1

Fluid motion in the posterior chamber of the eyeEnrico Bagnoli

Abstract

The human eye is the organ which gives us the sense of sight, allowing us to observe and learn moreabout the surrounding world than with any of the other four senses. The complete or partial loss of visionis a considerable handicap in a person’s life. The main cause of blindness in the world is glaucoma, adisease related to intraocular pressure gradient. Therefore it is essential to understand the physiological andpathological mechanisms that lead to changes in intraocular pressure. Among these one of the most importantis the flow of aqueous humor created in the posterior chamber through the pupil and finally to the anteriorchamber. In this work in particular is analyzed the pressure field and the flow of aqueous humor within theposterior chamber under various conditions.

I. THE EYE

The eye is not shaped like a perfect sphere, ratherit is a fused two-piece unit. The smaller frontal unit,more curved, called the cornea is linked to the largerunit called the sclera. The corneal segment is typicallyabout 8 mm in radius. The sclerotic chamber constitutesthe remaining five-sixths; its radius is typically about 12mm. The cornea and sclera are connected by a ring calledthe limbus. The iris, "the color of the eye", and its blackcenter, the pupil, are seen instead of the cornea due tothe cornea’s transparency. The fundus (area opposite thepupil) shows the characteristic pale optic disk, papilla,where vessels entering the eye pass across and opticnerve fibers depart the globe.

The eye is made up of three coats, enclosing threetransparent structures. The outermost layer, known as thefibrous tunic, is composed of the cornea and sclera. Themiddle layer, known as the vascular tunic or uvea, con-sists of the choroid, ciliary body, and iris. The innermostis the retina, which gets its circulation from the vesselsof the choroid as well as the retinal vessels, which canbe seen in an ophthalmoscope. Within these coats are theaqueous humour, the vitreous body, and the flexible lens.The aqueous humour is a clear fluid that is contained intwo areas: the anterior chamber between the cornea andthe iris, and the posterior chamber between the iris andthe lens. The lens is suspended to the ciliary body bythe suspensory ligament, Zonule of Zinn, made up offine transparent fibers. The vitreous humour is the cleargel that fills the space between the lens and the retina ofthe eyeball. It is often referred to as the vitreous bodyor simply "the vitreous".

II. THE POSTERIOR CHAMBER

The posterior chamber is a narrow chink behind theperipheral part of the iris of the lens, and in front

Figure 1. Anatomical structure of the eye, showing the position ofthe posterior chamber. The dimensions differ among adults by onlyone or two millimeters; it is remarkably consistent across differentethnicities. The vertical measure, generally less than the horizontaldistance, is about 24 mm among adults. The typical adult eye hasan anterior to posterior diameter of 24 millimeters, a volume of sixcubic centimeters and a mass of 7.5 grams.

of the suspensory ligament of the lens and the ciliaryprocesses. It consists of small space directly posterior tothe iris but anterior to the lens. The posterior chamberhas an approximately spherical shape, and is filled withaqueous humor, a transparent material with viscoelasticproperties. Besides providing an unhindered path forlight to reach the retina, the vitreous has the importantmechanical roles of supporting the eye shape, promotingthe adherence between the retina and the choroid (thevascular layer between the retina and the sclera), and

2IRIS STRUCTURE, AQUEOUS FLOW, AND GLAUCOMA RISK

JOHNS HOPKINS APL TECHNICAL DIGEST, VOLUME 28, NUMBER 3 2010 225

Figure 2. Blue arrows indicate the flow of aqueous fluid. (Image courtesy of the National Eye Institute, National Institutes of Health.)

Figure 3. Bright light causes pupil constriction, accompanied by a thinner iris and an open angle.

For further information on the work reported here, see the references below or contact david.m.silver@jhuapl.edu.

1Silver, D. M., and Quigley, H. A., “Aqueous Flow Through the Iris-Lens Channel: Estimates of Differential Pressure Between the Anterior and Posterior Chambers,” J. Glaucoma 13, 100–107 (2004).

2Quigley, H. A., Silver, D. M., Friedman, D. S., He, M., Plyler, R. J., Eberhart, C. G., Jampel, H. D., and Ramulu, P., “Iris Cross-Sectional Area Decreases with Pupil Dilation and Its Dynamic Behavior Is a Risk Factor in Angle Closure,” J. Glaucoma 18, 173–179 (2009).

