Download A Boussinesq Model of Natural Convection in the Human Eye and

A Boussinesq Model of Natural Convection in the
Human Eye and the Formation of Krukenberg's Spindle
Jeffrey J. Heys1 and Victor H. Barocas2*
1
Department of Chemical Engineering
University of Colorado, Boulder, CO 80309-0424
2
Deparment of Biomedical Engineering
University of Minnesota, Minneapolis, MN 55455
*corresponding author (phone:612-626-5572, fax:612-626-6583, email:[email protected])
2
Abstract
The cornea of the human eye is cooled by the surrounding air and by evaporation of the tear
film. The temperature difference between the cornea and the iris (at core body temperature),
causes circulation of the aqueous humor in the anterior chamber of the eye. Others have
suggested that the circulation pattern governs the shape of the Krukenberg spindle, a distinctive
vertical band of pigment on the posterior cornea surface in some pathologies. We modeled
aqueous humor flow the human eye, treating the humor as a Boussinesq fluid and setting the
corneal temperature based on infrared surface temperature measurements. The model predicts
convection currents in the anterior chamber with velocities comparable to those resulting from
forced flow through the gap between the iris and lens. When paths of pigment particles are
calculated based on the predicted flow field, the particles circulate throughout the anterior
chamber but tend to be near the vertical centerline of the eye for a greatest period of time.
Further, the particles are usually in close proximity to the cornea only when they are near the
vertical centerline. We conclude that the convective flow pattern of aqueous humor is
consistent with a vertical pigment spindle.
3
Introduction
Loss of pigment from the iris occurs in many pathological conditions (e.g., pigmentary
glaucoma) and in all irises with advancing age18. Liberated pigment is carried by the aqueous
humor circulating through the posterior and anterior chambers and is deposited on the iris, lens,
and posterior surface of the cornea. The pigment deposited on the posterior cornea surface is
in the form of a vertical line called Krukenberg’s spindle (Figure 1). Damage to the cornea may
occur as a result of heavy pigmentation of its endothelial surface, including blockage of the
corneal dehydration mechanism leading to edema2.
The prevailing hypothesis18 is that thermal convection currents of aqueous humor cause
the pigment to be deposited in the characteristic spindle pattern. The cornea is cooled by the
surrounding air and by evaporation of the tear film to a temperature less than 37oC17. The
cooling of the cornea creates a temperature gradient in the eye, which causes thermal
convection currents in the anterior chamber that are hypothesized to be responsible for the
formation of a spindle of pigment. There are, however, no experimental or theoretical studies of
the aqueous humor that quantify the supposed convection currents in the anterior eye, nor is
there any evidence that natural convection currents could cause the formation of the spindle.
The objective of this work is to develop a model of the anterior eye that can predict the
magnitude of the natural convection flow of the aqueous humor and use that flow field to predict
the movement of pigment particles in the anterior chamber of the human eye.
A number of models of heat transport in the eye have been proposed, motivated by the
development of cataracts after exposure of the eye to infrared radiation. Early models focused
on predicting the temperature rise of the lens due to infrared radiation, and the models
consequently contained assumptions that prohibit their application to the current problem of
interest. For example, Al-Badwaihy and Youssef1 assumed that the eye was spherical and
uniform, allowing an analytical solution to the 1-D heat transfer problem. A more recent model23
assumes the eye is axisymmetric with different material properties for the lens, cornea, aqueous
humor, and vitreous. The latter model assumes that the aqueous humor is stagnant, ignoring
the effects of natural convection, and it neglects the effect of warm blood circulating through the
iris.
Despite the differences between them, previous models of heat transport in the eye have
predicted similar temperature change across the anterior chamber. The models of Emery et al.5
and Lagendijk11 both predicted a temperature difference of approximately 1.5o C across the
anterior chamber. The model of Scott23 predicted a 2oC temperature drop across the anterior
chamber under normal conditions.
4
Model
Although the eye is symmetric about its central axis, the action of gravity breaks the
symmetry and requires a three-dimensional model.
