Modeling tumour progression and invasion using spatial

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Modeling tumour progression and invasion using
spatial evolutionary games
James Gui, Spencer Sheen
August 16, 2013
Abstract
Tumor progression has been a topic for extensive exploration in the
field of computational biology. Numerous models have been proposed for
the growth of different kinds of tumors, such as ones based on cellular
automata [6] and evolutionary game theory [3]. Here, we explore the spatial aspect of tumors that was neglected in Basanta’s game between three
different tumor cell phenotypes in gliomas–autonomous growth (AG), invasive behavior (INV), and glycolytic production (GLY). We use two different approaches to modeling this spatial component, one based on a
discrete continuous model and one based on a two-dimensional lattice
simulation model.
1
Introduction
In general, cancer cells display an increased reliance on glycolysis for energy
production, a phenomenon known as the Warburg effect [4]. These cells will
utilize glycolysis even in an oxygen-rich environment–normal cells in such an
environment would use the Krebs cycle in addition to glycolysis in order to
more efficiently generate ATP. The Krebs cycle, also known as the citric acid
cycle, takes pyruvate (a byproduct of glycolysis) and converts it into ATP using
oxygen. Regular cellular respiration can produce up to 38 molecules of ATP,
which is 19 times what glycolysis alone can produce. Furthermore, the glycolytic
cells are able to change the acidity of the area around them, causing normal cells
to undergo programmed cell death. In our study, we analyze the game between
two other phenotypes and the glycolytic phenotype to discover what role the
GLY cells play in tumor progression and invasion. The first of these phenotypes
is the AG phenotype, which in essence describes a cell that acts independently
of what the body tells it to do. This phenotype is seen in benign (or primary)
tumors, and are characterized by uncontrolled proliferation. The second phenotype is the INV phenotype, which is characterized by increased cell motility.
The possession of this phenotype allows cancerous cells to invade and become
malignant tumors. Usually, this happens when some tumor cells move from the
original tumor to another organ or tissue and form another tumor, known as
1
a metastasis. These cells generally work their way through the bloodstream to
create metastases in different areas of the body. The evolutionary game between AG, INV, and GLY has already been analyzed, but the analysis done so
far neglects an important aspect in the tumor: space [3]. We hope to enhance
our understanding of tumor cell behavior through creating and analyzing the
game with a spatial component.
2
The Discrete Continuous Model
We assume that all tumors have some percentage of GLY cells and consider three
types of tumors in our simulations: one with a random distribution of AG, INV,
and GLY cells, one with a high percentage of AG cells everywhere except for a
corner with high INV, and one with a high percentage of GLY cells everywhere
except for a corner with high INV. We will discuss first the discrete continuous
model (which uses a partial differential equation (PDE) that is discretized in
two dimensions) and then discuss our cellular automaton (CA) model which
models the game on a 10 by 10 grid but uses a different method of changing the
morphology of the tumor. These simulations use the same payoffs described in
the Basanta model, but include the spatial aspect of tumors. We begin with the
payoff matrix in the game between the three phenotypes. The full payoff in an
interaction between two cell is represented by 1, while the different parameters
c, n, and k denote the benefits and costs of interaction. Parameter c is the cost
in motility of the INV cells. Parameter n represents both the cost for an AG cell
to live near a GLY cell due to the GLY cell increasing acidity in the area and
the benefit for a GLY cell to increase the acidity of the environment. Finally,
parameter k represents the cost of using the less efficient glycolytic metabolism.
AG
AG
INV
GLY
1
2
1
2
1−c
+n−k
IN V
1
1 − 2c
1-k
GLY
1
2 −n
1−c
1
2 −k
Table 1 displays the payoffs for the interaction between the cell on the left and
the cell on the top. For example, the payoff gained by and AG cell interacting
with another AG cell is 12 because they share resources. When an INV cell
interacts with anything besides an INV cell, it gains the base payoff minus c
because it moves to another location. When the GLY cell interacts with the
AG cell it share the resources and gains fitness for changing the environment,
but loses k because of its glycolytic metabolism.
