Fluid Mechanics of the Eye

Transcription

1 FLUID MECHANICS OF THE EYE 1 Fluid Mechanics of the Eye Jennifer H. Siggers and C. Ross Ethier Department of Bioengineering, Imperial College London, London SW7 2AZ Key Words porous medium, buoyancy-driven flow, deformed sphere, glaucoma, drug delivery. Abstract Fluid mechanical processes are intrinsically bound up in several aspects of the physiology and pathology of the human eye. In this article we describe selected phenomena that are amenable to particularly interesting mathematical, experimental or numerical analyses. We initially focus on glaucoma, a condition often associated with raised intraocular pressure. The mechanics in this disease is by no means fully understood, but we present some of the modeling work that provides a partial explanation. We next focus on other features of the dynamics of the two specialized ocular fluids: the aqueous and vitreous humors. In the aqueous humor we discuss problems concerning transport of heat and proteins and the hydration of the cornea. In the vitreous humor we discuss the possibility of flow, which occurs primarily as a result of saccades or motions of the eyeball. Finally, we describe a model of the degradation of Bruch s membrane in the retina. CONTENTS INTRODUCTION FLUID MECHANICS OF GLAUCOMA Increase in resistance across the trabecular meshwork Flow within Schlemm s canal

2 Annu. Rev. Fluid Mech /97/ Mechanism of death of retinal ganglion cells in glaucoma FLOW IN THE ANTERIOR CHAMBER Thermal transport Fluid structure interaction models of the iris and aqueous humor Transport of proteins Dynamics of the cornea FLUID MECHANICS OF THE VITREOUS HUMOR Flow in spherical models of the vitreous chamber Flow in models that account for the real shape of the vitreous chamber Protective mechanisms in the eyes of woodpeckers Models of partially liquefied vitreous humor Mass transport in the vitreous humor TRANSPORT ACROSS BRUCH S MEMBRANE DISCUSSION SUMMARY POINTS KEY TERMS FUTURE ISSUES ANNOTATED REFERENCES INTRODUCTION The eye is a remarkable organ, capable of transducing photons into neural signals with high efficiency under a wide range of operating conditions. The retina, containing the specialized cells that carry out this transduction process, is aided in its function by many supporting tissues. The development and proper function 2

3 FLUID MECHANICS OF THE EYE 3 of this complex system depends critically on biomechanical factors, as has been summarized elsewhere (Ethier et al. 2004a). Here we focus specifically on the fluid mechanical aspects of the eye, of which there are many. We begin with an overview of relevant anatomy and physiology. There are a number of processes within the eye in which fluid flow is important. Perhaps the most evident of these is the production, circulation and drainage of aqueous humor, a clear, colorless fluid which is secreted at a flow rate of 2 to 2.5 µl/min (Brubaker et al. 1989) by a specialized tissue known as the ciliary processes (Figure 1), located just posterior to the iris (the colored part of the eye). In view of the very small flow rates and modest dimensions, the flow of aqueous humor is creeping and inertia can be neglected. The aqueous humor itself is a very dilute protein solution, and thus can be treated as Newtonian with viscosity nearly identical to that of saline (Beswick & McCulloch 1956, Moses 1979). It flows into and fills a small region anterior to the lens but behind the iris, known as the posterior chamber, see Figure 2a, and then passes anteriorly through the pupil (the aperture in the central part of the iris) and enters the anterior chamber, where it circulates while bathing the iris and the inner surface of the cornea (the clear part of the eye). Eventually the aqueous humor drains from the eye via specialized tissues located in the angle of the anterior chamber, where the iris, cornea and sclera (the white part of the eye) meet, see also Figure 2 and Sections 2.1 and 2.2. These tissues have a significant hydrodynamic flow resistance, and the drainage of the aqueous humor out of the eye therefore requires that there be a positive pressure within the eye itself, the so-called intraocular pressure (IOP). The flow of the aqueous humor performs two important physiological functions.

4 4 SIGGERS ETHIER First, the positive pressure that it generates stabilizes the otherwise flaccid eye, ensuring accurate positioning of 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, without which the cells in these tissues would die. Some models of aqueous humor flow in health are presented in Section 3. However, unfortunately as will be explained later, impairment in the drainage of this fluid leads to an elevation in IOP, which is a major risk factor for the disease known as glaucoma, the second most cause 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 damaged and 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 we shall see 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 transport effects in the retinal ganglion cell axons may be a promising new way to understand the pathogenesis of glaucoma. The cornea combines the attributes of mechanical strength and optical transparency, which is 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 hydration state of the cornea. There is a continuous flux of fluid in and out of the cornea and the body has therefore developed sophisticated mechanisms to control this transport; specifically, water is actively and continually pumped out of the corneal stroma (the central layer of the cornea) by the corneal endothelium, a specialized layer of cells lining the interior corneal surface. The net effect is that the stroma is in a continual state of

5 FLUID MECHANICS OF THE EYE 5 thermodynamic disequilibrium with respect to its adjacent bathing fluids, namely the tear film anteriorly and aqueous humor posteriorly, see Section 3.4. The majority of the ocular globe is filled by a clear, colorless, gel-like material known as vitreous humor, which occupies the vitreous chamber of the eye. The chamber is surrounded by two tissues: the retina and the choroid, each of which is composed of many layers, see Figure 4. Vitreous humor has complex viscoelastic properties, and, although there have been several attempts to characterize its properties experimentally (Lee et al. 1992, Nickerson et al. 2008, Swindle et al. 2008, Zimmerman 1980), its rheology is not fully understood. It is known that the vitreous becomes progressively liquefied with age; about 20% of the vitreous is liquid in year-olds, and this rises to more than half liquid in subjects aged years (Bishop 2000). In about 25 30% of subjects, liquefaction can lead to a process in which the retina detaches, risking loss of sight. In this context, it is also important to note that the eye is not normally stationary, even when apparently focused on a fixed point. Instead, the eye constantly executes a series of extremely rapid angular rotations (300 /sec or more) known as saccades (Rayner 1998). These have the effect of creating flow patterns within the vitreous humor, particularly if the vitreous has liquid characteristics, as will be 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 transport across Bruch s 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,

6 6 SIGGERS ETHIER the eye is richly supplied by a complex network of blood vessels, leading to many interesting physiological problems 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 recently investigated 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 is itself 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, though not always, characterized by an increase in intraocular pressure (IOP), and lowering the pressure is the only treatment currently available. Therefore there is significant interest in understanding the factors that control IOP. In almost all cases of glaucoma, the cause of the pressure increase is known to be an increase in the hydrodynamic resistance to aqueous humor drainage, necessitating a higher pressure to drive the outflow. Glaucoma can be further classified into closed angle glaucoma and open angle glaucoma. In closed angle glaucoma the iris moves anteriorly from its normal position, reducing or eliminating the gap between it and the cornea. This is an interesting fluid structure interaction problem, and frequently leads to a complete occlusion of outflow with attendant dramatic increases in IOP; a mathematical model of this is briefly discussed in Section 3.1. In open angle glaucoma, the drainage tissues remain accessible to the aqueous humor, but they present an elevated flow resis-

