Modeling of Drug Delivery to the Human Eye

Home , Human eye

MODELING OF DRUG DELIVERY TO THE HUMAN EYE

A Thesis

Presented to the

Faculty of

California State Polytechnic University, Pomona

In Partial Fulfillment

Of the Requirements for the Degree

Master of Science

Mechanical Engineering

Withanage D. Mahathanthila

2017

SIGNATURE PAGE

ii ABSTRACT

The human eye can mainly be divided into two chambers in numerical modeling: anterior

and the posterior lobes. The treatments for diseases in each of these sections differ. For

instance, the posterior segment, being more inward involves highly invasive treatments

such as surgery, whereas anterior eye diseases rely on topical drug applications. Even

though it appears to be complicated, surgery can assure the direct and efficient drug

delivery. Delivery of a necessary amount of topically applied drugs to the targeted site in

the anterior lobe of the eye can depend on multiple factors including the temperature

profile, anatomy of the sub segments as well as the fluid flow rate in the eye. In the current

study, the drug transportation in the anterior eye has been studied using numerical

modeling. The model considers the spatio-temporal behavior of the drugs in towards the

trabecular meshwork (TM), iris and the lens when applied to the corneal surface. The

aqueous humor (AH) plays a major role in the drug delivery procedure through natural

convection. These movements can depend on the temperature difference between the

environment (surface of the eye) and the targeted eye segment. The current study has been

able to depict the temporal drug concentrations changes in the TM, iris and the lens. These predictions display that the peak average drug concentration is found within one hour after application whereas it will be completely removed after five hours. Additionally, these

concentrations depend on the temperature differences, for instance the minimal peak

concentration in the eye is observed when the temperature of the environment is the same

as the ocular temperature.

iii

TABLE OF CONTENTS

Signature page ...... ii

Abstract ...... iii

List of Tables ...... vi

List of Figures ...... vii

Nomenclature ...... viii

1. Introduction ...... 1

1.1 Eye components considered in the model ...... 4 1.1.1 Cornea ...... 4 1.1.2 Sclera...... 5 1.1.3 Aqueous Humor (AH)...... 7 1.1.4 Trabecular Mesh-work ...... 8 1.1.5 Ciliary Body ...... 9 1.1.6 Iris ...... 9 1.2 Why topical drug delivery? ...... 9 1.3 Common diseases treated by topical drugs ...... 10 1.3.1 Glaucoma ...... 10 1.3.2 Cataract ...... 11 1.3.3 Uveitis ...... 13 2. Model selection ...... 15

3. Numerical model ...... 17

3.1 Eye Geometry ...... 17 3.2 Governing equations and initial/boundary conditions ...... 18 3.2.1 Transport of drug concentration...... 18 3.2.2 Aqueous Humor (AH) Flow ...... 21 3.2.2.1 The Boussinesq approximation ...... 22

3.2.3 Temperature profile ...... 22 4. Numerical results ...... 24

4.1 Steady State Velocity Distribution...... 24 4.2 Steady State Temperature Distribution of entire eye ...... 25 (Ambient Temperature 298K)...... 25 4.2.1 Temperature Distribution respect to different ambient temperatures ...... 26 4.3 Drug concentration distribution ...... 29 5. Validation of the results ...... 33

6. Conclusion ...... 35

References ...... 36

Appendix ...... 39 Convergence plot for transient solver ...... 39 Convergence plot for non-linear solver ...... 39

LIST OF TABLES Table 1. Eye model Geometry Dimensions ...... 17

Table 2. Diffusion coefficients of ECA in different domains ...... 19

Table 3. Thermo-physical properties in different domains ...... 19

Table 4. Parameters to solve steady state problem ...... 21

Table 5. Parameters used to find temperature profile ...... 23

LIST OF FIGURES

Figure 1.Structure of the eye and the cornea ...... 5

Figure 2.Front and side view of sclera ...... 6

Figure 3.Aqueous humor flow through trabecular meshwork ...... 8

Figure 4.Types of Cataracts ...... 12

Figure 5.Schematic of the eye ...... 13

Figure 6.Simulation domains of drug concentration...... 18

Figure 7.Steady State velocity distribution of the anterior eye ...... 24

Figure 8.Steady State Temperature profile ...... 25

Figure 9.Temperature variation along pupillary axis of the eye ...... 25

Figure 10.Temperature and velocity profile at ambient temperature 308K ...... 26

Figure 11.Temperature and velocity profile at ambient temperature 310K ...... 27

Figure 12. Temperature and velocity profile at ambient temperature 314K ...... 28

Figure 13.Temporal evolution of the drug at 3 different drug targets ...... 32

Figure 14.Comparison of results with previous reports ...... 33

Figure 15. Comparison of velocity profiles ...... 34

Figure 16.Comparison of Drug concentration at a point (TM, Iris, Lens) ...... 34

vii

NOMENCLATURE

c specific heat (J kg-1 K-1)

C ECA concentration (µM)

D diffusion coefficient (m2 s-1)

g gravitational acceleration (m s-2) h heat transfer coefficient (W m-2 K-1)

k thermal conductivity (W m-1 K-1)

T temperature (K)

v velocity vector (m s-1)

Q volume flow rate of the aqueous humor (m3/s)

p Pressure (Pa) s Non-dimensional parameter

E evaporation rate (W m-2)

viii

Greek symbols

β volume expansion coefficient (K-1)

µ dynamic viscosity (kg m-1 s-1)

-3 ρ0 reference density (kg m )

-3 ρ density (kg m )

σ Stefan-Boltzmann constant (W m-2 K-4)