We then replicate the results shown by Basanta et al. with the following
system of ordinary differential equations (ODEs):
dy
= y.(Ay − y T Ay)
dt
(1)
where A is the payoff matrix in Table 1 and y is a 3 by 1 vector with the
population fractions of the AG, INV, and GLY cells going from top to bottom.
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The sum of the elements in y must be 1, and the period represents element-wise
multiplication.
Plot of Population Fractions over Time
0.9
AG
INV
GLY
0.8
Population Fractions
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
50
100
Time Step
150
200
Figure 1: This graph shows how the population fractions change as time goes on with
the parameters c = .5, n = .4, and k = .2.
Inputting various values for each parameter in both our original ODE model
and the Basanta model returned similar outputs. For example, the parameter
values c = .5, n = .4, and k = .2 (later discussed as our “baseline” parameters)
returned a steady state of 18.75% AG, 50% INV, and 31.25% INV, the same as
seen in Basanta’s model if we plug the values of c, n, and k into the equations
in the paper. These equations are p = 1 − nk and p′ = 2kn+k−ck−cn
, where p
2n2
is the population fraction of INV cells and p’ is the population fraction of AG
cells [3]. Thus, our results from the ODE model are the same as the results
gleaned from the Basanta model: conditions favoring GLY cells also promote
the proliferation of INV cells.
2.1
The One-Dimensional Model
Now, we proceed to incorporating the spatial aspect of the game. We first
do this in one dimension by making the interactions take place in ten spatial
“cells” in a line. Each spatial “cell” has its own morphology of the three tumor
phenotypes. Furthermore, the “cells’ individual morphologies affect those of the
cells directly adjacent to them. In essence, each spatial “cell” tries to match
their morphologies with the “cells” next to them. The rate at which this occurs
is denoted by a diffusion constant a. By multiplying the diffusion constant to
the Laplacian matrix of the 1 by 10 grid, we can model our original ODE with a
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one-dimensional spatial aspect. We can think of the new equation as a reactiondiffusion equation, with the original ODE being the reaction component and the
Laplacian matrix being the diffusion component [5]. The following simulations
are run with “baseline” parameters c = .5, n = .4, and k = .2 and a random
initial condition.
Plot of INV Population in Cell r at Time t
Plot of AG Population in Cell r at Time t
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1
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Population
Population
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(b) INV populations
Plot of GLY Population in Cell r at Time t
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Population
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(c) GLY populations
Figure 2: These figures depict the steady states in each point on the line.
2.2
The Two-Dimensional Model
Finally, we move on to the spatial game in two dimensions. This time, the
game is played on a 10 by 10 grid, with each spatial “cell” trying to match its
morphology with the “cells” directly above, below, to the left, and to the right of
it. Although the process of applying the diffusion constant to the Laplacian and
solving the ODE is similar in the two-dimensional and one-dimensional cases,
the Laplacian matrix in the two-dimensional game is much more complicated
to compute. Furthermore, the plot of our results is difficult to interpret if we
use the same graph that we did in the one-dimensional case because the cell
numbers are labelled 1 to 100 but are in a 10 by 10 grid instead of a 1 by 100
grid 3.
4
Plot of AG Population in Cell r at Time t
Plot of INV Population in Cell r at Time t
0.8
0.7
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Population
Population
0.5
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100
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Time Step
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(b) INV populations
Plot of GLY Population in Cell r at Time t
1
Population
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150
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Time Step
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(c) GLY populations
Figure 3: Each point on the 10 by 10 grid is numbered from 1 to 100, and these
graphs display the steady states in each numbered point. The simulations are run
with baseline parameters and an initial condition of 80% AG and 20% GLY everywhere
except for the corner with 80% INV and 10% everything else. This initial condition
models the invasion from a specific point in the tumor.
As these graphs may be difficult to interpret, we included another visualization which plots the populations at each location on the 10 by 10 grid, with the
x- and y-axes representing locations on the grid. However, we could not plot
the solution against time with this method, so the population at time steps 100,
200, and 300 are shown.