7 FLUID MECHANICS OF THE EYE 7 tance. There are a number of different types of open angle glaucoma, the most common of which is known as primary open angle glaucoma. This condition has puzzled researchers for many years, since there are no evident structural changes in the drainage tissues that could explain their elevated resistance. Here we discuss some of the modeling work that has attempted to shed light on the source of the resistance: in Section 2.1 we discuss changes in the trabecular meshwork and in Section 2.2 we discuss changes in the mechanics of flow into and through Schlemm s canal. In Section 2.3 we discuss a possible mechanism for the 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, it is necessary to provide further details on the anatomy of the drainage tissues located in the angle of the anterior chamber, see also Figure 2. As the aqueous humor leaves the eye, it first passes through the trabecular meshwork, which can be represented as a biological porous material. It then enters into a collecting duct known as Schlemm s canal, notable for its unusual endothelial cellular lining containing a number of small micron-sized openings ( pores ). As will be discussed, the hydrodynamic interaction of the flows through the pores and that through the trabecular meshwork is of great interest. The aqueous humor then flows along the canal and out through a drainage structure known as a collector channel, from which it eventually mixes with venous blood in the sclera and returns to the right heart. A significant amount of work has been devoted to understanding the drainage of aqueous humor through these tissues. Originally, attention focused on analysis of

8 8 SIGGERS ETHIER the trabecular meshwork as a porous material (Ethier 1986, McEwen 1958, Seiler & Wollensak 1985). Micrographs of normal and diseased tissue were analyzed to estimate tissue porosity, ε TM, and specific surface, S TM, and the hydraulic permeability K TM, of the trabecular meshwork was estimated using classical Carman Kozeny theory (Bear 1988): K TM = ε3 TM kstm 2, (1) for suitable values of the Kozeny constant, k. The resulting computed permeabilities were compared 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 data. The conclusion was that the trabecular meshwork either had little hydraulic flow resistance, or, more likely, the fundamental assumptions underlying the calculation were incorrect. Some evidence indicates that the apparently open spaces within the trabecular meshwork are actually filled with a gel-like biopolymer consisting of proteins and long-chain carbohydrates. It was therefore appropriate to include the hydrodynamic effects of this gel material, achieved through the use of the fiber matrix model (Weinbaum 1998), in which the individual biopolymer strands are approximated by long, randomly-oriented cylinders. Using classical low Reynolds number results (Happel 1959, Spielman & Goren 1968), it was possible to make good estimates of the hydraulic permeability of a pure biopolymer gel or of tissues with well-characterized microstructrure (Ethier 1986). Application of this theory to the trabecular meshwork required consideration of a two-level hierarchical porous medium, in which a material (the gel) with characteristic length scales on the order of Angstroms to nanometers and hydraulic permeability K gel was embedded within a second material (the trabecular meshwork), having pores

9 FLUID MECHANICS OF THE EYE 9 with dimensions on the order of microns to tens of microns. Accounting for steric hindrance and tortuosity effects leads to a prediction of the overall permeability of the composite material as ε TM K = K gel. (2) 2 ε TM (Ethier 1986). 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 has called 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 hydraulic resistance than would be expected (Hubbard et al. 1997). The fluid mechanics of aqueous humor drainage through the small pores that make up the cellular lining of Schlemm s canal is also important (Bill & Svedbergh 1972, Eriksson & Svedbergh 1980). By exploiting the fact that these pores are small and relatively isolated, it suffices to consider flow through a single pore, and to treat the entire cellular lining as a parallel network of hydrodynamically non-interacting pores. The flow resistance of a single pore of radius R can be calculated from Sampson s theory in low Reynolds number hydrodynamics (Happel & Brenner 1983), which relates the pressure drop, p, across a thin surface to the flux, q, through the surface via q p = R3 3µ, (3) where q is the volumetric flow through the pore, µ is the dynamic viscosity of the aqueous humor, and p is the pressure difference across the surface between + and. The surprising finding from this calculation was that it predicts an extremely low flow resistance of the endothelial lining of Schlemm s canal,

10 10 SIGGERS ETHIER certainly much lower than the observed resistance of the entire system. This led to a paradox: neither the lining of Schlemm s 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 an interesting hydrodynamic interaction between the endothelial lining of Schlemm s canal and the upstream porous material of the trabecular meshwork. Specifically, because the pores are few and account for only a small fraction of the total area of the inner wall of Schlemm s canal, they must act to hydrodynamically focus or funnel aqueous humor drainage through the trabecular meshwork, see Figures 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, the hydraulic resistance of the ensemble structure can be calculated in a straightforward manner. The theory predicts an overall flow resistance which is generally consistent with experimental measurements (Overby et al. 2009); further, it reconciles observations that both pore density and the composition of the trabecular meshwork have an effect on overall resistance to flow in this tissue. 2.2 Flow within Schlemm s canal In a different study aimed at identifying the source of outflow resistance, Johnson & Kamm (1983) developed a model to consider the hydrodynamic effects of partial or total collapse of Schlemm s canal, caused by IOP-induced deformation of the trabecular meshwork. They modeled the inner wall as a permeable mem-

11 FLUID MECHANICS OF THE EYE 11 brane supported by linear Hookean springs with constant Young s modulus E, see Figure 6(a). Thus the height, h(x), of the canal the distance between the inner and outer walls as a function of position, x, along the canal is given by h = h 0 (1 (IOP P sc )/E), where P sc (x) is the pressure in Schlemm s canal and h 0 (assumed constant) is the height of the canal when the transmural pressure, IOP P sc, equals zero. The resistance to transmural flow is R w (assumed constant), that is a flux of 1/R w times the transmural pressure drop crosses the inner wall per unit length of wall. They modeled the aqueous humor as an incompressible Newtonian fluid, and assumed that dh/dx 1 and s h 0, where s is the half-distance between collector channel ostia, thus allowing them to approximate the flow using lubrication theory. The governing equations reduced to a single equation for h: 12µ (h h 0 ) = d ( wr w dx h 3dh dx ), (4) where w is the width of the cross-section of Schlemm s canal in the anterior posterior direction. Boundary conditions arise from prescribing the pressure at the collector channels, which, using the spring condition given above, leads to ( h = h 0 1 IOP P ) cc, (5) E at x = s, where P cc is the pressure in the collector channels. They solved the governing equation numerically over half the distance between neighboring collector channels, that is 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, IOP P cc, due to an increase in resistance to outflow. This is because the height of Schlemm s canal reduces when the pressure drop is large, increasing the canal s resistance, and hence the overall resistance. Thus, in order