Cornea

Open angle

Thinner iris

Lens

We note that tonometry, the standard measurement of intraocular pressure, measures the anterior chamber pressure but does not estimate the true pressure in the posterior chamber and vitreous cavity. Hence, the oper-ative pressure at the site of glaucoma interest, the retina and optic nerve head, may be incorrectly estimated in some eyes with narrow channel heights, thereby obscur-ing a risk factor.

Our iris model predicts that changes in pupil size affect the geometry of the iris–lens channel and the

Cornea

Blocked angle

Thicker iris

Lens

Figure 4. Darkness causes pupil dilation, accompanied by a thicker iris and a blocked angle.

anterior chamber angle. For instance, if the volume of the iris is conserved upon dilation, the iris can be expected to increase up to 50% in thickness, causing the iris to block the anterior chamber angle. Blocking the angle would impede aqueous flow and lead to higher intraocular pressure. Therefore, the iris must act like a sponge and ultimately lose volume upon dilation to maintain healthy aqueous flow.

We used ultrasound biomicroscopy to image the eye in conditions of bright light (Fig. 3) and darkness (Fig. 4). Upon dilation in darkness, we found immedi-ate iris thickening with anterior chamber narrowing, as predicted. In further work, we used anterior seg-ment ocular coherence tomography and found that, for normal healthy eyes, the iris loses volume within ~10 s after dilation by releasing fluid, thereby returning the iris to a thinner conformation and opening the exit angle. However, we found that eyes with angle-closure glau-coma do this much more slowly.2 It turns out that those with acute attacks of angle-closure glaucoma lose no iris volume upon pupil dilation.

Our research collaboration has uncovered an impor-tant new concept for the field of angle-closure glaucoma. Pupil dilation is accompanied by iris thickening and angle closure, giving rise to constraints on aqueous flow and changes in intraocular pressure. For healthy aque-ous flow, the iris needs to lose volume upon pupil dila-tion. This discovery translates into an additional new risk factor in angle-closure glaucoma damage and, in the future, may lead to a predictive, noninvasive test.

Figure 2. The flow of the aqueous humor, produced by the ciliaryprocesses, throughout the pupil from the posterior to the anteriorchamber. It finally leaves the eye through the intricate system ofoutflow channels located in the corneoscleral limbus.

also acting as a diffusion barrier between the anteriorand posterior segments of the eye. The aqueous humoris secreted by the ciliary bodies in the posterior chamber,which is bound anteriorly by the pigment epitheliumof the posterior iris; anterolaterally by the junctionalzone of the iris and ciliary body; and anteromediallyby the contact of the iris with the lens. Aqueous humorgains entry into the anterior chamber from the posteriorchamber via pupil. The equatorial portion of the lensforms the medial boundary of the posterior chamber.Posteriorly, the anterior face of the vitreous limits it.Laterally, the chamber is bounded by the ciliary bodywith its processes and valleys, and it may extend backto the point of contact between the anterior face of thevitreous.

In some cases the posterior chamber may be filledwith a fluid with almost Newtonian properties. This mayhappen after vitrectomy (removal of the vitreous humor)and refilling of the posterior chamber with silicone oil,which is an increasingly routine surgical procedure.Alternatively, during the aging process, the vitreousoften undergoes a liquefaction process called synchysis,whereby it progressively loses its elastic properties.

III. GLAUCOMA

Glaucoma is a condition that results in slow progres-sive damage to the optic nerve. Damage to the opticnerve leads to a slow loss of vision. Risk factors forglaucoma include elevated eye pressure, increased age,and previous ocular injury. The most important andmost treatable risk factor for glaucoma is elevated eyepressure. Inside the eye, there is a constant production offluid that normally flows out of the eye through a verysmall drain. In certain individuals, this drain can become

Figure 3. The two different types of glaucoma.

blocked for various reasons. The result is an increase ineye pressure, therefore increasing your risk of glaucoma.Glaucoma is a leading cause of blindness in the world,especially in older people. Early detection and treatmentby your ophthalmologist are the keys to preventing opticnerve damage and vision loss from glaucoma. There aretwo different types of glaucoma:

• Open-angle glaucoma: occurs slowly as thedrainage area in the eye becomes clogged. Pressurebuilds up when the fluid inside the eye is unable todrain. Side (peripheral) vision is damaged gradually.Open-angle glaucoma is the most common kind ofglaucoma.

• Closed-angle glaucoma: eye pressure builds uprapidly when the drainage area of the eye suddenlybecomes blocked. Blurry vision, rainbow halosaround lights, headaches or severe pain may occurwith closed-angle glaucoma. This type of glaucomais less common than open-angle and may causeblindness, if it is not treated immediately. Thisform of glaucoma occurs more frequently in peopleof African and Asian ancestry, and in certain eyeconditions.

