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1 TEAR DYNAMICS By HENG ZHU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007
2 Heng Zhu
3 To my parents, Quanshou Zhu and Yimei Zhang, and my wife, Jia Cai
4 ACKNOWLEDGMENTS First I thank my advisor, Professor Anuj Chauhan, for his guidance, inspiration and patience. His guidance on my research and on ot her aspects of life has been essential to the completion of my degree and will benefit me th roughout my life. From the beginning through the end of my PhD study, he has also offered generous help on preparing me for my future career. I thank my committee members, Professors Ra nga Narayanan, Richard Dickinson and Roger Tran-Son-Tay, for their valuable discussion and advice on my research. Especially I thank Dr. Subir Bhatia for his time and generous help on canaliculus anatomy and basic eye physiology. I also thank my group colleagues, Derya Gulse n, Marissa Fallon, Zhi Chen, Chi-Chung Li, Yash Kapoor, Jinah Kim, Brett Howell and Chhavi Gupta, as well as all the undergrad students in my group, for their enjoyable company and help. I al so thank other faculties and the staff in Chemical Engineering Department for their advice and help. Finally, I thank my parents for their unc onditional love, encouragement and support through all these years. I also give my gratitude to my wife, who made great sacrifice for my study abroad and stands behind me through difficultie s. Last but not the l east I am thankful to my other family members for their support and help.
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........9 LIST OF FIGURES................................................................................................................ .......10 ABSTRACT....................................................................................................................... ............12 CHAPTER 1 INTRODUCTION................................................................................................................. .14 Tear Physiology and Tear Balance.........................................................................................14 Tear Drainage.................................................................................................................. .......18 Tear Drainage Mechanism..............................................................................................18 Mechanical Properties of Lacrimal Canaliculi................................................................20 Conjunctiva Epithelium Transport of Solutes and Ions..........................................................21 Effect of Viscosity on Tear Drai nage and Ocular Residence Time........................................23 Tear Mixing Under the Lower Eyelid....................................................................................25 Application of the Tear Dynamics Model..............................................................................28 2 MATERIALS AND METHODS...........................................................................................34 Tear Drainage Model............................................................................................................ ..34 Model Development........................................................................................................34 Mechanical Properties of Lacrimal Canaliculi................................................................38 Tissue Samples................................................................................................................3 8 Dynamic mechanical an alyzer (DMA) setup...........................................................39 Frequency dependent rh eological response..............................................................39 Simulation of blink-interblink cycles.......................................................................40 Statistical methods....................................................................................................41 Conjunctiva Epithelium Transport Model..............................................................................41 Incorporation of the Tear Drainage Model and the Conjunctiva Transport Model into the Tear Balance............................................................................................................... ...42 General Tear Balance Model...........................................................................................42 Modified Model for an Ussing Chamber.........................................................................45 Algorithm...................................................................................................................... ..47 Model Parameters and Initial Conditions...............................................................................47 Application of the Tear Dynamics Model..............................................................................48 Tear Film at Steady State................................................................................................48 Tear Film at Transient States...........................................................................................49 Balance of tear fluid.................................................................................................49 Balance of solutes.....................................................................................................50 Effect of Viscosity on Tear Drai nage and Ocular Residence Time........................................53
6 Drainage of a Ne wtonian Fluid.......................................................................................53 Incorporation of Tear Drai nage into Tear Balance..........................................................54 Non-Newtonian Fluid......................................................................................................56 Tear Mixing Under the Lower Eyelid....................................................................................57 Physical Description of the System.................................................................................57 Mathematical Modeling...................................................................................................58 Velocity profiles.......................................................................................................59 Mass balance of the tear film...................................................................................60 3 RESULTS...................................................................................................................... .........72 Tear Drainage through Canaliculi..........................................................................................72 Mechanical Properties of the Lacrimal Canaliculi..........................................................72 Frequency dependent rh eological response..............................................................72 Simulation of blink-interblink cycles.......................................................................73 Canaliculus Radius and Pressure Transients...................................................................73 Conjunctiva Transport Model for the Simulation of a Ussing-chamber................................74 Cellular Concentrations and Cell Size.............................................................................74 Short-circuit Current ISC..................................................................................................75 Sodium (Na) and Chloride Cl Fluxes..............................................................................76 Transepithelial Potential Difference (PD).......................................................................76 Fluid Secretion under Open-circuit Condition................................................................76 Incorporation of the Tear Drainage Model and the Conjunctiva Transport Model into the Tear Balance............................................................................................................... ...77 Model Prediction of SteadyState Tear Variables...........................................................77 Normal tear parameters............................................................................................77 Steady-state tear film thickness as a function of tear evaporation rate....................77 Steady-state tear film thickness as a function of mechanical properties of canaliculi...............................................................................................................78 Steady-state tear film thickness as a function of tear surface tension......................78 Model Prediction of Dynamic Tear Variables.................................................................78 The effect of ion channel modulation on tear film...................................................78 The effect of evapora tion rates on tear film.............................................................79 The effect of osmolarity in dry eye medications on dry eye....................................79 The effect of punctum occlusion an d moisture chambers on dry eye......................80 The effect of drop volume on clearance time...........................................................81 Bioavailability of drugs delivered by drops with low viscosity...............................81 Effect of Viscosity on Tear Drai nage and Ocular Residence Time........................................81 Effect of Viscosity on Tear Drainage..............................................................................81 Effect of Viscosity on Residenc e Time of Instilled Fluids..............................................82 Bioavailability of Instilled Drugs in Hi gh Viscosity and Non-Newtonian Vehicles......82 Tear Mixing Under the Lower Eyelid....................................................................................83 Concentration Transients and Mixing Time....................................................................83 Effect of Instillation Volume on Tear Mixing Time.......................................................84 4 DISCUSSION................................................................................................................... ....119
7 Mechanical Properties of Po rcine Lacrimal Canaliculi........................................................119 Validation of the Tear Drainage Model................................................................................122 Steady State Assumption...............................................................................................122 Comparison of the Predicted Pressure Changes with Literature Results......................123 Comparison of the Predicted Drainage Rates with Literature Results..........................125 Relationship Between Tear Film Thickness a nd the Meniscus Radius of Curvature...127 Validation of the Conjunctiva Transport M odel by the Simulation of Ussing-chamber experiments.................................................................................................................... ...128 Application of the Tear Dynamics Model............................................................................130 The Effect of Surface Tension on the Tear Film Thickness..........................................130 Comparison of Predicted Residence Time of Instilled Fluid and Solutes with Experiments...............................................................................................................130 Overview of the residence time experiments.........................................................131 Comparison with experiments................................................................................132 Values of Ocular Bioavailability and the Effect of Drop Size on Ocular Bioavailability............................................................................................................134 The Effect of Ion Channel Modulation.........................................................................135 The Effect of Evaporation.............................................................................................136 The Effect of Osmolarity in Dry Eye Medications.......................................................137 The Effect of Punctum Occlusion and Moisture Chambers..........................................138 Implication on Basic Tear Physiology..........................................................................139 Can conjunctiva secretion account for all the normal tear secretion?....................139 Water transport mechanism....................................................................................139 Effect of Viscosity on Tear Drai nage and Ocular Residence Time......................................140 Drainage Rates...............................................................................................................14 0 Comparison of Residence Time for Ne wtonian or Non-Newtonian Fluids..................142 The Effect of Viscosity on Bioavailability....................................................................146 Drainage of Tears..........................................................................................................148 Tear Mixing Under the Lower Eyelid..................................................................................148 Comparison with Experiments......................................................................................148 Horizontal Shearing.......................................................................................................150 Implications on the Mathemati cal Model of Tear Dynamics........................................151 Implication of Tear Mixing for Drug Application........................................................151 5 CONCLUSIONS.................................................................................................................. 160 APPENDIX A DERIVATION OF EQUATIONS for TEAR DRAINAGE.................................................162 Pressure-Radius Relationship...............................................................................................162 Determination of Rb and Rib.................................................................................................162 Derivation of the Equation for the Drainage of Newtonian Fluids......................................163 Derivation of the Equation for the Dr ainage of Non-Newtonian Fluids..............................164 B RELATIONSHIP BETWEEN THE ME NISCUS CURVATURE AND TOTAL OCULAR FLUID VOLUME...............................................................................................165
8 C DERIVATIONS IN THE CONJ UNCTIVA TRANSPORT MODEL.................................168 Flux Equations................................................................................................................. .....168 Kinetic Parameters............................................................................................................. ...171 D DERIVATION OF THE TAYLOR DISPERSION COEFFICIENT D*..............................173 LIST OF REFERENCES............................................................................................................. 176 BIOGRAPHICAL SKETCH.......................................................................................................186
9 LIST OF TABLES Table page 2-1 Model parameters based on literature................................................................................65 2-2 Model parameters obtained by fittin g data from experiments on rabbits..........................67 2-3 Parameters obtained from fitting the rheolo gical data in literature using the powerlaw equation = 0n...........................................................................................................68 3-1 The storage moduli and loss moduli of porcine lacrimal canaliculi..................................85 3-2 Parameter values obtained by fitting the strain recovery to a double exponential............86 3-3 Comparison of cellular concentrations for Ussing-chamber simulation............................87 3-4 Comparison of steady state ISC after different maneuvers.................................................88 3-5 Comparison of t1/2 of ISC after different maneuvers...........................................................89 3-6 Comparison of net Na and Cl fluxes..................................................................................90 3-7 Comparison of the apical composition and volume...........................................................91 3-8 Bioavailability for instilled Timolol using different vehicles............................................92 3-9 The predicted Tapp and Tmax for shearing...........................................................................93 3-10 The predicted Tapp and Tmax for squeezing.........................................................................94 4-1 Decays of tear volume, concentration and solute quantity after instillation....................158 4-2 Predicted ocular bioavailabi lity for different drop volumes............................................159
10 LIST OF FIGURES Figure page 1-1 The structure of the tear film............................................................................................. 29 1-2 Tear production and elimination pathways........................................................................30 1-3 The lacrimal canalicul us during the blink phase................................................................31 1-4 The lacrimal canaliculus during the in terblink phase........................................................32 1-5 Taylor dispersion due to sh earing. A) A pulse of solute is introduced at t=0. B) The solutes are transported laterally due to c onvection. C) Meanwhile the solutes diffuse transversely. D) The pulse widens due to the combination of convection and transverse diffusion........................................................................................................... .33 2-1 DMA clamp setup............................................................................................................ ..69 2-2 Simplified epithelium structure and transport mechanisms...............................................70 2-3 Spreading of instilled dye due to shearing and squeezing motion of the lid and the globe.......................................................................................................................... .........71 3-1 Stress and strain transients during loading-recovery cycles..............................................95 3-2 Sample data for one recovery cycle along with the best fit dou ble-exponential curve.....96 3-3 Radius-axial position profiles during the blink phase at four different times....................97 3-4 Radius-axial position profiles during the in terblink phase at four different times............98 3-5 Pressure-time transients during the blink phase at thr ee different location along the canaliculus.................................................................................................................... ......99 3-6 Pressure-time transients during the interblink phase at three different locations along the canaliculus................................................................................................................ ..100 3-7 Time course of ISC in Ussing-chamber experiments........................................................101 3-8 The effect of tear evaporation and absorption on tear film thickness..............................102 3-9 The effect of canaliculus properties on tear film thickness.............................................103 3-10 The effect of surface tens ion on tear film thickness........................................................104 3-11 The effect of channel m odulation on tear volume...........................................................105 3-12 The effect of evaporation on tear volume osmolarity and conjunctiva secretion rate....106
11 3-13 The effect of isosmolar and anisosmolar (osmolarity 40 mM) fluid instillation on A) Tear volume. B) Tear osmolar ity. C) Conjunctiva secretion rate...............................108 3-14 The effect of punctum o cclusion on fluorescence clearan ce with different rates............109 3-15 Clearance of instilled drops of is osmolar fluid with different volumes...........................110 3-16 The effect of viscosity on the drainage rate through canaliculi for Newtonian fluids.....111 3-17 The effect of viscosity ( 0) and the exponential parameter (n) on the drainage rate through canaliculi for power-law fluids...........................................................................112 3-18 The transients of solute quantity (I) af ter the instillation Newtonian fluids with different viscosities.......................................................................................................... 113 3-19 The transients of solute quantity (I) after the instillation 0.2% and 0.3% sodium hyaluronate.................................................................................................................... ...114 3-20 Concentration of fluorescence in the exposed tear film after instil lation into the lower fornix for shearing amplitude of 1-5 degrees...................................................................115 3-21 Concentration of fluorescence in the exposed tear film after instil lation into the lower fornix for squeezing amplitude of 1-5 m.......................................................................116 3-22 The predicted Tapp and Tmax for shearing for different blink cycle duration (Tc)............117 3-23 The predicted Tapp and Tmax for squeezing for different blink cycle duration (Tc)..........118 4-1 Pressure and strain transients obtained by previous experiment and model, and the current study.................................................................................................................. ...153 4-2 Dependence of time scale and tear drainage rate on bE..................................................154 4-3 Effect of punctum o cclusion on tear volume...................................................................155 4-4 Comparison of the predicted and experime ntal transients of ocular surface solute quantity for 0.3% HPMC and 1.4% PVA........................................................................156 4-5 Comparison of the predicted and experi mental transients of precorneal solute quantity for 0.3% HPMC.................................................................................................157 A-1 Simplified meniscus cross-sectional area geometry........................................................167
12 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy TEAR DYNAMICS By Heng Zhu August 2007 Chair: Anuj Chauhan Major: Chemical Engineering The quantity and quality of tear fluid in an eye under normal circumstances is an important factor in maintenance of comfort and health of oc ular tissue. It is thus important to assess the effect of various physiological parameters on th e tear volume and composition, and to determine the factors that can increase them. We developed mathematical models for various aspects of tear dynamics such as canalicular drainage, conjuncti val absorption/secretion, and the tear mixing under lower eyelids. We performed a tear mass balance on an ey e. The mass balance requires mathematical models for the flows from or into the tear film such as canalicular drainage, evaporation and transport across cornea and conjunctiva. The tear drainage model developed is based on the lacrimal pump mechanism, according to which the muscle action during a blink drives the tear drainage. We also developed a mathematical mo del for the transport of ions and water through the conjunctival epithelium. The drainage and co njunctiva transport models were incorporated into a tear mass balance, which can predict the steady state tear film thickness and the residence time of fluids and drugs after dr op instillation. The tear balance is based on the assumption that tears on the ocular surfa ce is well mixed, which is supported by our mathematical model of tear mixing under lower lids based on Taylor disp ersion mechanism. In addition, models are
13 developed for the drainage of both high viscos ity Newtonian and non-Ne wtonian fluids through canaliculi, and these models are us ed to study the effect of fluid vi scosity on its drainage rate and ocular residence time. We found that the mechanical properties of th e lacrimal canaliculi, which are responsible for tear drainage, are essential to determining th e tear drainage rates. The mechanical properties of porcine canaliculus were measured in compre ssion mode while submerged in isotonic fluid by using a dynamic mechanical analyz er (DMA). Both storage modul us (E) and loss modulus (E) were measured at physiologically relevant freque ncies. When subjected to repeated loadingrecovery cycles to simulate the blink cycles, th e canalicular tissue relaxes at the same time scale as the pressure relaxes in a human canaliculi suggesting that the drainage of tears through the canaliculi is controlled by the mechanical properties of the canaliculi. The model predictions agree w ith various physiologi cal experiments. The model also helps resolve the differences between various tear dr ainage experiments and can be used to design more effective dry eye treatments and also more efficacious ophthalmic dr ug delivery vehicles.
14 CHAPTER 1 INTRODUCTION Tear Physiology and Tear Balance A number of eye-related issues such as dry eye syndrome, oc ular drug kinetics and contact lens fitting are related to the composition and th e quantity of tears in th e precorneal tear film. The preocular tear film that lies on the surface of the cornea is very important to ocular health. Its smooth surface is necessary for vision and it provides lubrication betw een eyelids and ocular surface and also helps in imm unization of the corneal tissue.1 The preocular tears can be divided into three compartments: the precorneal tear film, the conjunctival sac and the tear menisci containing a majority of the tears.2 In each blink, the tears in the three compartments are mixed and re-distributed.3 The tear film is composed of thr ee layers ( Figure 11). Contacting the corneal epithelium is a hydrophilic mu cus layer approximately 0.02 to 0.05 m in thickness. 4 As the underlying cornea itself is extremely hydr ophobic, this layer is believed to provide a wetting substrate for the aqueous laye r above it, which is about 6 to 9 m thick.4, 5, 6, 7 Outside the aqueous layer is a 0.1 to 0.2 m thick lipid layer that reduces tear evaporation.4, 8 In my study I focus mainly on the aqueous layer of the tear film, which consists most of the tear film volume. Tears function cooperatively with the ocular surface epithelium to maintain the health of ocular surface and the abnormal quantity or qua lity of tears may result in ocular disease.9 Additionally, the thickness of the tear film is a key indicator of ocular problems like the dry eyes. Thus, it is important to unders tand and quantify the effect of various physiological and anatomical features on tear volume and tear film thickness. It is clear that the balance between the production and elimination of tears determin es the quantity of preo cular tears, and an increase in the production or a reduction in elim ination will increase tear volume and tear film thickness. It is also well known that an increase in tear eliminati on rates can result in a reduction
15 in tear volume. For example, it is known that a reduction in tear produc tion or an increase in evaporation due to destruction of the lipid layer covering the tear film results in thinner tear films.1 However, there is no mathematical model in literature that relates the tear volume and tears film thickness to tear production and elim ination rates. Quantitative assessment of the factors that control the tear film thickness has not received much attention in the literature. To investigate such issues, we analyzed the main contributors to the produc tion and elimination of aqueous tears, which are the lacr imal glands, lacrimal canaliculi, the ocular surface epithelia and evaporation. Tears are produced mainly by the secretion of various glands and are eliminated by evaporation from mainly the pr ecorneal tear film, the drai nage through the canaliculi and possibly by absorption through the epithe lia of the conjunc tiva (Figure 1-2).10, 11 Each of these routes of tear elimination has been extensively studied experimentally. Lacrimal glands secrete about 1-4 L/min and account for most of the tear production, 12,13 active drainage through the lacrimal canaliculi is re sponsible for about 60% of the tear elimination,10 and evaporation rate is about 0.10 L/min.14 The ocular surface epithelium, most of which is conjunctival epithelium, can actively transport solutes and water from and in to the preocular tears. The secretion rate of fluid through rabbit conjunctiva is measured to be comparable with the tear turnover rate of rabbits,15 and therefore epithelium secretion might also be important for the tear dynamics. However, the mechanisms that relate the tear pr oduction and elimination to the total tear volume have not been addressed at a quantitative level. A mathematical tear balance model requires quantitative models for tear production and tear eliminati on (drainage, evaporation and absorption), which requires the detailed understanding of the physiology of these processes. It is believed that the tear evaporat ion is a passive process, wh ile the production, absorption and
16 drainage are active processes. Due to the relati vely simple mechanism, the theory for tear evaporation is relatively well established16 while the quantitative theore tical approaches for other routes of tear circulation remain limited. The production of tears is regulated by hormonal and neural control, and the mechanisms of the regulation processes are not yet completely understood, thus a mathematical descripti on of the tear production is difficult.17 Tear absorption is an active process involving th e active transport of sodium, chlo ride and potassium ions as well as water, glucose etc, and these mechanisms are also not fully understood.18 The mechanism of tear drainage is still not completely cl ear in spite of hundred s of years of study.19 In addition to experimental studies, there ha ve also been theoretical studies on certain aspects of the tear dynamics. Wong et al.20 showed that the tear film thickness can be related to the radius of curvature of the meniscus. This re sult only shows that a re lationship exists between the meniscus curvature and the tear film thic kness and Wongs model cannot predict either one of them. Based on previous experimental and theo retical work, Levin et al developed an ocular surface transport model in which they treated the conjunctival epithelium and the corneal epithelium as a single layer of uni formly distributed cells. The pr edictions of this model agreed well with the experimental data for the transepithelial potential difference.21 The theoretical work cited above cannot be used to predict the effect of various physiological parameters on tear volume and composition because these studies only modeled individual processes relevant to tear dynamics, such as ev aporation, and th e corneal and conjunctival epiehtlium transport. Therefore, we hope to develop a comprehensive model that can relate various physiologi cal parameters to the key variables of the tear film such as the tear volume and tear thickness. The proposed model is a dynamic tear balance in the eye, which states that the rate of accumulation of tears in the eyes is the difference between the rates of
