Biomedical Engineering Reference
In-Depth Information
d
M
d
t
¼
M
M
M
1
;
(3.3)
d
½
p27
d
t
¼
M
M
c
2
c
02
B
c
1
1
w
c
02
½
p27
;
(3.4)
þ
d
½
npRB
(3.5)
¼
d
2
ð
d
2
þ
d
1
½
Þ½
;
cycCDK
npRB
d
t
,
M
*
,
c
1
,
c
2
,
B
,
where
b
3
,
J
3
,
b
4
,
J
4
,
a
1
,
a
2
,
a
3
,
a
4
,
Z
w
,
d
1
and
d
2
are constants,
specified in [
22
].
In (
3.1
)-(
3.5
), when
M
is small, the cell is maintained in a state corresponding to
G
1
for which levels of Cdh1 are high and levels of cycCDK are low. Growth in the
cell mass increases Cdh1 degradation and reduces p27 production, so that cycCDK
increases. This leads to inhibition of npRB and Cdh1 and, hence, positive feedback
on cycCDK. At a certain point, corresponding to the
G
1
-
S
transition, the state with
high Cdh1 and low CDK is lost, and a state with low Cdh1 and high cycCDK is
attained. Finally, when Cdh1 levels are sufficiently low and CDK levels sufficiently
high, cell division occurs. The external
O
2
concentration
c
02
couples the subcellular
and diffusible scales by influencing progress through the cell cycle. Decreasing
c
02
reduces p27 degradation, and the resulting increase in levels of p27 counteracts the
effect of the increasing mass on cycCDK. In particular, if
c
02
levels are sufficiently
low, the
G
1
-
S
transition cannot occur. Further details about the model can be found
in [
2
,
3
].
The intracellular concentration of p53 regulates normal cell apoptosis, and that
of VEGF controls VEGF release by normal cells. The dynamics of p53 and VEGF
are coupled to one another and to the extracellular oxygen concentration, as
described by the following differential equations:
d
½
p53
d
t
¼
c
02
K
p53
þ
k
0
7
k
7
c
02
½
p53
;
(3.6)
d
½
VEGF
k
0
8
½
p53
½
VEGF
c
02
K
VEGF
þ
k
0
8
¼
k
8
þ
c
02
½
VEGF
;
(3.7)
d
t
J
5
þ½
VEGF
with the constants
k
7
,
k
0
7
,
K
p53
,
k
8
,
k
0
8
,
k
0
8
,
J
5
and
K
VEGF
(see [
22
]). The ODEs
(
3.1
)-(
3.7
) are solved subject to the initial conditions specified in [
22
], using the
open source CVODE library.
1
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