Subscripts

ah aqueous humor

amb ambient bl blood

cor cornea

ref reference

1. INTRODUCTION Current day biological and physiological research studies require model systems to depict

the safe and efficient use of suggested experiments on humans. Therefore, the bottleneck

in this human physiology studies lies in selecting a suitable animal, fly or virtual model to

display the initial results before moving to the clinic [1, 2]. For instance, most of the current

pharmacological studies depend on computational modeling and stimulations to calculate

the solubility as well as permeability of the millions of drugs that are being synthesized by

big pharma each day [3]. The utilization of these computational models provides an

interphase to closely look at the interactions of the drugs with the target tissues and proteins

of humans, thereby ensuring the safety of them to be consumed by living systems. One of

the most important benefits of computational modeling experiments involve the cost

efficiency due to the ability to avoid setting up wet lab experiments with sophisticated

techniques and equipment. Therefore, a computational model based initial screening allows

to optimize the basic parameters prior to setting up the actual experiments [4].

While these stimulations are extremely helpful for researchers in the natural sciences,

modeling of engineering processes will be highly cost efficient in optimizing the essential

parameters. In the current study, we aim to present a model system for drug delivery to the human eye. Eyes are very important features for humans as well as most animals in providing visual aids thereby giving an idea of the surroundings. The vision is essential for self-defense mechanisms contributing to the survival of life. For example, our eyes provide clues as to the obstacles in front of us thereby sending signals to the brain to avoid them as well as to see the beauty of the world [5]. Therefore, hurting eyes will be a great loss for anyone and the main goal of any patient with damaged eyes will be to regain their vision

1 through medication. In addition to accidents, there are several common diseases associated with eyes.

The eye has several chambers and if the ailments/diseases are in the anterior part, the main type of treatments will include topical drugs (i.e. ointments/eye drops). Any damages to posterior eye will involve more complicated procedures such as intravenous drug delivery and surgery [6]. However, the current drug treatments have their own drawbacks, including dilution of the compounds with tears or simply by turnover [7]. In addition, passing many barriers to reach the target tissue is a problem when achieving required drug concentrations.

Therefore, current research studies in the field focus on developing novel drug delivery methods to the eye. The major problem with developing these methods is the inability to go through a ‘trial and error’ approach with the delicate and precious organ of human eye

[2]. Computational modeling will be highly useful and applicable under these circumstances to determine the parameters of the drug delivery methods thereby enabling the optimization of the drug delivery methods without invading the eye. These methods enable the determination of drug distribution along the different segments of the eye, concentration profiles as well as the dependency of these factors on temperature and pressure differences in the chambers of the eye. Current mathematical modeling studies on drug delivery to the eye have been able to perform detailed analysis on specific drugs, exploring the transport mechanisms at micro particle and Nano-structure level [8]. For instance, a model developed for the drug brimonidine allowed to validate the predicted results of Nano-structured microparticles of the drug being the most effective than other forms across the ophthalmic delivery route [9].

The development of computational methods to investigate drug delivery and distribution in the eye requires thorough understanding of the anatomy of the eye, spatial-temporal profiles of the eye as well as the target tissues corresponding to the diseases related to the eye.

1.1 Eye components considered in the model 1.1.1 Cornea Cornea is the transparent outer cover of the eye. It does not contain any blood vessels, rendering the transparent quality [10]. The lack of nutrition supply due to the absence of blood flow is compensated by tears and a fluid that is in located behind the cornea in the front part of the eye known as aqueous humor. Cornea consists of highly organized structure with total of five layers that consists of three basic layers separated by two thin layers. The outer most layer as well as the first basic layer is called the epithelium. As its location implies, the major function of this layer is to protect the eye from external materials such as dust, bacteria or any other foreign material, while maintaining the surface for the absorption of oxygen and other nutrients. The reason for high sensitivity of any physical interactions on the eye (rubbing/scratching/hitting) is due to the presence of large number of nerve endings in the epithelium [11]. Epithelium is separated from stroma through Bowman’s membrane, which is mainly constituted from collagen. Any damages to this layer can cause scarring on this membrane, which can ultimately lead to loss of vision. The next basic layer stroma majorly comprises of water and collagen generating the thickest layer of the cornea. Even though it is a thick layer, the assembly of collagen fibers is able to retain the light transparency of cornea. The next thin layer is known as Descemet’s membrane, which provides protection from injuries and infections. This membrane also mainly comprises of collagen, yet the differences in the arrangement of these fibers lead to difference in thickness and properties than stroma. The next basic layer is thin and known as the endothelium. There are various fluids flowing through the inside of the eye to the stroma and these fluids need to be pumped out to avoid swelling of the stroma. Endothelium takes are of removing this excess fluid. The cornea surface contains a thin layer of an oily substance that prevents evaporation of the eye’s tear film which is produced by the Meibomian glands.

Figure 1. Structure of the eye and the cornea [1] 1.1.2 Sclera

Sclera is the tough outer covering of the eyeball identified as ‘white’ of the eye[12].

Conjunctiva is responsible for covering the sclera as well as providing lubrication to the eye. Extending from cornea to the optic nerve, sclera provides around 80% of surface of the eye and it has three divisions. Episclera is the outer most connective tissue that is directly connected to the conjunctiva. The dense area that provides the white color is known

5 as sclera proper. Inner most zone of sclera is made up of elastic fibers and is identified as lamina fusca [13].

Sclera contributes with several important functions to the eye. It is mainly involved in maintaining the shape of the eye ball as well as the eye pressure. The collagen and elastin fibers can provide a protective surface towards external damages. Due to the make-up of the sclera it can be involved in attachment to the external muscles thereby controlling the movement of the eye. Due to these important functions as well as the large surface area, sclera can be vulnerable to several ailments. E.g. yellow sclera, blue sclera as well as inflammation incidents. As implied by the functional properties, sclera becomes important in maintaining eye pressure, which can cause severe conditions when not controlled [14].

