫺ Vblink
`tc
`
`(8)
`
`The above procedure can be used to calculate the effect of vis-
`cosity on tear drainage. To determine the effect of tear viscosity on
`the residence time of eye drops, the tear drainage rates are incor-
`porated in a tear mass balance.
`
`Incorporation of Tear Drainage into Tear Balance
`A mass balance for the fluids on the ocular surface yields
`
`dV
`dt
`
`⫽ qproduction ⫺ qevaporation ⫺ qdrainage
`
`(9)
`
`where V is the total volume of the fluids on the ocular surface,
`including tears and the instilled fluids, qproduction is the combined
`tear production rate from the lacrimal gland and conjunctiva se-
`cretion, qevaporation 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
`
`where the second term accounts for the fluid in the meniscus and
`Vfilm accounts for the remaining tear volume, i.e., fluid in the
`exposed and the unexposed tear film. In the above equation Llid is
`the perimeter of the lid margin. The volume Vfilm depends on the
`tear film thickness (h), which in turn is related to viscosity through
`h ⫽ 2.12Rm(␮U/␴)2/3, where U is the velocity of the upper lid;
`and ␴ is the tear surface tension.12 The relationship yields unreal-
`istically large value of film thickness for large viscosities and there-
`fore is likely invalid. Therefore in this study Vfilm is assumed to be
`independent of viscosity and based on our earlier calculations11
`and the measurements by other researchers,13 its value is fixed at
`5.37 ␮l. This yields a total normal tear volume of 7 ␮l for a
`
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`E718 Viscosity and Tear Drainage—Zhu and Chauhan
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`meniscus radius value of 0.365 mm.14 It is noted that the instilla-
`tion 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.
`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
`
`d(cV)
`dt
`
`⫽ ⫺c共KconjAconj ⫹ KcorneaAcornea兲 ⫺ cqdrainage
`
`(14)
`
`where the constants Kcornea and Kconj are the permeabilities of
`cornea and conjunctiva to timolol, respectively, and Acornea and
`Aconj are the areas of cornea and conjunctiva, respectively. By com-
`bining the above mass balance with Eq. 9, bioavailability (␤), i.e.,
`the fraction of the instilled drug that permeates into cornea can be
`calculated as
`
`␤ ⫽
`
`KcorneaAcornea
`
`exp冋⫺冕
`V0冕
`
`⬁
`
`0
`
`0
`
`␶(KconjAconj ⫹ KcorneaAcornea) ⫹ (qproduction ⫺ qevaporation)
`V(t)
`
`dt册d␶
`
`(15)
`
`where V0 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 sys-
`temic circulation. The derivation and the details of Eq. 15 are given
`elsewhere.11
`
`relation between the shear stress ␶ and the shear rate ␥ can be
`written as
`
`␶ ⫽ K␥n
`(16)
`where ⌲ and n are rheological parameters that can be obtained by
`fitting the viscosity vs. shear rate data. Here, the constant ⌲ is
`assumed to be related linearly to the instantaneous polymer con-
`centration 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
`
`⫽ a冉⫺
`
`⳵R
`⳵t
`
`冊冉 1
`
`n
`
`⫺ 1冊 ⳵2R
`⳵ x2
`
`⳵R
`⳵ x
`
`(17)
`
`(18)
`
`where a is a constant that is defined as
`
`冉 bE
`
`2KR0
`
`2冊 1
`
`2n ⫹ 1
`n R0
`n
`3n ⫹ 1
`
`a ⬅ 1
`2
`
`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 quan-
`tity 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 ⌲ 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, “Rfitting”
`represents the correlation coefficient.
`
`TABLE 1.
`Physiological parameters used for the model
`
`Parameter
`
`Value
`
`Source (reference #)
`
`15
`7
`6
`16
`17
`18
`7
`19
`20
`21
`21
`22
`22
`
`1.2 ⫻ 10⫺2 m
`6 s
`0.04 s
`2.5 ⫻ 10⫺4 m
`43.0 ⫻ 10⫺3 N/m
`400 Pa
`0 (atmospheric)
`1.5 ⫻ 10⫺3 Pa 䡠 s
`57 ⫻ 10⫺3 m
`1.5 ⫻ 10⫺7 m/s
`5.2 ⫻ 10⫺7 m/s
`1.04 ⫻ 10⫺4 m2
`17.65 ⫻ 10⫺4 m2
`
`L
`tc
`tb
`R0
`␴
`p0
`psac
`ear
`Llid
`Kcornea
`Kconj
`Acornea
`Aconj
`
`␮t
`
`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 treat-
`ment 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
`
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`TABLE 2.
`Rheological parameters of typical fluids used in ocular
`studies. The parameters were obtained by fitting the
`rheological data in literature to the constitutive equation
`for a power law fluid, i.e., ␮ ⫽ ⌲␥n⫺1
`
`Solution
`
`⌲ (mPa 䡠 sn)
`
`n ⫺ 1
`
`0.2% NaHA2
`0.3% NaHA2
`CMC (low MW)10
`CMC (high MW)10
`Human tears19
`
`323.3
`884.2
`194.8
`194.4
`5.578
`
`⫺0.329
`⫺0.3913
`⫺0.09201
`⫺0.2943
`⫺0.264
`
`Rfitting
`
`2
`
`0.973
`0.9788
`0.9285
`0.933
`0.9963
`
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`Viscosity and Tear Drainage—Zhu and Chauhan E719
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`FIGURE 2.
`The effect of rheological parameters (⌲ and n) on the drainage rate
`through canaliculi for power-law shear thinning fluids.
`
`
`
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`FIGURE 1.
`The effect of viscosity on the drainage rate through canaliculi
`Newtonian fluids.
`
`for
`
`RESULTS
`Effect of Viscosity on Tear Drainage
`The effect of viscosity on tear drainage rate qdrainage immediately
`after instilling 25 ␮l of fluids is shown in Fig. 1 for a Newtonian
`fluid. The drainage rates for shear-thinning fluids depend on ⌲
`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 ␮l.
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`FIGURE 3.
`The transients of solute quantity (I) after the instillation of Newtonian fluids
`with different viscosities.
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`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.
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`Similar data is compared with experiments in Fig. 4, 5. It is noted
`that for fluids with viscosities lower than 4.4 mPa 䡠 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 ␮l.
`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 ⌲ 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
`
`Optometry and Vision Science, Vol. 85, No. 8, August 2008
`
`Eye Therapies Exhibit 2036, 5 of 11
`Slayback v. Eye Therapies - IPR2022-00142
`
`

