Biomedical Engineering Reference
In-Depth Information
=
downstream side (Fig.
17.1
,
t
10 s); as time progresses, fluid flux on the upstream
side increases until, at steady-state, the fluid flux becomes uniform throughout the
tissue thickness (
t
10
3
s). In FEBio, it is possible to request a full transient anal-
ysis, or to restrict the analysis only to the steady-state response. At steady-state, as
expected, it is found that the fluid flux
w
along the thickness of the sample is di-
rected from the high pressure fluid bath to the atmospheric fluid bath. This is the
classical phenomenon known as permeation. Furthermore, despite the fact that the
solute concentration returns to a uniform distribution at steady-state, the fluid pres-
sure gradient continues to drives the solute from upstream to downstream, because
of the frictional interactions between solvent and solute. This mechanism is generi-
cally known as the convective effect of the solvent on the solute; more specifically
in this problem, this phenomenon may be described as barophoresis.
=
17.4.2 Diffusion and Osmosis
Diffusion represents the flux of solute in response to a gradient in its concentration.
Consider a canonical problem of diffusion of a neutral solute across a tissue sample,
from an upstream bath at a concentration
c
0
to a downstream bath with negligible
concentration. Both upstream and downstream baths are maintained at zero ambient
pressure. Therefore, upstream boundary conditions are
c
u
=
˜
c
0
,
p
u
=−
˜
Rθc
0
(under
ideal conditions,
Φ
∗
=
0,
and zero axial displacement. Initial conditions may assume that the upstream bath
also starts out with zero solute concentration, so that
1),
t
u
=
0
, and downstream conditions are
c
d
=
˜
0,
p
d
=
˜
c
˜
=
0 and
p
˜
=
0 throughout
the tissue sample initially.
Upon raising the solute concentration to
c
0
upstream, a transient response ensues
whereby the solute begins to diffuse into the tissue on the upstream side. Over time,
the solute concentration within the tissue evolves into a linear profile, producing
a uniform solute flux across the entire thickness (Fig.
17.2
). Simultaneously, it is
observed that the solvent flows from the downstream side toward upstream (from the
low concentration bath to the high concentration bath). This mechanism is known
as osmosis.
17.4.3 Electrophoresis and Electroosmosis
To examine the phenomena of electrophoresis and electroosmosis, consider a tissue
described by a triphasic material, where the ions represent Na
+
(with
z
Na
=+
1)
and Cl
−
(
z
Cl
1) and the solid matrix is negatively charged. As in the above ex-
amples, the tissue is located between an upstream and a downstream bath. In these
baths, due to electroneutrality and the absence of a charged solid matrix, the anion
and cation concentrations must be the same. The salt concentrations in the upstream
and downstream baths may also be taken to be the same, to prevent diffusion and
=−
