Finite Element Modeling of Drug Transport

Processes after an lntravitreal Injection

- A Study of the Effects of Drug-Phase Geometry on

Bioavailability and Toxicity

by

Hai-Hui Lin

A thesis submitted in confomity with the requirements

for the Degree of Master of Applied Science in the

Department of Chernical Engineering and Applied Chernistry

University of Toronto

O Copyright by Hai-Hui Lin, 1 997

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Finite Element Modeling of Dmg Transport Processes affer an lntravitreal Injection

-A Study of the Effeds of Drug-Phase Geometry on Bioavailability and Toxicity

1997

Hai-Hui Lin

Department of Chernical Engineering and Applied Chemistry

University of Toronto

Intravitreal injection of drugs is the most effective treatment of ocular diseases associated

with the posterior segment of the eye . The efficacy of the treatment is related to the way the dnig is

distnbuted in and eliminated from the vitreous. Due to the physical and technical difficuîties in

obtaining in vivo data, especially in the human eyes, cornputer simulation of drug transport has

become a valuable tool for predicting the transient distribution of drug in the vitreous.

The objective of the project is to modify a simulation model previousfy developed by Friedrich

[1996], so that the modified model is more realistic in terms of the boundary conditions and geometry.

The focus of the study is on the effects of dnig-phase geometry on bioavailability and toxicity.

Various irregular geometries of dnig phase caused by the viscous fingering effect were investigated.

The resuits indicate that Friedrich's model with a cylindrical/sphencal drug phase is adequate to

predict the general bioavailabil'rty profile of the dmg accurately, but not adequate to evaluate the local

toxicity. It is proposed to use the single annlfinger model developed in this project for ocular toxicity

studies.

Acknowledgments

First and Foremost, I wouid Iike to thank my supervisors, Dr. Yu-Ling Cheng and Dr.

Bradley Saville for their exceltent guidance and continuous encouragement. I would Iike to

express my appreciation to Dr. David Kuhn for his generosity in providing the computing facility

necessary for this work. I would like to thank Xintong Lu, Eric Leung, Ted Mao, Karen Liu, Chns

Ho, Mimi Chen, Jeniffer Smith, Josh Markham, Henry Peng and Charlene Ng for their friendship

and hefp. I would also like to express my gratitude to my parents and my sister, Li-Hui, for their

love and support, and for always having faith in me, Finally, I would Iike to thank NSERC of

Canada for the financial support.

Table of Contents

2. Background

2.1 Physiochemical Properties of the VÎreous Body 2.1 -1 A Brief Overview of Major Structures in an Eye 2.1 -2 Composition of the Vïttreous Body 2.1 -3 Functions of the wtreous Body 2.1 -4 Eye Tissues as Bamers in Ocular Dnig Delivery to the Vireous Body 2.1.5 Transport within the Vireous Body

2.2 Concerns Regarding lntravitreal Injection Techniques

2.3 Brief Ovewiew of the Models by Other Researchers

3. Methods

3.1 Experirnental Obsewation of lnjedion Shapes 3.7 -1 Experirnental Setup and Scheme 3.1 -2 The "Viscous Fingeting" Phenomenon 3.1.3 The Effect of Injection Speed on the Geometry of Drug Phase

3.2 Developrnent of a New Model lncorporating the Realistic Geometry 3.2.1 An Introduction to Finite Element Method and FIDAP 3.2.2 Mathematical Charaderkation of the Drug Transport Process 3.2.3 Modifications Made to Friedrich's Model: Description of Transport

through the Ciliary Body 3.2.3.1 Modification of the Boundary Conditions on Ciliary Body Surface 37 3.2.3.2 Evaluation of Mass Transfer Coefficient and the Use

of a "Stagnant Film" to Replace Posterior Chamber 44 3.2.4 Simulation Plan 47

3.2.4.1 The Study of the Effect of Drug Phase Geornetry on Bioavailability Profile 47

3.2.4.2 The Study of the Effect of Drug Phase Geometry on Toxicity Profile 51

4. Resuits and Discussions 54

4.1 The Effect of Dnig Phase Geornetry on Bioavailability Profile 4.2 The Effect of Drug Phase Geometry on Toxicity Profile

5. Concfusions and Recommendations 75

6. References 77

Appendix 1 : Experimental Setup A1 Appendix 2: Justification for the Scale-up of Mode1 Vitreous and the Injection Volume A2 Appendix 3: Goveming Equation Used in FIDAP Software A3 Appendix 4: Logical Division and Meshing of Computational Domains A6 Appendix 5: Mathematicai Treatrnent of Retinal Domain A l 1

List of Figures

Figure 2.1 Cross Section of a Human Eye Figure 2.2 Molecular Structure of Hyaluronic Acid Figure 2.3 Schematic Diagram of the Ultra-Structure of Hyaluronic Acid and Collagen Figure 2.4 Eariy Simulation Models of Vtreous Figure 2.5 Ohtori and Tojo's Model Figure 2.6 Friedrich's Model

Chapter 3

Figure 3.1 Viscous Fingering Phenomenon Figure 3.2 The Effect of Injection Speed on the Geometry of an Aqueous Drug Phase Figure 3-3 The Effect of Injection Speed on the Geometry of a Viscous Drug Phase Figure 3.4 Meshing of an Eye Figure 3.5 Schematic Diagram of the Pmposed Model Figure 3.6 Perforated Ciliary Surface (2-0 View) Figure 3.7 Velocity Contour Plot of Posterior Chamber Figure 3.8 Velocity Plots of Tube A and Tube B Figure 3.9 Sensitivity of Hyaloid Elimination Rate to the Area of Perfusable

Surface of Cifiary Body Figure 3.10 lmposing Equivalent Stagnant Film vs. lmposing Velocity Field Figure 3.1 1 Six Cases of Injection Shapes (3-0 View) Figure 3.12 The Single A M i n g e r Model

Figure 4.1 Mean Vireal Drug Concentration-TÏme Profiles of Three Model Drugs Figure 4.2 Retinal EIirnination Rate-Time Profiles of Three Model Dwgs Figure 4.3 Hyaloid Elimination Rate-Tirne Profiles of Three Model Dwgs Figure 4.4 Central Cylinder Model Figure 4.5 Eight-Case Approach with the Modified Friedrich's Model Figure 4.6 Schematic Diagram of 'Operating region" of a Drug's Bioavailability Profile Figure 4.7 fietinal Elimination Rate-Time Profiles for Two Groups of Models Figure 4.8 Local AUGTime Profiles for Two Groups of Models Figure 4.9 Local -,-Tirne Profiles for Two Groups of Models

List of Tables

Chaoter 3

Table 3.1 Major Subroutines in Model Building Table 3.2 Table of Stagnant Film Thickness

Table 4.1 Fdd, C h and Tln Tables of Three Model Dfugs 56 Table 4.2 Local Table 68 Table 4.3 FretiMlr TjRi AUC and Cm for Single-Finger and Single-Am Models 69

1. Introduction

The vitreous body is the largest structure within a human eye. it is located between the

lens and the retina, and fills the center of the eye. It has a volume of approxirnately 4 ml and

constitutes about 80% of the globe, It is a sernisoIid hydmgel whi& is 98% to 99.7% water and

has a pH of 7.5 [Sebag, 19921. The moderate pH and high degree of hydration make the vitreous

body an excellent culture media for bacteria and viruses such as herpes simplex virus (HSV) and

cytomegalovinrs (CMV), which lead to various infections (e.g., CMV retinitis) associated with the

posterior segment of the eye.

Vitreolretinal diseases such as endophthalmitis are treated with antibacterial or antiviral

drugs. It would be desirable to have therapeutic dnig concentrations in the vitreous body to

suppress baderial or virai activity. Unfortunately, the access of dmgs to the vitreous body is

usually poor because there is virtually no penetration through the crystalline lens or between

ciliary processes and the crystalline lens. There is no sufficient access from the blood vessels in

the retina or from the choroid either, due to the bamer properties of the pigment epithelium

[Stjemscahntz, 19931. Therefore, intravitreal injection is one of the few, and offen the only,

effective means to deliver drug to the vitreous body. For example, in the treatment of

cytomegalovirus (CMV) retinitis, an AIDS-related vitreo-retinal viral infection, gancyclovir must

be adrninistered intravenously in large doses in order for the drug to reach the therapeutic

concentratiori in the vitreous body [Peyman, 19941. Unfortunately, gancyclovir is toxic to the

bone marrow (myelotoxic), thus, intravitreal injection becornes the only rernaining option for the

patient who has shown hematological abnomalities [Henry, l98n.

In spite of the danger of complications associated with repeated administration,

intravitreal injection has long been a part of the standard treatment of endophthalmitis. This

technique has assurneci an increasingly prominent role in ophthalmologic practice as the

incidence of ocular infections has significantly increased in recent years with widespread use of

parenteral corticosteroids and immunosuppressive agents [Sarkies and Blach, 1985). Moreover,

the number of immunocornpromised patients associated with acquired immunodeficiency

syndrome (AIDS) has reactied epidemic proportions [Freeman and O'Connor, 19841. It is

estimated that CMV retinitis is the first manifestation of AlDS in 1.8% of patients [Sison and

Holland, 19911, and eventually occurs in 15% to 40% of patients with this immunodeficiency

disorder [Henderiy and Freeman, 1987. Therefore, there is a strong incentive to evaluate

various intravitreally injeded antiviral drugs, to obtain vitreal drug concentration tirne profiles,

and to detemine ocular toxicity levels [Peyman and Schulman, 19861.

The efficacy of intravitreally injeded dmg is directly related to how the dnig is distributed

in and eliminated from the vitreous body. The fundamental understanding of the drug transport

process is a crucial piece of information in the design of a drug administration regime. Many

dmgs injected intravitreally are known to have a very narrow therapeutic concentration range,

and are often toxic at higher concentrations [Stainer and Peyman, 1977; Talamo and D'Arnim,

1985 1. If the disease is to be properly treated and damage to tissues by high concentrations of

dnig is to be avoided, comprehensive knowledge of drug distribution following administration is

required [Friedrich et al., 19971.

Unfortunately, data on dnig distribution in the vitreous body, especially for the hurnan

eye, are scarce, due to the many physical and technical difficulties in collecting such data. For

instance, the acquisition of transient drug distribution data requires repeated sampling of vitreous

humour at various positions inside the vitreous body. Since the vitreous humour can not be

regenerated quickty and the sample size is often not small compared to the total volume of

vitreous humour frequent sampling would influence dmg distribution. In addition, the vitreous

body is a very small physical structure and is surrounded by many delicate tissues; the precise

positioning of a sampling needle is technically very difficult, even by an expert. Moreover, due to

the danger of complications associated with tissue damage in the eye, there are few volunteers

for this type of expenment. ln the face of al1 these in vivo experirnental diffÏculties, cornputer

simulation of dnig transport has become a valuable tool to develop an understanding of the key

processes governing intraocular dnig distribution and elimination.

Computer simulation of ocular dnig transport processes was pioneered by Araie and

Maurice [1991], followed by Yoshida et al. f19921, Ohton' and Tojo [1994], and Friedrich et al.

[1997J. For the sake of mathematical simplicity, a set of simplified assumptions were made.

However, some of these assumptions are not fully justifid. For instance, a spherical or

cylindrical geometry was used to describe the shape of the vitreous body [Araie and Maurice,

1991, Yoshida et al., 1992, OhtoB and Tojo, f 9941, whereas the realistic shape is more like a

'bitten peach". A spherical or cylindrical geometry was also used to descnbe the shape of the

injeded drug phase [Araie and Maurice, 1991, Yoshida et al., 1992, Ohton' and Tojo, 1994,

Friedrich et al., 1996) wtiile ophthatmologists have obsewed an irregular dmg phase geometry

for years [Maurice, 19961. Moreover, some of the boundary conditions imposed on the models

[Araie and Maurice, 1991, Yoshida et al, 1992, Ohtori and Tojo, 19941 are not realistic. The

over-simplification has, to varying degrees, hurt the generality of these models. The work of

these researchers will be reviewed in greater detail in section 2.3.

Because of the shortcomings of the previous models, it is the objective of this projed to

deveIop a more refined model, which incorporates the complex geornetry of the vitreous body

and the injected dwg phase, as well as more realistic boundary conditions. The possible shapes

of the injected drug phase have been previously investigated by the present author [Lin, 19961.

The proposed new model will be created using FlDAP 7.6 (Fluid Dynamic Analysis Package)

Iicensed frorn Fluid Dynarnics International. Inc. Evanston, IL, USA. The numerical approach

adopted to solved m a s transfer problems in the vitreous body is the 3-0 finite element method

(FEMI.

2. Backaround

2.1 Phvsiochemical Promrties of the Vitmous Bodv

2.1 .1 A Bnef Ovenriew of Major Structures in an Eve

ANTERIOR CHAMBER -

MEMBRANE

-4 REINA

Figure 2.1 Cross Section of a Human Eye [Vale and Cox, 19851

A horizontal section through the eye is show in Figure 2.1 D(ale and Cox, 19851. There

are three layers which enclose the transparent media through which light passes before reaching

the retina. The outer layer is protective in fundion- It is predominantly white in color and opaque

(the sclera) with a transparent anterior portion (the comea). The middle layer is mainly vascular

and is made up of the choroid, ciliary body and iris. The innemost layer is the retina,

predominantly newous tissue. Within the three layers, the eye is divided into h o sections by the

lens. The frontal section contains the aqueous humour, and is itself divided into anterior and

p~sterior chambers by the iris. The section behind the lens, referred to as the posterior segment,

contains the vitreous humour and retina.

2.1.2 Com~osition of the V i i o u s Bodv

The vitreous is 98O& to 99.7% water and has a pH of 7.5. Dehydration often induces a

profound aiteration of the interna1 morphology [Sebag, 19921. The rest of the vitreous consists of

some high-molecular-weight compounds such as collagen, hyaluronic acid (HA), glycoproteins,

and some low-molecular-weight substances such as ascorbic acid, lipids, lactic acid, phosphate,

bicarbonate, sodium, calcium and sorne trace metals. Among them, collagen and hyaluronic acid

have the most profound impact on the morphology of the vitreous.

Collagen is the major structural protein of the vitreous. The average concentration of

collagen in the vitreous is only 0.09%, [Maurice, 19791. Throughout the body, collagen is

composed of three individual polypeptides. known as alpha chains, organized in a triple helix

configuration foming fibrils [Sebag, 19921. CoIlagen was first discovered as Wrosin", a protein

residue that was retained by the filter after passage of fresh vitreous. 'Virosin" was later found

to contain 18%M glycine, 8.4O& proline, 15.4Ob hydroxyproline, and a small amount of cystein

[Young and Williams, 19541. The composition is most similar to cartilage collagen which is

composed of alpha 1 type II chains. Collagen foms the ultrastrudure of the vitreous. Under a

microscope, it appears to be a randorn network of thin, uniform filaments.

Hyaluronic acid is a high-malecular-weight biological polymer, made of repeating

disacchande units of giucuronic acid and N-acetyl-$-glucosamine. The chernical structure of the

molecule is show in Figure 2.2. This glycosaminoglycan is a ubiquitous component of

extracellular matrix in connective tissue, skin, tendons, muscles, and cartilage [Bematchez and

Camber, 19921. In the eye, hyaluronic acid is present in the vitreous body, and in a lower

concentration, the aqueous humour. In the human vitreous, it consists of 0.02% of the total m a s

[Sebag, 19921. It has been suggested by some researchers to use the sodium sait of hyaluronic

acid (SM) as a replacement of hurnan vitreous and aqueous humour due to its noninflarnrnatory

nature [Stenkula, 19861. Hyaluronic acid (HA) is a long-chain unbranched polyrner with a

random coi1 conformation. HA in the human vitreous has an approximate molecular weight of 106

Da, and acts as a filter for mammolecules [Bematchez and Camber, 19921. It exhibiis a

nonideal colloid osrnotic pressure, thereby acting as an osmotic buffer [Bernatchez and Camber,

1992). The large domain of hyaluronic acid spreads the anionic charge of the molecule over a

wide space. Due to its entanglement and immobilization in the tissue, hyaluronic acid acts much

like an ion-exchange resin in that an electrostatic interaction occurs between the small charges

of mobile ions in the tissue and the electrostatic envelope of the stationary polyelectrolyte

[Sebag , 1 9921.

