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Fluid Mechanics of the EyeJennifer H. Siggers and C. Ross Ethier

Department of Bioengineering, Imperial College London, London SW7 2AZ,United Kingdom; email: [email protected], [email protected]

Annu. Rev. Fluid Mech. 2012. 44:34772

First published online as a Review in Advance onOctober 17, 2011

The Annual Review of Fluid Mechanicsis online atfluid.annualreviews.org

This articles doi:10.1146/annurev-fluid-120710-101058

Copyright c 2012 by Annual Reviews.All rights reserved

0066-4189/12/0115-0347$20.00

Keywords

porous medium, buoyancy-driven flow, deformed sphere, glaucoma,delivery

Abstract

Fluid mechanical processes are an intrinsic part of several aspects

physiology and pathology of the human eye. In this article we descrlected phenomena that are amenable to particularly interesting math

cal, experimental, or numerical analyses. We initially focus on glauc

condition often associated with raised intraocular pressure. The mecin this disease is by no means fully understood, but we present some

modeling work that provides a partial explanation. We next focus on

features of the dynamics of the two specialized ocular fluids: the aqand vitreous humors. With regard to the aqueous humor, we discusslems concerning the transport of heat and proteins and the hydrat

the cornea. With regard to the vitreous humor, we discuss the possibflow, which occurs primarily as a result of saccades or motions of the e

Finally, we describe a model of the degradation of Bruchs membraneretina.

347

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Intraocular pressure(IOP): the pressurewithin the eye, whichis generated by theproduction anddrainage of the

aqueous humor

1. INTRODUCTION

The eye is a remarkable organ, capable of transducing photons into neural signals with hi

efficiency under a wide range of operating conditions. The retina, containing the specializcells that carry out this transduction process, is aided in its function by many supporting tissu

The development and proper function of this complex system depend critically on biomechanifactors, as has been summarized elsewhere (Ethier et al. 2004a). Here we focus specifically o

the fluid mechanical aspects of the eye, of which there are many. We begin with an overview

relevant anatomy and physiology.There are a number of processes within the eye in which fluid flow is important. Perhaps t

most evident of these are the production, circulation, and drainage of aqueous humor, a clecolorless fluid that is secreted at a flow rate of 2 to 2.5 L min1 (Brubaker et al. 1989) byspecialized tissue known as the ciliary processes, located just posterior to the iris (the colored paof the eye) (Figure 1). In view of the very small flow rates and modest dimensions, the flow

aqueous humor is creeping and inertia can be neglected. The aqueous humor itself is a very diluprotein solution and thus can be treated as Newtonian with viscosity nearly identical to that

saline (Beswick & McCulloch 1956, Moses 1979). It flows into and fills a small region anteriorthe lens but behind the iris, known as the posterior chamber (see Figure 2a), then passes anterio

through the pupil (the aperture in the central part of the iris), and enters the anterior chambe

where it circulates while bathing the iris and the inner surface of the cornea (the clear part of teye). Eventually the aqueous humor drains from the eye via specialized tissues located in the ang

of the anterior chamber, where the iris, cornea, and sclera (the white part of the eye) meet (salso Figure 2 and Sections 2.1 and 2.2). These specialized tissues have a significant hydrodynam

flow resistance, and the drainage of the aqueous humor out of the eye therefore requires that thebe a positive pressure within the eye itself, the so-called intraocular pressure (IOP).

Vortex vein

Sclera

Choroid

RPE

Retinal vessels

RetinaLens

Iris

Cornea

Opticnerve

Zonules

Ciliary processes

Ciliary muscle

Anteriorchamber

Figure 1

Overview of a human eye with major anatomical structures identified. Abbreviation: RPE, retinal pigmentepithelium. Figure modified from Krey & Brauer (1998), copyright cMSD SHARP & DOHME GMBHGermany with kind permission.

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a b

Trabecularmeshwork

Ciliarybody

Iris

Posteriorchamber

Ciliaryprocesses

Ciliarymuscle

Trabecularmeshwork

Schlemmscanal

Schlemms canal

Collector channelsArterioles

Aqueous veins

Schlemmscanal

Cornea

Figure 2

Illustration of aqueous humor flow patterns in the anterior chamber and key drainage tissues. (a) Cross-sectional view through tanterior eye. The arrows show typical thermal convection patterns of aqueous humor in the anterior chamber for an upright suban ambient temperature less than 37C and also drainage pathways from the anterior chamber into Schlemms canal and thencethe aqueous veins/collector channels (bottom-most and top-most arrows). (b) Anterior-posterior view of Schlemms canal (thick greecollector channels (thin green structures), aqueous veins (light blue), and adjacent arterioles. Figure modified from Krey & Brauer copyright cMSD SHARP & DOHME GMBH Germany with kind permission.

Glaucoma: anophthalmic conusually characteby raised intraopressure, whicheventually to blby the death of ganglion cells

Mass transporttransport of a dspecies within aor solid that is dby convection,diffusion, anddestruction/production (thrchemical reactio

The flow of the aqueous humor performs two important physiological functions. First, thepositive pressure that it generates stabilizes the otherwise flaccid eye, ensuring accurate positioningof the optical elements of the eye and hence clarity of vision. Second, aqueous humor supplies

nutrients and removes waste products from the avascular lens and the central cornea, withoutwhich the cells in these tissues would die. Some models of aqueous humor flow in health are

presented in Section 3. However, unfortunately (as explained below), impairment in the drainageof this fluid leads to an elevation in IOP, which is a major risk factor for the disease known as

glaucoma, the second-most-common cause of blindness in the world (Quigley & Broman 2006)

(see Section 2). In glaucoma, a specialized type of cell known as the retinal ganglion cell is damagedand eventually dies (Qu et al. 2010). These cells are responsible for carrying visual information

from the retina to the brain, and therefore any insult to them can result in vision loss. As shown in

Section 2.3, mechanical factors are believed to play a central role in this disease (Burgoyne et al.2005), and consideration of flow and mass transport effects in the retinal ganglion cell axons maybe a promising new way to understand the pathogenesis of glaucoma.

The cornea combines the attributes of mechanical strength and optical transparency, whichis achieved by an extremely regular planar arrangement of collagen fibers (see Figure 3); in

particular, corneal transparency depends sensitively on fiber spacing and hence on the hydrationstate of the cornea. There is a continuous flux of fluid in and out of the cornea, and the body

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0.5 m

Figure 3

Normal human cornea of a 62-year-old male patient, showing the regular arrangement of collagen fiberswithin the corneal stroma. Figure taken from Langham, Maurice E., ed. The Cornea: MacromolecularOrganization of a Connective Tissue. Papers from a Symposium Held in Kyoto, Japan, 1967, under the Auspices of tDepartment of Ophthalmology, Osaka University, p. 124, figure 7-1. Copyright c 1969 by The Johns HopkiPress, reprinted with permission.

therefore has developed sophisticated mechanisms to control this transport. Specifically, wateractively and continually pumped out of the corneal stroma (the central layer of the cornea)

the corneal endothelium, a specialized layer of cells lining the interior corneal surface. The neffect is that the stroma is in a continual state of thermodynamic disequilibrium with respect

its adjacent bathing fluids, namely the tear film anteriorly and aqueous humor posteriorly (sSection 3.4).

