Biomedical Engineering Reference
In-Depth Information
Fig. 2.4
(
a
) 2D illustration of the coating unit cell. Charges are distributed into the coating region
Ω
c
and the non-coating region
Ω
i
\
Ω
c
.(
b
) Swelling pressure computed by the coating model
under overall charge density
ρ
eff
/F
55 mM. The coating radius is set to be 18 nm, and the charge
fraction
λ
for the bridging GAGs varies from 0
.
0to0
.
6. The optimal fitting occurs at
λ
=
=
0
.
6
We propose a coating model in which the total unit cell charge
Q
is partitioned
into two GAG-based classes: one based on charge derived from long interfibrillar
bridging GAGs and one based on charge derived from short GAGs that form the
coating. As in the previous model, we exclude the volume of the collagen fibrils in
our analysis. A sequence of unit cell domains is defined by symmetry of the lattice
and is denoted
Ω
i
with volume
V
i
.Here
i
is a configuration index corresponding
to tissue thickness (and thus hydration). The coating subregion of the unit cell is
denoted
Ω
c
and has volume
V
c
, which is
independent
of
V
i
in accordance with
the findings of Fratzl and Daxer (
1993
). The setup is shown in Fig.
2.4
(a). Letting
λ
denote the charge fraction parameter, the unit cell has two charge densities
computed as follows
∈[
0
,
1
]
λ
Q
V
i
ρ
i
s
=
(2.18)
over
Ω
i
, and
λ)
Q
V
c
ρ
c
=
(
1
−
(2.19)
over
Ω
c
. Clearly, the total charge in the unit cell
Ω
i
is conserved. Further, the coat-
ing charge density
ρ
c
is invariant with respect to hydration (i.e. index
i
). If the
coating subdomains overlap at low thickness values, we assume that the charge den-
sity is additive in the overlap region in order to conserve total charge. The boundary
value problem to be solved on each unit cell domain
Ω
i
is then
sinh
Fϕ
RT
,
in
Ω
i
,
ρ
ε
−
2
FC
0
ε
2
ϕ
−∇
=
(2.20)
