MATHEMATICAL MODELING OF TUMOR GROWTH
(Last updated August 2009)
Cancer is a
highly complex and heterogeneous set of diseases. Dynamic changes
in the genome, epigenome, transcriptome and proteome that result in the
gain of function of oncoproteins or the loss of function of tumor
suppressor proteins underlie the development of all cancers.
While the underlying principles that govern the transformation of
normal cells into malignant ones is rather well understood (Hanahan and
Weinberg, 2000). a knowledge of the changes that occur in a given
cancer is not sufficient to deduce clinical outcome.
The difficulty in predicting clinical outcome arises because many
factors other than the mutations responsible for oncogenesis determine
tumor growth dynamics. Multidirectional feedback loops
occur between tumor cells and the stroma, immune cells, extracellular
matrix and vasculature (Kitano, 2004). Given the number and
nature of these interactions, it becomes increasingly difficult to
reason through the feedback loops and correctly predict tumor
behavior. For this reason, a better understanding of tumor growth
dynamics can be expected from a computational model.
The holy grail of computational tumor modeling is to develop a
simulation took that can be utilized in the clinic to predict
neoplastic progression and response to treatment. Not only must
such a model incorporate the many feedback loops involved in neoplastic
progression, the model must also account for the fact that cancer
progression involves events occurring over a variety of time and length
scales. By developing individual models that focus on certain
tumor-host interactions, validating these models and merging the
individual pieces together, I aim to build one algorithm with
the potential to address a variety of questions about neoplastic growth
My Ph.D. thesis advisor, Professor Salvatore Torquato, along with
others, has developed
a novel cellular automaton (CA) model to simulate the mechanistic
solid tumor growth. The model takes into account four cell types:
cells, proliferating tumor cells, non-proliferating tumor cells
and necrotic tumor cells. The algorithm they developed was able
to successfully predict three-dimensional tumor growth and composition
using a simple set of automaton rules and a set of four microscopic
parameters that account for the nutritional needs of the tumor,
cell-doubling time and an imposed spherical symmetry term (Kansal et
al., 2000a). This simple model of tumor growth was applied
to consider the effect that an emerging clone of novel genotype has on
neoplastic progression (Kansal et al., 2000b, Figure 1) and to study
the response of a tumor mass to surgical resection followed by
chemotherapy (Schmitz et
Vascular Growth Model
The success of the original CA model is in part related to its
simplicity, and one of the simplifying assumptions is that the
vasculature is implicitly present and evolves as the tumor grows.
In order to incorporate more biological detail in the model, I worked
to modify the original algorithm to study the feedback that occurs
between the growing tumor and the evolving host blood vessel network
(Gevertz and Torquato, 2006).
The computational algorithm is based on the co-option/regression/growth
experimental model of tumor vasculature evolution. In this model,
as a malignant mass grows, the tumor cells co-opt the mature vessels of
the surrounding tissue that express constant levels of bound
Angiopoietin-1 (Ang-1). Vessel co-option leads to the
upregulation of the antagonist of Ang-1, Angiopoietin-2 (Ang-2).
In the absence of the anti-apoptotic signal triggered by vascular
endothelial growth factor (VEGF), this shift destabilizes the co-opted
vessels within the tumor center and marks them for regression (Holash
et al., 1999). Vessel regression in the absence of vessel growth
leads to the formation of hypoxic regions in the tumor mass.
Hypoxia induces the expression of VEGF, stimulating the growth of new
We developed a system of reaction-diffusion equations to track the
spatial and temporal evolution of the aforementioned key factors
involved in blood vessel growth and regression. Based on a set of
algorithmic rules, the concentration of each protein and bound receptor
at a blood vessel determines if a vessel will divide, regress or remain
stagnant. The structure of the blood vessel network, in turn, is
used to estimate the oxygen concentration at each cell site, and the
oxygen concentration determines the proliferative capacity of each
The model proved to quantitatively agree with experimental observations
on the growth of tumors when angiogenesis (growth of new blood vessels
from existing vessels) is successfully initiated and when angiogenesis
is inhibited. In particular, the model exhibits an "angiogenic
switch" such that in some parameter regimes the tumor can grow to a
macroscopic size whereas in other parameter regimes tumor growth is
thwarted beyond a microscopic size (Gevertz and Torquato, 2006, Figure
Growth in Confined,
original CA model and the vascular growth model limit the effects of
mechanical confinement to one parameter that imposes a maximum radius
on a spherically symmetric tumor. However, tumors can grow in
organs of any shape with nonhomogeneous tissue structure, and in a
study performed by Helmlinger et al (1997), it was demonstrated that
neoplastic growth is spherically symmetric only when the environment in
which the tumor develops imposes no physical boundaries on
growth. In particular, it was shown that human adenocarcinoma
cells grown in a 0.7% gel inside a cylindrical glass tube develop to
take on an ellipsoidal shape, driven by the geometry of the capillary
tube. However, when the same cells are grown outside the
capillary tube, a spheroidal mass develops. This experiment
highlights that the assumption of radially symmetric growth is not
valid in complex, non-spherically symmetric environments.
