Modelling aqueous humor outflow through trabecular meshwork

doi:10.1016/j.amc.2006.11.109

Applied Mathematics and Computation

Volume 189, Issue 1, 1 June 2007, Pages 734-745

https://doi.org/10.1016/j.amc.2006.11.109 Get rights and content

Abstract

A simple model of aqueous outflow through the trabecular meshwork in eye has been developed. The model considers the meshwork as an annular cylindrical ring with uniform thickness of homogenous, isotropic, viscoelastic material, swollen with continuously percolating aqueous humor through it. It incorporates a strain-dependent permeability function into Darcy’s law which is coupled to the force balance for the bulk material. A simple analytical expression relating aqueous flux to pressure differential is developed which shows how strain-dependent permeability can lead to reduction in hydraulic conductivity (aqueous outflow facility) with increasing intraocular pressure (IOP) as observed in experiments by ophthalmologists. Analytical expressions for the displacement, fluid pressure and dilatation in the trabecular meshwork have been obtained.

Introduction

Aqueous humor occupying the anterior chamber leaves it via the uveoscleral drainage system and the conventional outflow system. Most of the fluid, approximately 85%, drains away to the blood streams through the conventional outflow pathway which consists of in series: the uveal and corneoscleral meshwork, the juxtacanalicular meshwork (JCM), the inner endothelial wall of the Schlemm’s canal, the Schlemm’s canal and the 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 aqueous humor in the anterior chamber raising the intraocular pressure (IOP). Increased IOP, sustained for a long time, 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 resistance in the aqueous outflow system and to the investigations of its origin and its possible role in the development of a pathological state known as glaucoma [2], we still do not know how the majority of aqueous outflow resistance is generated which implies that we do not understand the fundamental factors controlling IOP. For those glaucoma therapies directed towards lowering the intraocular pressure through the enhancement of aqueous humor outflow, a better understanding of the outflow network, changes in which can lead to ocular 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 aqueous drainage mechanism in various components of the outflow network.

There have been numerous attempts to identify the primary site and the underlying cause of outflow resistance in the aqueous outflow system. McEwen’s [3] calculations of flow resistance based on the dimensions of the flow passage in the inner aspects of the meshwork indicated that both the uveal meshwork and corneoscleral meshwork do not generate significant resistance consistent with Grant’s [4] finding that removal of uveal meshwork had little effect on the outflow facility. Grant [5] later demonstrated that 75% of the resistance resided between the anterior chamber and aqueous veins. In humans, the pores in the inner wall endothelium of Schlemm’s canal generate perhaps 10% of the total flow resistance. Schlemm’s canal itself only generates significant flow resistance when it is substantially collapsed, and the collector channels and aqueous veins have been shown to have negligible flow resistance. It has been suggested that the apparently open spaces in the JCM are filled with an extracellular matrix gel, and that such a gel filled JCM is likely to be the major source of 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 to determine whether they might generate significant flow resistance. Under the circumstances of confusing conclusions regarding the aqueous outflow mechanism derived from experimental studies, a mathematical analysis of the aqueous outflow may facilitate the definite conclusion. Several mathematical models of the aqueous outflow in the trabecular meshwork, the juxtacanalicular meshwork and Schlemm’s canal have been developed to explore the outflow mechanisms in the respective components of the outflow network. But these models have limitations due to many assumptions and simplifications.

Despite numerous studies [4], [5], [9] of the aqueous outflow, the origin of outflow resistance in the meshwork and increase in it with IOP elevation in some cases is not clearly known. In order to select an ideal therapy for reducing the outflow resistance increase observed in open angle glaucoma, knowledge of causes of abnormalities 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 meshwork. Tandon and Avtar [7] presented a biphasic continuum model for aqueous flow through the meshwork using small strain expansion and obtained approximate solution by an iterative technique. The models proposed earlier are not sufficient to describe the outflow phenomenon.

The microstructure of trabecular meshwork can be described as composed of cellular conglomerates embedded 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 elastin and collagen fibres. Johnson and Grant [8] and several other investigators [9], [10] observed from their experimental studies that striking structural changes occurred in the meshwork in response to alterations in the intraocular pressure. An elevation in the IOP above the normal level may produce compression in the meshwork 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 forces more aqueous fluid to flow out through the meshwork, which, in turn, exerts a greater drag force on the solid matrix causing compression in the meshwork. The compression caused by the increased IOP may decrease the intrinsic permeability, which depends upon the dilatation of the solid phase. This concept is supported by Lai and Mow [11]. Once a fluid pressure is applied, the viscous drag caused by the fluid flowing through the tissues causes 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 of the solid matrix and the motion of the interstitial aqueous fluid. From a fluid mechanical point of view, there are 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.
(ii)
The deformation of elastic solid matrix caused by the drag force directly affects the aqueous outflow phenomenon.

A measure of the ease with which the flow of aqueous humor occurs through the aqueous outflow system is the hydraulic conductivity which is interpreted as ‘outflow facility’ in ophthalmology. It has been well established that aqueous outflow facility decreases with a rise in the IOP [12]. As mentioned earlier, most of the outflow resistance is encountered in the trabecular meshwork. The outflow facility of the aqueous outflow system is, therefore, mainly determined by the meshwork. The factors involved in the process of aqueous percolation through the meshwork play a role in determining the outflow facility, which is important determinant of the IOP.

