Biomedical Engineering Reference
In-Depth Information
3.2.3. Search Process
At each time period
t
= 0, 1, 2, ..., a tumor cell that is eligible to migrate
then "screens" the surrounding regions in its neighborhood to determine whether
there is a more attractive location. The biological equivalent for this mechanism
(and the cell behavior it induces) is cell-surface "receptor-ligand" interaction.
Each eligible tumor cell then ranks the
attractiveness
of a neighboring location
based on the following real-valued function:
.
LG
(, , , )
GU
p
=+
G
q
¸ ¸ ¸
G U
q
q p
+
G
[2]
j
j
j
j
j
G
j
U
j
p j
i
i
{ 's neighborhood}
j
Equation [2] states that the value of location
j
depends on (i) the function
G
j
(see
Eq. [3] below), which represents this location's attractiveness, which is due to its
cell population, as well as on (ii) the onsite environmental factors. Specifically
for the latter, the parameters
q
G
,
q
U
, and
q
p
capture the contributions of nutrient
supplies G
j
, toxic metabolites U
j
, and mechanical pressures
p
j
, respectively. The
last term in Eq. [2] captures the "neighborhood effect" due to cells that exert
influence, and are influenced by other cells located in adjacent locations. As we
have detailed in §3.1.4, our previous works have utilized the concept of the
"Moore neighborhood," which includes only those adjacent locations at most
one unit of distance away. The explicit form of
G
j
here is specified as a non-
monotonic function of the population density, I
j
, discounted by the distance of
location
j
from the evaluating tumor cell:
G
j
= (I
j
-
c
I
j
2
)exp(-S#
d
j
2
/2).
[3]
According to Eq. [3], a tumor cell is attracted to locations that already accom-
modate a number of other tumor cells, implementing the biological concept of a
"paracrine" attraction. However, there is also a negative "crowding out" effect
represented by the parameter
c
, such as if the location
j
's population of tumor
cells expands beyond a maximum (i.e., beyond the point where 0
G
j
/0I
j
= 0),
then the attractiveness of that location starts to decline due to, for example, lim-
ited carrying capacity and spatial competition. Because of such a maximum
threshold cell density,
G
j
is nonmonotonic: first, it increases, then at the maxi-
mum
it starts to decrease as the population of tumor cells grows
II
ss=
=
j
max
G I
/
0
j
j
further. Importantly, Eq. [3] also contains a geographical dimension since the
value of
G
j
is discounted at an increasing rate proportional to its squared dis-
tance
d
j
2
from the evaluating tumor cell's current location. The parameter S thus
captures the metabolic energy required for a single cell to move across regions.
