Modelo Drenaje Humor Acuoso Trabecular

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Modelling aqueous humor outflow throughtrabecular meshwork

Ram Avtar, Rashmi Srivastava *

Department of Mathematics, Harcourt Butler Technological Institute, Kanpur 208 002, India

Abstract

A simple model of aqueous outflow through the trabecular meshwork in eye has been developed. The model considersthe meshwork as an annular cylindrical ring with uniform thickness of homogenous, isotropic, viscoelastic material, swol-len with continuously percolating aqueous humor through it. It incorporates a strain-dependent permeability function intoDarcys law which is coupled to the force balance for the bulk material. A simple analytical expression relating aqueousflux to pressure differential is developed which shows how strain-dependent permeability can lead to reduction in hydraulicconductivity (aqueous outflow facility) with increasing intraocular pressure (IOP) as observed in experiments by ophthal-mologists. Analytical expressions for the displacement, fluid pressure and dilatation in the trabecular meshwork have beenobtained.2006 Elsevier Inc. All rights reserved.

Keywords: Aqueous humor; Trabecular meshwork; Intraocular pressure; Biphasic

1. Introduction

Aqueous humor occupying the anterior chamber leaves it via the uveoscleral drainage system and the con-ventional outflow system. Most of the fluid, approximately 85%, drains away to the blood streams through theconventional outflow pathway which consists of in series: the uveal and corneoscleral meshwork, the juxta-canalicular meshwork (JCM), the inner endothelial wall of the Schlemms canal, the Schlemms canal andthe collector channels and the aqueous veins. The part comprised of first three layers altogether is known

as the trabecular meshwork. An obstruction to the aqueous outflow results in the accumulation of aqueoushumor in the anterior chamber raising the intraocular pressure (IOP). Increased IOP, sustained for a longtime, can damage the optic nerve in the eye and can lead to blindness [1,24].

Despite many years of research, mainly devoted to the identification of principal site of the outflow resis-tance in the aqueous outflow system and to the investigations of its origin and its possible role in the devel-opment of a pathological state known as glaucoma [2], we still do not know how the majority of aqueousoutflow resistance is generated which implies that we do not understand the fundamental factors controlling

0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.11.109

* Corresponding author.E-mail address:[email protected](R. Srivastava).

Applied Mathematics and Computation 189 (2007) 734745

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IOP. For those glaucoma therapies directed towards lowering the intraocular pressure through the enhance-ment of aqueous humor outflow, a better understanding of the outflow network, changes in which can lead toocular hypertension and glaucoma and that of mechanism of aqueous outflow resistance increase is required.Both experimental and theoretical approaches should be adopted by researchers to elucidate the aqueousdrainage mechanism in various components of the outflow network.

There have been numerous attempts to identify the primary site and the underlying cause of outflow resis-tance in the aqueous outflow system. McEwens[3] calculations of flow resistance based on the dimensions ofthe flow passage in the inner aspects of the meshwork indicated that both the uveal meshwork and corneo-scleral meshwork do not generate significant resistance consistent with Grants [4] finding that removal ofuveal meshwork had little effect on the outflow facility. Grant[5]later demonstrated that 75% of the resistanceresided between the anterior chamber and aqueous veins. In humans, the pores in the inner wall endotheliumof Schlemms canal generate perhaps 10% of the total flow resistance. Schlemms canal itself only generatessignificant flow resistance when it is substantially collapsed, and the collector channels and aqueous veins havebeen shown to have negligible flow resistance. It has been suggested that the apparently open spaces in theJCM are filled with an extracellular matrix gel, and that such a gel filled JCM is likely to be the major sourceof flow resistance in the normal eye[6].

The source of outflow resistance in the aqueous outflow system of normal or glaucomateous eyes has not

been definitely established although each of the components of the outflow pathways has been analyzed todetermine whether they might generate significant flow resistance. Under the circumstances of confusing con-clusions regarding the aqueous outflow mechanism derived from experimental studies, a mathematical anal-ysis of the aqueous outflow may facilitate the definite conclusion. Several mathematical models of the aqueousoutflow in the trabecular meshwork, the juxtacanalicular meshwork and Schlemms canal have been developedto explore the outflow mechanisms in the respective components of the outflow network. But these modelshave limitations due to many assumptions and simplifications.