Student projects Project 1: Fluid motion in the posterior chamber of the eye

1. A steady flow model based on the lubrication theory I

Modelling assumptions

Newtonian fluid.

Rigid iris.

Steady and axisymmetric flow.

Thin domain. Let h(0) = h0 be the thickness of the domain at r = Ro . This assumptionimplies ϵ = h0/(Ro − Ri ) ≪ 1.

We neglect the flow in the region of the pupil.

Rodolfo Repetto (University of Genoa) Biofluid dynamics Academic year 2013/2014 300 / 345Figure 4. Schematic model for the posterior chamber in cylindricalcoordinates.

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IV. MODEL AND METHODS

To construct the model of the flow of aqueous humour in the posterior chamber of the eye the following assumptionswill be considered:

• The aqueous is modelled as a Newtonian fluid;• Rigid iris;• Steady and axisymmetric flow;• Thin domain. Let h(0) = h0 be the thickness of the domain at r = R0. This assumption implies ε = h0

R0−Ri�

1;• Neglect the flow in the region of the pupil.

Other considerations could be made:• Geometry of the domain: Ri < r < R0

• Use of cylindrical coordinates (z, r, ϕ) with corresponding velocity components (uz, ur, uϕ)• According to the geometry of the system, there is no motion in ϕ direction and also the velocity is not a

function of ϕ. So uϕ = 0 and ∂uϕ

∂ϕ = 0.First of all we consider the Navier-Stokes equations in cylindrical coordinates (1), (2), (3) and the ContinuityEquation (4):

∂uz∂t

+ uz∂uz∂z

+ ur∂uz∂r

+∂uϕr

∂uz∂ϕ

+1

ρ

∂p

∂z− ν(∂

2uz∂z2

+1

r

∂

∂r(r∂uz∂r

) +1

r2

∂2uz∂ϕ2

) = 0 (1)

∂ur∂t

+ uz∂ur∂z

+ ur∂ur∂r

+∂uϕr

∂ur∂ϕ−u2ϕ

r+

1

ρ

∂p

∂r− ν(∂

2ur∂z2

+1

r

∂

∂r(r∂ur∂r

) +1

r2

∂2ur∂ϕ2

− urr2− 2

r2

∂uϕ∂ϕ

) = 0 (2)

∂uϕ∂t

+ uz∂uϕ∂z

+ ur∂uϕ∂r

+∂uϕr

∂uϕ∂ϕ

+uruϕr

+1

rρ

∂p

∂ϕ− ν(∂

2uϕ∂z2

+1

r

∂

∂r(r∂uϕ∂r

) +1

r2

∂2uϕ∂ϕ2

+2

r2

∂ur∂ϕ− uϕr2

) = 0

(3)∂uz∂z

+1

r

∂

∂r(rur) +

1

r

∂uϕ∂ϕ

= 0 (4)

Since uϕ = 0 and ∂uϕ

∂ϕ = 0 the equation (3) is all zero. The other equations can be greatly simplified by usingthe so-called lubrication theory. This technique provides a good approximation to the real solution as long as thedomain of the fluid is long and thin, i.e. when the domain has two very different spatial scales. In this case thespatial domain is much longer than thick. Assuming L = Ri − R0 and h = h0 for the posterior chamber of the

eye we have that ε =h

L� 1. Can be useful work with dimensionless variables and scaling as follows:

r∗ =r

Lz∗ =

z

h0u∗z =

uzUz

u∗r =urUr

(5a, 5b, 5c, 5d)

Equation (5c) and (5d) imply that:Uzh0

vUrL⇒ Uz = εUr (6)

A reasonable measure of Ur is given by:

Ur =F

2πRh0(7)

where F is the aqueous humour flow produced by the ciliary processes. A typical value for F in human eye isF = 2.5µl/min = 4.2 · 1011m3/s; It is also very convenient to use a pressure scale that balances the gradientpressure term with that viscous. In other terms

P =ρνUrL

h20

(8)

Introducing now the Reynolds Number Re = UrLν the equation (1), (2), (4) take the form:

Reε4u∗z∂u∗z∂z∗

+∂u∗z∂r∗

ε4(Reu∗r −1

r∗) +

∂p∗

∂z∗− ε2∂

2u∗z∂z∗2

− ε4∂2u∗z∂r∗2

= 0 (9)

