Biomedical Engineering Reference
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indicating that after a long time the population stabilizes at some equilibrium
level K , i.e., the carrying capacity. The carrying capacity is here fixed and would
constitute a predefined limit to tumor growth. During tumor formation, such a
limit is likely to be imposed by physical or nutritional constraints, but, particu-
larly during the early stages of tumor formation, it may be overcome by the
adaptive capacity conferred by mutation. Constant mutation, favored by the mu-
tator phenotype, introduces competition among diverse cell populations and
requires modification of the previous model. The fact that a tumor is composed
of neoplastic cells and of stromal cells leads to the notion of competition be-
tween these two groups (15).
The consideration of a two-species competition model brings to the fore
how spatial constraints modify the expectations derived from nonspatial models.
Within the ecological context, two species will compete for given resources. In
the case of a malignant tumor the competition will take place among different
populations of cells. Gatenby (15) used the standard Lotka-Volterra system to
model tumor progression:
¬
dN
N
-
B
N
=
rN
1
-
® ,
[4]
1
1
2
-
11
--
dt
K
1
¬
dN
N
-
C
N
=
rN
1
-
® .
[5]
2
2
1
-
22
--
dt
K
2
Here r 1 and r 2 are the intrinsic growth rates of the tumor population and normal
host stromal population, respectively. The corresponding carrying capacities are
denoted K 1 and K 2 , and are defined as the maximum allowed population size that
could occupy the tissue space and be supported by the environment. Here B and
C are the interspecific competition coefficients. They measure the effects on
stromal cells caused by the presence of the tumor cells and the effect of the
stromal cells on the neoplastic cells, respectively. In this type of scenario, alter-
native models proposed by Gatenby incorporate different types of functional
responses and consider availability of resources as a separate variable (16).
Before further exploring the relevance of space considerations within the
context of tumor growth, we wish to consider one particular instance of the pre-
vious model that is defined by symmetric species competition, where the coeffi-
cients are the same, i.e., B = C. If we consider these to be tumor cell populations
with similar biological characteristics, the coexistence point is given by
+
==
(1
C
C
)
K
NN
*
*
.
[6]
1
2
1
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