Biomedical Engineering Reference
In-Depth Information
TABLE 6.9 Peak Convection Rates of Blood
in the Vasculature
Compartment
cm/sec
Aorta
140
40
Common carotid
100
20
Vertebral
a
36
9
Superficial femoral
90
13
Liver sinusoid
b
10
2
10
3
Data from DeWitt and Wechsler (1988).
a
Data from Jager et al. (1985).
b
Data from McCuskey (1984).
environment conducive to the differentiated state. The dominant mechanism of transport of
low molecular weight nutrients (e.g., oxygen and glucose) within tissue or cell aggregates is
diffusion. The length scale for diffusive transport (i.e., the distance over which oxygen pene-
trates into the tissue from nutrient stream before it is completely consumed by the cells)
depends on the volumetric concentration of metabolite in the nutrient stream,
C
0
; the rate
at which the cells consume the nutrient,
Q
i
; the diffusion coefficient for the metabolite in the
tissue,
D
t
; and the system geometry. The metabolic consumption rate
Q
i
is generally a function
of the nutrient concentration in the cell mass,
. The most common rate expression is
the Michaelis-Menton type, which reduces to zero-order kinetics for high concentrations
of nutrient (i.e.,
C
C
K
m
) and to first-order kinetics for low nutrient concentrations (i.e.,
C
K
m
). It is reasonable to consider the zero-order limit because the condition
C
K
m
is
met for many important nutrients under normal physiological conditions.
An estimate of the length scale for nutrient diffusion in a 3-D cell mass can be obtained
by mathematical modeling. The general equation governing the balance between steady-
state diffusion and metabolic consumption is
2
D
i
C
¼
Q
i
ð
6
:
18
Þ
where
is the concentration of the nutrient within the cell mass. Three simple geometries
amenable to analytical solutions are shown in Figure 6.24: a slab, cylinder, or sphere of cells
bathed in medium containing the nutrient at concentration
C
. Expanding the gradient
operator for each of these geometries, Eq. (6.18) can be written as
C
0
2
:
D
t
d
C
dx
slab
¼
Q
i
ð
6
:
19a
Þ
2
¼
Q
i
1
r
d
dr
rdC
dr
cylinder
:
D
t
ð
6
:
19b
Þ
¼
Q
i
2
1
r
d
dr
r
dC
dr
sphere
:
D
t
ð
6
:
19c
Þ