12 12 SIGGERS ETHIER to maintain a constant outflow to meet physiological demands, the IOP must increase more than it would if the dependence were linear. If, on the other hand, the wall resistance, R w, increases, while the other parameters maintain constant values, the IOP must also increase to maintain the outflow. For a normal or slightly raised IOP (up to about 25 mmhg), and, using parameter values suitable for the human eye, the predicted height of the canal is approximately spatially uniform and the total resistance to outflow is approximately constant. This shows that the majority of total outflow resistance does not derive from the resistance to flow within Schlemm s 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 complete collapse is unrealistic because Schlemm s canal contains septae, short structures, modeled as being of height h s, which protrude into Schlemm s canal from the outer wall to prevent complete collapse. With a normal or slightly raised IOP, these protrusions make no difference to the model results, because the minimum width of the canal is greater than h s. For higher IOPs there is partial collapse of the canal with the septae supporting the collapsed region, see Figure 6(b). For still higher IOP the channel is completely supported by the septae and has constant height h s, and under these conditions the total outflow resistance is constant. In a further extension the authors considered compliant septae, meaning the height drops below h s in the collapsed region, increasing the resistance of Schlemm s canal compared with the rigid septae model. A comparison of model results with experimental data suggests that Schlemm s 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 Schlemm s canal, as

13 FLUID MECHANICS OF THE EYE 13 opposed to that across the endothelial lining of the canal. In the vascular system, which shares many biological similarities with Schlemm s canal, the caliber of vessels is strongly influenced by the shear stress exerted by the blood flowing within the vessel. However, since the volumetric flow rates in the lumen of Schlemm s canal are minuscule, shear stresses might, a priori, also be expected to be too small to have physiological impact. However, a simple calculation in which the cross-section of Schlemm s canal was treated as an ellipse showed that the estimated shear stresses were similar to those observed in the vascular system (Ethier et al. 2004b), suggesting that the biological mechanisms for caliber regulation in the two systems could be similar. This is potentially important, as Schlemm s canal is observed to be shorter in the anterior-posterior direction in glaucomatous eyes, even though its height h is unaffected (Allingham et al. 1996). This increases outflow resistance, and thus dysregulation of the mechanisms controlling canal caliber may 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 al. 2004a, Fechtner & Weinreb 1994, Schumer & Podos 1994), and several mechanisms have been proposed. These include mechanical insult to optic nerve head tissues and/or a failure in vascular autoregulation to the nerve (Burgoyne et al. 2005, Morgan 2000, Pillunat et al. 1997, Riva et al. 1997, Yamamoto & Kitazawa 1998, Yan et al. 1994). Here we consider another, more fluid mechanically-based, mechanism, proposed by Band et al. (2009). Retinal ganglion cells require axonal transport to remain viable, in which cargo-containing structures, the vesicles, are transported along the axons by motor proteins. These

14 14 SIGGERS ETHIER motors require energy for their task, which they obtain from adenosine triphosphate (ATP) molecules, which are in turn released from mitochondria located along the axon. ATP is distributed along the axon by a combination of diffusive and, in the presence of flow, convective effects. If the supply of ATP is sufficiently depleted then active axonal transport will be reduced or stopped. This has been shown to lead to death of ganglion cells in primates (Anderson & Hendrickson 1974, Balaratnasingam et al. 2007, Minckler et al. 1977). Retinal ganglion cells contain axoplasm, a fluid that has approximately Newtonian properties. The walls of these cells are permeable to the axoplasm, and therefore, in the presence of a pressure gradient, it is possible for the axoplasm to flow along the axon, since it can be replenished by transmural flow. In the proposed mechanism, the rise in IOP leads to a significant axial flow of the axoplasm, and this causes convection of the ATP towards the brain. If the convection of ATP is stronger than diffusion, it will prevent ATP from diffusing in the upstream direction, leading to a region of washout along the axon. In their paper, Band et al. (2009) developed a mathematical model of the flow in the axons and used it to estimate the relative strengths of convection and diffusion, characterized by the Péclet number. They demonstrate that their suggested mechanism is plausible, since the flow is likely to begin to occur for elevations of IOP on the scale of those observed in glaucoma. The model is illustrated in Figure 7. Each ganglion cell axon lies partly within the eye globe and partly within the optic nerve bundle. In the eye globe, the flux of fluid through the axon 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 the axons

15 FLUID MECHANICS OF THE EYE 15 is also proportional to the pressure difference between the fluid in them. Flow along the length of the axon is driven by the axial pressure gradient. Using these assumptions, a relationship between the axial flux and pressure gradient along the axons can be found, leading to a solution for the flow and pressure in the form of sums of an infinite series of Bessel functions. If the IOP is elevated to levels commonly seen in glaucoma, the Péclet number for ATP predicted 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 potential importance 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 meshwork in the normal course of its drainage from the eye. In addition to this flow, there is also a thermally driven flow, since 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 is close to body temperature, but the anterior surface is closer to atmospheric temperature (usually cooler). 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. They treated the fluid as incompressible and Newtonian, and used

16 16 SIGGERS ETHIER the Boussinesq approximation to describe the variations in the density of the fluid. The posterior surface consisted of a disc representing the pupil in the centre and an annulus surrounding it representing the iris, which was assumed to be at a fixed temperature T 1 close to body temperature. The anterior surface of the chamber was assumed to be at a fixed cooler temperature T 0. The velocity at the pupil was assumed to be purely normal and given by a prescribed function w 0 (x,y), with no-slip velocity applied on the other boundaries, except that drainage occurs at the angles of the domain. Canning et al. (2002) used experimental measurements to argue that the aspect ratio of the model is small (h a), see Figure 8, and also estimated the reduced Reynolds number to be small. In the formal mathematical limit in which these quantities are small and also neglecting viscous dissipation, time-dependence and convection of heat they derived a simplified system of equations. They were able to manipulate these and reduce them to the single differential equation ( ) H h 3 H P = 12µw 0, (6) where H = e x / x + e y / y in Cartesian coordinates, P is the deviation in the pressure from the hydrostatic pressure profile and µ is the viscosity at the temperature T 0. The authors estimate the size of w 0 and conclude that, typically, it is likely to be small compared to the convective velocity, and therefore they consider the case w 0 = 0, which allows an exact solution of (6). Estimation of the stress induced by this flow suggests it is unlikely to be strong enough to cause particle or cell detachment from the iris. Comparing the transit time of the solution with the time it would take a particle in the aqueous humor to settle under gravity shows that that the convective velocity is several times faster than the settling

17 FLUID MECHANICS OF THE EYE 17 velocity. Calculation of the Stokes drag allows a full model of particle transport to be developed. The model showed that particles remain in the vertical plane that contains them and that is also perpendicular to the iris. The authors solved the model numerically and used their results to comment on features that would be present in a number of clinical conditions. This work was extended by Fitt & Gonzalez (2006) to include inflow (w 0 0), to consider other directions of gravity with respect to the model (for example a supine patient), to investigate vibrations of the lens as the head or eye moves (phakodenesis) and to investigate the flow produced by the rapid eye movement (REM) during sleep. Their results showed that the buoyancy-driven flow typically exceeds the flow driven by other mechanisms by orders of magnitude, and plays a dominant role in several medical conditions of 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 & Barocas (2002b). The authors of these papers developed an axisymmetric model of the flow in both the anterior and posterior chambers, in which they modeled the aqueous humor as a Newtonian viscous fluid and the iris as an incompressible neo-hookean solid. The iris deforms as a result of the stress exerted by the flow, and the authors calculated the steady-state position of the iris tissue. Heys & Barocas (2002a) considered a fully three-dimensional model and included thermal convection. Their results showed that convection effects in the flow are dominant, that is the calculated velocity magnitude predicted by the equations with the convection terms added is many times larger than

