Biomedical Engineering Reference
In-Depth Information
5.
BASIC MODEL SETUP
In our previous works, our space of observation is a torroidal square grid
lattice representing a virtual, two-dimensional slice of brain parenchyma. On
that grid lattice we introduce fields of environmental variables: nutrients, toxic
metabolites, and mechanical confinements. As we have discussed briefly above
in ยงยง3.2.1, 3.2.2, and 3.2.4, we impose dual thresholds in the levels of nutrients
(G
L
,G
U
) and toxicities (U
L
,U
U
), which in turn trigger the onset of migration, prolif-
eration, and cell death. To summarize, a tumor cell in location
j
proliferates if G
j
> G
U
and U
j
< U
L
, i.e., sufficiently high levels of nutrients (above upper threshold)
and low levels of toxicity (below lower threshold) lead to a positive probability
of proliferation, the extent of which in turn is proportional to the levels of nutri-
ents. If these optimal conditions for growth are not satisfied, then a tumor cell
can migrate whenever it is located on the tumor surface, and one of these three
conditions hold: (i) U
j
U
L
but G
L
< G
j
< G
U
, or (ii) G
j
G
U
but U
L
< U
j
< U
U
, or (iii)
G
L
< G
j
< G
U
and U
L
< U
j
< U
U
. Whether the cell actually migrates depends on
whether there is a more attractive location in its local neighborhood as deter-
mined by the combined impact of nutrients, mechanical resistance, and toxicity.
Finally, if either the nutrient levels are below the lower limit, G
j
< G
L
, or the
toxic levels are above the upper threshold, U
j
> U
U
, then cell death is imminent
with a probability proportional to the levels of toxic metabolites. In the follow-
ing, we detail how the environmental variables are initialized in our numerical
model.
5.1. Initial Levels of Environmental Variables
5.1.1. Nutrients
The initial distribution of nutrients is modeled as follows. We assume that
there are three distinct types of nutrient sources that can be distinguished based
on their geographical distribution. Specifically, there are two
replenished
pri-
mary sources of nutrients in the initial setup: (i) at the center of the grid, and (ii)
at the center of the northeast (NE) quadrant. Everywhere else, nutrients are ini-
tially
non
-replenished. In order to add to the site's growth-permissive environ-
mental conditions, the first primary source is placed next to the "crater" of
mechanical confinements and bell-shaped distributed with the peak located at
the tip of the pressure crater. The second source can be found to the NE of the
grid center and is also bell-shaped distributed with the peak level located at the
center of the NE quadrant. This second nutrient source exhibits significantly
higher levels than the first one such that the peak of this second source is five
times (5+) the peak of the first one. This arrangement ensures a chemoattractive
gradient where the two sources can be thought of as representing two distinct-
