Biomedical Engineering Reference
In-Depth Information
where
C
is the concentration of O
2
in the tissue,
R
t
is a given position in the
tissue, and
R
is the blood vessel/tissue interface. The first boundary condition
indicates that the concentration of oxygen in the blood stream is constant. De-
termine the flux of oxygen that enters the tissue cylinder at
R
t
. Assume steady-
state and that the concentration of O
2
in the tissue cylinder is very small.
Tissue space
Blood
vessel
R
t
R
10.28 Consider a capillary-cylinder tissue model to describe transport of glucose (
A
)
taking into account the thickness of the capillary wall and the axial variation
of solute concentration inside the capillary. The concentration of
A
decreases
along the capillary owing to the solute diffusion from the capillary into the
surrounding tissue.
(a) Assume that radial variations in solute concentration inside th
e
capillary
are small, and use an average concentration inside the capillary
Cz
. The
()
average velocity in the capillary is
V
, and the capillary wall thickness is
R
0
-
R
1
.
Th
e flux (
N
) across the capillary wall is given b
y
NKC C
=
0
(
−
)
A
A
R
0
where
C
is the concentration in the capillary,
C
is the solute
AR
0
concentration in the tissue at the outer capillary wall and
K
0
is a mass
transfer coefficient. Do a solute balance on a thin slice of capillary
o
f thick-
ness
Δ
z
. Take the limit
Δ
z
→
0 to find a differential equation for
Cz
.
()
(b) Solve the diffusion equation for
C
A
, the solute concentration in the tissue
c
y
linder. Use as a boundary condition that the concentration at
R
=
R
0
is
C
, which depends on
z
. What values do you need to calculate the con-
AR
0
centration of gluco
s
e in the tissue?
(c) To show how
C
varies with
z
, do a macroscopic mass bal-
ance over a length
z
of the capillary as shown next. The rate at
which
A
enters the capillary at
z
=
0 minus the rate that
A
consumption per
unit time per unit volume of tis
su
e is
co
nstant. Write an algebraic equation
for the mass balance. Assume
C
is
C
at the capillary entrance.
0
(d) Take the derivative with respect to
z
of the result from (
C
) and substitute it
into the result from (b) to find the solute concentration in the capillary.
(e) Find the solute concentration in the tissue.
R
2
C
A
Tissue
V
R
1
R
0
Capillary
C
A
0
(z)
