Biomedical Engineering Reference
In-Depth Information
*
represents a cluster if and only if
of contiguous, nonempty locations
I
jt
jC
. For example, in (7,8), we fixed
I
= 5, which means an agglomera-
I I
jt
jC
tion of more than five adjacent cells (i.e., their locations share a common border)
qualifies as a cluster. A tumor cell is defined to be on the surface of a cluster if
(i) its location
j
belongs to a cluster, i.e.,
j
C
, and at the same time (ii) there
must exist an empty location
I
in
j
's neighborhood, formally: I
i,t
= 0 and
i
j
's
neighborhood.
3.1.3. Measure of Distance
Since our model explicitly takes the geography of a brain tumor into con-
sideration, a distance measure is necessary. We chose the L-infinity metric of
distance for the simple reason that it can be conveniently implemented in a
computer algorithm. Specifically, given two points
A
and
B
in the two-
dimensional grid lattice with coordinates (
x
A
,
y
A
) and (
x
B
,
y
B
), respectively, then the
distance between these two points is computed as
d
AB
= Max[abs(
x
B
-
x
A
),abs(
y
B
-
y
A
)].
3.1.4. Local Neighborhood
The set of locations, which are adjacent to a tumor cell's current site, consti-
tute the local neighborhood of that cell. In our previous works, the local
neighborhood consists only of those locations sharing a common border with the
tumor cell's current location
j
, i.e., those locations
i
that are within one unit of
distance away:
d
ij
1. Typically, we adopt the notion of a
Moore
neighborhood,
which includes locations in the north, south, east, and west of the tumor cell's
current location, as well as in the NE (northeast), NW (northwest), SE (south-
east), and SW (southwest) directions.
3.2. Cell Behavior and Environment
The local environmental variables in our model are captured by nutrient
supplies, mechanical confinements, and deposits of detrimental toxic metabo-
lites. The generic term "nutrients" can be interpreted as representing glucose, as
it is the principal source of energy for the brain. Indeed, (32) demonstrated that
the extent of glioma malignancy is highly correlated with the expression of the
GLUT3 glucose transporter. However, nutrients here can be also interpreted, for
example, as epidermal growth factor (EGF), which has been shown to stimulate
and guide the invasion of glioma tumor cells in vitro (33). The active migration
