of Melanoma Growth

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Dec 1, 2013 (3 years and 6 months ago)

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Incorporating Adhesion in a Cellular Automata Model

of Melanoma Growth

Chris DuBois (’06)
, Ami Radunskaya*

Dept. of Mathematics, Pomona College, Claremont, CA

Abstract


Accurate models of tumor growth could guide new experiments, give direction for
new therapeutic approaches, reduce the guesswork in clinical trials, and bring
new success for cancer treatment. In this work, we present a cellular automaton
model of early melanoma growth where the behavior of each cell within our
simulation is dependent on its surroundings (e.g. nutrient availability, nearby cells,
pH, etc.). Evidence suggests that local conditions such as these can have drastic
effects on tumor growth, malignancy, and response to treatment
6
. Evidence also
suggests cell
-
cell adhesion plays a role in tumor growth, affecting a tumor's
progression towards a solid spheroid or fragmentation and metastasis. In order
to study this variation, we incorporate a model for the potential energy between
cells due to adhesive and elastic forces. We propose this provides a more
accurate model of solid tumor growth in vivo that could help explore the
progression of early melanoma.

Introduction

Accurate models of tumor growth could guide new
experiments, give direction for new therapeutic
approaches, reduce the guesswork in clinical trials, and
bring new success for cancer treatment.

Evidence suggests that
local conditions

(e.g. nutrient
availability, pH, etc.) play a key role in tumor behavior,
possibly having drastic effects on tumor growth, malignancy,
and response to treatment
6
. New evidence argues that
low
oxygen

conditions force tumor cells to metabolize glucose
less efficiently and in turn consume more, challenging the
common perception that tumor cells are always glycolytic
7
.

In this work, we present a cellular automaton model of early
melanoma growth from an energy budget perspective,
where
fuel consumption and ATP production are dependent
on local oxygen concentration, available fuel concentration,
and pH
.

We propose that this provides a more accurate model of
solid tumor growth in vivo that could help explore the role of
tumor metabolism and hypoxia in tumor growth.

Future Directions

Future work includes:



Incorporate angiogenesis



Incorporate immune response and chemotherapy agents

References

1. Araujo.


Casciari, J.J., Sotirchos, S.V., and Sutherland, R.M. (1992). Variations in Tumor Cell
Growth Rates and Metabolism With Oxygen Concentration, Glucose Concentration, and
Extracellular pH.
Journal of Cellular Physiology

151
:386
-
394.

2. Gatenby, R.A. and Gawlinski, E.T. (2003). The Glycolytic Phenotype in Carcinogenesis and
Tumor Invasion: Insights through Mathematical Models.
Cancer Research

63
: 3847
-
3854.

3. Preziosi, L.
Cancer Modelling and Simulation.
Chapman & Hall/CRC (2003). Mathematical
Biology and Medicine Series.

5. Subarsky, P. and Hill, R.P.. (2003). The hypoxic tumour microenvironment and metastatic
progression.
Clinical & Experimental Metastasis
.
20
(3):237
-
250.

6. Turner

6. Vaupel, P., Thews, O., Kelleher, D.K. and Hoeckel, M. (1998). Current status of knowledge
and critical issues in tumor oxygenation
-

Results from 25 years research in tumor
pathophysiology.
Oxygen Transport to Tissue XX
,
454
:591
-
602.

7. Zu, X.L. and Guppy, M. (2004). Cancer metabolism: facts, fantasy, and fiction.
Biochemical
and Biophysical Research Communications
.
313
:459
-
465.

Cellular Automata


Cellular automata (CA) are common discrete implementations
because of their ability to replicate the complexity of biological
systems
3
.
The simulation space

is a multi
-
layered NxN grid which
represents a thin layer of tissue where each grid element
represents a physical volume of 175x175x40 microns, or 1e
-
6 ml.
Each element contains information on local cell populations and
chemical concentrations.

Sample Diffusion Simulation








Growth and Invasion

Cell Growth:

Cellular growth rate depends on how much energy
cells have available for growth, which we calculate for an element
(i,j)

from a population’s current ATP turnover
C
ij
(t)
.
V
ij

represents the
tumor population.
M

represents the energy required for cellular
maintenance functions, and
g
represents the energy needed for
mitosis.



Tumor Invasion:

At the tumor edge, proliferating tumor cells are
able to invade nearby tissue by causing low pH or a degraded
extracellular matrix, conditions favorable for tumor growth
2
, as seen
in Figure * below.

ATP Turnover Rates vs Local Conditions


We calculate the rate of cellular energy (ATP) production from
nutrient consumption equations above. Tumor cells have a
competitive advantage in adverse conditions (e.g. low pH).

Research Supported
by the HHMI Grant

Diffusion of Nutrients

Tumor cells compete with normal cells for nearby nutrients such as
oxygen and glucose, both of which are delivered by nearby blood
vessels. One way to model the movement of small particles is to
average the concentration with a random neighboring element.
After many steps, this is equivalent to averaging the concentrations
of a particular grid element and its four neighbors. We
systematically do this across the entire simulation space.


Vascular Collapse and Hypoxia

Once the tumor reaches a critical size, the pressure at the tumor
center compresses the blood vessels. This inhibits both nutrient
flow to cells within the tumor as well as the delivery of blood
-
borne
therapies
1
. The resulting low oxygenation at the tumor's center
provokes cell
-
death (via apoptosis or necrosis), forming what is
clinically known as the necrotic core.

Research also shows correlations between hypoxia and metastatic
progression, treatment resistance, and patient survival
.
6


In our simulation, when blood vessels are surrounded by a
sufficient number of tumor cells, the flow of molecules in and out of
the vessel is restricted in order to model this effect.




Metastasis and Tumor Models

Metastasis is the spread of cancer from one part of the body to
another. The occurrence of metastases is the leading cause of
death among cancer patients
3
. Metastasis begins when
cancerous cells detach from the primary tumor and invade the
surrounding tissue, becoming more severe when invading cells
reach a blood or lymphatic vessel. While both genotype and
environmental conditions affect the prognosis of similar sized
tumors, various molecular mechanisms also facilitate invasion
into the surrounding tissue
3
.

Cellular adhesion molecules, such as integrins, play a
particularly large role in the life and mobility of tumor cells,
affecting their interactions with neighboring cells and the
surrounding extracellular matrix
3
. When modeling the
progression of a tumor through its different stages, it is
necessary to consider the effects of cellular adhesion between
cells and the extracellular matrix to gain insight into the
progression of metastases.

Cell Adhesion

Turner() provides a method for estimating the diffusion
coefficient for biological cells modeled as adhesive,
deformable spheres by considering the ``potential energy of
interaction" between individual cells. Turner shows that the
diffusion coefficient is proportional to the second derivative of
the cells' energy density,
e(n),

which can be derived as a
function of cell density,
n
, in terms of the biological parameters
of individual cells such as elasticity and adhesiveness.

In our model, tumor cells move in the direction that results in
the greatest decrease in potential energy as defined above.
More specifically, we allow proliferating cells in a grid element
to move to the neighboring grid element
i

where
de(n
i
)/dn

is
least; if this amount exceeds
n
eq
,

then the remaining cells
move into the neighbor with the next smallest
de(n)/dn

value.