Figure 2 shows a slice of the model
domain, which is bounded by the cornea, iris, lens, ciliary bodies, vitreous, and trabecular
meshwork. The dimensions of the anterior eye and the position of the iris are based on
ultrasound
measurements 6 .
The aqueous humor is modeled using the Boussinesq
approximation, in which the effect of temperature on density enters into the momentum balance
but not the mass balance. Under the Boussinesq approximation, the following steady-state
equations arise:
ρv ⋅ ∇v = −∇p + µ∇2v + ρgβ( T − Tref )
(1)
∇⋅v = 0
(2)
ρCPv ⋅ ∇T = k∇2 T
(3)
where v is the velocity, p is pressure, ρ is the density, µ is the viscosity, β is the volumetric
expansion coefficient, g is gravitational acceleration, k is the thermal conductivity, Tref is a
reference temperature (the corneal surface temperature in this case), and C p is the heat
capacity. Table 1 summarizes the values for the physical parameters used in the model.
Equations (1, 2) are the Navier-Stokes equations for incompressible fluid flow with an
additional term accounting for the thermal change in fluid density. Equation (3) describes the
convective and diffusive transport of energy by the aqueous humor.
dimensionless quantities are defined,
ρ 2CPβg( Tiris − Tref )b3
Ra ≡
µk
Pr ≡
CPµ
k
x* ≡ x / b
T* ≡
( T − Tref )
( Tiris − Tref )
The following
5
v* ≡ v / vref where vref ≡
p* ≡
k Ra Pr
ρCPb
p ⋅b
µ vref
allowing equations (1-3) to be simplified to the following form:
Ra
Ra
v * ⋅∇v* = −∇p * +∇2v * +
T*
Pr
Pr
∇ ⋅ v* = 0
Ra Pr v * ⋅∇T* = ∇2 T *
(4)
(5)
(6)
The Prandtl number (Pr) is 4.7 based on the data in Table 1, and the Rayleigh (Ra) number,
which is a function of the anterior cornea surface temperature, Tref, is 87 when the anterior
cornea surface is assumed to be 34oC.
Boundary Conditions
The lens, anterior vitreous surface, and iris are modeled as stationary, and the velocity is
set to zero along the surface (i.e., no slip). The temperature of these surfaces is set to 37oC
based on the assumption that blood flow through the iris and sclera is sufficient to maintain the
iris, vitreous, and lens at the core body temperature. Previous models of heat transport in the
eye have frequently simplified the geometry by assuming the iris did not exist or by assuming
that there was negligible blood flow in the iris, although no justification for either assumption was
given. One notable exception is the work of Patil20, who assumed the iris was at core body
temperature during studies of drug-muscle interaction.
The ciliary bodies and the trabecular meshwork represent the source and outflow route
respectively for aqueous humor. Along both boundaries the velocity tangential to the surface is
set to zero, and the velocity normal to the surface is specified so that the flow is parabolic and
the total flow is equal to 2.4 µL/min
24.
The aqueous humor excreted into the eye by the ciliary
bodies is assumed to be at 37oC, and the trabecular meshwork is assumed to be at 37oC based
on the blood flow in the nearby choroid, sclera, and ciliary bodies19.
6
The cornea is modeled as a stationary tissue, so the velocity is set to zero along the
posterior cornea surface. Previous models of heat transport in the eye have modeled the
cornea by estimating a convective heat transfer coefficient and an evaporation rate for the
surface of the cornea. Since accurate experiments to measure the convective heat transfer
coefficient have not been conducted, their use in the proposed model would introduce
unacceptable variability into a critical boundary condition. Therefore, the cornea is modeled by
using experimental measurements of the temperature of the anterior cornea surface available in
the literature. We used the average (34oC) of the following measurements: 35.4oC16, 34.3oC4,
33oC10, 31.9oC14, 34.8oC12, and 33.9oC15. We also considered a corneal surface temperature of
30oC to examine the potential effects of living in a cool and dry climate.