5
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Figure 4: AG cell populations on the 10 by 10 grid
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Figure 5: INV cell populations on the 10 by 10 grid
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Figure 6: GLY cell populations on the 10 by 10 grid
Incorporating the spatial aspect to the ODE model produced steady states
that were similar to the ones produced by the original ODE, and it also changed
how the population fractions of each cell reached the steady state. For instance,
choosing a higher diffusion constant made the cells reach their steady states
much faster. This happens because cells near each other achieve the same
morphology more quickly, making the entire tumor reach the same steady state
at a higher rate.
6
3
The Cellular Automaton Model
Like the Discrete Continuous Model, the Cellular Automaton Model runs on
a 10 by 10 grid. However, this model runs differently using the same base
parameters c=0.5, n=0.4, and k=0.2. The Cellular Automata Model works in
four steps.
The first step is to initialize all the cells on the grid. This will assign each cell
to a certain type: Autonomous, Invasive, or Glycolytic cells. In our case, there
are three main types of initial conditions. They will be explained more in detail
in the results section.
The next step is the find the fitness value for each cell on the board. It
is calculated by the sum of the fitness payoff of every adjacent cell to obtain
the fitness of the center cell. The payoff is calculated by using the table in
the Bastanta model. For example, to calculate the center AG cell, you would
need to add the sum of all the payoff values based on the aforementioned payoff
matrix.
The third step is to decide which cells need to be updated. Each cell has
a 10 percent chance of being updated. You can see this process in the truefalse table. True represents the cells that need to be updated, and they are
highlighted in yellow.
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The final step is to update each cell. In this process, the neighboring cell
with the highest fitness value will replace the old cell. This is how certain cells
can change type (autonomous to invasive or invasive to glycolytic). Once the
final step is finished, a full generation has been created. Steps 2 through 4 will
be repeated to create future generations.
4
Results
Both the Discrete Continuous Model and the CA model used the payoff matrix
that Basanta created, but they returned different outputs. Because tumors
always have a percentage of GLY cells present, we chose a set of “baseline”
parameters that returned a steady state with a sizable GLY cell population.
Thus, the values we chose to conduct simulations for different initial conditions
and treatments were as follows: c = .5, n = .4, and k = .2. The steady state
that the original ODE model produces is 18.75% AG, 50% INV, and 31.25%
INV. With these parameters, we will run simulations of both the continuous
model and the CA model with three different initial conditions and explore how
the morphology of the tumor changes. Finally, we will simulate treatments by
running the simulations with changed parameters. To simply the process, the
same initial condition will be used in each of the treatment simulations, and the
diffusion constant will be assumed to be .01. Furthermore, we assume that the
treatment is spatially homogeneous–that is, the change in parameters will occur
in every “cell” of the 10 by 10 grids.
4.1
The Discrete Continuous Model
We will first explore the results of the continuous model. Running the simulation
with a random initial condition returned the steady state in 50 time steps. The
steady state from the random initial condition was 20.21% AG, 48.96% INV,
and 30.83% GLY. We ran the next simulation with 80% AG cells, no INV cells,
and 20% GLY cells in every spatial “cell” of the 10 by 10 grid except for a
corner with 10% AG cells, 80% INV cells, and 10% GLY cells. We found that
the population fractions of each phenotype, when visualized in a 10 by 10 grid,
exhibited a wave-like pattern extending from the corner with the INV cells
before settling into a steady state. Furthermore, a large increase in GLY cells
was immediately followed by a large increase in INV cells, suggesting that the
INV phenotype can invade more easily in a tumor with almost exclusively GLY
cells. The average of each population fraction at 300 time steps was 18.58% AG,
8
50.09% INV, and 31.33% GLY. This initial condition took much longer to reach
the steady state, almost six times the time steps it took to reach the steady
state from a random initial condition. We have already provided visualization
of this simulation in Figures 4 through 6 4.
We ran the final simulation with 20% AG cells, no INV cells, and 80% GLY
cells in every spatial “cell” of the grid except with a corner with 10% AG cells,
80% INV cells, and 10% GLY cells. As expected, the simulation was similar to
the one with mostly AG and INV in a corner, producing a wave-like pattern
emanating from the corner with the high INV concentration. However, this
simulation essentially starts at a ”later” point of the second simulation, since
GLY is already high. Because of this fact, the steady state for this initial
condition is the same as that of the second simulation.