17 inflow (lacrimal gland secretion) and the outflow (evaporation a nd drainage through the canaliculi). For the lacrimal dr ainage through the canaliculi, (Fi gure 1-2) we first developed a mathematical model22 that was based on the active drai nage mechanism suggested by Doane,23 which is described below. Then we develope d a mathematical model for the ion and water transport through ocular epithelia, which incl udes corneal and conjunctival epithelia. The mechanisms of transport across corneal and conjunctival epithelia involve both active and passive transport of solutes. Due to the larg e surface area of the conjunc tiva, the ion and water transport through the conjunctiva is expected to play a more impor tant role in the tear dynamics. 15, 24 Even though transport across corneal ep ithelium has been studied and modeled25, it was decided to neglect the corneal epithelium in this pa rt of the study partly because of the small area and permeabilities of cornea, and partly to avoid making the mode l too complex. It should be pointed out that the corneal transport of solute s and water may have considerable effects on the pre-corneal tear film, as during the interblink, th e thinned tear film near the tear meniscus regions (black lines) may impede the exchange of solutes and water between the pre-corneal tear film and other regions of tear film. This issu e is neglected in the current model, but can be explored later. Because the lacrimal gland secr etion mechanism is relatively complicated, in my study we assume that the secretion rate of the la crimal gland is constant. A mathematical model of the ion and water secretion by the lacrimal gland can be devel oped in the future work, and it can be easily incorporated into th e tear dynamics model. The tear evaporation rate is assumed to be constant in this study, and the effect of its va riation on the tear film is explored. Based on the drainage mechanism described above, we also de veloped a mathematical model of the drainage of Newtonian fluids with viscosity higher than tear viscosity as well as non-Newtonian fluids
18 through canaliculi. In addition, we also developed a mathematical model of the mixing of tears under the lower eye lid based on Ta ylor dispersion mechanism. Tear Drainage Tear Drainage Mechanism It is estimated that tear drainage contri butes to more than 50% of the normal tear elimination,10 and thus investigation of tear drainage mechanism is valuable for understanding the factors that may lead to tear depletion in the eyes. There have been a number of experimental studies on tear drainage but no theoretical model has been developed for the process of tear drainage through the canaliculi. We first develop a mathematical model to predict the tear draina ge rates from the canaliculi. Th e model can serve to validate the mechanisms proposed in the literature and can al so be used to estimate the effect of various parameters on tear drainage rates. It is widely accepted that the tear drainage th rough the canaliculi is an active process, in which blinking plays an important role. Based on anatomical studies, J ones proposed that during a blink the canaliculi are shortened and compresse d by pretarsal muscles, and the lacrimal sac expands, leading to tear flow from the canaliculi to the sac. The cyclic actions of the canaliculi and the lacrimal sac together result in tear drainage.26 Rosengren inserted catheters into the canaliculi to measure the pressure during the blink and the interblink27 and Maurice and Wright10 used dark particles as visual tracers to observe the flow of tears in the eyes. Based on the direct visualization, they concluded that most of the tears in the meniscus flow along the lower meniscus, and go into the lower punctum. This observation was later supported by the studies by Nagashima et al.28 In addition, Rosengren concluded that there must be a valve mechanism between the canaliculi and the la crimal sac during the interblin k, which was later confirmed by studies that used radioactive tracers.29 Both Maurice and Rosengren also concluded that the
19 canaliculi are compressed during a blink and that the tears are squeezed from the canaliculi into the lacrimal sac,10, 27 which agrees with the high-speed photography observation of Doane.23 However the physiological studies of Rosengren27 also show that the pre ssure in the lacrimal sac increases during blinking, which contradicts the finding of Jones.30 The main difference between the conclusions of the studies listed above concer ns the role of the lacr imal sac in drainage. Because it was reported that tear drainage was not greatly affected even with a disabled lacrimal sac,27 it is reasonable to assume the lacrimal sac does not play any important role in the tear drainage. Our mathematical model is based on the drainage mechanisms proposed by Doane, which are described in detail below. At the beginning of the lid-cl osing period, the canaliculi are f illed with tears and the lids begin to move towards each other. Because the punc ta are at the medial end of the lids, they meet tightly and are closed when the lids are 1/3 to 1/2 closed. During the rest of the lid-closing period, the canaliculi are compressed by the musc les, and with closed puncta, the tears are squeezed out of the canaliculi a nd into the lacrimal sac. At the beginning of the lid-opening period, during which the lids are moving apart fr om each other, the canaliculi are no longer compressed. The puncta continue to be firmly attached until the lids are about half-open, and during this time the pressure inside the canalicul i drops as the force on th e canaliculi is removed and the elastic canaliculus wall tends to expand. Then the puncta burst open and the tears in the tear menisci are drawn into th e puncta by the pressure differen ce between the canaliculi and the menisci. This blink-interblink cycle acts as a pu mp, which drives the drainage of tears (Figure 13 and 1-4).23 While there are minor discrepancies in the proposed mechanism of tear drainage, it is accepted that canaliculi play a key role in the process of active tear drainage. Therefore,
20 understanding the anatomy of the can aliculi is essential to the a ssessment of the tear drainage rate. According to anatomical studies, each canalic ulus has a vertical part that is about 2 mm long and a horizontal part that is about 10 mm long.23 The diameter of the vertical and the horizontal parts are about 0.3 mm and 0.5 mm, respect ively. The joint between these two parts is called the ampulla and its diameter can be up to 2 to 3 mm.30 Before the superior and the inferior canaliculi open into the lacrimal sac, they merge and form the common canaliculus, which has an inner radius smaller than each of the other cana liculi. The walls of the canaliculi are made of tissue that consists of elastic fibers which allo ws the canaliculi to stretch and shrink during each blink cycle, which is essential for tear drainage.31 Anatomic studies show that the Horners muscle applies contraction force on both the verti cal and the horizontal parts of the canaliculi. The force is largest for the vert ical part and it decreases along the horizontal part towards the lacrimal sac.32 It is noted that while the mechanism deta iled above is commonly accepted, several questions related to the tear dr ainage remain unanswered. For in stance, it is unclear how tears drain during sleep. Also the composition of the lacr imal part of the orbicularis oculi muscle is unknown and in particular it is not clear whether the lacrimal part of the orbicularis oculi muscle have specialized muscle spindl es like the outer eye muscles33, 34. The exact structure of the muscles will effect the distribution of the appl ied force on the canaliculi and thus impact the drainage. Mechanical Properties of Lacrimal Canaliculi According to the mechanisms proposed by Do ane, the blink-interb link cycle leads to repeated compression and expansion of the canaliculi that pushes tears from the tear film to the lacrimal sac.23 According to this mechanism, the m echanical properties of the canaliculi are expected to be a key factor to th e drainage rate of tears. Our preliminary studies showed that both
21 the rate and the time scale of t ear drainage depend on the modulu s of the canaliculus. However, there had been no reports in li terature regarding the mechani cal properties of the lacrimal canalicuil. In our preliminary studies, the modul us of the canaliculus is estimated by matching the inflow and outflow of tears under physiolo gical conditions. However by using this fitted value of the modulus, the model pr edicted a much smaller time scale for pressure relaxation in the canaliculi compared to the experiments, especi ally for the interblink pe riod. The errors in the model prediction likely arose from the assumpti on that the canaliculus is a linearly elastic material. The structure of the lacrimal canalicu li is complex and thus it is expected to be viscoelastic, and so its mechanical properties can be described using the storage modulus (E), which represents the elastic property, and the lo ss modulus (E), which represents the damping ability of the tissues. The goal of this part of the study is to measure the viscoelastic behavior of the canaliculi and compare it with the estimated va lues obtained by matching the tear inflow with tear outflow. Specifically, we measured the dependence of dynamic mechanical properties of porcine canaliculi on frequency and age by using a Dynamic M echanical Analyzer (DMA). Typically, in DMA a sinusoidal stress is applied and the resulting strain is measured, and the storage modulus and the loss modulus are calculat ed from the in-phase and out-of-phase portions of the strain, respectively. Also, the canaliculi were subjected to repe ated loading-recovery cycles to simulate blinking, and the results were compared with reported measurements on pressure relaxation in the canalic uli and also with prediction of our tear drainage model. Conjunctiva Epithelium Transport of Solutes and Ions Other than the tear drainage through cancliculi, the transpor t of water and ions through conjunctival epithelium is also believed to play an important role in tear dynamics. The conjunctival epithelium consists of three to six la yers of epithelial cells, which transport ion and water, and goblet cells, which secret e the mucus layer of the tear film.35 In this study we focus on
22 the aqueous portion of tears, and therefore on the transport of i ons and water by normal epithelial cells. The cell membrane of the epithelial ce lls contains ion cha nnels, water channels, cotransporters and pumps that transport solutes a nd water. Solutes and water are also transported through paracellular pathways. Since the movement of electric charges is involved, transport processes depend both on concentration and poten tial difference across the membranes. For the conjunctival epithelium, the apical side contacts tears and the basolate ral side contacts blood through the fenestrated capillaries. Before developing a model for conjunctiva tr ansport it is essential to understand the transport mechanisms of ions and water from the in vitro and in vivo studies reported in literatures. In the in vitro st udies on conjunctival epithelium, typically the excised rabbit conjunctival epithelia are mounted between Ussing-chambers, where tissues are mounted between two chambers with controllable medi a compositions, and the s hort-circuit current (SCI ) or the open-circuit potential difference (PD) is measured. By controlling the solute concentrations in the Ussing chambers and by usi ng specific pathway inhibitors Kompella et al.36 and Shi and Candia37 showed that the conjunctival epitheli um transport mainly occurs through Cl channels and Na-Glucose cotransporters at the apical side, an d Na-K pump, Na-K-Cl cotransporters and K channels at the basolate ral side. To simplify the notation, the sign and magnitude of the electrical charge associated w ith each ion is not explicitly included in this study. The studies cited above also suggested that the conjunc tival epithelium could either secrete or absorb water, depending on the solute concentrations on the apical and basolateral sides. These experiments also quantified the contribution of Na and Cl transports to SCI. The basic mechanisms proposed by Kompella et. al and Shi and Candia were supported by other experiments in Ussing-chambers38,39,40,41,42,43,44and also by immulocalization techniques.45,46
23 Other studies focused on the changes in water tran sport as a result of a ddition of chemicals or changes in osmolarities.15,24 In addition to the in vitro studies listed above, Levin and Verkman47,48, 21 measured the water permeabilities of the conjunctival tissue and the cell membrane in living mice, and also studied the e ffects of modulation or deletion of CFTR (Cystic Fibrosis Transmembrane conductance Regulator) a nd the Na transport pathways on the potential difference between the apical and basolateral sides. Effect of Viscosity on Tear Drai nage and Ocular Residence Time Eye drops are commonly instilled to treat a va riety of ocular problems such as dry eyes, glaucoma, infections, allergies, etc. The fluid in stillation results in an increase in tear volume, and it slowly returns to its steady value due to te ar drainage through the canaliculi, and also fluid loss through other means such as evaporation or tran sport across the ocular epithelia. In fact, if the instilled fluid has a viscosity si milar to that of tears, which is about 1.5 cp, the instilled fluids or solutes are eliminated from the tears in a few minutes. As a result, the fl uids or solutes have a short contact time with the eye surf ace, which results in reduced eff ects for artificial tears or low bioavailability for ophthalmic drugs. To increase the duration of comfor t 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 inst illed fluid leads to an increase in the retention time. Zaki et al.49 showed that increasing the viscos ity of instilled fluid leads to a minor effect on the residence time unless the vi scosity of instilled so lution increases above 10.2-3Pas. The study by Wilson50 also supports the notion that the viscosity needs to be above a certain value in order to slow down the fluid clearance si gnificantly, and that critical value is larger than 8-3Pas. However little was known about the reason for the existence of such critical viscosity. Additio nally, although increasing fluid visc osity increases the residence
24 time, it may also cause discomfort and damage to oc ular epithelia due to an increase in the shear stresses during blinking. Shear th inning fluids such as sodium hya luronate (NaHA) solutions can be used to obtain the beneficial effect of an increase in reten tion and yet avoid excessive stresses during blinking. The likely reason is that the shear ra tes during blinking are very high and at such high shear rates these shear -thinning fluids exhibit low vi scosity but during the interblink which is the period during which te ar drainage occurs, these flui ds act as high vi scosity fluids leading to reduced drainage rates and a concurrent increase in residence time. While the mechanisms of the impact of vi scosity 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 an d 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 ef fect of viscosity on tear dynami cs, and also aid as a tool in development of better dry eye treatments and drug de livery vehicles. The goal of this part of the study is to develop a mathematical model to predic t the effect of viscosity on drainage rates and the residence time for both Newtonian and non-Newtoni an fluids. This part of the study is based on the tear drainage mechanism proposed by Doane,23 and it is an extension of our tear drainage model that focused on modeling drainage of tears, which were considered to be Newtonian fluids with a viscosity of 1.5 cp. In this study, the dr ainage model will be modified to calculate the drainage rate of instilled fluids with viscosities that are larger th an the critical viscosity, and so the system does not reach steady state in the blin k phase. Additionally, the drainage rates will be calculated for power-law fluid, which is a typical type of non-Newtonian fluid used for ocular instillation. Finally, the modified tear drainage model will be in corporated into a tear balance model to predict the effect of viscos ity on residence time of eye drops.
25 Tear Mixing Under the Lower Eyelid In order to develop a tear dynamics model, it is important to understand whether the preocular tears are well-mixed, since if they ar e indeed well-mixed it w ill greatly simplify the tear dynamics model. The preocular tears can be divided into three compartments: the precorneal tear film, the conjunctival sac and the tear menisci.2 The fluid in these compartments gets mixed due to the convection generated in the eyes by the blink. However it is not completely known whether these compartments are perfectly well-mixed or whether there are concentration differences between these compartments. This issue is of fundamental importance to the understanding of tear dynamics and it is also re levant to applications such as ophthalmic drug delivery. For instance it has been suggested that the residence tim e of ophthalmic drugs could be increased if these are injected into the lower conjunctival sac because of limited mixing between the sac and the tear film. During blinking the upper lids move downwards about 10 mm,51, 52, 53, 54 covering the area of the exposed tear film, and the lower lids m ove upwards but the amplitude is only about 1 mm or less.55, 56 It is thus expected that the tears in th e upper conjunctiva sac and the exposed tear film are better mixed than those in the lowe r conjunctiva sac. To understand the mixing of the whole tear film, it is essential to first understand the mixing of th e tears in the lower conjunctiva sac. It has been shown that after in stilling a dye solution into the lower conjunctiva sac, it takes about a few minutes for the dye to emerge on the lower lid edge, which is shorter than the time expected for purely di ffusion-driven mixing.55 To elucidate the mixing mechanism under the lower lids, Macdonald and Mauric e instilled fluorescein into the lower conjunctiva sac and studied the time of fluorescein appearance on the lid edge with the subjects blinking normally. The time of fluorescein appearan ce was observed to be much shorter that the estimated time for
26 the mixing driven only by diffusion. It had also been shown in earlier research that the rate of the fluorescein loss through the conjunctiva is slow compared to the tear mixing.57 Thus it was suggested that the mixing might be caused by th e periodic tear flow generated during the blink cycle. 55, 58 The mechanism that was suggested by M acdonald and Maurice is commonly called Taylor dispersion, and this mechanism is illust rated in Figure 1-5. To illustrate this mechanism, let us consider a case in which a fl uid is contained in betw een two flat plates, and the top plate oscillates with a periodic veloc ity. This situation is analogous to the lower conjunctival sac. Let us now consid er that a pulse of solute that is uniform in the y direction is introduced at some axial location at t = 0. The goal of Taylor dispersion modeling is to quantify the spread of this pulse. The periodic movements of the eyelids create a velocity field of tears and the solutes move in the x direction driven by convection. The time scale for the diffusion of the solutes in the y direction is comparable to that for the periodic motion, and so as axial velocity stretches the initial rectangular pulse to a trapezoidal shape, lateral diffusion reestablishes the rectangular shape with a larger than original width. Due to the combination of temporal (y direction) diffusion and axial (x dire ction) convection, the pulse will be spread and the solutes will be transported in the along the lowe r lids from the fornix to the edge of the lower lids. Although the experiments by Macdonald an d Maurice provided data that supports such mechanism,55 the quantitative relationship between lid /globe motion and the solute transport was not available. While the above researchers discussion is only based on the movement of the lower lid, the same mechanism similarly applies for the mixing related to the movement of the globe. If the tear mixing in the lower conjunctiva s ac is indeed driven by Taylor dispersion, the mixing time is expected to depend on the relati ve motion between the lower lid and the globe.
27 During a blink the lower lid moves horizontally towards the nose by about 3 mm57,59 and also moves up but with much smaller amplitude. It has been shown that the globe rotates horizontally towards the nose and downward s vertically by about 1~5 each and then returns to the original position within the first 0.05 s of a blink.56 In addition to the verti cal and horizontal shearing described above, during blinking the lids also press against the globe and the globe retreats in the posterior direction for about 1-2 mm,51, 59, 60 which suggests that the tear film may also be squeezed during the blinking. Because the time course for the globe retreat is in sync with the eyelid motion,56 if the squeezing of the tear film exis ts, the squeezing motion would be finished within the first 0.2 s of a blink cycle. The purpose of this part of the study is to develop a mathematical model to relate the mixing in tears to the cyclic movement of the globe and the lower eye lid during blinking. Based on the experimental studies listed above, we deve lop models for two different velocity profiles: (1) shear flow generated by verti cal (inferior-superior) motion of either the globe or the lower eyelid, (2) squeeze flow generated by the motion of the globe perpendicu lar to the eyelid. In reality both of these motions o ccur together but since there is insufficient data regarding the exact kinematics of the globe and eyelid movement and the two kinds of motion are not in sync, we construct mathematical models for each of th e two cases separately. In addition, although it is likely that the horizontal lower-eyelid motion in th e nasal-temporal direction also contributes to the mixing of tears, it is not included into the model due to the relative complex mixing mechanism for this scenario. However this case will qualitatively discussed. In this part of the study, we first mathemati cally analyze the mixing in the lower lid, and develop a model which can predict the concentrat ion of tracers both under the lower eyelid and in the exposed tear film after administration of tracers in the t ears below the lower eyelid. We
28 compare the predictions with experimental data to validate the model. In addition to elucidating the mechanism of tear mixing, the study can also be helpful in the interpretation of various experimental studies in which tra cer concentrations are monitored in various parts of eyes after fluid instillation. Application of the Tear Dynamics Model Since the tear dynamics model is physiologybased, it intrinsically contains many of the parameters that are relevant to understanding an d alleviating problems re lated to tears. The mathematical model can be solved to either predic t the time course of the parameters such as tear volume and tear film thickness, or predict the steady state of such parameters. The model is expected to predict the tear osmo larity, and other issues related to the effect of salt concentration on tear volume, conjunctival secretion, and the transport of ions across the conjunctiva. The model is also able to predict the dependence of the tear film thickne ss on various parameters such as tear viscosity, surface tension, tear ev aporation rates, canaliculi elasticity, etc. Additionally the model is able to predict the rates of changes in tear volu mes after instillation of a specific volume of tear fluid and the residen ce time of tracers and drugs in the tear film, and thus it can be incorporated into an ocular phar macokinetic model. Such a model can effectively be used to develop an improved understanding of the tear dynamics and also help in evaluating and designing treatments for t ear-related alignments such as dry eye syndrome, a common but poorly understood disease.
29 Figure 1-1 The structure of the tear film
30 Figure 1-2 Tear production and elimination pathways
31 Figure 1-3 The lacrimal canal iculus during the blink phase
32 Figure 1-4 The lacrimal canalic ulus during the interblink phase
33 Figure 1-5 Taylor disper sion due to shearing. A) A pulse of solu te is introduced at t=0. B) The solutes are transported laterally due to c onvection. C) Meanwhile the solutes diffuse transversely. D) The pulse widens due to the combination of convection and transverse diffusion.
34 CHAPTER 2 MATERIALS AND METHODS In this part of the study, first the mathemati cal models for tear dr ainage and conjunctiva transport were developed, and then they were in corporated into a tear dynamics model. Due to the lack of information about the mechanical pr operties of the lacrimal canaliculi, which are responsible for the tear draina ge, dynamic mechanical analysis (DMA) was conducted during the development of the tear drainage model. Then the drainage model was extended to describe the drainage of Newtonian fluids w ith high viscosity as well as n on-Newtonian fluids. At last, a mathematical model was developed for the tear mixing under the lower eyelids. Tear Drainage Model Model Development Based on the tear drainage mechanisms de scribed in the introduction, we develop a mathematical model for tear drainage in the cana liculi. According to the mechanism described by Doane,23 the blink-interblink cycle can be divided into four periods, which are detailed in Figure 3 of the reference cited above. The first period begins with the downward motion of the upper eyelids and ends when the puncta are occluded due to the contact of the lid margins. In the second period, the upper eyelid continues to mo ve down, the puncta stay occluded, and the lid closure acts to squeeze the canaliculi to drain th e fluid. The third peri od begins with the lid opening and ends when the punctual papillae pop ap art to open the canalic uli to the meniscus. In this period the compressive force on the canalic uli is removed leading to creation of suction. In the fourth period, the lids open up completely a nd then remain stationary till the next blink. During this period, the vacuum creat ed in the canaliculi draws flui d from the menisci. In our model, periods 1 and 3 do not lead to drainage an d thus are neglected. In the following sections, period 2 is called the blink phase, period 4 is called the interb link phase, and a cycle of one
35 blink and one interblink is referred to as a b link cycle. To summarize the mechanism, during the blink phase, the puncta are occluded and th e canaliculi are compressed, and this causes a flow towards the sac. This flow stops when the pr essure in the canaliculi equals that in the sac, where the pressure is assumed to always be atmospheric. During the interblink phase the pressure in the canaliculi is reduced due to the re moval of the muscle forc e, and the valve at the sac end of the canaliculi closes. Since the meniscus is at a higher pressure than the canaliculi at this time, tears flow into the canaliculi till the pressure equalizes. The mechanism described above is illustrated in Figures 1-4 and 1-5 and also by Doane.23 A recent study has suggested that the valve at the sac end of the canaliculi is not a real valve but is caused by the swelling of the cavernous body of the lacrimal passage.61 It is noted that whether the valve at the sac end is real or is based on different swelling states does not affect our proposed model. In our mathematical model, the canaliculus is simplified as a straight pipe of length L with an undeformed radius R0 and wall thickness b. The stress-stra in relationship is complicated for soft tissues like the lacrimal canaliculi, but due to lack of data and to simplify the model, it is assumed that the canalicular wall is linearly elastic with a modulus E. The canaliculus is considered to be a thin shell, i.e., its thickness is negligible in comparison to the radius, and thus, axial deformation is neglected and the length of the canaliculi L is assumed to be constant. During the blink phase, the muscles around the canaliculus apply a pressure p0, which is assumed to be constant and uniform, to compress the canaliculus, and this pressure is balanced by the hoop stress. Additionally, axial pressure gradients drive fluid flow inside the canaliculus. The fluid flow equations are combined with the solid deformation model to yield a single partial differential equation that can be so lved to predict the radius of the canaliculus as a function of
36 the axial position and time. The details of the de rivation are in Appendix A. The linearized form of the partial differential equation is t R x R 16 bER2 2 tear 0 (1) where E and b are the elastic modulus and the th ickness of the canaliculus wall, respectively, tear is the viscosity of tears or fluids instilled on the ocular surface, R is the canaliculus radius, x is the axial coordinate originating at the puncta and t denotes time. Equation (1) describes the variation of canalicular radius as a function of time and axial position during the blink cycle and from the radius variation the drainage rate and pressure can be predicted. In order to solve the above partial differe ntial equation, boundary c onditions and initial conditions are needed. Based on the photography observation of Doane,23 the following assumptions are made. During the blink phase, the can aliculus is occluded at the punctal side and it opens into the sac on the other si de. Thus, the flow rate is assumed to be zero at x = 0, and the pressure at x = Lcanaliculi is assumed to stay constant at the sac pressure, where Lcanaliculi is the length of canaliculi. Since it is assumed that th e sac pressure stays zero (i.e., atmospheric), the pressure at x=L remains zero during the blink pha se. During the interblink phase, the canaliculus is occluded at the sac side and is open at th e punctal side, so there is no flow at x = Lcanaliculi. We further assume that the pressure at the punctal end i.e., at x = 0 equals the pressure in the tear menisci. Due to the cyclic nature of the blinkinterblink process, the initial conditions for the blink phase are the same as the conditions at th e end of the interblink. Similarly, the conditions at the end of the blink are used as the in itial conditions for th e interblink phase. Thus, the boundary conditions and initial conditions for the blink phase (from t = 0 to t = tb) are
37 ib canaliculiR 0) t R(x, 0 t) L p(x 0 t) 0, q(x (2) where q is the tear flow rate, p is the internal pressure, Rib is the radius of the canaliculus at the end of the interblink phase and tb is the duration of the blink phase. The boundary conditions and initial conditions for the interblink phase (from t = tb to t = tc) are b b canaliculi mR ) t t R(x, 0 t) L q(x R t) 0, p(x (3) where is the surface tension of tears, Rm is the radius of curvature of the meniscus, Rb is the radius of the canaliculus at th e end of the blink phase, and tc is the duration of the blinkinterblink cycle. It is noted that even though the menisci pressure is negative due to the tearmenisci curvature, tears flow into the canaliculi during the interblink because the pressure in the canaliculi is even more negative. Furthermore due to the presence of the valve at the sac end, the pressure in the canaliculi at x = L is no longer forced to be equal to the zero pressure in the sac. The steady state radii, i.e., th e radii obtained by the method described in the Appendix are: b E R p p 1 R R0 sac 0 0 b (4) bE R R 1 R R0 m 0 ib (5) where p0 is the pressure outsi de the canaliculus, psac is the pressure in the lacrimal sac, and Rb and Rib are the steady state radii at the end of th e blink and the interbli nk phases, respectively.