I

Figure 2.2 Molecular Structure of Hyaluronic Acid

The vitreous is composed of interpenetrating networks of hyaluronic acid molecules and

collagen fibrils. The collagen fibrils provide a solid structure to the vitreous, which is "inflated" by

the hydrophilic contribution of hyaluronic acid. The interaction between these two components is

veiy strong. The physicochemical properties of the vitreous in vivo are not simply the resutt of a

combination of these two molecular elements. If the collagen is removed from the vitreous, the

remaining hyaluronic acid will fom a viscous solution. If hyaluronic acid is removed, the gel will

shrink [Comper and Laurent, 19781. Without the hyaluronic acid, the collagen ultrastructure will

not be stable; this is evidenced by a depression of the collagen melting point from 46% to 38OC

in the absence of hyaluronic acid [Snowden, 19821. A schematic diagram of the ultrastnidure of

hyaluronic acid and collagen interaction is shown in Figure 2.3 [Peyman, 19941.

\ col

Figure 2.3 Schematic Diagram of UftraStructure of Hyaluronic Acid and Collagen

[Peyman, 19941

2.1.3 Functions of the Vitreous Bodv

It has been suggested that the vitreous serves as a metabolic repository for the retina,

hyalocytes, and the neighboring tissues due to the presence of galactose, glucose, mannose,

fructose and hyaluronic acid precursors glucuronic acid and glucosamine walker and Patrick,

19671. The vitreous rnay also serve as a depository for metabolic waste such as lactic acid.

Vitreous ascorbic acid may a d as a scavenger for free radicals generated by retinal and lens

metabolism and photochernical reactions. Due to the anionic nature of hyaluronic acid, HA in the

vitreous foms an 'anionic shield" for the retina and lens against the hydrated electrons produced

from the water irradiated by the ionizing radiation [Cornper and Laurent, 1978).

2.1.4 Eve Tissues as Barriers in Ocular D N ~ Deliverv to the Vireous Bodv

There are four different methods of administering dnigs for therapy of the eye: topically,

oraliy , intravenously and intravitreally. Most ocular dmgs are adrninistered topically to the eye

due to convenience. However, this administration method is only good for treating superficial

infections or inflammations of the eye. It is totally ineffedive for treating intraocular diseases

such as retinitis due to the poor penetration of dnig into the eye. The cornea, conjunctiva and

sclera form the most essential baniers to dnig penetration [Stjemschantz and Astin, 19931.

Wihin the comea, tight jundions around the epithelial cells make it difficult for water soluble

substances to p a s through. The conjunctival epithelium also constitutes a relatively tight bamer

to drug penetration, although not as tight as the corneal epithelium [Maurice and Mishima, 19841.

However, the vasculature of the conjunctiva will absorb a substantial part of the drug that has

passed through the epithelium, The sclera is the weakest barrier to dnig transport among the

three mentioned above. It generally does not f o m a tight bamer to the penetration of dmgs,

even of relatively large molecular weight [Maurice and Mishima, 19841. After passing through

these barriers, the amount of topically instilled dnig that reaches the vitreous body is negligible ,

and insuffident for any therapeutic purpose.

It is also difficult to access the vitreous humour by administering drugs systernically.

either through oral or intravenous routes. The low bioavailability is due to the blooâ-retinal

bamer consisting of the pigrnented epitheiiurn and vascular endothelium in the retina, wtiich

prevent dmgs from enteting the extravaswlar space of the retina and into the vitreous

[Stjernschantz and Astin, 19931.

2.1.5 Transport W i n the V i i o u s Bodv

Vireous humour is precîominantly made of water and its transport property is not much

different from that of water. This statement is supported by the finding that, in some animais'

(e-g., bovine, monkey) as well as human vitreous humours, small molecules such as fluorescein

spread as if freely diffusing in water [Cunha-Vaz and Maurice, 1967). Although there is a

collagen-HA netwock inside the vitreous humour to Wtef large molecules, this network does not

appear to hinder the movement of injeded dmg molecules, since the molecular sizes of most

intravitreally injected drugs are close to that of fluoresein (Le., 256 Da). It has also k e n

reported that hyaluronic acid of molecular weight 1 . 5 ~ 1 0 ~ Da migrates to the anterior chamber

after being injeded into the vitreous humour, and its rate of elimination from the vitreous humour

seems to indicate the absence of a bamer [Hultsch and Balazs, 19751. This lack of resistance to

the rnovement of even large molecules can be accounted for by the fad that. the 2 pm spacing

between collagen fibn'fs is orders of magnitude larger than molecular dimensions of dmgs, which

are usually measured in Angstroms (go4 pm) [Maurice. 19591. Therefore, it is conduded that the

filtering effed can be neglected as far as the transport of intravitreal injected drugs is

concemed.

Vitreous humour plays an important role in the movement of solutes and solvents within

the eye. Several mechanisms can influence transvitreal movernent of molecules. These include

diffusion, forced convection due to hydrostatic pressure, osmotic pressure driving forces, and

active transport across surrounding tissues. Among them, diffusion and adive transport are the

most important whereas convection and other forms of transport are considered insignificant

[Sebag, 19891. It is reported that in rabbits, and presumably in humans, when the vitreous body

is not significantly liquefied, the convedive flow through the gel is negligible in cornparison with

the rate at which solutes can spread by their own molecular motion [Maurice, 19791. This

statement is supported by the finding that, the estimated flow of Iiquid through the vitreous body

is only 0.5 @min in a human eye, and the superficial bulk flow velocity toward the sciera is about

7.25E-07 cm/s [Fatt, 1975, Fatt and Hedbys, 19701. Knowing the physical dimension of the

vitreous body and the diffusivity of the drug, the Peclet nomber can therefore be calculated (see

Equation 2.1). For a srnall molecule such as fluoresein, the Peclet number is about 0.12 (Le.,

&-n = 6.OE-06 crn/s). Therefore, it is concluded that, for low-rnolecular-weight substances,

the velocity of transvitreal bulk flow is too small to influence the transport [Sebag, 19891. Thus, it

is justifieci to simply view the vitreous humour as a body of stagnant water.

Equation 2-1,

where Pe is the Peclet number; u is the velocity of the bulk fiow; i is the characteristic length,

which can be taken as the radius of the vitreous body (1.e.. for a human eye, the vitreous radius

is about 0.8 cm to 1 .O cm); and D is the diffusivity of the drug in vitreous humour.

When a drug is injected into the vitreous body, it spreads through the vitreous humour by

diffusion, and then Ieaves the eye via two pathways. The first pathway is through the antenor

hyaloid membrane into the posterior chamber and out of the eye with aqueous humour drainage.

The second pathway is directly across the retinal surface. For most dnigs injected intravitreally,

the retinal pathway is usually the major elirnination route due to its high penneability to small

molecules, and its large surface area. For a human eye, the average surface areas of the retina

and hyaloid membrane are estimated to be 10.5 cm2 and 1.2 cm2 [Friedrich et al.. 199q,

respedively. The dmg can be eliminated across the retina via either passive or adive transport,

and very often both. Once the drug reaches the choroid, a highly vascularized tissue that

underlies the retina, the dnig is quickly removed by the blood circulation. Therefore, the

concentration of dnig in the blood Stream right behind the eye bal1 is oflen negligibly fow.

The passage through the hyaloid membrane can sometimes become the major

elimination route when the drug has a very low retinal pemeability. Molecules comparable in

size to albumin (69,000 Da) are known to have Iow retinal permeabilities [Maurice, 19791. lt is

evident that there are advantages in choosing a drug that leaves the vitreous body by the

anterior rather than the retinal route. These advantages are (1) its therapeutic action will be

prolonged and (2) its concentration within the vitreous will be more unifom [Maurice, 19793.

Once the drug diffuses across the hyaloid membrane, which imposes little resistance to drug

transport, it is camed away by the flow of aqueous humour secreted from the ciliary body

surface. The aqueous humour production rate in a human eye is similar to that of a rabbit eye

(2.2 flmin) [Kinsey and Reâdy, 19641. The volume of the posterior chamber in a human eye is

estimated to be 0.1 5 ml [Friedn'ch et al., 19961. Both convedive and diffusive m a s transfer

occur inside the posterior chamber.

2.2 Concems Recrardina lntravitreal Injection Techniaues

Aithough intravitreal injection is a highly effective technique for treating ocular diseases

in the posterior segment of the eye, there is a reludance to adopt such a technique. This cm be

attributed to the possible danger from two major sources, trauma and toxicity [Maurice, 19791.

These cuncerns will be addressed in sequence.

Due to its high risk, intravitreal injedion can only be perforrned by an experienced

ophthalrnologist in a clinic with the proper equipment. Dmgs are usually injected by hand, in a

srnall volume ranging from 10 pl to 200 pl [Peyman, 19941. The site of injection is preferably the

supenor quadrant of the eyeball [Straatsrna and Foos, 1969). However, if the infection is

sedonal, injection can be made near the affected region [Maurice, 19791. A Ziegler knife is

inseRed into the midvitreous, 2.5 mm behind the limbus in children, 4.0 mm in phakic eyes, and

3.5 mm in aphakic eyes; a needle on a tuberculin synnge containing the drug is then placed

through the sdera knife tract into the midvitreous [Peyman, 19941. To rninimize tissue damage,

intravitreal injection is perfomed using a thin needle (e-g., a 30G needle) and by holding the

point under gentle pressure near the ora semta with the shaft perpendicular to the eyeball, and

rotating the synnge backward and forward between finger and thumb [Kelly, 19481. If the injected

volume is large (e-g., 100 pi), paracentesis, the withdrawal of an equaI volume of vitreous

humour from the vitreous body before injeding the dnig, is often necessary to avoid an

excessive increase in the ocular pressure dunng the injection,

Trauma may occur during an intravitreal injedion. The sharp tip of the needle, the rapid

backfiow of the injected fluid, and excessive increase in intraocular pressure can ause

mechanical damage to the retina, vitreous humour, and lens [Maurice, 1979J. Hemonhage is

often one of the hannful biologic responses to the immediate trauma [Maurice, 1979, Sebag,

19891. The trauma is usually more severe if a large volume is injeded. Paracentesis followed by

the injection of a large drug volume can cause the severe disruption of the delicate physical

structures within the eye, such as the HA-collagen network inside the vitreous body. The high

ocular pressure resuiting from this procedure can displace the lens forward by as much as 2 mm,

and rnechanically compromise the comeal endothelium, and the ciliaty body which is ngM in

front of the lens [Maurice, 19791. To minimize the trauma, the use of a thin needle, a small

injection volume and a low frequency for repeated administrations are recommended.

Toxicity is another major and probably most important concem in intravitreal injections.

Ocular toxicity is caused by the exposure of tissues to a high concentration of drug for certain

period of time. Both exposure concentration and exposure time can be important. The toxic

effect is directly related to the transient drug distribution inside the vitreous body. The

consequence of a toxic effect can be so severe that it resufts in loss of vision [Peyman, 19941.

The retina and optic nerve are two tissues which are most sensitive to the drug concentration in

the adjacent vitreous. A high vitreal dnig concentration against the retina can damage the retinal

photo-receptors, and possibly even the optic nerve [Peyman, 19941. The toxic Ievel of drug is

established by making histologic slides of eye tissues and observing morphologie changes of

retinal tissue (Karacorlu et al., 1992, Dolnak et al., 19921. Due to the extrerne difficutties in

evaluating the toxic effed of dmgs in the human eye, animal data such as rabbit data are used

by the ophthalmologist as a guideline for designing an administration regime, For example, in a

study conducted by Karacorlu et al. to detemine the ocular toxic level of a new antiviral agent ,

ara-M. various doses of drug ranging from 20 pg to 800 pg were injeded into twelve rabbits'

eyes; the rabbits were then sacrificed 10 days after receiving the injection [i992]. Histologie

slides of the injected eyes showed toxicity at a dose between 400 pg and 800 pg (i.e., a mean

vitreal drug concentration between 285 Wrnl and 571 pg/mC); the outer layers of the rabbits'

retinas that received these concentrations showed various degrees of damage to photoreceptor

outer segments [Karacorlu, 19921. No such degeneration was found in rabbits that had received

a dose less than 400 pg; and no complications such as hemorrhage, catarads or retinal

detachments were observed [Karacorlu, 19921. It was therefore concluded by Karacorlu et al.

that, the ocular toxic concentrations of ara-M are 285 @ml and above (1 9921.

The counterpart of toxicrty is the effmcy of drug, which is also diredly Iinked to the dmg

distribution inside the vitreous body. One of the important parameters for evaluating the effmcy

of an intravitreally injected antiviral agent (e.g., acyclovir, ara-M) or antibiotic (e-g., penicillin,

gentamicin) is the lCs0, 50% inhibitory drug concentration, at wtiich 50% of viral or bacterial

activity is suppressed. This value is specific for each type of virus and baderium. 1CW and ocular

toxic concentration f o m the Iower bound and the upper ceiling of the therapeutic window of an

antiviral agent or an antibiotic. For example, m e n acyclovir is used against cytomegalovirus

(CMV), the lCso is 28 Mml , and the toxic concentration of acyclovir is 2000 crg/ml [Peyman,

19941. Therefore, the therapeutic window of acydovir in the treatment of CMV retinitis is located

between 28 pg/ml and 2000 pglml.

In a well designed administration regime, the drug concentration should be able to

remain above the minimum therapeutic concentration (e-g., lCso for an antiviral agent or

antibiotic) for a substantial period of time (e.g., a few days or weeks), whereas the peak

concentration against the retina is well below ocular toxic level. However, it should be realized

that maintaining therapeutic effectiveness and avoiding ocular toxicity are two competing

objectives. To maintain a therapeutic concentration for a long pend of time, the drug can be

injected in erther small multiple doses or a high concentration single dose. For patients with a low

risk of trauma complications such as a heaithy young patient, the former is preferred, For the

patient who reacts unfavorably to physical trauma, a high dose regime is often adopted to avoid

frequent administration.

A high dose always resuits in a higher nsk of toxicity, especially for the case of a small

injected volume. As rnentioned previously, a smaller injeded volume (e-g., 10 pl) is often

favored over larger one (e.g., 200 pl) to avoid excessive ocular pressure. A high dose and a

srnall volume can push the injected drug concentration well above the toxic level. If the drug

phase is properly placed (e.g., nght at the center of the eye), the risk of ocular toxicity is minimal

due to steep concentration gradients formed between dmg phase and retina, resufting from the

long transport distance and the low vitreous diffusivity of drug. However, in the case of

rnispositioning or treating sectorial infection, the drug phase can come in close proximrty to the

retina, and the risk of toxicity then increases. Besides the above two situations, drug leakage

created by backfIow along the needle during the injection can be a real danger as well, since a

single stream of drug solution can protmde from the bulk phase of drug al1 the way to the site

where the needle penetrates the eye.

There is indeed no unique administration regirne good for al1 patients. It is up to the

ophthalmologist to evaluate the patient's conditions and find the optimum between various

cornpeting factors.

2.3 Brief Ovewiew of the Models bv Other Researchers

Due to its highly wupled relationship with dmg efficacy and toxicity, the information on

transient drug distribution is important to dnig evaluation and the design of administration

regimes. There has been a strong incentive to predict dnig distribution via cornputer modeling.

Numerous such studies have k e n conduded with various degrees of sophistication. In this

section, the models developed by Araie and Maurice (1991], Yoshida et al, (19921, OMori and

Tojo [1994], and Friedrich et al. [A9961 are briefly reviewed.

Araie and Maurice [1991] adopted a very simple approach in their study; the vitreous

body was modeled as a sphere with the entire outer surface representing the retina [Figure 2.41.

However, in reality, the vitreous body is far from being a perfed sphere. Its geornetry is best

describecl by the shape of a 'bitten peach" [see Figure 2-11. The anterior surface of the vitreous

body is right against the lens and the hyaloid membrane. The posterior surface is covered by the

retina. All three tissues, Iens, hyaloid membrane and retina, have different dnig perrneabilities.

The lens is made of densely packed collagen-like material, which is generally impermeable to

dnigs. The hyaloid membrane is highly perfusable, even for large molecules. The retinal

pemeability is usually somewhere in between, depending on the physical and chernical

properties of the dnig. Therefore, there are at least three distinctive boundary conditions which

must be applied to the surfaces of vitreous body. Therefore, the model created by Araie and

Maurice is not only inaccurate in the geometry, but also in the applied boundaiy conditions,

which are inconsistent with the different degrees of pemeability located around the vitreous.

