Biomedical Engineering Reference
In-Depth Information
must be able to suppress the theoretically predicted global effects of competitive
replacement. Within the context of tumor growth, the space constraints are par-
ticularly important since cell movement within the tumor tissue is limited and
cell-cell interactions, including paracrine effects, are short range.
We review here some key results and consider the theoretical arguments
and computer simulations that support the view that the expected population
structure to be found in a tumor is a heterogeneous spatial distribution of geno-
types and phenotypes. At a time when molecular therapies are being imple-
mented, the understanding of tumor heterogeneity in general theoretical terms
can contribute to the design and understanding of multi-drug molecular therapies
and to the prediction of treatment response.
3.
COMPETITION IN TUMOR CELL POPULATIONS
A rich literature explores the growth of tumors assuming a homogeneous
structure of the cancer cell population, without detailed reference to competition
among the different cell types that compose a tumor (e.g., neoplastic cells and
stromal cells). Under such approximation, the growth in the total number of cells
N
can be represented by a one-dimensional differential equation:
dN
=
G
()
N
,
[1]
dt
where the right-hand side would contain a density-dependent behavior (i.e., size-
dependent growth) and such parameter(s) as replication rate N. A first approxi-
mation is given by a linear function G
N
(
N
) = N
N
and leads to an exponential in-
crease in the number of tumor cells. Under conditions of limited resources, this
model has to be modified by a term expressing the effect of a population control
mechanism. The standard example is provided by a logistic equation:
N
¬
-
dN
N
NN
==
G
()
1
-
®
,
[2]
--
dt
K
which can be solved, giving the time-dependent solution
K
Nt
()
=
e
N
t
,
[3]
¬
-
KN
N
-
0
-
-
®
0