38 The details of the derivation of Rb and Rib are also given in Appendix a. Based on equations (4) and (5) and the fact that the radius of the cana liculus reaches steady states during the blink and the interblink phase for normal t ear fluid, the average tear drai nage rate can be written as 2 0 0 0 2 0 m 0 c canaliculi drainagebE R p 1 R bE R R 1 R t L q (6) Mechanical Properties of Lacrimal Canaliculi Tissue Samples Porcine eyes with eyelids were purchased from Animal Technologies Inc (Tyler, TX). Eyes from pigs belonging to two age groups (6 to 9 months old, n=25 and over 2 years old, n=23) were used for the measurement of E and E. The loading-recovery tests were done on a separate batch of eyes from 6 to 9 months old pigs (n=8). The tissues were kept at 5 C until the extraction of the canaliculi. The porcine eyes ha ve only a single canaliculus and it was extracted using surgical blades and a lacrimal probe within 38 hours postmortem. The mechanical properties of the tissue that surr ounds the canaliculus are significantly different from that of the canaliculus. Accordingly, the pr esence of even a very small am ount of surrounding tissue in the samples is manifested in significant deviations in the measured mechanical properties. Therefore extreme care was taken to ensure adequate removal of surrounding tissues and the reproducibility in the measurements suggest that the samples utilized in the study reported here did not contain any tissue other than the canaliculus. Each cana liculus sample was cut along the length to obtain a flat sheet. The samples we re kept in the isotoni c Dulbeccos phosphate buffered saline (Sigma, St. Louis, MO) under room temperature until the measurements. The
39 samples were numbered and were photographe d with a digital camer a (Coolpix 5600, Nikon, Japan). The sample areas, which are required as input parameters for DMA, were measured from the digital photos using ImageJ image anal ysis software available on the NIH website. 62 Dynamic mechanical analyzer (DMA) setup The rheological measurements were conducted on a Dynamic Mechanical Analyzer (DMA Q 800, TA Instruments, New Castle, DE) in compression mode while submerged in Dulbeccos phosphate buffered saline at ambient temperature ( 24.0.3C). Briefly, the flat-sheet samples were mounted between the moving upper clamp a nd the fixed chamber, which contained saline, and the strains in the samples under programmed st ress were recorded (Figure 2-1). All the measurements were conducte d within 40 hours postmortem. Frequency dependent rheological response The rheological response was measured in a co mpression mode with a periodic force. At the beginning of each run, a static preload force of 0.01 N was applied to ensure adequate contact between the clamps and the sample, and then the sample thickness was measured automatically by DMA. During these measurements, the DMA applies a periodic compressive strain and measures the force required to achieve the desi red strain. The magnitude of strain needs be sufficiently small so that the response is in th e linear range, which implies that the viscoelastic properties do not depend on the strain magnit ude. Pilot experiments were conducted with the frequency fixed at 1, 10 or 25 Hz and th e oscillation amplitude changing between 5 m to 100 m to determine the linear range of viscoelasticity. A plateau region of 5 to 50 m for both the storage modulus and the loss modulus was observed, within which the variation of the moduli is less than 15% and beyond which the moduli rose to over 900% of the plateau values. The oscillation amplitude of 40 m, which is in the linear range was eventually chosen to ensure that
40 the corresponding stress was comparable to the physiological stress in a canaliculus during blinking, which is about 400 Pa. A Force Track of 115% was used in these experiments, which means that an additional time independent compressi ve force that equals to 15% of the amplitude of the dynamic force was imposed to ensure ad equate contact between the samples and the clamps. Physiologically, the duration of the can aliculi compression is about 0.04 s (blink), and the duration of the canaliculi expansi on is several seconds (interblink).23 Therefore, the frequency range of interest should be between 1 Hz and 25 Hz. Accordingly, the storage modulus (E) and loss modulus (E) were obtai ned under three different frequencies, 1 Hz, 10 Hz and 25 Hz. Simulation of blink-interblink cycles To simulate the blink-interblink cycles, the canaliculus sample was exposed to repeated cycles that comprised of a lo ading phase during which a consta nt stress of 400 Pa was applied (blink) and a recovery phase during which the stat ic force was removed (int erblink). Ideally, to simulate a real blinking cycle, the durations of blink and interb link should be about 0.04 s and 6 s, respectively. However, it was found that due to equipment limitation, the shortest duration of the loading period is 0.01 min. Due to this lim itation, the duration of th e loading (blink) period was set to be 0.01 min (0.6 s). Experiments we re conducted with a recovery (interblink) phase duration of 5.4 s so that the dur ation of one blink-interblink cycl e is equal to the physiological value of 6 s. Similar to the frequency sweep prot ocol, the sample thickness was measured at the beginning of each test with a preload force of 0.01 N. In these experiments, this static force of 0.01 N was applied at all times to ensure sample-clamp contact. Application of this static force is equivalent to using Force Track, which wa s used in the frequency dependent rheological measurements, except that if a Force Track is applied, the magnitude of the static force is determined by the equipment. The experiments comprised of measuring the dynamic stress and
41 strain as the samples were subjected to the repeat ed blink-interblink cycles. It is noted that each sample was first subjected to 21 cycles with 6 s recovery duration and then 11 cycles of 57.6 s recovery duration. Statistical methods Unless otherwise stated, the results are expr essed as meanS.D.. A two-way Analysis of Variance (ANOVA) was used to analyze the de pendence of the moduli on frequency and age, and the significant level was chosen to be 0.01. Conjunctiva Epithelium Transport Model The epithelium typically consists of about 3~6 cell layers, but there is little quantitative data on transport of solutes and water between adj acent cell layers. It has been suggested that the layer of cells in contact with th e tears offers a majority of the resistance to transport, and so subsequent cell layers can be neglected.21 Therefore, in this model the epithelium is assumed to consist of a single cell layer, with a unifor m distribution of homogeneous cells. With these simplifications, the system consists of three compartments: the apical compartment, the cellular compartment and the basolateral compartment. In th is part of the study, we include the transport of Na, K, Cl ions, and glucose and water mo lecules, both across the cellular compartment boundaries (transcellular) and through the space in between adjacent cells (paracellular). The main transport mechanisms included in the model are the Na-Glucose cotransport, the Cl channel and water transport at the apical membrane, th e Na-K pump, Na-K-2Cl cotransport and the K channel and water transport at the basolateral membrane,36,37 as well as the paracellular transport of Na, K, Cl, Glucose and water. The flux of water is assumed to be the sum of the osmotic flow through the transcellular and paracellular pathways and th e electro-osmotic flow through the paracellular pathway, for which the water flow is proportional to the paracel lular electric current.
42 The simplified scheme of the epithelium structure, along with the transport mechanisms is shown in Figure 2-2. It is possible that some unknown transport pathwa ys are not included in the model. In fact, there is experimental evidence that one kind of apical Na-amino acid cotransport exists in rabbit conjunctiva epithelium.63,64 This pathway is not included in th e model due to lack of adequate information for this cotransport mechanism. Furthermore, the role of acid-base transport in tear dynamics is not clear. It is also speculated that there may be more than one kinds of Cl channel on the apical surface of the epith elium. We hope to include these mechanisms into the tear dynamics model as adequate information a bout these mechanisms is developed from experimental studies. Incorporation of the Tear Drainage Model and the Conjunctiva Transport Model into the Tear Balance General Tear Balance Model Below we perform mass balances for ions, gl ucose and water in the apical, cellular and basolateral compartments, and use the electroneut rality condition in each co mpartment. In the model developed below, we consider each compartm ent to be ideally mixed, which is reasonable in view of the small sizes of the cells and the la rge convection during the blink in the tear film. Since the basolateral compartment is in contact with blood through fenestrated capillary vessels35 the solute concentrations in th is compartment are assumed to be known constants. Solutes other then Na, K, Cl and glucose are lumped together into a single concentration variable (COthers) and it is assumed that these other solutes are no t transported across the conjunctiva. Based on Figure 1 and the above assumptions, the model contains 12 unknowns, including 8 concentrations, which are the concentrations of Na, K, Cl and glucose in the apical (CNa,a,CK,a,CCl,a and CGlu,a)and the cellular (CNa,c,CK,c,CCl,c and CGlu,c)compartments, 2 volumes,
43 which are the volumes of the apical and the cellular compartments (Va and Vc), and 2 electric potential differences, which are the potential di fferences across the apical and the basolateral membranes. To obtain these 12 unknowns, 12 equa tions can be written using the mass balance and electroneutrality conditions and these are li sted below. The mass balances of Na, K, Cl, glucose and water in the apical compartm ent (tear side) yield equations (7)-(11): conj ar paracellul Na Glu Na a Na drainage 0 tear Na retion sec a a NaS J J 2 C q C q dt V C d (7) conj ar paracellul K, a K, drainage 0 tear K, secretion a a K,S J C q C q dt V C d (8) conj ar paracellul Cl, channel Cl, a Cl, drainage 0 tear Cl, secretion a a Cl,S J J C q C q dt V C d (9) conj ar paracellul Glu, Glu Na a Glu, drainage 0 tear Glu, secretion a a Glu,S J J C q C q dt V C d (10) conj a lar, transcellu w, ar paracellul w, n evaporatio drainage secretion aS J J q q q dt dV (11) Equations (12)-(16) are the mass balance equati ons of Na, K, Cl, glucose and water in the cells, respectively. conj Cl K Na pump Glu Na c c Na,S J 3J 2J dt V C d (12) conj channel K, Cl K Na pump c c K,S J J 2J dt V C d (13) conj channel Cl, Cl K Na c c Cl,S J 2J dt V C d (14) conj c Glu, Glu Glu Na c c Glu,S C r J dt V C d (15) conj b lar, transcellu w, a lar, transcellu w, cS J J dt dV (16)
44 Equations (17) and (18) are th e electroneutrality conditions fo r the total fluxes entering the apical and cellular compartments, respectively. 0 J J J J 2 Jar paracellul Cl ar paracellul K ar paracellul Na Glu Na channel Cl (17) 0 J J J J 2channel K pump channel Cl Glu Na (18) In the above equations, the Js represent the fl ow rates of solutes, which are functions of the 12 unknowns and are normalized by the surface area of conjunctiva, Sconj. Jpump, JNa-K-Cl and JNa,Gluare the turnover rates of the Na-K pump, th e Na-K-Cl cotransport and the Na-Glucose cotransport, respectively. JNa,paracellular, JK,paracellular, JCl,paracellular, JGlu,paracellular and Jw,paracellular are the paracellular flow rates of Na, K, Cl, Glucose and water, respectively. JCl,channel and JK,channel are the flow rates of Cl and K th rough their respective channels. Jw,transcellular,a and Jw,transcellular,b are the rates of water transport through the apical and the basolateral cell membranes, respectively. CNa,tear 0 ,, CK,tear 0, CCl,tear 0 and CGlu,tear 0 are the concentration of Na, K, Cl and Glu in the lacrimal gland secretion. rGlu is the rate of glucose consumption in the cell. We note that fluxes in the direction of apical to basolatera l are defined to be positive, and the turnover rates are always positive numbers. The fluxes of ion, glucose and water through channels, transporters or the paracellular route are functions of concentrations and membrane potentials, and the expressions for these fluxes are detailed in the appendix. In the above equations, qsecretion and qevaporation are the flow rates of lacrimal tear secretion and ev aporation, respectively, and these are assumed to be constants. Furthermore, qdrainage is the flow rate of tear drai nage through canaliculi, which is given by equation (6) In the above equations, the value of CGlu,tear 0 is assumed to be zero, and thus the tear glucose is contributed solely by transport from the blood thr ough the conjunctival epithelium. This assumption is consistent with the fact that the tear glucose level is correlated with the blood
45 glucose level, which has been exploited as a method to monitor the blood glucose concentration.65 In the above equations, the glucose cons umption rate in the cell is normalized by the surface area of conjunctiva (Sconj). We note that in equation (15), the function rGlu(CGlu,c) represents the glucose consump tion rate inside the cell, and it is assumed to be linearly dependent on the cellular gl ucose concentration, i.e., 0 c Glu, c Glu, 0 Glu c Glu, GluC C r C r (19) where rGlu 0 is a fitting constant whos e value was obtained by matching the model prediction with experiments for the time scale of the changes in the ISC after Na-K-Cl cotransporter blockade36 In the above expression CGlu,c 0 is arbitrary, and for convenience, it can be set equal to the normal glucose concentration in cells which is typically 0.5 mM. 66 Results of model simulations indicate that the time scales for changes in i on transport rates are very sensitive to the glucose consumption rate. The fitted value of rGlu 0 is close to the glucose consumption rate measured for corneal epithelium cells,67 which suggests that this value is pe rhaps reliable. However, it is likely that the expression for glucose consumption is more complex than the assumed linear kinetics. The metabolism of glucose in cells is expected to involve actively-contro lled processes, and the activities of enzymes will definitely affect the glucose consumption. If additional information about cellular glucose consumption is available, it can be incorporated into the model to yield more reliable predictions. Modified Model for an Ussing Chamber The above equations couple all the routes of inflow and outfl ow of solutes and water into an eye, and thus represent a comprehensive model for tear dynamics. To validate the model, the set of equations given above can be modified to simulate in vitro experiments in Ussing-
46 chamber. Below we modify the equation set developed above to simulate Ussing-chamber experiments, and compare the model predictions with experiments. In the Ussing-chamber experiments, the cham ber volumes are sufficiently large so that only negligible changes in con centration can occur during the e xperiment. Accordingly, the concentrations in both the apical and the basolateral chambers can be treated as fixed constants. For the cellular compartment, equations (12) to (16) account for the mass balances and equation (18) accounts for electroneutrality. For the sh ort-circuit experiments, the 6 unknowns (cellular concentrations of Na, Cl, K, glucose, cell vo lume and the potential across the cell membrane on either the apical or the basolateral sides) can be obtained by solving equa tions (12) to (16) and equation (18). For the open-circuit case ther e are 7 unknowns because the potential differences across the apical and the basolate ral sides are different, and thes e 7 unknowns can be obtained by solving equation (12) to (16), which account for cellular mass balances, and equations (17), (18), which account for el ectroneutrality. In Ussing-chamber experiments, typically the ISC or the transepithelial potential difference is measured for a variety of different com positions in the apical and the basolateral compartments. In some cases, ions that are tr ansported through specific pathways are replaced by other solutes, and in some other cases, chemi cals are added to block some specific transport pathway. To simulate the ion channel blocka de conditions in our model, the corresponding channel permeability or maximal flux was changed to 1/1000 of the normal value. Similarly, to simulate the stimulation of Cl channel by certa in chemicals, the Cl channel permeability was increased to 10 times of the normal value. The factor of 10 was chosen so that simulations conducted with the increased permeability correctly predicted the increase in ISC in the Ussingchamber experiments. 15 To simulate the Cl free condition on the apical side, Cl concentration in
47 apical compartment was set to zero, and the change in osmolarity was compensated by increasing the concentration of the inert ions in the same compartment. Other ion substitution cases were simulated similarly. Algorithm A procedure similar to that used by Fischbarg and Diecke25 and Novotny and Jakobsson68 was used to solve the equations. Briefly, in each time step, the concentrations and membrane potentials from the previous time step were used in equations (A8) to (A12) to compute the ion and glucose fluxes. Next, the computed fluxes and the volumes from the previous time step were utilized in equations (7)-(10) a nd (12)-(15) to obtain the concen trations at the new time step. The computed concentrations were then used to compute the fluxes of water, and then the volumes were computed by using equations (11) and (16). Finally, the concentrations were updated again by using the computed volumes, and the potentials were then calculated using the electroneutrality conditions, i.e ., equations (17-18). The time s cales of the solute and water transport suggested that the time steps in th e simulations should be on the order of 0.1 s. However, calculations showed that changing the ti me step from 0.1 s to 1 s led to undetectable differences in the results, and thus a time step of 1 s was used in all the calculations. Model Parameters and Initial Conditions A number of parameters are required in the tear dynamics model. Some of these are available in literature (Table 2-136, 69, 70, 71,66,72,66, 26 ,30 ,69 ,70 ,73,74,75,47,76, 22, 2, 10, 16), and some were estimated based on reported studies (Table 2-2). We note that the surface tension value used in the current study is the static surface te nsion of the air-liquid (tear) interface.69 In Table 2-1, CNa,b, CK,b, CCl,b and CGlu,b are the basolateral concentratio ns of the respective solutes; COthers,a, COthers,b, and COthers,tear 0 are the concentrations of the inert ions in the apical and basolateral compartments and the lacrimal gland secretion; PCl,paracellular, PK,paracellular, PNa,paracellular and
48 PGlu,paracellular are the paracellular permeabilities of the respective solutes; and Pw,paracellular and Pw,transcellular are the paracellular and transcellular wate r permeabilities. The remaining parameters (Table 2-2) were obtained by fo llowing the procedure described in Appendix C. In this study, kinetic parameters were based on findings from pigmented rabbit conjunctiva. These parameters could be substituted with more accurate measuremen ts from humans as these become available, e.g. Na-Glucose cotransport varies across species and so the rates obtained from rabbits may be inaccurate.77 Application of the Tear Dynamics Model The tear dynamics model can provide quantitat ive information for the steady state and the transient state of the tear film. Below some of the examples of the model application will be presented, and additional examples will be presented in the discussion chapter. Tear Film at Steady State At steady state, i.e., under normal conditions, th e tear volume is relatively constant and thus the left hand side of equati on (11) can be equated to zero. The right hand side of equation (11) includes the drainge rate, which depends on Rm and various physiological parameters L, tc, R0, bE, p0 and psac. These parameters are either known or have been estimated, and the values of them are listed in Table 2-1. Thus, e quation (11) can be used to determine Rm at steady states by setting the left hand side of equation (11) to be zero. The precorneal tear film thickness is related with the radius of curvature by equation (20),20 3 2 tear m U 2.12R h (20) where h denotes the tear film thickness; tear is the tear viscosity; U is the velocity of the upper lid; and is the tear surface tension. By using equa tion (11) one can determine the steady state meniscus radius of curvature and then equation (20) can be used to determine the tear film
49 thickness. By following this procedure, the depe ndence of the steady state tear film thickness on various physiological parameters such as Lcanaliculi, tc, R0, bE, p0, psac and the tear evaporation and production rates can be determined. Tear Film at Transient States Balance of tear fluid Now let us consider the case when a volume of fluid is instilled into an eye and as a result the tear volume suddenly deviates from the steady value. In this case the left hand side of equation (11) is no longer zero. Therefore it is necessary to develop a relationship between Rm and the tear volume (Vtotal) so that equation (11) can be solv ed to determine the dynamic changes of Rm and the total ocular fluid volume Vtotal. By anatomical considerations, Mishima et al 2 approximated the preocular tear volume as the sum of the tear volume in the precorneal tear film, the tear menisci and the conjunctival sac, a nd calculated the tear volume of three parts separately. Using a similar approach, the relationship between the total ocular fluid volume and the meniscus curvature can be written as lid 2 m 2 globe 3 2 m m totalL )R 4 1 (1 4 2 1 U 2.12R R V R (21) where Llid is the perimeter of the lid margin and Rglobe is the radius of the globe, which has been assumed to be spherical in shape. The details of the derivation of equation (21) are given in Appendix B. The instillation of ex tra fluid will increase the tear vol ume and also the tear radius of curvature. For the normal drainage, tears ar e mainly drained through the lower canaliculus,78 however, when the ocular fluid volume is significantly larger than normal, as for instillation, the upper canaliculus also c ontributes to drainage.79 Therefore, for the case of extra fluid instillation, the term corresponding to the drainage rate thro ugh the canaliculi in equation (11) should be multiplied by a factor, the value of which is between 1 and 2, depending on the fluid volume
50 remaining on the ocular surface. In the result s described below for the dynamic tear volumes after instillation, a factor of 2 has been used. The relationshi p between tear volume and tear meniscus radius depends on geometric factors and is not expected to change significantly after fluid instillation. Thus, if no spillage occurs afte r instillation, substitution of equation (6) into the modified equation (11) gives th e following differential equation 2 0 sac 0 0 2 0 m 0 c canaliculi conj a lar, transcellu w, ar paracellul w, n evaporatio secretion totalbE R p p 1 R bE R ) ( R 1 R t L 2 S J J q q dt ) ( dV t Rm (22) where the total volume Vtotal now is the sum of the tears and the instilled fluids. This equation can be integrated to yield the dynamic radius of cu rvature. After the dyna mic radius of curvature is obtained, the dynamic tear volume can be dete rmined from equation (21). In all the following calculations and discussions for th e drainage of tears or fluids with viscosities close to tear viscosity, the density, viscosity and surface tensi on of the ocular fluid af ter the instillation are assumed to be the same as those of tears. The effect of viscosity for Newtonian fluids with high viscosity or non-Newtonian fluids will be studied later in this study. Balance of solutes Now let us consider the balance of the solute that is present in the instilled fluid. This solute could either be a tracer that does not penetrate the cornea sclera and the conjunctiva and has been added only to measure the drainage rate or it could be a drug that can pass through the epithelium to enter the ocular tissues and the bl ood stream. Assuming that that the solute is always uniformly mixed in the tear fluid, below we utilize the tear model shown above to
51 develop solute balances for each of these tw o cases: non-penetrating solutes and penetrating solutes. In the first case it is assumed that the tr acers are not absorbed by cornea, sclera or conjunctiva and are eliminated only by canalicular drainage. If the instilled fluid is mixed ideally with tears, the concentration of the solute is uni form in the tears and the decrease in the total mass of solute in the tears is equal to the solute that flows out with the drained tears, i.e., drainage totalcq dt cV d (23) where c is the concentration of the tracer or dr ug in the bulk tears. After combining equations (23) and (10) and integrating, we get the followi ng expression for the tracer concentration in the tears: t t total conj a lar, transcellu w, ar paracellul w, n evaporatio secretion 00dt ) ( V S J J q q exp c c(t)t (24) where c(t) is the concentration of tracers in the bulk tears at time t after instillation, c0 is the tracer concentration in the bulk tears just after instillation, and Vtotal(t) is the total volume of tears and the instilled fluid at time t. Since Vtotal(t) can be calculated with e quation. (21) and c(t) can be found from equation. (24), the to tal amount of tracer left in the tear fluid can be calculated as cVtotal. In this part of the study, the secretion and evaporation rates are assumed to remain unchanged after the instillation of fluid. This is perhaps an inaccurate assumption because the instillation may change the salt c oncentrations in the te ars and thus the tran sport of tears through the conjunctiva may be altered. Furthermore, the instillation of the extra fluid impairs the integrity of the lipid layer, therefore the rate of evaporation will also change. However, due to lack of data for these effects, and also to k eep the model simple, the secretion and evaporation rates are treated as fixed.