The majority of the ocular globe is filled by a clear, colorless, gel-like material known vitreous humor, which occupies the vitreous chamber of the eye. The chamber is surrounded

two tissues, the retina and the choroid, the former of which comprises many layers (see Figure Vitreous humor has complex viscoelastic properties, and although there have been several attemp

to characterize its properties experimentally (Lee et al. 1992, Nickerson et al. 2008, Swindle et

2008, Zimmerman 1980), its rheology is not fully understood. It is known that the vitreous humbecomes progressively liquefied with age; approximately 20% of the vitreous humor is liquid

1418-year-olds, and this rises to more than 50% in subjects aged 8090 years (Bishop 2000In approximately 25%30% of subjects, liquefaction can lead to a process in which the reti

detaches, risking loss of sight. In this context, it is also important to note that the eye is nnormally stationary, even when apparently focused on a fixed point. Instead, the eye constan

executes a series of extremely rapid angular rotations (300 s1 or more) known as saccades (Rayn

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Viscoelastic flua fluid whose strtensor depends both the deformand the rate ofdeformation of particles

50 m

Vitreous humor

Ganglion cell layer

Inner plexiorm layer

Outer plexiorm layer

Inner nuclear layer

Outer nuclear layer

Inner segments o photoreceptors

Outer segments o photoreceptorsRetinal pigment epithelium

Choroid

Sclera

Direction of

light travel

Figure 4

Cross section through the retina and choroid within the macula (the part of the retina with the greatestconcentration of rod and cone cells). The asterisk denotes the choriocapillaris, and the white arrowheadspoint to Bruchs membrane. Reprinted from Curcio et al. (2009) with permission of Elsevier.

1998). These have the effect of creating flow patterns within the vitreous humor, particularly if

it has liquid characteristics, as explained in Section 4. A common cause of vision loss in elderly

subjects is age-related macular degeneration, which is thought to be caused by impaired transportacross Bruchs membrane, the membrane situated at the base of the retina, due to accumulation

of lipid particles deposited within the membrane over a period of years (see Section 5).With the notable exception of the lens, central cornea, and the vitreous humor, the eye is

richly supplied by a complex network of blood vessels, leading to many interesting physiologicalproblems associated with the regulation of blood flow in the network. For example, the retina has

a remarkably high metabolic rate and a correspondingly large need for blood, which was recentlyinvestigated by Liu et al. (2009) in a reconstructed network model to compute flow and mass

transport. Moreover, because the ocular vasculature is contained within the eye globe, which isitself pressurized, the ocular veins can experience collapse, behaving as a Starling resistor. The

physiology of ocular blood flow has been much studied (e.g., Kiel & van Heuven 1995, Reitsamer

& Kiel 2002) but has received little attention from the fluid mechanics community.

2. FLUID MECHANICS OF GLAUCOMA

Glaucoma is often, although not always, characterized by an increase in IOP, and lowering the

pressure is the only treatment currently available. Therefore, there is significant interest in un-derstanding the factors that control IOP. In almost all cases of glaucoma, the cause of the pressure


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increase is known to be an increase in the hydrodynamic resistance to aqueous humor drainag

necessitating a higher pressure to drive the outflow. Glaucoma can be further classified inclosed-angle glaucoma and open-angle glaucoma. In closed-angle glaucoma, the iris moves ant

riorly from its normal position, reducing or eliminating the gap between it and the cornea. This an interesting fluid-structure interaction problem and frequently leads to a complete occl

sion of outflow with attendant dramatic increases in IOP; a mathematical model of this is briediscussed in Section 3.1. In open-angle glaucoma, the drainage tissues remain accessible to t

aqueous humor, but they present an elevated flow resistance. There are a number of different typof open-angle glaucoma, the most common of which is known as primary open-angle glaucom

This condition has puzzled researchers for many years, as there are no evident structural changin the drainage tissues that could explain their elevated resistance. Here we discuss some of th

modeling work that has attempted to shed light on the source of the resistance. Section 2.1 d

cusses changes in the trabecular meshwork, and Section 2.2 focuses on changes in the mechanof flow into and through Schlemms canal. Section 2.3 discusses a possible mechanism for t

death of retinal ganglion cells, which is the cause of vision loss in glaucoma.

2.1. Increase in Resistance Across the Trabecular Meshwork

To understand the fluid mechanics of aqueous humor drainage, we must provide further detaon the anatomy of the drainage tissues located in the angle of the anterior chamber (see al

Figure 2). As the aqueous humor leaves the eye, it first passes through the trabecular meshworwhich can be represented as a biological porous material. It then enters a collecting duct known

Schlemms canal, notable for its unusual endothelial cellular lining containing a number of smmicrometer-sized openings (known as pores). As discussed below, the hydrodynamic interacti

of the flows through the pores and that through the trabecular meshwork are of great intere

The aqueous humor then flows along the canal and out through a drainage structure known acollector channel, from which it eventually mixes with venous blood in the sclera and returns

the right heart.A significant amount of work has been devoted to understanding the drainage of aqueo

humor through these tissues. Originally, attention focused on analysis of the trabecular meshwoas a porous material (Ethier 1986, McEwen 1958, Seiler & Wollensak 1985). Micrographs

normal and diseased tissue were analyzed to estimate the tissue porosity, TM, and the specisurface, STM, and the hydraulic permeability, KTM, of the trabecular meshwork was estimat

using classical Carman-Kozeny theory (Bear 1988),

KT M =3T M

kS2T M,

for suitable values of the Kozeny constant, k. The resulting computed permeabilities were com

pared with estimates from experimental measurements, with the surprising finding that the computed permeability was one to two orders of magnitude higher than the best experimental da

The conclusion was that either the trabecular meshwork had little hydraulic flow resistance o

more likely, the fundamental assumptions underlying the calculation were incorrect.Some evidence indicates that theapparently open spaces within thetrabecular meshwork area

tually filled with a gel-like biopolymer consisting of proteins and long-chain carbohydrates. It w

therefore appropriate to include the hydrodynamic effects of this gel material, achieved throuthe use of the fiber matrix model (Weinbaum 1998), in which the individual biopolymer strands a

approximated by long, randomly oriented cylinders. Through the use of classical lowReynolnumber results (Happel 1959, Spielman & Goren 1968), it was possible to make good estimat

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Porous mediuma solid material small-scale strucharacterized byfilled with fluidin such materia

usually governethe Darcy equat

of the hydraulic permeability of a pure biopolymer gel or of tissues with well-characterized mi-

crostructure (Ethier 1986). Application of this theory to the trabecular meshwork required theconsideration of a two-level hierarchical porous medium, in which a material (the gel) with charac-

teristic length scales on the order of angstroms to nanometers and hydraulic permeabilityKgel wasembedded within a second material (the trabecular meshwork) having pores with dimensions on

the order of micrometers to tens of micrometers. Accounting for steric hindrance and tortuosityeffects leads to a prediction of the overall permeability of the composite material as (Ethier 1986)

K= Kgel T M2 T M . (2)

This produced a much more satisfactory agreement between theory and result, leading to what

is now known as the gel-filled meshwork theory. However, subsequent experimental data havecalled this model into question; for example, treatment of the trabecular meshwork with enzymes

that are known to degrade the biopolymer gel seems to have a much smaller effect on hydraulicresistance than would be expected (Hubbard et al. 1997).