18 18 SIGGERS ETHIER that predicted by the equations without convection included. Their results were consistent with clinical observations of Krukenberg s spindle, a condition in which pigment from the iris becomes attached to the posterior surface of the cornea in characteristic vertical stripes. A similar model was used by both Huang & Barocas (2006) and Amini & Barocas (2010), who respectively investigated the flow and deformation produced by small oscillations of the position of the iris, and the recovery from an indentation in the cornea or sclera, 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 into its stress tensor to represent contraction of the sphincter muscle in the normal circumferential direction. They tuned the geometry in order to model both normal eyes and eyes with features that are thought to be risk factors in closed angle glaucoma. Their results predict that the further forward the lens position, the greater the likelihood of iris lens contact, which leads to a greater pressure difference between the posterior and anterior chambers. On the other hand, decreasing the diameter of the 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 for different 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 conditions when the pupil is large. 3.3 Transport of proteins As noted above, the aqueous humor is produced behind the iris, passes through the pupil, and fills the anterior chamber before draining from the eye. It is

19 FLUID MECHANICS OF THE EYE 19 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 whose details have only been elucidated over the past decade or so, despite the fact that the circulation of aqueous humor 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 of the 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 in the 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. This mechanism 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 protein mass transport whose predictions 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 was 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 in the tangent plane of the stroma. They developed an optical pachometer able to measure dynamic 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, re-

20 20 SIGGERS ETHIER lating corneal hydration to the local swelling pressure, and using Darcy s law allowed them to deduce stromal permeability 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 tangent plane, 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. Using classical results for flow in porous media in the limit of vanishing fiber solid fraction, the permeability is predicted to be approximately two fold lower for flow in the normal direction than that in the plane of the fibers (Happel & Brenner 1983), a result that is quantitatively in agreement with the experimental data. 4 FLUID MECHANICS OF THE VITREOUS HUMOR The vitreous cavity has an approximately spherical shape and contains vitreous humor, which is subject to mechanical forces as a result of motion, and possibly deformation, of the eyeball (due primarily to motion of the head and rotation of the eyeball within the socket). In the absence of deformation of the vitreous chamber, translational motion alone does not result in any relative motion of the humor within the vitreous chamber, since the accelerations involved can be balanced by a pressure gradient. Rotational motion on the other hand does induce relative motion of the humor. The fastest motions occur when the vitreous humor is liquefied, which can be the case either following the process of liquefaction described in the Introduction, or following vitrectomy, a surgical procedure

21 FLUID MECHANICS OF THE EYE 21 in which some vitreous humor is removed and replaced with another fluid, often silicone oil or a gas bubble. In this case the fluid filling the vitreous chamber is approximately Newtonian. Several authors have developed mathematical and experimental models of the flow in the vitreous humor, which we discuss in this section. We first discuss models of the dynamics that approximate the vitreous chamber as a rotating sphere, including a viscoelastic model, see Section 4.1, and then consider extensions of this work to account for the effects of the geometry of the chamber, while also simplifying to the case of a Newtonian fluid, Section 4.2. We then discuss the potential effects of dynamic deformation of the vitreous chamber by considering woodpeckers, whose eyes are subjected to enormous accelerations during pecking, and which appear to have a number of special protective adaptations, Section 4.3. This has potential applications to understanding the mechanism of damage in shaken baby syndrome. We then discuss models of partially liquefied vitreous humor, Section 4.4, and finally discuss the implications for mass transport in the vitreous humor, which has important implications for drug delivery to the retina, Section Flow in spherical models of the vitreous chamber David et al. (1998) investigated the periodic flow produced during small torsional oscillations of the eyeball, modeling the vitreous chamber as a rigid sphere. They used the Maxwell Voigt viscoelastic model proposed by Lee et al. (1992) to characterize the rheological behavior of the vitreous. The angular displacement was modeled as ǫ cos ωt, while the assumption of small oscillations allowed them to linearize the model and seek solutions proportional to e iωt. This led to a linear relationship between the shear stress and the shear strain, whose constant of pro-

22 22 SIGGERS ETHIER portionality was the complex modulus, G, dependent on ω. In terms of spherical polar coordinates, (r,θ,φ), centered on the sphere and with axis parallel to the axis of the oscillations, the velocity field is: u = iǫr3 ω(sin(ar/r) (ar/r)cos(ar/r)) 2r 2 (sin a acos a) sin θ e iωt e φ + c.c., (7) where R is the radius of the sphere, a = α c e iπ/4, α c is the complex Womersley number, given by α c = iρω 2 R 2 G, (8) ρ is the fluid density, e φ is the unit vector in the direction of increasing φ. and c.c. denotes the complex conjugate. Thus for small values of α c the fluid moves almost as a rigid body, while for large α c the motion becomes confined to a Stokes boundary layer of width α c 1 and the fluid in the centre 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 non-myopic 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 amplitude and frequency, the behavior was characterized by two dimensionless parameters: the Womersley number, α, and the angular amplitude of the oscillations, ǫ. Their results 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, that is a single

23 FLUID MECHANICS OF THE EYE 23 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 component of 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, they compared 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 used their results to show that the shear stress is not strongly dependent upon the angle through which the eye moves in a saccade, and thus, since saccades of small angle are much more frequent than those of large angle, small-angle saccades are responsible for generating the majority of the shear stress on the retina when integrated over time. In addition to the behavior just described, there is also a steady component of flow in the vitreous ( steady streaming, see for example Riley 2001). For smallamplitude 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 tend to cancel over a period of the oscillatory motion. Therefore Repetto et al. (2008) studied this steady 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 ǫ. They formulated the solution as a series expansion in powers of the small parameter ǫ: u = ǫu 1 + ǫ 2 u , p = ǫp 1 + ǫ 2 p The leading-order solutions u 1 and p 1 have frequency ω and were also given by (7), but with α c replaced by the Womersley number α. The first corrections u 2 and p 2 are driven by the nonlinear term u 1 u 1 in the Navier Stokes equation, and are thus

24 24 SIGGERS ETHIER a superposition of a solution with frequency 2ω and a steady solution, denoted u (2) 2, p(2) 2 and u (0) 2, p(0) 2, respectively (thus u 2 = u (0) 2 + u (2) 2 and p = p (0) 2 + p (2) 2 ). Since the steady solution is more important in terms of its implication for mass transport, the authors calculated u (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 whose dependence on θ and φ was found exactly, but whose dependence on r took the form of an integral that had to be computed numerically. In the limit α 1 the integral could be calculated analytically, in which case the velocity can be shown to be proportional to α 6, and thus grows very slowly as α increases. The integral can also be found analytically in the limit α 1, and in this case the velocity tends to a constant value. The authors also performed experiments using the same apparatus as Stocchino et al. (2007), but taking images only once per period to reveal the average, rather than the instantaneous, velocity. The theoretical and experimental results showed good agreement for small amplitudes ǫ (within 10% for ǫ = ) over the whole range of α (from 3.1 to 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 indentation into the chamber caused by the presence of the lens. In order to investigate the effect of the shape, Stocchino et al. (2007) used a similar experimental model to the spherical model of Repetto et al. (2005), but with a modified shape. Based on their analysis of several ultrasound and magnetic resonance scans, the authors modeled the lens as a spherical indentation into the sphere, both spheres having the same radius. This introduces a further non-

