Biomedical Engineering Reference
In-Depth Information
8. If at a node,
C
t
change the state of the cellular automaton element at
that node to the necrotic state, i.e., both proliferating and quiescent cells will be
dead.
9. Repeat 6-8 for 10 time steps.
10. After 10 time steps, if dead cells are present in the tumor mass, compute the
average radius of the necrotic core.
11. Since dead cells do not consume oxygen, calculate the new distribution of
oxygen concentration,
C
t
ð <
x
;
y
C
n
;
at all nodes in the lattice starting from the
boundary of the necrotic core to the edge of the tumor where it meets the
healthy tissue.
12. Repeat 6-8 for the next 10 time steps. In addition, if at a node,
C
t
ðÞ
x
;
y
C
p
;
and if the cellular automaton element at a node in the lattice is in the quiescent
state, change the state of that element to the proliferating state.
13. Repeat 10-12 and proceed until the edge of the grid is reached.
ð >
x
;
y
7 Results and Discussion
We performed growth simulation for both homogeneous and heterogeneous tumors
for 400 time steps on a 800
800 lattice. In the case of the former, clear symmet-
rical layers of proliferating, quiescent, and necrotic cells were visible, Fig.
2a
. In the
case of the latter, the shape of tumor was asymmetric and the boundaries of the
three layers were irregular, Figs.
2b-d
. This is corroborated by the fact that in
heterogeneous tumor, mutated cancer cells contribute to different velocities of
growth of cells in different directions because of dissimilar proliferation and
nutrient consumption rates by the various phenotypes. In our simulation, the
tumor remains heterogeneous throughout its growth, Fig.
3b
. Absence of mutated
phenotypes in homogeneous tumor allows it to grow at the same rate in all
directions making the tumor spherical.
Our model establishes that the size of an avascular tumor is limited by the
diffusion of nutrients [
2
], in this case, oxygen. The number of proliferating cancer
cells increase rapidly initially and the count stabilizes later in the growth as is seen
in Fig.
3a
where the stability in our growth model is achieved after about
72 iterations when the number of proliferating cells is about 474 cells. In other
words, in 72 20-hour time steps the tumor reaches the size limited by diffusion.
At this stage, there are 474 proliferating tumor cells.
We conclude that our CA model incorporating heterogeneous cell population is
able to capture the tumor growth dynamics at the cellular level that is comparable
with theory. We will extend the model to three dimensions in the future. To obtain a
bigger size of tumor that is not achievable here due to computational limits, we will
implement the model in GPU.
