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
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