`

`E720 Viscosity and Tear Drainage—Zhu and Chauhan
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`FIGURE 4.
`Comparison of the predicted (solid lines) and experimental transients
`(overlapping dashed lines) of solute quantity on the ocular surface for
`0.3% HPMC and 1.4% PVA.
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`FIGURE 5.
`Comparison of the predicted (solid lines) and experimental (dashed lines)
`transients of precorneal solute quantity for 0.3% HPMC.
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`lines after rescaling. The details of the rescaling will be given in the
`discussion section.
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`Bioavailability of Instilled Drugs
`The bioavailability of timolol instilled as an eye drop is calcu-
`lated 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 tb (⬃0.04 s), the time of the inter-
`blink period tib (⬃6 s) and the time scale for achieving steady
`canaliculus deformation ␶. Based on the relative magnitude of ␶
`with respect to tb and tib, there are three different scenarios for
`the tear drainage. When the viscosity is ⬍4.4 mPa 䡠 s, ␶
`⬍ tb
`s
`and ␶
`⬍ tib and so the canaliculus radius can reach steady state
`s
`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 䡠 s but smaller than
`⬍ ␶
`⬍ tib, and so the canaliculus radius cannot
`654 mPa 䡠 s, tb
`s
`reach steady state during the blink phase but still can reach
`steady state during the interblink phase. In this case the canal-
`iculus 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 䡠 s, ␶
`⬎ tib
`⬎ tb, and the canaliculus radius cannot
`s
`reach steady state during both in the blink phase and the inter-
`blink 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 䡠 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 mea-
`sured, it is not possible to compare the predictions with experi-
`ments. It is noted that the data shown in Fig. 1 has three distinct
`regions. In the first region (␮ ⬍4.4 mPa 䡠 s), there is no effect of
`viscosity on drainage rates. In the second region (4.4 ⬍ ␮ ⬍100
`mPa 䡠 s), the viscosity has the maximum impact on the drainage. In
`the last region (␮ ⬎100 mPa 䡠 s), the effect of viscosity on drainage
`rates becomes small. These trends qualitatively agree with observa-
`tions 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 䡠 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 ⌲ because of the smaller viscosities. The
`data in Fig. 2 suggests that the best range of ⌲ for non-Newtonian
`eye drops should be below 400 mPa 䡠 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 resi-
`dence 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
`
`Optometry and Vision Science, Vol. 85, No. 8, August 2008
`
`Eye Therapies Exhibit 2036, 6 of 11
`Slayback v. Eye Therapies - IPR2022-00142
`
`

`

`Viscosity and Tear Drainage—Zhu and Chauhan E721
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`FIGURE 6.
`Comparison of model predictions (solid lines) with experiments (dash lines) for the transients of solute quantity (I) after the instillation of 0.2% and 0.3%
`sodium hyaluronate for (a) bE ⫽ 2.57 Pa 䡠 m, (b) bE ⫽ 1.29 Pa 䡠 m and (c) bE ⫽ 0.64 Pa 䡠 m.
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`TABLE 3.
`Predicted timolol bioavailability for different fluids as
`drug vehicles
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`Fluid
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`Bioavailability (%)
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`below 4.4 mPa 䡠 s Newtonian
`10 mPa 䡠 s Newtonian
`40 mPa 䡠 s Newtonian
`70 mPa 䡠 s Newtonian
`100 mPa 䡠 s Newtonian
`0.2% NaHA
`0.3% NaHA
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`1.16
`1.16
`1.18
`1.19
`1.20
`1.20
`1.21
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`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 䡠 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 䡠 m, are used.
`
`Optometry and Vision Science, Vol. 85, No. 8, August 2008
`
`Eye Therapies Exhibit 2036, 7 of 11
`Slayback v. Eye Therapies - IPR2022-00142
`
`

`

`imental 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 differ-
`ences between the two studies.
`For non-Newtonian fluids, the model predictions can be
`compared to experiments for 0.2% NaHA, 0.3% NaHA con-
`ducted by Snibson et al.2 The predicted transient profiles for
`these two cases are plotted in solid lines Fig. 6 after rescaling
`⬘ ⫽0.80 I 0 and Ir
`⫽ 0.30 I0, together with
`using Eq. 19 with I0
`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 solu-
`tions, 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 resi-
`dence 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