Computations for Twersky's theory of corneal transparency based on Hart-Farrell's swelling pressure

Steven B. Berger* Harvard-MIT Program in Health Sciences and Technology and Department of Physics, Massachusetts Institute of Technology, Cambridge,

Massachusetts 02139 (Received 24 November 1975, revision received 27 May 1976)

We have calculated corneal transparency using a theory given by Twersky and a model of Hart and Farrell. The results do not correspond with experimental observations.

Recently, Twersky1 has presented a simple approxima­tion for the attenuation of light by pair-correlated ran­dom distributions of lossless sca t te rers with small r e ­fractive-index contrast, and with average center spac­ing small compared to wavelength. The purpose of this let ter is to apply this treatment to a calculation of cor­neal transparency that takes into account hydration ef­fects based on the model of Hart and Farre l l 2 of the corneal s troma.

To review briefly the theory, we note that the t r ans ­mission of the cornea at a given thickness t=q/q1, where q1 is normal thickness, is given by (following Twersky's development)

and W(w) is the packing factor. W(w) is given by

where

and v1 =0.44 (in accord with the data of Hedbys and Dohlman3).

Thus,

where

One may verify that

and from standard Taylor ser ies expansions of E and F, we find

p is the pressure in the fluid of particles, kB is Boltz-mann's constant, TA is the absolute temperature, ρ is the number of particles per unit volume, v is the vol­ume per particle, and w = ρv is the volume fraction.

The swelling pressure of corneal s t roma in the p r e s ­ence of saline has been calculated on the basis of a structural theory of the stroma by Hart and F a r r e l l . 2

Hart and Far re l l give, for a poly electrolyte gel in 0.15 M NaCl solution,

p = swelling pressure

Setting T(t = 1.0) = normal transmittance = 0.90, 4 we find, for the packing factor,

This gives

and

where

with

F(k) is the complete elliptic integral of first kind

Apparently, the transmittance calculated in this mod­el does not fall very much as the thickness increases. We may explain this discrepancy with experimental ob­servation by noting that as the thickness increases r e ­gions where collagen fibers a r e absent ("lakes") form in the stroma which are effective sca t terers of light. This may point to a defect in Twersky's analysis. Al­ternatively something may be wrong with the Har t -Fa r -rel l swelling pressure model. It is clear that further work must be done to resolve this issue.

E(k) is the complete elliptic integral of second kind lV. Twersky, J. Opt. Soc. Am. 65, 524 (1975). 2R. W. Hart and R. A. Farrel l , Bull. Math. Biophys. 33, 165

(1971).

864 J. Opt. Soc. Am., Vol. 66, No. 8, August 1976 Copyright © 1976 by the Optical Society of America 864

3B. O. Hedbys and C. H. Dohlman, Exp. Eye Res. 2, 122 (1963).

4We do not enter into a discussion here of the problems of passing from two to three dimensions and the questions posed

by introducing a scaling length which is assumed to be con­stant, independent of the parameters of interest . The physi­cal significance of such a scaling length is unclear from Twersky's work.

865 J. Opt. Soc. Am., Vol. 66, No. 8, August 1976 Copyright © 1976 by the Optical Society of America 865