Moreover, by imposing a uniform pemeability on the entire surface of the sphere, the arug

concentration profile in the anteflor sedor is assumed to be the same as the posterior sedor,

and there is no flux across the central plane that divides the sphere into anterior and posterior

sedors. If this assumption is true, the sum of Ioss across hyaloid membrane and the loss across

the retina in the anterior sector (in front of the central plane) must equal the loss across the

retina in the posterior sedor. By coincidence, this condition happened to be tnie for fluoresein

[Friedrich et al., 19961, but can not be generalized. Due to oversimplification, the generality of

Amie and Maurice's model is poor. The mode! has very lirnited use in prediding the elimination

profiles of drugs other than those having similar transport charaderistics to fluoresein.

~ o d e l of Araie and Maurice

Model of Yoshida et al.

Figure 2.4 Early Simulation Models of V'ireous [Araie and Maurice, 1991, Yoshida et al., 1994

Realizing that the boundary conditions are different in different parts of the vitreous

surface, Yoshida et al. [1992] proposed a new model to address that issue, wtiile retaining the

spherical geometry for the vitreous body due to its mathematical simplicity. The vitreous body

was divided into an anterior and posterior hemisphere. Each hemisphere was further subdivided

into eight compattrnents, and a separate penneability was assigned to the outer surface of each

hemisphere [Yoshida et al., 1989, 19921. The concentration was assurned to be uniform inside

each cornpartment. Despite its improvement over Amie and Maurice's work, this mode1 still

suffered from oversimplification. The concentration data calculated from the eight-cornpartment

model were far too few and far too crude to constitute a smooth three dimensional dmg

concentration profile within the vitreous body. Therefore, this mode1 is inadequate to predid the

dnig concentration distribution accurately, especially for those having a steep gradient. At best,

the results yielded by this model can only be qualitatively correct.

Ohtori and Tojo [1994] proposed a model with improved generality h i l e retaining a

simplifieci geometry [see Figure 2.51. The proposed madel is cylindrical in shape with one end of

the cylinder and its curved surface representing the retina, and the opposite end of the cylinder

divided into an outer sedion representing the hyaloid membrane and an inner section

representing the lens [OMori and Tojo, 19941. The injeded dnig phase is spherical in shape and

placed at the center of the cylinder, In this modei, al1 three possible elimination routes, retina,

hyaloid membrane and lens were inciuded and distindively tfeated. This was a big improvement

over the two previous rnodels. However, the geometry of the vitreous body is still not realistic

enough. The boundaries of the model do not closely match the physiological boundafies of the

vitreous. Therefore, the Iargest inaccuracies in drug concentration predidion are expeded near

the boundaries. Unfortunately , the dmg ciistribution near the boundaries (e.g., retina) is usually a

crucial piece of information, for instance in the evaluation of ocular toxicity. In this regard,

Ohtori and Tojo's modei is clearly inadequate.

to lens

to posterior chamber t to posterior ctiarnber

A 4

to retina to retina

to retina

Figure 2.5 Ohtori and Tojo's Model [OMori and Tojo, 19941

Friedrich et al. f1996, 1997] proposed a new mode1 to accurately describe the geometry

of the eye [see Figure 2.61. The mode1 included fmed-boundary entities such as the vitreous

humour, retina, hyaloid membrane and dnig phase, and mobile domains such as the aqueous

humour flowing inside the posterior chamber, and boundary surfaces such as outer-retinal

surface, posterior and anterior surface of lens, ciliary body surface, and posterior surface of iris.

All possible elimination routes were wnsidered. Appropriate boundary conditions were imposed

on each surface. 60th diffusive and wnvedive m a s tmsfer were taken into account in the

calculation. The mode1 utilized the finite element method (FEM). The 3-0 continuous

concentration profiles of dnig in the vitreous humor, retina. hyaloid membrane and posterior

chamber of the eye were solved numerically by discretizing the domain into hundreds of

thousands of brick elements (well-mixed compartments). Compared to al1 previous models.

Friedrich's model is by far the most sophisticated and elegant, in ternis of its generality and

potential accuracy.

Hyaloid Membrane ( thWn~r=0.01 un) --\

Humour 1

C radius = 0.53 cm

-ior *

-. ,.S

Figure 2.6 Friedrich's Model Friedrich et al., 19961

Friedrich et al. [1997a, 1997b] thomughly investigated the fadors that affect dnig

distribution including the physiological fadors (Le., vitreal diffusivity, retinal permeability), and

injection parameters (Le., injection position, injection volume). However, one important variable,

the shape of the dnig phase upon injection, was left untested. The interface between the dnig

phase and vitreous humour is one of the boundary surfaces essential in the calculation of vitreal

dnig distribution. Once the geometry of this boundary surface has changed, the solution of the

entire domain (Le., drug concentration profile in vitreous) will change. A major criticism from the

ophthalmologists who reviewed Friedrich's work is that the geometry of dnig phase specified in

the model is unrealistic [Maurice, 19961. The shapes of the injected dnrg phase obsewed are

more likely to be irregular rather than the well-defined spheres or cylinders assumed by

Friedrich. It is the objective of this project to determine the sensitivity of dnig

distribution to the geornetry of the injected drug phase. The impact of injection shape on

the dnrg bioavailability profile and toxicity profile will be studied in detail.

3. Methods

3.1 -1 Exoerimental Setun and Scheme

To properly investigate the effect of drug phase geometry, it is necessary to gain some

qualitative and quantitative understanding of the fluid dynamic behavior of the injeded dnig

solution, so that the geometry of drug phase (injection shapes) can be predided for a variety of

injection conditions. An experimental study was conduded previously by the present author,

examining the possible fadors (i-e., the injection speed, the size and shape of a needle, the

viscosity of drug delivery vehicle) that affect drug phase geometry [Lin, 1996). A summary of

this study follows.

A syringe pump (SAGE INSTRUMENT MODEL 3418) was used to control the injection

speed. The operating range is from 0.07 mumin to 1.5 mumin. Various sites of needles, ranging

from very small (30G) to very large (16G), were tested. The mode1 dn~g (0.15% fluoresein)

injeded was prepared in both aqueous medium (distilled waterwhich has a viscosity of 1 cP) and

viscous medium (carbopol gel which has a viscosity of 90,000 cP), intended to simulate both

aqueous and viscous formulation of an intravitreally injected drug. 1 S%wt carbopol (polyacrylic

acid. 4x10~ Da) was used as the modal systern for the vitreous humour, for its rheologiml

sirnilarities to the vitreous humour [Lin, 19961. Fluoresein (bright yellow in a neutral and acidic

medium, a rnolecular weight of 256 Da) was chosen as the mode1 drug because of its

sirnilarities in physical size and transport properties with respect to intravitreally injected dmgs,

and its sharp color in contrast to the background, which facilities the acquisition of a high quality

image [Lin, 19961. The images of injeded drug phase were taken using an image capture device

[see Appendix 11 and post-processeci Wh an image analysis software (MOCHA 1.2, Jandel

Scientific). The volume of human vitreous body is roughly 4 ml, and the volume of injected drug

is generally l e s than 200 pi (0.2 ml). In the study, the volume of the sirnulated vitreous humour

and the injeded drug were scaled up, to 10 ml (sometimes 25 ml) and 0.5 ml (sometimes 1 ml),

respectivefy, so that a ciear image of the injected dmg phase coufd be obtained. The justification

of such a scale-up has k e n d'iscussed in the B.A.Sc ths is by the p m n t author [Lin, 1996).

The crucial arguments are summarized in Appendix 2.

3.1.2 The Viscous Finaenna" Phenomenon

The most common formulation for intravitreal dmgs is the aqueous solution. At a low

injection speed, the resuiting geometry of the aqueous drug phase right after the cornpletion of

the injedion is highly irregular and asymmetric. The shape of the dnig phase looks like a piece

of feather [see Figure 3.11. The water-gel interface is extremely rough. If one part of the

interface is magnified, it can be clearly seen that the water front is saw-toothed. It loob as if

there are thousands of small Yïngers" growing from the aqueous phase into the gel phase. This

phenornenon is called the viscous fingering effed, which is caused by the high mobility ratio

between injeded phase (aqueous solution) and the displaced phase (vitreous gel) pavarier,

19911. It is much more difficult to defom the highly viscous gel phase than the less viscous

aqueous phase. Thus, the aqueous solution will have a much higher mobility than the gel, under

the same pressure head, Therefore, when the aqueous solution is injeded into the viscous gel,

the front of the aqueous phase will push aside the gel, rush ahead and leave the gel entrapped

[Lin, 19961 [Figure 3.11. The greater diffefence between the viscosity of two phases, the more

severe the fingen'ng effed Mil be.

(a) Schematic Diagram of Fingering Effed @) Image of Injection Shape

Figure 3.1 Viscous Fingering Phenomenon

3.1.3 The Effect of lniection Speed on the Geometrv of Druo Phase

Injedion speed was found to be the most crucial factor affeding the geometry of the

dnig phase, especially for dnigs in aqueous formulations mn, 19961. There were three important

observations made:

The degree of fingering (the degree of irregularity) increased as the injedion speed

decreased [see Figure 3-21. At a low injection speed (e.g., 0.1 rnWmin), the drug phase was

spread extensively across the vitreous, and the surface to volume ratio of the drug phase

was high. At a much higher injection speed (e-g., 1 .O mWmin), the drug phase was

concentrated around the center of the vitreous, and the surface to volume ratio was low

(e.g., like that of a sphere).

Over a wide range of injection speed, the drug phase was irregular; and only within a very

namw speed range (Le., 0.9 - 1.1 ml/min), the drug phase geometry was neariy a cylinder

or a sphere. Therefore, the injeded dmg phase is most likely to assume an irregular shape.

This finding is in agreement with what many ophthalmologists have obsenred in practice

[Maurice, 19961.

At an extrerneiy high injedion speed (e-g., 15 mumin), the dnig solution was shot across the

vitreous humour and deposited on the surface of the retina. The drug phase usually has a

long tail in the direction of needle penetration.

The various injection shapes of aqueous solution observed here will be incorporated into the

simulation models.

Unlike that of an aqueous solutioci, the injection shape of a viscous solution is l e s

sensitive to ifijedion speed. It was obsewed that the injedion shape of a viscous solution was

close to being a sphere or cylinder over a wide range of injedion speeds (i-e., O S mumin - 10

mumin). Only at an ultra-low injedion speed, did the injedion shape deviate fmm the spherical

or cylindrical geometry, becoming more dish-like [Figure 3-31. Therefore, ï t is concluded th*,

for a viscous-fonnulation drug injected at a moderate or high speed, it is justified to

assume spherical or cylindrical geomefry for the resutting drug phase in the cornputer

simulatÎon. However, for an aqueous-fomulation dwg, this assumption is invalid.

(a) 0.1 mVmin by pump, extensive fingering @) 0.34 mUmin by pump, moderate fingerÏng

(c) 1.0 mumin by pump, minimal fingering (d) Rapid injedion by hand . -15 mVmin

Figure 3.2 The EîTect of Injection Speed on the Geometry of an Aqueous Dnrg Phase

(a) Rapid Injedion, 10 milmin (b) slow injection, 0.1 mumin

Figure 3.3 The Effect of Injection Speed on the Geometry of a Viscous Dntg Phase

3.2 ûevelo~ment of a New Mode! Incomoratina the Realistic Geometrv

3.2.1 An Introduction to Finite Etement Method and FIDAP

Two of the main goals of engineering analysis are to be able to identify the basic

physical principle(s) that govem the behaviour of the system and to translate those principles

into mathematical model involving equation(s) that can be solved to accurately predid the

qualitative and quantitative behavior of the system [Bickford. 19941. The resuking mathematical

models offen wnsist of a set of differential equations with appropriate hundary conditions. In a

relatively simple system, the mathematical model may only involve elementary ordinary and

partial differential equations, which can be solved exadly with some classical methods.

However, far more frequently, the goveming differential equation(s) or the regions in Wieh the

solution is sought are so complex that it is necessary to use an approximate or numerical

method for extracting the desired information regarding the behavior of the system (Bickford,

19941. In the last quarter century, with the increasingly capability of computers, the finite

element method (FEM) has becorne an increasingly popular numerical technique for solving

such problems. Applications of FEM have been extended fmm Îts traditional domain of solid and

fiuid mechanic into the new field of biomedical engineering, where the problerns often involve

difficulties with geometric and matenal non-linearity [Davies, 1 980).

The fundamental idea of the finite element method is the replacement of continuous

fundions by piecewise approximations, usually polynornials [Davies, 19801. lt converts the

goveming differential equations of a system into a set of linear algebraic equations. The

popularity of FEM rests largely on the ease with which these equations can be assembled and

solved on a cornputer.

When applying FEM to a problem, the first step is to discreüze the region of interest. For

2-D or 3-0 problems, this invoives the division of a surface or volume into a finite number of

small regions (elements) of simple geometry such as a triangle or rectangle in the 2-0 case, or a

pyramid or box in the 3-D case. This process is calied mesh generation [Figure 3.41, which can

take up to 80% of the work hours in building the model. In the past, m e n the mesh was

generated by hand, FEM was restricted to small size problems only (e-g., a few thousand

elements at most). Even in the design of the steaJth fighter F-117 (Night Ha*) in the early

seventies, the various simulations done with FEM were 2-0 only Wao, 19951. Now, with the

power of modem computers. a large 3-0 mode1 with several million elements can be solved in a

few days to a few weeks. Moreover, the precision and accuracy of solution are rnuch better than

before, due to a more refined mesh grid and optimized element dimensions, and the higher

number of iterations possible.

(a) 2 4 Meshing (b) 3-0 Meshing

Figure 3.4 Meshing of an Eye

The FEM package used in this projed is FlDAP 7.6 (Fluid Dynamics Anaiysis Package)

licensed from Fluid Dynamics International, Inc. (Evanston, IL, U.S.A_). The package is mn on a

UNlX platfom with a Silicon Graphic IRIS Indigo workstation. The minimum size of RAM

required to run a 3-0 problem is 64 MB. The hard disk space required to store the start-up and

resulting data file for a typical problem of 100,000 elements ranges from 300 MB to 900 MB,

depending on the complexity of the problem.

FIDAP software is primarily designed to solve cornplex fluid fiow problems such as

moving boundary pmblems, two-phase flow, turbulent flow, porous medium flow, and Newtonian

as well as Non-Newtonian viscous flow, ali of which can be coupteci with heat transfer, m a s

transfer (up to 15 species) and simple chernical readions. The pnnciple goveming equations

used in the package are the equations of momentum conservation, m a s conservation, energy

consetvation and an equation of state (for both compressible and incompressible fluid), which

typically relates density, temperature, pressure and species concentration [Appendix 31 [FIDAP

theory rnanual, 19931. The package serves to sirnplify repetitive tasks such as mesh-

generation, matrix decoupling, and numerical iteration, thus allowing the user to focus on the

design of a grand scheme for solving the pmblern.

FIDAP is operationally organized into modules, and each module consists of a collection

of subroutines; each subroutine can perfom a specific task- To construd a rnodel, the user

writes a main program which issues a string of cornrnands to cal1 these subroutines. There are

five principle modules which were used to build the models in this project, namely FI-GEN, FI-

BC, FI-PREP, FI-SOLV and FI-POST. The first three modules are responsible for creating the

rnodel. Definition of model geometry and mesh genelation in al1 domains are done in FI-GEN.

Definition of the boundary surface as well as boundary and initial conditions are perfomed in FI-

8C. Physical properties of domains and solution scheme for the problern are given in FI-PREP.

FI-SOLV is a compiler and solver for performing numerical calculations in accordance to the

scheme set in FI-PREP. FI-POST is a data pst-pmcessing module h i c h extra& the data from

the result database to plot various graphs defined by the user. There are various subroutines

called in each modules. Some of the major subroutines in the first three modules are listed in

Table 3.1 .