Reε2u∗z∂u∗r∂z∗

+∂u∗r∂r∗

ε2(Reu∗r −1

r∗) +

∂p∗

∂r∗− ∂2u∗r∂z∗2

− ε2∂2u∗r∂r∗2

− ε2 u∗r

r∗2= 0 (10)

∂u∗z∂z∗

+u∗rr∗

+∂u∗r∂r∗

= 0 (11)

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Considering ε � 1 the first two previous equation can be further simplified neglecting the terms of order ε2, ε4,and assuming also Reε� 1, neglecting Reε2, Reε4 :

∂p∗

∂z∗= 0 (12)

∂p∗

∂r∗− ∂2u∗r∂z∗2

= 0 (13)

These equations have to satisfy the following boundary conditions:ur = uz = 0 in z = 0

ur = uz = 0 in z = h(r)

p = p0 in r = R0

F = −∫ h0

0 urdz = Urh0 in r = Ri

(14)

The first two boundary equations are the conditions of adhesion of the fluid to the extreme of the domain. Asregards the other two they are required for the final solution . In fact, in order to calculate the trend of pressure,the initial pressure p0 and the flow F must be known. In dimensionless terms we can write:

u∗r = u∗z = 0 in z∗ = 0

u∗r = u∗z = 0 in z∗ = h∗(r∗)

p∗ = p0h20

ρνUrLin r∗ = R0

L

F ∗ = −∫ 1

0 u∗rdz

∗ = 1 in r∗ = Ri

L

(15)

From equation (12) we obtain p∗ = p∗(r) whereas integrating twice equation (13) with respect to z∗ we obtain:

u∗r =1

2

∂p∗

∂r∗z∗2 + c1z

∗ + c2 (16)

Applying the boundary conditions we can find c1 and c2 and the above equations reads:

u∗r = −1

2

∂p∗

∂r∗(z∗2 − z∗h∗) (17)

Integrating now the continuity equation from 0 to h∗(r∗) with respect to z∗ we obtain:

− 1

12

∂

∂r∗(r∗

∂p∗

∂r∗h∗3) = 0 (18)

Going forward and knowing that ∂p∗

∂r∗ = 12, which derives from∫ 1

0 u∗rdz

∗ = −1, it results:

∂p∗

∂r∗=

12

h∗3r∗⇒ p∗ =

∫ R1

R0

12

h∗3r∗dr∗ (19)

Substituting the above expression into equation (17) we obtain the velocity distribution:

u∗r = −6

h∗3r∗(z∗2 − z∗h∗) (20)

Note that up to this point no assumptions were made for the thickness of the posterior chamber h∗(r∗). If it isknown analytically then also the pressure and velocity are uniquely determined. For simplicity we consider:

1) h∗ = h∗0 = 1 = constant and so a straight profile of the chamber

2) h∗ = 1 + a(r∗ − 1) where a =1− h∗i1−R∗

i

and so a linear profile

That lead to the following pressure distibution:

1) p∗ =12

h∗3ln r∗ + costant

2) p∗ =6

(a− 1)3

(− (a− 1)2

(a(r − 1) + 1)2+

2(a− 1)

(a(r − 1) + 1)+ 2 log(

(a(r − 1) + 1)

ar)

)One last interesting case of study concerns the analysis of pressure and velocity during the movements of the pupil.This implies that h also depends on time: h = h(t, r). In particular the pupil compresses or expands the spatial

domain, acting with a velocity v along the z axis. Considering T as the characteristic time scale we assume: v =h0

T.

So by imposing the continuity of flows vL v Urh0 ⇒ v = εU and scaling as previously we have respectively overz, r :

5

Reε4L∂u∗z∂t∗

+Reε3h0u∗z

∂u∗z∂z∗

+Reε4Lu∗r∂u∗z∂r∗

+Ph2

0

Urµ

∂p∗

∂z∗− ε3Ur

∂2u∗z∂z∗2

= 0 (21)

Reε2∂u∗r

∂t∗+Reε3Lu∗z

∂u∗r∂z∗

+Reε2u∗r∂u∗r∂r∗

+ ε2 PL

Urµ

∂p∗

∂r∗− ∂2u∗r∂z∗2

= 0 (22)

Again the motion can be considered stationary if Reε� 1 and the equation (21), (22) can be simplified considering

P =ρνUrL

h20

getting the equation (13) and (14) as the previous case. However what changes are the boundaries

condition, we have:

u∗r = u∗z = 0 in z∗ = 0

u∗z =uzv

= 1 in z∗ = h∗(r∗)

p∗ =p0h

20

ρνUrLin r∗ = R0

L

F ∗ = −∫ 1

0 u∗rdz

∗ = 1 in r∗ = Ri

L

(23)

The result is a system very similar to that previously studied, but the velocity uz cannot be neglected, having alinear trend uz = v

h(r)z. Imposing the flux continuity we obtain the relationship between uz and ur. The final resultimplies an increase of the velocity and pressure than when the pupil is at rest. This increase is dependent by thevalue of v.

