Biomedical Engineering Reference
In-Depth Information
phase requires,
ρ
f
F
=
C
Na
+
−
C
Cl
−
+
0
,
(2.2)
where
F
is the Faraday constant. Equations (
2.1
) and (
2.2
) can be solved for the
equilibrium mobile ion concentration (Buschmann and Grodzinsky,
1995
)giving
ρ
f
ρ
f
2
F
+
C
Na
+
/
Cl
−
=∓
C
0
.
4
F
2
+
(2.3)
The osmotic pressure in the two phases is given by
RT( C
Na
+
−
C
Cl
−
),
P
poly
=
P
bath
=
2
RTC
0
,
(2.4)
where
R
is the gas constant and
T
is the absolute temperature. The osmotic pressure
difference
P
os
between the two phases is then computed as
2
RTC
0
ρ
f
1
.
P
os
=
P
poly
−
P
bath
=
4
F
2
C
0
+
1
−
(2.5)
In Fig.
2.2
we depict experimental measurements and fitting curves for the equi-
librium swelling pressure of human corneal stroma with a bath concentration of
C
0
=
0
.
15 M as reported by Hedbys and Dohlman (
1963
) and Olsen and Sper-
ling (
1987
). Letting
ρ
0
represent the fixed charge density at physiological sample
thickness
t
0
=
0
.
5 mm, and letting
ρ
f
represent the charge density at tissue sample
thickness
t
, we find by conservation of fixed charge that
ρ
f
=
ρ
0
(t
0
/t)
. Using this
result in Eq. (
2.5
), the osmotic pressure difference at thickness
t
may be estimated
in terms of the physiological fixed charge density
ρ
0
. It has been shown that
ρ
0
de-
pends on the salt concentration in the bath through a process of ion binding. Hodson
(
1971
) has estimated
ρ
0
/F
for human stroma at physiological hydration and bath
ionic concentration
C
0
=
0
.
15 M to be approximately 48 mM. Values of
ρ
0
/F
for
bovine cornea have been measured at around 36 mM; see Elliott and Hodson (
1998
)
for a review. The osmotic pressure difference
P
os
based on Donnan theory Eq. (
2.5
)
with
ρ
0
/F
48mMisshowninFig.
2.2
. The prediction agrees well with the ex-
perimental data at physiological thickness
t
=
0
.
5 mm but deviates significantly for
all other values, particularly for
t<
0
.
5 mm. We conclude that Donnan theory is
incapable of predicting swelling pressure accurately for lower thicknesses, which
correspond to hydration levels lower than physiological.
=
2.2.2 Poisson-Boltzmann Theory
We now consider a thermodynamic framework for describing the swelling pressure
experiment on corneal stroma. Consider a sample of isolated corneal stroma of area