Table 3.la

FI-GEN POINT

C U R E

SURFACE

MESHLLOOP

MESH-EDGE

MESHSHELL

MESHSOLID

Give the coordinate of the points that & t h e the structure of the model (e.g., 3 points to define a triangle). Conned points into lines and curves that form the Irvire-frame" of the model geometry. Define the planes where al1 interior and exterior surfaces of mode1 reside. Define the loops that fom the inner and outer boundaries of meshable region on a surface (e.g., to rnesh a ring-Iike region requires two circies as mesh-loops) [see Appendix 41. Discretire the meshloops into tiny fragments. User has the uftirnate control of grid sire and number. Define a meshable surface ( Le., a 2-D entity) in conjundion with SURFACE and MESH-LOOP subroutines; map meshing or pave meshing the defined region. A unique name is assigned to such region by the user to distinguish it from other meshable surfaces in the model. If it is a 2-0 rnodel, FI-GEN will end here [see An~endix 41. Define a meshable 3-0 domain ( i.e., a rnesh-shell) fully enciosed by six meshfaces, which can be IogicalIy describecl as front, back and four sides. The number of gnds for the front and back surface must be same [see Appendix 41. Mesh the 3-0 domain defined by the MESH-SHELL subroutine. A meshed 3-0 domain is called an 'entity", for which a unique name must be given by the user to distinguish one entity frorn another in the model,

Table 3.1 b

FI-BC

L

IC-NODE

BC-GROUP BC-NCDE

BC-FLUX Give initial conditions to the surfaces definedin BGGROUP or to ' the entities defined in FI-GEN

Name the boundary surfaces Give boundary conditions to the boundary surfaces defined in BC- GROUP subroutine. Give flux-like boundaw conditions to boundaw surfaces

Table 3 . 1 ~

solid or fluid entities, it is permeable or impermeable to certain species. Various user-defined property models such as viscosity model (e.g., funàion of temperature), density model (e.g., rnixing of two species) are also given here.

PROPERTY Give concrete number to propenies or the parameters of the property rnodel defined in ENTlTY subroutine.

PROBLEM Select a set of goveming equations which will be solved in the model (e.g ., choosing the equation of state for incompressible fuid over compressible fluid in the case of liquid ffow, ignoring the momentum equation in the case of stagnant fluid).

TIME- Set the time-step and its variation scheme for transient problems INTEGRATION SOLUTION Design the solution scheme such as the selection of solver (e-g.,

cho&ing segregated solver over highly-coupied solver), setting convergence criteria and the relaxation factor. A well designed solution scheme can ensure the solution converges quickly, with accurate resuits.

Table 3.1 Major Subroutines in Model Building

3.2.2 Mathematical Characterization of the Dniq Trans~ort Process

Due to the complexi€y of the human eye, it is impossible as well as undesirable to re-

create every little detail of this biological system in a computer model. A good simulation model

should be sophisticated enough to adequately represent the general transport characteristics of

the eye, and yet simple enough to be mathematically and computationally manageable. The first

step in constmcting a computer model is to transfomi qualitative physiological features into

mathematical expressions, which can be solved quantitatively. To do so requires reasonable and

objective abstraction of the transport process associated with the vitreous body. The vitreous

body is the main compartment of the eye mode(. The vitreous compartment is bounded by the

lens, the retina, and the hyaloid membrane; the anterior surface of the hyaloid membrane is

exposed to the posterior chamber of the aqueous humour. lnside the vitreous compartment,

there is an injeded dnig phase. Based on the physiology and anatomy of the eye, a set of

assumptions were made, and the appropriate goveming equations and boundary conditions for

various tissues were written. The mode! equations presented below were developed to describe

the transpor! and elimination of drug following injedion. Thus, the equations focus on the time

period after the injedion of drug is completed (Le., t = O corresponds to completion of the

injection).

Vïtreous Body:

The vitreous humour is a stagnant fiuid (see sedion 2.1 -5); therefore, the linear momentum

equation can be ignored, and only the species continuity equation needs to be considered. Mass

transfer of drug occurs only via molecular diffusion; the convective terni in the species continuity

equation can thus be ignored (Equation 3.1). There is very little metabolic degradation of drug

inside the vitreous humour due to the lack of enzymes [Sebag, 1989). Therefore, the tem for

material generation can also be eliminated. The species continuity equation for the vitreous

domain is then

where C, and D, are the drug concentration and diffusivity in the vitreous humour, respectively;

the diffusivity of drug in the vitreous is the same as in water. The vitreous humour is assumed to

be initially free of drug, giving the initial condition:

IC 1 for Equation 3.1: C, = O. ut c = O (3.2).

The vitreous body is bounded by four bou~dary surfaces, the vitreotdrug-phase, vitreolretinal ,

vitreodens, and vitreolhyaloid interfaces. The boundary conditions on these four interfaces,

which are necessary for solving equation 3.1, will be discussed in detail below.

Drug Phase:

It is assurned that after the injection, the patient dues not undergo significant physical activity.

Therefore, the aqueous drug phase is stagnant, and there is no bulk motion (no mixing). Thus,

the linear momenturn equation does not have to be considered, and the species continuity

equation contains no bulk convection terni :

-- zd D,v'c, = O a (3.3).

wtiere Cd and Dd are the drug concentration and diffusivity in the aqueous drug phase,

respedively; and the diffusivity of drug in the drug phase is the same as in water. The drug

phase is assumed to have a unifonn drug concentration initially:

IC 2 for Equation 3.3: Cd = Drug Llosdnjection Volume, at t = O (3.4).

The drug phase is treated as a part of the vitreous humour. It is assumed an equal concentration

and equal flux at the vitreo/drug-phase interface:

BC' I for Equation 3.3 and 3.1: C,=C* ut the vitreo/'drug-phase interface (3.5);

a7, X d BC 2fir Equation 3.3 and3.1: Ndn = -Dv- = -Dd - al '

at the vitreo/drugphase interface (3.61,

where rVdn is the dnig flux out the dnig phase, in the diredion normal to the vitreojdrug-phase

interface. The subscript 'n" stands for the normal direction.

Retina:

The retina bounds roughly 90% of the vitreal surface. The appropnate boundary condition at the

vitreolretinal interface is,

where ,\;., is the drug fiux into the retina from the vitreous humour, in the direction normal to the

retinal surface; Pr is the retinal permeabiiity that includes both active and passive transport, and

also accounts for the retina/vitreous partitioning and retinal thickness; C,,, and Co,, are the

vitreal concentration at the interface and the equivalent vitreal concentration at the outer surface

of the retina, respedively. Since the choroid, the tissue underiying the retina, is highly

vascularized, the drug is quickly removed by the blood circuiation after crossing the retina [see

section 2.1 -51. As a result, the drug concentration on the outer surface of retina is negligibly low.

Thus, C-, can be set to zero.

However. FlDAP does not allow a pemeabiliRy value to be diredly imposed on the vitreal

surface. A retinal cornputational domain has to be computationally constnicted, and placed

against the vitreal surface [see Appendix 51. The retina is a very thin tissue, with a thickness of

0.01 cm. only .1% of the radius of the vitreous [Friedrich et al., 19961. Thus, it is assurned that a

Iinear concentration profile across the retina exists at al1 times. The justification for this pseudo

steady state assumption is presented in detail, in Appendix 5. The species continuity equation

for the retinal domain is therefore,

D,V'C, = O (3.9),

where Cr is the retinal drug concentration of vitreal equivalence which is obtained by dividing real

retinal concentration with the retinolvitreal partition coefficient; Dr is a 'lurnped" diffusion

coefficient back calculated from the retinal perrneability value (P,),

Dr = P,L (3.1 O),

where L is the thickness of the retina. As a summary, there are three boundary conditions for

solving equation 3.9:

BC I for Equation 3.9: Cv = C, at viireo/retinal interface (3.1 1);

a7, BC 7 /or Equation 3.9: N, , = -D,, - - .r=, - -0,- , at vitreo/?etinal interface (3.1 2) ; a? al BC 3 for Equation 3.9: Cr = O (sink condition), at the outer surface oyretina (3.13).

Lens:

The lens is made of densety packed collagen-like material. which is generally impermeable to

dnrgs [Sebag, 19921 (also see section 2.3). Therefore, the posterior surface of the lens which is

also the vitreoflens interface is set to have a zero flux at the direction normal to the lens:

BC 4 for Equation 3.1: Ni,, = O, al the posterior surface of the lem. (3.14).

Hyaloid Membrane and Posterior Chambec

The hyaloid membrane is a highly petfusable tissue, and poses no resistance to dmg transport

[see sedion 2.1.5]. Therefore, the diffusivity of dnig in hyaloid membrane is set to be same as

that in water. It can be simply treated as a part of the vitreous. The species continuity equation

for the hyaloid-membrane domain is

where Ch and Dk are the drug concentration and diffusivity in the hyaloid membrane. The hyaloid

membrane is assumed to be free of drug initially,

1C 1 for Equatton 3.15: Ch = O, ut t =O. (3.16).

It is also assumed an equal concentration and an equal flux at the vitreo/hyaloid interface (the

posterior surface of the hyaloid membrane),

BC I for Equation 3.15, BC 5for Equation 3.1: Ch = C, at the vitreo/hyaloid interface (3.1 7);

.1=, X h BC 2 for Equation 3-15. BC 6/or Equation 3.1: N, , = - D, - = - 4 - a2 a2 '

at the vitreo/hyaloid interface (3.1 8).

where .Chn is the dmg flux into the hyaloid membrane from the vitreous. The anterior surface of

the hyaloid membrane is exposed to the posterior chamber which is constantly flushed by the

aqueous humour secreted from the surface of the ciliary body. The fiow in the posterior chamber

gives rise to a convective fiux of drug away from the anterior surface of hyaloid:

x v BC3for Equation 3.15: N,, = kchC, = -D,, - a ' at the anterior surface ofhyaloid membrane (3.1 91,

where Aip,, is the fiux into the posterior chamber from the hyaloid membrane; and kCvn is the m a s

transfer coefficient away from the hyaloid membrane. Since KCph is a function of the flow field, to

estimate the value of kc., requires the correct modeling of the flow field in the posterior chamber.

The steady state velocity field of the posterior chamber is calculated with the Navier-Stokes

equation,

where p. p and Cr are the density, viscosity and velocity of the aqueous humour in the posterior

chamber. Together with the surface of the ciliary body, the anterior surface of the hyaloid

membrane, anterior surface of the lens and the posterior surface of the iris forrn the boundary of

the posterior chamber. All the above boundary surfaces have a zero fiuid velocity (non-slip

surface assumption), except the part of the ciliary body surface which sewes as the

passageways (flow inlets) of aqueous humour.

BC 1 for Equation 3.20 : Ut = O and = O, at the anterior d a c e of the lem (3.21);

BC 2 for Equation 3.20 : Ut = O and Li, = O. at the posterior W a c e ofthe iris (3 -22) ;

BC 3for Equation 3.20 : Ut = O and Grn = O, at the anterior surface of the hyufoid membrane (3.23);

BC 4 for Equation 3.20 : Ut = O and Un = O, at the "'non-passage " surface of the ciliary body (3.24),

where C; and U, are the tangential and normal velocity of aqueous humour to the surface. The

secretion rate of aqueous humour into the posterior chamber is 0.15 mVhr [see section 2.1 SI.

The momentum boundary condition at the fiow inlets of aqueous humour is,

BC 5 for Equation 3.20 : & = O, Lr,, = Qa ,. A. at thej7ow inlets of aqueous humour (3.25),

where .4 is the total cross sectional area of aqueous humour passageways on the ciliary body

surface [also see 3.2.3.11, and Qu is the flow rate of aqueous humour.

Having calculated the velocity field, the drug concentration profile in the posterior chamber c m

thus be deterrnined using the species continuity equation taking the form of

where Ca and D, are the drug concentration and diffusivity in the aqueous humour o f the posterior

chamber. 80th convection and diffusive m a s transfer of drug are considered. It is assumed no

degradation of drug occurs inside the posterior chamber. The species boundary conditions and

the initial condition for solving equation 3.26 are as following:

Lens is made of densely packed collagen-like material. thus it is impenneable to drug [Sebag,

1 9921,

BC l for Equation 3.26: NLn =O, ut the anterior surface of lem (3.271,

where .LrL,, is the flw normal to the lens surface.

The diffusional exchange of drug between the iris and the aqueous humour is usually negligibly

low [Maurice, 19791,

BC 2for Equation 3.26: N,, = O, at the mterior surface of iris (3.281,

where .Xe, is the dnig flux normal to the iris surface.

Most surface of ciliary body is impermeable to drug due to the tight junctions between the

epithelial layers (also see section 3.2.3.1),

BC 3 for Equation 3.26: .Vc.. = O, at the non-passage surface of the ciliary body (3.291,

where is the flux normal the ciliary body surface.

Since the freshly secreted aqueous humour is always free of drug, a sink condition is imposed on

the flow inlets of aqueous humour, which occupy only a srnall fraction of ciliary body surface (see

sedion 3.2.3.1),

BC 4for Equation 3.26. Ca =O, at thejlow inlets ofaqueous humour (3.30).

Since both hyaloid membrane and aqueous humour are predominantly made of water, there is

no partition between these two phases. An assumption of equal concentration and flux at

hyaloidlaqueous-humour interface is made.

BC 5 for Eqmion 3.26: Np," = - Dh - . at the anterior surfoe of the hvaioid membrane (3 .SA); a BC 6 for Equation 3.26: Ca = CI, ut the anterior surface ofthe hyaloid membrane (3-32),

where XP,, is the dnig flux into the posterior chamber from the hyaloid membrane.

It is also assumed that the posterior chamber is initially free of drug,

IC I for Equation 3.26: Ca =O, at t =O, for the entire posterior chamber (3.33).

After successfully rnodeling the transport processes in the posterior chamber, the mass transfer

coefficient away from the anterior surface of hyaloid membrane (k, ,) was estimateci. The k,

value at the steady state was then transfomed into a stagnant film which gives the same mass-

transfer resistance. lnstead of imposing the velocity field of the posterior chamber upon the

anterior surface of the hyaloid membrane as done by Friedrich [1996], the stagnant film was

used [Figure 3.31. By doing so, the CPU time required to solve the problem can be significantly

reduced. The detailed explanation for adopting such a aapproach is in section 3.2.3.1.

The vitreous body is the central cornpartment of the eye model. The rnodeling of the

interior entity, the dnig phase, and the surrounding tissues, the retina, lens, hyaloid membrane

and posterior chamber of aqueous humour, al1 sewe to correctly describe the boundary

conditions about the vitreous, so that the species continu@ equation of the vitreous

cornpartment (equation 3.1) can be solved properly. As a summary, the dnig distribution in the

vitreal domain is solved with equation 3.1 and 7 initial or boundary conditions, 3.2, 3.5-3.7, 3.14,

3.17 and 3.1 8. The drug distribution in the drug-phase domain is solved with equation 3.3 and 3

initial or boundary conditions, 3.4-3.6. The drug distribution in the retinal domain is solved with

equation 3.9 and 3 boundary conditions, 3.11-3.13. The drug distribution in hyaloid-membrane

domain is solved with equation 3.15 and 4 initial or boundary conditions, 3.16-3.19. The velocity

field of the posterior chamber is solved with equation 3.20 and 5 momentum boundary

conditions, 3-24 -3.25. The dnig distribution in the posterior chamber is solved with equation 3.26

and 7 initial or boundary conditions, 3-27-3.33.

Equivalent Stagnant F

Hyaloid Membrane 1

t lrregular Orug Phase

Figure 3.5 Schematic Oiagram of the Proposed Model

3.2.3 Modifications Made to Friedrich's Model: Description of Transport throuah the

Cilianr Bodv

This projed is a continuation of Friedrich's work. Besides incorporating the realistic

geometry of drug phase into the model, there are a few other minor modifications made to his

model. All the modifications are associated with modeiing of the transport processes in the

posterior chamber.

3.2.3.1 Modification to the Boundaw Conditions on the Surface of Ciliaw Body

In Friedrich's mode1 [1996], the surface of the ciliary body is viewed as the 'inletw for the

flow of aqueous humor and the inlet fiow velocity is uniform everywhere (Le., the entire surface

is open to the passage of aqueous humour). To ensure that the freshly secreted aqueous humour

is free of drug, a zero-concentration boundary condition was imposed on the entire surface.

However, by doing so. a mathematical artifad was created. It implied that 100% of the ciliary-

body surface is perfusable to the drug from the posterior chamber. The dn~g can freely diffuse

out of the posterior chamber via this route as if the ciliary body poses no resistance to drug

transport (Le., the ciliary body is a perfect 'sink" for dnrg). By re-ninning Friedrich model, it was

found that, when a constant flu of fluoresein is imposed across the surface of the hyaloid

membrane (the one facing the posterior chamber), 80% of fluoresein entering the posterior

chamber is eliminated via Mary body readsorption, and remaining 20% is removed convectively

via the aqueous humour drained away at the narrow exit fomed between the iris and the lens.

This result indicates that for Friedrich's model, ciliary-body readsorption is dominant over

convective removal, which is not consistent with the physiological properties of the ciliary body.

