A New Mathematical Model of Corneal Topography

Sight is the most important sense that we posses. It is crucial to

understand the mechanics of vision in order to treat various diseases

that may occur and disturb regular seeing. The eye's main part

responsible for about two-thirds of refractive power is the cornea.

Cornea is transparent, shell-like structure situated in the frontal part

of the eye. It is one of the most sensitive parts of human body and its

various irregularities can cause many seeing disorders. Precise

knowledge of corneal shape is very important and accurate

mathematical models are necessary to fully understand biomechanics

of cornea. We present a new model of corneal geometry based on a

nonlinear membrane equation. We establish existence of solution and

provide some estimates. When fitting with data we use its simplied

form and find that mean error is of order of a few percent. Also, we

are concerned with determining some unknown parameters when the

solution is known (usually with a noise). This is one example of so

called Inverse Problems. They are usually more difficult to solve and

analyze than direct ones. Moreover, they often are ill-posed, that is,

not necessarily have unique solution which is continuous with respect

to the initial data. We propose some regularization methods and apply

them to the real corneal data. We struggle with two types of

different inverse problems. The first one concerns constant parameter

case. It turns out that determination of these constants is a nonlinear

problem in two unknowns to which solution we develop an iteration

scheme and prove rates of convergence. The second problem is linear

one and concerns the case when one of unknown parameters is not

necessarily constant.
We propose some regularization method based on classical ones. In

the end we obtain a stable method of determining unknown

parameters of our differential equation from the knowledge of

corneal shape. These parameters may have direct relation to some

biomechanical properties of the eye and can be used to provide some

insights of corneal structure.