52 In the second case, the solute can be absorb ed by the cornea and the conjunctiva. If the instilled drops contain a drug, a fraction of the drug is absorbed into the cornea and conjunctiva, and some drug is drained through the lacrimal ducts into the nasal cavity where it is absorbed systemically through the na sal pathway. Also some drug may be lost due to spillage. Below we develop a model that can be used to predict th e ocular bioavailability which is defined as fraction of the drug that is absorbed by the cornea. In this study I assume that the drug that is drained through the can aliculi and that is absorbed by conjunctiva enter the blood str eam and do not contribute to the ocular bioavailability. Also the sclera absorption is neglected since it is not a major elimination pathway for a number of topically applied drops.80 In addition, we neglect any drug loss by spillage. The transport mechanism of drug molecules through ep ithelia is complicated, but in this study we assume the drug transport through th e ocular tissues to be passive, and calculate the transport rate using the permeability coefficient given in literatures.81 It is further assumed that the concentration of the drug is uniform in the ocular fluid, and that the concentration at the inner side of the cornea and conjunctiva is zero. Ther efore, the decrease in th e amount of the drug in the tears is the sum of the drug amounts that is contained in the drained tears and also the amount that is absorbed by the cornea and the conjunctiva, i.e., drainage cornea cornea a conjunctiv a conjunctiv totalcq S K S K c dt cV d (25) where Ks and Ss are the permeab ilities and surface areas, respectively of the subscripted ocular tissues. The value of them are liste in Table 21. After combining equati ons (25) and (10) and integrating, we obtain the fo llowing expression for the tracer concentration in the tears
53 t t total n evaporatio secretion cornea cornea a conjunctiv a conjunctiv 00dt V q q S K S K exp c c (26) The ocular bioavailability ( ) can be expressed as: 0 0 drop tear cornea corneadt c c V V S K (27) where Vdrop is the drop volume and Vtear is the total tear volume before the application of eye drops. Note that c0 is the concentration in the bulk tears immediately after inst illation and not the solute concentration in the drop. Effect of Viscosity on Tear Drai nage and Ocular Residence Time Drainage of a Newtonian Fluid It can be shown by the tear drainage model that for tears w ith a viscosity of 1.5 cp, the canaliculus radius will reach steady states during both the blin k and the interblink phase. The canaliculus reaches a steady state in the bli nk phase when the stresses generated by the deformation of the canaliculi balance the pressure applied by the muscles. The steady state is reached in the interblink when th e canaliculi has relaxed to an exte nt at which the pressure in the canaliculi equals that in the tear film. Achievi ng 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 pe r 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 0 2 2 canaliculibER L 16 tear where Lcanaliculi, b and R0 are the length, thickness and the undeformed radius of the canaliculi, E is the elastic modulus of canaliculi, and tear 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 r eaches a steady state in both the
54 blink and the blink, there is no ch ange in the total amount of fluid drained in a blink, and so there is also no change in the draina ge rates. This explains the observation of Zaki that below a critical viscosity, increasing viscosity does not lead to enhanced retention.49 However, as the viscosity increases to a critical value at whic h the time to achieve steady state becomes larger than the duration of the blink phase, the cana liculus does not deform to the fullest extend possible, and so the amount of tears that drain into the nose during the interblink decreases. In this case the equations (1) to (3) are still valid and the only difference is that the canaliculus radius cannot reach steady state during the blink phase. The radius of the canaliculus can be solved analytically as a function of axial position and time from equations (1). The volume of fluid contained in the canaliculus at any instant in time can be computed by using the following equation: L 0 2 canaliculi(x)dx R V (28) 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 th e canaliculus can be computed as c blink interblink drainaget t V t V t q (29) The above procedure can be used to calculate the effect of viscosity on the drainage of Newtonian fluids. In order to determine the eff ect of fluid viscosity on the residence time of eye drops, the tear drainage rates are in corporated in a tear mass balance. Incorporation of Tear Drainage into Tear Balance To simplify the model, in this part of the study the conjunctival tran sport is assumed to have a constant rate and thus a mass balan ce for the fluids on the ocular surface yields
55 drainage n evaporatio production totalq q q dt dV (30) where Vtotal is the total volume of the fluids on th e ocular surface, including tears and the instilled fluids, qproduction is the combined tear production rate from the lacrimal gland and conjunctiva secretion. Combining equation (30) and ( 23) yield the following equati on for the total quantity of solutes dt V q q exp V V I Itotal n evaporatio production 0 total 0 (31) where I (= cVtotal) 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 de pends on the total tear volume, and the drainage rate is affected by the curvature through boundary condition Error! Reference source not found.. The tear volume can be related to the meniscus curvature by the following equation modified from equation (21): lid 2 m film m totalL )R 4 1 (1 V R V (32) where Vfilm is the combined volume of fluid in the exposed and the unexposed tear film and Llid is the perimeter of the lid marg in. Equation (21) shows that Vfilm can be written as a function of Rm for fluids with a viscosity close to that of tear s. But when the fluid viscosity is high, as is the case in the current study, it can be shown that the function Vfilm(Rm) used in our previous study yields unrealistic tear film th ickness and therefore is not likel y to be applicable for high viscosity. Therefore in this part of the study Vfilm is assumed to be a constant, and the instillation of any extra fluid only contri bute to the volume of tear menisci. The value of Vfilm is assumed to
56 be the combined volume of the exposed and the unexposed tear film before any instillation, which can be calculated to be 9.9 L. By solving equations (1), (29), (30) and (31) simultaneously using finite difference method, the transient quantity of the solutes in the ocular fluids can be obtained as a function of time. For ocular drugs delivered via drop the mass balance needs to be modified to include drug transport through the ocular tissu e. The bioavailability of such drugs can be obtained as the following using the same deriva tion process for equation (27) dt dt V q q A K A K exp V A K 0 t t total n evaporatio production cornea cornea conj conj 0 cornea cornea0 (33) Non-Newtonian Fluid Using similar methods to the above, we can calculate the residence time of nonNewtonian fluids that are instilled onto the ocular surface, with only equation Error! Reference source not found. modified. Unlike Newtonian fluids, whic h have a linear relationship between the shear stress and the shear rate, non-Newtoni an fluids have more complicated relation between the shear stress and the shear rate. On e 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 ocul ar instillation it can be appr oximated as power-law (shearthinning) fluid, i.e. the rela tion between the shear stress and the shear rate can be written as n 0 (34) where 0 is a constant with the unit of viscosity a nd n is a constant (0
57 literatures. Using equation (34) the equation for the deformation of canaliculi as a result of blinking can be derived as 2 2 1 n 1x R x R a t R (35) where a is a constant that is defined as 1 3n R R 2 bE 2 1 an 1 2n 0 n 1 2 0 0 The derivation of equation (35) is described in detail in Appendix A. It is noted that Newtonian fluid is a special case of a powe r-law fluid with n = 1, and equa tion (35) correctly reduces to equation (1) for this case. By solving equations (35), (29), (30) and (31) 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 bioava ilability can be also calculated using equation (34) af ter obtaining the volume transient from equations (35), (29) and (30). The rheological data for nonNewtonian fluids used in the calculations in this study are obtained from literature and fitted using equation (34) and the fitting results, which are listed in Table 2-3. Tear Mixing Under the Lower Eyelid Physical Description of the System The geometry of the system and the motions considered in the tear mixing model are shown in Figure 2-3. In this study we simplify the geometry of the tear film under the lower eyelid as a thin film of thickness h sandwiched be tween the eyeball (y = 0) and the lower eyelid (y = h), both of which are treated as flat plates This assumption is valid because the tear film thickness, which is several microns, is much smal ler than the radius of the eyeball, which is
58 about 12 mm. The positions x = 0 and L (= 10mm fo r this study) correspond to the lower fornix and the position at which the lower sac meet s the exposed tear film, respectively. As described in the previous section, only th e vertical shearing be tween the globe and the lid (motion (a) along x direction in Figure 2-3) and the squeezing of the globe against the lid (motion (b) along y direction in Fi gure 2-3) are modeled in this study, and they are considered separately. In the vertical sheari ng motion, the globe rotates inferior ly at the beginning of a blink, then it quickly returns to its original position, at which it remains still until the beginning of the next blink. In this case the tear film is c onsidered to have a cons tant thickness of about 7 m. In the squeezing motion, the lid move s interiorly against the globe at the beginning of a blink, then it quickly returns to its original position, at wh ich it remains still until the beginning of the next blink. In this case, the thickness of the tear film changes as a result of the squeezing. Mathematical Modeling According to Figure 2-3, the characteristic le ngths in x and y directions are L and h, respectively, and thus dimensi onless lengths are defined as L x and h y It can be shown that the problem contains a short time scale corresponding to blinking a nd the diffusion along the y direction (s), and a long time scale corresponding to the diffusion along the x direction (l). Therefore the nondimensionalization of time variables is given in the appendix in the context of detailed derivations instead of in the method section for clarity. The current study is based on a mass balance of the solutes instilled into the t ear film in the lower conjunctiva sac. Since the mixing of the solutes is depends on the lid/globe motion, the veloci ty functions of the lid/globe and the velocity profiles in the tear film are first given below.
59 Velocity profiles Vertical shearing. The periodic displacement in x direction of the li d with respect to the globe () for vertical shearing can be de scribed by the following equation: t f 0 (36) where 0 is the amplitude of the shearing motion, and f(t) is a periodic function with an agular velocity corresponding to blinking. According to the above physical description of the shearing motion, the function f(t) is assumed to have the following form: t T 2 cos 2 1 t fshearing (0
60 0 ) v(t, (38) where u is the velocity in x, v is the velocity in y direction, which is zero at all times, and t f is the derivative of t f It can be seen that =2 /Tc. Squeezing. The periodic displacement in y direction of the lid with respect to the globe ( ') for vertical shearing can be de scribed by the following equation: t g 0 (39) where 0 is the amplitude of the squeezing motion, and g( t) is a periodic function with an agular velocity corresponding to blinking. Similar to the shearing motio n, the function g( t) is assumed to have the following form: t T 2 cos 2 1 t gsqueezing (0
61 2 sac 2 2 sac 2 sac sac sacy c x c D y c v x c u t c (42) in which csac is the concentration of the solute in the lower conjunctival sac (under the lower eyelid), u and v are the fluid velo city components in the x and the y directions, respectively, D is the diffusivity of the solute in the fluid, and t, x, y are time and position c oordinates. For Taylor dispersion, it can be shown that the solute concentration in the above equation can be approximated as a function of l and and it can be obtaine d by solving the equation 2 sac 2 l sac c D c (43) in which D* is the Taylor dispersion coefficient. In th is model, it is assumed that the instilled solute does not penetrate into the epithelium b ecause of the short duration of the nixing time in comparison to the conjunctival uptake times,57 and therefore there is no flux of solutes at the fornix, or at the y = 0 (eyeball) and y = h (palpebr al conjunctiva) surfaces. It is also assumed that the solution drops are instilled to the bottom of fornix at the begi nning. It is also assumed that there is no mass transfer ba rrier at the edge of th e lower eyelid and so the concentration at x = L equals the concentration in the tear film Based on the above assumptions, the boundary conditions and initial conditions can be expressed as tear sacc c at 1 0 csac at 0 sacc c from 0 to at 0 t where is the dimensionless width of the initially instilled solute plug and *c is the concentration in the plug. These two pa rameters are related through the equation
62 drop dropV c Wh L c *, where cdrop and Vdrop are the concentration and volume of the instilled drop, h is the distance between the lower lid and the eyeball, and W is the width of the lower eyelid from the medial canthus to the lateral canthus, which is approximated as Rglobe. Equation (43) can be solved for the solute co ncentration under the lower lid if D* and ctear are known. The dispersion coefficient D* is related to the fluid velocity u and therefore depends on the lid/globe kinematics, it needs to be derived for the two possible scenarios separately. It is shown in the appendix that for the vertical sheari ng case, the ratio between the Taylor dispersion coefficient D* and the molecular diffusion coefficient D can be expressed by 1 n n n n n n 2 2 2 0 2 n n *cos cosh sin sinh 1 2h N n d 4d 1 D D (44) where N is defined as Tc/Tshearing, dn an d-n are the Fourier series coefficients for the function t f and D h nn22 The Fourier series coefficients can be obtained for arbitrary periodic function F(t) with a period T by T t t i n ndt e t F T 1 d (45) Similarly, the dispersion coefficient D* for the squeezing case can be given by the following expression 1 n n n n n n 2 2 2 2 0 2 n n *cos cosh sin sinh 3 1 h x h N 6 n d 4d 1 D D (46) where dn and d-n are the Fourier series coefficients for the function t g. It is noted that although in this study the velocity function is approximated with the simplified forms, the method presented in this study can be used for arb itrary periodic velocity functions to derive D*.
63 The concentration profile in the tears under th e lower lid is coupled to the concentration transients in the exposed tear film (ctear) through the boundary condition of equation (43). Thus in order to solve the problem, one also needs to solve the mass balance equation for the exposed tear film. It is assumed that the fluorescence does not penetrate into the ocular surface, and therefore the fluorescence is eliminated from the ocular surface only by the tear drainage into the lacrimal canaliculi. It is also assumed that the instillation of fl uorescence does not cause significant reflex tearing, and thus the tear secr etion rate is a constant. Because the instilled volume is small compared to the total tear volume, it is assumed that the tear volume remains in a steady state, i.e. the tear production and elimina tion rates are equal and they remain constant. In addition, the much velocity of th e upper lid during blinki ng is much larger than that of the lower lid. Thus it can be shown that the time scale fo r tear mixing in the upper conjunctiva sac and the exposed tear film is much smalle r than the tear mixing under the lo wer lid. Therefore the tears in the upper conjunctiva sac and the exposed tear film are assumed to be well-mixed. Based on these assumptions, a mass balance of the fluorescence on the ocular surface yields drainage tear dye tear tearq c j dt V c d (47) The initial condition of equation (47) is 0 ctear at 0 t In equation (47) Vtear is the volume of the tears in th e exposed tear film and under the upper lid, and jdye is the total flow rate of the fluorescen ce from the lower conjunctiva sac to the tear film, which is given by the following expression, A dx dc D jsac dye (48) where A (=Wh) is the total area of contact betw een the lower lid and th e exposed tear film.
64 The concentrations csac(x,t) and ctear(t) can now be obtained by numerically solving equations (43) and (47) using the expression for D* given in equations (44) and (46). To compare with the experiments by Macdonald and Maurice55, it is useful to extract from the simulated profiles for ctear the time for appearance of fluorescence in the tears (Tapp) and the time for the fluorescence to reach maximal concentr ation in the exposed tear film (Tmax). Since it can be seen from equations (47) and (43) that ctear and csac are linearly related to c* and so below the results for concentrations are normalized by c*.
65 Table 2-1 Model parameters based on literature Parameter name Parameter Value CNa,a (=CNa,b) (mM) 141 36 CK,a (=CK,b) (mM) 5 36 CCl,a (=CCl,b) (mM) 118.5 36 CGlu,a (=CGlu,b) (mM) 5 36 COthers,a (=COthers,b) (mM) 30.5 36 CNa,b (mM) ** 139.9 69 CK,b (mM) ** 3.92 69 CCl,b (mM) ** 107.1 69 CGlu,b (mM) ** 5 66 COthers,b (mM) ** 44.08 CNa,tear 0 (mM) 135 72 CK,tear 0 (mM) 46 72 CCl,tear 0 (mM) 123 72 COthers,tear 0 58 CGlu,tear 0 (mM) 0 qevaporation ( L/min) 0.8 qproduction ( L/min) 2 qsecretion ( L/min) 0.6 bE (Pa s) 2.57 22 Lcanaliculi (m) 1.2-2 26 Llid (m) 5.7-2 2 (Pas) 1.5-3 69 tc (s) 6 22 R0 (m) 2.5-4 30 (N/m) 43-3 69 p0 (Pa) 400 70 psac (Pa) 0 (atmospheric) PCl,paracellular (m/s) 9.5-9 73 (rabbit) PK,paracellular (m/s) 9-9 (rabbit) PNa,paracellular (m/s) 8.6-9 73 (rabbit) PGlu,paracellular (m/s) 5-8 (rabbit) Kcornea (m/s) 1.5-7 81 Kconj (m/s) 5.2-7 81 KNa,pump (mM) 8.3 74 KK,pump (mM) 0.92 74 KNa,Na-K-Cl (mM) 105 75 KK,Na-K-Cl (mM) 1.22 75 KCl,1,Na-K-Cl (mM) 103 75 KCl,2,Na-K-Cl (mM) 23.9 75 Pw,transcellular (m/s) 2.5-4 47 (mouse) Pw,paracellular (m/s) 1.1-5 47 (mouse) Rglobe (m) 1.2-2 2 Rm,0 3.65-4 101 Scornea (m2) 1.04 10-4 76 Sconj (m2) 17.65 10-4 76
66 U 5-2 m/s 16 Parameters for the Ussing-chamber experiments. ** Parameters for the tear dynamics model. The value of COthers,b is determined by assuming that the blood osmolarity is 300 mM. The range of reported values for tear secretion is 1-4 L/min12,13 but these measurements represent the total input of tears into the eyes, and thus include the lacrimal and the conjunctival secretions. Thus we choose qsecretion, which is the contribution from la crimal secretions, to be 0.6 L/min. For the tear evaporation rate, we adopt the value from our previous study89, so that the portion of the tear drainage in the total tear elimination is still 60%, in accordance with a previous report10. The value of COthers,tear 0 is set by using the reported concentrations for all ions and th en using the electroneutrality condition.
67 Table 2-2 Model parameters obtained by fitting data from experiments on rabbits File name This file contains PCl,channel (m/s) 1.68-8 PK,channel (m/s) 3.15-8 Jpump,max (mols-1m-2) 1.56-6 JNa-K-Cl,max (mols-1m2) 2.65-6 CT (mol of NaGlucose transporter.m-2) 7.22-8 rGlu 0 (mols-1m-2) 2.2-7
68 Table 2-3 Parameters obtained from fitting the rh eological data in literature using the power-law equation = 0n Solution 0 (cp) n R2 0.2 % NaHA 323.3 -0.329 0.973 0.3 % NaHA 884.2 -0.3913 0.9788 CMC (low MW) 194.8 -0.09201 0.9285 CMC (high MW) 194.4 -0.2943 0.933 Human tears 5.578 -0.264 0.9963
69 Figure 2-1 DMA clamp setup.
70 Figure 2-2 Simplified epithelium st ructure and transport mechanisms
71 Figure 2-3 Spreading of instilled dye due to shearing and squeezing motion of the lid and the globe
72 CHAPTER 3 RESULTS Below, first the results of the tear drainage model and the conjunctiva transport model for Ussing-chamber experiments under short-circuit a nd open-circuit conditions are presented. The comparison between the predicted results and those observed in chamber experiments helps justify the choices of parameters. Then the results of integrating both the tear drainage model and the conjunctiva model into tear dynamics are presen ted. The effect of visc osity on tear drainage and residence time and the tear mixing under the lo wer eyelids are presented at the end of the section. Tear Drainage through Canaliculi Mechanical Properties of the Lacrimal Canaliculi Frequency dependent rheological response The frequency dependent storage moduli (E) and the loss moduli (E) are listed in Table 1 for both age groups. An unbalanced two fact or ANOVA shows that E and E depends significantly on frequency (p ~ 0 and p ~ 0, re spectively), but not signi ficantly on age (p = 0.0379, p = 0.2056, respectively). The product of the storage modulus and the tissue thickness, which is corresponding to the parameter bE in th e tear drainage model, varies from 4.33 to 7.51 Pam, which agrees reasonably well with the value (2.57 Pam) estimated earlier by balancing the tear inflow and outflow. In view of the possibility of the variation of bE across species and subjects, throughout th e whole study we still used th e bE value of 2.57 Pam, and studied the dependence of tear film variable s on bE by varying bE around the normal value. Additionally, although the DMA study showed that the canalic uli are viscoelas tic rather than elastic, for the current study the linear el asticity assumption can still provide a good approximation for tear drainage rate, and the reason will be discussed in the following chapter.
73 Simulation of blink-interblink cycles Figure 3-1 shows typical stress and strain transients during the loading-recovery cycles Individual data points are separate d by time of 0.1 s. The data show s that stress and strain reach periodic steady states in about 2 cycles. To quantify the recovery of the canaliculus, we fitted the strain transient during the recovery period with exponential functions. It was found that the strain recovery occurred on two different time scales (1 and 2), i.e., a double-e xponential function (2 1/ t / tbe ae ) fits the data much better than a singl e exponential function. A sample recovery curve along with the best fit curve is plotted in Fi gure 3-2. The values of a, b, 1 and 2 are listed in Table 3-2 for each sample, along with the R2 value for the fits. It is noted that data for all the periods after the first two were used in the fit. Canaliculus Radius and Pressure Transients Equation (1) is solved analyt ically with the boundary condi tions and initial conditions given in (2) and (3) and the results are given below. For the blink, the radius change is expressed as t n 2 1 canaliculi 0 n n b ib b2e L x n 2 1 cos n 2 1 1 2 R R R t x, R (49) and for the interblink, the ra dius change is expressed as 0 n t n 2 1 canaliculi b ib b2e L x n 2 1 sin n 2 1 2 1 R R R t x, R (50) Figures 3-3 and 3-4 show the radi us profiles at different times for the blink and the interblink phases, respectively. In these figures the dime nsionless radius is pl otted as a function of
74 x/Lcanaliculi for four different values of the dimensionless time t/ where =16 tearL2 canaliculi/( 2bER0). By using the solid mechanics equation (A1), whic h relates the radius with the pressure, the pressure transients can be determined for the blink phase and the interb link phase. The pressure transients at x = Lcanaliculi/4, Lcanaliculi /2 and 3 Lcanaliculi /4 are plotted in Figures 3-5 and 3-6 for the blink and the interblink, respectively. In these figures the pressure is plotted as a function of dimensionless time t/ In the figures described above, time t = 0 denotes the beginning of the blink phase in Figure 3-3 and 3-5, and it denotes the beginning of the in terblink phase in Figure 3-4 and 3-6. As the figures show, at the beginni ng of the blink phase, the internal pressure suddenly increase to 233 Pa, and then quickly dr ops to a steady state value of zero, i.e. atmospheric, pressure. The pressure drops to -400 Pa immediately at the beginning of the interblink, and increases to a steady st ate value of about -167 Pa, which is /Rm. The time to get to the steady states for blink and interblink phase is about L2/(bER0). For bE is in the range of 0.649 to 34.022 Pam, the time to reach stea dy state is expected to be between 0.0546 to 0.0010 second. Conjunctiva Transport Model for the Simulation of a Ussing-chamber In order to validate the conjunctiva transpor t model, the computed values of cellular concentrations, fluxes, cellular volume and potential differences under both short-circuit and open-circuit conditions are comp ared with experiments. Cellular Concentrations and Cell Size In the Ussing-chamber experiments, the cellula r concentrations were not measured, and so we can not directly compare the model predictions with experiments. However cellular compositions have been measured for the cornea endothelial cells, which may be comparable to
75 the cellular concentrations in c onjunctiva and so in Table 3-325,66,82 we compare the predicted cellular concentrations with cornea l concentrations. In Table 3-3, CNa,c 0, CK,c 0, CCl,c 0 and COthers,c 0 are the reference cellular concentrations of Na, K, Cl and the inert ions, and Vc 0 is the reference volume of the cellular compartment. Short-circuit Current ISC The short-circuit current (ISC) can be computed by solving eq uations (12) to (16) and (18), and then imposing electroneutrality on the apical and the basolate ral compartments. This gives the following equation for ISC: channel Cl, Glu Na scJ 2J F I (51) The ISC obtained this way is the current normaliz ed by the surface area of conjunctiva. In addition to measuring ISC under physiologically relevant cond itions, experimentalists typically vary ion concentrations and add drugs to construc t situations that correspond to apical Cl free, both sides Cl free, apical Na free, pump inhibiti on, Cl channel inhibition, K channel inhibition, and Na-K-Cl cotransporter inhibition. In these experiments, the time course of changes in ISC is measured after performing one of the maneuvers described above. The simulated time courses of ISC after various maneuvers are plotted in Figure 3-7 below. We note that the curves for both sides Cl free and Na-K-Cl inhi bition almost overlap. The cont rol represents the current under normal conditions, and other curves correspond to different maneuvers that are indicated on each curve. The comparison between the calculated values and those measured in experiments is shown in Table 3-4 36,37 for ISC and in Table 3-5 36 for t1/2, i.e., the time required for ISC to achieve 50% of the total change.
76 Sodium (Na) and Chloride Cl Fluxes The net fluxes of Na and Cl have been m easured experimentally by isotope method in Ussing-chamber experiments under both short-circu it and open-circuit conditions, and these can also be computed by the model. For the apical and basolateral compositions listed in Table 2-1, the comparison of calculated and measured Na and Cl net fluxes is shown in Table 3-6. 44,42,39 Transepithelial Potential Difference (PD) To check whether the values of paracellula r permeabilities of ions are reasonable, we compare the predicted PD under open-circuit co ndition with experimental values. The PD reported in experiments on rabbit conjuncti va with Ussing-chambers is 17.7.8 mV36 or 14.6.5 mV 37, compared to the mode l prediction of 15.7 mV. Fluid Secretion under Open-circuit Condition To further compare model predictions with experiments, the effects of Na-K pump inhibition and the apical additi on of 20 mM glucose on fluid secr etion were simulated. Addition of 20 mM glucose to the apical compartment incr eases the fluid secretion from basolateral to apical compartment by about 2.7 times, mainly due to changes in osmotic flow. This is contradictory to the observati on that adding 20 mM glucose apically decreased the water secretion.24 The reason for this discrepanc y will be discussed later. In addition, when Na-K pump maximal flux is decreased to 0.001 of the norma l value, the model pred icts undetectable fluid secretion, which agrees w ith experimental observation.24 While the predictions for the conjunctiva m odel are not in perfect agreement with the experiments, there is sufficient qualitative and quantitati ve agreement to suggest that the values of model parameters are reasonable, and that the model developed above has captured the essential mechanisms, and thus it can be included into the comprehensive tear dynamics model.