The ciliary body is covered by a double layer of epithelial cells. The outer Iayer, facing

the stroma, is called the pigmented Iayer. The inner layer, facing the posterior chamber, is called

the non-pigmented layer, and is responsible for the secretion of aqueous humour [Stjemschantz

and Astin, 19931. Between the epithelial layers, the jundions are tight and therefore not easily

traversed by drug [Schoenwald, 19931, As a resuft, the arnount of drug eliminated from the

posterior chamber via ciliary readsorption should be negligible (minor) compared to that via

convedive removal by aqueous humour, unless the drug is strongly protein bound.

Although the exact mechanisrn of aqueous humour production is not known, it is

generally accepted that pradically al1 aqueous humour is fomed by an active pump mechanism

(e.g., by active purnping of ions such as Na' and Cr. it mates an osmotic pressure to drive

aqueous humour across the cell membrane into the posterior chamber) [Sears, 19811- From the

nonpigmented epithelial cells, aqueous humour is secreted into posterior chamber via thousands

of water passageways. The size, the shape and total number of these passageways are not

known. However, based on the fact that the ciliary body is nearly impemeable to the drug from

the posterior chamber, the size and fractional surface occupied by these passages should be

rather small so that the passive diffusion of drug into the ciliary body along water passageways is

insignificant In this regard, it is certainly inappropriate to set entire surface of ciliary body to be a

"sink", like what has been assumed in Friedrich's model.

The ciliary body surface is best described as a surface perforated with thousands of tiny

holes (exits of water passageways) which are permeable (Le,, sink condition), while the large un-

perforated area is impemeable to species transport. The flow of aqueous humor enters the

posterior chamber through the perforated surface only, and the fluid is stagnant on the

unperfomted surface. However, such a situation is nearly impossible to describe

computationally, even if the size, shape and configuration of these water passages were known.

Each water passageway is an individual flow regime. For finite element analysis, the mesh

element size should be much smaller than the domain in order to have reasonable computational

precision. Therefore, millions of mesh elements would be required to represent these water

passageways. It is impractical to represent the irislciliary body in this fashion considering the

amount of CPU tirne and disk memory required- Moreover, if the passageways are on a

molecular scale, the goveming equations of conventional fluid mechanics and m a s transfer will

no longer be valid since the continuum hypothesis will no longer apply.

To resolve this problern, a simplified description is adopted. For each 2-0 cross-section

of the ciliary body, only three holes are created on the cilliary body surface at equal spacing

[Figure 3.6). For 3-D modeling, these three hofes are projeded 360 degrees to yield three 'rings"

[Figure 3.71. The total cmss sectional area for the inlet flow (the perfusable surface) is only 3%

of the total surface of the ciliary body. The justification for using this 3-ring-3%-perfusablearea

model is based on the following two arguments:

The insensitivity of the velocity profile over the bulk of the posterior chamber to the

inlet velocity, when the total flow rate is fixed: The velocity of aqueous humour is the

pnmary factor affecting the m a s transport of drug inside the posterior chamber. Therefore, it

is important to have the velocity field calculateci correctly. For a steady, fully developed

flow, the bulk velocity is only a fundion of total flow rate and the geornetry of the confined

boundary (e.g.. for a laminar flow in a mund tube, V(r) = ( ~ ~ / A ) [ ~ - ( z ~ / D J ~ J , where V is the

velocity, Q is the flow rate, A is the cross-sedional area, r is the distance from the central

line, and D is the diameter of the tube). To illustrate this concept, a simple 2-0 simulation of

larninar pipe flow is show in Figure 3.8. There are two sections of tubes, each with identical

geometry. The entrance of flow is at the right end of the tube. The flow rate for both tubes is

identical. The only difference between them is that tube A has the entire right-end cross

section open to the fiow, whereas tube B only has 20% of the right-end cross-sedion open to

flow (Le., the flow enters the tube through two narrow slits). Thus, tube B has an inlet velocity

five times higher than that of tube A. The velocity vector plots and the speed contour plots

show that, despite the huge difference in the inlet velocity, the velocity fields of both tubes

are virtually identical except at the small region amund the inlet, where 'entrance effects"

prevail. The sarne thing cm be said about the ciliary body surface. Whether the aqueous

humour enters the posterior chamber via thousands of tiny passages or via three large ring-

shape slits does not affect the bulk velocity field. This conclusion is further supported by the

speed contour plots of the posterior chamber shown in Figure 3-7. Based on the above

evidence, the choice of 3 rings for the inlet is justified.

The insensifivity of total hyaloid elimination rate to the inlet cross-sectional area,

when the inlet cross-sectional afea is less than 5% of the total surface of the ciliary

body: To compare with Friedrich's model, the same simulation with a constant fiuoresein

flux imposed on the hyaloid membrane was run again using the newly developed 3-ring-3%-

petfusable-surface model. The resuit shows that over 80°h of fluoresein entering the

posterior is eliminated by the aqueous humour drainage, compared to 20% elimination

predicted by Friedrich's model. Therefore, the proposed new rnodel is more consistent with

reality than Friedrich's model. Theoretically. the new model can be further improved by

additional redudion of perfusable area (e-g., to be < 1°h of total ciliary body surface, so that

the dnig that diffuses through the ciliary body surface can be reduced to a negligibly low

level. The disadvantage of such an approach is that it will leacl to a significant reduction in

mesh size, and consequently, a significant increase in the number of elements required.

Since the primary interest of this project is placed on the elimination of dwg from the

vitreous body, only the elimination rate across hyaloid membrane and the elimination rate

across the retina are considered relevant. As long as the perfusable area is small enouyh,

so that any further reduction of the perfusable surface area no longer impacts the total

elimination rate across the hyaloid membrane, the model is acceptable. A sensitivity test of

the perfusable surface area shows that there is little difference between the elimination rate-

time profiles for the cases of 5% and 3% perfusable area [Figure 3.91. Therefore, the

settings with 04% perfusable surface are within the robust region. Based on the above

evidences, the choice of 3% perfusable area for the inlet is justified.

Figure 3.6 Perforated Cilliary Surface (2-0 View)

I surface)

(a) Friedrich's Model (b) Proposed New Model

Figure 3.7 Velocity Contour Plot of Posterior Charnber

I

(a) Sketches of Tubs

I Tube A: Full Inlet, 100% Passage Area

(b) Velocity Vedor Plot

Tube B: 2-Hole Inlet, 20% Passage Area

I

(c) 40 Lines Speed Contour Plot 1

The fluid in the tube is water. nie flow rate is I m m . The inlet width and height are 7 cm. The lengffi of tube is 10 cm.

Figure 3.8 Velocity Plots of Tube A and Tube B

COLOR CODE: VELOCITY

8:511E31 0.601E+01 0.561E+01 @.4@1E+01 0.300E+@ 1 0.200€+@ 1 0 .10E+@1

Hyaloid Elimination Rate vs. Time

+ Çnedrich's Model. 100% Perfusabie Area

+ Three "Rings". 50% Peffusabie Area

+ lhree "Rings". 25% Perfusable Area

-- Three "Rings". 12.5% Perlusabk Area - Three "Rings*. 5% Perfusaôie Area

+ Three '%ings", 3% Perfusabk Area

w

IC: C=15O uglml. Total Drug Mass = 9.2475 ug

5 I O Time (hr)

Figure 3.9 Sensitivity of Hyaloid Elimination Rate to the Area of Perfusable Surface of

Ciliary Body

The above elimination profiles are h m a problem where a quarter of posterior chamber is

attached to a taction of vitreous humor. That fiaction of vitreous humor has a volume 0.061 7 ml.

7he initial dmg concentration is 150 &l, unifotm inside the m o u s humour. The geomew of

the mode1 is shown in Figure 3.76.

3.2.3.2 Evaluation of a Mass Transfer Coefficient and Use of a "Staanant Film" to

Replace the Posterior Chamber

It is generally very time consuming to solve convedive m a s transfer problems,

especially wtien the problem size is big (Le., over 100,000 mesh elements). Because of the

inclusion of convedive terms (al1 three directions) in the goveming equations, the global matrix

of the problem is often so big that the solution takes days to converge, and often with large

truncating errors. Although the posterior charnber is not the primary region of interest in this

project, the çomputation in this domain takes a disproportionately high 80°h of the CPU time

required for solving the entire problem. At the beginning of this projed, Friedrich's approach

(1 9961 for calculating the velocity profile of the posterior chamber was followed; the velocity field

of the postefior chamber was calculateci separately and then irnposed on the surface of hyaloid

membrane. In al1 sirnuIations, both diffusive and convective terms were included in the

goveming equations. This approach usually required 50 to 70 hours of CPU time to solve a

problem having a size of 100,000 elements or so. To be more cost-effective in terrns of CPU

tirne, a simpler approach was Iater adopteci. lnstead of diredly irnposing the velocity field on the

anterior surface of hyaloid membrane, an empirically estimated mass transfer coefficient was

used.

To estimate the mass transfer coefficient for drug transport away from the hyaloid

membrane (kd), a 3-0 model of the posterior charnber (Le., 3-ring-3%-perfusable-surfa=

model) was built. A constant drug concentration was irnposed on the hyaloid membrane surface

facing the posterior charnber. From the steady state resutt of this model, the overall m a s

transfer coefficient across the hyaloid membrane is computed according to Equation 3.1 9. Based

on this value, a stagnant film which gives same resistance is placed against the hyaloid

membrane to replace the posterior chamber, Since other domains of the model (Le., the

vitreous, retina, hyaloid membrane) are also stagnant, the corivedive term is completely

eliminated from the transport equation of the model; thus, the size of the global matrix is

significantly reduced. By doing so, the CPU time required to solve the problem above has been

reduced from 70 hours to less than 24 hours. The values of the stagnant film thickness for

various postenor chamber models descnbed in section 3.2.3-1 have been tabulateci in Table 3.2.

Q = &dellrlal = @A)(A)(AC& (3.1 9)

where Q is the total elimination rate across the hyaloid membrane; kCvh is the m a s transfer

coefficient; CMOrd is the drug concentration at anterior surface of the hyaloid membrane; A is the

surface area of hyaloid membrane; D is the diffusivity of dnig in water; L is the equivalent

thickness of the stagnant film; defib, is the concentration gradient across the stagnant film. By

applying an infinite sink condition on the other side of the stagnant film, ACfih is equal to Cwd;

thus, t = D/kCeh. For each specific dnig, there is a specific value for the thickness of the

equivalent stagnant film in the posterior chamber fl ow field.

Table 3.2 Table for Stagnant Film Thickness

Condition on Cilliary SuHace Film Thickness

Friedrich's model, 100% Perfusabte Area 0.1000 cm

50% Perfusable Area 0.1 138 cm

25% perfusable Area 0.1234 cm

6.25% PerfusaMe Area 0.1600 cm

5% PerfusaMe Area 0.1690 cm

4% FerfusaMe Area

I 1 t

Note: D=O-O216 d / h r , which is Iluoresein"~ diffusnrify in water.

0.1735 cm

3% Perfusable Area

To test the validity of this approach, a small scale transient simulation was run. 80th the

equivalent stagnant film and the velocity field calculated from 3-ring-3%-perfusable-area model

were imposed on a hyaloid membrane which is also attached to a small fraction of vitreous

humourl the initial drug concentration in the vitreous is 150 pglml and the posterior chamber is

0.1760 cm

initially free of dnig (Le., the same problem described in section 3.2.3.1, Figure 3.9). The

resuiting dnig elimination profiles of both cases are nearly identical [see Figure 3.101. Thus, it is

justified to replace the ROW field of posterior chamber with a stagnant film of equivalent

resistance to m a s transfer. In this projed, for any pmblern which has over 100,000 elements,

the stagnant film model will be used.

Hyaloid Elimination Rate vs. Time

- with the velocity field of the 3-ring model

If + with equivaient stagnant film

Three Rings, 3% Perfusable Ciliary Surface

IC: C=150 uglml, total mass =9.2475 ug

O 5 10 15 20 25 30 35

Time (hr)

Figure 3.10 lmposing Equivalent Stagnant Film vs. lmposing Velocity Field

3.2.4 Simulation Plan

3.2.4.1 The Studv of the Efïect of Dmo Phase Geometw on Bioavailabilitv Profile

The bioavailability profile of a drug consists of a set of description about the drug

distribution characteristics It is the most crucial piece of information about a dnig and its

administration regime, since the way the drug is distributed largely determines the efficacy of the

treatment. The bioavailability profile of a drug is expressed in tems of parameters such as Ti0

(half life of drug), AUC (area under concentration curve), CL (clearance of dnig), and other

variable-time profile such as dnrg elimination rate vs. time profile, and mean vitreal dnig

concentration vs. time profile. The above parameters and the variable-time profiles will be

corn puted from simulation resuits.

To cover the shapes of the dnig phase at various position and under a wide range of

injection speed, six different cases were established based on the shapes of drug phase

observed in the experiments [see Figure 3.2 and 3.111. The overall geometry of the eye and

boundary conditions are the same in al1 these rnodels. Only the shape of the drug phase has

been changed. In al1 models, the volume of dnig phase is 50 4, a cornmon injection volume. ln

ophthalmologic practice, the injection volume is usually between 25 pl to 100 pl [Peyman,

19941, and rarely exceeds 200 pl [Maurice, 19791. The amount of injeded dnig is set to be 50 pg

in al1 models for the sake of cornpanson. The effeds of injection volume and injection position

on the dnig elimination profile have been studied by Friedrich et al. [1996, 1997bj; thus they are

not repeated in this project. In Friedrich's study, four different positions of the drug phase (Le.,

central placement, against-retina placement, against-hyloid placement and against-lens

placement) were investigated [Friedrich et al., 1996, 1997bJ- In this project, only the two most

common placements of dnrg phase are incorporateci into the models, namely central placement,

and off-central placement which is 0.5 cm away from the center of vitreous body and on the axis

parallel to the lens [see Figure 3.lla, cl. Central placement of the drug phase is usually the

"target" injection location since it gives a more unifom dmg distribution in the vitreous humour,

and is far enough from the retina to avoid toxicity. mcentral placement of drug phase is usually

a consequence of imprecision during injection. However, it is sometimes deliberately chosen to

treat sedorial infection [Maurice, 1979). It is also possible that off-central placement might enjoy

some advantages over central placement under certain special circumstances. For instance, if

the injected dmg is eliminated slowiy (e-g., Tfn > 10 days) and the drug phase is "turbid* (1-e., not

clear), central placement of drug will block the optic pathway for a rather long period of time,

wtiich is a discornfort to the patient. If the toxicity of drug is very low, off-central placement can

be chosen in such a case.

To reflect the transport characteristics for a wide range of dnigs, three different model

dmgs with hypothetical transpoft properties have been simulated. The vitreal diffusivity values

of al1 three model drugs are set to be the same as fluoresein in water (0.0216 m2/hr). Since

most of the drugs intravitreally injeded have a molecular weight in the same magnitude of

fluoresein (Le., < 1000 Da), their diffusivity in vitreous humour should al1 be in the same

magnitude. The retinal permeabilities for the three model drugs are set to be 2.6E-04 crn/s, 2.6E-

05 cm/s and 2.6E-07 crn/s, respedively. 2.6E-05 cm/s is the mean retinal penneability of

fluoresein found in various studies [Friedrich et al., 19961. 2.6E-04 cm/s is a typical permeability

value for the drug which is mainly eliminated through retinal pathway; 2.6E-07 cm/s is a typical

pemeability value for a drug for which the hyaloid membrane is the major elimination route.

central placement, (cylindrical shape, Friedrich's approach)

(b) central placement, moderate fingering -

(c) off-central placement, no fingering (cylinder) -

(d) offcentral placement, moderate fingering

(e) off-central placement, extensive fingenng (slow injection)

(d) retinal deposson (fast injection)

Nofe: The lînes and cuwes oufline me structure of the eye mode/. The meshed dark region is the

drug phase.

