Biomedical Engineering Reference
In-Depth Information
the polyelectrolyte system that recognizes the spatial heterogeneity of charge den-
sity that exists in the tissue. A similar thermodynamic approach was employed by
Hart and Farrell (
1971
), but the present work uses an entirely different description
of the electrostatic free energy. The electrostatic free energy of a polyelectrolyte
found through a mean-field approximation can be expressed as a functional of the
electrostatic potential, fixed charge density and local ionic concentrations (Che et
al.,
2008
). The electrostatic potential is determined from solution of the Poisson-
Boltzmann equation over a unit cell and the swelling pressure is found as the gra-
dient of the free energy with respect to the swelling volume. By considering the
volumetric domains of polyelectrolyte excluded by the collagen fibrils, the model
finds excellent agreement with the experimental swelling pressure data.
In order to improve the model for low levels of hydration, we were lead by ex-
perimental observations to postulate that the stromal PGs are partitioned into two
sets. One set is associated with PGs that bridge (perhaps by duplexing of the longer
DS and CS GAGs) between neighboring collagen fibrils; these supply the charge
density responsible for the osmotic pressure at physiological hydration. A second
set produces a charge-rich coating around the collagen fibrils (perhaps formed pri-
marily by the shorter KS GAGs). At physiological hydration, the coatings do not
interact and add very little to the osmotic pressure. As hydration is reduced, the col-
lagen fibrils come into closer proximity and the coatings will overlap producing a
significant increase in local charge density and a concomitant increase in swelling
pressure and electrostatic repulsion. We conclude that the PG-coatings may rep-
resent a mechanism to order the collagen fibril lattice as required in order for the
cornea to be a good transmitter of light.
2.2 Comparison of Donnan and Poisson-Boltzmann Theories
Applied to the Cornea
2.2.1 Donnan Theory
If a polyelectrolyte phase is in equilibrium with an external bath ionic solution, os-
motic pressure will result from the polyelectrolyte fixed charges and the disparity of
ionic concentrations in the two phases. Donnan theory may be employed to model
the osmotic pressure under the assumption that the fixed charge density is spatially
invariant. Consider a sample of isolated corneal stroma placed in a NaCl bath. As-
suming ideal Donnan equilibrium, the distribution of mobile ions satisfies
C
Na
+
C
Cl
−
=
C
0
,
(2.1)
where
C
Na
+
and
C
Cl
−
are the mobile ion concentrations in the stroma and
C
0
the
ionic concentration in the bath. The GAG disaccharide units provide a fixed (non-
mobile) negative charge density
ρ
f
and electroneutrality within the polyelectrolyte
