Problem 4.

A method for introducing drugs into the eye in a controlled manner is to use drugs encapsulated into therapeutic contact lenses. Model the eye as a one-dimensional space from x = L to x = 0. Starting from the mean escape time equation D2w = 1, we can calculate the average time for a drug molecule diffusing into the aqueous humour of the eye to travel from the contact lens at x = L to the back of the aqueous humour at x = 0.

By integrating twice, and using the boundary conditions, we can obtain that the average time for a molecule to pass from the contact lens to the back of the aqueous humour is w(L) = L2/2D. If the diffusivity of the drug in the eye is D = 5 105cm2s1 and L = 2.25mm, we find the time to be 506.25s, or approximately 8.5 minutes. Note that in this simple calculation, we ignored the time taken to traverse the cornea.

To reach the aqueous humour, the drug must in fact first cross the cornea. Model the cornea as 0.5mm thick and suppose the diffusivity of the drug in the cornea is 1 107cm2s1. The same calculation as in the previous question (but with the new parameters) gives the average time to cross the cornea as 12500s, or approximately 3.5 hours. This indicates that the main delay for the drug delivery would be caused by its low diffusivity in the cornea. The full answer to the problem requires us to solve D(x)w(x) = 1 where D(x) is the function describing the diffusion coefficient as a function of distance through the eye. Do this. Make a labelled plot of w(x) valid on 0 < x < 2.3mm and calculate the average time for a drug molecule to reach the back of the aqueous humour after passing through the cornea and the aqueous humour.