Once the anterior surface temperature is known, the heat flux across the cornea can be
calculated as a function of posterior surface temperature. The cornea is assumed to be a solid
with a thermal conductivity close to that of water (see Table 1). There is a small flow from the
cornea into the anterior chamber, but this flow is sufficiently slow (approximately 0.1 µL/min13,
implying Pé = 1x10-4) that it can be neglected. It is assumed that the cornea is thin enough that
tangential heat transport is negligible compared to radial heat transport. The cornea is also
assumed to be a section of a spherical solid shell, giving the following flux equation:
qcornea =
k ( Tp − Ta )
 r 
rp 1 − p 
 ra 
(7)
where q is the heat flux (in kg/s3) and the subscripts p and a refer to the posterior and anterior
surfaces respectively. The radius of curvature for the cornea is 8.3 mm 6, and the thickness is
0.5 mm 9. Defining the dimensionless heat flux as:
q* ≡ q / qref
where
qref ≡
k ( T2 − Tref )
b
and substituting in the values for qref, Ta, ra and rp, gives:
q *cornea = −2.06 ⋅ T *cornea
where T*cornea is the dimensionless temperature of the posterior cornea surface at a given point.
(8)
7
Particle Motion
The movement of pigment particles within the aqueous humor is modeled by assuming
the particles are independent. For a 1-µm diameter particle21 in water, the Stokes-Einstein
equation gives an estimated diffusion coefficient of 4.5 x 10-9 cm2/s. Combining this result with
a characteristic velocity of 0.01 cm/s (see results section) and a length scale of 1 µm, we
calculate a particle Péclet number of 222, indicating that diffusion of the particle is negligible.
Thus, we analyze the particles in terms of classical Newtonian mechanics. For the pigment
particles in the anterior chamber, the two forces under consideration are the drag force exerted
by the aqueous humor and the net gravitational force if the density of the particle is sufficiently
different from that of the aqueous humor. Under creeping flow conditions, the drag force on a
sphere is 3πµDp(vAH - vP), where Dp is the particle diameter, vP is the particle velocity, and vAH is
the aqueous humor velocity. The gravitational force is πDp3(ρp-ρAH)g/6, where ρP is the particle
density, ρAH is the aqueous humor density, and g is the gravitational constant. The ratio of the
gravitational force to the drag force is equal to gDp2(ρp-ρAH)/(18µv0 ).
Based on a particle
diameter of 1.0 µm and a characteristic aqueous humor velocity of 0.01 cm/s, we calculate that
the pigment particle specific gravity would have to be greater than 130 for the gravitation force
to equal the drag force. Since the particles are highly hydrated, the density is not significantly
different from that of water. Therefore, only the drag force on the particle will be considered in
determining particle motion. The particle motion is described by the equation:
ρP
dvP 18µ
= 2 vAH − vP
dt
DP
(
)
(9)
This equation was solved separately after equations (4-6) had been solved by assuming the
particle phase is dilute and the interfacial forces between the pigment particles and the aqueous
humor are negligible.
Solution Method
Equations (4-6, 9) were solved using the FIDAP computer package (Fluent Incorporated;
Lebanon, NH). The discrete problem was formulated using the finite element method and
dividing the domain into 27-node brick elements.
On each element, the velocity and
temperature were approximated using triquadratic interpolation functions, and the pressure was
approximated using continuous trilinear interpolation functions.
The large matrix problem
resulting from the finite element formulation was solved using a segregated algorithm and
8
Gaussian elimination. Equation (9) was subsequently solved using an implicit time stepping
algorithm with a variable time step.
Results
The first set of results addresses the question of whether significant thermally-driven
natural convection exists within the anterior chamber. These results were obtained by solving
equations (4-6) over the 3-dimensional domain of interest. Figure 3 shows the velocity profile of
the aqueous humor in the anterior and posterior chambers for a sagittal plane through the
central axis of the eye. Gravity causes the cooler, more dense fluid near the cornea to move
downward, while the warmer fluid near the iris rises. A significant vertical circulation is thus
established throughout the anterior chamber. The velocities generated by natural convection
are the same order of magnitude as those caused by forced flow from the posterior chamber
into the anterior chamber. If the average temperature of the anterior cornea surface is reduced
from 34oC to 30oC, the flow profile is nearly identical, but the velocities due to natural convection
increase by approximately 50%.