The fluid mechanics of aqueous humor drainage through the small pores in the cellular liningof Schlemms canal is also important (Bill & Svedbergh 1972, Eriksson & Svedbergh 1980). By

exploiting the fact that these pores are small and relatively isolated,one needs only to consider flow

through a single pore and treat the entire cellular lining as a parallel network of hydrodynamically

noninteracting pores. The flow resistance of a single pore of radius R can be calculated fromSampsons theory in lowReynolds number hydrodynamics (Happel & Brenner 1983), whichrelates the pressure drop, p, across a thin surface to the volumetric flow through the pore, q,

through the surface via

q

p= R

3

3, (3)

where is the dynamic viscosity of the aqueous humor, and p is the pressure difference across

the surface between locations infinitely far from the surface on either side of the surface. The

surprising finding from this calculation was that it predicts an extremely low flow resistance ofthe endothelial lining of Schlemms canal, certainly much lower than the observed resistance of

the entire system. This led to a paradox: Neither the lining of Schlemms canal nor the trabecular

meshwork alone seemed to offer sufficient flow resistance to agree with experimental evidence.A possible resolution of this paradox was put forward by Johnson et al. (1992), who noted aninteresting hydrodynamic interaction between the endothelial lining of Schlemms canal and the

upstream porous material of the trabecular meshwork. Specifically, because the pores are few andaccount for only a small fraction of the total area of the inner wall of Schlemms canal, they must

act to hydrodynamically focus (or funnel ) aqueous humor drainage through the trabecularmeshwork (see Figure 5). The situation was modeled as a porous slab bounded on one surface by

a plate pierced by hydrodynamically isolated pores. By considering a simplified unit-cell model,

consisting of a single pore and the upstream region of the porous medium drained by that pore,one can calculate the hydraulic resistance of the ensemble structure in a straightforward manner.

The theory predicts an overall flow resistance that is generally consistent with experimental

measurements (Overby et al. 2009). Furthermore, it reconciles observations that both pore densityand the composition of the trabecular meshwork have an effect on the overall resistance to flowin this tissue.

2.2. Flow Within Schlemms Canal

In a different study aimed at identifying the source of outflow resistance, Johnson & Kamm (1983)developedamodeltoconsiderthehydrodynamiceffectsofthepartialortotalcollapseofSchlemms


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20

30

27

24

Edge o JCT(z= L)

Inner wall(z= 0)

Pore center(x= 0)

Zoomed view: pore area

Edge o domain(x= b/2)

x

x

zz

10

0

0

18

24

21

0 1 2

1

2

10 20

21

69 12

15

3

Pore

Inner wall

Unit cell

Flow

a b

Figure 5

(a) Normalized pressure contours obtained by numerical simulation of flow in the juxtacanalicular tissue ( JCT), treated as a porous

medium, in the neighborhood of a fenestration (pore) in the inner wall of Schlemms canal. The length scales are normalized by thepore radius, the pressure is normalized by the pressure drop that would be needed to force the same volume flux through the JCT alo(i.e., without the inner wall), b is a typical distance between neighboring pores, and L is the thickness of the JCT. (b) Illustration of thsetup considered in the model. Figure taken from Johnson et al. (1992).

canal, caused by IOP-induced deformation of the trabecular meshwork. They modeled the inn

wall as a permeable membrane supported by linear Hookean springs with constant stiffness(see Figure 6a). Thus the height, h(x), of the canalthe distance between the inner and out

walls as a function of position, x, along the canalis given byh = h0 [1 (IOP Psc)/E], whePsc(x) is the pressure in Schlemms canal, and h0 (assumed constant) is the height of the can

when the transmural pressure, IOP Psc, equals zero. The resistance to transmural flow is R

(assumedconstant),whichimpliesthatafluxof1/Rw times thetransmural pressure drop crossestinner wall per unit length of wall. The authors modeled the aqueous humor as an incompressib

Newtonian fluid and assumed that|d h/d x| 1 and s h0, where sis the half-distance betwecollector channel ostia, thus allowing them to approximate the flow using lubrication theory. T

Collectorchannel

Schlemms canal

TM

Outer wall

Inner wall

Intraocular pressure (IOP)

x= 0 x= s

Schlemms canal

Intraocular pressure (IOP)

x= 0 x= sa bCollectorchannel

Figure 6

Schematic diagrams of the model for flow entering and within Schlemms canal developed by Johnson & Kamm (1983): ( a) originalmodel and (b) model with septae included. Abbreviation: TM, trabecular meshwork. Reprinted with permission of ARVO.

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governing equations reduced to a single equation for h:

12

wRw(h h0) =

d

d x

h3

d h

d x

, (4)

where w is the width of the cross section of Schlemms canal in the anterior-posterior direction.

Boundary conditions arise from prescribing the pressure at the collector channels, which, usingthe spring condition given above, leads to

h = h0

1 IOP Pc cE

, (5)

at x = s, where Pcc is the pressure in the collector channels. They solved the governingEquation 4 numerically over the half-distance between neighboring collector channels (i.e., the

region 0 x s in Figure 6), imposing a symmetry condition at x = 0.The model predicts a nonlinear dependence between the total outflow and the pressure drop,

IOPPc c, due to an increase in resistance to outflow. The nonlinearity occurs because the heightof Schlemms canal decreases when the pressure drop is large, increasing the canals resistance

and hence the overall resistance. Thus, to maintain a constant outflow to meet physiologicaldemands, the IOP must increase more than it would if the dependence were linear. Another cause

of increased IOP could be that the inner wall resistance, Rw, increases, while the other parametersmaintain constant values, which would mean the IOP must also increase to maintain the outflow.

For a normal or slightly raised IOP (up to approximately 25 mmHg), and using parameter valuessuitable for the human eye, the predicted heightof the canal is approximately spatially uniform, and

the predicted resistance to outflow is approximately constant as IOP increases. The approximately

uniform height of Schlemms canal implies an approximately spatially uniform luminal pressure,which shows that the majority of total outflow resistance does not derive from the resistance to flow

within Schlemms canal. For higher IOPs, the canal begins to collapse near the collector channels,and at IOP = 29 mmHg, there is complete collapse. However, complete or almost-completecollapse is unrealistic because Schlemms canal contains septae, short structures modeled as beingof heighths, which protrude into Schlemmscanalfrom the outer wall to prevent complete collapse.

With a normal or slightly raised IOP, these protrusions make no difference to the model resultsbecause the minimum width of the canal is greaterthanhs. For higher IOPs, there is partial collapse

of the canal with the septae supporting the collapsed region (see Figure 6b). For still-higher IOPs,the channel is completely supported by the septae and has constant height hs, and under these

conditions the total outflow resistance is constant. In a further extension, the authors consideredcompliant septae, meaning that the height can drop below hs in the collapsed region, increasing

the resistance of Schlemms canal compared with the rigid septae model. A comparison of modelresults with experimental data suggests that Schlemms canal collapse does not occur in glaucoma,

implying that glaucoma cannot be caused by weakening of the trabecular meshwork alone.

A related question concerns the flow within the lumen of Schlemms canal, as opposed to thatacross the endothelial lining of the canal. In the vascular system, which shares many biological

similarities with Schlemms canal, the caliber of vessels is strongly influenced by the shear stress

exerted by the blood flowing within the vessel. However, because the volumetric flow rates inthe lumen of Schlemms canal are minuscule, shear stresses might, a priori, also be expectedto be too small to have physiological impact. However, a simple calculation in which the cross

section of Schlemms canal was treated as an ellipse showed that the estimated shear stresseswere similar to those observed in the vascular system (Ethier et al. 2004b), suggesting that the

biological mechanisms for caliberregulation in the two systems could be similar. This is potentiallyimportant, as Schlemms canal is observed to be shorter in the anterior-posterior direction in


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glaucomatous eyes, even though its heighth is unaffected (Allingham et al. 1996). This increas

outflow resistance, and thus dysregulation of the mechanisms controlling Schlemms canal calibmay play a role in glaucoma.