4.1.1
Treatments
Next, we ran simulations of treatments that would change one parameter, in
order to discover ways to prevent the INV phenotype from invading. We ran
these simulations with only one initial condition (where AG is high everywhere
except for a corner with high INV) to simplify analysis. By running the simulation with three different c values (.6, .65, .7) and baseline n and k values, we
explored the effect that a treatment increasing the cost of motility would have
on the tumor’s morphology. The treatment simulated when we change this parameter is one that can increase the cell-cell adhesion in the tumor, therefore
increasing the cost in motility. At c = .6, the tumor’s final steady state was
half INV and half GLY. Again, the INV phenotype remained low until the GLY
phenotype became predominant, at which point the INV phenotype increased
its population fraction to 50%. It took about 300 time steps to reach this steady
state. The wave-like pattern seen before is also present in this scenario, likely
because of the initial condition. It took about 300 time steps to reach this
steady state. At c = .65, the steady state took 250 time steps to reach. The
final morphology was 28.39% INV and 71.61% GLY. This time, the GLY cells
spent much longer as the predominant cell type (over 90%) in the tumor before
it ceded to the INV phenotype.
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Figure 7: INV cell populations on the 10 by 10 grid when c = .65
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Figure 8: GLY cell populations on the 10 by 10 grid when c = .65
Finally, at c = .7, the GLY phenotype took over the entire tumor relatively
quickly–a little over 50 time steps. From this result, it seems that if the cost of
motility is too high, the GLY phenotype will invade as long as n is greater than
k (as in this scenario the GLY cells have the highest relative fitness). Since GLY
cells are also seen in many malignant tumors, we deem this type of treatment not
as desirable as one that helps AG cells atke over the tumor.Curiously enough, if
the cost of motility is actually lowered, the GLY phenotype invades for a certain
period of time, then becomes almost nonexistent. The AG and INV cells then
take up most of the tumor, with the INV to AG ratio becoming larger as c
becomes smaller. For example, at c = .4, the final steady state at 300 time
steps was 37.40% AG, 50.10% INV, and 12.50% GLY. This probably happens
because the INV cells become predominant when c is low, making the fitness
advantage that the GLY cells have less effective (the INV cells are not affected
by the lowered pH of the local microenvironment because they move away from
other cells when they interact).
The next parameter we changed to simulate treatments was n, in order to
make the GLY cells less fit to survive. We make c = .5 and k = .2, and run
the simulations with n = .3, .2. This represents a treatment that increases the
pH in the tumor microenvironment in order to eliminate the advantage GLY
cells receive when they change pH through glycolysis. In all of the following
simulations, the wave-like pattern is observed, because of the initial condition.
At n = .3, the AG phenotype becomes more fit, so its population fraction increases. The final steady state is 38.51% AG, 33.58% INV, and 27.91% GLY.At
n = .2, we see a dramatic change in the steady state. It takes around 250 time
steps to reach this steady state at every point in the 10 by 10 grid, and there is
a very noticeable increase in AG cells. The steady state is 95.78% AG, 2.78%
INV, and 1.44% GLY, which is almost exclusively AG cells. Because the tumor
changes from having an equal distribution of the phenotypes to having mostly
AG with such a small change in the parameter n, we hypothesize that changing
the microenvironment to favor GLY less is an effective treatment for tumors.
10
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Figure 9: AG cell populations on the 10 by 10 grid at n = .2
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Figure 10: INV cell populations on the 10 by 10 grid at n = .2
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Figure 11: GLY cell populations on the 10 by 10 grid at n = .2
Lastly, we explore the efficacy of a combination of the two treatments by
setting c to .65 and n to .2. At our original initial condition, the average
population fractions appear to be 80.09% AG and 19.91% GLY. However, at
a random initial condition, the average population fractions is 76.35% AG and
23.65% GLY. This is an example of the spatial model returning different steady
states with different initial conditions, something we cannot see with the regular
ODE model.