Figure 2. Front and side view of sclera

1.1.3 Aqueous Humor (AH)

There are three chambers in the eye. The anterior chamber that is located between

cornea and iris, posterior chamber situated in between the iris and eye lens [15]. The vitreous chamber the large space in the back of the eye. Aqueous humor is present in both anterior and posterior chambers of the eye. As the word implies aqueous humor a clear, light fluid that is made up of water and nutrients.

The nutrients in aqueous humor including amino acids, glucose and vitamins nourishes the cornea and the lens. These liquids and solutes are also able to maintain eye pressure as well as the shape of the eye. In addition, aqueous humor can also be understood as a blood surrogate to the eye, providing necessary nutrients as well as removing the metabolites from the eye [16]. Aqueous humor production and drainage are controlled under an equilibrium, thus any influence on this control can lead to abnormal IOP levels, which is one of the main reasons of glaucoma.

Formation of aqueous humor can be under active processes as well as passive processes. Two passive processes, diffusion and ultrafiltration contributed by lipid soluble and water soluble substances respectively contribute to between 10-20% of aqueous humor formation. Active secretions, majorly through non-pigmented epithelial cells contribute to 80-90% of aqueous humor formation [17].

1.1.4 Trabecular Mesh-work The trabecular meshwork is a tissue located in the anterior chamber of the eye. It is

important for the drainage of aqueous humor from the eye and therefore plays and

important role in determining the IOP [18].

Trabecular Mesh work considered as the outlet of the anterior chamber which

passes AH in the anterior part of the eye to the back of the eye. The conventional method

or route of AH exit is through the well-organized tissue network of the trabecular meshwork flowing in to the Schlemm’s canal (Figure 3). Developed resistance for the drainage of AH through the TM or Schelmm’s canal ultimately leads to lack of control of intraocular pressure, causing initiators for diseases such as glaucoma [19].

Figure 3. Aqueous humor flow through trabecular meshwork [18] 8

1.1.5 Ciliary Body Ciliary body is located on the posterior side of the iris facing the choroid and arranged as a

ring of tissues around the eyeball. It provides a platform for the ciliary muscle, which is responsible

for the curving of the lens [20]. Ciliary body is part of the uvea, which is responsible for the delivery

of oxygen and nutrients to the eye. The ciliary epithelium is the site of aqueous humor production;

thus, it has influence on the AH dynamics.

1.1.6 Iris The iris is located at the anterior segment of the eye in between lens and the cornea. It is responsible

for assigning eye color. Iris contains micro vessels, which gives the property of high absorbency of

drugs. These absorbed drugs can be micro circulated efficiently and it can be assumed that the drugs

will not reach the posterior side of the eye during delivery mechanisms [21].

1.2 Why topical drug delivery? Topically applied drugs have two main purposes. It can be a treatment to an infection such as conjunctivitis, keratitis sicca etc. or to treat to ocular diseases such as Glaucoma, cataract and Uveitis. Once the drug is applied to the eye it will penetrate through the cornea towards anterior of the eye. Topically applied drugs have the ability to reach target locations such as TM, Iris, CB and lens.

1.3 Common diseases treated by topical drugs 1.3.1 Glaucoma

Glaucoma is mainly identified as a group of diseases that lead to increase intraocular

pressure (IOP), there by leading to damages to the optic nerve [22]. These conditions can

ultimately lead to blindness and there are around 200,000 Glaucoma cases reported every

year in the United States. The increased pressure in the eye has been reported to be the

major cause of this ailment. Since it leads to fiber loss on optic nerve head (ONH), the

disease is more specifically known as glaucomatous optic neuropathy (GON). However, it

cannot be generalized that all IOP conditions lead to ONH or it is essential for all GON

patients to have IOP. Additionally, there are several other forms of glaucoma such as low-

tension or normal-tension glaucoma, angle-closure glaucoma, congenital glaucoma,

secondary glaucoma, pigmentary glaucoma and pseudoexfoliation glaucoma.

Identification of the disease at an early stage is the most effective way to obtain successful

treatment, even though this detection has been known to be the most difficult task. This is

due to the delay in functional changes such as loss of vision, which starts with a tunnel

vision condition and do not become observable until the morphological effects get severe.

Current treatment mainly includes lowering the optic pressure, thereby controlling the

damages to the optic nerve. There are several methods of glaucoma treatments. Medicines

such as eye drops and pills can help in lowering the eye pressure mainly by decreasing the

fluid production in eyes. Radiation based therapies can be used to aid in fluid drainage

from the eye. However, these therapies such as laser trabeculoplasty can cause

10 inflammation, which will need to be treated with additional medication. In addition, conventional surgery can be used to create a novel opening in the eye for the fluid to exit.

However, the drawback is this approach cannot be used for patients who have already gone through eye surgery such as cataracts [23].

1.3.2 Cataract

Cataracts can be identified as the major reason for impairment of eyesight around the world, accounting for over 22 million Americans and around 50% of blindness worldwide. It can be addressed as an age-related condition, where there is higher tendency to develop cataracts for people over the age of 40, while majority of the people over the age of 65 have shown to develop cataract like conditions at least on one of the eyes. In addition to gage, diabetes, smoking, over-exposure to the sun, as well as special medication requirements (e.g. steroids) can increase the risk of developing cataracts [24]. The location of the cataract on the eye can be nuclear, cortical or posterior subcapsular

(Figure 4). These differences as well as the stage of the cataract development can give rise to various symptoms including unclear vision, sensitivity to lights, decrease in color and contrast appearance, etc. However, like most of the eye ailments, cataracts can be difficult to be detected at an early stage unless tested at a professional eye clinic.