2.3. Mechanism of Death of Retinal Ganglion Cells in Glaucoma

In glaucoma, the cause of death of the retinal ganglion cells is not fully understood (Ethier et 2004a, Fechtner & Weinreb 1994, Schumer & Podos 1994), and several mechanisms have be

proposed. These include mechanical insult to optic nerve head tissues and/or a failure in vasculautoregulation to the nerve (Burgoyne et al. 2005, Morgan 2000, Pillunat et al. 1997, Riva et

1997, Yamamoto & Kitazawa 1998, Yan et al. 1994). Here we consider another, more fluid mchanically based, mechanism proposed by Band et al. (2009). Retinal ganglion cells require axon

transportto remain viable, in which cargo-containing structures, thevesicles, aretransported alothe axons by motor proteins. These motor proteins require energy for their task, which they o

tain from adenosine triphosphate (ATP) molecules, which in turn are released from mitochondlocated along the axon. ATP is distributed along the axon by a combination of diffusive and, in t

presence of flow, convective, effects. If the supply of ATPis sufficiently depleted, then active axontransport will be reduced or stopped. This has been shown to lead to the death of ganglion ce

in primates (Anderson & Hendrickson 1974, Balaratnasingam et al. 2007, Minckler et al. 1977

Retinal ganglion cells contain axoplasm, a fluid that has approximately Newtonian propertiThe walls of these cells are permeable to the axoplasm; therefore, in the presence of a pressu

gradient,itispossiblefortheaxoplasmtoflowalongtheaxon,asitcanbereplenishedbytransmuraflow. In the proposed mechanism, the rise in IOP leads to a significant axial flow of the axoplasm

and this causes convection of the ATP toward the brain. If the convection of ATP is strongthan diffusion, it will prevent ATP from diffusing in the upstream direction, leading to a regi

of washout along the axon. Band et al. (2009) developed a mathematical model of the flow in taxons and used it to estimate the relative strengths of convection and diffusion, characterized b

the Peclet number. They demonstrated that their suggested mechanism is plausible because tflow is likely to begin to occur for elevations of IOP on the scale of those observed in glaucom

The model is illustrated in Figure 7. Each ganglion cell axon lies partly within the eye glo

and partly within the optic nerve bundle. In the eye globe, the flux of fluid through the ax

Intraocular space atpressurepe

CSF atpressurepc

Pressurep(z;r)

Pressurep+(r,z)

r

Site o lamina cribrosa

Sclera

Synapseatz= L

Cell bodies atz= M

Flux o axoplasm

Retrograde AAT

Orthograde AAT

Axis osymmetry

Optic nervehead

z

Figure 7

Mathematical model to analyze flow in retinal ganglion cells. Abbreviations: AAT, active axonal transportCSF, cerebrospinal fluid. Figure taken from Band et al. (2009).

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Homogenizatia technique useanalyze multipleproblems; quanare averaged ovsmall scale, lead

simplified largeequations

wall is assumed to be proportional to the local pressure drop across the axon wall. In the optic

nerve, the axons are treated as a bundle of fibers whose cross sections form a hexagonal lattice,and flux between neighboring axons is also proportional to the pressure difference between the

fluids in them. Flow along the length of the axon is driven by the axial pressure gradient. Theseassumptions were used to homogenize the model, leading to a relationship between the axial flux

and pressure gradient along the axons, which can be solved to find the flow and pressure in theform of sums of an infinite series of Bessel functions.

If the IOP is elevated to levels commonly seen in glaucoma, the P eclet number for ATPpredicted by the model is greater than one within substantial extraocular regions of the axons.

This suggests there will be significant depletion of ATP in these regions, illustrating the potentialimportance of the proposed mechanism.

3. FLOW IN THE ANTERIOR CHAMBER

3.1. Thermal Transport

In the anterior chamber, the aqueous humor flows radially outward toward the trabecular mesh-work in the normal course of its drainage from the eye. In addition to this flow, there is also a

thermally driven flow, as temperature gradients exist between the anterior and posterior surfaces

of the anterior chamber (the back of the cornea and the front of the iris). The posterior surface isclose to body temperature, but the anterior surface is closer to atmospheric temperature (usuallycooler). Convection is thought to increase the efficiency of nutrient delivery, but it is also likely

to give rise to significant clinical effects if there is particulate matter within the aqueous humor,

such as blood cells or pigment particles (Canning et al. 2002).Canning et al. (2002) and Fitt & Gonzalez (2006) developed a model of the fluid flow in

the anterior chamber of the eye, using the simplified geometry illustrated in Figure 8. Theytreated the fluid as incompressible and Newtonian and used the Boussinesq approximation to

describe the variations in the density of the fluid. The posterior surface consisted of a disc rep-resenting the pupil in the center and an annulus surrounding it representing the iris, which was

assumed to be at a fixed temperature T1 close to body temperature. The anterior surface of the

chamber was assumed to be at a fixed cooler temperature T0. The velocity at the pupil was assumed

z

T= T0

T= T1

x

y

a

a

a

aPupilaperture

Iris:z= 0

Cornea:z= h(x,y)

Flow w0(x,y)

Figure 8

Sketch of the model developed by Canning et al. (2002) to investigate flow in the anterior chamber.Reprinted with permission of Oxford University Press.


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Buoyancy-drivenflow: flow driven byspatial densitygradients in the fluid,frequently due totemperature variations

in the fluid

to be purely normal to the plane of the iris and given by a prescribed functionw0(x,y), with no-s

velocity applied on the other boundaries, except that drainage occurs at the angles of the domaCanning et al. (2002) used experimental measurements to argue that the aspect ratio of th

model is small (h a) (see Figure 8) and also estimated the reduced Reynolds number to small. In the formal mathematical limit in which these quantities are small, and also neglectin

viscous dissipation, time dependence, and convection of heat, they derived a simplified systemequations. They were able to manipulate these and reduce them to the single differential equati

H h3H P = 12w0,where H = ex/x + ey/y in Cartesian coordinates, Pis the deviation in the pressure frothe hydrostatic pressure profile, and is the viscosity at the temperature T0.

The authors estimated the size ofw0 and concluded that, typically, it is likely to be sm

compared to the thermally driven velocity, and therefore they considered the case w0 = 0, whiallows an exact solution of Equation 6. Estimation of the stress induced by this flow suggests it

unlikely to be strong enough to cause particle or cell detachment from the iris. A comparison the transit time of the solution with the time it would take a particle in the aqueous humor to set

under gravity shows that the convective velocity is several times faster than the settling velocitCalculation of the Stokes drag allows a full model of particle transport to be developed. The mod

showed that particles remain in the vertical plane that contains them and that is also perpendicuto the iris. The authors solved the model numerically and used their results to comment o

features that would be present in a number of clinical conditions. This work was extended bFitt & Gonzalez (2006) to include inflow (w0 = 0), to consider other directions of gravity wirespect to the model (e.g., a supine patient), to investigate vibrations of the lens as the head

eye moves (phakodenesis), and to investigate the flow produced by rapid eye movements durinsleep. Their results showed that the buoyancy-driven flow typically exceeds the flow driven

other mechanisms by orders of magnitude and plays a dominant role in several medical conditioof the anterior chamber.