Despite numerous studies[4,5,9]of the aqueous outflow, the origin of outflow resistance in the meshworkand increase in it with IOP elevation in some cases is not clearly known. In order to select an ideal therapy forreducing the outflow resistance increase observed in open angle glaucoma, knowledge of causes of abnormal-ities in the tissue and that of mechanism of outflow resistance increase is required. Poiseuille flow[3]models as

well as various empirical models have been applied to describe aqueous flow through the trabecular mesh-work. Tandon and Avtar[7] presented a biphasic continuum model for aqueous flow through the meshworkusing small strain expansion and obtained approximate solution by an iterative technique. The models pro-posed earlier are not sufficient to describe the outflow phenomenon.

The microstructure of trabecular meshwork can be described as composed of cellular conglomeratesembedded in a porous fibrous extracellular matrix and interstices of the matrix are filled with aqueous fluid.It is like a bi-phasic material, comprised mainly of aqueous humor enclosed within an elastic matrix of elastinand collagen fibres. Johnson and Grant[8] and several other investigators [9,10]observed from their experi-mental studies that striking structural changes occurred in the meshwork in response to alterations in theintraocular pressure. An elevation in the IOP above the normal level may produce compression in the mesh-work diminishing the flow regions.

Aqueous humor flowing through the meshwork will interact with its solid phase giving rise to a drag force,i.e. a force exerted by aqueous humor on the solid phase as it passes through the tissue. A rise in IOP forcesmore aqueous fluid to flow out through the meshwork, which, in turn, exerts a greater drag force on the solidmatrix causing compression in the meshwork. The compression caused by the increased IOP may decrease theintrinsic permeability, which depends upon the dilatation of the solid phase. This concept is supported by Laiand Mow[11]. Once a fluid pressure is applied, the viscous drag caused by the fluid flowing through the tissuescauses it to compact in a non-uniform manner, thereby decreasing the permeability.

The behavior of aqueous humor, thus, depends upon the intrinsic interaction between the deformation ofthe solid matrix and the motion of the interstitial aqueous fluid. From a fluid mechanical point of view, thereare two primary mechanisms for the transport of aqueous humor across the meshwork.

(i) The interstitial aqueous fluid may transport through the poroelastic, permeable meshwork under the

influence of a pressure differential across the meshwork.

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(ii) The deformation of elastic solid matrix caused by the drag force directly affects the aqueous outflowphenomenon.

A measure of the ease with which the flow of aqueous humor occurs through the aqueous outflow system isthe hydraulic conductivity which is interpreted as outflow facility in ophthalmology. It has been well estab-

lished that aqueous outflow facility decreases with a rise in the IOP [12]. As mentioned earlier, most of theoutflow resistance is encountered in the trabecular meshwork. The outflow facility of the aqueous outflow sys-tem is, therefore, mainly determined by the meshwork. The factors involved in the process of aqueous perco-lation through the meshwork play a role in determining the outflow facility, which is important determinant ofthe IOP.

In addition to the pressure differential drop across the meshwork, and elasticity of the meshwork, severalother physiological factors such as the ciliary muscle contraction, the intrinsic contractility of the trabecularmeshwork and scleral spur cells, the cellular volume, extracellular matrix status, protein concentration, nervouscontrol have been observed to affect the aqueous outflow facility. An increase in the ciliary muscle contractionresults in distension of trabecular meshwork with subsequent reduction in outflow[21]. But the mechanism ofthe ciliary muscle contraction in reducing outflow resistance is not well understood[12]. The contraction of thetrabecular meshwork cells and scleral spur cells, decreases the permeability of the trabecular meshwork because

the size of the intercellular space is reduced. Similarly when trabecular meshwork cells and scleral spur cellsrelax, the opposite effect appears and the permeability of tissue increases[21]. Whether contractility of trabec-ular meshwork and scleral spur cells also play a role in old and glaucomatous human eyes is not known[19].Similarly, volume regulatory properties would be affected as well. The survey of ophthalmic literature showsthat there is a lot of confusion regarding the importance of protein concentration in generating flow resistancein the aqueous outflow pathway. Some studies [1618,20] have shown that soluble protein in the aqueoushumor of bovine eye can influence the aqueous outflow resistance. In 1901, Troneoso[22]suggested that glau-coma is caused by an excess of protein in aqueous humor. Johnson et al. [23]confirmed that plasma-derivedproteins in the aqueous humor of the trabecular meshwork can generate a significant fraction of aqueous out-flow resistance. In 1997, Sit et al. [18] concluded that outflow facility would be controlled by a resistance causingmechanism other than the bulk level of aqueous humor proteins. Finally some aspects of trabecular meshwork