Figure 4 shows temperature contours in the same plane as shown in Figure 3. The total
temperature drop between the cornea and lens is approximately 2oC, but the natural convection
currents cause the posterior cornea temperature to vary. If there were no natural convection in
the eye, the temperature contours would be symmetric, as they are even with natural convection
if the temperature contours in the horizontal plane passing through the central axis are plotted.
When a particle is placed in the velocity field shown in Figure 3, the particle circulates
throughout the anterior chamber with the fluid. Figure 5 shows the path of two different particles
circulating with the aqueous humor for 500 sec. and starting in the vertical plane passing
through the central axis of the eye. Because of the symmetry of the eye, these particles remain
in the vertical plane allowing their motion to be examined by using a simpler 2-dimensional plot.
One of the particles is placed near the tip of the iris. The particle circulates through the anterior
chamber and eventually comes close to the iris anterior iris surface where the velocity is nearly
zero (due to the no slip condition on the iris surface). The second particle is introduced closer to
the center of the eye and circulates in the anterior chamber with increasing circulation diameter,
until this particle also comes in close proximity to the anterior iris surface.
Figure 6 shows 300-sec paths of two particles that were not introduced in the same
sagittal (vertical) plane as Figure 5. The paths of these particles, which are typical of other
particles released near the inner edge of the iris, are very complex. In most cases examined,
the particles did not come sufficiently close to a surface for the surface to have a significant
9
effect on the velocity of the particle, even when particles were tracked for over 2 hrs. In a few
cases, the particles came close to the anterior iris surface and only traveled a short distance
during the 2 hrs. that their paths were predicted. Most particles remained in the anterior
chamber for the 2 hrs simulated, which is not surprising since the residence time of aqueous
humor is nearly 2 hrs. A few particles, however, escaped in as few as 40 minutes.
Figure 7 shows the path of a single particle for 33 minutes. While the path is very
complex, for the majority of the time interval the particle is circulating near the vertical plane
through the center of anterior chamber. This particle path is typical for particles not released in
the central vertical plane. To quantify the position of particles during the time they circulate in
the anterior chamber, we constructed a plot showing the time spent by particles as a function of
position in the coronal plane (Figure 8a). Thirty-six particles were released from points evenly
spaced around the inner circumference of the iris tip (indicated by the inner circle) and tracked
for 2 hrs. The two dark bands in the center of the eye represent the location where the particles
circulated for the longest time. Two outer bands represent an additional region occupied for a
significant time. Figure 8b is obtained from the same data as Figure 8a, except that in this case
only the time when the particles were within 200µm of the cornea was included.
Figure 9 is two plots that are similar to Figure 8, except the two plots in Figure 9 were
generated by assuming the average temperature of the cornea surface was 30oC. Figure 9a has
two distinct bands that represent the most frequent location of circulating pigment particles. The
bands are more distinct and wider spaced than the bands shown in Figure 8a. The particles
also spend less time overall near the cornea (Figure 9b) compared to the results for the warmer
cornea surface (Figure 8b).
Discussion
Simulation results show that the temperature gradients within the anterior chamber of
the human eye are sufficient to create significant natural convection currents of aqueous humor.
The velocities in the anterior chamber are of the same magnitude as the highest velocities
observed due to forced flow from the posterior chamber into the anterior chamber.
Free
convection and forced convection can be compared by calculating the Grashof number and the
Reynolds number in the anterior eye. The Grashof number (Gr), which is equal to the Rayleigh
number divided by the Prandtl number, provides a measure of the ratio of the buoyancy force to
the viscous force acting on the fluid.
The Grashof number plays the same role in free
convection that the Reynolds number (Re) plays in forced convection. Both free and forced
10
convection must be considered when (Gr/Re2) ~ 1 8 , and the ratio (Gr / Re2 ) is 0.96 in the
anterior eye.
