Biomedical Engineering Reference
In-Depth Information
the probability of a tumor cell at
(
x
i
,
y
j
,
z
k
)
remaining at that location is given by
P
0,0,0
P
T
, and the probability of that tumor cell moving, for example, one volume
in the positive
x
directions is given by
P
+
1,0,0
/
/
P
T
.
4.2.2 Tissue Environment
The tissue, including the components and mechanisms listed above, is described by
¶
O
¶
t
=
2
O
D
o
∇
−
−
+
k
oo
O
k
oc
OC
k
oe
E
(4.11)
¶
M
¶
t
=
2
M
D
m
∇
+
k
mc
C
−
k
mm
M
(4.12)
¶
E
¶
t
=
−
k
em
EM
(4.13)
(
)
where
C
is 1 if a given location
is occupied by a tumor cell, and 0 oth-
erwise. It is important to note that two of the parameters in the tissue model----the
oxygen consumption rate,
k
oc
, and the matrix degradative enzyme production rate,
k
mc
---- v a r y f r om p o i n t
x, y, z
t o p o i n t d e p e n d i n g upon the state of the tumor cell at that
location.
The tissue equations are solved using an explicit Euler method:
O
n
+
1
i,j,k
O
i,j,k
+
O
i,j,k
)
=
l
D
o
L
(
k
oo
O
i,j,k
D
t
k
oc
O
i,j,k
C
i,j,k
D
t
k
oe
E
i,j,k
D
t
−
−
+
(4.14)
M
n
+
1
i,j,k
M
i,j,k
+
M
i,j,k
)+
k
mc
C
i,j,k
D
t
k
mm
M
i,j,k
D
t
=
(
−
l
D
m
L
(4.15)
E
n
+
1
i,j,k
E
i,j,k
−
k
em
E
i,j,k
M
i,j,k
D
t
=
(4.16)
where
V
i,j,k
)=
V
i
−
1
,
j
,
k
+
V
i,j
−
1
,
k
+
V
i,j,k
−
1
−
6
V
i,j,k
L
(
V
i
+
1
,
j
,
k
+
V
i,j
+
1
,
k
+
V
i,j,k
+
1
+
(4.17)
10
3
≈
For stability, this method requires lmax
(
,
we solve
M
and
E
on the same mesh as that used for tumor cells, but solve
C
on a coarser mesh with spacing 32 times larger (since 32
2
D
c
,
D
m
)
<
1
/
6. Since
D
c
/
D
m
O
(
)
10
3
). This requires
averaging and interpolating values between fine and coarse meshes, but allows the
use of a much larger timestep.
≈
4.2.3 Processes Controlling Individual Tumor Cells
The effects of tumor cell heterogeneity are included in the model by assigning
to each cell an immutable phenotype, or state, chosen at birth from any of 100
predetermined phenotypes. The phenotype determines certain aspects of the cell's
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