Figure 5. Pressure distribution for h∗ = constant.

V. RESULTS

The model has been implemented with the followingdimensionale values:

• h0 = 0, 7mm = 7 · 10−4m, thickness of theposterior chamber;

• h(Ri) = 5µm = 5 · 10−6m, height of iris-lens-channel;

• Ri = 2mm = 2 · 10−3m, radius of pupil aperture;• R0 = 4, 5mm = 4, 5 · 10−3m, radius of posterior

chamber;• F = 2, 5µl/min = 4, 2 · 10−11m3/s, aqueous

humour production;

Figure 6. Pressure distribution for h∗ = 1 + a(r∗ − 1)

• IOP = 15mmHg.

The results obtained with the two assumptions on h∗

are very different. The pressure distribution of the twomodel adopted are shown in Fig.5 and Fig.6. While in thefirst case also the trend of the pressure is almost linear,in the other one we notice a hyperbolic course and aconsiderable rise in pressure values. However variationsin pressure are far to be compared with the intraocularpressure IOP = 15mmHg, infact introducing the ratio∆pIOP it results almost always < 0, 06%. The maximumvalue of pressure occurs at r = r0 and has been set top(r0) = 0 in both cases. Infact, acqueous humor goes

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Figure 7. Flow distribution for h∗ = 1 + a(r∗ − 1). The result isq = 1

r

Figure 8. Flow distribution for h∗ = 1 + a(r∗ − 1). The result isq = 3h∗−2

rh∗3

from the ciliary processes to the anterior chamber wherethe pressure has its minimum. The values shown arenegative because we consider relative and not absolutechanges in pressure. To compute the absolute pressure aconstant must be added to our results.

We can also define the quantity:

q∗ =∫ h∗

0

0u∗rdz (24)

that represents the flow of aqueous humor related to thechange of r∗.

The differences between the two models are clear fromthe graphs. The first one shows a logarithmic trend ofpressure. Both flux and pressure variations are smoothand uniform along the radius of the chamber. Changingthe assumptions on h(R) there is a significant increase ineither pressure and flux. Pression enhances with a factorof 104 while flux has a gain factor of 107. Moreover thevariations take place in a much smaller spatial domainin proximity of Ri. The course of pressure is hyperbolicand is then reflected in the flux. Results are plausibleand consistent with that reported by other studies.

VI. CONCLUSION

Knowing the pressure distribution is very important instudying the glaucoma. In the severe glaucoma (closed-angle glaucoma), eye pressure builds up rapidly when thedrainage area (trabecular meshwork) suddenly becomesblocked, in this case the high amount of pressure in theposterior chamber results in increase of the fluid pressurewithin the inner eye, which can damage the optic nerveand lead to vision loss.

Pupil dilation is accompanied by iris thickening andangle closure, giving rise to constraints on aqueous flowand changes in intraocular pressure. For healthy aqueousflow, the iris needs to lose volume upon pupil dilation.

REFERENCES

[1] CANNING, C., GREANEY, M., DEWYNNE, J., AND FITT, A.Fluid flow in the anterior chamber of a human eye. MathematicalMedicine and Biology 19, 1 (2002), 31–60.

[2] FITT, A., AND GONZALEZ, G. Fluid mechanics of the humaneye: aqueous humour flow in the anterior chamber. Bulletin ofmathematical biology 68, 1 (2006), 53–71.

[3] REPETTO, R. Notes on Biofluid Dynamics, 2014.[4] SILVER, D., AND QUIGLEY, H. Iris structure, aqueous flow, and

glaucoma risk. Johns Hopkins APL technical digest 28, 3 (2010),224.

[5] STOCCHINO, A., REPETTO, R., AND SIGGERS, J. H. Mixingprocesses in the vitreous chamber induced by eye rotations.Physics in medicine and biology 55, 2 (2010), 453.

[6] WIKIPEDIA. Human eye.