25 FLUID MECHANICS OF THE EYE 25 dimensional parameter, δ, equal to the maximum depth of the indentation divided by the vitreous cavity radius R. Again they subjected this apparatus to periodic, torsional rotations, and, approximately at each of the times when the angular velocity reached its maximum absolute value, observed a circulation structure generated at the back of the indentation. This structure then traveled towards the center of the sphere and was annihilated. The path taken by the structure depended on the value of the Womersley number, α. For small α, it traveled approximately in a straight line to the center of the vitreous cavity. For large α, the circulation structure initially took the same path as in the low-α case, but then diverged from the low-α track as it moved away from the lens. This experimental work was extended by Stocchino et al. (2010) who used particle image velocimetry with images separated by a multiple of the oscillation period to find a steady streaming flow on the plane of symmetry orthogonal to the axis of rotation. For moderate α, this revealed two large, counter-rotating steady circulation cells. As α was increased, a complicated sequence of topological changes took place in the flow, and, for the largest value of α considered (α = 45.7), the most obvious circulations were a counter-rotating pair with the opposite sense of rotation to those visible for small α. There has also been analytical progress on this problem. Repetto (2006) assumed the flow to be incompressible and irrotational. Thus the governing equations reduce to Laplace s equation for the velocity potential, subject to nopenetration boundary conditions, and time enters the problem only as a parameter. In a perfect sphere the velocity equals zero, since there is no stress at the boundary to drive a flow. Motivated by this, the author assumed the indentation to be small, δ 1, and linearized the problem. He found the potential as

26 26 SIGGERS ETHIER a sum of the spherical harmonic functions each multiplied by a function of the radial coordinate. The linearized unsteady Bernoulli equation was used to find the pressure. However, this solution did not reproduce the circulations seen in the experiments, since these are not irrotational. To model these, Repetto et al. (2010) dropped the assumption of potential flow and considered Newtonian viscous flow in an indented sphere. They also assumed that δ is small and expanded the velocity as a double series u = ǫ(δ 0 u 10 +δu )+ǫ 2 (δ 0 u 20 +δu )+..., and similarly for the pressure. The component u 10 and the steady streaming component of u 20, denoted u (0) 20, equal the components u 1 and u (0) 2 of the solution for the flow in a true sphere described in Section 4.1. The calculation of u 11 and u (0) 21 (the steady streaming component of u 21 ) is performed in terms of vector spherical harmonics, which are a basis of pairwise orthogonal, vector-valued functions of θ and φ. The components u 11 and u (0) 21 are written as a sum of an unknown function of r times a vector spherical harmonic times a known function of t. The analysis also shows that u 11 and u (0) 21, which arise as a result of the deformed geometry, grow rapidly as α increases, and become increasingly important in the overall flow structure. Thus the method is not expected to predict the velocity accurately for large α. Plotting u 10 + δu 11 reveals a circulation that forms every half period behind the indentation, moves to the centre of the sphere and is annihilated. This reproduces the features of the experimentally observed circulations for low α, but not the path of the circulations for high α, which is to be expected since the series expansion is not accurate for large α. Examination of the steady component arising due to the deformation, u 21, reveals that there are two large steady

27 FLUID MECHANICS OF THE EYE 27 circulations 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, and has two additional smaller maxima 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 undergo very large accelerations and decelerations that, if scaled up correctly to the human eye, would cause significant damage and loss of sight, yet the woodpecker eyes seem to be unharmed. They therefore aimed 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, which is usually characterized by retinal hemorrhage, subdural hematoma and acute encephalopathy. The mechanisms causing retinal hemorrhage are currently unknown, but investigation of the protective 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 would presumably confer protection against large accelerations. This work highlights the fact that dynamic motion of the eye will lead to deformation of the eye globe, which has not been incorporated into previous studies of vitreous flow, but which will undoubtedly lead to much interesting fluid mechanics.

28 28 SIGGERS ETHIER 4.4 Models of partially liquefied vitreous humor Repetto et al. (2004) considered a spherical model of the vitreous chamber of radius R containing an elastic membrane dividing the chamber into two equal hemi-spherical 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 the sphere performs torsional oscillations about a diameter, whose endpoints are points of attachment of the membrane. In both cases they assume the membrane displacement and amplitude of the velocity are small, allowing them to linearise the system. They also assume that the membrane displacement from equilibrium, η(r,φ,t), is proportional to sin φ, where (r,θ,φ) is a system of spherical coordinates that has its axis normal to the equilibrium plane of the membrane (note that, in the forced case, these coordinates rotate in time). Assuming a separable solution allows them to expand the membrane displacement as η = m=1 ( ) αm r J 1 sin φe m (t), (9) R where α m is the mth positive zero of the Bessel function J 1 of the radial coordinate and e m (t) are functions to be determined. The velocity potential satisfies Laplace s equation, and they expand it as ϕ = m=1 ψ m (r,θ)sin φ de m(t) dt ( ) + χ(r,θ)sin φe iωt + c.c., (10) where the second term involving the function χ is only needed in the case of forced oscillations. The analysis for free motions yields the natural frequencies of the system, which are the frequencies associated with each of the functions e m. These are found to be substantially lower than the natural frequencies of the membrane in the absence of fluid. With forced oscillations there is an infinite response at

29 FLUID MECHANICS OF THE EYE 29 each of the natural frequencies, suggesting that, for a viscous fluid, there will be a large but finite response at the natural frequencies. Such a response could in turn lead to the generation 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 partially liquefied vitreous humor. They modeled the liquefied component as a Newtonian incompressible fluid and the gel component as a homogeneous isotropic viscous elastic incompressible solid, characterized by a Mooney Rivlin strain energy function, and assumed that the two components were separated by an elastic membrane. They solved a numerical model to find the solid deformation and fluid flow. Their results showed oscillations of the vitreous humor for sufficiently large values of the elastic modulus of the solid. The stresses were particularly high near to the points of attachment of the membrane to the retina, which could account for the increased risk of retinal 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 drug to the retina (Maurice 2001). The instantaneous distributions at various times after injection and the timescales associated with uptake of the drug have been investigated by a number of authors. Xu et al. (2000) investigated the distribution of a drug after injection using a numerical model. They included both diffusion of the drug, and convection due to the slow flow that exists because of a pressure drop between the anterior and posterior of the vitreous chamber and/or by active uptake by the retina. The