physiology and their significance in tissue permeability and outflow facility are still unknown.The model considers coupled dynamical interaction between the aqueous outflow and mechanical deforma-

tion in the meshwork. The aqueous outflow in the meshwork is described by the Darcys law. The strain-dependent permeability is incorporated in the Darcys law. In this work, the trabecular meshwork is modelledas a thin wall of a cylinder surrounding the anterior chamber. The inner side of the cylinder represents theanterior chamber. The trabecular meshwork as a whole consists of a series of flat perforated sheets or lamellaelying one on the top of the other, connective tissues containing collagen and elastin fibres, a ground substancesand the endothelial cells. Thus, the trabecular meshwork filled with aqueous humor may be described as abiphasic matrix comprising the solid and fluid phases.

2. Mathematical model of filtration flow through trabecular meshwork tissue

The mathematical model is based on the theory for consolidation of porous elastic materials due to Kenyon[13]and a phenomenological equation for strain-dependent permeability due to Lai and Mow[11]. The anal-ysis considers the trabecular meshwork to be cylindrical annular ring (Fig. 1) with thickness Hcontaining afluid inside it at pressurePm. The polar coordinates of the cross section are taken asr andh whilezis the axialcoordinate. The external pressurePois set at zero for reference. To develop a mathematically tractable modelof the meshwork, the following assumptions are introduced.

2.1. Assumptions

(i) The trabecular meshwork is an isotropic, homogenous, porous, deformable matrix swollen with contin-uously flowing aqueous humor.

(ii) The permeability of the meshwork is a function of the dilatation.

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(iii) Only the fluid mechanical aspects of aqueous flow phenomenon are considered.(iv) Both the constituents of the system are isothermal.(v) The aqueous fluid and solid matrix are intrinsically incompressible.

(vi) Inertial and body forces are neglected and diffusional couples do not exist.(vii) The displacement of the tissue depends only on radial position.

(viii) Aqueous flow is steady, slow, laminar, Newtonian, viscous and incompressible.(ix) The protein accumulation, the muscular contraction, the nervous control and the ciliary muscle traction

are neglected.(x) Due to fluid flow in the lumen, the axial wall shear stress is negligible in comparison to other stresses

acting on the wall.

Using the symbols SijEijfor the components of the extra stress (strain) tensor in the tissue matrix, Pforthe interstitial fluid pressure, Urfor the radial displacement of the tissue,Wfor the filtration velocity,k and Gfor the Lame constants of the tissue, lfor the fluid viscosity and kfor the permeability, the governing equa-

tions are as follows:dP

dr

1

r

d

drrSrr

1

rShh; 1

dP

dr

l

jW; 2

d

drrW 0: 3

Eq.(1)represents the force balance describing equilibrium in the bulk material; Eq. (2) is Darcys law whichrelates the interstitial fluid flow to the fluid pressure gradient; and Eq.(3)is the continuity equation assuming a

stationary solid phase.The trabecular meshwork tissue is assumed to be isotropic and linearly elastic so that

Sij kEijdij2GEij; 4

where dijis the Kronecker Delta Function. The only non-vanishing components of the strain tensor are

Err dUr

dr and Ehh

Ur

r : 5a; b

Substituting Eqs.(4) and (5)into Eq.(1)and replacing Ur(only non-vanishing displacement) by U, we find

dP

dr k2G

d

dr

1

r

d

drrU

; 6

where the term (k 2G) is referred to as the aggregate elastic modulus.

Fig. 1. Schematic diagram of the proposed model of the trabecular meshwork.