The treatment for cataracts is not considered emergency procedure. Therefore, it gives patients to diagnose and plan accordingly depending on the effect of this ailment to their day-to-day life. Cataracts surgeries are reported to be almost 100% successful with very little complications. During the cataract removal surgery, the affected lens will be replaced with an intraocular lens (IOL) [25].

Figure 4.Types of Cataracts

1.3.3 Uveitis

Uveitis is generally considered a rare disease due its low persistence of 38 persons out of 100,000.

However, it is considered the 4th common reasons for blindness around the world mostly affecting people younger than the age 40. It is caused by continuous inflammation inside the eye, mainly in the uveal tract and neighboring parts [26]. The rareness of the disease can cause failures in diagnosing the condition at an early stage, thus leading to increased inflammations, which can ultimately cause permanent loss of vision.

Figure 5.Schematic of the eye

Three types of Uveitis have been identified. Incidents of inflammation and swelling of the uveal tract near the front of the eye is identified as anterior uveitis. This condition usually appears and gets better within 2 months, but it can continue for a longer period as well as appear intermittently. Occurrence of symptoms in the middle part of the eye is known as intermediate uveitis and this condition can linger for years, going through cycles of various lengths. The third type which is known as posterior uveitis occur in the back part of the eye, which can occur slowly yet lasting for longer time periods.

Unfortunately, medical professionals have not been able to define specific reasons for uveitis. There are several possible hypotheses as uveitis enhances. For instance, people who have encountered severe viral infections such as herpes simplex virus, lime disease, etc and parasites are thought to be at high risk. In addition, inflammatory diseases and injuries to the eye are also thought to be enhancing the chances of getting uveitis. This ailment is usually identified by symptoms such as red eyes without pain, unclear vision, increase of the sensitivity of the eye as well as sudden appearance of ‘floating’ objects in front of the eyes. Since uveitis is thought to be connected to other diseases, multiple tests may be required in addition to eye tests. However, the treatment is mainly topical with eye drops helping to relieve inflammation as well as dialate pupils to aid in decreasing the pain [27].

2. MODEL SELECTION The drug distribution across the eye can depend on various parameters including the temperature difference across the eye, the flow rate of the aqueous humor (AH), anatomical details of the segments of the eye as well as the application form of drug (liquid/ointment).

Therefore, an ideal numerical model to predict the drug distribution in the eye as well as the general AH flow will have to consider these essential factors into their calculations.

Prior to initiating drug delivery calculations, the eye related numerical models focused on determining the temperature distribution of the eye [28, 29]. The successful calculation of the temperature profiles of the eye, corresponding to experimental results rendered the hallmark of mathematical modeling studies for ophthalmic drug delivery. However, the main drawback in the early calculations was the assumption of the stagnant nature of the

AH and therefore despite the successful match with the experimental results these numerical models were questioned due to unsubstantiated assumptions.

Deriving from these doubts the later methods [30, 31] took into consideration the AH flow in the eye and the results displayed unsymmetrical nature of the temperature profiles in contrast to initial results. However, these second-generation methods mainly reported in these studies failed to derive the corneal and total eye ball temperature distributions.

Therefore, progressing from these caveats, two dimensional and three dimensional models

[32-34] were developed assuming a correlating relationship between AH flow and temperature profiles. These studies could confirm the temperature dependency of the natural convection of AH. Most of these temperature distributions have been analyzed for detailed circumstances of extreme ambient temperature (-10 °C, 60 °C) conditions [35, 36],

laser-thermokeratophasty [37], eye tumor [38], as well as laser surgery. The results of these 15

mathematical modeling studies correlated with the experimental results further confirming

the adaptability of calculations to predict wet-lab experiments.

The discoveries on the correlating properties of AH flow and the ocular temperature,

rendered the adaptation of these mathematical models for the determination of drug

distribution in the eye. Early studies on drug distribution models focused on the posterior

chamber due to the highly invasive nature of the treatments applied for this segment of the

eye [39-41]. Numerical modeling applications for the topical drug distribution in the

anterior chamber of the eye has not been popularly used until later. Ferreira et al. developed

a 1D mathematical model under the assumption of stagnant aqueous humor to predict the

drug distributions of the anterior lobe when using therapeutic contact lenses. One of the

later studies by Lin and Yuan calculated the drug flow from the pre-corneal segment to the

trabecular meshwork of the eye [42]. Even though they considered the AH flow towards

their calculations, the natural convections that play an equal role were neglected. Therefore,

an ideal model for drug distribution predictions should include of flow profile as well as

temperature profiles. Considering the essential parameters, Wyatt et al proposed a numerical model including the temperature distribution [43], yet the difference in the temperature across the eye was limited to 1K (Kelvin).

In the current study, we aim to use a 2D model considering the total structure of the eye together with AH flow, temperature distribution profiles as well as the natural convections to obtain versatile numerical predictions of the drug distribution of the anterior chamber of the eye.

3. NUMERICAL MODEL 3.1 Eye Geometry Commercial Finite Element analysis software COMSOL Multiphysics was used to analyze the drug delivery to the anterior portion of the eye. The eye 2-D axisymmetric geometry

was created using the data in [35]. Dimensions are tabulated below.