77 Therefore, the conjunctiva model is integrated in to the tear dynamics model and the results for the tear dynamics are presented below. Incorporation of the Tear Drainage Model and the Conjunctiva Transport Model into the Tear Balance Model Prediction of Steady-State Tear Variables Normal tear parameters The tear dynamics model can be solved to predict the tear volume and composition, drainage rate, and transport rate s of water and various ions ac ross the conjunctiva. The steady state tear composition and volume obtained by the simulations are compared with experimentally measured values in Table 3-7. 83,84 The model predicts a value of 1.10 L/min for the water secretion rate from the conjunctiva, which is within the range of the tear turnover rate of about 1~4 L/min. The potential difference between the apical and basola teral sides has been measured in vivo to be about -15mV on rabbits85 and about -23 mV on mice,21 with the basolateral side as the reference. This compares well with the predicted value of -15.1mV. The model pr edicts values of 7.1 L and 297.6 mM for the tear volume and osmolarity, respectively. In addition to the steady state values under the normal conditions reported abov e, the model can be used to compute both steady and transient compositions, volumes, and potentials of tears and the conjunctival cells, and also the canalicular drainage rate, and the fluxes of water and ions and solutes across the conjunctiva. Some of these model predictions are discussed below and these predictions are compared to experimental data, if they are available. Steady-state tear film thickness as a function of tear evaporation rate It has been reported that when the lipid layer in the tear film is impaired, the evaporation rate may increase to up to 17 times the normal value86 and therefore the parameter which is
78 defined as the ratio of the evaporation rate to th e secretion rate, can easily reach 1.0. In Figure 38 the parameter is varied from the normal value 0.4 to 1.0, which is a possible range based on the experimental results. As shown by Figure 3-8, when varies from 0.4~1.0, the steady state tear film thickness varies from 11.2 m to 3.3 m. Steady-state tear film thickness as a func tion of mechanical pr operties of canaliculi The parameter bE is estimated to be about 2.57 Pam under normal conditions22. Due to the possibility of cross-subject vari ation of bE mentioned above, in this part of the study we vary bE between 0.1 Pam and 3 Pam. Figure 3-9 sh ows the steady state tear film thickness for different values of bE. When bE is varied between 0.1 Pam and 3 Pam, the steady state tear film thickness changes from 4.0 m to 17.4 m. Steady-state tear film thickness as a function of tear surface tension Figure 3-10 shows the steady state tear film thickness for different values of surface tension. As shown in the figure, as is varied from 20-3 N/m to 60-3 N/m, the steady state tear film thickness changes from 8.7 m to 12.5 m We note that this beha vior is contradictory to the commonly accepted observation that lower su rface tension is desirable for a more stable tear film, and this issue will be discussed in the next chapter. Model Prediction of Dynamic Tear Variables The effect of ion channel modulation on tear film When the Cl channel permeability (PCl,channel) is increased, the secr etion of water through the conjunctival epithelium is expected to in crease due to the coupli ng between water and ion transport. As shown in Figure 3-11, when PCl,channel is increased to 10 times its original value, the predicted tear volume increases to a maximum of 8.1 L, and then decreases to a steady state of about 7.3 L in about 50 min. The prediction for the conjunctival fluid secr etion increases from
79 1.10 L/min to 1.83 L/min, and then reaches a steady state of 1.12 L/min. Additionally, when the Cl channel permeability is increased, the mode l predicts hyperpolari zation and there is an immediate change in PD from -15.1 mV to -20.1 mV and then a gradual ch ange to a stabilized value of -15.5 mV, which agrees qualitatively with experimental observation.21 Furthermore, since it is suggested that the permeabili ties of Cl channel and the K channel (PCl,channel and PK,channel) can both be increased by cAMP,41 calculations were also condu cted for this situation. As shown in Figure 3-11, when PCl,channel and PK,channel are both increased to 10 times the original values, the steady state tear volum e increases to a maximum of 10.0 L, and then decreases to a steady state of about 7.3 L in about 43 minutes. In this case, the conjunctival fluid secretion increases from 1.10 L/min to a maximum of 5.44 L/min and then reaches a steady state value of 1.12 L/min. Thus, as expected, increasing the permeability of both K and Cl channels leads to a larger increase in the conjunctival fluid se cretion and tear volume compared to increasing the permeability of only Cl channels. The effect of evaporatio n rates on tear film It is commonly believed that dry eye sympto ms are alleviated or reduced if the subjects are exposed to controlled humidity conditions, possi bly due to an increase in the tear volume. To understand the potential benefits of reduction in evaporation on dr y eye symptoms, we used our model to simulate the effect of changes in ev aporation rates on the quantity and composition of the tear film. These results are shown in Figure 3-12. The effect of osmolarity in dry eye medications on dry eye A typical dry eye treatment comp rises of instilling about 25-30 l of solution into the eyes. One may expect that hypoosmolar tear solutions may be more e fficient at relieving dry eye symptoms but there are contradi ctory claims in literature.87 To explore this issue, we simulated
80 the dynamic changes in tear volumes and osmolarities after instilling 25 l drops of varying osmolarities. The results for the dynamic tear volumes, tear osmolarities and the conjunctival secretion are shown in Figures 3-13. The effect of punctum occlusion a nd moisture chambers on dry eye Punctum occlusion can be simula ted by setting the canalicular dr ainage rate to be zero. The model predicts that the tear osmolarity and conj unctival secretion reach a new steady state after punctum occlusion, but the tear volume increases con tinually, and thus tears are expected to roll off the eyes. While there are some reports of tear overflow (epiphora) caused by punctum occlusion, punctum plugs do not always lead to overflow, and this discrepancy will be discussed later. To further investigate th e change in tear dynamics by punctum occlusion, the fluorescence clearance after the occlusion of both puncta was simulated. In accordance with the experiments, the fluorescence in the eyes was computed for 15 minutes afte r instillation of 5 L of fluid with 2% fluorophore concentration.88 In these calculations, it was assumed that the fluorophore does not penetrate into the ocular surf ace. It was also assumed that tear spillage does not occur, which is a reasonable assumption due to the small in stillation volume and the short duration of the experiment (15 min). In Figure 3-14, the pred icted fluorophore concentration is plotted as a function of time. Besides punctum occlusion, moisture chambers which can decrease the tear evaporation rate, have also been explored as potential treatments for dry eyes. As shown above, if evaporation rate is increased, the tear film vol ume decreases and osmolarity increases, and these symptoms are similar to those experienced by dr y eye sufferers. To determine the impact of moisture chambers on dry eye sufferers, we cond ucted simulations in which the initial condition corresponded to the steady stat e results with four times th e normal evaporation rate, and
81 accordingly a thin, hyperosmolar tear film, and th en we reduced the evaporation rate to the normal level. The model predicts that the tear volume and the tear osmolarity go back to their respective baseline values in about 13 minutes, a nd thus it is expected that moisture chambers should bring a rapid relief to dry eye sufferers. The effect of drop volume on clearance time The effect of drop volume on clearance time wa s studied with our preliminary study of a tear dynamics model, which did not include the active transpor t through conjunctival epithelium.89 In this part of the study, the time cour se of ocular fluid after instillation was recalculated after incorporation of the conjunctiva model. As show n in Figure 3-15, the new tear dynamics model predicts that after instilling 15 L and 25 L of isosmolar fluid, the ocular fluid volume returns to its baseline in about 18 minutes and 24 minutes, respectively. As in the previous study, the baseline was defined as the value within 1% of the eventual steady state. Bioavailability of drugs delivered by drops with low viscosity The bioavailability of ocular drugs that are applied through drops that have viscosities close to that of tears was com puted in our preliminary study, 89 and we recalculated the bioavailability with the current tear dynamics model. In these simulations, an isosmolar drop of timolol was instilled, and it was assumed that the drug diffuses only passively. The current model predicted a bioavailability of 1.27. Thes e predictions agree with the reports that only about 1% of the instilled drug is absorbed by the cornea.80 Effect of Viscosity on Tear Drai nage and Ocular Residence Time 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 Figure 3-16 for a Newtonian fluid. The drainage rates for shear-thinning fluids depend on 0 and n, and the results for shear thinni ng fluids are shown in Figure 3-17. We
82 note that since the drainage rate increases when the volume of ocular fluids increases, the drainage rates in Figures 3-16 and 3-17 are generally larger than the tear drainage rate without instilling extra fluids. Effect of Viscosity on Reside nce 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 of fluorescence. The transients of the total tracer mass in the tear volume solute quantity, I(t), which is a measure of the total signal from the tracer, are plotted in Figure 3-18 for Newtonian fluids for a range of viscosities. It is noted that for fluids with viscosities lower than 4.4 cp, the transients of I overlap. In these and all other calculations reported below, the volume of all the instilled drops is set to be 25 L. Additionally, the viscosity of the tear fluid draining through the canaliculi is assumed to be changing linearly from the viscosities of the instilled fluids to the viscosity of tears (1.5 cp) to account for the dilution due to tear refreshing. For non-Newtonian fluids, the transients of I ar e calculated for sodium hyaluronate acid of 0.2% and 0.3% w/v concentrations which are commonly used for ocul ar instillation. The initial values of 0 and n for these and all other shear-thinning fluids that are discu ssed in this study are listed in Table 2-3. To account for the dilution due to tear refreshing, 0 is assumed to change linearly from the initial values of 0 to the viscosity of tears, and the values of n is assumed to be unchanged. The solute quantity transients I(t) are plotted in Figure 3-17 for these two fluids. Bioavailability of Instilled Drugs in Hig h Viscosity and Non-Newtonian Vehicles The bioavailability of instilled drugs is calculated for both Newt onian and non-Newtonian fluids and is listed in Table 3-8.
83 Tear Mixing Under the Lower Eyelid Concentration Transients and Mixing Time Below we present the predicted concentrati on of fluorescence in the tear film for the shearing and squeezing cases, respectively. The amplitude varies for different gaze positions56 and additionally, for the squeezing motion the ex act value of the amplitude is not clear. Therefore, we will explore the effect of the am plitude and frequency of the lid/globe movement on the tear mixing. According to the data re ported by Riggs et al., th e shearing amplitude is calculated by assuming that the eyeball rotati on varies from 1 to 5 degrees depending on different gaze positions.56 The amplitude of motion is simply the product of the eyeball radius and the angular rotation in radians. For the s queezing case, the amplitude has not been reported in literatures, but it should not exceed the tear film thickness of about 7 m. Accordingly, in this part of the study, the squeezing amplitude is varied from 1 m to 5 m. It is noted that throughout this study, the appearance time of the solute in the exposed tear film is defined as the time when the concentration reaches 1% of th e maximal concentration. In all the results discussed below, unless specified the value of is fixed at 0.04. The effect of variations in this parameter is discussed later. Figures 3-20 and 3-21 show the concentration transients in th e exposed tear film for the cases of shearing and squeezing, respectively for Tsqueezing = 0.2 s, Tshearing = 0.05 s and Tc= 6s. In Figure 3-20, the amplitudes of shearing varies from 1 to 5 degree, and Figure 3-21, the amplitude for squeezing varies form 1 to 5 m. The values of Tapp and Tmax can be obtained from each curve in Figures 3-20 and 3-21, and these are list ed in Table 3-9 for shearing and Table 3-10 for squeezing.
84 To investigate the effect of blinking freque ncy on tear mixing, the period for a blinkinterblink cycle (Tc) is varied from 1 s to 8 s while th e amplitudes of the shearing and squeezing are fixed at 1 degree and 3 m, respectively. In these calculations, Tc is varied but the actual time period for lid/globe movement, i.e. Tshearing or Tsqueezing, are kept fixed at 0.05 and 0.2s, respectively. The Tapp and Tmax are plotted as a function of Tc in Figures 3-22 and 3-23 for shearing and squeezing, respectively. Effect of Instillation Volume on Tear Mixing Time In the experiment the instilled drop volume could vary between 0.05 to 0.2 l, while in the model it is fixed at 0.1 l. The change in the instilled volume will affect the width of the fluorescent band in the initial cond ition of the model, and both Tapp and Tmax will decrease if the band width increases, and vise versa. Since the 2D problem in this study essentially assumes the concentration is uniform in the th ird (nasal-temporal) direction, th e initial condition used in this study does not accurately describe the actual situa tion when a drop of dye is instilled into the fornix. To validate the above assumption, the effect of on Tapp and Tmax is calculated. If the instilled volume is changed from 0.1 l to 0.2 l in the model, the predicted Tapp and Tmax for the shearing amplitude of 1 will decrease to 51.1 min and 279.0 min, respectively. If the instilled volume is changed from 0.1 l to 0.05 l in the model, the predicted Tapp and Tmax will increase to 52.6 min and 280.6 min, respectively. For the squeezing mode with the amplitude of 3 m, if the instilled volume is changed from 0.1 l to 0.2 l, the predicted Tapp and Tmax for the shearing amplitude of 1 will decrease to 0.8 min and 9.2 min, respectively. If the instilled volume is changed from 0.1 l to 0.05 l, the predicted Tapp and Tmax will increase to 1.3 min and 10.0 min, respectively.
85 Table 3-1 The storage m oduli and loss moduli of porcine lacrimal canaliculi. Variable 1Hz 10 Hz 25 Hz E young* (kPa) 3.070.94 3.361.17 4.961.54 E young (kPa) 0.820.28 2.080.65 4.541.14 E old** (kPa) 2.701.08 2.841.45 4.312.23 E old (kPa) 0.680.23 1.840.85 4.391.21 6~9 months old (n=25) ** >2 years old (n=23)
86 Table 3-2 Parameter values obtained by fitting the strain recovery to a double exponential. Sample # a (%) b (%) 1 (s) 2 (s) R2 1 13.4 2.15 0.115 1.28 0.998 2 12.6 2.82 0.0921 2.02 0.987 4 15.1 2.10 0.0542 1.37 0.998 5 7.83 1.29 0.163 1.92 0.998 6 14.0 2.13 0.106 2.01 0.987 8 13.5 1.19 0.192 4.95 0.987 9 13.4 1.07 0.194 2.97 0.993 10 9.39 1.49 0.179 2.64 0.984
87 Table 3-3 Comparison of cellular concen trations for Ussing-chamber simulation. Measured Values Model Predictions CNa,c 0 (mM) 13 25 12.56 CK,c 0 (mM) 132 25 132.69 CCl,c 0 (mM) 40 25 47.90 CGlu,c 0 (mM) 0.5 66 (pig) 0.68 COthers,c 0 (mM) 114.50 106.32 Vc 0/Sconj (m3/m2) 11.9 10-6 82 12.84 10-6 In Table 3-4, COthers,c 0 represents the concentration of inert solut es, and its value is set so that the initial cellular osmolarity is 300mM. Accu rate measurements of conjunctival epithelium cell height are not reported, but the cell height and the dimension of cell surface are expected to be comparable. So, in Table 3-4, the calculated value for Vc 0/Sconj, which in fact represents the dimension of the cell at the apical side, is compared with the cell height reported in the lite rature. It is also confirmed by our calculation that changing the value of Vc 0/Sconj to a value of 10 m, which is a typical value for cell height, does not make significant differences in the predictions.
88 Table 3-4 Comparison of steady state ISC after different maneuvers. Condition Experimental data (rabbit) Model prediction Control 0.144A/m2 37 0.151A/m2 Apical Na free 20%~50% decrease 37 34.3% decrease Apical Cl free 10% 37 or 180% 36 increase 2.6% increase Apical and basolateral Cl free 75% decrease 36 51.6% decrease K channel blockade 80% decrease 36 87.5% decrease Cl channel blockade 80% decrease 36 49.8% decrease Na-K pump blockade 100% decrease 36,37 98.9% decrease Na-K-Cl cotransporter blockade 42% 36 or 44% 37 decrease 51.5% decrease
89 Table 3-5 Comparison of t1/2 of ISC after different maneuvers. Manuvers Experimental data36 (rabbit) Model prediction Na-K pump inhibition 41.56.06 min 18.57 min Na-K-Cl cotransporter inhibition 8.330.98 min 8.86 min Cl channel blockade 0.92 0.14min ~0 min K channel blockade 5.5 0.8 min ~0 min
90 Table 3-6 Comparison of net Na and Cl fluxes. Net flux (basolateral to apical being positive) Experimental Data (rabbit) Model Predictions Cl (molm-2s) (6.941.67)-7 ( short-circuit)44 (4.171.39)-7 (open-circuit)42 9.69-7 (short-circuit) 1.73-7 (open-circuit) Na (molm-2s) -4.17-7 (short-circuit)39 -(3.891.39)10-7 (shortcircuit)44 -5.99-7 (short-circuit)
91 Table 3-7 Comparison of the apical composition and volume. Experimental Data Model Predictions CNa,a (mM) 133.2 83 128.3 CK,a (mM) 24.0 83 20.2 CCl,a (mM) 127.2 (estimated) 109.9 CGlu,a (mM) 0.56 84 0.63 COthers,a (mM) 34.21 (estimated) 38.6 Va ( L) 11.8 7.1
92 Table 3-8 Bioavailability for insti lled Timolol using different vehicles Fluid Bioavailability below 4.4 cp Newtonian 1.16 % 10 cp Newtonian 1.16 % 40 cp Newtonian 1.18 % 70 cp Newtonian 1.19 % 100 cp Newtonian 1.20 % 0.2 % NaHA 1.20 % 0.3 % NaHA 1.21 %
93 Table 3-9 The predicted Tapp and Tmax for shearing Amplitude (degree) Tapp (min) Tmax (min) 1 52.3 280.3 2 25.2 133.6 3 14.0 78.2 4 8.9 53.5 5 6.2 40.2
94 Table 3-10 The predicted Tapp and Tmax for squeezing Amplitude ( m) Tapp (min) Tmax (min) 1 6.7 37.8 2 2.3 16.5 3 1.1 9.8 4 0.7 6.7 5 0.4 4.8
95 Figure 3-1 Stress and stra in transients during lo ading-recovery cycles.
96 Figure 3-2 Sample data for one recovery cycle along with the best fit double-exponential curve.
97 Figure 3-3 Radius-axial positi on profiles during the blink phase at four different times.
98 Figure 3-4 Radius-axial position profiles during the interblink pha se at four different times.
99 Figure 3-5 Pressure-time transi ents during the blink phase at three different location along the canaliculus
100 Figure 3-6 Pressure-time transien ts during the interbli nk phase at three different locations along the canaliculus.
101 Figure 3-7 Time course of ISC in Ussing-chamber experiments.
102 Figure 3-8 The effect of tear evapora tion and absorption on t ear film thickness.
103 Figure 3-9 The effect of canaliculus properties on tear film thickness.
104 Figure 3-10 The effect of surface tension on tear film thickness.
105 Figure 3-11 The effect of cha nnel modulation on tear volume.
106 Figure 3-12 The effect of evaporation on tear vo lume, osmolarity and conjunctiva secretion rate.
108 Figure 3-13 The effect of isosmolar and anis osmolar (osmolarity 40 mM) fluid instillation on A) Tear volume. B) Tear osmolarity. C) Conjunctiva secretion rate.
109 Figure 3-14 The effect of punc tum occlusion on fluorescence clea rance with different rates.
110 Figure 3-15 Clearance of instilled drops of isosmolar fluid with different volumes.
111 Figure 3-16 The effect of viscosity on the draina ge rate through canaliculi for Newtonian fluids.
112 Figure 3-17 The effect of viscosity ( 0) and the exponential parameter (n) on the drainage rate through canaliculi for power-law fluids.
113 Figure 3-18 The transients of solute quantity (I ) after the instillation Newtonian fluids with different viscosities.
114 Figure 3-19 The transients of solute quantity (I ) after the instillation 0.2% and 0.3% sodium hyaluronate.
115 Figure 3-20 Concentration of fl uorescence in the exposed tear f ilm after instil lation into the lower fornix for shearing amplitude of 1-5 degrees.
116 Figure 3-21 Concentration of fl uorescence in the exposed tear f ilm after instil lation into the lower fornix for squeezing amplitude of 1-5 m.
117 Figure 3-22 The predicted Tapp and Tmax for shearing for different blink cycle duration (Tc).
118 Figure 3-23 The predicted Tapp and Tmax for squeezing for different blink cycle duration (Tc).
119 CHAPTER 4 DISCUSSION Mechanical Properties of Po rcine Lacrimal Canaliculi DMA has been routinely used to study th e mechanical properties of non-biological materials,90, 91, 92 and there have been report s of DMA measurement of bones,93 but there are few reports on the application of DMA on soft tissu es. Our study shows that DMA is a powerful tool for the mechanical measurement of soft tissues su ch as the lacrimal canal iculi, especially since these tests can be conducted under p hysiologically relevant conditions. In the current study, compressive load was app lied on the canalic uli that were cut open into a flat sheet, and the storage moduli, loss m oduli and the stress-strain curve under repeated loading-recovery were measured. Results show th at the magnitude of E and E are comparable and that they both depend significantly on fr equency. The relevant frequencies differ significantly between the blink and the interbli nk phase ranging between 1 Hz and 25 Hz during each blinking cycle, and so the frequency depend ence of the rheological properties must be taken into account while trying to understand any issue related to tear drainage In particular, these results suggest that our previous tear drainage model can be improved by using a more complex constitutive equation for the lacrimal canaliculi, which include both the elastic and the damping properties of the tissue. The current study did not find any significant differences in the rheological properties between the 6~9 months old and the >2 years old groups. However, we note that the typical lifespan of pigs is about 12~16 years,94 and therefore the differences in ages between these two groups is not significant. Theref ore, further studies are needed with more widely spaced age groups to determine the effect of age on rh eological properties of the canaliculus.