30 30 SIGGERS ETHIER flow was assumed to be governed by Darcy s law. The authors performed in vitro experiments with a small sample of bovine vitreous humor, in order to determine the diffusion coefficient of a model compound representing the drug. They also determined the hydraulic conductivity by performing compression experiments on a sample of vitreous humor, and then numerically solved the equation for mass conservation and a governing equation for the network phase. They used their results to estimate the Péclet numbers in human and mouse eyes, finding these to be approximately 0.41 and 0.024, respectively. Thus they concluded that the slow anterior posterior flow does not typically play the dominant role in transport in the vitreous humor, at least for the model compound considered. Once injected, various mechanisms can lead to non-delivery of the drug to the retina. These include convection due to choroidal blood flow, active transport by the retinal pigment epithelium, and convective losses due to collecting vessels outside the sclera. Balachandran & Barocas (2008) developed a model to investigate typical loss rates due to these three mechanisms. They considered a model consisting of three regions: the vitreous chamber, the retinal pigment epithelium (surrounding the vitreous chamber on its posterior surface), and the choroid (surrounding the retinal pigment epithelium). They used Darcy s law and the convection diffusion equation to model the fluid flow and the drug transport, respectively (with different diffusivities in each region). 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 k act c, where c is the drug concentration and k act is a vector in the radially outward direction. In the choroid there was no additional transport, but they added a rate-of-uptake term γ(c bl c), where c bl is the drug concentration in the blood

31 FLUID MECHANICS OF THE EYE 31 and γ 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 hyaloid membrane (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 rate of uptake proportional to the amount of drug available, but with different constants of proportionality in the two regions. The authors solved the system numerically to obtain concentration profiles of the drug, and compared the loss rates by each of the three mechanisms. Repetto et al. (2010) also used their calculated flows to estimate a Péclet number quantifying the degree of mixing that occurs due to convective mass transport in the vitreous. Since the flow components u 10, u 11 and u (0) 20 all consist of closed streamlines, these components do not induce mixing, and thus the estimate of the Péclet number is based on the maximum magnitude of u (0) 21. For fluorescein, a commonly used tracer in ophthalmology, this gives an estimated Péclet number of around 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 associated with 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 mass transport, with flows at high Womersley numbers transporting the fluid significantly further after a fixed number of periods, see Figure 10.

32 32 SIGGERS ETHIER 5 TRANSPORT ACROSS BRUCH S MEMBRANE Among the elderly of the industrialized world, age-related macular degeneration is the most common cause of loss of vision. Bruch s 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 build up of lipids within Bruch s 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 transport leads 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 modeled by McCarty et al. (2008) both theoretically and experimentally. In the theoretical model they assumed the fluid crossing the membrane is Newtonian and incompressible and treated the membrane as a porous medium with specific hydraulic conductivity K m. Thus the mechanics was governed by Darcy s equation and the continuity equation, which together reduce to Laplace s equation for the pressure, 2 P = 0. They treated the lipid as being composed of identical rigid spheres each of radius r a, and developed two models to estimate the effective specific hydraulic conductivity, K, of the porous medium when embedded, approximately uniformly, by lipid spheres with volume fraction ϕ (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

33 FLUID MECHANICS OF THE EYE 33 medium, such that the volume fraction of the rigid sphere equalled ϕ. They assumed the velocity on the outer surface of the porous sphere was the average velocity in the medium. In this assumption, the outer surface is sufficiently far from the rigid sphere that the velocity on it is approximately uniform, and therefore ϕ must be small for it to be valid. The resulting model can be solved exactly to find the pressure field, and comparing the spatially averaged pressure gradient with Darcy s law yielded the effective hydraulic conductivity K = K m 1 ϕ 1 + ϕ/2. (11) The second way to estimate K started with the rigid sphere embedded in the porous sphere of the first model and used the calculated pressure distribution to find the total force on the rigid sphere, which was ( 4 1 3ϕ πr3 aµv 0 K 1 ). (12) K m Comparing this with the formula derived by Brinkman for the force on a sphere in a porous medium they obtained the relationship 1 K = 1 K m + ( 9ϕ 2ra r ) a + r2 a, (13) (1 ϕ) K 3K which agrees with (11) to first order in ϕ in the limit K r 2 a (which was relevant for their experiments). The authors tested these theoretical results by conducting experiments. They used Matrigel, a material that has similar properties to Bruch s membrane. After the addition of latex nanospheres to the Matrigel, the measured values of the effective hydraulic conductivities agreed well with those predicted by the theory. However, with embedded spheres of LDL instead of latex nanospheres, the effective hydraulic conductivity decreased significantly more than the theory would predict, a phenomenon that has not yet been satisfactorily explained.

34 34 SIGGERS ETHIER 6 DISCUSSION Our aim in writing this article was to show that the eye presents a wealth of interesting and challenging problems in fluid mechanics. Several of these problems have been tackled; however, there remain many outstanding unsolved fluid mechanical problems. We recommend this area to the reader as a source of interesting and accessible research questions that have potential impact on our most important sense. 7 SUMMARY POINTS 1. Mechanics of glaucoma: Resistance to aqueous outflow: The combined resistance of the trabecular meshwork filled with biopolymer together with the inner lining of Schlemm s canal is estimated to be sufficient to be the primary source of resistance to outflow of aqueous humor in health. The observed increase in IOP during glaucoma could be partially due to faulty caliber regulation in Schlemm s canal, but is not due to collapse of the inner wall of Schlemm s canal. Cell death in glaucoma: 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 cells to cause washout of energy-providing ATP in the cells, which could promote cell death and vision loss. 2. Flow and transport in the anterior chamber: Thermal convection is typically the dominant mechanism driving flow

35 FLUID MECHANICS OF THE EYE 35 in the anterior chamber. Position of the iris: The iris deforms as a result of the mechanical forces acting on it, leading to a complicated fluid structure interaction problem, which is relevant for closed angle glaucoma and recovery after a transient deformation of the iris position. Transport of proteins within the anterior eye does not follow the same path as the flow of aqueous humor itself. Fluid transport in the cornea is anisotropic due to the arrangement of fibers within the corneal stroma. 3. Flow and transport in the vitreous chamber due to eye movements: If the vitreous humor is treated as viscoelastic, the chamber is assumed to be spherical and the movements 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 streaming component of flow can be found semi-analytically. Shape of cavity: 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 the primary flow and in the steady streaming. Transient temporal deformations in the shape of the vitreous cavity are likely to have a big effect on the flow and pressure, which is not fully understood. Nonhomogeneous properties of the vitreous can lead to additional stresses. A membrane separating the cavity into two regions could lead to the possibility of resonance at particular frequencies of oscillation.