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Lai and Mow [11] proposed the following non-linear relationship between permeability and strain forcartilage;

k k0 expM!; 7

wherek0and Mare material constants and is the dilatation ( = Ekk). Although Eq.(7) was determined forcartilage, it is reasonable to expect k=k() for the trabecular meshwork tissue since the meshwork and car-tilage are both porous, fluid filled solids, consisting largely of collagens fibres. The change in the water contentper unit volume of the porous sample is equal to the dilatation (), if the density of the solid is assumed to beconstant. Since the deformation of the solid matrix of the soft meshwork induces a change in its microstruc-ture, the permeability of the material is, in general a function of deformation. Lai and Mow[11]have dem-onstrated the validity of the following two- parameter model in their study of small strains of articularcartilage:

1

k

1

k01M!: 8a

This was later used by Klanchar and Tarball [14]while studying water transport through arterial walls forwhich for small M is equivalent to

k k01M!: 8b

Eq.(8b)is, of course, the small strain expansion of Eq.(7)and is a general approximation to anyk() for suf-ficiently small .

The dilatation in cylindrical coordinates is represented by

!dU

dr

U

r : 9

Substitution of Eqs.(8a) and (9) in Eq.(2)yields

dP

dr

l

j01M

dU

dr

U

r

W: 10

Eqs.(3), (6) and (10)constitute a system of equations governing the aqueous flow in the trabecular meshwork.

3. Boundary conditions

The tissue is assumed to be homogenous and the pressure continuous in the radial direction. The pressuresare specified at the inner and outer walls

Pr R Pm Pr RH 0: 11i

But at these boundaries the extra stresses are zero

Srrr R 0; Srrr RH 0: 11ii

4. Dimensionless scheme

We non-dimensionalize the governing Eqs.(3),(6) and (10)with respect to the radius of anterior chamberRand the Lame constant (2G+k) and the following non-dimensional variables are introduced:

p P

2Gk; r

r

R; u

U

R; w

W

U0; h

H

R; sij

Sij

2Gk:

Introducing these variables, Eqs.(6), (10) and (3), respectively, are converted into the following normalizedforms (Dropping the bars for convenience):

dp

dr

d2u

dr2

1

r

du

dru

r2; 12

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dp

dr

lRU0

2Gkj0M

du

dru

r

1

w; 13

d

drrw 0: 14

Boundary conditions(11i) and (11ii) in dimensionless forms become as follows:

pr 1 Pm

2Gk pr 1h 0; 15i

srrr 1 0 srrr 1 h 0: 15ii

Eqs.(8a), (9), (12), (13) and (14)must be solved subject to appropriate boundary condition which will be dis-cussed subsequently.

Integrating of Eq.(14)gives

wA1

r : 16

whereA1is a constant to be determined. Substituting the above value ofw in Eq.(13), and using the resultingequation in, Eq.(12)we have:

lRU0A1

2Gkrk0M

du

dru

r

1

d2u

dr2

1

r

du

dr

u

r2; 17

which is linear equation for u(r), Eq. (17)can be cast in the form of an Euler equation:

r2d2u

dr2 1A1BCr

du

dr A1BC1u

A1BCr

M ; 18

where dimensionless parameters are

B lMk02Gk

; C RU0:

The solution of Eq.(18)is

uA2

r A3r

A1BC1 r

2M; 19

where A2 and A3 are new integration constants.The pressure distribution is obtained by substituting Eq. (19) into Eq. (12) and integrating the resulting

equation.

p A3A1BC2rA1BC A4; 20

where A4 is the final integration constant. The extra stress component, srrgiven in non-dimensional form:

srr du

dr

k

2Gk

u

r 21

is calculated by substitution of Eq.(19)into Eq.(21)with the result:

srr A3 A1BC 1 k

k2G

rA1BC

A2

r

k

k2G1

1

M: 22

Solving for the integration constants in Eqs. (20) and (22)using above boundary conditions, we find the fol-lowing approximate relationship for w in terms of pressure and radial position:

Pm2Gkh2 2hGkhwrBC

MhwrBC2Gk 2Gk : 23

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Now the filtration velocity can be expressed as

w 2GkPmk0

rlh2 2hGk MPmC: 24

Notice that the hydraulic conductivity (w/Pm) now depends on the hydrostatics pressurePmand the new mate-

rial parameter Mmust be a negative quantity. At r= 1, it can be expressed as

Lp 2Gkk0

lh2 2hGk MPmC: 25

An expression for the permeability variation within the tissue can be derived by substituting Eq. (9) and (19),into Eq.(8a), in the form:

k

k0

rA1BC

MA3A1BC2: 26

The deviation of the ratio k/k0from unity is an indication of the extent of strain induced permeability vari-ation within the material.