Since many organs, like the brain, impose non-radially symmetric
physical confinement on tumor growth, we modified the original CA
algorithm to incorporate boundary and heterogeneity effects on
neoplastic progression. The CA evolution rules are comparable to
the rules used in the original model, except that all assumptions of
radial symmetry are removed and replaced with rules that account for
tissue geometry and topology (Gevertz et al., 2008). In doing
this, we showed that models that do not account for the structure of
the confining boundary and organ heterogeneity lead to inaccurate
predictions on tumor size, shape and spread. In Figure 3, this is
illustrated by studying tumor growth in a 2D representation of the
cranium and comparing the size of the tumor as a function of time when
the original and modified algorithm is employed.
Heterogeneous Tumor Cell
Population: Genetic Mutations
It is well known
that each patient's tumor has its own genetic profile, and that this
profile is important in determining growth dynamics and response to
treatment. To account for tumor cell heterogeneity, we begin with
the assumption that tumors are monoclonal in origin; that is, a tumor
mass arises from a single cell that accumulates genetic and epigenetic
alterations over time (Fialkow, 1979). In our algorithm, a tumor
cell of one genotype/phenotype initiates tumor growth. As a tumor
cell undergoes mitosis, we assume that there is a 1% chance that an
error occurs in DNA replication. When such an error occurs, the
daughter cell will take on a mutant phenotype that differs in one
respect from the mother cell. Given the tumor cell properties
considered in the model, we allow malignant cells to acquire altered
phenotypes related to cell proliferation rates (which can correspond to
cells altering the production rate of growth factor such as PDGF and
TGF-α or tumor suppressor proteins such as pRb or p53) and the length
of time a cell can survive under sustained hypoxia (which can
correspond to the expression levels of the X-box binding protein
1). We have implemented the model such that a tumor cell can
express one of eight mutant phenotypes (Gevertz and Torquato, 2008),
all of which are defined in Figure 4.
Using the genetic mutations model, we have explored the probability of
emergence of both beneficial and deleterious mutations in the tumor
cell population. We have found that, as expected, mutations which
give cells a proliferative advantage arise frequently in our
(78% of the time). More surprisingly, we have also observed that
the obviously deleterious mutation that decreases the proliferation
rate of a cell emerges in 7% of our simulations. Furthermore, the
mutation that determines the hypoxic lifespan of a cell is not strongly
selected for or against, provided the changes in the lifespan are not
too drastic in either direction (Gevertz and Torquato,
submiited). Using this model, we
can explore how growth dynamics are impacted by the emergence of one or
more mutant strains.
Merged Model of Tumor Growth
Each of the
previously discussed algorithms were designed to address a particular
set of questions and successfully served their purpose. Much can
be gained, however, by merging the evolving vasculature, confined
growth and genetic mutations models. The resulting multiscale
simulation tool will not only consider multiple forms of feedback and
heterogeneity, it will also likely have emergent properties not
identifiable prior to the integration of the three models.
To pinpoint the effects that the environment has on tumor growth, here
I show environment in which the tumor grows (bottom row of Figure 5),
along with the shape and spread of a tumor after approximately four
months of growth in this environment (top row of Figure 5). This
figure allows us to see how and to what extent the structure of the
environment alters tumor shape and spread.
Currently, this merged model is being used
to test the impact that various
vascular-targeting therapies have on tumor progression.
- Generalize the model to three dimensions. Currently, the
original CA, confined and heterogeneous growth model, and the genetic
mutations model have been implemented in both 2D and 3D. However,
the vascular algorithm has only been implemented in 2D, limiting the
merged model to be two-dimensional as well.
- Develop a more realistic model of the underlying capillary
network in tissue. We only focus on modeling the capillaries
because this is the level of the vascular tree where oxygen and
nutrient exchange occurs. The capillary network, it should be
noted, does not exhibit the
same branching structure that is seen in higher-order vessels.
- Expand upon the confined-growth algorithm to examine not only how
the host deforms the tumor, but also to consider the reciprocal
deformities induced in the host by the growing neoplasm.
- Incorporate other forms of tumor-host interactions: extracellular
matrix, immune system...
- Include the process of single cell invasion in which individual
cancer cells break off the main tumor mass and invade healthy tissue
(particularly important in brain tumor growth dynamics)
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