Biomedical Engineering Reference
In-Depth Information
equations for the oxygen concentrations in the tissue while cellular automata (CA)
are used to model tumor growth at the cell level. CA is a collection of cells on a grid
of specified shape that synchronously evolves through a number of discrete time
steps, according to an identical set of rules applied to each cell based on the states of
its neighboring cells [ 11 , 17 , 18 ]. The grid can be implemented in any finite number
of dimensions, and neighbors are a selection of cells relative to a given cell.
2 Previous Work
Mathematical modeling of tumor growth dates back to as early as 1972 when
Greenspan [ 3 ] modeled simple tumor growth by diffusion to study growth
characteristics from the most easily obtained data. Growth in terms of movement
of the outer radius of tumor as a function of time was studied along with the steady
state histology. Other studies that employ the continuum models include reaction
diffusion model by Gatenby et al. [ 4 ] to describe spatial distribution and temporal
development of tumor tissue. Ward et el. [ 5 ] modeled avascular tumor by using
nonlinear partial differential equations that took into account two types of cells—
cancer and dead. Ferreira et al. [ 6 ] extended the reaction diffusion model by
including cell motility in their model. Ambrosi et al. [ 7 ] modeled tumor using
continuum mechanics framework. They described growth as an increase of the
mass of the particles of the body and not as an increase of their number. Later,
Byrne et al. [ 8 ] modeled tumor using the theory of mixtures and Cristini et al. [ 9 ]
performed nonlinear simulations of tumor using the mixture model. An earlier
review on mathematical modeling of tumor is by Araujo et al. [ 10 ].
Cellular automata modeling of tumor growth is relatively young compared to
continuum modeling. One of the early CA models of tumor was developed by Qi
et al. [ 11 ]. They modeled tumor growth using two-dimensional CA. Immune
system surveillance against cancer was taken into account. The model was based
on the assumptions that cell division occur only in the presence of an empty space in
one of its nearest neighbors and that dead cells dissolve and disappear instead of
forming a necrotic core as seen in real tumors. Kansal et al. [ 19 ] modeled growth to
reproduce the macroscopic structure of a tumor arising from microscopic processes.
However, the transition rules used in the model are neither local nor homogeneous.
Moreover, the nutrient gradient is always considered originating from the center of
the tumor mass and directed outwards toward the tumor boundary. This does not
resemble a biological growth situation since the necrotic core, which is a mass of
dead cells, does not consume nutrients; so if nutrients are not consumed whilst still
diffusing, the proposed gradient would only last a short time after the cells have
become necrotic. Dormann et al. [ 12 ] employed lattice-gas cellular automata
(LGCA) to model self-organized avascular tumor that showed formation of a
layered tumor structure in a 200
200 lattice. Their model does not include the
phenotypical evolution—presence of mutated, more aggressive cancer cells—of
tumor. Study by Vermeulen et al. [ 20 ] provides evidence that a single cancer
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