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4.2
The Cellular Automaton Model
In the CA model, we ran the simulation using the same baseline parameters:
c=0.5, n=0.4, and k=0.2. The results are calculated based on an average of 5
simulations. Each simulation consists of 50 generations. During each simulation,
there are three different initial conditions. The first condition is where all the
cells are randomly generated in different parts of the grid. The next condition
is when the grid is filled with about 2/3 autonomous cells, 1/3 glycolytic cells,
and 4 invasive cells in the bottom right hand corner. The final condition is the
same as the second condition except that there are 2/3 glycolytic cells and 1/3
autonomous cells.
When the grid was randomly initialized, the simulation averaged out to be
22.8% AG, 72.4%INV, and 4.8%GLY. When the grid had 2/3 Autonomous cells
with Invasive Cells in a corner, on average, there were 5.6% AG, 51.6% INV,
and 42.8% GLY. Finally, when there were 2/3 Glycolytic cells with Invasive
Cells in a corner, there averaged to be 1.6% AG, 49% INV, and 49.4% GLY.
4.2.1
Treatments
Like the Discrete Continuous Model, we ran a series of treatments to prevent
the Invasive cells from spreading throughout the grid. We dud this by changing
parameters ”c” and ”n” one at a time.
We started by changing the value of parameter ”c” from 0.5 to 0.65 and 0.7.
This increases the motility cost for invasive cells, making it cost more fitness for
them to move around. As a result, the loss of invasive cells will be replaced by
more autonomous and glycolytic cells. When the cells were randomly generated,
there were 15.4% AG, 78.4% INV, and 6.2% GLY when c=0.65. There were
10.6% AG, 82% INV, and 7.4% GLY when c=0.7. The graph for this simulation
is displayed below.
When there were 2/3 Autonomous cells and Invasive cells in a corner, there
were 0.6% AG, 37.6% INV, and 61.8% GLY when c=0.65. There were 2.4%
AG, 37% INV, and 60.6% GLY.
When there were 2/3 Glycolytic cells and Invasive cells in a corner, there
were 1.6% AG, 42.4% INV, and 56% GLY when c=0.65. There were 0.2% AG,
43.2% INV, and 56.6% GLY.
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Another treatment we used was to lower the ”n” parameter from 0.4 to 0.3
and 0.2. The treatment will decrease the cost of fitness for glycolytic cells to
produce energy resulting in more glycolytic cells. When the cells were randomly
generated, there were 20.6% AG, 77.8% INV, and 1.6% GLY when n=0.3. There
were 5.4% AG, 94.6% INV, and 0% GLY when n=0.2.
When there were 2/3 Autonomous cells and Invasive cells in a corner, there
were 4.2% AG, 45% INV, and 50.8% GLY when n=0.3. There were 3.2% AG,
42.6% INV, and 54.2% GLY when n=0.2.
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When the cells were randomly generated, there were 0.4% AG, 48.8% INV,
and 50.8% GLY when n=0.3. There were 1.2% AG, 50.2% INV, and 48.6% GLY
when n=0.2.
4.3
4.3.1
Analysis
Randomly Generated Cells
When the cells were randomly generated, there were mostly Invasive cells, even
when the treatments were made. This is largely because the grid had 1/3
invasive cells in comparison to the 1/25 invasive cells in the other grid formats.
In other words there were more invasive cells and it was easier for the cancer
cells to spread throughout the grid. Also, when ”n” was decreased, 95% of the
grid was Invasive cells. This is due to the fact that Invasives can easily take over
the large number of glycolytic cells. In conclusion, the treatments didn’t work
with the randomly generated cells because there still was a very large percentage
of Invasive cells.
4.3.2
2/3 Autonomous Cells and Invasive Cells in a Corner
Unlike the randomly generated cells, both treatments mostly worked. However,
there was a low number of Autonomous cells, so it wasn’t a 100% cure. The
invasive cells went down making the Glycolytic cells the largest percentage of
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cells in the grid. This is because it is a lot harder for the Invasive cells to take
over the high number of Autonomous cells in the beginning of the simulation.
Eventually, the Glycolytic cells replaced the Autonomous cells making it easier
for Invasive cells to spread.
