The mathematical modelling of flow and deformation in the human eye

Abstract

Modelling the human eye provides a great challenge in both the field of mathematical medicine and in engineering. Four different problems regarding flow and deformation in the eyeball are considered, showing how changes in both the fluid and solid mechanicsof the human eye contribute to the development of pathological states. Firstly,a mathematical model is presented for the flow of aqueous humour through the trabecular meshwork and into the Schlemm canal. This predicts the intraocular pressure and investigates how this influences primary open angle glaucoma. Secondly, paradigm problems concerning the development of rhegmatogeneous retinal detachment are presented. A two-dimensional model of pressure driven fluid flow between rigid walls, and
between one rigid and one moving wall is presented and followed by a three-dimensional model concerning liquefied vitreous humour flow induced by saccadic eye motion. The purpose of these models is to examine the flow behaviour and the deformation of the detached retina. Thirdly, a mathematical model of aqueous humour flow, driven by buoyancy effects through the detached descemet membrane in the anterior chamber, has been developed to analyse the fluid mechanics concerning the progression of descemet membrane detachment. Lastly, mathematical models studying the effects of a tonometer and a scleral buckle on the shape of the eyeball membrane are presented. The modelling of fluid flow in these studies is based on the lubrication theory limit of the Navier-Stokes equations. However, the full Navier-Stokes equations have been used in the development of a three-dimensional model of retinal detachment. In the modelling of the tonometry and scleral buckling the membrane theory of spherical shells has been used. The results of these models predict changes in the intraocular pressure as well as examining the fluid flow behaviour and the deformation of the detached retina. The modelling of descemet membrane detachment is shown to explain the progress of the spontaneous reattachment and redetachment of descemet membrane may be controlled under the correct conditions. The results of the modelling of the tonometer cast doubt on the Imbert-Fick law, but the results of the scleral buckle may prove useful to predict changes in the focal length of the eye when a scleral buckle is present.

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