All previous models of the eye used an overall heat transfer coefficient of 20 W/(m2 oC)
for the cornea surface based on measurements by Lagendijk11 on rabbit eyes. Assuming a
room temperature of 20oC and an evaporative cooling rate of 100 W/m2
23,
the previous models
of the eye typically predicted an average heat flux of 500 W/m2 through the cornea. The
average heat flux across the cornea in our new model is 800 W/m2. The difference between
previous models and our model is due to the higher temperature gradients in the anterior
chamber because we assumed that the iris was at 37oC and, to a lesser degree, the inclusion of
both natural and forced convection. In our model, the total dimensionless heat transfer across
the cornea is 75 with natural convection, and 72 with the AH velocity set to zero everywhere.
The small change indicates that the previous models of heat transport in the eye do not have
significant errors from neglecting aqueous humor flow and natural convection.
The paths of particles floating within the aqueous humor have been plotted based on a
number of different starting positions. For particles starting near the iris tip and in the vertical
plane passing through the center axis, the projected paths had increasing diameter until the
particles came sufficiently close to a boundary that the velocity became very small.
The
aqueous currents were sufficient to bring the pigment particle close to the cornea, but we never
observed the pigment particle coming within a particle diameter of the wall in these simulations.
Most particles released from points near the iris tip circulated without coming into close
proximity with a domain boundary for over 2 hrs. Particles did circulate near the horizontal
center of the anterior chamber for the majority of the time modeled. This effect was especially
pronounced during periods when particles were within 200 µm of the cornea surface, a situation
that primarily happens in the two vertical bands near the center of the eye shown in Figure 8a.
While not conclusive, these results suggest that natural convection contributes to the spindle
shape of pigment deposits on the posterior surface of the cornea.
Of the parameters used in the model, the density, viscosity, specific heat, and thermal
conductivity of the aqueous humor, as well as the thermal conductivity of the cornea, are
accurately known and are not believed to be a significant source of error in the model. The only
parameter in the model that varies significantly between individuals is the cornea surface
temperature. We explored the sensitivity of the model to changes in cornea surface temperature
when the average temperature of the cornea surface was reduced from 34o C to 30o C. The
lower cornea temperature caused the velocities due to natural convection to increase by 50%
and the paths taken by circulating pigment particles were affected. In the case of the cooler
11
cornea, the particles did not circulate as close to the center of the eye, and the particles were
not as close to the posterior surface of the cornea. This suggests that people living in cooler,
dryer climates might not develop a spindle as distinct and heavily pigmented as the general
population. We are not aware of any experimental evidence to compare to this result, which
would of course depend on individual variations as well as on the climate.
The projected paths of pigment particles never reached within ten particle diameters (10
µm) of the cornea in all simulations conducted. A few possible explanations for why the model
fails to predict the contact and adherence of the pigment granules that are observed on the
posterior cornea surface follow. First, the surface of the cornea was assumed to be smooth in
the model, but the surface of the cornea is covered with the endothelial cells, causing small
(~10 µm) protrusions22. Second, external motion of the head, eyeball, lens and iris will have
some effect on the movement of fluid within the anterior chamber. Finally, particles will diffuse,
causing some to come closer to the cornea.
For the previously calculated diffusivity of
4.5x10-9 cm2 s-1 and an exposure time of 100 sec, the characteristic diffusion distance ( 2
Dt )
is 13.4 µm for a 1-µm pigment particle. Thus, a significant fraction of particles that are within
10-20 µm of the cornea for 100 sec. can diffuse to the posterior cornea surface.
In conclusion, the model predicted significant natural convection currents in the anterior
chamber due to the temperature gradient between the cornea and iris. The circulating aqueous
humor caused pigment particles released near the iris tip to float near the central cornea area,
where Krukenberg’s spindle is observed, confirming that natural convection currents likely
determine the shape of the spindle.
Acknowledgements
This work was supported by grants from the Whitaker Foundation and the National Institutes of
Health (R01-EY12291-01) and by support for JJH from the NSF VIGRE grant DMS-9810751.
Simulations were possible through the support of University of Minnesota Supercomputing
Institute for Digital Simulation and Advanced Computation.