3.2. Fluid-Structure Interaction Models of the Iris and Aqueous Humor

Other studies of aqueous humor flow include those by Heys et al. (2001) and Heys & Baroc

(2002b). These authors developed an axisymmetric model of the flow in both the anterior anposterior chambers, in which they modeled the aqueous humor as a Newtonian viscous flu

and the iris as an incompressible neo-Hookean solid. The iris deforms as a result of the streexerted by the flow, and the authors calculated the steady-state position of the iris tissue. He

& Barocas (2002a) considered a fully three-dimensional model and included thermal convectioTheir results showed that convection effects in the flow are dominant; that is, the calculated veloc

magnitude predicted by the equations with the convection terms added is many times larger th

that predicted by the equations without convection included. Their results were consistent wiclinical observations of Krukenbergs spindle, a condition in which pigment from the iris becom

attached to the posterior surface of the cornea in characteristic vertical stripes. A similar model w

used to investigate the flow and deformation produced by small oscillations of the position of tiris (Huang & Barocas 2006) and the recovery from an indentation in the cornea or sclera (Ami& Barocas 2010), which was done by imposing an initial rotation in the iris position at its root

In other work, Huang & Barocas (2004) adapted the model by adding an active term inits stress tensor to represent contraction of the sphincter muscle in the normal circumferent

direction. They tuned the geometry to model both normal eyes and eyes with features that athought to be risk factors in closed-angleglaucoma.Theirresults predictedthat thefurther forwa

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the lens position is, the greater the likelihood of iris-lens contact, which leads to a greater pressure

difference between the posterior and anterior chambers. Conversely, decreasing the diameter ofthe anterior chamber leads to a smaller angle between the iris and the cornea, which is likely to

increase the flow resistance of the aqueous outflow pathway. However, testing their model fordifferent pupil diameters suggests that the condition is most severe when the pupil is small. This

contradicts clinical tests that suggest the condition is most serious in dark environments when thepupil is large.

3.3. Transport of Proteins

As noted above, the aqueous humor is produced behind the iris, passes through the pupil, andfills the anterior chamber before draining from the eye. It is therefore natural to assume that the

proteins within the aqueous humor would follow the same route. However, this is not the case;protein mass transport in the anterior chamber proceeds in a manner for which the details have

only been elucidated over the past decade or so, despite the fact that the circulation of aqueoushumor has been reasonably well understood for a century.

The essential fact driving the difference between water and protein transport in the anterior

eye is that the lining epithelium of the ciliary processes, the tissue responsible for secretion ofthe aqueous humor, is highly impermeable to proteins. Therefore, proteins naturally present

in the ciliary body are prevented from entering the posterior chamber and instead build up inthe extravascular space of the ciliary body to create a reservoir of plasma proteins. Proteins then

diffuse anteriorly from this reservoir, leaking into the anterior chamber from the anterior iris. Thismechanism has been confirmed by an elegant series of tracer studies in various species (Barsotti

et al. 1992, Bert et al. 2006, Freddo et al. 1990), complemented by theoretical models of proteinmass transport with predictions that show a reasonable agreement with experimental data (Barsotti

et al. 1992).

3.4. Dynamics of the Cornea

Hedbys & Mishima (1962) carried out early quantitative studies of fluid transport in the cornea.

Their work is notable because it investigated water transport both across and in the plane of

the corneal stroma and because of their clever experimental design for measuring transport inthe tangent plane of the stroma. They developed an optical pachometer capable of measuringdynamic thickness profiles of corneas and applied this to corneal samples in which the water had

been partially expelled from part of the cornea (see Figure 9). Conserving mass, relating corneal

hydration to the local swelling pressure, and using Darcys law, they were able to deduce stromalpermeability as a function of hydration. They observed that the transport properties of the cornea

are anisotropic, with a lower permeability for flow normal to the stroma than for that in the tangentplane, and this difference becomes more pronounced as the corneal hydration decreases.

These phenomena are understandable when the ultrastructure of the cornea is considered,in which arrays of collagen fibers are oriented largely parallel to the plane of the cornea (see

Figure 9). With the use of classical results for flow in porous media in the limit of vanishing

fiber solid fraction, the permeability is predicted to be approximately twofold lower for flow inthe normal direction than that in the plane of the fibers (Happel & Brenner 1983), a result thatis quantitatively in agreement with the experimental data.

4. FLUID MECHANICS OF THE VITREOUS HUMOR

The vitreous cavity has an approximately spherical shape and contains the vitreous humor, whichis subject to mechanical forces as a result of the motion of the eyeball (due primarily to the motion


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xc

hc

h

xdh/dx

Position along strip,x

Thickness,

h

hc

A

xc

1 mmmm1 mm

Loss o fuid

Gain o fuid

1.20

1.000.80

0.60

0.40

0.20

00 1 2 3 4 5 6

Position along strip (mm)

Thickness(mm

)

7 8 9 10 11

48 h24 h7 h0 h

a b

Figure 9

Experimental work using a pachometer by Hedbys & Mishima (1962) to investigate water transport in the cornea. ( a) Illustration ofwater movement along the corneal strip. The upper panel depicts the sample, and the lower panel shows the thickness of the sample

over time (the darker area represents the loss of fluid from the swollen part, and the lighter area the gain by the dry part).(b) Experimentally measured heights of the corneal strip at different times, which indicate that there is a point, xc, at which the heighdenoted hc, is approximately constant in time. The fluid movement is calculated across the area A, shown in the upper panel of (a),which is located at the pointxc. Reprinted with permission of Elsevier.

of the head and rotation of the eyeball within the socket). Deformation of the vitreous chamb

due for example to a head impact, lens movement during focusing or pulsation of retinal blovessels, also gives rise to forces. However, in the absence of any deformation, purely translation

motion does not result in any relative motion of the humor within the vitreous chamber becau

the accelerations involved can be balanced by a pressure gradient, whereas rotational motion doinduce the relative motion of the humor. The fastest motions occur when the vitreous humor

liquefied, which can be the case either following the process of liquefaction described in Sectio

1 or following vitrectomy, a surgical procedure in which some vitreous humor is removed anreplaced with another fluid, often silicone oil or a gas bubble. In this case, the fluid filling tvitreous chamber is approximately Newtonian. Several authors have developed mathematical an

experimental models of the flow in the vitreous humor, which we discuss in this section. Wfirst discuss models of the dynamics that approximate the vitreous chamber as a rotating sphe

including a viscoelastic model (Section 4.1), and then consider extensions of this work to accoufor the effects of the geometry of the chamber, while also simplifying to the case of a Newtoni

fluid (Section 4.2). We then focus on the potential effects of dynamic deformation of the vitreochamber by considering woodpeckers, whose eyes are subjected to enormous accelerations duri

pecking andwhich appear to have a number of special protectiveadaptations (Section 4.3). This h

potential applications to understanding the mechanism of damage in shaken baby syndrome. W

then discuss models of partially liquefied vitreous humor (Section 4.4) and finally the implicatiofor mass transport in the vitreous humor, which has important implications for drug delivery the retina (Section 4.5).

4.1. Flow in Spherical Models of the Vitreous Chamber

David et al. (1998) investigated the periodic flow produced during small torsional oscillatioof the eyeball, modeling the vitreous chamber as a rigid sphere. They used the Maxwell-Voi

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Steady streamithe time averagfluctuating flowarising because nonconservativforce, Reynolds

stresses, or bouneffects

viscoelastic model proposed by Lee et al. (1992) to characterize the rheological behavior of the

vitreous humor. The angular displacement was modeled as cost, and the assumption of smalloscillations allowed them to linearizethemodel andseek solutions proportionalto e it.Thisledtoa

linear relationship between the shear stress and the shear strain, whose constant of proportionalitywas the complex modulus, G, dependent on . In terms of spherical polar coordinates (r, ,), centered on the sphere and with the axis parallel to the axis of the oscillations, the velocityfield is

u = iR3[sin(ar/R)

(ar/R)cos(ar/R)]