36 36 SIGGERS ETHIER Alternatively, if the vitreous cavity is occupied by a hemispherical region of elastic solid and a hemispherical region of viscous fluid, with the two parts separated by a membrane, then the stress is particularly high at the points of attachment of the membrane. Mass transport in the vitreous chamber: In the case of liquefied vitreous, mass transport is typically primarily due to convection 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 is significantly affected by both the shape of the chamber and the frequency of oscillation. 4. Impaired transport through Bruch s membrane, thought to be responsible for macular degeneration, can be partially understood by considering a homogenized mathematical model of lipid particles embedded in a membrane. References Allingham RR, de Kater AW, Ethier CR Schlemm s canal and primary open glaucoma: correlation between Schlemm s canal dimensions and outflow facility. Exp. Eye Res. 62: Amini R, Barocas VH Reverse pupillary block slows iris contour recovery from corneoscleral indentation. J. Biomech. Eng. 132: Anderson DR, Hendrickson A Effect of intraocular pressure on rapid axoplasmic transport in monkey optic nerve. Invest. Ophthalmol. Vis. Sci. 13: Balachandran RK, Barocas VH Computer modeling of drug delivery to

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39 FLUID MECHANICS OF THE EYE 39 Fechtner RD, Weinreb RN Mechanisms of optic nerve damage in primary open angle glaucoma. Surv. Ophthalmol. 39:23 42 Fitt AD, Gonzalez G Fluid mechanics of the human eye: Aqueous humour flow in the anterior chamber. B. Math. Biol. 68:53 71 Freddo TF, Bartels SP, Barsotti MF, Kamm RD The source of proteins in the aqueous humor of the normal rabbit. Invest. Ophthalmol. Vis. Sci. 31: Happel J Viscous flow relative to arrays of cylinders. AIChE J. 5: Happel J, Brenner H Low Reynolds number hydrodynamics. The Hague: Martinus Nijhoff Publishers Hedbys BO, Mishima S Flow of water in the corneal stroma. Exp. Eye Res. 1: Heys JJ, Barocas VH. 2002a. A Boussinesq model of natural convection in the human eye and formation of Krunberg s spindle. Ann. Biomed. Eng. 30: Heys JJ, Barocas VH. 2002b. Computational evaluation of the role of accommodation in pigmentary glaucoma. Invest. Ophth. Vis. Sci. 43: Heys JJ, Barocas VH, Taravella MJ Modeling passive mechanical interaction between aqueous humor and iris. J. Biomech. Eng.-T. ASME 123: Huang EC, Barocas VH Active iris mechanics and pupillary block: Steadystate analysis and comparison with anatomical risk factors. Ann. Biomed. Eng. 32: Huang EC, Barocas VH Accommodative microfluctuations and iris contour. J. Vision 6:

40 40 SIGGERS ETHIER Hubbard WC, Johnson M, Gong H, Gabelt BT, Peterson JA, et al Intraocular pressure and outflow facility are unchanged following acute and chronic intracameral chondroitinase ABC and hyaluronidase in monkeys. Exp. Eye Res. 65: Johnson M, Shapiro A, Ethier CR, Kamm RD Modulation of outflow resistance by the pores of the inner wall endothelium. Invest. Ophth. Vis. Sci. 33: Johnson MC, Kamm RD The role of Schlemm s canal in aqueous outflow from the human eye. Invest. Ophth. Vis. Sci. 24: Kiel JW, van Heuven WAJ Ocular perfusion pressure and choroidal blood flow in the rabbit. Invest. Ophthalmol. Vis. Sci. 36: Krey HF, Bräuer H Chibret Augenatlas: Eine Repetition für Ärtze mit Zeigetafeln für Patienten. Munich: Chibret Medical Service Lee B, Litt M, Buchsbaum G Rheology of the vitreous body. Part I: Viscoelasticity of human vitreous. Biorheology 29: Liu D, Wood NB, Witt N, Hughes AD, Thom SA, Xu XY Computational analysis of oxygen transport in the retinal arterial network. Curr. Eye Res. 34: Maurice D Review: Practical issues in intravitreal drug delivery. J. Ocul. Pharmacol. 17: McCarty WJ, Chimento MF, Curcio C, Johnson M Effects of particulates and lipids on the hydraulic conductivity of Matrigel. J. Appl. Physiol. 105:

41 FLUID MECHANICS OF THE EYE 41 McEwen WE Application of Poiseuille s law to aqueous outflow. AMA Archives of Ophthal. 60: Minckler DS, Bunt AH, Johanson GW Orthograde and retrograde axoplasmic transport during acute ocular hypertension in the monkey. Invest. Ophthalmol. Vis. Sci. 16: Morgan JE Optic nerve head structure in glaucoma: astrocytes as mediators of axonal damage. Eye 14: Moses RA Circumferential flow in Schlemm s canal. Am. J. Ophthalmol. 88: Nickerson CS, Park J, Kornfield JA, Karageozian H Rheological properties of the vitreous and the role of hyaluronic acid. J. Biomech. 41: Overby DR, Stamer WD, Johnson M The changing paradigm of outflow resistance generation: towards synergistic models of the JCT and inner wall endothelium. Exp. Eye Res. 88: Pillunat LE, Anderson DR, Knighton RW, Joos KM, Feuer WJ Autoregulation of human optic nerve head circulation in response to increased intraocular pressure. Exp. Eye. Res. 64: Qu J, Wang D, Grosskreutz CL Mechanisms of retinal ganglion cell injury and defense in glaucoma. Exp. Eye Res. 91:48 53 Quigley HA, Broman AT The number of people with glaucoma worldwide in 2010 and Br. J. Ophthalmol. 90: Rayner K Eye movements in reading and information processing: 20 years of research. Psychol. Bull. 124: Reitsamer HA, Kiel JW A rabbit model to study orbital venous pressure,

42 42 SIGGERS ETHIER intraocular pressure, and ocular hemodynamics simultaneously. Exp. Eye Res. 43: Repetto R An analytical model of the dynamics of the liquefied vitreous induced by saccadic eye movements. Meccanica 41: Repetto R, Ghigo I, Seminara G, Ciurlo C A simple hydro-elastic model of the dynamics of a vitreous membrane. J. Fluid Mech. 503:1 14 Repetto R, Siggers JH, Stocchino A Steady streaming within a periodically rotating sphere. J. Fluid Mech. 608:71 80 Repetto R, Siggers JH, Stocchino A Mathematical model of flow in the vitreous humor induced by saccadic eye rotations: effect of geometry. Biomech. Model. Mechan. 9:65 76 Repetto R, Stocchino A, Cafferata C Experimental investigation of vitreous humour motion within a human eye model. Phys. Med. Biol. 50: Repetto R, Tatone A, Testa A, Colangeli E Traction on the retina induced by saccadic eye movements in the presence of posterior vitreous detachment. to appear Biomech. Model. Mechan. Riley N Steady streaming. Annu. Rev. Fluid Mech. 33:43 65 Riva CE, Hero M, Titze P, Petrig B Autoregulation of human optic nerve head blood flow in response to acute changes in ocular perfusion pressure. Graefes Arch. Clin. Exp. 235: Schumer RA, Podos SM The nerve of glaucoma! Arch. Ophthalmol. 112:37 44 Schwartz W, Keyserlingk DG Electron microscopy of normal and opaque human cornea (in The Cornea: Macromolecular Organization of a Connective