120 It is noted that while the ma gnitude of loads were adjust ed to simulate physiological situations, the loading conditions in the tests were still significantly different from those in vivo. The first major difference is that the canaliculus is a cylindrical tube and the stress is applied is mainly in the radial direction bu t the tests were conducted with fl at sheets. In fact the major component of the stress in vivo is the hoop stress, which is absent in the in vitro experiments. It is perhaps most relevant to measure the viscoelas tic response of the canaliculus in experiments in which the canaliculus filled with tears is submerged in a bath of tears, and is then subjected to pressure oscillations. Such experime nts have been conducted with blood vessels95 but because of the small length, low stiffness, and the complicated shape of the canaliculi, these experiments are extremely difficult to conduct. It is noted that th e mechanical properties of an isotropic system will not depend on the direction of loading but a canaliculus is likely to be anisotropic. Furthermore, the properties of the canaliculi are likely to va ry depending on the location, and also the magnitude of the applied force also varies with position. Anatomic studies show that the Horners muscle applies contra ction force on both the vertical and the horizontal parts of the canaliculi, and the force is largest for the vertic al part and it decreases along the horizontal part towards the lacrimal sac.96 Finally, in the current study porcine canaliculi were used rather than human canaliculi. The blinking rate of pig is about 30~36 blinks/minute,97 which is similar to that in humans, but it is unlikely that the mechanical properties of the human canaliculus are the same as that of porcine canaliculus. Despite such limitations, it is instructive to compare the results of the measurements reported above with t hose reported in literatu re for the value of the modulus and for the time scale of pressure relaxation. It has been shown that the parameter bE, which is the pr oduct of the canaliculus wall thickness and the Youngs modulus, is the most impor tant parameter to the model of canalicular
121 tear drainage. Due to the lack of data, in our previous tear drainage model we assumed the canaliculi to be linearly elastic and estimated bE to be about 2.75 Pam by equating the outflow of tears under normal condition with the tear production obtained in experiments. According to the current study, the product of the storage modulus and the sample thickness (in Pam) is 4.51.11 (1 Hz), 4.96.48 (10 Hz) and 7.51 2.67 (25 Hz) for the 6-9 months group, and 4.33.01 (1 Hz), 4.56.54 (10 Hz) and 7.11.45 (25 Hz) for the +2 years group. Therefore, the bE value estimated in our earlier model is comparable with the measured values of the corresponding parameter, bE, particularly considering the fact that the measurements correspond to porcine canaliculus. We note that the storage modulus of lacrimal canaliculi is much smaller than those of cornea and arteries, both of which are of the order of 106 Pa.98,99 The recovery of the canaliculus is directly related to the pressure changes inside the canaliculus which have been measured experimentally by Wilson and Merrill.70 In Figure 4-1, we plot a typical strain tran sient during the recovery (int erblink) period along with the measurements for pressure that were obtained by inserting a poly ethylene tubing (inner diameter 0.27mm; outer diameter 0.6mm) 8 mm into the lower punctum.70 In the same figure, we also plot the pressure profiles predicted by our tear drainage model. It is noted that in Figure 4-1, t = 0 is defined as the start of the in terblink period, the solid line co nnecting the stars is an exponential fit to the strain data. The time scale for the strain recovery in the current study is comparable to the time scale of pressure relaxation obtained by direct measur ement (Figure 4-1), and it is much longer than the pressure relaxation time scale predicted by our previous model. The good agreement between the time required for pressure relaxation in in vivo experiments a nd that for the strain recovery in our measurements strongly suggests that the recovery of canaliculi during the
122 interblink period is controlled by th e rheological properties of the ti ssue. In our previous tear drainage model, the tissue was assumed to be pur ely elastic which implies that any stress on the tissue will result in an instantaneous strain res ponse. Therefore the rate of canaliculi recovery was controlled by the viscous resistance offered by the fluid (tears) as it flows from the tear film into the canaliculi to relieve the vacuum. However the m easurements reported in this study show that the canaliculus is viscoelastic, and the time scale for the recovery is longer than the time scale for the tears to flow into the canaliculi. Accordingly the shape and the pressure changes in the canaliculi are cont rolled by the canaliculus rheology and not by the tear viscosity. The good fitting of the strain recovery tran sient using the double exponential function suggests that there are two time scales for the stra in recovery. The short ti me scale of about 0.1 to 0.2 second is corresponding to the physiologica l recovery of the cana liculi, as suggested by Figure 3-2 and the fact that the fitting parameter a accounts for almost all the strain recovery. There is a small portion of the st rain recovery, which corresponds to the fitting parmeter b and the long time scale of about 2 second. Validation of the Tear Drainage Model The tear dynamics model is based on a number of assumptions that are stated in the method and result sections. Belo w we first discuss the validity of some of the main assumptions and then compare the model predictions with experi mental data available in literature and finally discuss some of the model predictions. Steady State Assumption The tear film is reformed in each blink and it stays relatively stable during the interblink period.100 However, during this period the tears are drained through the cana liculi and the film also deforms due to the effects of gravity. Thus, the tear film is never truly at a steady state. However, the tear flow rate under no rmal conditions is about 1~4L/min13 and the normal tear
123 volume is about 10L, 2 and thus the tear tur nover time of about 2.5~10 min is much larger than the interblink period of a bout 6 seconds. Therefore, the change of the volume of the preocular tears during the interblink is negligible compared with the total volume itself, and it is valid to assume a time independent Rm and set the left hand side of equa tion (11) to zero. This is in accordance with the observation by Yokoi et al. that the meni scus radius does not have significant variation duri ng most of the interblink period un less the gaze direction is changed.101 Comparison of the Predicted Pressure Changes with Literature Results Wilson and Merrill70 inserted a polyethylene tubing (inner diameter 0.27mm; outer diameter 0.6mm) 8 mm into the lower punctum and connected the tube to a pressure transducer to measure the dynamic pressure in the canalic uli. According to their measurements, during voluntary blinks, the pressure in th e canaliculi starts from an initia l value of zero before a blink, and then goes through a rapid increase of about 3-4 mm Hg (about 400 to 533 Pa) in about 0.1~0.2 second after the blinking begins. The pre ssure then drops quickly to 3-4 mm Hg below zero (about -400 to -533 Pa) while the eyelids are opening in a bout 0.1~0.2 second, and then slowly returns to the initial value of zero during the interbli nk in another 0.5~0.6 second. 70 The pressure transients obtained from the model are similar to those in the studies of Wilson and Merrill, but they differ quantitatively. During the blink phase, the pressure in the canaliculi starts from an initial value of -167 Pa before a bli nk, and then goes through a rapid increase to about 233 Pa immediately after the blink begins. Subsequently, the pressure drops from about 233 Pa to zero in about 0.0010 to 0.0546 second, depending on the model parameters. The pressure then drops immediately to -4 00 upon the removal of p0, and then slowly return s to -167 Pa during the interblink in another 0.0010 to 0.0546 second. The major difference between the pressure transients from the experiments and the model are the following:
124 (1) The model predicts that the time to attain steady state is the same for the blink and the interblink, but in the experiments the time taken fo r the pressure to stabilize in the interblink is about 3-5 times that during the blink. The discre pancy can be explained by noticing that in the experiments the outer diameter of the tube was la rger than the canalicular diameter, and thus the lower canaliculus was not directly connected to the tear menisci. It was however connected indirectly through the up per canaliculus (Figure 1 of Wilson and Merrill70) Consequently the length over which the tear fluid had to flow in the interblink was twice the length over which fluid has to flow during the blink. According to the model, the equilibration time scales as L2, and thus doubling of length is ex pected to increase the time by a f actor of 4. Additionally, in our model it is assumed that the lacrimal sac does not contribute to the dr iving of tear drainage, but recent research suggested th at the role of the lacrimal s ac in the tear drainage may be underestimated.102 Thus the time scale in the pressure m easurement may include the time for the expansion of the lacrimal sac as well as the expansion of the canaliculus. (2) The time scale for equilibration during the blink is about 0.1 s in experiments but the equilibration time scale predicted by the mode l is about 0.0010 to 0.0546 second. The upper limit of the prediction is close to the experimental value but the lower limit is far below. There are a number of factors that may contribute to an under-prediction of the equilibration time. In the proposed model the pressure applied on the cana liculi is assumed to be uniform and constant. However, the actual pressure applied on canalicul i in the blink phase is neither uniform nor constant.32 Also in the model we have assumed the cana liculus to be a straight pipe, and thus we have neglected the bend in the canaliculi. The presence of the bend causes additional resistance, and thus neglecting the bend is expected to contribute to a lower prediction of the equilibration time. Furthermore, the time for the pressure increase from 0 to the maximal pressure during the
125 blink phase (period 1) and for the pressure dr op from 0 to the minimal pressure during the interblink phase (period 3) are neglected in the model as such pressure changes are assumed to be instantaneous. Period 1 and 3 last about 0.04 and 0.06 s, respect ively, and thus inclusion of these periods will increase the pr ediction of the equilibration time by about 0.1s. Finally, we note that the presence of the tube in the canalicul us could have led to changes in the equilibration time because the tube offers extr a resistance to deformation. Comparison of the Predicted Drainage Rates with Literature Results Rosengren27 inserted a tube through the lower punc ta into the canalic uli and obtained the flow rate through the cana liculi. This horizontal tube was connected with a vertical cylinder, the fluid level in which was at the same height as the puncta. A bubble was intentionally trapped in the horizontal tube to detect the direction and measure the magnitude of the fluid flow inside the tube during a blink and an inte rblink. (Figure 4 of Rosengren27) During a blink the fluid was expected to flow both into the t ube and into the lacrimal sac. It was found that during the blink phase, the bubble moved quickly away from th e punctum by about 0.5 mm, which corresponded to 0.29 mm3 of fluid flow out of the punctum into the gl ass tube. Since the pre ssure at either side of the lower canaliculus (sac and the glass tube) is about zero fo r the blink phase, the total tears squeezed is about 0.58 mm3 for the lower canaliculus. The squeezed tear volume per blink by single canaliculus predicted by our model w ith equation (29) can vary between 0.4001 mm3 and 0.0100 mm3 for bE between 0.649 Pam and 34.022 Pam. Again the model predictions for the lowest value of bE are consistent with experime nts. Rosengrens experiments also showed that during the interblink phase, 0.87 mm3 of fluid flows into the lowe r canaliculus, which is about three times as much as the amount of fluid s queezed out during the blink phase. However, according to the mathematical model developed in this study, the flow rates in the blink and the interblink should be equal. This difference be tween the model predictions and the experiments
126 can be resolved by noting that in Rosengrens ex periments, the pressure at the entrance of the lower canaliculus is zero, i.e. atmospheric, and the pressure at the entrance of the upper canaliculus is /Rm, since it is connected with the upper te ar meniscus. Therefore the pressure difference between the entrances of the lower an d upper canaliculus will drive flow into the lower canaliculus and out of the upper punctum. Fr om a rough Poiseuille flow estimation, the fluid flow driven by such a pressure difference du ring the interblink phase ma y be of the order of a few microliters. Thus, this flow may be the cause of the difference between the flows in the blink and the interblink in Rosengrens experiments. Sahlin et al. measured the tear drainage cap acity with an overloaded tear film by the drop test and concluded the tear drainage could be as much as 1.11~4.03 mm3 per blink for different voluntary blink freque ncies and age groups.103 Our model can be used to predict the tear drainage rates from an overloaded tear film by assuming that the radius of curvature of the tear menisci is close to infinity, which makes the pressure ou tside the puncta larger than that in normal conditions, and increases the tear drainage. Addi tionally, in an overloaded tear film, both the upper and the lower canaliculus are used for tear drainage. Thus, the volume squeezed per blink of an overloaded tear film is larger than that of a normal tear film. It is noted that in the experiments by Sahlin et al., the blink is volunta ry and the frequency is about 30 and 60 blinks per minute, which is much higher than the norma l blink frequency of 10 blinks per minute. The drainage rate is dependent on the duration of th e blink cycle as suggest ed by equation (29) and therefore also dependent on the bl ink frequency. Thus, the volume drained per blink is a more appropriate choice for comparison with the m odel than the volume drained per minute. The dependence of the tear draina ge rate for both normal and overl oaded tear films and the time scale for achieving steady states on the product bE is depicted in Figure 4-2. According to curve
127 1 in Figure 4-2, as bE increases from 0.649 to 34.022 Pam, the time to reach steady states of the blink phase and the interblink phase ( 0) decreases from 0.0546 to 0.0010 s. According to curves 2 and 3, when bE increases from 0.649 to 34. 022 Pam, the drainage ra te decreases from 4.00 L/min to 0.10 L/min for the normal tear film and from 11.74 L/min to 0.28 L/min for the overloaded tear film. We note that for the same bE value, the drainage rate of the overloaded tear film is always larger than the drainage rate of the normal tear film. Specifically, if bE=0.649 Pam, the drainage rates from an overloade d tear film can be as large as 1.174 mm3 per blink. The difference in drainage rate of a single canali culus due to the overloaded tear film, as well as the participation of the upper can aliculus, explains the large drai nage rate in the measurements by Sahlin et al.103 Relationship Between Tear Film Thickne ss and the Meniscus Radius of Curvature In normal conditions the lower canaliculus is mainly responsible for the active tear drainage,28 and thus the meniscus radius that is used to calculate the drainage should be that of the lower meniscus. However the meniscus radius that is used in equation (20) which relates the tear film thickness to the radius of curvature sh ould be that of the upper meniscus because tears are deposited by the dragging action of the upper eyelid.20 As shown by Creech et al.,16 the radii of the upper and lower menisci may not be the sa me. According to their observation, for one specific subject the Rm for the upper meniscus is 60% smalle r than that for the lower meniscus after instilling fluorescein dye16. In our model we have treated bot h of the menisci radii as equal and this will lead to some errors. We also note that the meniscus radius and lid velocity are not constants during the depo sition of tear film,20 and thus the deposited tear film thickness is not expected to be uniform, which is contrary to th e assumptions of our mode l. Furthermore, the tear film thickness obtained by equation (20) is the thickness immediately after the upward
128 movement of the lid.20 It has been verified that after the ey elid movement, further change of tear film thickness is possible due to the dragging of the aqueous tears by the lipid layer flow driven by the surface tension gradient,104 which may cause the tear film thickness change that is not included in equation (20). The a bove assumptions may affect the quantitative model predictions. It is noted that the detailed behavior of the tear film deposition can easily be incorporated into the proposed model. However, these inclusions will make the model much more complex and thus have been avoided. Finally it is noted that King-Smith et al.104 suggested that consideration of the extensibility of the lipid layer modifies the right-hand side of e quation (20) by a constant factor of about 0.63. If this modification is appl ied to the current study, the predicted tear film thickness values will decrease and will also match the recent experimental measurements obtained by King-Smith et al.105 and Wang et al.106 Validation of the Conjunctiva Transport Mode l by the Simulation of Ussing-chamber experiments In the Ussing-chamber experiments, typically the time course of ISC is measured after performing maneuvers such as ch anging the composition of the api cal or the basolateral or both compartments, and adding compounds that prefer entially block certain transport pathways. Below we first compare the magnitude of the change in ISC and then the time scales for the changes in ISC after such maneuvers. The predicted value for control ISC is within 10% of the experi mentally measured value. Also the values of predicted ISC after K channel blockade, Na-K pump blockade, and apical Na elimination are within 10% of the measured valu es. The predictions for other cases listed in Table 3-4 are within 30% of the measured va lues except the case corre sponding to apical Cl elimination. In this case ther e are significant differences betw een the values reported in two different experimental studies and one of the study reports a negligible increase in ISC, which is
129 consistent with the prediction. The differences between the predictions and the measurements are about 40-50% for the net Na and Cl fluxes thro ugh the epithelium under both short-circuit and open circuit conditions (Table 3-6). Although there are quantit ative difference between the predictions and the experimental values, such difference could partially be attributed to experimental errors, and uncertain ties in the model parameters. It can be shown by the model that the time scale for the conjunctival transport to equilibrate depends strongly on th e cellular glucose consumption ra te. In view of this strong dependency, it was decided to obtain rGlu 0 by matching the predicted time scale for the changes in ISC after blockade of Na-K-Cl cotransport with th e experimentally measured value. Therefore, the good agreement shown in Table 3-5 for the Na-K-Cl inhibition case is expected. The predictions listed in Table 3-5 agree qualitatively in each case, but the quantitative agreement is poor for Cl channel blockade and K channel bloc kade. In all these ca ses, the predicted t1/2 is smaller than the experimental t1/2. One of the reasons for the disc repancy could be that it takes a finite period of time for the chemicals to be mixed in the compartments and then to be fully effective on the corresponding transport pathways; however, the concentrations change instantaneously in the simulations. It is also possible that when the transport pathways are modulated, the kinetics of cellular glucose cons umption is changed. Since glucose consumption is essential to the prediction of time scale, such change in the consumption kinetics may cause changes in the predicted time scale. In addition, the time scale prediction may also be different if the epithelium is modeled as a multilayer of cells instead of a single layer. We note that as shown in Figure 3-7, the model predicts that mane uvers such as Cl, K channel blockade, pump inhibition and Cl free conditions lead to an instantaneous change of ISC in the beginning, and a slower change after that, while in experiments the ISC change has a single time scale. This could
130 also arise due to the instantaneous change in th e kinetic parameters in the model compared to gradual changes in the experiment s. For the simulation of Na-K-Cl cotransport inhibition, there is no instantaneous change of ISC because Na-K-Cl cotransporter is electroneutral, and therefore its inhibition impacts ISC only indirectly through the ch ange in ion concentrations. Application of the Tear Dynamics Model The Effect of Surface Tension on the Tear Film Thickness Figure 3-10 shows that lower surface tension will lead to thinner tear film, which is contradictory to the expectation that a smaller va lue of surface tension in dicates a thicker lipid layer and is thus expected to correspond to sm aller evaporative losses and thicker films. The reason for the discrepancy is that the evapora tion rate is assumed constant when the surface tension is varied in Figure 3-10, and since larger surface tension values leads to smaller drainage rates, the film thickness increases. If the incr ease in evaporative losses due to an increase in surface tension is included in our model, a larg er surface tension will pr esumably correspond to thinner films. Although the contradiction between Figure 310 and clinical observations show that the role of surf ace tension on tear film thickness can not be fully captured by the current model, it clarifies the role of surface tension on the drainage rate and also encourages further study on the effect of tear evapora tion on the tear film thickness. Comparison of Predicted Residence Time of Instilled Fluid and Solutes with Experiments The experimental studies on t ear and solute dynamics have focused on three classes of measurements. These studies used a variety of techniques to measure the dynamic tear volumes Vtotal(t) or the solute concentration c(t) or the tota l solute quantity in the tears, which is simply c(t)Vtotal(t). Below we first describe the experiment s reported in literature for each of these classes and then compare the results of our model with the experiments.
131 Overview of the residence time experiments Dynamic tear volumes. Yokoi et al.107 used a photographic system called videomeniscometer to follow the radius of curvatur e of tear menisci and therefore to obtain the dynamic change in tear volume. They observed the swelling of the tear meni scus after instillation and the subsequent shrinking due to drainage. They found that af ter instilling 15L artificial tear consisting of 0.1%KCl and 0.4%NaCl, the radius of curvature of tear meniscus returned to the value which is not distinguishable from that be fore the instillation after about 5 minutes. Also with video-meniscometer, Ishibashi et al.108 observed that after instil ling 15L of 0.5% aqueous timolol, the tear meniscus radius returned to th e value which is not distinguishable from that before the instillation after about 10 minutes. Total tracer amount. The quantity of tracer present in the tears can be measured without invasive sampling by measurement of fluores cence or radioactivity. Snibson et al.109 divided the ocular region into five regions of interest and measured the dynamic radioactivity in each region of interest after instillation of solutions (sodium hyaluronate solutions an d saline) that contained radioactive technetium-99m labeled di ethylene-triamine-pentacetic acid (99mTc-DTPA). For the region of ocular surface (see Figure 2 of Snibson et al.109), after the instillation of 25L aqueous fluid, the remaining radioactivity of la beled saline reached the steady state level within about 1000 seconds. Wilson et al.110 studied the ocular residence time of a carbomer gel in humans using radiolabeled sali ne as control. The residence time of 25L saline was about 900 seconds. Also with a similar technique, Meseguer et al111 studied the ocular residence of pilocarpine eyedrops with and without viscosity enhancers. Th ey found that 1 minute after the instillation of 25L control saline without viscosity enhancers, th e remaining radioactivity on the ocular surface is 23%; and 21 minutes after the instillation the remaining radioactivity is 12%. Additionally, the radioactivity is reduced to an undetectable value after about 6 minutes.
132 Meadows et al112 used non-penetrating fluorescent tracers and a biomicroscope as a fluorometer to study the ocular residence time for solutions with different visc osities. In their study with the saline solution, a conservative estimate of the re sidence time of tracers afte r the instillation of 25 L solution was reported to be about 15 minutes. Tracer concentration in the tears. In contrast to the a bove non-sampling methods, the measurement of the solute concentration in the tear s requires withdrawal of small tear samples at various instances in time. Scuderi et al113 instilled one drop (~40L) of 3mg/mL netilmicin and then withdrew 0.5L samples at various times, which are analyzed with high performance liquid chromatography (HPLC). The netilmic in concentration returned to a value that was less than 5% of the initial concentration after about 80 minutes. Comparison with experiments The comparison between the model predictions and the reported experimental results is summarized in Table 4-1.107,108,113,109,110,112 Table 4-1 shows that the predicted residence times are longer than the measured values for the cas es of volume and tracer quantity measurements, and are shorter than the reported values for the concentration measuremen ts. The overprediction for the case of volume and intensity can perhaps be attributed to the fact that in the model the rate of evaporation is assumed to be unaffected by fluid instillation. This assumption is contrary to the reports which show that weakening of the lipid layer, which may be caused by the instillation of extra fluids, c ould increase the evaporation rate by as much as 17 times.110 If the amount of evaporation is increased, the model prediction for the residence times will become smaller. For example, if the rate of evaporation is 4 times the normal value at the instillation and decreases linearly with ocular fluid volume afte r that, the predicted time for remaining tracer
133 quantity to become 1% of the in itial value after 25 L insti llation decreases from 1566 to 1426 seconds. Contrary to the cases of the intensity and volume measurements, the predicted values for the concentration decay are smaller than those m easured experimentally. The results in Table 41 show that the predicted concen tration decay time is only about half of the experimental value obtained by the sampling method. A possible reason for this discrepa ncy could be the stimulated tearing caused by the invasive sampling of the te ars, which is needed for the concentration measurement. Additionally, it was pointed out by th e researchers that the drug used in their study is irritant and therefore it may cau se more reflex tearing than the solutes used in the studies that measured the intensity decay. In fact it was repo rted that in experiments there was an 11-fold dilution after the in stillation of drug,113 while in the other studies such a dilution is not reported. The proposed model can be modified to include the effect of stim ulated tearing by using larger tear secretion rate. It was verifi ed that if the secretion rate in the model is increased by a factor of 3, the predicted concentration decay time matches that reported in the experiment. In addition to quantitative discrepancies disc ussed above, there are a few other differences between the model results and the experiments based on the radioactivity measurements. The experiment by Meseguer et al. shows that after about 6 minutes following the instillation of saline, the residual act ivity reached a steady state of about 15% of the initial value.111 Similar residual radioactivities are also obtained in other experiments using the same technique. In contrast, the model does not predic t such residual radioactivity si nce no absorption of tracers is included. The residual activity may be due to permea tion of the tracer into th e ocular tissue. It is suggested that the cut-off molecular weight for corneal penetration is larger than 400 114 and that for conjunctival penetrati on is larger than 20,000 .115 Since the molecular weight of the
134 radioactive tracers is lower than these cut-off va lues, these tracers may penetrate the corneal and conjunctival epithelia. Another difference between th e model and the experimental results is that the radioactivity studies show a bi-exponential decay with very fast decay rates within the first 1~2 minutes, which is not predicted by the model. The radioa ctivity experiments of Snibson et al.109 have shown that the remaini ng radioactivity drops to 50% of the initial value within 1 minute, while the model prediction is about 7.0 minute. Furthermore, the time for the tracer quantity to decrease to 23% of the initial value is 1 min according to Mesegure et al.111 while the model prediction is about 12.5 min. The fast cl earance during the firs t 1~2 minutes can be explained by noticing the dynamic changes in differe nt regions in the experiments of Snibson et al.109 The decay time in the radioactivity experi ments cited above is for the ocular surface region, which excludes the area near the medi al canthus. (Figure 2 of Snibson et al.109) Therefore, it is possible that the initial fast decay of tracer quantity for the ocular surface described in the radioactivity studies is mainly caused by the movement of tracers to the medial canthus area instead of the drainage through th e canaliculi. This explanation is supported by Figure 3-13a of Snibson et al.,109 which shows that the radioactivity in the medial canthus & lacrimal sac area undergoes fast increase almo st at the same time during which the fast clearance in ocular surface area occurs, suggesting the possibility of movement of tracers from the latter area to the former area. Values of Ocular Bioavailability and the Eff ect of Drop Size on Ocular Bioavailability The model predicts that for a 40 l drop of timolol, the ocular bioavailability, which is defined as the fraction of the instilled drug th at enters the cornea is 1.27%. This model prediction is in agreement with th e results quoted in the literature.80 Also, the model predicts that the ocular bioavailability decreases on re duction in the permeabilities. The model results
135 shown in Table 4-2 also demonstrat e that the ocular bioavailability is expected to be relatively independent of the drop volume. This is in di rect contrast with the experiments of Patton116 on rabbits using pilocarpine nitrite which showed that when the instilled volume decreases from 25 L to 5 L, the ocular bioavailability increases by 160%. This discrepancy could be partially due to the larger spillage for larger dr ops. Furthermore, increase in drop size is expected to lead to an increase in the conjunctival area for absorption of drug and a larger area will lead to a larger flux into the conjunctiva and thus reduce the ocular bi oavailability. In fact, if the effective surface area of the conjunctiva is decreased to a value of 1/3 of its total surface area, the predicted ocular bioavailability for a 5 L drop of timolol solution increases from 1.3% to 3.4%. The dependence of bioavailability on other parameters will be discussed later. The Effect of Ion Channel Modulation A ten fold increase in Cl channel permeability le ads to a large transient increase in the tear volume and the conjunctival secr etion rate followed by a gradua l decrease. Both tear volume and conjunctival secretion reach steady states in about 50 minutes at values that are only 2.7% and 1.8% higher, respectively than the control. The initial rapid increase in tear volume after Cl channel stimulation could be attrib uted to the instantaneous increase in the Cl flux that leads to an increase in the paracel lular current, and which in turn causes an increased electro-osmosis driven water secretion. The increased Cl secr etion reduces the cellular Cl concentration, and therefore the increased permeability is more or less balanced by the decreased Cl concentration difference. As a result, at the steady state, change s in permeability cause small differences in tear volumes and conjunctival secretion. When the K channel is stimulated in addition to the Cl channel, the channel associated K flux out of the cell increases, a nd this leads to an increase in the Na-K pump turnover rate. As the pump is the primary driving force for the electric current in the system, the paracellular current from the basolateral side to the apical side increases, leading
136 to higher elecro-osmosis driven water secretion. Thus the system settles at a higher tear volume than that in the case of only Cl channel stimul ation. We note that some experimental studies show a much larger increase in water secretion than the predicted increase,15,24 and this suggests that some of the ion and water transport mechanis ms are not accurately take n into account in the current model. The Effect of Evaporation As shown in Figure 3-12, an increase in the evaporation rate reduces the tear volume and increases the tear osmolarity, and both of th ese factors could potenti ally lead to dry eye symptoms. Interestingly, the increase in osmola rity leads to an increased secretion from the conjunctiva, which partially cancel s the effect of the increased evaporation, leading to only a weak dependence of tear volumes on evaporation rate s. Specifically, if the tear evaporation rate is increased to 4 times its normal value, the mode l predicts that the steady state tear osmolarity increases by about 9.0 mM, which is slightly larg er than the reported increase in osmolarity from 304.4 mM to 310 mM in dry eye sufferers.83 However, some other researchers have measured a much larger difference in osmolarities between normal subjects and dry eye patients.87 This suggests that in addition to increased evaporation due to deterioration in the lipid layer, there may be other mechanisms that increase the osmolarity in dry eye patients such as an increase in the osmolarity of the secreted tears. Our simulati on shows that if the osmolarity of the lacrimal gland secretion increases, the syst em equilibrates at a larger tear osmolarity, a larger osmotic secretion from the conjunctiva, a nd consequently a larger tear vol ume. These results suggest that the increased osmolarity in dry eye patients resu lts from a combination of the above mentioned factors.