An expression for the dilatation variation in the meshwork can be obtained by substituting Eq. (19)into

Eq.(9).

! A3A1BC 2rA1BC

1

M: 27

On Comparison Eq.(24)with Eq.(16), we find

A1 2GkPmk0

lh2 2hGk MPmC:

Using the boundary conditions, we find the following integration constants:

A2 PmA1BC2Gk 2Gk Gk

2GkA1BC21 1hA1BC2G

;

A3 Pm

2GkA1BC21 1hA1BC

;

A4 Pm1h

A1BC

2Gk1 1hA1BC

:

5. Results and discussion

Computational results for a representative human eye have been obtained from the above solutions byintroducing appropriate values of the physiological parameters listed in Table 1. The elastic properties of

Table 1Typical values of the parameters used in the model computations

Parameter Description Typical value

Po Pressure at the outer wall of the meshwork 0Pm Pressure at the inner wall of the meshwork 5 mm HgE Elastic modulus of the meshwork 4.5 Nm2

R Radius of the anterior chamber 5.62 103 mH Thickness of the trabecular meshwork .01 103 mU0 Characteristic velocity of aqueous fluid 3.03 10

6 m s1

k0 Permeability of the meshwork 1.99 1012 m2

M Material constant 4.5

m Poissons ratio .35



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the material are specified by the Youngs modulus, E, and Poissons ratio, m. The Lame constants Gand k,defining the elastic properties of the meshwork, are related to parameters by

G E=21m k Em=1m12m:

The curves depicted in Figs.2ac and3ac represent the characteristic profiles of the normalized solid phasedisplacement and the fluid pressure, respectively, in the trabecular meshwork for different values of intraocularpressure differential (Pm).

These profiles demonstrate that there is a continuous decrease in the displacement along the radial direc-tion. Evidently, the displacement is the maximum at the inner wall and approaches to zero at the outer wall ofthe meshwork. The characteristic graphs also show the effect of intraocular pressure differential across themeshwork on the dimensionless displacement. It is evident from theFig. 2ac that an increase in the pressuredifferential increases the displacement. These observations provide support to the fact that raised pressure dif-ferential causes greater compression in the meshwork, reducing interstitial space available for the aqueous pas-sage. Increased pressure differential causes larger local displacement in the meshwork with greatercompression.

Fig. 3ac displays the effect of pressure differential on the pressure distribution in the meshwork. The curvedemonstrates that increasing pressure differential induces an increase in the fluid pressure in the meshwork.

The curves depicted inFig. 4show the effect of intraocular pressure differential on the normalized permeabilityof the meshwork. As the pressure differential increases, the permeability of the meshwork decreases. Thisresult provides support to the fact that the decreased interstitial space of aqueous outflow pathways due toraised pressure differential-induced compression offers greater resistance to aqueous outflow and so the per-meability decreases. Decreased permeability retards aqueous flow in the meshwork with concomitant increasein the fluid accumulation in the anterior chamber leading to further rise in IOP. This result is supported by aobservations[21].

Fig. 5, shows the distribution of the dilatation of the solid matrix in trabecular meshwork as a function ofthickness of the meshwork and the effect of the pressure differential (IOP) across the meshwork on the dila-tation. Increased pressure differential enhances the dilatation. The change in volume of the meshwork materialand change in aqueous content in the meshwork increases with a rise in the pressure differential. A raised IOP

forces more aqueous to percolate into the meshwork enhancing the aqueous content and volume of the mate-rial in the meshwork.The curve inFig. 6depicts the variation in filtration velocity of aqueous humor in response to the changes

in the pressure differential. Evidently, the filtration velocity or volume flux of the aqueous humor in the mesh-work at its inner wall increases with a rise in the pressure differential.

The curve depicted inFig. 7shows variation in the hydraulic conductivity (aqueous outflow facility) as afunction of the pressure differential. The curve shows the significant reduction of aqueous outflow facility withan increasing pressure differential. Elevated, IOP causes greater compaction between the intratrabecular sheetsand lamellae with reduced intratrabecular fluid space which decreases the aqueous outflow facility of themeshwork. This result is supported by a number of experimental observations [15,21].

