Biomedical Engineering Reference
In-Depth Information
Figure 10.12
The Krogh tissue cylinder showing the capillary and the tissue space.
capillary. It assumes a cylindrical layer of tissue surrounding each capillary with
the solute transferred only from that capillary. The capillary is assumed to be cylin-
drical and of constant radius,
R
. As the solute move moves along the capillary, its
concentration
C
decreases because of solute transport through the capillary wall.
By considering the tissue cylinder surrounding each blood vessel to have a uniform
transport behavior, Krogh proposed that the diffusion of oxygen away from the
blood vessel into the annular tissue was accompanied by a zero-order reaction, that
is
R
O
2
= −
m,
where
m
is a constant. At steady state, (10.57) reduces to
dC
m
⎛
⎞
dr
A
=
rdr
.
(10.60)
⎜
⎟
⎝
⎠
dr
D
AB
Assuming that the concentration of oxygen in the blood stream is constant, a
boundary condition is
dC
rR
dr
=
,
A
=
0
R
t
where
C
is the concentration of
O
2
in the tissue, and
R
t
is a given position in the tissue. Integrating (10.60) twice and
substituting boundary conditions, and rearranging,
Another boundary condition is
C
=
C
t
at
r
=
(
)
rR
2
−
2
m
m
R
2
R
⎛⎞
t
CC
D
=+
+
ln
t
(10.61)
⎜
⎝⎠
A
t
4
D
2
r
AB
AB
Equation (10.61) is used to understand the concentration change within the
tissue and has served as the basis of understanding oxygen supply in living tissue.
Due to its simplicity and agreement with some observations, it is extensively used
and successfully extended to other applications such as hollow-fiber bioreactors
used for the production of monoclonal antibodies, bioartificial liver, to predict and
optimize the oxygen exchange performance of hollow-fiber membrane oxygena-
tors, drug diffusion, water transport, and ice formation in tissues.
