Biomedical Engineering Reference
In-Depth Information
we solve the diffusion equation after every 10 time steps. Each time step is equal to
the time taken for a cell division to occur (20 h). After the first 10 time steps, we
check for the presence of a necrotic core in the tumor lattice. Since dead cells do not
consume oxygen, thereafter we solve the diffusion equation after every other
10 time steps to acquire a new distribution of oxygen concentration,
C
t
ðÞ
x
;
y
,
starting from the boundary of the necrotic core to the edge of the tumor where it
meets the healthy tissue. We differ here from previous models [
14
,
16
,
19
], where
concentration gradient is always computed from the center of tumor mass. Since
dead cells in the necrotic core do not consume oxygen, here we calculate the oxygen
concentration gradient from outside the average radius of the necrotic to the tumor
boundary.
5 Automaton Rules
1. The state of a CA element determines the type of cell in that element. The
applied CA rules depend on the type of cell.
2. If an automaton element is a cancer cell, it can divide into daughter cells if
(a) The level of oxygen concentration
C
t
ðÞ
x
;
y
in its neighborhood is greater
10
6
gcm
3
[
16
,
24
],
(b) A normal cell occupies one of its neighborhood and,
(c) The cell age has not exceeded its lifespan.
than the proliferative threshold value
C
p
¼
1
When (a), (b), and (c) are true, tumor growth is permitted. An empty place for
the daughter cell is created in one of its neighboring sites by shifting outward
the surrounding cells. However, the position in which the daughter cell will
move into is not known yet and therefore, is evaluated first. If cell C can
reproduce, the daughter cell can take at random one of four positions 1, 2, 3,
or 4 (gray sites in Fig.
1b
) with equal probability. Once the position the daughter
cell is going to occupy is determined, the normal cell occupying that position is
pushed to one of the sites in the neighborhood of the normal cell (dark sites in
Fig.
1b
) with equal probability. Therefore, growth is a result of two processes—
propagation of a normal cell into its neighborhood to create a space for the
daughter cell followed by the proliferation of cancer cell into this vacant position
from where a normal cell was displaced.
Recent models [
11
,
19
,
23
], assume presence of at least one pre-existing
empty space in the neighborhood of a CA element (to be occupied by a daughter
cell) as a necessary condition for a proliferating cell to divide. While this may
simplify the modeling process, this is not an accurate description of the way in
which proliferation occurs. Rather, biologically, an empty space is created and
proliferation occurs. This is accounted for in our model as described in Automa-
ton Rules (
2
) above.
3. If the level of oxygen concentration
C
t
ðÞ
x
;
y
falls below the proliferative thresh-
10
6
gcm
3
, but is greater than the necrotic threshold value
old value
C
p
¼
1