6. Concluding remarks

The prime objective of the present work was to construct a simple mathematical model for describing theoutflow of aqueous humor in the trabecular meshwork of eye, to provide an estimate of effect of intraocularpressure on the flow characteristics of the aqueous and a simple framework for interpreting ophthalmologistsexperiments on the aqueous outflow through the meshwork. The results presented above that a rise in theintraocular pressure causes an increases in the fluid pressure, the displacement, the dilatation and aqueous fil-tration (percolation) velocity in the meshwork whereas it decreases the permeability as well as the hydraulicconductivity of the porous trabecular meshwork. Thus, intraocular pressure has a outflow facility decreasingeffect which is well established by experimental observation[12]. The effect occurs through the compression inthe trabecular meshwork matrix. From a clinical point of view, glaucoma therapy should be evolved whichattempts to minimize these secondary effects. A therapy which tends to counteract the compression by neu-

tralizing the intraocular pressure rise may be operative in treatment of glaucoma. It may be more appropriate



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1.3

1.302

1.304

1.306

1.308

1.31

1.312

1.314

1.316

0.998 1 1.002 1.004 1.006 1.008 1.01 1.012

r

ux10-11(

displacement)

ux10-11(

displacement)

ux10-11(

displacement)

Pm=3mmHg

2.04

2.045

2.05

2.055

2.06

2.065

2.07

0.998 1 1.002 1.004 1.006 1.008 1.01 1.012

r

Pm=5mmHg

2.785

2.79

2.795

2.8

2.805

2.81

2.815

2.82

0.998 1 1.002 1.004 1.006 1.008 1.01 1.012

r

Pm=7mmHg

Fig. 2. (a)(c) Effect of intraocular pressure on the normalized displacement distribution.



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0.59

0.6

0.61

0.62

0.63

0.64

0.65

0.66

0.67

0.68

0.69

0.998 1 1.002 1.004 1.006 1.008 1.011 1.012

r

px104(press

ure)

px104(pressure)

px104

(pressure)

Pm=3mm Hg

1.58

1.59

1.6

1.61

1.62

1.63

1.64

1.65

1.66

1.67

1.68

0.998 1 1.004 1.006 1.008 1.01 1.012

r

Pm=5mmHg

1.002

5.57

5.58

5.59

5.6

5.61

5.62

5.63

5.64

5.65

5.66

5.67

5.68

0.998 1 1.002 1.004 1.006 1.008 1.01 1.012

r

Pm=7mmHg

Fig. 3. (a)(c) Effect of intraocular pressure on the normalized pressure distribution.



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0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8 9 10

pm

W(

filtrationvelocity)

Fig. 6. Variation of filtration velocity with intraocular pressure.

0.9855

0.986

0.9865

0.987

0.9875

0.988

0.9885

0.998 1 1.002 1.004 1.006 1.008 1.0 1.012

r

k/ko(permeab

ility)

Pm=3mmHg

Pm=5mmHg

Pm=7mmHg

Fig. 4. Effect of intraocular pressure on permeability distribution.

0.0026

0.0027

0.0028

0.0029

0.003

0.0031

0.0032

0.0033

0.998 1 1.002 1.004 1.006 1.008 1.01 1.012

r

dilatation

Pm=3mmHg

Pm=5mmHg

Pm=7mmHg

Fig. 5. Effect of intraocular pressure on dilatation distribution.



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and realistic for future researchers to conduct and unsteady state analysis of the aqueous flow through the

biphasic meshwork, by incorporating more generalized stressstrain relationship into the theoretical framework of the present study.

References

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Ophth. Vis. Sci. 34 (13) (1993) 35493557.[24] Ram Avtar, Rashmi Srivastava, Aqueous outflow in Schlemms canal, Appl. Math. Comput. 174 (2006) 316328.

1.541

1.542

1.543

1.544

1.545

1.546

1.547

1.548

1.549

1.55

0 1 2 3 4 5 6 7 8 9 10

pm

Lp(outflow

facility)

Fig. 7. Variation of hydraulic conductivity (outflow facility) with intraocular pressure.


Modelo Drenaje Humor Acuoso Trabecular

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