Figure 3.11 Six Cases of Injection Shapes (3-0 View)

3.2.4.2 The Studv of the Effect of Dnia Phase Geometw on Toxicihr Profile

It is conceivable that due to the fingering effed, the drug phase can come in close

pmximity to the retina, even in the case of centrai placement. A dnrg phase with fingering is

spread extensively across the vitreous humour, and covers (not occupying) much targer space

than that without fingering. Therefore, some Yingers" can protnide from bulk drug phase and

approach the retina, which can be dangerous. A Vinger' near the retina cxtn lead to an

exceptionally high local dnig concentration, which may be toxic to the retina- To examine the

impact of fingenng, the vitreal drug concentration at the site (Le., a spot at vitreousretinal

interface) closest to the drug phase (Le., finger) should be wnsidered and monitored. There are

two important parameters regarding toxicity that can be obtained from the concentration-time

profile of the site. One is the local peak vitreal concentration (local C,, -&, the other is

local AUC, wtiich is the area under the local concentration-time curve. If the retinal toxicity is the

direct result of acute exposure to a high vitreal drug concentration (1-e., concentration is

important, but exposure time is not necessarily important), the local Cm -, can senre as

an indication of local ocular toxic level, To avoid toxic effed, the local Cm should be always

kept below Cmb, (e.g., < 2000 pg/ml for acydovir [Peymen, 19943). If the retinal toxicity is

largely due to the cumulative exposure (e.g., can be an exposure to a high concentration for a

short tirne or an exposure to a fow concentration for a long time), local AUC will be a better

indication of local ocular toxic level. Since both local Cm, -mm and local AUC are a strong

function of the finger's distance from the retina (Le., the greater distance, the lower values of

local Cm and AUC) , it was decided to set up a series of simulations with Tmgersm of different

sizes and various distances to retina, so that a 'minimum safe distance" from the retina can be

established for each dnrg administration regime. The minimum safe distance can be taken as a

gros parameter in the evaluation of the 'safety level" of a particular drug administration regime.

The lower value of minimum safe distance, the higher safety margin of the administration

regime. The detail about how to obtain this value is explained in section 4.2.

For illustration purposes, six cases divided into two groups were studied. The skeletons

of these six cases are shown in Figure 3.13, The first group consists of three 'single-finger

cases", each with a different finger length. The finger is pointing toward the optic nerve. The size

of the finger and its distance to retina are tabulated in Table 4.3a. The retinal site corresponding

to the optic newe is of particular concem in a toxicity study. A high local vitfeal drug

concentration at that site can damage photo receptors and cause blindness. The second group

consists of three 'single-am casesn, each with a different a m length. The a m is pointed along

the diredion of needle penetration. The specifications of these models are tabulated in Table

4.3b. The 'leakage arma is formed by backfiow of drug along the outer surface of needle during

the injection [Maurice, 19791, as shown expenmentally in Figure 3.4. The higher the injection

speed, the longer the leakage a m wili be. Although like a viscous finget, the leakage ami is a

protnrsion from the bulk phase of dmg, the reason for its formation ba rs no relationship to the

physical pn'nciples underîying viscous fingering phenomenon. Therefore, the name of leakage

a m is given to distinguish it from other type of drug phase protnision.

The drug chosen to be simulated in these six rnodels is model drug 2 (fiuoresein), which is

the most common tracer material used in the ocular phatmacokinetics studies. As in previous

simulations, the injection volume and the dose of drug are set to be 50 pl and 50 pg respectively

(Le., initial drug phase concentration = 1000 pg/ml). The model drug, fluoresein, itself is known

to be non-toxic even at very high concentration. However, for illustrating how to detemine the

minimum safe distance of a particular administration regime (Le., fluoresein, 50pg/50pi in this

case), fluoresein is set to have a hypothetical acute toxic Ievel of 100 pg/ml.

. , .- - a - .. (a) Short Finger

1

I (b) Medium f inger I

Group 1 : Single-Finger Cases

(a) Short A m

I - , -

, V

(b) Medium A m

1 /A-- \/'-

(c) Long A m

Group 2: Single-Am Cases

Figure 3.12 The Single A M i n g e r Model

4.1 The Effect of Drua Phase Geometrv on Bioavailabilitv Profile

The simulation resuits to determine the bioavailability profiles of three model drugs are

summarized in ternis of Tin, A U k i and F- [Table 4-11, mean vitreal drug concentration-

time profiles [Figure 4.11, retinal elirnination rate-time profiles [Figure 4-21, and hyaloid

elimination rate-time profiles [Figure 4-31.

The half life. Tln. is the time required for 50% of injeded drug to be eliminated from the

vitreous body. For an intravitreally injeded dmg, a long half life is desired because having vitreal

dnig concentrations in the thenpeutic range for a longer period of time reduces the frequency of

required injections. AUC is a parameter that is often used as a measure of cumulative exposure

of certain tissue to a dnig. AUC of the mean vitreal drug concentration-time curve will reflect the

cumulative exposure of vitreous humour to a drug. It is generally preferred to have a high

AU- value because it means prolonged drug action inside the vitreous humour. Clearance

(CL) of dnig fmrn the vitreous cornpartment is defined as dose/AU&, with a dimension of

volurne/time. A high ocular clearance value means fast elirnination of drug from the vitreous

body. CL and AUC are similar parameters dewibing the dmg elimination processes, but CL

values are more frequently quoted by researchers than AUC values, That is because CL values

tend to be dose independent whereas AUC is certainly dose dependent.

The three model dnigs, 1. 2. 3, al1 have a same vitreal diffusivity of 0.021 6 cm2/hr, while

their retinal penneabilities are 2.6E-07 cm/s, 2.6E-05 crn/s, and 2.6E-04 cm/s, respedively. All

discussion will be made in sequential order according the drug's retinal peneability, starting with

model dnig 1.

and the dose is 50 pg.

Fraction of Dnig Eliminated through Retins

(a) FdmI Table

Modet Dnig 1 h - 0 . 0 2 1 6 cm2/hr

Note: Model drug 2 is actually fluoresein. The injection volume for al! fhree model drugs is 50 pi,

Model Dmg 2 ~ 0 . 0 2 1 6 cm2/hr P -2.6E-O5 c m l ~

0.856

0.865

0.879

Central, Cylinder Centtal, Moderate Fingering Off-central, Cy linder Offkentral, Moderate Fingering Off-~entrai, Extensive Fingering Retinat Deposition, Ra@ Injection

Central, Cyiinder Centrai,

Note: The total volume of vifreous humour is 4.0 ml.

Mudel Dnig 3 -0.0218 cm2/hr

0.908

0.91 6

0.932

P -2.6E-07 cm/s 111 0.148

0.151

0.151

0.1 50

0.1 60

0.1 73

Cylinder Off-central, Moderate Fingering Off-cenhai, Exiensive Fingering Retinal Deposition, Ra~ id lniection

(b) CLtogl Table

Total Clearance of Drug (mUhr)

0.076 1 0.930

04638

0.051 4

0.0527

0.902

0.957

Model Drug 3 -0.0216 m2mr Pntid2,6E-04 WS

0.41 7

0.445

Mode1 Drug 1 -0.0216 crn2/hr Pmd2.6E-U7 CWS

0.0494

0.0509

0.944

0.987

Mode1 Drug 2 h - 0 . 0 2 1 6 m2/hr P-?2.6€-05 cm/~

0.308

0.320

I 0.373

0.402

0.605

0.S58

0.635

1.57

Haif L i i of Drug (Hour) 1 Mdef-2- 1 Mode1 Orug 3-

(c) Tt@ Table

Table 4.1 FritiMIi C h I and Ttn Tables of Three Model Dnigs

Mean Vitreal Drug Concentration vs. Time h y 0 J 2 1 6 tm2nir. P,#E07 cmh. tow R.thol P.mwiblUW Cam

O 50 1 0 0 1 6 0 m 2 6 0 3 0 0 3 6 0

Time (hr)

(a) Model Drug 1

Mean V i l Drug Concentration vs. T i m h a * 3 2 l s emZlhr. P d W M o d . r r t . Rith.l- - m-1

O 1 O 20 30 40 50

Time (hr)

(b) Model Drug 2

Mean V i e a i Drug Concentration vs. Time M . 0 2 1 6 cm2lhr. P e 6 6 û 4 cds, Htgh Ratinai PvmrbüityC8se

(c) Model Dnig 3

Figure 4.1 Mean Vitraal Dnig Concentration-Thne Profiles of Three Model Drugs

Retinal Elimination Rate vs. Time n m t i QI2*, a34 ~ . r il ~ i r r a ) ~ ~ . r

O 20 40 60 80 100 120 140 160 180

l i m e (hr)

(a) Model Drug 1

Retinal Elimination Rate vs. Time

O 1 O 20 30 40 50

Tirne (hr)

(b) Model Dnig 2

Retinal Elimination Rate vs. l ime -.û218 cnQRr. P- anh. Hbb R.(hil Ehdndon

O 6 10 16 A) 26 30 36 4û

nme (hr)

(c) Model Dnig 3

Figura 4.2 Retinal Elimination Rate-Time Profiles of Three Model Drugs

Hyaloid Elimination Rate vs. Time ü,&.&.O216 cm'fhr, P e 6 E 4 7 cmh. Law R.tkrrl PmnlMBLy Case

O 60 100 160 200 260

Time (hr)

(a) Model Drug t

O 10 20 30 10 W 60

Time (hr)

(b) Model Dmg 2

Hyaloid Elimination Rate vs. T h e h y 0 . 0 ; 2 1 6 cdnir. Pwm+Q1 cmlr. Hlgh Retinal Pmmwabiiky Case

(c) Model Dnig 3

Figure 4.3 Hyloid Eliminatîon Rate-Time Profiles of Thme Model Dmgs

Mode1 dmg 1 has a very Iow retinal pemeability. Less than 20% of the drug injeded is

ultimately eliminated through the retinal pathway, even in the case of retinal deposition [Table

4.la). The drug elimination profile [Figures 4.1a. 4.2a. 4.3aJ shows very Iittle sensitivity to the

shape or position of the dmg phase. All six plots of mean vitreal dnig concentration vs. tirne

collapse ont0 one cuwe (Figure 4.laI. An injedion producing extensive fingering gives the

almost same result as an injection leading to a cylindrical injedion shape. There is l e s than a

5% difference in Tln values for central cylinder, off-central cylinder and retinal deposition cases

[Table 4.1~1. This phenomenon is due to the fact that, for this drug, the transport process is

retina controlled. The value of vitreal diffusivity over the radius of vitreous body is 6.OE-06 cm/s

(i-e., D = 0.0216 m2/hr, R = 1 cm). The retinal pemeability of drug is 1.OE-07 cm/s. Thus, the

value of P/(D/R), a dimensiontess group refleding the ratio between the drug-transfer resistance

imposed by vitreous humour and retina, is 0.01 7. As a general nile of thumb, when PRID value

is l e s than 0.1 (i-e,, the vitreal resistance is only 10% of the retinal resistance to the dnig

transport), it is a dear case of a retina-controlled process. Due to the high retinat resistance, a

steep concentration gradient across the retina is required to induce a significant retinal flux. The

time required to build up a signifiant gradient across the retina is very large compared to the

time required for the dmg to diffuse through the vitreous. The injected dmg should be well

distributeci throughout entire vitreous humour before significant retinal pemeation occurs. As a

resuit, the effects of dnig-phase volume, position and geometry have diminished.

The insensitivity of the bioavailability profile to the injection volume, position [Friedrich et

al, 1996, 1997b1, and the geometry of drug phase in the case of low retinal permeability can be

effectively utilized to simplify the simulation model. When the retinal pemeability of the dnig is

IO-' cmls rnagnlude or les, the model does not need to account for the volume, position and

geometry of the dmg phase. Instead, one simple 'central cylinder" rnodel is adequate [see

Figure 4.4).

It would be ideal if aH the intravitreally injeded drugs have a very low retinal permeability

(e.g., Pnuorescei~n g w m i d e = 7.62E-07 c m [Friedrich, 19Qôi). A low retinal pemeability leads to a

much more unifom intravitreal distribution and much longer half life of dnig in the vitreous

humour [Maurice, 19791. Computationally, by eliminating several parameters (e.g., volume,

position and geometry factor of drug phase), the simulation of the drug's transport process is

much easier; and as a rule of ttiurnb, the resutt is more robust if fewer sources of variation are

introduced into the model. Unfortunately, many intravitreally injected drugs are known to have a

high or moderate retinal pemeability (e.g., P-, = 4.6E-06 cm/s [Friedrich, 19963).

Therefore, it is premature to exclude the effect of dwg phase geornetry from the mode1 at this

point.

Figure 4.4 Central-Cylinder Model

Model drugs 2 and 3 were assigned retinal pemeability values of 2.6E-05 cm/s and

2.6E-04 cm/s, respedively, hivo and three orders of magnitude higher than that of model dnig 1.

The PRtD values for model dnig 2 and 3 are t .7 and 17, respectively. Their bioavailability

profiles [Table 4.1, Figure 4.1 b,c, Figure 4.2b,c, Figure 4.3b,c) have indicated an increasing

sensitivity to drug phase geometry and position. Therefore, for moderate or high retinal

pemeability (Le., 1 .OE-06 cm/s - 1 .O€-04 cmls) drugs, the position factor of drug phase must be

included in the model. The geometry fador of drug phase should be inciuded in the mode] at

least for the case of extrerne injection speed (Le., as low as 0.1 mumin wtiich causes extensive

fingering, and as high as 15 mumin which causes retinal deposition). The above conclusions are

based on the following obsenrations:

1. The mean vitreal drug concentration-time profile of off-central cylinder case deviates more

and more from the central cylinder case as the retinal permeability value increases [see

Figure 4.11. For rnodel drugs 1.2 and 3, the percentage difference in total clearance vaIues

between off-central cylinder model and central cylinder rnodel is 6%, 19% and 30%

respectively [see Table 4.1bI. Thus, it is concluded that the position fador is significant for

high or moderate retinal pemeability (Le., 1 .OE-06 cm/s - 1 .OE-04 cm&) dmgs, which also

confimis the finding in Friedrich's work [1996].

2. At a low retinal permeability, the retinal elirnination rate-tirne curves of al1 three off-central

placement models (Le., cylinder, moderate fingering and extensive fingenng) overfap one

another [Figure 4.2aI. However, at a moderate and high retinal pemeability, clear

differences in retinal elimination rate can be observed between the cases of off-central

cylinder and off-central extensive fingering [Figure 4.2b,cJ. For model dnigs 1, 2 and 3,

the difference in total clearance ( C m between off-central cylinder model and off-central

extensive fingering cases is 2%, 10% and 17%, respectively [Figure 4.1bI. Thus, it is

concluded that, at an extremely low injection speed (e.g., 0.1 ml/min), which leads to

extensive fingenng of the drug phase, the resulting injection shape is different enough from

the cylindrical shape to significantly affect the bioavailability profile, in the event of

moderate or high retinal permeability (Le., 1 .OE-06 cm/s - 1 .OE-O4 cmls).

For model drug 1 (Pm = 2.6E-07 cm/s), the TIR values of central cylinder case and retinal

deposition case are 53.4 hours 53.3 hours, respedively. For model drug 3 (Pmd = 2.6E-

04), the TIQ values of central cylinder case and retinal deposition case are 7.36 hours and

0.386 hours. In each case, the percentage difference in Tln value between these two cases

is 0.2% and 94.8%. respedively. Evidently, at an extremely high injection speed (eg. 15

mWmin), wtiich leads the retinal deposition of the drug phase, the resulting injection shape is

different enough from the cylindrical shape to significantly affect the bioavailability profile,

in the event of a modemte or high retinal pemeability (Le,, 1 .OE-Oô cm/s - 1 .OE-04 crnk).

4. Over a wide range of retinal penneabil'm (Le. 1.OE-07 cm/s - 1.OE-04 cm/s), the drug

elimination-time cuwes of moderate fingering cases are indistinguishable from the ones of

cylinder cases in both central placement and off-central placement situations [Figure 4.1,

4.2 and 4.31. For al1 three model drugs, there is l e s than a 5 O r 6 difference in total clearance

between the cylinder and moderate fingenng cases [fable 4.lbI. Clearly, at a moderate

injection speed (e.g. 1 .O mumin), which leads to the moderate fingenng of the dwg phase,

the resulting injection shape does not affect the bioavailability profile.

The results have ctearly showr, that an extreme injection speed can result in a drug

phase geometry that is so different from being cylindrical that when retinal pemeability is

moderate or high, Friedrich's model can not accurately predict the bioavailability profile. In such

circumstances, the aciual shapes of the dwg phase must be incorporated into the model.

However, in pradice, neither extremely high nor extremely low injection speeds are encouraged.

Assuming the injeded dmg volume is 100 pl, the injedion process will take 0.4 second at a

speed of 15 mumin, and 60 seconds at a speed of 1.0 mumin. At a high injedion speed, due to

the high rnornentum, the injeded drug will rush through the vitreous humour, and be stopped by

the Mina. The high risk of ocular toxicity and complications associated with excessive ocular

pressure and the backfîow of injeded drug have made high injedion speed dearly undesirable.

