Biomedical Engineering Reference
In-Depth Information
The tumor does so in the second phase through angiogenesis. The new vessels
enhance the supply of nutrients allowing the tumor to enter the third phase, the
vascular phase. At this stage, tumor cells proliferate aggressively and metastasize
thus invading the surrounding tissue.
The avascular phase is also called the primary growth phase and is considered
relatively benign. The detection and treatment of tumor at this stage has a greater
probability of having the disease cured. On the other hand, the vascular phase, also
called the secondary growth phase, is more malignant and treatment becomes far
more difficult at this stage, on most occasions leading to serious complications.
In this paper, we model tumor growth in the avascular phase. The size and shape
of a tumor at this stage is predominantly determined by its cellular composition,
time required for cell division (mitosis), cell mutation (phenotypical evolution),
intercellular adhesion, concentration of vital nutrients, and mechanical stresses
from surrounding tissue, for example, in the case of a brain tumor, mechanical
stresses due to confinement in the skull.
Tumor heterogeneity contributes to its irregular shape. A tumor mass consists of
three types of cells—proliferating, necrotic, and quiescent. Cells, mostly on the
tumor boundary, that are exposed to high levels of oxygen concentration undergo
cell division and lead to tumor proliferation. In contrast, cells at the center of tumor
suffocate due to lack of oxygen and die (necrosis) forming a necrotic core. More-
over, some cells die after naturally exceeding their lifespan (apoptosis) and are seen
scattered in the tumor mass. Some cells in the mass are exposed to nutrient levels
that are higher than suffocation levels but insufficient to promote proliferation.
Such cells are dormant and are called quiescent cells. They neither die nor undergo
cell division. However, they participate in the normal cell cycle once sufficient
oxygen level is restored. In addition, some tumor cells mutate and give rise to a
different phenotype that survive at smaller nutrient concentrations and proliferate
faster. This heterogeneous population of cells leads to different velocities of growth
in different directions forming an asymmetric irregular tumor volume.
To date, tumor growth modeling approaches include the continuum [
3
-
10
],
discrete, and hybrid continuum-discrete approaches [
11
-
16
]. Continuum models are
based on balance laws—balance of mass of the several components of tissue, balance
ofmomentumand balance of energy—for the description of cell population [
7
]whilea
set of reaction diffusion equations are devised for nutrients and chemicals that influ-
ence growth. However, growth description through such modeling is phenomenologi-
cal and it does not reflect the microscopic mechanisms of cancerous growth, such as
proliferation, necrosis and apoptosis as well as the mechanical pressure inside tumor.
Continuum models, therefore, are not sensitive to small fluctuations in the tumor
growth system. This is a significant shortcoming as in some cases such small changes
can be the leading cause in driving a nonlinear complex bio-system to a different state.
Discrete models, on the other hand, can represent individual cells in space and
time and can incorporate biological rules to define behavior at the level of cells.
Such models better respond to small changes in the tumor system. In this paper, we
make use of a hybrid discrete-continuum approach in a bid to take advantage of the
strengths of both of these approaches. In particular, we solve partial differential