12
References
1.
Al-Badwaihy, K.A. and A.B.A. Youssef. "Biological thermal effect of microwave radiation
on human eyes". In: Biological Effects of Electromagnetic Waves, edited by C.C.
Johnson and M.L. Shore. Washington DC: DHEW Publication, 1976, pp. 61-78.
2.
Bergenske, P.D. Krukenberg's spindle and contact lens-induced edema. Am. J. Optom.
Physiol. Opt. 57:932-935, 1980.
3.
Beswick, J.A. and C. McCulloch. Effect of hyaluronidase on the viscosity of the aqueous
humour. Br. J. Ophthalmol. 40:545-548, 1956.
4.
Efron, N., G. Young, and N.A. Brennan. Ocular surface temperature. Curr. Eye Res.
8:901-905, 1989.
5.
Emery, A.F., P. Kramar, A.W. Guy, and J.C. Lin. Microwave induced temperature rises
in rabbit eyes in cataract research. J. Heat Transfer C 97:123-128, 1975.
6.
Fontana, S.T. and R.F. Brubaker. Volume and depth of the anterior chamber of the
normal aging human eye. Arch. Ophthalmol. 98:1803-1808, 1980.
7.
Hartmann, C. Studies on the development of extra-endothelial and intra-endothelial
pigment deposits by means of direct and indirect contact specular microscopy of the
cornea. Graef. Arch. Clin. Exp. Ophthalmol. 218:75-82, 1982.
8.
Incropera, F.P. and D.P. DeWitt. Fundamentals of Heat and Mass Transfer - Second
Edition. New York: John Wiley & Sons, 1985, 802 pp.
9.
Kobayashi, A.S., S.L.Y. Woo, C. Lawrence, and W.A. Schlegel. Analysis of the corneoscleral shell by the method of direct stiffness. J. Biomech. 4:323-330, 1971.
10.
Kocak, I., S. Orgul, and J. Flammer. Variability in the measurement of corneal
temperature using a noncontact infrared thermometer. Ophthalmologica 213:345-349,
1999.
11.
Lagendijk, J.J.W. A mathematical model to calculate temperature distributions in human
and rabbit eyes during hyperthermic treatment. Phys. Med. Biol. 27:1301-1311, 1982.
12.
Mapstone, R. Measurement of corneal temperature. Exp. Eye Res. 7:237-243, 1968.
13.
Maurice, D.M. The location of the fluid pump in the cornea. J. Physiol. 221:43-54, 1972.
14.
Morgan, P.B., A.B. Tullo, and N. Efron. Infrared thermography of the tear film in dry eye.
Eye 9:615-618, 1995.
15.
Mori, A., Y. Oguchi, E. Goto, K. Nakamori, T. Ohtsuki, F. Egami, J. Shimazaki, and K.
Tsubota. Efficacy and safety of infrared warming of the eyelids. Cornea 18:188-193,
1999.
16.
Mori, A., Y. Oguchi, Y. Okusawa, M. Ono, H. Fujishima, and K. Tsubota. Use of highspeed, high-resolution thermography to evaluate the tear film layer. Am. J. Ophthalmol.
124:729-735, 1997.
17.
Moses, R.A. "Intraocular pressure". In: Adler's Physiology of the Eye. 6th Edition, edited
by R.A. Moses. St. Louis: C.V. Mosby Company, 1975, pp. 179-191.
18.
Moses, R.A. "The iris and the pupil". In: Adler's Physiology of the Eye, 6th Edition, edited
by R.A. Moses. St. Louis: The C. V. Mosby Company, 1975, pp. 320-323.
13
19.
Okuno, T. Thermal effect of infra-red radiation on the eye: a study based on a model.
Ann. Occup. Hyg. 35:1-12, 1991.
20.
Patil, P.N. Enhanced sensitivity of the iris sphincter to the muscarinic agonist carbachol
at lower temperature. J. Ocular Pharmacol. Therap. 15:65-71, 1999.
21.
