Biomedical Engineering Reference
In-Depth Information
incompressible fluid (with constant cell density) moving in a porous medium—the
ECM. We used a sharp interface approach, where X
ð
t
Þ
denoted the moving tumor
volume with boundary R
ð
t
Þ
; we denoted the surrounding host tissue by X
H
.In[
60
],
we set X
H
to enclose X in an L
100
200 lm ring of tissue:
X
[
X
H
¼
x : x
x
center
ð
t
Þ
f
j
j
R
ð
t
Þþ
L
g;
ð
1
Þ
where
R
ð
t
Þ¼
max
f
j
x
x
center
ð
t
Þ
j;
x
2
X
ð
t
Þ
g;
ð
2
Þ
and where x
center
is the center of mass of X
ð
t
Þ
. We scaled space by L (the nutrient
diffusion length scale) and time by a mechanical relaxation time scale k
R
. The
time is rescaled in all plots to correspond to the cell mitosis time scale k
M
24 h.
See [
51
,
58
-
60
] for more details.
We introduced a single nondimensional ''nutrient'' r which was required for
cell survival and drove growth. The nutrient was released by the host vasculature
at o X
[
X
ð Þ
, diffused through the non-vascularized nearby host tissue X
H
to the
tumor, and was then consumed by tumor cells in X. Following [
18
] and as
described in [
58
], we make the quasi-steady assumption: nutrient transport and
consumption occur on much faster time scales than cell proliferation and tissue
deformation, and so on the time scale of simulation, or
=
ot
0. Thus, r satisfies
0
¼r
D
H
r
r
ð
Þ
x
2
X
H
ð
3
Þ
0
¼r
D
T
r
r
ð
Þ
r
x
2
X
subject to boundary and matching conditions
½
R
¼
0
½
D
r
r
n
R
¼
0
r
ð
x
Þ
o X
H
[
X
ð
4
Þ
Þ
¼
1
;
ð
where for any x
2
R, the jump function
f
ð½
R
is defined as
f
ð
x
½
R
¼
lim
X
3
y
!
x
f
ð
y
Þ
lim
X
H
3
y
!
x
f
ð
y
Þ:
ð
5
Þ
In [
60
], D
T
¼
1 as a result of nondimensionalization. The nutrient is used to
implicitly define viable and necrotic regions (X
V
and X
N
, respectively) of the tumor:
X
V
¼
x
2
X such that r
ð
x
Þ
r
N
f
g
ð
6
Þ
X
N
¼
x
2
X such that r
ð
x
Þ
\r
N
f
g;
where r
N
is the necrotic threshold value of r. Note that X
¼
X
V
[
X
N
.
Within the tumor's viable rim, cells were assumed to proliferate at a rate propor-
tional to r and apoptose at a constant background rate. In X
N
, the model degraded
necrotic debris and released volume, acting as a biomechanical stress relief. We
assumed the host tissue was in homeostasis (proliferation and apoptosis were in
balance), but cells and tissue could be displaced by forces generated by the tumor.
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