In addition to the pressure differential drop across the meshwork, and elasticity of the meshwork, several other physiological factors such as the ciliary muscle contraction, the intrinsic contractility of the trabecular meshwork and scleral spur cells, the cellular volume, extracellular matrix status, protein concentration, nervous control have been observed to affect the aqueous outflow facility. An increase in the ciliary muscle contraction results in distension of trabecular meshwork with subsequent reduction in outflow [21]. But the mechanism of the ciliary muscle contraction in reducing outflow resistance is not well understood [12]. The contraction of the trabecular 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 cells relax, the opposite effect appears and the permeability of tissue increases [21]. Whether contractility of trabecular 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 shows that there is a lot of confusion regarding the importance of protein concentration in generating flow resistance in the aqueous outflow pathway. Some studies [16], [17], [18], [20] have shown that soluble protein in the aqueous humor of bovine eye can influence the aqueous outflow resistance. In 1901, Troneoso [22] suggested that glaucoma is caused by an excess of protein in aqueous humor. Johnson et al. [23] confirmed that plasma-derived proteins in the aqueous humor of the trabecular meshwork can generate a significant fraction of aqueous outflow resistance. In 1997, Sit et al. [18] concluded that outflow facility would be controlled by a resistance causing mechanism 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 deformation in the meshwork. The aqueous outflow in the meshwork is described by the Darcy’s law. The strain-dependent permeability is incorporated in the Darcy’s law. In this work, the trabecular meshwork is modelled as a thin wall of a cylinder surrounding the anterior chamber. The inner side of the cylinder represents the anterior chamber. The trabecular meshwork as a whole consists of a series of flat perforated sheets or lamellae lying one on the top of the other, connective tissues containing collagen and elastin fibres, a ground substances and the endothelial cells. Thus, the trabecular meshwork filled with aqueous humor may be described as a biphasic matrix comprising the solid and fluid phases.

Section snippets

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 analysis considers the trabecular meshwork to be cylindrical annular ring (Fig. 1) with thickness H containing a fluid inside it at pressure P_m. The polar coordinates of the cross section are taken as r and θ while z is the axial coordinate. The external pressure P_o is set at zero for

Boundary conditions

The tissue is assumed to be homogenous and the pressure continuous in the radial direction. The pressures are specified at the inner and outer walls $P (r = R) = P_{m} P (r = R + H) = 0 .$ But at these boundaries the extra stresses are zero $S_{rr} (r = R) = 0, S_{rr} (r = R + H) = 0 .$

Dimensionless scheme

We non-dimensionalize the governing Eqs. (3), (6), (10) with respect to the radius of anterior chamber R and the Lame constant (2G + λ) and the following non-dimensional variables are introduced: $\bar{p} = \frac{P}{(2 G + λ)}, \bar{r} = \frac{r}{R}, \bar{u} = \frac{U}{R}, \bar{w} = \frac{W}{U_{0}}, \bar{h} = \frac{H}{R}, {\bar{s}}_{ij} = \frac{S_{ij}}{(2 G + λ)} .$ Introducing these variables, Eqs. (6), (10), (3), respectively, are converted into the following normalized forms (Dropping the bars for convenience): $\frac{d p}{d r} = \frac{d^{2} u}{d r^{2}} + \frac{1}{r} \frac{d u}{d r} - \frac{u}{r^{2}},$ $\frac{d p}{d r} = \frac{μ {RU}_{0}}{(2 G + λ) κ_{0}} [M (\frac{d u}{d r} + \frac{u}{r}) - 1] w,$ $\frac{d}{d r} (rw) = 0 .$ Boundary conditions (11i), (11ii) in

Results and discussion

Computational results for a representative human eye have been obtained from the above solutions by introducing appropriate values of the physiological parameters listed in Table 1. The elastic properties of the material are specified by the Young’s modulus, E, and Poisson’s ratio, ν. The Lame constants G and λ, defining the elastic properties of the meshwork, are related to parameters by $G = E / [2 (1 + ν)] λ = E ν / [(1 + ν) (1 - 2 ν)] .$ The curves depicted in Figs. 2a–c and 3a–c represent the characteristic

Concluding remarks

The prime objective of the present work was to construct a simple mathematical model for describing the outflow of aqueous humor in the trabecular meshwork of eye, to provide an estimate of effect of intraocular pressure on the flow characteristics of the aqueous and a simple framework for interpreting ophthalmologists experiments on the aqueous outflow through the meshwork. The results presented above that a rise in the intraocular pressure causes an increases in the fluid pressure, the

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Modelling aqueous humor outflow through trabecular meshwork

Abstract

Introduction

Section snippets

Mathematical model of filtration flow through trabecular meshwork tissue

Boundary conditions

Dimensionless scheme

Results and discussion

Concluding remarks

Exp. Eye Res.

Exp. Eye Res.

Surv. Ophthalmol.

Bull. Math. Bio.

Am. J. Ophthalmol.

Exp. Eye Res.

Appl. Math. Comput.

Ocular biomechanics and biotransport

Ann. Rev. Biomed. Eng.

Glaucoma

Application of Poiseuille’s law of aqueous outflow

Arch. Ophthalmol.

Facility of flow through the trabecular meshwork

Arch. Ophthalmol.

Further studies on facility of flow through the trabecular meshwork

Arch. Ophthalmol.