Biomedical Engineering Reference
In-Depth Information
Fig. 7 Left: Cell position x, maximum adhesion interaction distance R
A
, volume V (light gray
area), nuclear volume V
N
(dark gray area), and equivalent radii R and R
N
. Right: Key forces in
the model, labeled for cell 5. Figures reprinted with permission from [
56
]
0
1
cell
cell interactions
z}|{
cell
BM interactions
@
A
z}|{
X
N
ð
t
Þ
1
m
þ
c
i
cma
E
F
i
cca
þ
F
i
ccr
þ
F
i
cba
þ
F
i
cbr
þ
F
loc
v
i
¼
; ð
10
Þ
j
¼
1
j
6¼
i
|{z}
cell
medium interactions
where N
ð
t
Þ
is the number of simulated cells/agents at time t. For this discussion, we
set E
0 and F
loc
¼
0 to model nonmotile cells contained in a lumen without ECM.
See [
56
] for the specific forms of the forces, which were modeled using potential
functions with finite interaction distances, consistent with the maximum adhesion
interaction distance R
A
. These forces are labeled for cell 5 in Fig.
7
(right).
Each cell has a phenotypic state
Sð
t
Þ2 A;P;Q;f g
, where
A
cells are
apoptosing,
P
cells are proliferating (in non-G
0
),
Q
cells are quiescent (in G
0
), and
N
cells are necrotic. Transitions between phenotypic states are governed by microen-
vironment and signaling-dependent stochastic processes. For example, quiescent
cells enter the cell cycle with a (nondimensionalized) O
2
-dependent probability:
!
Z
t
þ
Dt
t
a
QP
O
2
ð
s
Þ
O
2
;
hypoxic
1
O
2
;
hypoxic
Prob
ðSð
t
þ
Dt
Þ¼PjSð
t
Þ¼QÞ¼
1
exp
ds
Dt
;
O
2
ð
t
Þ
O
2
;
hypoxic
1
O
2
;
hypoxic
a
QP
ð
11
Þ
where a
QP
is the normoxic
Q!P
transition rate (when O
2
¼
1), and O
2
;
hypoxic
is
the hypoxic oxygen threshold. The
Q!A
transition is similar but does not
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