2r2(sin a a cos a) sin eit e + c.c., (7)where R is the radius of the sphere; a = cei/4; c is the complex Womersley number, given by

c =

i2R2

G; (8)

is the fluid density; e is the unit vector in the direction of increasing ; and c.c. denotes the

complex conjugate. Thus for small values of |c|, the fluid moves almost as a rigid body, whereasfor large |c|, the motion becomes confined to a Stokes boundary layer of width |c|1 andthe fluid in the center of the sphere remains stationary. Their results show that for myopic eyes,which usually have a larger radius, the shear stress generated by the vitreous humor on the retina

is typically larger than for nonmyopic eyes.Repetto et al. (2005) studied vitreous fluid dynamics experimentally by creating an enlarged

model of the vitreous chamber in the form of a perspex cylinder containing a spherical cavity.They mounted the cylinder on a motor that could perform prescribed torsional rotations about

the vertical axis and observed the resulting motion of the fluid on the horizontal mid-plane of the

model using particle image velocimetry. Under periodic forced rotations of prescribed amplitudeand frequency, the behavior was characterized by two dimensionless parameters: the Womersley

number,, and the angular amplitude of the oscillations, . Theirresults were shown to agree well,both qualitatively and quantitatively, with the theoretical predictions of David et al. (1998). The

authors also considered angular displacements based on measurements of realistic saccades, thatis, a single rotation through a fixed angle starting from an initially stationary fluid. At each point

in space, they measured the maximum over time of the absolute value of the azimuthal componentof the fluid velocity and also the timescale over which it was achieved. By decomposing the

time dependence of the angular displacement into a linear superposition of Fourier modes, theycompared these measurements with the theory of David et al. (1998), finding good agreement

even though the flow is not periodic, whereas David et al.s model assumes periodicity. They usedtheir results to show that the shear stress is not strongly dependent on the angle through which

the eye moves in a saccade. Thus, because small-angle saccades are much more frequent than

large-angle saccades, small-angle saccades are responsible for generating the majority of the shearstress on the retina when integrated over time.

In addition to the behavior just described, there is also a steady component of flow in thevitreous humor [steady streaming (e.g., see Riley 2001)]. For small-amplitude oscillations, this

component is much smaller in magnitude than the leading-order oscillatory flow, but even so

it can play an important role in mass transport because the transport it induces does not tendto cancel over a period of the oscillatory motion. Therefore, Repetto et al. (2008) studied thissteady streaming flow analytically in a similar system, i.e., a torsionally oscillating sphere filled

with a Newtonian viscous and incompressible fluid, assuming rotations of small angular amplitude. They formulated the solution as a series expansion in powers of the small parameter : u =u1 + 2u2 + , p = p1 + 2p2 + . The leading-order solutions u1 and p1 have frequency and are also given by Equation 7, but with c replaced by the Womersley number . The first


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corrections u2 and p2 are driven by the nonlinear term u1 u1 in the Navier-Stokes equatiand are thus a superposition of a solution with frequency 2 and a steady solution, denoted up (2)2 and u

(0)2 , p

(0)2 , respectively [thus u2 = u(0)2 + u(2)2 and p2 = p (0)2 + p (2)2 ]. Because the stea

solution is more important in terms of its implication for mass transport, the authors calculatu(0)2 and p

(0)2 , but neglected u

(2)2 and p

(2)2 . The solution took the form of a sum of terms who

dependence on and was found exactly, but whose dependence on rwas an integral that hto be computed numerically. In the limit 1, the integral could be calculated analyticalin which case the velocity can be shown to be proportional to 6 and thus grows very slowas increases. The integral can also be found analytically in the limit 1, and in this cathe velocity tends to a constant value. The authors also performed experiments using the samapparatus as Stocchino et al. (2007) but taking images only once per period to reveal the averag

rather than the instantaneous, velocity. The theoretical and experimental results showed go

agreement for small amplitudes (within 10% for = 0.0885) over the whole range of (from 3to 15.9).

4.2. Flow in Models that Account for the Real Shape of the Vitreous Chamber

The vitreous chamber is not perfectly spherical, and the most prominent feature is an indentati

into the chamber caused by the presence of the lens. To investigate the effect of the shapStocchino et al. (2007) used an experimental model similar to the spherical model of Repet

et al. (2005), but with a modified shape. Based on their analysis of several ultrasound and magneresonance scans, the authors modeled the lens as a spherical indentation into the sphere, with bo

spheres having the same radius. This introduces a further nondimensional parameter, , equal

the maximum depth of the indentation dividedby the vitreous cavity radiusR. Again they subjectthis apparatus to periodic, torsional rotations and, approximately at each of the times when th

angular velocity reached its maximum absolute value, observed a circulation structure generatedthe back of the indentation. This structure then traveled toward the center of the sphere and w

annihilated. The path taken by the structure depended on the value of the Womersley numb. For small , it traveled approximately in a straight line to the center of the vitreous cavity. F

large , the circulation structure initially took the same path as in the low- case but then divergfrom the low- track as it moved away from the lens. This experimental work was extended

Stocchino et al. (2010), who used particle image velocimetry with images separated by a multipof the oscillation period to find a steady streaming flow on the plane of symmetry orthogonal to t

axis of rotation. For moderate , this revealed two large, counter-rotating steady circulation celAs was increased, a complicated sequence of topological changes took place in the flow, and f

the largest value of considered ( = 45.7), the most obvious circulations were a counter-rotatipair with the a sense of rotation opposite to those visible for small .

There has also been analytical progress on this problem. Repetto (2006) assumed the flow

be incompressible and irrotational. Thus the governing equations reduce to Laplaces equatiofor the velocity potential, subject to no-penetration boundary conditions, and time enters t

problem only as a parameter. In a perfect sphere, the velocity equals zero because there is no str

at the boundary to drive a flow. Motivated by this, the author assumed the indentation to be sma 1, and linearized the problem. He found the potential as a sum of the spherical harmonfunctions each multiplied by a function of the radial coordinate. The linearized unsteady Bernou

equation was used to find the pressure.However, this solution did not reproduce the circulations seen in the experiments, as they a

not irrotational. To model these circulations, Repetto et al. (2010) dropped the assumption potential flow and considered Newtonian viscous flow in an indented sphere. They also assum

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that is small and expanded the velocity as a double series u = (0u10 + u11 + )+ 2(0u20 +u21 + ) + , and similarly for the pressure. The component u10 and the steady streamingcomponent of u20, denoted u

(0)20 , equal the components u1 and u

(0)2 of the solution for the flow

in a true sphere described in Section 4.1. The calculation of u11 and u(0)21 (the steady streaming

component of u21) is performed in terms of vector spherical harmonics, which are a basis of

pairwise orthogonal, vector-valued functions of and . The components u11 and u(0)21 are written

as a sum of an unknown function of rtimes a vector spherical harmonic times a known function

oft. The analysis also shows thatu11

and u(0)

21, which arise as a result of the deformed geometry,

grow rapidly as increases and become increasingly important in the overall flow structure. Thus

the method is not expected to predict the velocity accurately for large .Plotting u10 + u11 reveals a circulation that forms every half-period behind the indentation,

moves to the center of the sphere, and is annihilated. This reproduces the features of the exper-imentally observed circulations for low but not the path of the circulations for high , which

is to be expected as the series expansion is not accurate for large . Examination of the steady

component arising because of the deformation, u21, reveals that there are two large steady circu-lations on the horizontal mid-plane (the plane of symmetry perpendicular to the axis of rotation)

just inside the indentation. The wall shear stress is maximal on the apex of the indentation andhas two additional smaller maxima on either side of this point.

4.3. Protective Mechanisms in the Eyes of Woodpeckers

Wygnanski-Jaffe et al. (2007) observed that, during pecking, the eyes of woodpeckers undergovery large accelerations and decelerations that, if scaled up correctly to the human eye, would cause

significant damage andloss of sight, yet the woodpecker eyes seemto be unharmed. They thereforeaimed to understand the physiological adaptations protecting the woodpecker eye. This could be

relevant to shaken baby syndrome, a condition caused by violent shaking of a small child, whichis usually characterized by retinal hemorrhage, subdural hematoma, and acute encephalopathy.

