Generalized Mathematical Modeling of Aqueous Humour Flow in the Anterior Chamber and Through a Mesh Channel in the Human Eye
Chapter One
OBJECTIVES OF THE STUDY
This study is undertaken on generalized mathematical modeling of aqueous humour flow in the anterior chamber and through a mesh channel in the human eye. The objectives of the study are to:
- formulate a mathematical model that describes the fluid flow in the human eye when ambient temperature is higher than core body temperature,
- investigate the dynamics of the model and compare with that of existing models,
- describe the pressure and flow velocity in a healthy/glaucomatous eye,
- describe the velocity streamlines and pressure contours in healthy/glaucomatous eye and
- analyze the effect of resistance of the drainage system on the flow distribution and intraocular pressure (IOP).
CHAPTER TWO
LITERATURE REVIEW
Modeling the aqueous humour flow in the human eye and obtaining an insight in the pattern of outflow and the pathological conditions of some eye diseases is a complicated problem. Several models have been previously reported that explore the temperature distribution and causes of aqueous humour flow in the anterior chamber of the eye. In particular, most of them dwell on the aqueous humour flow driven by temperature gradient, but these models are limited due to some assumptions. None of the models reported in the literature have attempted to consider the ambient or room temperature in excess of the body temperature as the driven force for the flow between the iris and the inner chamber of the cornea.
Canning et al (2002) developed a mathematical model of buoyancy driven flow in the anterior chamber of the human eye. He used it to investigate the motion and distribution of particulates in the anterior chamber. He concluded that the buoyancy effect is driven by the temperature gradient between the anterior surface of the cornea and the plane formed by the pupil aperture and the iris. That this difference exists because the temperature at the back of the anterior chamber is close to core body temperature (370C) whilst the outside of the cornea which forms the front wall of the chamber is exposed to ambient conditions (Say 250C during the day when the eye is open). His principal aim of the study was to show that even relatively small temperature differences could give rise to the flows that are observed. He however, completely ignored the climatic conditions of places (deserts or equatorial regions) where the ambient or room temperature is in excess of the body temperature.
In their review, El-shhed and Abdslmba (2005) extended the model of Canning et al by considering a slip flow boundary condition for the velocity along the cornea. They too considered the ambient or room temperature to be less than the body temperature in their assumption for the convective flow. This model was further extended by Avter and Srivastava (2006) wherein, along with a slip-flow boundary condition for the velocity, they also assumed a convection heat conduction boundary on the cornea.
In their study, Fitt and Gonzalez (2006) studied fluid flow in the anterior chamber caused by buoyancy-driven flow arising from the temperature difference between the anterior surface of the cornea and the iris, flow generated by the aqueous production of the ciliary body, flow generated by the interaction between buoyancy and gravity while sleeping in face-up position, flow generated by lens tremor (Phakodenesis) and flow generated by Rapid Eye Movement (REM) during sleep. While maintaining the same assumptions on the body temperature and ambient temperature as proposed by Canning et al, they considered the buoyant flow to be the most powerful force to drive flow in the anterior chamber, producing flow speeds that are an order of magnitude greater than any other mechanism.
Crowder and Ervin (2013) investigated numerically fluid flow through the anterior cavity (AC) and trabecular meshwork (TM) of the human eye. They based this on the dependence of the pressure difference between the anterior chamber and the trabecular meshwork on the flow rate of the aqueous humour, the viscosity of the aqueous humour, and the permeability of the trabecular meshwork. They assumed the fluid flow to be axissymmetric, enabling the fluid flow (through a change or variable to cylindrical coordinates) to be modeled as 2-D problem. Though they assumed no temperature variation across the eye, the body temperature was taken as 370C and ambient temperatures still taken to be less.
In a more recent study, Zuhaila (2013) build on the work of Canning et al to develop a new mathematical model of the aqueous humour flow in the anterior chamber during descement membrane detachment (DMD). They considered this to be driven by buoyancy effects in rectangular shaped region. They also assumed a thin small flap, which is elastic and deformable and is kept in contact with the bottom plate. They applied the general theory of beam bending under axial load and used asymptotic analysis to analyze the flow behavior and the deformation of the flap. Both simplified models were solved numerically using COMSOL multiphysics. In their analysis, the results showed that buoyancy effects may be responsible for driving the flow in the anterior chamber during Desment Membrane Detachment (DSM).
CHAPTER THREE
MODEL FORMULATION AND SOLUTION
THE MODELS
A Model for Thermally-Driven Flow in the Anterior Chamber of the Eye
Schematic Diagram of the Anterior Chamber of the Eye
This model is for the flow in the front portion of the eye between the cornea and pupil.
Reasons for Changing the Model
Even though the thickness of the cornea is assumed to insulate the content of the anterior chamber (fluid) from fluctuating in areas where the ambient (or room) temperature is in the excess of the body temperature (370C). we find that this insulation action may not be true and thus, may not be constant (Gokul et al, 2013). However, which is the temperature at the chamber interface with the body may then be seen to be less than. If this is the case, then we find that is negative and will not generally give the required convective flow as required. Therefore, to generally care for various ambient temperatures, we need to modify the temperature variations in the eye which will in general modify the given equation for the flow in the anterior chamber of the eye.
CHAPTER FOUR
RESULTS AND DISCUSSION
Result and Discussion on the Models on Aqueous Humour Flow Driven by
Temperature Gradient
CHAPTER FIVE
SUMMARY AND CONCLUSION
SUMMARY
We considered and discussed two related models in the anterior chamber of the human eye; a generalized mathematical model of aqueous humour flow driven by temperature gradient and a mathematical model of fluid flow through a mesh channel.
Canning et al (2002) studied the model in situations where the exterior temperature is lower than the normal human internal temperature of 370C. In tropical Africa, there are regions where temperature rise beyond 370C. This work attempts to address this peculiar situation, the model of Canning et al (2002) is modified to address this situation. Analytical solutions were obtained; the MATLAB Software Package was used to present graphical profiles of the solution.
In the second model, we considered the effect of particles “seeding” or constrictions in the trabecular mesh channel and the subsequent increased pressure in the chamber as it may result to the pathological state of primary open angle glaucoma (POAG). The model equations are also solved and analyzed using MATLAB. The effects of resistance to flow and wall shear stress on the flow parameters were presented using graphs.
CONCLUSION
A mathematical model for the fluid flow through a mess channel has been presented. Solutions to the model equations are obtained using the slip boundary condition. It was observed that resistance to flow increases with the height and number of stenoses, Darcy’s number and slip parameter but decreases with viscosity ratio, half width ratio and couple stress parameter. Also, the wall shear stress increases with the height of the stenoses, half width ratio, couple stress parameter and the slip parameter but decreases with the viscosity ratio and Darcy’s number. Also, an elevation in IOP as a result of the stenoses narrows the meshwork thereby increasing the wall shear stress exerted by the flow of aqueous humour. This is a great risk factor for primary open angle glaucoma (POAG).
Furthermore, the model presented above shows that thermally driven flows in the anterior chamber are inevitable, and indicates that only very small temperature difference (high ambient temperatures than core body temperature) are required to drive such flows.
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