Biomedical Engineering Reference
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of a phase transition leading to two distinct spatiotemporal patterns depending
on the dominant search mechanism (related to the cells' energy expenditure re-
quired by the particular environmental conditions). In brief, if global search is
dominant, the result is an invasive tumor system operating with a few large clus-
ters that expands rapidly but also dies off quickly. By contrast, if local search is
dominant, the result is many small cell clusters with a longer lifetime but much
slower spatiotemporal velocity. Building on this work, (7) focuses on search
precision
and implements local search only. The main finding there is that a less
than perfect search can yield the fastest spatiotemporal expansion, thus indicat-
ing that multicellular tumor systems might be able to exploit the in vivo, pre-
sumably (chemically and mechanically) rather "noisy" microenvironment. In the
following section we describe the mathematical model in more detail.
3.
MATHEMATICAL MODEL
In our model, a tumor cell can proliferate, migrate, become quiescence, or
undergo cell death depending on (a) its own specific location, (b) other tumor
cells sharing the same location and competing for resources, (c) the onsite levels
of microenvironmental variables, and (d) the state of neighboring regions. To
discuss the nature of the relationships among tumor cells and between tumor
cells and their environment in our model, we first introduce a number of theo-
retical notions.
3.1. Definitions
3.1.1. Population Dynamics
Let I
j,t
represent the (discrete) number of tumor cells residing in location
j
at
time
t
. In our model, both space and time are discrete, i.e., the
x
-
y
coordinates of
j
and time
t
are nonnegative integers. The population of tumor cells in any loca-
tion changes due to (i) proliferation of new offsprings, (ii) net migration (i.e., in-
migration minus out-migration), and (iii) cell death.
3.1.2. Clustering
A cluster is the
spatial agglomeration
of virtual tumor cells in contiguous
sites. These virtual clusters represent cell aggregates observed in actual experi-
ments involving malignant brain tumors (3,31). More formally, we define a clus-
ter of tumor cells,
C
, as a federation of contiguous regions, each of which
contains at least one viable tumor cell I
j,t
> 1. To this, we add another qualifica-
tion that for a group of regions to be a legitimate cluster
C
it must be the case
that their collective population size exceeds a certain minimum size: the union