43 FLUID MECHANICS OF THE EYE 43 Tissue, Langham ME ed.), chap. 7. Baltimore: The Johns Hopkins Press, Seiler T, Wollensak J The resistance of the trabecular meshwork to aqueous-humor outflow. Graefe s Arch. Clin. Exp. Ophthalmol. 223:88 91 Spielman L, Goren SL Model for predicting pressure drop and filtration efficiency in fibrous media. Environ. Sci. Technol. 2: Stocchino A, Repetto R, Cafferata C Eye rotation induced dynamics of a Newtonian fluid within the vitreous cavity: the effect of the chamber shape. Phys. Med. Biol. 52: Stocchino A, Repetto R, Siggers JH Mixing processes in the vitreous chamber induced by eye rotations. Phys. Med. Biol. 55: Swindle KE, Hamilton PD, Ravi N In situ formation of hydrogels as vitreous substitutes: Viscoelastic comparison to porcine vitreous. J. Biomed. Mater. Res. 87A: Weinbaum S Whitaker Distinguished Lecture: Models to solve mysteries in biomechanics at the cellular level; a new view of fiber matrix layers. Ann. Biomed. Eng. 26: Wygnanski-Jaffe T, Murphy CJ, Smith C, Kubai M, Christopherson P, et al Protective ocular mechanisms in woodpeckers. Eye 21:83 89 Xu J, Heys JJ, Barocas VH, Randolph TW Permeability and diffusion in vitreous humor: Implications for drug delivery. Pharm. Res. 17: Yamamoto T, Kitazawa Y Vascular pathogenesis of normal-tension glaucoma: A possible pathogenetic factor, other than intraocular pressure, of glaucomatous optic neuropathy. Prog. Retin. Eye Res. 17:

44 44 SIGGERS ETHIER Yan DB, Coloma FM, Metheetrairut A, Trope GE, Heathcote JG, Ethier CR Deformation of the lamina cribrosa by elevated intraocular pressure. Brit. J. Ophthalmol. 78: Zimmerman RL In vivo measurements of the viscoelasticity of the human vitreous humor. Biophys. J. 29: KEY TERMS 1. Buoyancy-driven flow: Flow driven by thermal spatial gradients in the fluid. 2. Glaucoma: An ophthalmic condition, usually characterized by raised intraocular pressure, which leads eventually to blindness by death of retinal ganglion cells. 3. Homogenization: A technique used to analyze multiple-scales problems. Quantities are averaged over the small-scale, leading to simplified largescale equations. 4. Intraocular pressure: The fluid pressure of the aqueous humor of the eye. 5. Mass transport: Transport of another species of particles within a fluid. Driven by convection (especially due to steady streaming), diffusion, and uptake/production. 6. Porous medium: A solid material whose small-scale structure is characterized by pores filled with fluid. Flow in such materials is usually governed by the Darcy equation. 7. Steady streaming: The time average of a fluctuating flow, arising due to a nonconservative body force, Reynolds stresses or boundary effects.

45 FLUID MECHANICS OF THE EYE Viscoelastic fluid: A fluid whose stress tensor depends on both the deformation and the rate of deformation of its particles. 9 FUTURE ISSUES 1. Problems in the study of glaucoma: Open angle glaucoma is known to be caused by increased resistance in the outflow pathway of the aqueous humor. However, the exact locations and causes of the change in resistance are not understood. Closed angle glaucoma is due to the iris physically blocking the outflow of aqueous humor. The mechanisms underlying this condition have not been fully resolved. Glaucoma results in the death of retinal ganglion cells and subsequent vision loss. The mechanism of cell death has not been conclusively proven. 2. Problems in modeling the vitreous humor: Characterization of the rheological properties of vitreous humor and incorporation into a model of vitreous flow. Effects of transient deformation of the vitreous cavity on the vitreous pressure and flow, which may be important for understanding the effect of impacts and retinal hemorrhage in shaken baby syndrome. Possible mechanical causes of retinal detachment and strategies for treatment. Drug transport in the vitreous, in particular understanding the timescales involved and locations of delivery.

46 46 SIGGERS ETHIER 3. Transport across Bruch s membrane: the reasons for the increase in resistance to transport when the membrane contains embedded lipids are not fully resolved. 10 ANNOTATED REFERENCES 1. Band et al. (2009): Demonstrates the plausibility of a proposed mechanism for cell death in glaucoma. 2. David et al. (1998): Calculated a closed form solution for the flow of vitreous in a model of eye movements. 3. Ethier (1986): Improved estimates of the resistance of the trabecular meshwork to aqueous outflow using a hierarchical model. 4. Fitt & Gonzalez (2006): Demonstrated that thermal convection is typically the dominant driver of flow in the anterior chamber. 5. Hedbys & Mishima (1962): Developed a new experimental technique and a theoretical model to investigate flow in the cornea. 6. Johnson & Kamm (1983): Developed a mathematical model of flow in and through Schlemm s canal to investigate the source of increased aqueous outflow resistance in glaucoma. 7. Johnson et al. (1992): Developed a model to investigate the combined resistance of the trabecular meshwork and inner wall of Schlemm s canal to the outflow of aqueous. 8. McCarty et al. (2008): Developed a mathematical model to investigate the effect of embedded lipid particles on the transport through Bruch s membrane.

47 FLUID MECHANICS OF THE EYE 47 Figure 1: Overview of a human eye with major anatomical structures identified. Modified from Krey & Bräuer (1998).

48 48 SIGGERS ETHIER (a) (b) Figure 2: Illustration of aqueous humor flow patterns in the anterior chamber and key drainage tissues. (a) Cross-sectional view through anterior eye. The tissue labeled Meshwork, representing the trabecular meshwork, is shown extending exterior to Schlemm s canal. This is somewhat atypical, as the trabecular meshwork is usually only found interior to Schlemm s canal. (b) Anterior posterior view of Schlemm s canal (thick green ring), collector channels (thin green structures), aqueous veins (light blue) and adjacent arterioles. Modified from Krey & Bräuer (1998).

49 FLUID MECHANICS OF THE EYE 49 Figure 3: Normal human cornea of a 62-year-old male patient, showing the regular arrangement of collagen fibers within the corneal stroma. From Schwartz & Keyserlingk (1969).

50 50 SIGGERS ETHIER Figure 4: Cross-section through the retina and choroid within the macula (the part of the retina with the greatest concentration of rod and cone cells). V, vitreous; GCL, ganglion cell layer; INL, inner nuclear layer; ONL, outer nuclear layer; IS/OS, inner and outer segments of photoreceptors; RPE, retinal pigment epithelium; Ch, choroid; asterisk, choriocapillaris; white arrowheads, Bruch s membrane; S, sclera. Bar, 50 mm. From Curcio et al. (2009).

51 FLUID MECHANICS OF THE EYE 51 Figure 5: Normalized pressure contours obtained by numerical simulation of flow in the juxtacanalicular tissue, treated as a porous medium, in the neighborhood of a fenestration (pore) in the inner wall of Schlemm s canal. The length scales are normalized by the pore radius. The right-hand inset illustrates the set up considered in the model. From Johnson et al. (1992). (a) (b) Figure 6: Schematic diagrams of the model for flow entering and within Schlemm s canal developed by Johnson & Kamm (1983). (a) Original model; (b) model with septae included.

52 52 SIGGERS ETHIER Sclera Cell bodies R p (z;r) Site of lamina cribrosa z = M PNIFF p + (r,z) r 2a Axis of symmetry z Orthograde AAT Retrograde AAT Synapse z = L Optic nerve head Intraocular space, p e CSF, p c Figure 7: Mathematical model to analyze flow in retinal ganglion cells: CSF cerebrospinal fluid; AAT active axonal transport; PNIFF passive neuronal intracellular fluid flux. From Band et al. (2009) Figure 8: Sketch of the model developed by Canning et al. (2002) to investigate flow in the anterior chamber.