Biomedical Engineering Reference
In-Depth Information
with the boundary conditions:
|
|
=
|∇
|
=
ϕ
0
,
ϕ
·
n
0
,
on
Γ
inner
,
(2.21)
∇
ϕ
·
n
=
0
,
on
Γ
outer
,
(2.22)
and where
Γ
inner
and
Γ
outer
denote the inner and outer boundaries of the unit cell,
respectively. The electrostatic free energy takes the form
2
RTC
0
cosh
Fϕ
RT
1
d
Ω
i
,
ε
2
|∇
ϕ
|
2
F
el
=
−
+
ρϕ
−
−
(2.23)
Ω
i
where the charge density distribution
ρ
in Eqs. (
2.20
) and (
2.23
) is given by
⎧
⎨
ρ
i
s
,
in
Ω
i
\
Ω
c
,
ρ
(i)
ρ
i
s
+
=
(2.24)
ρ
c
,
on
Ω
c
,
⎩
2
(ρ
i
s
+
ρ
c
),
on
Ω
c
-
c
.
Here
Ω
c
-
c
refers to the subdomain of
Ω
c
resulting from the possible geometric
intersection of coating domains as
Ω
i
is mapped to correspond to lower thickness
values with increasing index
i
.
In this model, the coating radius is taken to be
r
c
=
18
.
0 nm. The effective charge
density is again defined by
ρ
eff
=
Q/V
0
and used for reporting results. The influ-
ence of the charge fraction
λ
on the swelling pressure is shown in Fig.
2.4
at the
physiologically plausible
ρ
eff
/F
=
0, all charge is concentrated in
the coatings and the swelling pressure is zero until the coatings begin to interact at
around
t
55 nm. At
λ
=
0
.
3mm.As
λ
is increased the swelling pressure increases monotonically
toward the experimental curve. At a value of
λ
=
0
.
6, the computed swelling pres-
sure finds excellent agreement with the experimental curve and further improves on
the result of the previous section.
=
2.4 Molecular-Level Unit Cell Model
Here we present a molecular-level unit cell model which explicitly considers the
GAG chains that are bridging neighboring collagen fibrils. In this model, the bridg-
ing GAGs domains are approximated by an effective cylindrical volume (Hart and
Farrell,
1971
; Jin and Grodzinsky,
2001
). The non-bridging GAGs that constitute the
collagen fibril coating are modeled, as above, with a continuum description of the
charge density. The charge density within the cylindrical GAG domain is denoted
ρ
g
and is determined by three parameters: the half length of the GAG disaccharide
unit
b
0
.
64 nm (Jin and Grodzinsky,
2001
), the cylinder radius
r
g
and a molecular
ratio factor
α
=
=
L
c
/L
d
, where
L
c
is the contour length of the GAG chains and
L
d
is end-to-end distance. Then
ρ
g
may be computed from
αe
πbr
g
ρ
g
=
,
(2.25)