Table 1. Eye model Geometry Dimensions

Eye model Geometry Dimensions

Domain Geometric Parameter Value

Sclera Thickness 1.0 mm Inner half-axis (HA) in the r direction 11.5 mm Inner (HA) in the z direction 12.0 mm

Anterior chamber HA in the r direction 7.8 mm HA in the z direction 7.8 mm

Cornea Thickness 0.52 mm

Lens HA in the r direction 4.5 mm HA in the z direction 1.5 mm

Iris Thickness 0.4 mm Angle between iris and the r-axis 6.45º

Laminar flow module, heat transfer module and transport of diluted species module were

used as physics of the problem in COMSOL Multiphysics.

Initially, following equations were solved to obtain the steady state velocity and

temperature distributions. Since the problem involves natural convection the momentum

equation is coupled with the temperature in the energy equation. Boussinesq approximation

is used to simplify the variations of the density of the AH.

3.2 Governing equations and initial/boundary conditions 3.2.1 Transport of drug concentration Drug concentration distribution with respect to time and space is considered in the highlighted area of the eye domains.

Figure 6. Simulation domains of drug concentration

Changes in drug concentration in the anterior portion of the eye has been numerically calculated through the unsteady convection-diffusion equation

+ = 𝜕𝜕𝜕𝜕 2 𝑣𝑣⃗∙ ∇𝜕𝜕 𝐷𝐷∇ 𝜕𝜕 𝜕𝜕𝜕𝜕

Where C denotes the drug concentration while denotes the velocity field of the AH of the anterior chamber and posterior chamber. Mass𝑣𝑣⃗ diffusion coefficient D, for different domains have been tabulated in table 2 taken from [37,38]

Table 2. Diffusion coefficients of ECA in different domains

Diffusion coefficients of ECA in different domains

Domain Cornea AH Iris Lens Sclera CB

Diffusion Coefficient 1.31 × 10-6 7× 10-6 1.31× 10-6 4.85× 10-6 4.85× 10-6 4.85× 10-6 (cm2 s-1)

Table 3. Thermo-physical properties in different domains

Thermo-physical properties in different domains

Cornea AH Vitreous Iris Lens Sclera TM CB

Thermal 0.58 0.58 0.603 1.0042 0.4 1.0042 1.0042 1.0042 Conductivity k (W m-1 K-1) Specific Heat c 4178 3997 4178 3180 3000 3180 3180 3180 (J kg-1 K-1) Density ρ 1050 996 1000 1100 1050 1100 1100 1100 (kg m-3)

Three different cases of drug delivery through cornea been considered. Initial drug concentration applied on top of cornea ( ) is assumed to be a constant value at all time

𝑐𝑐𝑐𝑐𝑐𝑐 steps which is an idealized situation. In 𝜕𝜕this case the constant concentration ( ) is taken

0 as 75µM [40] 𝜕𝜕

𝜕𝜕𝑐𝑐𝑐𝑐𝑐𝑐 𝜕𝜕0

Two other drug concentrations analyzed in the analysis were mentioned below. Both equations represent different half-lives of the drug. First order kinetic equations.

= × . ( ) −0 0039𝑡𝑡 𝜕𝜕𝑐𝑐𝑐𝑐𝑐𝑐 𝜕𝜕0 𝑒𝑒 𝜇𝜇𝜇𝜇 = × . ( ) −0 00038𝑡𝑡 𝜕𝜕𝑐𝑐𝑐𝑐𝑐𝑐 𝜕𝜕0 𝑒𝑒 𝜇𝜇𝜇𝜇 =

𝜕𝜕𝑐𝑐𝑐𝑐𝑐𝑐 𝜕𝜕𝐶𝐶𝐶𝐶𝐶𝐶𝑒𝑒𝐶𝐶𝐶𝐶 𝑆𝑆𝑆𝑆𝐶𝐶𝑆𝑆𝐶𝐶𝑆𝑆𝑒𝑒 𝐷𝐷𝐶𝐶𝑆𝑆𝐷𝐷 𝜕𝜕𝐶𝐶𝐶𝐶𝑆𝑆𝑒𝑒𝐶𝐶𝜕𝜕𝐶𝐶𝐶𝐶𝜕𝜕𝐶𝐶𝐶𝐶𝐶𝐶 =

𝜕𝜕0 𝐼𝐼𝐶𝐶𝐶𝐶𝜕𝜕𝐶𝐶𝐶𝐶𝐶𝐶 𝜕𝜕𝐶𝐶𝐶𝐶𝐶𝐶𝑒𝑒𝐶𝐶𝐶𝐶 𝑆𝑆𝑆𝑆𝐶𝐶𝑆𝑆𝐶𝐶𝑆𝑆𝑒𝑒 𝐷𝐷𝐶𝐶𝑆𝑆𝐷𝐷 𝜕𝜕𝐶𝐶𝐶𝐶𝑆𝑆𝑒𝑒𝐶𝐶𝜕𝜕𝐶𝐶𝐶𝐶𝜕𝜕𝐶𝐶𝐶𝐶𝐶𝐶

3.2.2 Aqueous Humor (AH) Flow Continuity, energy and Navier-Stokes equations are simultaneously solved for steady

state condition in anterior and posterior chambers.