The mechanisms causing retinal hemorrhage are currently unknown, but investigation of theprotective features of the woodpecker eye could give insight into the particular mechanism of

failure in human eyes when subjected to large accelerations and decelerations.The authors identified a number of anatomical specializations in the woodpecker eye that pre-

sumably confer protection against large accelerations. This work highlights that dynamic motion

of the eye will lead to deformation of the eye globe, which has not been incorporated into previousstudies of vitreous flow, but which will undoubtedly lead to much interesting fluid mechanics.

4.4. Models of Partially Liquefied Vitreous Humor

Repetto et al. (2004) considered a spherical model of the vitreous chamber of radius R containingan elastic membrane dividing the chamber into two equal hemispherical parts. They considered

both free membrane motions, in which the sphere remains stationary but the membrane and

fluid start from a nonequilibrium configuration, and periodically forced motions, in which thesphere performs torsional oscillations about a diameter whose end points are points of attachmentof the membrane. In both cases, they assume the membrane displacement and amplitude of the

velocity are small, allowing them to linearize the system. They also assume that the membranedisplacement from equilibrium, (r, , t), is proportional to sin , where (r, , ) is a system of

spherical coordinates that has its axis normal to the equilibrium plane of the membrane (in theforced case, these coordinates rotate in time). The assumption of a separable solution allows them


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to expand the membrane displacement as

=

m=1J1

mrR

sin em(t),

where m is the m-th positive zero of the Bessel function J1 of the radial coordinate, and em(t) afunctions to be determined. The velocity potential satisfies Laplaces equation, and they expan

it as

= m=1

m(r, )sin d em(t)d t

+ [ (r, )sin eit+ c.c.], (1

where the second term involving the function is only needed in the case of forced oscillationThe analysis for free motions yields the natural frequencies of the system, which are the frequenc

associated with each of the functions em. These are found to be substantially lower than the natufrequencies of the membrane in the absence of fluid. With forced oscillations, there is an infini

response at each of the natural frequencies, suggesting that, for a viscous fluid, there will be a larbut finite response at the natural frequencies. Such a response could in turn lead to the generati

of large shear stresses on the retina, potentially leading to damage and subsequent detachment

Repetto et al. (2011) studied a circular model of the vitreous chamber filled with partialliquefied vitreous humor. They modeled the liquefied component as a Newtonian incompre

ible fluid and the gel component as a homogeneous isotropic viscous elastic incompressible solcharacterized by a Mooney-Rivlin strain energy function, and assumed that the two componen

were separated by an elastic membrane. They solved a numerical model to find the solid deformtion and fluid flow. Their results showed oscillations of the vitreous humor for sufficiently lar

values of the elastic modulus of the solid. The stresses were particularly high near the points attachment of the membrane to the retina, which could account for the increased risk of retin

detachment at these locations.

4.5. Mass Transport in the Vitreous Humor

Direct injection into the vitreous humor is commonly used to deliver large quantities of a drto the retina (Maurice 2001). The instantaneous distributions at various times after injection a

the timescales associated with uptake of the drug have been investigated by a number of authoXu et al. (2000) investigated the distribution of a drug after injection using a numerical mod

They included both diffusion of the drug and convection due to the slow flow that exists becausea pressure drop between the anterior and posterior of the vitreous chamber and/or by active upta

by the retina. The flow was assumed to be governed by Darcys law. The authors performed in vitexperiments with a small sample of bovine vitreous humor to determine the diffusion coefficie

of a model compound representing the drug. They also determined the hydraulic conductivi

by performing compression experiments on a sample of vitreous humor and then numericasolved the equation for mass conservation and a governing equation for the network phase. Th

used their results to estimate the P eclet numbers in human and mouse eyes, finding these to

approximately 0.41 and 0.024, respectively. Thus they concluded that the slow anterior-posteriflow does not typically play the dominant role in transport in the vitreous humor, at least for thmodel compound considered.

Once injected, various mechanisms can lead to nondelivery of the drug to the retina. Theinclude convection due to choroidal blood flow, active transport by the retinal pigment epith

lium, and convective losses due to collecting vessels outside the sclera. Balachandran & Baroc(2008) developed a model to investigate typical loss rates due to these three mechanisms. Th

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considered a model consisting of three regions: the vitreous chamber, the retinal pigment epithe-

lium (surrounding the vitreous chamber on its posterior surface), and the choroid (surroundingthe retinal pigment epithelium). They used Darcys law and the convection-diffusion equation

to model the fluid flow and the drug transport, respectively (with different diffusivities in eachregion). In the vitreous humor, there was assumed to be no source or uptake of the drug, while, to

model the active transport within the retina, there was an additional transport term kactc, wherec is the drug concentration and kact is a vector in the radially outward direction. In the choroid,

there was no additional transport, but they added a rate-of-uptake term (c

cbl), where cbl isthe drug concentration in the blood and is constant. The boundary conditions were as follows:

At the lens, they applied no penetration of fluid and no mass flux of drug; at the anterior hyaloidmembrane (the anterior surface of the vitreous humor immediately posterior to the lens) and at

the surface of the sclera, they set the pressures (with an approximate drop of 5 mmHg between

them driving the flow); and at both the hyaloid membrane and at the sclera, they assumed a rateof uptake proportional to the amount of drug available, but with different constants of propor-

tionality in the two regions. The authors solved the system numerically to obtain concentrationprofiles of the drug and compared the loss rates by each of the three mechanisms.

Repettoet al. (2010) also used theircalculated flows to estimate a Peclet number quantifying thedegree of mixing that occurs because of convective mass transport in the vitreous humor. Because

the flow components u10, u11, and u(0)

20

all consist of closed streamlines, these components do notinduce mixing, and thus the estimate of the P eclet number is based on the maximum magnitude

ofu(0)21 . For fluorescein, a commonly used tracer in ophthalmology, this gives an estimated Pecletnumber of approximately 1,000. This would suggest that the strength of the convection induced

by saccades is typically much greater than diffusion, and thus convection should not be neglected

in a model of drug transport. Stocchino et al. (2010) calculated the particle trajectories associatedwith the steady component of the flow and used these to find typical distances traveled by a particle

over time. They found that the value of the Womersley number has a significant effect on masstransport, with flows at high Womersley numbers transporting the fluid significantly further after

a fixed number of periods (see Figure 10).

5. TRANSPORT ACROSS BRUCHS MEMBRANEAmong the elderly of the industrialized world, age-related macular degeneration is the mostcommon cause of vision loss. Bruchs membrane is the innermost layer of the choroid, and it is

situated immediately outside the retinal pigment epithelium, which is the outer layer of the retina(see Figure 4). The macula is an approximately circular region of the retina situated close to the

optic nerve and has the highest density of photoreceptors.Age-related macular degeneration is thought to be caused by a buildup of lipids within Bruchs

membrane, which reduces mass transport across the membrane in a process that bears some

similarities to atherosclerosis, the main cause of arterial disease. The reduction in mass transportleads to injury to the photoreceptors because it both reduces the nutrients supplied and decreases

the removal rate of metabolites, which causes vision loss (Curcio et al. 2009).