4.3.3
2/3 Glycolytic Cells and Invasive Cells in a Corner
For this scenario, the treatments weren’t as effective. However, the percentage
of Invasive cells still went down, but it was only by about 7%. The treatments
didn’t work as much due to the large number of Glycolytic cells. This made it
easier for Invasive cells to spread around the grid. The Glycolytic cells still took
over the Autonomous cells and still managed to take up half of the grid.
4.4
Differences Between the Two Models
The two models differ greatly in their implementation, and as such the results
from each model also differ. While the discrete continuous model has population
fractions in each space of the 10 by 10 grid, the CA model has a single phenotype
in the spaces. Because of this fact, the results from the CA model are more
stochastic. In addition, the discrete continuous model is “larger” in scale than
the CA model because of the way it is implemented. We can think of each space
of the discrete continuous model’s 10 by 10 grid as a CA grid, since it deals with
population fractions as opposed to single cells. Finally, in the CA model the
cells with the lowest fitness will completely die out, whereas in the continuous
model, most cells with lower fitness will simply be at a lower population fraction.
This is due to the updating system used in the CA model–if one cell loses, even
by the tiniest fitness value, it will be replaced.
5
Conclusions
Our continuous model supports the hypothesis set forth by D. Basanta: a switch
to glycolysis for energy production in tumor cells makes invasive phenotypes
more likely to invade [3]. When we ran the simulations, a dominance of GLY
in the tumor always preceded an INV cell takeover. Furthermore, when the
parameters were set to give GLY cells a fitness advantage, the INV phenotype
was able to displace the AG cells. This happened most likely because the GLY
cells would make the environment unfit for AG cells, making room for the INV
cells to invade. The continuous model returned similar results to those seen in
Basanta’s model, but with slightly different steady states. We believe that our
use of diffusion with a Laplacian matrix is an effective method of incorporating
space into previously explored models. Other models have used similar systems
of partial differential equations to model tumor growth, but this is one of, if not
the first model that melds evolutionary game theory with a reaction-diffusion
model [1]. The continuous model of tumor progression is still quite simple,
however, since we generalize the types of tumors (gliomas) and the types of
15
cells in those tumors. To refine our model, we could find parameters with a
stronger biological basis as our “baseline” parameters. Such parameters would
reflect the tumor’s growth more realistically and enhance our ability to produce results seen in experiments. In addition, we assume that treatments and
parameters are spatially homogeneous–if we could instead have these parameters change as different morphologies are considered, the model would become
more realistic. Having treatments “diffuse” across the tumor would provide an
enriched simulation of actual therapy. Finally, we have not yet explored the biological significance of the time steps, i.e. what each time step actually means in
units. This is because the MatLab command (ode45) that we used to solve the
ODE system chooses time steps on its own. Furthermore, our cellular automata
model produced different results from the continuous model, and did not seem
to match results in past models [2]. We hope to refine the updating system and
grid so that the results will be more akin to what we expect. Despite its shortcomings, the cellular automata model gave us an invaluable new visualization
of the tumor.
References
[1] E. L. Newman R. J. C. Steele A. M. Thompson A. R. A. Anderson, M. A.
J. Chaplain. Mathematical modelling of tumour invasion and metastasis.
Journal of Theoretical Medicine, 2:129–154, 1999.
[2] Alsner J. Bach L. A., Sumpter D. J. T. and Loeschcke V. Spatial evolutionary games of interaction among generic cancer cells. Journal of Theoretical
Medicine, 5:47–58, 2003.
[3] Hatzikirou H. Basanta D., Simon M. and Deutsch A. Evolutionary game
theory elucidates the role of glycolysis in glioma progression and invasion.
Cell Proliferation, 41:980–987, 2008.
[4] Warburg O. The metabolism of tumors. 1930.
[5] Steven J. Ruuth. Implicit-explicit methods for reaction-diffusion problems
in pattern formation. Journal of Mathematical Biology, 34:148–176, 1995.
[6] Salvatore Torquato Yang Jiao. Emergent behaviors from a cellular automaton model for invasive tumor growth in heterogeneous microenvironments.
Plos Computational Biology, 7:1–14, 2011.
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