Biomedical Engineering Reference
In-Depth Information
depend upon O
2
. Cells become irreversibly necrotic (
S¼N
) when O
2
\O
2
;
hypoxia
.
The proliferative and apoptotic states have fixed durations s
P
and s
A
. Cell volume
and other key properties are controlled by a ''sub-model'' for each phenotypic
state. Proliferating cells in
P
divide in half after progressing through S, G
2
, and M;
their two daughters spend G
1
growing (linearly) to their mature volumes and then
return to
Q
. Apoptotic cells are removed from the simulation after s
A
. We do not
impose contact inhibition (a common feature for cellular automata models:
reduced
Q!P
transitions for cells when surrounded by neighbor cells); this is
because patient pathology for Ki-67 (a proliferation marker) frequently shows
proliferating cells that are completely surrounded by other cells. As we shall see, a
properly-calibrated mechanistic model can predict quantitatively-reasonable DCIS
growth without need for contact inhibition. See [
56
] for full details on the pro-
liferative and apoptotic sub-models.
4.1.1 Necrosis Sub-model
Let s denote the elapsed time spent in the necrotic state. Define s
NL
to be the
length of time for the cell to swell, lyse, and lose its water content, s
NS
the time for
all surface receptors to become functionally inactive, and s
C
, the time for calci-
fication to occur. We assume that s
NL
\s
NS
\s
C
.
We assume a constant rate of calcification, reaching a radiologically-detectable
level at s
¼
s
C
.IfC is the nondimensional degree of calcification (scaled by the
detection threshold), then C
ð
s
Þ¼
s
=
s
C
for 0
s
s
C
, and C
ð
s
Þ¼
1 otherwise. (We
do not track further calcification after s
C
). We model the degradation of any surface
receptor S (scaled by the non-necrotic expression level) by exponential decay with
rate constant log 100
=
s
NS
, so that S
ð
s
NS
Þ¼
0
:
01S
ð
0
Þ
. We set S
ð
s
Þ¼
0 for s [ s
NS
.
To model the necrotic cell's volume change, let f
NS
be the maximum percentage
increase in the cell's volume (just prior to lysis), and let V
0
be the cell's volume at
the onset of necrosis. Then we model:
(
V
ð
s
Þ¼
V
0
1
þ
f
NS
s
s
NL
if 0
s\s
NL
ð
12
Þ
V
N
if s
NL
\s
:
To model uncertainty in the cell morphology during lysis, we randomly perturb
its location x such that its new radius R
ð
s
NL
Þ
is contained within its swelled radius
R
ð
0
Þ
1
þ
f
NS
Þ
3
.
ð
4.1.2 Other Model Details and Numerical Implementation
As we described in [
56
], microenvironmental quantities are modeled with reac-
tion-diffusion equations throughout the computational domain. Uptake terms (e.g.,
for O
2
) are created by a coarse-graining technique: first construct a high-resolution
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