Biomedical Engineering Reference
In-Depth Information
Fig. 17.1
Permeation analysis for a biphasic-solute tissue. The finite element mesh is shown at
left
. For this analysis,
h
=
1mm,
p
0
=
0
.
2MPa,
c
0
=
1 mM. The solid matrix is neo-Hookean
with Young's modulus
=
1 MPa and Poisson's ratio
=
0; the initial solid volume fraction
is
ϕ
r
=
0
.
2. The hydraulic permeability is 10
−
3
mm
4
/
N
·
s; the solute diffusivity in free so-
lution is
d
0
=
10
−
3
mm
2
/
s, and the diffusivity in the tissue is
d
=
0
.
5
×
10
−
3
mm
2
/
s. At
steady-state, the fluid flux and solute flux are uniform and equal to 0
.
22
×
10
−
3
mm
/
sand
0
.
11
×
10
−
3
nmol
/
mm
2
·
s, respectively
p
u
=
to determine that the effective fluid pressure in the upstream bath is
˜
p
0
−
p
d
=−
RθΦ
∗
c
0
, and that in the downstream bath is
RθΦ
∗
c
0
. Since
˜
p
is a variable
˜
p
u
on the
that remains continuous across boundaries, we may thus prescribe
p
u
=˜
˜
p
d
on the downstream face. Similarly, for the solute, the
boundary conditions must be prescribed on the effective concentration
upstream face, and
p
d
=˜
˜
c
instead
˜
c
u
=
c
u
=
c
0
, and
c
d
=
c
d
=
c
0
, where
of
c
. This is done using Eq. (
17.13
), thus
κ
∗
=
we have made use of the fact that
1 for a neutral solute in a fluid bath under
κ
∗
=
1 and
ψ
∗
=
ideal conditions (
0). Finally, it is also necessary to enforce traction
and displacement boundary conditions: On the unconstrained upstream face,
t
u
=
−
p
0
n
, whereas on the constrained downstream face, the axial displacement is zero.
Initial conditions also need to be prescribed for this problem. It may be assumed
initially that the upstream and downstream bath pressures are both atmospheric, thus
˜
p
u
=˜
0. According to Eqs. (
17.15
), (
17.16
), equilibrium conditions (zero fluid
and solute flux, and zero solid velocity) are achieved when there is no gradient in
˜
p
d
=
p
and
c
. Thus, based on the boundary conditions and the requirement for a uniform
value of
˜
p
at equilibrium, it may be assumed initially that
˜
p
˜
=
0 throughout the
tissue. By a similar argument, it may be assumed initially that
c
˜
=
c
0
throughout the
tissue.
Upon the application of the fluid pressure upstream, a transient response ensues,
whereby the tissue slowly compacts as interstitial fluid begins to exude from its
