Biomedical Engineering Reference
In-Depth Information
is the averaging volume of the cell phase (m
3
); V is the averaging volume (m
3
); k
is a cell conversion factor (mol cell
-1
); d is the death rate (day
-1
); and n is a semi-
empirical parameter. This equation inhibits cell growth in case of an overpopu-
lation of available carrier space [
19
]. The Moser equation restricts cell growth in a
direct relation to the available oxygen concentration [
88
,
104
],
"
#
A
cell
c
O
2
ð
x
;
y
;
z
;
t
Þ
P
Mr
x
;
y
;
z
;
t
ð
Þ
c
cell
Þ
dx
;
y
;
z
;
t
ð
Þ
K
q
e
cell
þ
c
O
2
ð
x
;
y
;
z
;
t
The Moser equation reduces to the Monod equation for n equal to one. This
equation couples the OUR directly to the cell growth rate as,
A
cell
;
max
Y
CO
2
c
O
2
x
;
y
;
z
; ð Þ
K
q
e
cell
þ
c
O
2
x
;
y
;
z
;
t
P
Md
x
;
y
;
z
;
t
ð
Þ
c
cell
þ
m
cell
Þ
dx
;
y
;
z
;
t
ð
Þ
ð
where A
cell,max
is the maximal specific cell growth rate (day
-1
); Y
CO2
is the yield of
cells per unit oxygen (cells mol
-1
); m
cell
is the maintenance coefficient (mol cell
-1
day
-1
), the minimum oxygen consumption required to keep the cells alive.
The main difference between linear and non-linear systems is that the former will
produce a significant region of uniform proliferation, a phenomenon that is rarely
observed in practice [
70
]. This heterogeneity can also implicitly be implemented
using a custom-defined function derived from experimental data [
25
,
36
].
Finally new matrix production and carrier remodeling can also directly alter
oxygen delivery to the cells, though this effect is more pronounced for larger
solutes [
9
,
13
,
94
].
In the following paragraphs we will give a brief overview on how these oxygen
models can be efficiently applied to tackle some specific problems a tissue engi-
neer could encounter. Static in vitro culture of tissue substitutes generally gives
rise to heterogeneous cell growth, especially when substitute dimensions exceed a
critical size [
25
,
80
]. Enhanced proliferation of cells in the peripheral regions and
coupled increases in oxygen uptake, have thereby been speculated as factors
determining the incidence and severity of tissue hypoxia and associated cell death
[
10
]. To test this hypothesis and gain improved understanding of the mechanisms
which underlie these observations, mathematical models have been developed
describing the interactions between oxygen tension and cell density. Effectively
applying this strategy Demol et al. presented a model to describe in vitro behavior
of human periosteum derived cells cultured inside a fibrin hydrogel construct
(Fig.
2
)[
25
].
Necessary model input parameters were derived from dedicated in vitro
experiments that allowed to assess cell proliferation, the influence of oxygen
tension on cell death and proliferation, and the diffusivity of oxygen in fibrin. As
the constructs were cultivated over a period of 14 days, a significant region of dead
cells in the construct center could be detected which progressively expanded
outwards with longer cultivation times (Fig.
3
a). The observation was accompa-
nied by the formation of a multilayered cell sheet which had an average thickness
Search WWH ::
Custom Search