The effect of lipid accumulation on fluid flow was studied by McCarty et al. (2008) boththeoretically and experimentally. In the theoretical model, they assumed that the fluid crossing themembrane is Newtonian and incompressible and treated the membrane as a porous medium with

specific hydraulic conductivityKm. Thus the mechanics was governed by Darcys equation and thecontinuity equation, which together reduce to Laplaces equation for the pressure,2P= 0. Theytreated the lipid as being composed of identical rigid spheres, each of radius ra, and developed twomodels to estimate the effective specific hydraulic conductivity, K, of the porous medium when


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1.00 0.50 0 0.50 1.00

1.00 0.50 0 0.50 1.00

1.00

0.75

0.50

0.25

0

0.25

0.50

0.75

1

b2

b2

b2

b2

b2

b2

0.020

0.016

0.012

0.008

0.004

0

a

0.0800.064

0.048

0.032

0.016

0

b

0.50

0.40

0.30

0.20

0.10

0

c

0.050

0.040

0.030

0.020

0.010

0

d

0.1500.120

0.090

0.060

0.030

0

e

0.80

0.64

0.48

0.32

0.16

0

f

y/R0

1.00

0.75

0.50

0.25

0

0.25

0.50

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y/R0

1.00

0.75

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y/R0

1.00

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y/R0

1.00

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y/R0

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y/R0

x/R0

x/R0

1.00 0.50 0 0.50 1.00

x/R0

1.00 0.50 0 0.50 1.00

1.00 0.50 0 0.50 1.00

x/R0

x/R0

1.00 0.50 0 0.50 1.00

x/R0

Figure 10

Contour plots of the nondimensional absolute square particle displacement, b2 (which is scaled by the square of the radius of thedomain, R20), in two experiments by Stocchino et al. (2010). They used nondimensional maximum indentation depth = 0.3,amplitude of the sinusoidal rotations = 0.17 rad, and Womersley number = 3.8 in panels acand = 45.7 in panels df. Thepanels show the displacement after (a,d) 50, (b,e) 100, and (c,f) 500 cycles. Reprinted with permission of the IOP.

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embedded, approximately uniformly, by lipid spheres with volume fraction (volume of spheres

per unit total volume).In the first model, they used a unit-cell approach, in which a single rigid sphere was surrounded

by a larger concentric spherical volume of the porous medium, such that the volume fraction ofthe rigid sphere equaled . They assumed that the velocity on the outer surface of the porous

sphere was the average velocity in the medium. In this assumption, the outer surface is sufficientlyfar from the rigid sphere that the velocity on it is approximately uniform, and therefore must

be small for it to be valid. The resulting model can be solved exactly to find the pressure field,and a comparison of the spatially averaged pressure gradient with Darcys law yielded the effective

hydraulic conductivity, K:

K= Km1

1+ /2 . (11)

The second way to estimate Kstarted with the rigid sphere embedded in the porous sphere ofthe first model and used the calculated pressure distribution to find the total force on the rigid

sphere, which was4

3r3aV0

1

K 1

Km

. (12)

Comparing this with the formula derived by Brinkman for the force on a sphere in a porousmedium, they obtained the relationship

1

K= 1

Km+ 9

2r2a (1 )

1+ ra

K+ r

2a

3K

, (13)

which agrees with Equation 11 to first order in in the limitK r2a (which was relevant for theirexperiments).

The authors tested these theoretical results by conducting experiments. They used Matrigel,

a material that has similar properties to those of Bruchs membrane. After the addition of latexnanospheres to the Matrigel, the measured values of the effective hydraulic conductivities agreed

well with those predicted by the theory. However, with embedded spheres of low-density lipopro-tein instead of latex nanospheres, the effective hydraulic conductivity decreased significantly more

than the theory would predict, a phenomenon that has not been satisfactorily explained.

6. DISCUSSION

Our aim in writing this article was to show that the eye presents a wealth of interesting andchallenging problems in fluid mechanics. Several of these problems have been tackled; however,

there remain many outstanding unsolved fluid mechanical problems. We recommend this area tothe reader as a source of interesting and accessible research questions that have potential impacts

on our most important sense.

SUMMARY POINTS1. The combined resistance of the trabecular meshwork filled with biopolymer together

with the inner lining of Schlemms canal are estimated to be sufficient to be the primarysource of resistance to the outflow of aqueous humor in health. The observed increase in

IOP during glaucoma could result partially from faulty caliber regulation in Schlemmscanal but does not result from the collapse of Schlemms canal per se.


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2. Using a mathematical model, it is possible to show that typical raised IOP values in

glaucoma can drive a sufficiently large flow along the axons of the retinal ganglion cellsto cause washout of energy-providing ATP in the cells, which could promote cell death

and vision loss.

3. Thermal convection is typically the dominant mechanism driving flow in the anterior

chamber.

4. The iris deforms as a result of the mechanical forces acting on it, leading to a complicatedfluid-structure interaction problem, which is relevant for closed-angle glaucoma andrecovery after a transient deformation of the iris position.

5. The transport of proteins within the anterior eye does not follow the same path as theflow of aqueous humor itself.

6. Fluid transport in the cornea is anisotropic owing to the arrangement of fibers withinthe corneal stroma.

7. If the vitreous humor is treated as viscoelastic, the vitreous chamber is assumed to bespherical, and movements of the eyeball are assumed to be torsional and sinusoidal, the

linearized equations can be solved exactly to find the primary azimuthal component of

the flow. If the fluid is additionally assumed to be Newtonian, then a secondary streamingcomponent of flow can be found semianalytically.

8. The departure from perfect sphericity in the real shape of the vitreous cavity has a

significant effect on the flow, leading to additional circulation structures in both theprimary flow and in the steady streaming flow. Transient temporal deformations in the

shape of the vitreous cavity are likely to have a large effect on the flow and pressure,which is not fully understood.

9. Nonhomogeneous properties of the vitreous humor can lead to additional stresses. Amembrane separating the cavity into two regions could lead to thepossibilityof resonance

at particular frequencies of oscillation. Alternatively, if the vitreous cavity is occupied by

a hemispherical region of elastic solid and a hemispherical region of viscous fluid, withthe two parts separated by a membrane, then the stress is particularly high at the pointsof attachment of the membrane.

10. In the case of liquefied vitreous humor, mass transport typically primarily results fromconvection induced by flow due to eye movements. The steady streaming component

of the flow plays one of the dominant roles in transport. In addition, mass transport issignificantly affected by both the shape of the chamber and the frequency of oscillation.

11. Impaired transport through Bruchs membrane, thought to be responsible for maculardegeneration, can be partially understood by considering a homogenized mathematical

model of lipid particles embedded in a membrane.

FUTURE ISSUES

1. Open-angle glaucoma is known to be caused by increased resistance in the outflow path-way of the aqueous humor. However, the exact locations and causes of the change in

resistance are not understood.

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2. Closed-angle glaucoma results from the iris physically blocking the outflow of aqueous

humor. The mechanisms underlying this condition have not been fully resolved.

3. Glaucoma results in the death of retinal ganglion cells and subsequent vision loss. Themechanism of cell death has not been conclusively proven.

4. The rheological properties of the vitreous humor need to be characterized and incorpo-rated into a model of vitreous flow.

5. It is important to explore the effects of transient deformation of the vitreous cavity on thevitreous pressure and flow, which may be important to understand the effect of impacts

and retinal hemorrhage in shaken baby syndrome.

6. Possible mechanical causes of retinal detachment and strategies for treatment need to be

explored.

7. Convective drug transport in the vitreous humor needs to be investigated, in particular

understanding the timescales involved and locations of delivery.

8. With regard to the transport across Bruchs membrane, the reasons for the increase

in resistance to transport when the membrane contains embedded lipids are not fullyresolved.

DISCLOSURE STATEMENT

The authors are not aware of any biases that might be perceived as affecting the objectivity of thisreview.

ACKNOWLEDGMENTS

C.R.E. acknowledges financial support from a Royal Society Wolfson Research Merit Award.

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