Biomedical Engineering Reference
In-Depth Information
It is remarked that this expression defines a concave functional; it has a unique
equilibrium potential ϕ at which the functional is maximized (Fogolari and Briggs,
1997 ).
Specializing ( 2.13 ) and ( 2.14 ) to a binary electrolyte we find:
sinh Fϕ(r)
RT
,
ρ f
ε
2 FC 0
ε
2 ϕ
−∇
=
(2.15)
and
2 RTC 0 cosh
RT
1 d Ω.
ε
2 |∇
2
F el =
ϕ
|
+
ρ f ϕ
(2.16)
Ω
Returning to the equilibrium swelling pressure problem and taking (as in the Don-
nan solution) a uniform fixed charge density of ρ f /F
=
48 mM and bath ionic con-
centration of C 0 =
0 . 15 M, the swelling pressure was computed using Eq. ( 2.8 )
and the results are shown for varying sample thickness t in Fig. 2.2 . The deriva-
tive of the free energy
F el with respect to t was computed using central differ-
ence. As expected for this case of uniform charge density, the results are consistent
with the Donnan prediction and confirm the thermodynamic framework provided by
Eqs. ( 2.8 ) and ( 2.16 ).
2.2.3 An Unit Cell Model Based on Collagen Fibril Volume
Exclusion
Both the Donnan and Poisson-Boltzmann (PB) theories confirm that the assumption
of a spatially uniform charge distribution results in an unsatisfactory prediction of
stromal swelling, especially for low levels of hydration. However, it is observed that
the collagen fibrils occupy approximately 30 % of the stroma by volume (Meek and
Leonard, 1993 ). As the tissue is compressed during the swelling pressure experi-
ment, we argue that (i) the volume occupied of the collagen fibrils is unaffected by
the change in stromal thickness, and (ii) the collagen fibrils maintain very few net
ionized groups and contribute little or nothing to the electric balance within the tis-
sue (Elliott and Hodson, 1998 ). Therefore, the GAG charges, which are conserved,
must be restricted to the volumetric region between the collagen fibrils. Clearly, as
the tissue is compressed, there will be a nonlinear increase in charge density due to
the geometric effect of the collagen fibril volume exclusion.
Experimental estimation of the interfibrillar spacing l c and radius of collagen
fibril r f in the human cornea suggest they lie in the ranges of 45
60 nm and
11 . 5
15 . 0 nm, respectively (Fratzl and Daxer, 1993 ; Elliott and Hodson, 1998 ;
Muller et al., 2004 ). Consider a perfect hexagonal collagen lattice with l c =
53 . 0nm
and r f =
12 . 5 nm as shown in Fig. 2.3 (a). Charge is now assumed to be uniformly
distributed in the interfibrillar region Ω s ∈ R
3 only and is zero in the fibril domains.
The PB equation ( 2.13 ) was solved on a sequence of unit domains
Ω i f }
{
in which
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