137 The Effect of Osmolarity in Dry Eye Medications Recently, it has been reported that there are only minor differences in effectiveness of hypoosmolar and isosmolar soluti ons for dry eye treatment.87 In this study it was argued that the difference between the effects of hypoosmolar and isosmolar solutions is minor because the reduction in tear osmolarity after the applicati on of drops lasts only briefly. Also, studies by Candia et al. show that both hyperosmolar and hypoosmolar instillation d ecreases the diffusional water permeability of conjunctiva epithelium,117 and our model suggests that if these permeabilities are reduced, water secretion from conjunctiva decreases, which offsets the addition of fluid. This c ould further reduce the efficacy of hypoosmolar instillations. Our simulations predict that changes in osmola rity in tear film caused by instillation of 25 l of hyptertonic (+40 mM by adding NaCl) and hypoosmolar (-40 mM by subtracting NaCl) vanish after only about 3 minutes (Figure 3-13B). Our model also predicts that th e dynamic tear volumes for the case of hyperosmolar solution are only about 10% higher than those for the hypoosmolar solutions, and these differences vanish in about 25 min (Figur e 3-13A). Thus the model predictions are in agreemen t with the studies cited above. However as stated above, there are some other studies in the literature that claim that th e hypoosmolar solutions are more effective as dry eye treatments. One possible reason could be that the transient lowering of tear osmolarity by hypoosmolar solutions aids the epith elial cells in recoveri ng from inflammations that are common in dry eye patients. Furthermor e, in most studies with hypoosmolar solutions, sodium hyaluronate solutions were used instead of aqueous solutions, and this difference could enhance the effect of the hypoosmolar solution on ep ithelial cell recovery b ecause of an increase in the residence time. It is not ed that the current model does not include the regulation of water transport as a function of cell volume and other para meters such as osmolarity. It is observed that
138 the water permeability of cell membrane is subject to volume regulation.118 It is also reported that aquaporin-3 exists on the basolateral membrane of the conjunctival epithelium, and these may play a role in facilitating the transcellular water transport, and in regulating the paracellular water transport.47 Therefore, the neglect of the water transport regulation may make the model less accurate for conditions involving large changes in cell volume or osmolarity. The Effect of Punctum Occlusion and Moisture Chambers In the case where both puncta are occluded, the model pred icts that the tear volume increases monotonically (Figure 3-14), and is exp ected to lead to overflow. While there are some reports of tear overflow due to inserti on of punctum plugs, a monotonic increase of tear volume predicted by the model is not commonly obs erved in clinical practice. The reason could be that the sensation of ocul ar surface decreases after punctum occlusion, which leads to decreased lacrimal gland secretion88. As shown in Figure 3-14, when the lacrimal gland secretion rate is decreased, the rate of increase of tear volume after punc tum occlusion becomes slower. It is also possible that the punctum plugs do not comp letely block the tear flow, partly because the canaliculi deforms to a larger radius on punctum plug insertion.119 For the fluorescence clearance after punctum occlusion, the pred icted fluorophore concentrations at 15 min after instillation are 0.39% and 0.22%, for the occlusion of both puncta and normal drainage without punctum occlusion, respectively, which ag ree at least qualitatively with the measured mean values of 0.49% and 0.28%, respectively.88 The measured values after punc tum occlusion are larger than the prediction partly because punctum occlus ion leads to reduced lacrimal secretion,88 which slows down the decrease in fluoroph ore concentration, and this effect is not included in the current model. We note that in the simulation, it is assumed that the fluorophores does not penetrate into the ocular surface. If the fluorescence penetration is included, it will affect the
139 measurement of fluorescence clearance, but whether it will lead to overestimation or underestimation of the remaining fluorescence is not clear. Implication on Basic Tear Physiology In addition to predicting the effects of various physical and physiological parameters and conditions on tear dynamics, our model can also help in reso lving some outstanding issues related to mechanisms of tear dynamics. Two such issues are discussed below. Can conjunctiva secretion account fo r all the normal tear secretion? Since the fluid secretion rate of th e conjunctival epithelium can be 1~2 L/min, which is about the same as the estimated tear turnover rate, it has been speculated that under normal conditions the secretion from the conjuncti va accounts for all the tear fluid production.15 In the model, if the rate of lacrimal gland secretion is set to zero, the system r eaches a steady state with essentially zero drainage through th e canaliculi, which is not realis tic. This implies that without the supplement of electrolytes from lacrimal gl ands, the electrolyte secr etion by the conjunctiva is insufficient to balance the electrolyte loss th rough canalicular drainage. Therefore, the model suggests that under normal conditions where conj unctival secretion accounts for a large portion of tear fluid production, the secretion from the lacrimal glands are still necessary. Water transport mechanism There are controversies regarding the mechanism of water transport through epithelia.25 The current model includes bot h electro-osmosis and osmotic flows to account for the water transport in the epithelium. The electro-osm osis mechanism was included because Ussingchamber experiments show a net water secretion from the basolateral side to the apical side even in the absence of osmolarity gradient.15 There is however a possibility that the transport of water occurs through other routes such as thr ough a cotransporter or through water channels aquaporins. The electro-osmosis mechanism is shown to be valid in corneal endothelium25 and a
140 model of electro-osmosis has also been developed for a general leaky epithelium.120 However, it is not yet proven that the same assumption applies to the conjunctival water transport. To test whether a combination of osmotic flow and elect ro-osmosis correctly account for all transport routes for water, we simulated the Ussing-chambe r experiments related to water transport with our model, and below we compare the results with experiments. Simulations show that if the apical glucose concentration is increased by 20 mM, the fluid secretion increases. This is contradictory to the experiments, which re port a 77% decrease in fluid secretion.24 In fact, it has been reported that the permeability of the parace llular Na-Glucose cotransporter increases with an increase in glucose concentration,121 and if this fact is included in the model, the predicted secretion rate will increase further. This c ontradiction suggests that there may be other mechanisms responsible for the water transport in conjunctiva such as th e cotransport of water by the Na-Glucose cotransporter122 or through aquaporins. Once these mechanisms are experimentally determined, thes e can be included in the conj unctiva transport model developed in this study. Effect of Viscosity on Tear Drai nage and Ocular Residence Time Drainage Rates There are three important time scales releva nt to the drainage process; blink time tb (~0.04 s), interblink time tib (~6 s) and the time scale for the cana liculus radius to achieve steady state 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 less than 4.4 cp,
141 within this range will not change the drainage rate qdrainage. When the viscosity is larger than 4.4 cp but smaller than 654 cp, tb < < tib, and so the canaliculus radi us cannot reach steady state during the blink phase but still can reach steady state during th e 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 radi us is not uniform at the end of the blink phase and Vblink need to be calculated by determining the posi tion dependent radius at the e nd of the blink and then using equation (28). Increasing viscosity within this range will decrease the drainage rate, and this range is applicable to most of the high viscos ity Newtonian fluids that are used for ocular instillation. When the viscosity is larger than 645 cp, >tib>tb, and the canaliculus radius cannot reach steady state during either both the blink pha se and the interblink phase. However at this high viscosity the shearing to the ocular surface is likely to be high a nd may cause irritation. Therefore Newtonian fluids with viscosities higher than 645 cp are not likely to be used for ocular instillation and are not considered in this study. Since drainage rates through the canaliculi are not typically m easured, it is not possible to compare the predictions with experiments. It is noted that the data shown in Figure 3-16 has three distinct regions. In the first region ( < 4.4 cp), there is no eff ect of viscosity on drainage rates. In the second region (4.4 < < 100 cp), the viscosity has the maximum impact on the drainage. In the last region ( > 100 cp), the effect of viscosity on drainage rates becomes small. These trends qualitatively agree wi th observations noted in literatu re. Also, the data in Figure 316 suggests that it may be best to use eye drops with a viscosity of about 100 cp to increase the retention time, and yet not cause damage to oc ular epithelia due to excessive shear during blinking.
142 The trends shown in Figure 3-17 for shear-thi nning non Newtonian fluids are similar to those for Newtonian fluids. When the value of 0 is below 400 cp, the viscosity parameter 0 has a relatively large impact on the drainage, and when 0 is higher than 400 cp the impact on drainage becomes small. The trends for the thr ee typical n values are similar, with larger n yielding smaller drainage rates at the same 0. The data in Figure 3-17 suggests that the best range of 0 for non-Newtonian eye drops should be be low 400 cp. However, it has been reported that some polymers in such non-Newtonian eye dr ops have the ability to bind to the mucous ocular surface and thus increase the residence time of such eye drops. The binding effect is not included in the current study due to the insuffici ent information about the binding isotherms of such polymers, and for polymers with binding abili ty the actual drainage rate should be lower than that predicted by this model. Also from Figure 3-17 it seems that a larger n value is beneficial for lowering drainage ra tes. However, solutions with larger n values will likely cause excessive shear during blinking. Comparison of Residence Time for Newtonian or Non-Newtonian Fluids The transients of solute quantity predicte d by our model can be compared with the experimental studies in which a tracer-laden flui d is instilled and the transients of the tracer amount are measured.109, 123, 124, 125 These experimental studies measured the transients of tracer quantity in different regions on the ocular surface, and generally a fast decrease in tracer quantity was observed immediately after the instillati on. In some studies the precorneal area was measured, and in some other studies the ocula r surface was measured, which is defined as the whole ocular surface excluding the medial canthus ar ea. As discussed earlier, the data from these studies could suggest that such fast decrease in tracer quantity may simply due to the mixing of the instilled fluids on the ocular surface, which takes only a few blinks for low viscosity fluids,
143 instead of the drainage through canaliculi. In th e model presented above the fluids on the whole ocular surface is considered, which includes th e ocular surface defined in the above studies and the medial canthus area. As the tracer mixing on the whole ocular surface will not affect the total tracer quantity in the same region, the tr ansients plotted in Figure 3-18 earlier are not expected to show a immediate decrease in trac er quantity. Additionally, in the model presented above we have neglected transpor t of tracers through the ocular epithelia and also binding to the ocular surface. Such possibili ty is supported by the fact that the radioactivity measurements typically level off at finite values rather than going to zero, which w ould be expected in the absence of permeation and/or binding. Since th ese two factors may bring substantial difference in the tracer quantity prediction, the tracer quantity transients pr edicted by the above model are rescaled in order to compare with the experiment al studies that only measured part of the whole ocular surface, such as the precorneal or th e ocular surface area th at excludes the medial canthus region. It has been shown that instilled fluids are mixed with tears within a few blinks. Therefore it is assumed that af ter the instillation of the fluids they are well-mixed on the whole ocular surface immediately. If th e tracer quantity I decreases from I0 to I0 immediately due to the initial mixing, and the amount of tracers absorbed or binded to the ocular surface at the end of the measurements is Ir, the predicted tracer quantity can be re scaled using the following equation. rI ) I I(I Ir 0 (52) Equation (52) is applicable only when the instillati on of fluids with lower viscosities so that the mixing time is negligible compared to the drainage through canali culi. The transient profiles can also be used to determine the residence time, wh ich is defined as the time for I to decrease to within 0.01 I0 to the end value. In the model calculati on and the experimental studies mentioned below, the volume of the instilled fluids is 25 L.
144 The results for I(t) for viscosities of 5 cp and 8 cp Figure 4-4 can be compared with the experimental study by Snibson et al. using radioactive tracers (99Tcm),123 in which the viscosities of two kinds of Newtonian fluids 1.4% PVA and 0.3% HPMC were measured to be about 5 cp and 8 cp, respectively. The experimentally measured I(t)/I0 are also plotted in Figure 4-4 as dashed lines, with I0 =0.6I0 and Ir=0.131I0 according to the data by Snib son et al.. The theoretical profiles are in reasonable agreement with the experi ments, with a slower decrease in the first 300 seconds. The residence times for fluids with viscosities of 5 and 8 cp are 1344 s and 1362 s, respectively, and these are in reasonable agreement with the re sidence time of about 1900 s for both solutions calculated from the experimental data, which is taken as the time for the solute quantity reaches within 1% of th e final value. Greaves et al. measured the precorneal residence of 0.3% HPMC, which is Newtonian and has a viscosity of 6.649 cp.124 The solute quantity transients predicted by the current study after res caling using equation (52) is plotted in Figure 45 in solid line, together with the experimental da ta by Greaves el al. in dashed line. The residence time predicted by the model is about 1362 s, but the residence time from the experimental data is only about 550 s. For non-Newtonian fluids, the model predictions can be compared to experiments for 0.2% NaHA, 0.3% NaHA conducted by Snibson et al.109 The experiments showed that the clearance of NaHA was slower than that of the control saline solution, and that 0.3% NaHA was cleared more slowly than 0.2% NaHA. The predicted transi ent profiles for these two cases are plotted in solid lines Figure 3-19, together wi th 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 instantaneous, 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
145 difficult to judge whether the trac er quantity transient had reache d steady state at the end of the measurement. Therefore the rescaling in equati on (52) is not applicab le, and the predicted transients for NaHA solutions are plotted direc tly without rescaling. The predicted residence times for the two solutions are 4884 s and 26580 s. It is noted that in the experimental study lasted only 2000s after th e 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 model predictions are in reasonable agreement with the experiments for the two Newtonian fluids 1.4% PVA and 0.3% HPMC and the non-Newtonian NaHA and CMC solutions. There are several possible causes fo r the discrepancies between experiments and model predictions. First, the re scaling in equation (52) assumes the tracers are well-mixed instantaneously after instillation, while it is e xpected that such mixing is viscosity-dependent. The rescaling can be improved if quantitativ e information of the dependence of mixing on viscosity is available. Also a more realistic description of tracer permeation and binding would require permeability and binding isotherm measurements 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 changes in viscosity of the fluid due to tear turn over. After the solu tions 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 viscosit y is often not linear to the polymer concentration. This issue could also be more accurately in the model if the relationship between polymer concentration and solution viscosity is known for the solution of in terest. It is also noted that this issue will cause only a small difference for solutions with star ting viscosities of 10 cp or less. It is also
146 noted that the current model assumes th at the surface tension of tears (~43-3N/m) is not changed after the instill ation of extra fluids, which may not be good assumption particularly if the tracers are surface active. Fi nally, there are several assumpti ons involved in developing the tear drainage model, which are discussed earli er, 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 beca use some of them used rabbits as subjects in stead of human, and the others lack either the cl ear definition of the area being measured or the viscosity data of the exact solution used for inst illation. It is known th at rabbits have a much lower blinking rate than human. The drainage of instilled fluids through canaliculi is driven by blinking, and thus instil led fluids are expected to be cl eared differently from rabbit eyes. Although the effect of viscosity on the residence time showed a sim ilar 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. There can be considerable difference for the transients of tracer quantity in different regions on the ocular surface, and therefore it is inappropriate to compare the residence time without a clear definition of the ar ea of interest. The Effect of Viscosity on Bioavailability It has been shown that incr easing the viscosity of eye dr ops will increase the ocular uptake/bioavailability of the drug. Podder et al. showed that by using eye drops with viscosityenhancing polymers the ocular absorption could be increased two-fold for rabbits.126 Although the current model predicts that the timolol bi oavailability increases for higher viscosity, the increase is much smaller. One of the reasons fo r the small increase is th at the bioavailability depends not only on the viscosity of the eye drops but also on the permeabilities of cornea and
147 conjunctiva to the drugs. In an extreme case where punctal pulgs are applied and the drainage through the canaliculi is co mpletely blocked, the bioavailability will be conj conj cornea cornea cornea corneaS K S K S K (53) and according to the parameter values in Table 2-1, b would be 1.34%, which is the maximal bioavailability achievable for this drug. The time scale for drug absorption into the conjunctiva can be estimated by absorption=V/( KconjSconj), and for this study it is about 35 s, which is much smaller than the time scale of the drainage th rough canaliculi (~1000 s). This suggests that the systemic absorption predicted by the model is main ly due to the conjunctiva l absorption. Since in the current model any change in viscosity only a ffects the drainage throu gh canaliculi, significant change in bioavailability is not expected, whic h agrees with the theoretical study by Keister et al.127 It is noted that the permeability data in Table 2-1 is only for Timolol and were measured on rabbits. Based on the above estimation of absorption, viscosity is expected to have a larger impact on bioavailability if the permeability of a drug th rough conjunctiva is higher. Additionally, it has been suggested that some polymers that are used as viscosity enhancers can bind to the ocular surface. Increased binding will potentially increa se bioavailability, but such effect was not included in the calculation of bioavailability due to the insu fficient information on the binding isotherms of such polymers on the ocular surface. Fu rthermore, it has been suggested that at least part of the drug that is absorbed by the conjuncti va can reach the intraocular space instead of the systemic circulation. Therefore the current prediction of bioavailability may be an underestimation and it could be improved if mo re information about the mechanisms of conjunctiva absorption is available.
148 Drainage of Tears In our previous study of the tear drainage model, tears ar e assumed to be Newtonian and a constant viscosity of 1.5 cp is used. Howeve r tears are in fact non-Newtonian as shown the fitting in Table 2-3, possibly due to the presen ce of mucin other large molecules. The current study shows that any variance of viscosity below 10 cp leads to indistingui shable changes in the drainage rate through canaliculi, as evident fr om Figure 3-18. Therefor e the above assumption for the tear drainage model is valid, and simila r assumption can be used for any fluids with viscosity varying below 10 cp. Tear Mixing Under the Lower Eyelid Comparison with Experiments In the experimental study by Macdonald and Maurice,55 the mean values of the appearance time and the time for maximal concentration of fluorescence after applying less than 0.2 l of solution at the bottom of the lower cul-de-sac ar e 3.8 min and 8.0 min, respectively. In the current study, the pr edicted values of Tapp and Tmax vary significantly depending on the magnitude of lid and eyeball motion. The measur ed values are within th e range of theoretical predictions, and the significant variations in th e predictions are consistent with the large variations reported in the e xperiments (0.5-9 min for Tapp and 2-17 min for Tmax). The reasonable match between the model prediction a nd the observations in experiments supports the contention of Macdonald and Maurice that the mixing under the lid could be driven mainly by the combination of periodic convective moveme nt and diffusion. The closer match between the Tapp and Tmax predicted from squeezing than those from shearing suggests that under normal conditions, the squeezing motion may play a more im portant role in the tear mixing in the lower fornix.
149 While the model predicts time scales comparable to experimental measurements, there are still several possible causes for di screpancies between the model a nd the experiment. Firstly, the model is based on a 2D geometry which implies that it does not explicitly include the nasaltemporal motion. The potential contribution from the nasal-temporal motion to tear mixing is discussed later. The 2D model essentially assumes that the concentration is uniform in the third (nasal-temporal) direction. This assumption is certainly inaccurate, part icularly at short times after the drop instillation. It is noted that the weak dependence of the Tapp and Tmax on the value of is encouraging because it suggest that some inaccuracies in the initial conditions do not lead to significant variations in the concentration profiles. A 2D geometry has been used successfully to model ophthalmic problems such as tear mixing behind a contact lens and ophthalmic drug delivery by contact lenses.128, 129 Thus, while we expect the use of 2D geometry to introduce quantitative errors, it still preserves the important mech anisms and so the predictions of the model are expected to be in reasonable agreement with experiments. Secondly, in the model it is assumed that once the dye reaches the exposed tear film, it inst antly mixes within the exposed tear film due to the refreshing of tears and the blinking of the upper eyelid, and thus in the model the concentration in the exposed tear film is uniform. However, after the fluorescence reaches the exposed tear film from under the lower lid, it is likely that the distribution of the dye is not uniform. Since in the experiment the fl uorescence was measured in a small region on the cornea surface, the well-mixed assumption in the model may contribute to the discrepancies. To estimate whether the assumption leads to an overestimate or underestimate of Tmax, more information is needed about the mixing in the expos ed tear film, which is not within the scope of the current study. Furthermore, the current study only includes the shearing and squeezing motion of the lower eyelid and the globe, and there might be other modes of motion also
150 contributing to the tear mixing in this region, such as horizontal shearing, which will be discussed later, and the torsional eye movements.52 Additionally, in the current study the tears are assumed to be Newtonian fluid, while in fact tears are shown to be sh ear-thinning fluid. The effect of the non-Newtonian propert y of the fluid on mixing requires a more complex model. It is also noted that there are discrepancies in the literature regarding the values of Tshearing,130 and different values of Tshearing will lead to different predicted values of Tapp and Tmax. Finally, in the case of forceful, voluntary blinking, which ofte n happens in experiment al settings, the time periods for lid/globe movements and the blinking will be different,131 and this case is not included in the current study. Horizontal Shearing In addition to the shearing and squeezing moti on described by the model, it is reported that during blinking the globe moves in the nasal direction for about 1 and then moves back in the temporal direction. It is also observed that the lower eyelid moves in the horizontal direction during blinking.59 This horizontal motion of both the globe and the lid may also contribute to the tear mixing in this region. This horizontal shearin g could lead to a rapid spreading of the solute towards the canthus, and when the dye reaches the medial canthus region in the conjunctiva sac, the geometrical constraint could force the dye to be transported upward and reach the exposed tear film. If the amplitude of the horizontal shearing is 3 m, as suggested by Macdonald and Maurice, it can be estimated th at the dispersion coefficient D* is 3.56-7 m2/s, which corresponds to a time scale of about 17 min fo r the horizontal dispersi on of the fluorescent dye. Since such time scale is comparable to the e xperimental values and those predicted in the previous sections for shearing and squeezing, the horizontal shearing may also play a considerable role in the tear mixing in this region. Such mechanism is supported by the tendency
151 of the dye appearance near the inner canthus region obser ved by Macdonald and Maurice.55 However, the mechanism of the dye appearance due to horizontal shearing is more complex, and the kinematics of the torsional eye movements is not clear. Therefore more information is needed before these motions can be included in the model. Implications on the Mathematic al Model of Tear Dynamics In the mathematical modeling of tear dyna mics, an important assumption is that the solutes in tears are well-mixed, i.e. the concentration of a solute is uniform in the tear film. The current study and the observations by Macdonald and Maurice55 suggest that the tears under the lower eyelids are not well-mixed, and the typical mixing time is several minutes. However, it is noted that the current study only includes the cyclic motions of the eyelid and the globe when the gaze position of the subjects is fi xed at some certain angles, wh ich corresponds to the conditions in the experiments. However, under normal conditions the gaze position is not fixed but is likely to vary from time to time, and the shearing motion might be larger than that in the experimental condition and in the model. Additionally, under normal conditions there will also be more irregular, saccadic movements of the globe and the lids130 in addition to the periodic shearing and squeezing motion modeled in this study. The amp litude of such movement could be well above 1~5 and the frequency of such movement can also be higher than the normal blink frequency. It has been shown that with increased amplitude and frequency, the mixing time for instilled solutes can be reduced to below 1 minute. Because in the previous tear dynamics studies, the time scale of interest is a few minutes, the a ssumption of well-mixed tears is perhaps valid. Implication of Tear Mixing for Drug Application As tear mixing under the lower lids is slower th an that in other regions of the eyes, it has been suggested that the residence time of ophthalm ic drugs can be increased by instillation in the lower fornix.132 However this proposal is invalidat ed by some experimental observations.55 As
152 discussed above, in normal conditions the mixi ng time of tears under the lower lids could be below 1 min due to the changes in gaze positions and saccadic lid/globe movements. Therefore, the current study resolves the a pparent contradiction between th e studies that show a limited mixing beneath the lower lid and those which show no appreciable increase in residence time/bioavailability of drugs delivered in the lower fornix. However, if drugs are delivered using methods other than drops, such as ocular inserts into the lower fornix, the dispersion of drugs in the lower fornix could be important. If the disper sion is the rate-limiting step for drug transport, the model can be incorporated into a pharmacokinetic model for systems such as ocular inserts.
