Biomedical Engineering Reference
In-Depth Information
10
7
gcm
3
C
n
¼
5
[
16
,
25
],
the state of
the automaton element
(proliferating cell) changes to the quiescent state.
4. If the level of oxygen concentration
C
t
falls below
C
n
, the state of both the
proliferating cells and the quiescent cells change to the dead state.
5. In the case of quiescent cells, once the local level of oxygen concentration is
restored to the proliferative threshold value
C
p
, their state changes to the
proliferating state.
6. We model intercellular adhesion by considering the number of external
neighbors a cell is attached to, Q
e
[
14
]. If Q
e
2, a cell adheres to its neighbors
whereas if Q
e
2, the cell is allowed to migrate.
7. To model mutation, we consider four different phenotypes. Initially, all cells are
of phenotype I. A cell can mutate with a probability P_mut
ðÞ
x
;
y
0.1 to one of
phenotypes II, III, or IV. Phenotype II can proliferate at half the nutrient
concentration required by phenotype I and can reproduce at half the time
required by phenotype I. We proceed similarly to determine the time and oxygen
required by phenotype III and IV to proliferate. Anderson [
14
] used, in addition
to the above, a random mutation sequence where all the cells are initially
assigned to one of 100 phenotypes randomly and through mutation, another
phenotype is selected randomly. He concluded that while these two methods for
considering tumor cell heterogeneity are different, they ultimately produce
similar results.
8. When the age of a cell equals its lifespan, we check if it can mutate. If the cell
can mutate, it can acquire a different phenotype as mentioned in 7 above. The
age of the cell is then reset to zero. Otherwise the cell will die due to naturally
exceeding its lifespan (apoptosis).
¼
6 The Tumor Growth Algorithm
The tumor growth algorithm follows the steps listed below:
1. Load a two dimensional lattice with a grid size of N
N.
2. Load the boundary conditions.
3. Seed five nodes at the center of the lattice with proliferating cells.
4. Initialize time stepping.
5. Calculate the oxygen concentration level,
C
t
at all nodes in the lattice
using the finite difference method as described in Solution to the Diffusion
Equation.
6. If at a node in the lattice, the cellular automaton element is in the proliferating
state and, if
C
t
ðÞ
x
;
y
ð >
x
;
y
C
p
;
cell division occurs as described in (Automaton
Rules 2) above.
7. If at a node,
C
n
<
C
t
change the state of the cellular automaton
element at that node to the quiescent state.
ð <
x
;
y
C
p
;
