Biomedical Engineering Reference
In-Depth Information
Rather, a static probability distribution function is exogenously assumed to de-
termine whether events such as migration, proliferation, or cell death can occur.
Such external rules, imposed in a top-down manner, however, rule out the pos-
sibility for virtual cells to be true autonomous "decision-making" agents. All
these CA applications typically assume discreteness in time and space. The dis-
crete treatment of time and space is often desirable not only because it models
biological systems more realistically but, more importantly, because it allows for
examination of tumor progression over time and across space. The latter is in-
variably the basis of any clinician's prognosis in practice. However, there are
many variables of interest that are less discrete in nature, such as nutrient
sources, toxic metabolites, and mechanical confinements. For these types of
variables, their dynamic evolution in the extracellular matrix can be better de-
scribed using the continuous Navier-Stokes or reaction-diffusion equations. In
cases where the spatiotemporal evolution of the tumor is closely linked to envi-
ronmental conditions, an approach combining both the discrete (for time and
space) and continuum (for environmental conditions) elements of a tumor sys-
tem would offer a very sensible and promising alternative. For this reason, more
contemporary modeling efforts extend the CA framework into
agent-based
models that still simulate time and space discretely, yet treat many of the bio-
logical components of interest as continuous variables, thus avoiding the need to
transform these variables into unrealistic integer states as in a traditional CA
model. Recent contributions that have made use of discrete-continuum intersec-
tions include (6-9,22,23).
Another useful method for classifying existing models is determining
whether the focus is on the proliferation or migratory behavior of tumor cells, or
on both growth and invasion. Many previous studies have focused on either the
proliferative growth of the tumor (24,25) or on the invasive behavior (26,27).
More recent modeling efforts have attempted to place equal emphasis on both
cell proliferation and cell motility. The work that has been done in this area of
research with a dual focus on both growth and invasion includes (6-9,28,29).
Proliferation
-focused studies can be further subdivided into those that em-
ploy deterministic-continuum and those that make use of stochastic-discrete
models. In the former, the tumor system is viewed as an aggregation of (multi-
cellular) tumor lumps, and the variable of interest is invariably tumor volume,
which is a continuous variable. As an example of a continuum platform, (24)
develops a deterministic mathematical framework to generate the growth pattern
of multicellular tumor spheroids that follows the Gompertz law. On the other
hand, a stochastic-discrete approach typically employs a cellular-level agent-
based model to enable the explicit examination of chance elements in the behav-
ior of individual cells. For example, using a three-dimensional cellular automa-
ton model, (25) shows that macroscopic tumor behavior can emerge from local
interactions at the microscopic level. (22) presents an attempt to bridge these
two approaches by introducing random elements into a continuum model. Note