׏ήݒԦ =0 Continuity

Steady incompressible Navier– ଶ ߩ଴ (ݒԦή׏) ݒԦ = െ׏݌ + ߤ׏ ݒԦ Stokes equation using the + ߩ଴ ൣ1 െߚ൫ܶ െܶ௥௘௙ ൯൧݃Ԧ Boussinesq approximation

ߩܿ(ݒԦ׏ܶ) = ׏(݇׏ܶ) Energy

Boundary Conditions to solve momentum equation ݒԦ =0 No slip at the boundaries of anterior chamber other than inlet and outlet ߤή൫׏ݒ Ԧ + (׏ݒԦ) ൯ή ݊ሬԦ =0 Viscous stresses are assumed to be zero at the out let (TM)

ܳ௔௛ Velocity at the inlet െܵ)݊ሬԦ ௖௕ 1)ܵכ 6 כ ݒԦ஼஻ = െ ܣ௖௕ (Qah-Aqueous-humor Generation rate, Acb-Area of the ciliary body) Table 4. Parameters to solve steady state problem Parameters to solve steady state problem

Parameter Description Value

-3 ߩ଴ Reference density 996 kg m

ߤ Dynamic viscosity 0.00074 N s m-2

ߚ Volume expansion coefficient 0.000337 K-1

ܶ௥௘௙ Reference temperature 307 K

ߩ Mass density of the AH Variable

݃ Gravitational acceleration 9.81 m s-2

3.2.2.1 The Boussinesq approximation Boussinesq approximation is used in the problem to reduce the complexity of the behavior

of density change due to the temperature difference. This method of approximation

neglects the differences in density on the flow field except that they give rise to buoyancy

forces. Approximation of density is a reasonable consideration because the pressure

variation of the AH is trifle compared to the density change due to the temperature gradient

across the eye.

3.2.3 Temperature profile Temperature profile was in obtained by numerically solving the energy equation. Where k

denotes the thermal conductivity and c denotes the specific heat. Conductivity for different media are tabulated in table 3

( ) = ( )

𝜌𝜌𝑆𝑆 𝑣𝑣⃗∇𝑇𝑇 ∇ 𝑘𝑘∇𝑇𝑇 Convection and radiation assumed to be happened through the outer surface of cornea.

Parameters used to find temperature profile have tabulated in the table 5.

Table 5 . Parameters used to find temperature profile Parameters used to find temperature profile

Parameter Description Value

݄௕௟ Convection coefficient of the outer surface of sclera 65 [28] (W m-2 K-1) -2 -1 ݄௔௠௕ Ambient convection co-efficient (W m K ) 10 [44]

ܶ௕௟ Blood temperature (K) 310

ܧ Evaporation rate (W m-2) 40 [29]

ߪ Stefan-Boltzmann constant (W m-2 K-4) 5.67×10-8

ߝ Emissivity of the corneal surface 0.975 [45]

Boundary conditions for energy equation ߲ܶ Convection between sclera outer surface െ݇ = ݄ (ܶെ ܶ ) ߲݊ሬԦ ௕௟ ௕௟ and blood vessels.

߲ܶ Convection, radiation and tear evaporation െ݇ = ݄ (ܶെ ܶ ) ߲݊ሬԦ ௔௠௕ ௔௠௕ (E) through outer surface of Cornea ସ ସ + ߪߝ( ܶ െܶ௔௠௕) + ܧ

23 4. NUMERICAL RESULTS 4.1 Steady State Velocity Distribution

Figure 7. Steady State velocity distribution of the anterior eye Steady state AH flow field in the anterior eye is plotted in figure 7. Arrows show the direction of the

local AH while surface color indicate the magnitude of the velocity. Important parameter which is

ambient temperature, for this flow pattern is 298K. Since the corneal temperature increases from

surface to the posterior, warmer fluids with lesser density tend to move along the corneal surface

creating a counter clockwise direction of flow.

4.2 Steady State Temperature Distribution of entire eye (Ambient Temperature 298K)

Figure 8. Steady State Temperature profile

Figure 9. Temperature variation along pupillary axis of the eye

Figure 9 shows the full temperature profile at the steady state condition at the ambient temperature of

298K. The temperature which is 310K drops from posterior (inner eye) to the anterior surface of the eye. Significant changes of Isothermal lines can be seen at the anterior potion of the eye.

4.2.1 Temperature Distribution respect to different ambient temperatures Temperature profiles of anterior portion of the eye depends on the ambient temperature. Below

temperature distributions are calculated when the ambient temperature is 308K,310K and 314K

respectively. 310K is the body core temperature, which is the boundary temperature of the outer

surface of sclera. Arrow plot shows the direction and the magnitude of the velocities of the aqueous

humor in the anterior chamber. When the ambient temperature is lower than the body core

temperature (310K), aqueous humor (AH) shows a counter-clockwise circulation while the ambient

temperature is higher than the body core temperature AH changes its circulation direction.

(a) (b)

(c)

Figure 11.Temperature and velocity profile at ambi ent temperature 310K

Figure 11 (a) shows the tempera ture distribution when the ambient temperature is same as the body temperature (310K). Figure 11 (c) illustrates that the temperature drop across the entire eye is less than 1K. This temperature drop in dicates the heat loss at the cornea due to tear evaporation.

27 (a) (b)

(c)

Figure 12. Temperature and velocity profile at ambient temperature 314K

Figure 12 (a) indicates the temperature distribution across the pupillary axis. When the ambient temperature is 4 Kelvins above the body temperature the aqueous humor circulation direction changes and its direction to clockwise direction.

28 4.3 Drug concentration distribution

Initially normalized drug concentration was found using the idealized condition ܥ௖௢௥ = ܥ଴

is used to obtain the average concentration distribution. And the drug distribution of the

ି଴.଴଴ଷଽ௧ anterior portion of the eye analyzed afterwards using ܥ௖௢௥ = ܥ଴ × ݁ (ߤܯ) equation which simulates the case of drug application through the cornea in the form of eye drops. Below diagrams shows the temporal and special evolution of average drug concentration.

