The degree of surfactant penetration inside the ocular surface should also be proportional to the surface area available for the surfactant to diffuse into the epithelium. Similarly, when liposomes are used as an alternative, the amount of surfactant penetrating the lipid bilayer should be directly proportional to the surface area of the liposomes. This clearly suggests that experiments have to be designed with differences between liposomes and the corneal epithelium taken into consideration. To illustrate the importance of differences between liposome and corneal geometry, it is instructive to consider a mass balance on substances such as lysosomal enzymes, histamine, and inflammatory mediators that begin to leak from inside the corneal cells due to the toxic effects of surfactant penetration into the bilayer of the corneal epithelium. The mass balance yields
\[V_{\mathrm{Cornea}}\ \frac{dC_{S}}{dt}\ {=}\ {-}K_{\mathrm{Perm}}A_{\mathrm{Cornea}}C_{\mathrm{S}}\]
where
V Cornea is the cellular volume of the corneal epithelium,
K Perm is the permeability of the corneal epithelium to the species that leaks out,
A Cornea is the corneal area available for penetration, and
C S is the concentration of the species of interest inside the corneal cells. Similarly, a mass balance on a test component such as a dye present inside the liposomes can be obtained by
\[V_{\mathrm{Liposome}}\ \frac{dC_{\mathrm{Dye}}}{dt}\ {=}\ {-}K_{\mathrm{Perm,Lipo}}\ A_{\mathrm{Liposome}}C_{\mathrm{Dye}}\]
where
V Liposome is the volume of the liposomes,
K Perm,Lipo is the permeability of the liposomes,
A Liposome is the liposome surface area available for penetration, and
C Dy e is the concentration of the species that leaks out. The presence of surfactants is manifested in increased permeabilities for both the cornea and the liposomes. Based on these equations, the time scale for the leakage of the molecules is
KA/V. The surface area-to-volume (
A/
V) ratio is much larger for liposomes due to their small size. Thus, if the surfactant concentration for the liposome assay is chosen to be the same as that in the Draize test, or even the CMC, the time scale for dye leakage will be extremely small, and so the percentage leakage will be very large unless the measurements are done at extremely short intervals. Since short-time measurements are prone to artifacts due to issues such as mixing, it is more appropriate to ensure that the time scale for the leakage in the liposomes is comparable to that in the corneal cells. Since the time scale is
KA/V, the higher values of
A/
V can be compensated for by a lower
K for the liposome assay. The permeability
K is related to the amount of surfactant that binds to the liposomes, and so the value of
K in the liposome assay can be controlled by controlling the surfactant concentration in the assay. Based on these arguments, the following equation can be used to evaluate the effective concentration that should be tested in the liposome assay to correctly predict the irritancy of surfactants in vivo
\[C_{\mathrm{TEST}}{=}\ \frac{A_{\mathrm{Cornea}}V_{\mathrm{Liposome}}}{V_{\mathrm{Cornea}}A_{\mathrm{Liposome}}}\ \mathrm{CMC}\]