A low injection speed is not preferred either. It is very difficuit to keep the patient's head and

dodor's hand absolutely still, and the same time push the piston of the syrÏnge at a unifom rate

for over a minute. Any unwanted movement of the patient's head or the syringe can cause

severe trauma to the delicate ocular tissues- In practice, the injection process only takes a few

seconds; the injection speed should therefore be around 0.3 mllmin (Le., 20 seconds to inject

100 pi) to 1.5 mVmin (4 seconds to injed IO0 pl). Within this range, moderate fingering is

expeded [see Figure 3-21. Therefore, it is concluded fhat, over the pradical range of injection

speeds, Friedrich's cylindricaldmg-phase model, after some changes of boundary conditions to

the ciliary body surface, should accurately predid the bioavailability profile of a drug.

There are three important dimensional properties describing a drug phase, the shape,

the volume and the position. By leaving the complex geometry of dnig phase out of the pidure,

the simulation for determining the bioavailability profile of a drug will becorne much simpler

computationally, since there are only two variation sources needeâ to be considered. 60th the

volume and the position of a drug phase were found to have a strong impact on the

bioavailability profile of a drug Friedrich et al., 1996, 1997bJ. To account for possible variations

of the bioavailability profiles due to volume and position, an eight-case approach with the

modifiecl Friedrich's model is proposed here to predid the 'operating region" of the

bioavailability profile. In this rnethod, three extreme positions and one reference position are

considered, namely, against-retina placement, against-hyloid placement, against-Iens placement

and the central placement [Friedrich, 19961. For each position, two extreme volumes are

considered, 10 pl and 200 4. Four positions by two injedion volumes leads to eight cases

[Figure 4 4 . Once these eight cases are created, they can be repeatedly used for different

drugs. To nin these cases, the user only needs to input the intended dose, retinal pemeability of

dmg, and diffusivity of the drug in the aqueous humour, the hyaloid membrane and the vitreous

humour. Depending on the cornputer, a sketch of the 'operating region" of a dmg, bounded by

eight curves should be ready in a matter of several hours to a few days [Figure 4.q. This will

be a very useful piece of information in the design of drug administration regirne.

Before fuIIy embracing the eight-case approach, one precaution should be raised. The

eight-case approach is good for the evaluation of bioavailabilii profiles only, and the results

generated from the model can not be used to evaiuate toxicity. Bioavailability profites are l e s

sensitive to the local variation of drug distribution because they are al1 Tissue averaged" (e.g.,

mean vitreal concentration-time profiles, total retinal elimination rate-time profiles), whereas

toxicity profiles are much more sensitive to the uneven distribution of drug. For instance, a small

Yïngef' (e-g., only 5% of total injeded volume) near the retina can yield an exceptionally high

local dmg concentration which is toxic to retina, and at the same time, has very little impact on

the overall retinal elimination profile, since the region affected is a very small portion of retina.

The above conclusion is supported by the results tabulated in Table 4.2. The results

indicate that the local toxicity profiles of the cylindricaldnig-phase case and the moderate-

fingeringdnig-phase case are dissimilar, whereas their overall bioavaiiability profiles are not

significantly different. For example, for rnodel drug 3, at the vitreous-retina interface, off-central

fingering case yields a local maximum vitreal dnig concentration which is eight time higher than

the local C,, *, value of cylinder case. If a drug administration regime is designed and

clairnecl to be 'safe" based on the fad that local Cm-, value predicted by cylinder mode1

is well below the Cm& ,- listed in the Iiterature, such regime can be potentially dangerous to

the patient. Therefore, when it cornes to evaluation of local toxicity, fingering can not be

ignored. This also foms the rationale for the development of the single-fingerlam model for

local toxicity evaluation.

1 Injection volume 200 pl 1 Injection volume 10 pi

(a) central F

(b) against-retir

(c) against-h yall

ia placement

oidplacement -

(d) against-lens placement

Note: The dark region is the drug phase.

Figure 4.5 Eight-Case Approach with the Modïfïed Friedrich's Model

time

Figure 4.6 Schemafic Diagrarn of "Operating Region" of a Drug's Bioavailabilîty Profile

Central, Cylinder Central,

Moderate Fingering 1 1 1 1 Note: 1. Injected dmg volume is 50 pl, and dose is 50 pg (i.e. initiai C- = 1000 &mi).

2- Model drug 2 is actual/y fluoresein. 3. Local Cm. ,- rneans the maximum W a l drug concentration ever occum at any spot of entire vr'Zreous-retina intehce.

Cylinder 1 Off-central, 166

Table 4.2 Local C- -, Table

Local Maximum Vitreal Drug Concentration at V'rtreous-Retina Interface (pghnl)

Mode1 Dntg t -0.0216 cm2lhr P-2.6E-Of C ~ / S

10.4

10.9

144 57.8

Mode1 Dmg 2 k=O.O216 cm2lhr P-~2.6€-05 cm/s

3.58

6.1 5

Model Drug 3 +-O.OZ16 m h r PreH#2.6E-04 CWS

0.783

1 -53

4.2 The Effect of Drua Phase Geometrv on Toxicitv Profile

(a) Fm,,, Tin , AUC and Cm Table for Single-Finger Models

FingerjAnn Volume (Po Shortest Distance to the Monitored Retinal Site (cm) Retinaf Elirnination Fraction Haif Life (hr) Average A U k m (pg.hr/ml) Average Cm (pglrni) Local AUC ,, ,, (pg. hr/ml) Local Cm--- (@ml) Note: 1. The injeded wJume is 50 jd, and dose is 50 pg in al/ sjmu/ation.

2. 'Fingef r e m to the viscous finger pointed at optic nerve; %rmW refers fo the leakage arm fonned along the needle penefrafed Mo the m o u s humor. 3. œLocal Cm " means the maximum vifreal drug concenfrafion ever expefïenced at one particu/ar spot of vitreous-retina interface; -average Cmm refers to the peak value of mean vitreal drug concentfation of entire vibeous-retina inferface. Likewise is the definmon for local AUC and average AUC.

No FingerlAmi 0.0

0.76

0.866

9.90

43.1

3.17

47.6

3-64

Short Finger 3.0

0.52

0.863

9.80

43.8

3.26

62.3

4.21

T 1 No FingetIArm FingerfArm Volume 1 0.0 (4) Shortest Distance to the

Medium Finger 6.0

0.28

0.869

935

44.2

3.28

66.9

Monitored Retinal Site (cm) Retinal Elimination

Long Finger 9.0

0.04

0.879

9.25

44.7

3.32

144

0.75

Fraction Hatf Life

Long A m 9.0

Short A m 3.0

0.866

(h r) Average AU-

5.97

Medium Ann 6.0

0.51

9.90

(pg . hrlml) Average C,, vibeanweüna

(b) Fdm, Tln , AUC and C,, Table for Single-Am Models

Table 4.3 Fda, Tin , AUC and C,, Tables for Single-Finger and Single-Ann Models

176

0.851

43.1

W m i ) Local AUC - m. hr/ml) Local Cm,Mm*

l

0.27

9.86

3.1 7

0.03

0.839

43.2

36.9

2.75

0.822

9.76

3.1 6

9.58

42.6

39.3

3.36

41.8

3.1 0 2.98 I

$1 -1

5.86

109

104

Retinal Elimination Rate vs. Tme D e 0 1 1 6 cl12hr. P e ( E 9 5 4% Fhœndm

Tiie (hr)

(a) Single-finger Models

Retinal Elimination Rate vs. Time D d . 0 2 1 6 aiif Jhr. P A 6 E 4 S rih F b m &

n - shortam

+I- mrdlum um

+ kng a m i

(b) Single-Ann Models

Figure 4.7 Retinal Elimination Rate-ïïme Profiles for Two Gmups of Models

(a) Single-Finger Models

Local AUC, vs. Time

-16 - 2 ~ . P-*E95 aih. PbwirrC,

O 10 20 30 40 50 6û 70

Time (hr)

(b) Single-Ann Models

Figure 4.8 Local AUC-Time Profiles for Two Groups of Models

Local Viieal Drug Concentration , vaCm vs. Time

0.01 0.1 0 1 .O0 10.00 100.00

Time (hr)

(a) Single-Finger Models

Local Virtreal Dmg Concentration - - ., vs. Time

0.01 0.10 1.00 10.00 100.00

lime (hr)

(b) SingleAm Models

Figure 4.9 Local CvfbiomCW -Tirne Profiles for Two Groups of Models

As expeded, the simulation tesuits Figure 4 4 clearly indicate that fingering effed has

little impact on the general bioavailability of dnig over a wide range of finger sizes (ranging from

6% to 18% of total injected volume). The Fm, and TIn values of single-finger/arrn models are

al1 within range of 5% of the values from nu-fingerlarm mode1 [see Table 4.31. Their retinal

elimination rate vs. time curves can be virtually super-imposed on one another [see Figure 4.71;

even their average A U L , and average Cm, -, values are nearly identical. AUC

and Cm, wtiich can be loosely translated into 'cumulative exposure to dnig ' and ' acute

exposure to dnig", are two most commonly used parameters in the evaluation of toxicity. In this

case, the average Cm -, values are al1 around 3 pglrnl, well below the toxic level of 100

pgfml. However, the mean value does not tell anything about the variation of concentration

profile on vitreous-retina interface. The finger and a m wtiich are in close proximity to the retina

can yield a local dnig concentration higher than the toxic level. For instance. the local Cm ,,, values in both long-finger and long-am cases are al1 above 100 pg/mI [Table 4.31.

Moreover, the local AUC values in these two cases are about three times of the local AUC value

in no-ann/finger case [Table 4.3 and Figure 4.81. This rneans that roughly 200% more dnig is

delivered to that particutar site (e.g., optic newe) than the amount predided by no-finger/arm

model.

While the average Cm and A U L , values suggest that toxicity should not occur,

the local Cm -na and AU-, are much higher, indicating a danger of ocular toxicity

at the retinal site closest to the Iriscous fingef and 'leakage am". Therefore, it is important for

ophthalmologists to be aware of such danger, especially in the event of off-central injection for

treating sectonal infection. The administration regime, such as the initial concentration (intended

dose / injected volume), and the intended position of dnig phase should be adjusted to avoid

toxicity. It should be realized that there is an unique 'minimum safe distance" to be kept between

frontier of the drug phase and retina, for each individual drug and the associated administration

regime (e.g., dose, volume, position). In the event of a central injection of 50 pg of rnodel dnig 2

in 50p1 aqueous solution, the minimum safe distance for the viscous finger and leakage a m

from the retina to retina should be around 0.05 cm, based on the resuk tabulated in Table 4.3. A

distance of 0.05 cm is just about 5% of vitreous body's radius. This rneans only the outer 15%

space of vitreous body is the 'pmhibited region" for the Yingers" and "am". This particular

administration regime will be very "forgiving' in ternis of nsk of ocular toxicity. By properly

adjusting injection speed and carefully positioning the needle, an experienced ophthalmologist

should be able to keep "fingers" and 'am" within the central 85% space of vitreous humor.

The experimental studies showed that, usually, more than one finger protrudes from the

bulk phase of drug. These fingers, of different volume, length, growth diredion, will interact with

one another, and affect each othefs transport behavior. The local toxicity can resutt from a

combination of multiple fingers, although the closest finger should be the dominant factor. The

singlefinger/ann model obviously does not capture the essence of cornplex interadiori of

multiple fingers. It simulates the drug transport process as if al1 the fingers and anns a d

independently. However, it should also be realized that there is no way to preâict the exact shape

of dnig phase and describe it quantitatively. The volume, the length, the shape and the growth

direction of fingers are virtually random [see Figure 3.21, although some qualitative description

might be applied. Therefore, it is not mathematically meaningful to create a specific multiple-

finger model and daim the *minimum safe distance" value predicted by this model is the 'only

correct one". On the other hand, despite the shortcoming of being l e s realistic, the single-

fingerfarm model is much more cleariy defined and robust than the random muttiple fingenng

model. The *minimum safety distance" detemined by single-fingerlam rnodel can be used as a

'ballpark" figure or warse guideline.

5. Conclusions and Recommendations

Based on the experimental findings and simulation results, following statements of

conclusion and recommendation have been made:

Injection speed is the predominant factor that effeds the geometry of the injeded drug

phase. The viscous fingering phenornenon gradually diminishes as the injection speed

increases. By the delicate control of injection speed, certain types of dnig phase geometry

can be preferentially seiected over others (Le., choose moderate fingeflng over extensive

fingering).

Extreme injection speeds, either high or low (e-g., 15 mumin or 0.1 mVmin), are not

recommended due to the high risk of trauma, ocular toxicity and relative short haif life of

drug. If it is to be used, a realistic geornetry of dnig phase (e.g., extensive fingering case at

an extremely high injection speed, and retinal deposition case at an ultra low injection speed)

should be incorporatecl into the simulation mode1 to predid the bioavailability profile of the

dnig, accurately.

Over the pradical range of injection speeds (e-g. 0.3 rnUmin - 1.5 mumin), only a moderate

degree of fingering is observeci. The geometry of the dnig phase in such a case has little

impact on the general bioavailability profile. Friedrich's cylindrical-dmg-phase model after

some modifications (e-g., changes of boundary conditions applied to the surface of the ciliary

body), is adequate to predid the general bioavailability profile of the dmg.

lt is recommended to use the eight-case approach with the modified Friedrich's model to

establish the "operating region" of the bioavailability profile of a drug. The profiles obtained

can be used as an aid in the design of a drug administration regime. The eight cases can be

used for any drug by just inputting appropriate parameters (e.g., vitreous diffusivity value,

retinal permeability value).

In the event of an extremely low retinal pemeability (Le. c 1.OF-07 cmls), the "operating

region" will collape into a single cuwe. In such a case, a single model (Le., one central

cylinder mode0 is adequate to predid the general bioavailabil'rty profile of the drug.

6. The irregular geometry of the dnig phase caused by the viscous fingering effed has a

significant impact on the local toxicity. Although a good mode1 for prediding the general

bioavailability profile, the rnodified Friedrich's mode1 is inadequate for evaluation of the local

toxicity.

7. For the study of local toxicity, it is recornmended to use the single-finger/am mode1

developed in this projed. The 'minimum safe distance" value yielded by the mode1 can be

used as a g ros parameter to evaluate of the 'safety level" of a particular dnig administration

regime. A lower value of the 'minimum safe distance" generally means a lower risk of ocular

toxicity .

6. References

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Balazs €A: The introduction of elastoviscous hyaluronan for viscosurgery, In: Viscoelastic Material, edited by Rosen ES, Oxford, Pergamon Press, 1989, pp167-181.

Baviere M: Basic Concepts in Enhanced Oil Recovery Processes, New York, Elsevier Applied Science, 1991 : ppl -39.

Bematchez B and Camber O: Use of hyaluronic acid in ocular therapy, Boca Raton. CRC Press, 1992: pp106-115.

Bickford WB: A First Course in the Finite Method, 2nd Edition, Burr Ridge, 1RWIN. Inc, 1994: ppl -20.

Cornper W and Laurent T: Physiological functions of connedive tissues polysaccharides, Physiol Rev 1978; 58:255-260.

Cunha-Vaz JG and Maurice DM: The acitve transport of fluoresein by the retinal vessels and the retina, J. Physiol. (London)l967; 1 91 :467-468.

Davies AJ: The Finite Elernent Method, Oxford, Clarendon Press, 1980: ppl-10.

Dolnak DR, Munguia Dl Wley CA: Lack of retinal toxicity of the antocytomegalovirus drug (S)-1- (3-hydroxy-2-phosphonymethoxypropyl) cytosine. lnvest Ophthalmol Vis S u 1992; 33:1557- 1563.

Fatt 1: Flow and diffusion in vitreous body of eye, Bull Math Biol 1975; 37:85-90.

Fatt 1, Hedbys BO: Flow of water in the sciera, Exptl Eye Res 1970; 10: 243-249.

Fluid Dynamic International, Inc.: FiDAP Theory Manual 7.0, Evanston, Fluid Dynarnics International, 1993, chapter 2-5.

Freeman WR, O'Connor GR: Acquired immune deficiency syndrome retinopathy, pneurnocystis, and cotton-wool spots. Am J Ophthalmol 1984; 98:235-237.

Friedrich, SW: Ocular phanacokinetic modelling (a Ph.D. thesis), Department of Chernical Engineering, University of Toronto, 1996: pp70-140.

Friedrich SW, Cheng Y-L, Saville BA: Finite element rnodelling of drug distribution in the vitreous humor of the rabbit eye. Annals of Biomedical Engineering 1997; 25:3=314.