Time=5 mins Time=10 mins

Time = 15 mins Time = 20 mins

Time = 25 mins Time = 30 mins

Time = 35 mins Time = 40 mins

Time =45 mins Time =50 mins

Time = 55 mins Time = 60 mins

Figure 13. Temporal and special evolution of average drug concentration.

After penetration through the corneal surface by diffusion, drug is advected by AH circulation in the anterior chamber. In the anterior chamber drug is moved along optical axis towards lens and iris. At the mean time drug is diffused through sclera towards Trabecular meshwork. Average drug concentration change with respect to time is studied considering the targets as TM, Sclera and lens.

Figure 14. Temporal evolution of the drug at 3 different drug targets

Time dependent drug concentration changes at specific target locations are graphed in the figure 14.

It shows that the maximum average concentration reaches its peak values before one hour of application of the drug on cornea. Figure 14 shows that the maximum average drug concentration is received by the TM while iris and lens receives much lower concentration.

5. VALIDATION OF THE RESULTS Velocity distribution and temperature distribution produced in the study are compared with the numerical results in the literature. And the validity of the temporal and special evolution of the drug in the anterior portion of the eye is compared with the experimental data.

Figure 15.Comparison of results with previous reports

[28]

Figure 15 shows the comparison of the numeral results with the experimental data in [28].In rabbit

eye experiment being done and the considered rabbit body temperature to be 38.8 0 C (311.8 K).

Three temperature measurements have been collected on Cornea, behind the lens and retina (on

sclera). Temperature along pupillary axis matches with the temperature of experimental data

conforms the validity of the model.

Velocity profile of the AH is matching with the results of [32] although the geometric model is

somewhat different. Velocities in both models are in same magnitude with matches with the

velocities of previous studies [35].

Figure 16. Comparison of velocity profiles

Figure 17.Comparison of Drug concentration at a point (TM, Iris, Lens)

Figure 17 shows the comparison of average drug concentration between three important target in drug delivery. The order of magnitude of results and results in [42] are matching and it conforms the accuracy of the generated model.

6. CONCLUSION Numerical model of heat transfer of heat transfer, AH flow and drug transport are developed to study

delivery of topically applied drugs in the anterior human eye. Outcomes of the numerical simulation

are being compared with the data in the literature. Effects of the ambient temperature in steady state

are being studied. Main conclusions of the study are taken in to a summery below.

AH has the influence of the heat transfer in the to its flow. Though the transport of drug occurs by

diffusion through cornea convective transport play a dominant role in drug delivery to the target

locations. Changes in the ambient temperature effects the AH flow circulation direction as well as it

affects the effective drug concentration at target locations.TM has the chance of getting the highest

average concentration compared to other two target locations, iris and the lens. Peak concentration of

drug in each target location drops to zero when the ambient temperature increases. Peak concentration

become zero when the ambient temperature equals the body temperature and it rapidly increases with

further increase of the ambient temperature.

Results in this thesis helps to provide directions and improve the efficiency of the topically applied drugs. This model can be used in treating Glaucoma, Cataract and Uveitis. It gives an idea to enhance

the efficacy of the drug administration by changing the corneal temperature.

REFERENCES 1. Herwig, R., Computational modeling of drug response with applications to neuroscience. Dialogues Clin Neurosci, 2014. 16(4): p. 465-77. 2. Estlack, Z., et al., Microengineered biomimetic ocular models for ophthalmological drug development. Lab Chip, 2017. 17(9): p. 1539-1551. 3. Zoete, V., A. Grosdidier, and O. Michielin, Docking, virtual high throughput screening and in silico fragment-based drug design. J Cell Mol Med, 2009. 13(2): p. 238-48. 4. Brodland, G.W., How computational models can help unlock biological systems. Semin Cell Dev Biol, 2015. 47-48: p. 62-73. 5. McCaa, C.S., The eye and visual nervous system: anatomy, physiology and toxicology. Environ Health Perspect, 1982. 44: p. 1-8. 6. Agarwal, M., V. Jha, and J. Biswas, Rapidly blinding posterior tubercular uveitis. J Ophthalmic Inflamm Infect, 2014. 4: p. 13. 7. Shah, S.S., et al., Drug delivery to the posterior segment of the eye for pharmacologic therapy. Expert Rev Ophthalmol, 2010. 5(1): p. 75-93. 8. Fernández-Colino, A., et al., Development of a mechanism and an accurate and simple mathematical model for the description of drug release: Application to a relevant example of acetazolamide-controlled release from a bio-inspired elastin- based hydrogel. Materials Science and Engineering: C, 2016. 61: p. 286-292. 9. Park, C.G., et al., Mathematical modelling of brimonidine absorption via topical delivery of microparticle formulations to the eye. Journal of Industrial and Engineering Chemistry, 2016. 39: p. 194-202. 10. DelMonte, D.W. and T. Kim, Anatomy and physiology of the cornea. J Cataract Refract Surg, 2011. 37(3): p. 588-98. 11. Stiemke, M.M., H.F. Edelhauser, and D.H. Geroski, The developing corneal endothelium: correlation of morphology, hydration and Na/K ATPase pump site density. Curr Eye Res, 1991. 10(2): p. 145-56. 12. Jia, X., et al., Biomechanics of the sclera and effects on intraocular pressure. Int J Ophthalmol, 2016. 9(12): p. 1824-1831. 13. Strouthidis, N.G. and M.J. Girard, Altering the way the optic nerve head responds to intraocular pressure-a potential approach to glaucoma therapy. Curr Opin Pharmacol, 2013. 13(1): p. 83-9. 14. Patel, S.J. and D.C. Lundy, Ocular manifestations of autoimmune disease. Am Fam Physician, 2002. 66(6): p. 991-8. 15. Goel, M., et al., Aqueous humor dynamics: a review. Open Ophthalmol J, 2010. 4: p. 52-9. 16. Gong, H., R.C. Tripathi, and B.J. Tripathi, Morphology of the aqueous outflow pathway. Microsc Res Tech, 1996. 33(4): p. 336-67. 17. To, C.H., et al., The mechanism of aqueous humour formation. Clin Exp Optom, 2002. 85(6): p. 335-49. 18. Llobet, A., X. Gasull, and A. Gual, Understanding trabecular meshwork physiology: a key to the control of intraocular pressure? News Physiol Sci, 2003. 18: p. 205-9.