Rodrigues, M.M., G.L. Spaeth, S. Weinreb, and E. Sivalingam. Spectrum of trabecular
pigmentation in open-angle glaucoma: A clinicopathologic study. Tr. Am. Acad.
Ophthalmol. & Oto-laryngol. 81:258-276, 1976.
22.
Rodrigues, M.M., G.O. Waring, J. Hackett, and P. Donohoo. "Cornea". In: Ocular
Anatomy, Embryology, and Teratology, edited by F.A. Jakobiec. Philadelphia: Harper &
Row, 1982, pp. 153-166.
23.
Scott, J.A. A finite element model of heat transport in the human eye. Phys. Med. Biol.
33:227-241, 1988.
24.
Smith, S.D. Measurement of the rate of aqueous humor flow. Yale J. Biol. Med. 64:89102, 1991.
14
Table 1: Physical Parameters used in the model
Parameter
Value
Source:
AH viscosity, µ
7.1 x 10 Pa s
23
AH density, ρ
9.9 x 102 kg/m3
3
AH thermal conductivity, k
0.58 W/(m K)
5
AH specific heat, Cp
-4
3
4.2 x 10 J/(kg K)
(bovine)
(water)
Volume expansion coefficient, β
3.2 x 10
Thermal conductivity of cornea
0.58 W/(m K)
23
Cornea surface temperature, Tref
34 oC
12, 16
-4 o
C
-1
(water)
15
Figure Legends
Figure 1. Topography of the Krukenberg Spindle based on Hartmann7. Region I is the pigmentfree region of the cornea, region II is the border area, region III is the intermediate area, and
region IV is the central area.
Figure 2. A 2-D representation of the model domain and boundary conditions. The drawing is
to scale, and the distance from the lens to the cornea along the centerline is 3 mm.
Figure 3. Velocity vectors in the sagittal plane for the anterior eye assuming a cornea surface
temperature of 34o C. The flow rate through the trabecular meshwork and ciliary bodies is
2.5 µL/min resulting in velocities through these boundaries much less than those in the anterior
chamber.
Figure 4. Temperature contours (oC) in the sagittal plane for the anterior eye assuming the iris
and lens are at 37oC and the exterior cornea surface is at 34oC.
Figure 5. Particle paths of two particles circulating in the sagittal plane through the central axis
for 500 sec. and using the velocity field from Figure 3. The particles remain in the plane due to
the symmetry of the model eye.
Figure 6. Particle path for two particles, over a time interval of 300 sec, released by the iris. The
particles continued circulating in the anterior chamber for 2 hrs, never crossing the central
sagittal plane. The cornea surface temperature was assumed to be 34oC.
Figure 7. The particle path for a single particle released near the iris tip and circulating for 33
min. The lower plot is the same path as the upper 3-dimensional plot, only viewed from above
in 2-dimensions. This plot shows the particle circulating near the center of the anterior chamber
for the majority of the time interval shown.
Figure 8. (a) A plot of the time spent by particles at different positions in the coronal plane for
the entire anterior eye. The plot was created by releasing 36 particles spaced every 10o around
the inner circumference of the iris tip. The particles occupy the vertical bands near the pupil for
the greatest time. (b) The time spent by the particles within 200 µm of the cornea at different
points in the coronal plane.
Figure 9. (a) A plot of the time spent by circulating particles at different positions within the
coronal plane assuming a cornea surface temperature of 30oC. (b) The time different particles
were within 200 µm of the cornea surface as a function of position in the coronal plane. Both
plots were generated by tracking 36 particles release from equally spaced points around the
inner radius of the iris.
I
II
III
IV
Heys, Figure 1
Ciliary Body
v = v(z)
T= 37 oC
Trabecular
Meshwork
v = v(z)
T= 37 oC
Iris
v=0
T = 37 oC
Lens
v=0
T = 37 oC
Cornea
v=0
Heat flux
boundary
condition
Vitreous
v=0
T = 37 oC
Heys, Figure 2
Heys, Figure 3
Heys, Figure 4
Heys, Figure 5
Heys, Figure 6
Heys, Figure 7
Heys, Figure 8
Heys, Figure 9