Friedrich, SW, Cheng Y-L, Saville BA: The effects of intravitreal injection position and volume on drug distribution in the human eye. Current Eye Research 1997; 16:663-669.

Friedrich SW, Cheng Y-L, Saville BA: The effeds of aphakia and changes in retinal pemeability and vitreous diffusivity on dnig distribution in the vitreous humor of the human eye. Accepted for publication, Journal of Ocular Phamiacology, 1 997.

Gumy R, Ibrahim H: Design and evaluation of controlled telease systems for the eye, J. Pham Sci 1985; 74:399

Hendedy DE, Freeman WR, Causey DM: Cytomegalovirus retinitis and response to therapy with gancidovir. Ophthalmology 1987; 94:425-434.

Henry K, Cantrill H, Fletcher C: Use of intravitreal gancyclovir (dihydroxy propoxymethyl guanine) for cytomegalovinis retinitis in a patient with AIDS. Am J Ophthalmol 1987; 1 O3:17-23.

Hultsch E, Balazs EA: Transport of hyaluronic acid from the vitreous: mechanism and dynamics, ARVO Abs. 1975.

Karacorlu S, Peyman GA, Karacorlu NI: Retinal toxicity of 6-methoxypurine arabinoside (ara-M): A new selective and potent inhibitor of varicella-zoster virus. Retina 1992; 12:261-264.

Kelly TS: A note on intravitreous penicillin, Brit. J. Ophthalmol, 1948; 32:153.

Kinsey Veand Reddy DV: Chemistry and dynamics of aqueous humour. In: The Rabbit Eye in Research, edited by J.H. Prince. Springfield: C.C. Thomas, 1964, pp.218-319.

Lin H, Image Analysis of Dmg Distribution in the wtreous Body (undergraduate thesis), Department of Applied Chemistry and Chernical Engineering, University of Toronto, 1996: ppl 1- 27.

Livesley, RK: Finite Element - An Introduction for Engineers, 1st Edition, Cambridge, Cambridge University Press, 1 983: ppl -1 O.

Maurice DM: Injection of drugs into the vitreous body. In: Chapter 6 of Symposium on occular therapy, St Louis, C. V. Mosby Co, 1979; 959-72

Maurice DM: Protein dynamics in the eye studied with labelled proteins, Amer- J. Ophthalmol 1959: 47:461-463.

Maurice DM, Mishima S: Ocular pharmacokinetics, In: Pharmacology of the Eye, Edited by Searç ML, Berlin, Spnnger-Verlag, 1984: ppl9-35.

Maurice DM: Communication with Maurice MD through Cheng Y-L and Saville BA, 1996.

Nikoleit JR, Peyman GA, Lagouros P: Retinal toxicity of intravitreal ethyldeoxyuridine and zinc. Ann Opthalmol 1988; 20:385-387.

Ohtori A and Tojo K: In vivo/ in vitro correlation of intravitreal delivery of dmgs with the help of computer simulation, Biol. Pham. Bull 1994; 77(2)283-290.

Peyman G, Schulman J: Intravitreal surgery, Principle and Pradice. New York, Appleton- Century-Crofts, 1986:407-455

Peyman G, Schulman J: Intravitreal surgery, 2nd Edition, Norwalk: Appleton and Lange, 1994: pp 1-1 5, pp 951 -971.

Sarkies NJC, Blach RK: Ocular disease in immunosuppressed patients. Trans Ophthalmol Soc UK 1 985: 1 04:243-247

Schoenwald RD: Phannacokinetics in ocular dnig delivery In: Biopharmaceutics of Ocular Drug Delivery, edited by Edman P, 1st edition, Boca Raton, CRC Press, 1993; pp 1 SMgl .

Sears ML: The Aqueous, in Alder's Physiology of the Eye, Moses, R. A., Ed., C. V. Mosby, St. Louis. 1 981 : pp204-205.

Sebag J: The Vireous, Structure, Fundion and Pathobiology, 1st edition, New York, Springer- Verlag, 1989: pp 35-67. Sebag J: The Vireous, Chapter 9 in Mosby Year Book, (St. Louis, Mosby Year Book, Inc), 1992: pp270-345-

Sison RF, Holland GN, MacArthur W: Cytomegalovirus retinopathy as the initial manifestation of the acquired immunodeficiency syndrome. Am J Ophthalmol 1991 ; 112:243-249.

Snowden JM: The stablizatiori of in vivo assembled collagen fibrils by proteoglycansi glycosaminoglycans, Biochem Biophys Ada 1982; 706:153-156.

Stjemschantz J, Astin M: Anatomy and physiology of the eye - physiological aspects of ocular dnig therapy. In: Biopharrnaceutics of Ocular Dnig Delivery, edited by Edman P, 1st edition, Boca Raton, CRC Press, 1993; pp 1-25.

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Straatsma BR, Foos RY and Spencer LM: The retina - topography and dinical correlation, Trans New Orleans Acad. Ophthalmol, 1969.

Talarno JH, D'Amico DJ. Hanninen LA: The influence of aphakia and vitredomy on experimental retinal toxicity of aminog lycoside antobiotics. Ameri. J. Ophthalmol 1 985: 1 OO:840-847.

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Tojo K and Ohtori A: Pharmacokinetic mode1 of intravitreal dnig injection. Mathematical Biosciences, 1 994: l2:359-75.

Vale J and Cox 8: Drugs and the eye, (iondon, Butterworths Co.), 1985: ppl-20.

Walker F and Patrick RI Constituent monosaccharides and hexosamine concentration of normal vitreous humor, Exp Eye Res 1967; 6227-231.

Xiao G: F-117, the stealth fighter. In: HangKongZhiShi (the Knowledge of Aerospace), July edition, 1995, pp25-30.

Yoshida A, lshiko S, and Kojima M: Outward pemeability of the blood-retinal bamers. Graefes Archive of Clinicat and Experiemental Ophthalmology, 1992; 230:78-83.

Yoshida A, Kojima M, lshiko S: lnward and outward permeability of the blood-retinal bamer. In: Ocular Fluorophotometry and the Future. edited by Cunha-vaz J and Leite E, Amsterdam, Kugler and Ghedini Publication. 1989, pp89-97.

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Appendix 1 : Experimental Setup

An image capture device was set up to obtain images of injection shapes [Figure Al].

This device consists of seven components, the IigMing source, the sample-holding stand, a

monochrome video carnera (PULNIX TM-845 with a COSMICAR television lens, 25 mm. 1 :1.4),

a VCR (SHINTOM VCR460), a TV (14" SONY TRINITRON), an image freezer (PUWIX VF-

400), and a cornputer (IBM DXll 66). The lighting source is a ISOW 'soft-whitew GE lamp which

shines on the sample (Le., simulated vitreous humour) from the back to provide the uniform

lighting. A piece of silicon-coated g las is sometimes required to disperse the light, softening the

light to shine more uniformly on the sample. The sample was contained in a sphen'cal g l a s

flask (either 10 ml or 25 ml). The fi ask was held upright and the dnrg was injected from the top.

The mode1 compound was injeded at the des id site, and an image was taken. The resufting

images were viewed on a TV screen and recorded using a VCR conneded to the camera. The

subsequent post-pmcessing of the video images was perfomed using an image freezer, which

can capture still images, and using image analysis software (MOCHA t -2, Jandel Scientific).

Figure A1 Experimental Setup [Sin, 19961

Appendix 2: Justification for the Scale-up of Model Vïtreous and the Injection Volume

Since the rnobitity of the aqueous displacing fluid is greater than the gel rnobitity, the

flow is easier in the parts of the vitreous body where the vitreous gel has been displaced by the

more mobile aqueous solution of drug. The fluid velocity will thus tend to increase in these parts,

so that any initial irregularity in the displacement pattern will tend to be magnified [Bavarier,

19911- This means that the injeded aqueous fluid prefers to travel inside the existing aqueous

pocket; the shape of injeded phase (phase of displacing fiuid) will not change but the size will

grow, as the displacing fluid is k i n g injeded. This phenomenon of 'conserving initial injection

shape" is evident in the pidures show in Figure A 2 Figure A2 consists of three pidures taken at

various times during the injedion. It can be deariy seen that as the volume of the water phase

increases, the shape of water phase does not change much. Due to this specific charader of

viscous fingering phenomenon (Le., conservation of the shape), the injection shapes observed

in the scated-up system (e.g., injed 0.5 ml dmg solution into 10 ml simulated vitreous humour)

can adequately reflect the true injection shapes [Lin, 19961.

(a) 15 Seconds (b) 100 Seconds (c) 200 Seconds Note: The injection speed is 0.25 mumin.

Figure A2 Growth of lnjected Aqueous Phase

Appendix 3: Goveming Equations Used in FIDAP Software [FIDAP, 19931

Mornentum Conservation

d4 p ( â + u,u, ) = o,., + pf, Equation A1 .O

( i j = 1,2 for two-dimensional or axi-symmetric flows, ij = 1.2.3 for three dimensional flows),

where u, is velocity (mass-average velocity for mulü-component fluid), p is density, o;, is the

stress tensor and fi is the body force per unit mass. For a fluid, the stress tensor can be written as

0, = +, + r , Equation A l .1

where is p is pressure , q is the deviatoric stress tensor and 4, is the Kronecker delta.

Mass Conservation

Equation A 2 0

where C, is the speûes concentration (expressed as a mass fraction) of the nYh species. is

the diffusive mass flux, Rn is the chernical reaction rate, and qm is a general source term.

In FIDAP it is assumed that the diffusive m a s flux is due primanly to concentration gradients

(molecular diffusion); a contribution from thermal gradients (Soret diffusion) may also be taken

into account:

j:" = - ,ua:")c,, + a:) ( T, / T), n = 1,2,. . . Equation A21

where a,,'"' are m a s diffusMües (which may be tenson), and or?' are thennodiffusion

coefficients.

Energy Conservation

a (-+%TI) =-qlJ t H a Equation A3.0

where T is temperature, c, is specific heat at constant pressure, q, is the flux of themial energy,

and H is a general heat generation tenn.

It is assumed in FIDAP that heat is detennined by Fourier's law,

4, = -k, T, Equation A3.1

where k,, is the thermal conductivity tensor.

Equation of State

f ( P . P , T,c,,c2.--) = 0 Equation A4.0

which typically relates density, temperature, pressure and species concentration. In FIDAP, there

are two specific forms this equation can take.

Compressible Fluid

P MP ldeal gas law is used, p = - - - RT - R'T

Equation A4.1

where M is the molecular weight of the gas and R' is the universal gas constant: R-R~M.

For a gas mixture, p = N-i M~~ Equation A4.2

R * ~ I + ~ ( M , / M, - I)CJ

wtiere Mn and C, are the molecular weight and m a s concentration of the n'th cornponent; and in

this fomulation, cornponent N is frequently referred to as the m e r fluid.

Situations often arise in which the density of fluid varies with temperature and species

concentration, but is independent of pressure. For such cases. the density equation is expressed

as

Equation A4.3

for a mufti-component fluid, where po, To are reference values and PT is the coefficient of

volumetnc expansion associated with temperature variations.

Incompressible Fluid

f o r an incompressible fluid, the equation of state is simply

P = PO Equation A4.4

Appendix 4: Logical DMsion and Meshing of Computational Domains

Example 1: A Simple 2.9 FEM Model

(a) Schematic Diagram of the Model

4

I I 1 4

4

' 4

L b ' *

, 4

b .

(b) The Skeleton of the Model

(c) The Mesh of Computational Domains

The small redangular area e . lightly shaded area)

represents the dmg phase, wtiereas the darkly shaded area

represents the vitreous humour. The drug phase is located at

the center of the vitreous humour.

Using MESHLOOP subroutine, the boundaries of drug phase

and vitreous humour are defined. The drug phase is bounded

by meshloop 2, whereas vitreous humour is bounded by both

meshloop 1 and meshloop 2. 80th meshloops 1 and 2 are

divided into 80 intervals, using MESHEDGE command. This

means there are eighty nodes on each boundary.

Using the MESH-FACE subroutine, the mesh of dnig phase

and the mesh of vitreous humour are generated. MESH-FACE

subroutine can rnesh a surface bounded by more than one

meshloop (Le., vitreous humour). The size and shape of mesh

elements (finite eiements) are optimizecl.

Fiaure A3. A S im~ le 2-D Model

Example 2: A Simple 3-0 FEM Model

(a) Schematic Diagrarn of the Model

(b) Logical Division of the Vittreous Domain

This is the 3-0 case of Example 1. The small cube represents

the dmg phase- The space between large and small cube is the

vitreous humour.

To rnesh a 3-0 cornputational domain, the domain must be

defined (encfosed) by six surfaces, which are designated front,

back, and side surfaces. Since vitreous humour are bounded

by 12 surfaces, it is not rneshable by MESH-SOCID subroutine.

The only solution is to cut the vitreous humour into six 'mesh-

shells" ; and each rnesh-shell is perfectly defined by six

surface.

The meshes of dnig phase and h o of six vitreous pieces are

show in the Figure A4c. The rnost difficult part of 3-0

meshing is to divide the computational domain into mesh-

shells. The way of division must be 'logical" to ensure the

good quality of mesh. Any over-bent surface, or highly

distorted geornetry should be avoided.

1 /6 vitreousThumour (c) Meshing of Domains

Figure A4. A Simple 3-D FEM Model

Example 3: A Complex 3 4 Eye Model

+

1

1

I

1 I

7

moi pie

(a) Skeleton of the Eye Model (3 Different Views)

This model indudes various structures in an

eye, the posterior chamber, the hyloid

membrane, the vitreous, the retina, and the

drug phase (Figure A5b). A cylindrical drug

phase of 50 pi volume is placed at the

position that is 114 of vitreous diameter away

from the center of vitreous body.

Since the geometry of these structures are

often very cornplex and can not be describecl

by a simple mesh-shell, it is necessary to

further subdivide these structures so that they

can be meshed- As a resuh, vitreous, hyloid

membrane, posterior chamber and retina are

subdivided into 14, 2, 4 and 7 mesh-shells,

respedively.

The skeleton of the eye model viewed from

three different angles is shown in Figure Asa

on the left hand side. The lines and cuwes

outline the surfaces bounding 28 mesh-shells

in this eye model.

posterior chamber (4 mes hshells)

dnig phase (1 mesh-sheli)

-

(1 4 meshshells)

A @) Structures in an Eye

(c) Interior View

Surface

: the Eye ~ o d e l

View of the Eye

Figure AS. A Complex 3-D Eye Model

Appendix 5: Mathematical Treatment of the Retinal Domain

tetinal flux q

(sink)

*.-.*

Cv,i = vitreal drug concentration at interface

Cr,i = retinal drug concentration at i m

Cr,outer = reünal outer surface concentration T = thickness of the retina

Figure A6. Schematic Diagram of the Retina

FIDAP does not allow a peneabifity

value to be diredly imposeci on the vitreal

surface. Therefore, a retinal computational

domain has to be physically construded, and

placed against the vitreal surface. The

thickness of the retina is set to be 0.01 cm

[see Figure A6). The retina is so thin and the

retinal distribution volume is so small that very

little time is required for the drug to establish

the steady state concentration profile in

response to a change in concentration at the

vitreal boundary (C,,). For instance, the half

life of 50pg/50pl fiuoresein injeded at the

center of vitreous is about 10 hours (Chapter

4), whereas the time required to establish the steady state concentration profile across the retina

T2 T - 24 seconds, where the retinal permeability of is only in the order of seconds (Le., - = - - 160 I6P

fluoresein is 2.6E-05 cm/s, and the thickness of retina (T) is 0.01 cm). Therefore, it is justified to

assume that an instantaneous steady state is reactied for each change of the boundary

orthogonal bo retina concentration at the vitreal side. This implies a linear

concentration profile across the retina at al1 times (pseudo

node 1 steady state assumption).

# To irnplement a linear concentration profile, the

L Node 1 and node 2 are

retina domain is set to have only one layer of elements. This

hm adjacent nodes. Figure Al. Nodes of Retina means there are only two element nodes in the diredion

Domain orthogonal to the retinal surface [see Figure Al]. In a finite

elernent analysis, the relationship between two adjacent nodes is aiways linear. Therefore, the

concentration profile across the retina is implicitly forced to be linear.

In FIDAP, the species continuity equation for a stagnant domain takes fom of

az -- D v2C = O Equation A5.0. a Due to the pseudo state assumption, the transient terni is dropped. Equation A 5 0 is reduced to

o,v2c, = O Equation A5-1,

where Dr and Cr are the 'lurnped" diffusional coefficient and the dnig concentration in the retina.

respectively.

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