153 Figure 4-1 Pressure and strain transients obtained by previous experiment and model, and the current study
154 Figure 4-2 Dependence of time scale and tear drainage rate on bE
155 Figure 4-3 Effect of punctu m occlusion on tear volume
156 Figure 4-4 Comparison of the predicted and expe rimental transients of ocular surface solute quantity for 0.3% HPMC and 1.4% PVA
157 Figure 4-5 Comparison of the predicted and ex perimental transients of precorneal solute quantity for 0.3% HPMC
158 Table 4-1 Decays of tear volume, concentra tion and solute quantity after instillation. Experiment Model Volume decay (15L instillation) 300 s 107 >600 s 108 1115 s Concentration decay (~40L instillation) 4800 s 113 2401 s Quantity decay (25L instillation) ~1000 s 109 >900 s 110 ~900 s 112 1454 s
159 Table 4-2 Predicted ocular bioava ilability for different drop volumes (timolol) (small K) 5 L 1.28% 0.92% 15 L 1.28% 0.89% 25 L 1.27% 0.88% 40 L 1.27% 0.86%
160 CHAPTER 5 CONCLUSIONS A better understanding of tear dyna mics could be useful in a wi de variety of areas related to ocular drug kinetics, contact lens fitting, and treatment of di seases such as dry eyes. This study combines models for individual contributor s to the tear flows in the eyes into a comprehensive tear dynamics model. In additi on a mathematical model also developed for the tear mixing under lower eyelids. The tear dynam ics model can compute both the steady and the transient compositions, volume, an d potential of tears and the c onjunctival cells and also the canalicular drainage rate, and fl uxes of water and ions and solu tes across the conjunctiva. A majority of the model predictions are shown to be at least in qualitative agreement with experiments, and for predictions that do not agree with experiments, possible sources of discrepancies have been discu ssed. The model can be furt her improved by including more accurate values of parameters and by includi ng other transport pathways. Additionally, the mechanical properties of the lacr imal canlaliculi, whic h are responsible for tear drainage, have not been reported in the litera tures. Therefore in this study we also report the mechanical properties of porcine canaliculus that were measured in compression mode while submerged in fluid by using a dynamic mechanical analyzer (DMA). The model presented in this study can help deve lop a better understand ing of the effect of various physiological parameters on issues relate d to tear dynamics. Additionally, this model can be useful in understanding the causes of reduced tear volumes or increased osmolarity in dry eye patients, and furthermore, the model can help in preliminary evaluation of the effectiveness of various dry eye treatments. The tear dyna mics model can also help relate different experiments that focus on the same issue such as the experiments that measure either the volume decay or the tracer concentrati on decay or fluorescence/radioactivity decay after instillation of
161 fluid in the eyes. Finally this model can help the experimentalists in identifying the critical parameters and assumptions, and desi gning the most useful experiments. It is noted that this model presents a basic framewor k for tear dynamics, and it could be further improved by including several issues that may be important to tear dynamics such as corneal transport, effect of proteins, neural regulation a nd other feedback mechanisms, permeability regulation, lipid dynamics, tear film de position and breakup, mixing in the eye, etc. This model could also easily be adapted to incl ude presence of a contact lens or a drug eluding device or nanoparticles in the tear film. While this model is exp ected to be useful, it is not intended to serve as a replacement for experiments. It is expected to serve as a tool to enhance the mechanistic understanding of tear dynamics to design and understand experiments in a quantitative manner, and serve as a guide in design of various protocols fo r the treatment of dry eyes and the delivery of ophthalmic drugs.
162 APPENDIX A DERIVATION OF EQUATIONS FOR TEAR DRAINAGE Pressure-Radius Relationship The radius of the canaliculus R is less than the undeformed radius R0. The deformation of the canaliculus leads to a hoop strain of 0R R (0R R R ) and a concomitant hoop stress, which equals 0R R E. Neglecting the axial variations, the radial force balance on the canaliculi gives.133 0 0 oR R R E 2bL p p 2RL (A1) Determination of Rb and Rib If the time scale to attain steady state is shor ter than the duration of the interblink and the blink, canalicular radius will reach the steady st ate values in each cycle. If the steady state pressure is known, the steady state radius can be calculated from equa tion (A1). At the steady state of the blink phase, the pressure outside the canaliculus wall is p0 and the pressure inside is the pressure at the sac end of the canaliculus (psac). By equation (A1), the radius at the steady state of a blink (Rb) can be expressed with equa tion (4) in the main text. At the steady state of an interblink phase, th e pressure outside the canaliculus wall is zero as the canaliculus is not squeezed, and the pressure inside is equal to the pressure in the meniscus outside the puncta, which is given by /Rm. By equation (A1), the radi us at the steady state of an interblink (Rib) can be expressed with equa tion (5) in the main text.
163 Derivation of the Equation for th e Drainage of Newtonian Fluids The mass balance of fluid in a differential section of the canaliculus lying between axial locations x and x + dx is133 x q t ) R (2 (A2) The Reynolds number (Re), which is the ratio be tween inertial terms and viscous terms in the fluid mechanics equation, determines whethe r the flow is inertia dominated or viscosity dominated. According to the above definition, t R R ~ L R uR Re, where R is the magnitude of change in radius during the blink or the interblink phase. In normal conditions, the time scale for blinking is about 10-1s.70 During the blink cycle the radius changes between Rb and Rib; R is thus taken to be Rib-Rb and with the value of parameters given in the main text, R is at most 0.1R0. According to the above estimation, Re is about 10-2, which means the viscous term is much larger than the inertial term in normal conditions. Thus the inertial terms in the fluid flow equation can be ne glected and the flow is locally like a Poiseuille flow. The flow rate for Poiseuille flow in a cylindrical pipe is133 8 R x p q4 p (A3) where qp is the Poiseuille flow rate, R is the pipe radius, is th e fluid viscosity and x p is the pressure gradient driving the flow. Also, the Re stays small for the entire range of bE used in this study. By substituting q in equation (A2) with qp given by equation (A3) and substituting p with the pressure given by equation (A1), the foll owing equation of R as a function of x and t is obtained:
164 t R 8 R x R R R R bE p x2 4 0 0 0 (A4) As stated above R is small in comparison to R0, and thus equation (A4) can be linearized as equation (1) in the main text. Derivation of the Equation for the Drainage of Non-Newtonian Fluids The flow rate q through a cylindrical pipe of power-law fluids that obeys Equation (34) can be written as n n nR n n x p R q1 1 0 21 3 2 1 (A5) Equation (35) can be obtained by combing equations (A5) an d (A2) and some algebraic manipulation.
165 APPENDIX B RELATIONSHIP BETWEEN THE MENISCUS CURVATURE AND TOTAL OCULAR FLUID VOLUME The thickness of the exposed tear film is determined by the balance between the viscous drag force and the capillary suction force on the tear film, and the tear film thickness can be expressed by equation (20). Theref ore, the volume of the exposed precorneal tear film can be expressed as exposed 3 2 m exposedA U 2.12R V (A6) where Aexposed is the palpebral aperture area. The tear volume in the menisci can be calculated after assuming that the radii of curvature of th e upper and the lower menisci are the same and uniform along the eyelids. Because the height and ra dius of curvature values of tear meniscus are comparable,134 we approximate the geometry of the tear meniscus cross-sectional area to be the geometry shown in Figure A1 Therefore the cross section area of tear meniscus is (Rm 2Rm 2/4), and the tear volume in the meniscus (Vmeniscus) is approximated by lid 2 m 2 m meniscus)L R 4 1 (R V (A7) where Vmeniscus is the tear volume in the lacrimal menisci, and Llid is the perimeter of the lid margin. Finally, the tear volume in the conjunc tival sac can be estimated by assuming the thickness of the unexposed tear film that fills the conjunctival sac to be the same as the exposed tear film thickness.2 As the total area of the unexposed and exposed tear film is estimated to be about half of the surface area of the globe,2 the sum of Vsac and Vexposed can be expressed as globe m sac osedR U R V V2 3 2 exp4 2 1 12 2 (A8) where Rglobe is the globe radius, after approximating th e geometry of the globe to be a perfect
166 sphere. From equations (A6) and (A7), the total t ear volume can be estimated as a function of the radius of curvature of tear menisc us, equation (21) in the main text. It should be noted that in th e derivation of the above equation, the shape of the assumed cross-sectional of the tear menisci (Figure A1) is not completely accurate because the meniscus is assumed to be tangential to the lid. In fact it has been observed that near the lid surface, the meniscus shape could be convex. This discrepanc y may affect the estimation of the meniscus cross-sectional area and in turn the total tear volume. The calculated area of the tear menisci based on the geometry shown in Figure A1 and for the normal value for Rm listed in Table 2-1, is about 0.03 mm2 which is smaller than the value of 0.05 mm2 obtained by Mishita et al.2 However it is noted that Mishita et al2 instilled 1 L of fluorescein solution pr ior to the observation, and based on the eyelid perimeter listed in Table 2-1, this volume corresponds to an increase in cross section of about 0.02 mm2, which is very close to the difference between the computer area based on Figure A1 and the measured value. Therefore, we conclude that even though the geometry shown in Figure A1 is not precise, it serves as a reasonable approximation to predict the menisci area and the tear volume.
167 Figure A-1 Simplified meniscus cross-sectional area geometry
168 APPENDIX C DERIVATIONS IN THE CONJUNCTIVA TRANSPORT MODEL Flux Equations The flow of ions through the ion channels and the paracellu lar pathway driven by both the concentration gradients and the electric potentia l difference is modeled by the constant field equation.135 RT F zV RT F zV C C RT zFV P Jion ion ion ion 12 12 2 1 12 12 ,exp 1 exp (A9) where Jion,12 is the flow rate of ions from compartment 1 to compartment 2, Pion is the permeability of the ions through the channel or the paracellular pathway, z and F are the ion valence and the Faradays constant, R is the universal gas constant, T is temperature, V12 is the potential difference between compartment 1 and compartment 2, and Cion,1, Cion,2 are the concentrations of the ion in compartment 1 a nd 2, respectively. This equation is used for the apical Cl channel, basolateral K channel and the paracellular pathway of Na, K and Cl. The temperature is set to be 310 K and the ion permeab ilities are either calcul ated from experimental data (see below), as in the case of Cl channel a nd K channel, or adopted from literature, as in the case of paracellular ion transport. Alternatively, when the solute is passivel y driven by only concentration gradient, the following equation is used: 2 1 12 solute solute solute soluteC C P J (A10) where Jsolute,12 is the flow rate of the solute from compartment 1 to 2, Psolute is the permeability of the solute, and Csolute,1, Csolute,2 are the solute concentrations in compartment 1 and 2, respectively. Equation (A10) is applicable fo r the paracellular glucos e transport, and the
169 paracellular ion transport in the short-circu it case, where there are no potential difference between compartment 1 and 2 as a driving-force fo r ion flow. It is noted that for the case of short-circuit experiments, i.e., V12= 0, Equation (A9) reduces corre ctly to equation (A10). The permeabilities for these cases are adopted from literature. The kinetics of Na-K pump and the Na-K-Cl cotransporter are desc ribed using MichaelisMenten model. 25 1 10 53 2 3 max cb K pump b K b K Na pump c Na c Na pump pumpV K C C K C C J J (A11) ) (, 2 , 1 , , 2 , 1 , , max Cl K Na Cl c Cl c Cl Cl K Na Cl c Cl c Cl Cl K Na K c K c K Cl K Na Na c Na c Na Cl K Na Cl b Cl b Cl Cl K Na Cl b Cl b Cl Cl K Na K b K b K Cl K Na Na b Na b Na Cl K NaCl K NaK C C K C C K C C K C C K C C K C C K C C K C C J J (A12) In equation (A11), Jpump and Jpump,max are the turnover rate and the maximal turnover rate of the pump. Since this transpor t is not electroneutral, the a bove equation includes a potential dependent term68. In equation (A12), JNa-K-Cl and JNa-K-Cl,max are the turnover rate and the maximal turnover rate of the cotranspor ter. The Ks are the Michaelis constants. The values of the maximal turnover rate and the Michaelis cons tants are adopted form the literature. The kinetic equation of Na-Gluco se cotransporter is based on the study by Eskandari et al. on the rabbit intestinal Na-Glucose cotransporter that was cloned in Xenopus laevis oocytes.136 From a six-state transport model which assume s an ordered binding of Na and glucose and a stoichiometry ratio of 2:1 between Na and glucose, the turnover rate of the cotransporter can be expressed as a function of the c oncentrations of sodium and gluc ose in the apical and cellular compartments and the potential difference between the two compartments.
170 a Glu a Na a Na a Glu a Na a Glu a Na T Glu NaC C C C C C C C J, 2 2 2 2 (A13) where CT is a constant representing the density of th e cotransporter and can be obtained from the experimental data of JNa-Glu, and and are functions of CNa,c, CGlu,c and the potential difference, and contain kinetic constants directly adopted from the literatures.136 The same equation as (A13) was earlier developed by Parent et al.137, but Eskandari et al.136 used additional data from experiments done in the reverse mode, i.e. the turnover of th e cotransporters from inside to outside the cells. Both of these stud ies obtained kinetic parame ters for equation (A13) by fitting the experimental data to the equation. To confirm which parameters are more accurate, we simulated the Na-K pump inhibition in Ussing-chamber experiments by using kinetic parameters from both of these studies. Of these two sets of parameters only the ones obtained by Eskandari et al.136 correctly predict the decrease in tur nover rate of Na-Gluco se contransporters after pump inhibition.35,36 The reason for such decrease is that the inhibition of the pump leads to decreased Na gradient across the apical membra ne. The dependence of th e cotransporter turnover rate on intracellular Na concentration is weak for the parameters obt ained by Parent et al.,137 and therefore the change in cellular Na concentratio n due to Na-K pump i nhibition cannot lead to detectable difference in Na-Glucose cotranspor t turnover rate. The above comparison shows that the conjunctiva transport model can be also us ed to compare and evaluate different kinetic models for a specific transport pathway. The flux of water is assumed to be the sum of two parts: one part driv en by electro-osmosis through the paracellular pa thway, and the other part, osmotic fl ow, driven passively by osmosis difference through both the transcellular and pa racellular pathways. Th e first part can be described as: 25
171 ar paracellul ion ion eo w wJ z F r v J, (A14) where w is the partial molar volume of water, Jion,paracellular is the paracellular ion flux, zion is the valance of the ions and F is the Faradays constant, and reo is a constant denoting the coupling ratio between the water flow driven by electr o-osmosis and the paracellular current. This coupling ratio is estimated to be 0.0073 C-1mol. from the experimental data for water transport in the absence of transepithelial osmosis difference.15 The paracellular current is first calculated using the model under open-circuit condition with th e apical and basolatera l concentrations used in the experiment and cellular composition given in Table 2-1, and then the coupling ratio is calculated so that the resulting paracellular water tr ansport rate is the same as the fluid secretion rate measured in the experiment. The water tran sport driven by osmosis difference is described by: 1 2 2 1 2 1 ,Osm Osm v P Jw w w (A15) where Jw,1-2 is the water flux from compartment 1 to compartment 2, Pw,1-2 is the water permeability of the barrier between compartments 1 and 2, w is the partial molar volume of water, and Osm1, Osm2 are the osmolarities in compartments 1 and 2, respectively. It is assumed that the osmotic flow exists for both the trans cellular and the paracellular pathways. The water permeabilities are adopted from measurem ents on rat reported in literature.47 It is noted that the value used as the paracellular water permeability is actually that of the whole tissue, which may be an overestimation. But it can be shown by simulation that such an overestimation is not likely to introduce significant error in this study. Kinetic Parameters Of the required parameters in the above kineti c equations for transport pathways, some are available in literature, as listed in Table 2-1 in the main text. On the other hand, the parameters
172 listed in Table 2-2 need to be obtained by fitting experimental data. Briefly, according to the Ussing-chamber experiments,36,37 apical Cl channel accounts for 60% of the total short-circuit current (ISC) of 14.4 A/cm2 at the steady state. Therefore th e flux of Cl through the apical Cl channel must be 0.6ISC/F, where F is the Faradays constant. Using this flux and the apical and the basolateral compositions used in Shi and Candias experimental study37 and the cellular concentrations in Table 2-1 (meas ured values) in equation (A9), we compute the permeability of the Cl channel (PCl,channel), which is listed in Table 2-2. Based on the mechanisms shown in Figure 2-2, if Cl channel accounts for 60% of the ISC, the Na-Glucose cotransporter should account for remaining 40%. Therefore, the flux of Na through the Na-Gluco se cotransporter can be obtained as 0.4ISC/F, and the parameter CT can be calculated from equation (A13) by using the same apical, basolateral and cellular com positions as used above to determine PCl,channel. Using stoichiometrical relations that can be obtained from Figure 22, the steady state fluxes through the basolateral K channel, the Na-K pump and the Na-K-Cl cotransporte rs can be calculated. Using a similar approach as described above for the Cl channel and Na-Glucose cotransporters, the K channel permeability (PK,channel), the maximal turnover rate of Na-K pump (Jpump,max) and the maximal turnover rate of the Na-K-Cl cotransporters (JNa-K-Cl,max) can be obtained from their respective kinetic equations, which are (A9), (A12) and (A13), respectively. When using Equation (A12) to calculate Jpump,max, the transcellular potential difference is assumed to be -50 mV, with the extracellular compartment as refe rence, which is based on a typical value for cornea cells.25 The same apical, basolateral and cellular compositions are used in these calculations as used above in the estimation of PCl,channel.
173 APPENDIX D DERIVATION OF THE TAYLOR DISPERSION COEFFICIENT D* Below we derive D* for the shearing motion as an example. The 2D mass balance of the instilled solutes can be descri bed by equation (42) Three time scales are involved in this problem, the time scale of blinking (1/ ), the time scale of diffusion in x direction (L2/D) and the time scale of diffusion in y direction (h0 2/D), where is the angular velocity of blinking, L is the characteristic length in x directi on and h is the characte ristic length in y direction. For the current problem, the period of the blink cycles is assumed to be 6 s and therefore is about 1.05 rad/s, h is about several microns and L is about several millimeters. It can be shown that the first two time scales are comparable for this problem ( h2/D~1), and they are both much shorter than the time scale for diffusion in x direction. By using a multiple time scale analysis, the solute concentration can be treated as a function of positions ( L x and 0h y ), a short time scale (short) represented by t, and a long time scale (long) represented by Dt/ L2, i.e. 2, ,L Dt t c c As described in equation (38), the velocity of the tears in the lower fornix for the shearing scenario should be t f t, u' 0 (A16) in which =y/h0 and h0 is the normal thickness of the tear film. This periodic function can be wr itten in Fourier series form n 0 n n i n n 0e d t, u (A17) where i is the imaginary unit, dn is the Fourier series coeffi cients and can be obtained by
174 T t t i n ndt e t, u T 1 d (A18) Equation (A17) can be substituted into Equation (A15) and in this case the velocity in y direction is zero. The dimensionless form of the resulting equation is 2 2 2 2 2 n 0 n n i n n 0 l 2 s c c c e d D h d c c (A19) where =h2 0/D is the ration between the time scale of blinking and the time scale of the diffusion in y direction, and it can be shown that these two time scales are comparable. It can also be shown that d0h/D~1. Because h0/L<<1, the concentration can be expanded in as the following 2 2 1 0 c c c c (A20) After substituting Equation (A20) into (A19), the resulting equation can be analyzed at different order of 0 (A21) with 00 c at =0,1 This suggests that c0 does not depend on either short or 1 2 1 2 0 n 0 n n i n n 0 s 1 c c e d D h d c (A22) with 01 c at =0,1 2 0 2 0 c cs
175 Assuming c e c D h d c0 m 0 m m i m m 1, 0 1 Equation (A22) becomes n 0 n n i m 2 m 1, 2 n 0 n n i n n s im m 0 m m m 1,e c e d e c s (A23) with 0, 1 mc at =0,1 2 c c c e d D h d c c 2 2 2 0 2 1 n 0 n n i n n 0 l 0 s 2 (A24) After integration from 0~2 with respect to s and from 0~1 with respect to and some algebra, Equation (A24) beco mes Equation (43), with csac approximated by the leading order concentration c0 and with D* defined as 1 0 2 0 n 1 n s n 1, n 2 0 *d d c d 2 D h d 1 D D (A25) After solving for c1,-n() from Equation (A23) and the corresponding boundary conditions and substituting the result into E quation (A25), the expression for D* for the shearing scenario can be obtained. 1 n n n n n n 2 2 2 0 2 n n *cos cosh sin sinh 1 2h N n d 4d 1 D D (A26) which is Equation (44). Using a similar appro ach, the dispersion coefficient for the squeezing case in Equation (46) can also be obtained.
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186 BIOGRAPHICAL SKETCH Heng Zhu was born to Quanshou Zhu and Yimei Zhang on May 19, 1980 in Suzhou, China. He grew up in Beijing, China and gra duated from Beijing No. 5 High School in 1998. He was then admitted to Tsinahua University, China and obtained his bachelors degree in chemical engineering in 2002. After that he joined Dr. Anuj Chauhans group in the University of Florida to pursue his PhD in chemical engineering. His research focuses on the dynamics of tears and other fluids on the ocular surface, and its imp lications on ocular ailments and ophthalmic drug delivery.