19. Bill, A., Blood circulation and fluid dynamics in the eye. Physiol Rev, 1975. 55(3): p. 383-417. 20. Delamere, N.A., Ciliary Body and Ciliary Epithelium. Adv Organ Biol, 2005. 10: p. 127-148. 21. Gramatikov, B.I., Modern technologies for retinal scanning and imaging: an introduction for the biomedical engineer. Biomed Eng Online, 2014. 13: p. 52. 22. Grieshaber, M.C. and J. Flammer, Fundamental sciences in glaucoma. Introduction. Surv Ophthalmol, 2007. 52 Suppl 2: p. S99-S100. 23. Grieshaber, M.C. and J. Flammer, Blood flow in glaucoma. Curr Opin Ophthalmol, 2005. 16(2): p. 79-83. 24. Allen, D. and A. Vasavada, Cataract and surgery for cataract. BMJ, 2006. 333(7559): p. 128-32. 25. Brown, N.A., The morphology of cataract and visual performance. Eye (Lond), 1993. 7 ( Pt 1): p. 63-7. 26. Barisani-Asenbauer, T., et al., Uveitis- a rare disease often associated with systemic diseases and infections- a systematic review of 2619 patients. Orphanet J Rare Dis, 2012. 7: p. 57. 27. Ness, T., D. Boehringer, and S. Heinzelmann, Intermediate uveitis: pattern of etiology, complications, treatment and outcome in a tertiary academic center. Orphanet J Rare Dis, 2017. 12(1): p. 81. 28. Lagendijk, J.J., A mathematical model to calculate temperature distributions in human and rabbit eyes during hyperthermic treatment. Phys Med Biol, 1982. 27(11): p. 1301-11. 29. Scott, J.A., A finite element model of heat transport in the human eye. Phys Med Biol, 1988. 33(2): p. 227-41. 30. Fitt, A.D. and G. Gonzalez, Fluid mechanics of the human eye: aqueous humour flow in the anterior chamber. Bull Math Biol, 2006. 68(1): p. 53-71. 31. Kumar, S., et al., Numerical solution of ocular fluid dynamics in a rabbit eye: parametric effects. Ann Biomed Eng, 2006. 34(3): p. 530-44. 32. Karampatzakis, A. and T. Samaras, Numerical model of heat transfer in the human eye with consideration of fluid dynamics of the aqueous humour. Phys Med Biol, 2010. 55(19): p. 5653-65. 33. Ooi, E.H. and E.Y. Ng, Simulation of aqueous humor hydrodynamics in human eye heat transfer. Comput Biol Med, 2008. 38(2): p. 252-62. 34. Ooi, K. and R. Samali, Discrepancies in Gleason scoring of prostate biopsies and radical prostatectomy specimens and the effects of multiple needle biopsies on scoring accuracy. A regional experience in Tamworth, Australia. ANZ J Surg, 2007. 77(5): p. 336-8. 35. Shafahi, M. and K. Vafai, Human Eye Response to Thermal Disturbances. Journal of Heat Transfer, 2010. 133(1): p. 011009-011009-7. 36. Shafahi, M. and K. Vafai, Thermal Modeling of the Human Eye as a Porous Structure. 2009(43581): p. 709-714. 37. Ooi, E.-H., W.-T. Ang, and E.-Y.-K. Ng, A boundary element model of the human eye undergoing laser thermokeratoplasty. Comput. Biol. Med., 2008. 38(6): p. 727-737.

38. Ooi, E.H., W.T. Ang, and E.Y.K. Ng, A boundary element model for investigating the effects of eye tumor on the temperature distribution inside the human eye. Comput. Biol. Med., 2009. 39(8): p. 667-677. 39. Friedrich, S., Y.L. Cheng, and B. Saville, Finite element modeling of drug distribution in the vitreous humor of the rabbit eye. Ann Biomed Eng, 1997. 25(2): p. 303-14. 40. Park, J., et al., Evaluation of coupled convective-diffusive transport of drugs administered by intravitreal injection and controlled release implant. J Control Release, 2005. 105(3): p. 279-95. 41. Stay, M.S., et al., Computer simulation of convective and diffusive transport of controlled-release drugs in the vitreous humor. Pharm Res, 2003. 20(1): p. 96- 102. 42. Lin, C.W. and F. Yuan, Numerical simulations of ethacrynic acid transport from precorneal region to trabecular meshwork. Ann Biomed Eng, 2010. 38(3): p. 935-44. 43. Wyatt, H.J., Modelling transport in the anterior segment of the eye. Optom Vis Sci, 2004. 81(4): p. 272-82. 44. Emery, A.F., et al., Microwave Induced Temperature Rises in Rabbit Eyes in Cataract Research. Journal of Heat Transfer, 1975. 97(1): p. 123-128. 45. Mapstone, R., Measurement of corneal temperature. Exp Eye Res, 1968. 7(2): p. 237-43.

APPENDIX Convergence plot for transient solver

Convergence plot for non-linear solver