The Trees for the Forest

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The Trees for the Forest

• A Discrete Cell Model of Tumor Growth,
  Development, and Evolution

Craig J. Thalhauser
Ph.D. student in Mathematics/Computational
Bioscience Dept. of Mathematics
Statistics Arizona State University Workshop on
Mathematical Models in Biology Medicine
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Outline

• Biological Review of Cancer
• Structure, Genetics, and Evolution
• Model Systems in vitro
• Review of Mathematical Models of Cancer
• Models of the Multicellular Spheroid (MCS) Tumor
  The Greenspan Model and Beyond
• Continuous and Hybrid models Cellular Automata
• The Subcellular Element Model Approach
• Derivation of the MCS system
• Tumor-environment interactions

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What is Cancer?
Cancer is a class of diseases characterized by
uncontrolled division of cells and the ability of
these cells to invade other tissues, either by
direct growth into adjacent tissue or by
implantation into distant sites (from
Wikipedia.com)
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What makes a transformed cell
Cancer involves a collection of traits acquired
through mutation
Cancers are strongly heterogeneous many genetic
paths can lead to transformation
(Hanahan Weinberg, 2000)
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Structure of a tumor
(image from http//www.wisc.edu/wolberg/Insitu/in_
situ.html)
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Genetics Evolution in Cancer
(image from http//www.fhcrc.org/science/education
/courses/cancer_course/basic/molecular/accumulatio
n.html)
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The Multicellular Spheroid
The Multicellular Spheroid (MCS) is an in vitro
model of avascular tumor growth
(image from http//www.ecs.umass.edu/che/henson_gr
oup/research/tumor.htm)
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Greenspans Model of the MCS
R0(t) Outer radius of MCS Rg(t) Inner
radius of growth Ri(t) Radius of Necrotic
Core ?(r,t) Diffusible nutrient from
media ?(r,t) Diffusible toxin from tumor
(Nagy 2005) and (Greenspan 1972)
Assumptions

1. Perfect spherical symmetry
2. Necrosis caused by nutrient deficiency only
3. Toxin leads to decreased growth rate

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Moving Beyond the Greenspan System
Spatial Asymmetries in GBM (brain cancer)
Growth-Diffusion equation for cell density in
dura with spacially varying migration rates
(Swanson et al. Cell Proliferation.
33(5)317 (2000)
Model predicts tumor cell density far outside of
detection range for modern diagnostic procedures
Cellular Automata Hybrid of Nutrient
Reaction-Diffusion Equations Cellular Automata
cell densities (Mallet Pillis. J. Theo. Bio.
2005)
Model predicts tumor-host interface structure is
strongly dependent upon tumor growth rate
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The Subcellular Element Model (SEM)
An Agent-Based Model system
Agents (Cells) are not directly associated with a
lattice (a la cellular automata) agents live
in non-discretized 3-space.
Agent Construction
1. Each Agent is 1 cancer cell
2. An Agent is composed of 1-2N elements which
contain a fixed volume of cellular space
3. Elements within a cell behave as if connected
by a nonlinear spring
4. Elements between cells repel with a modified
inverse-square law
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The SEM and the MCS
Agent Actions
1. Reacts to external chemical fields
Ni(x,y,z,t) concentration of nutrient
I ?(x,y,z,t) interpolated density of tumor
cells f(N) absorption/utilization rate of
nutrient
2. Responds to nearest neighbor actions
Growth and/or movement of neighbors leads to
changes in local density, which leads to
interactions via contact laws
3. Attempts to grow at all costs
Assemble sufficient nutrients to allow for growth
Stochastic mutations to growth parameters allow
cells to adapt to a changing environment
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A Typical MCS Simulation
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Challenges with the SEM

1. Adaptation of non-discretized agents to
   discretized nutrient field

Solution take nutrient field grid to be smaller
than agent size and use linear interpolation
mapping between settings
2. Scalability
Solution Optimize for massively parallel
computers
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Concluding Thoughts

1. Current models of avascular tumor development,
   while mathematically useful, do not capture the
   extremely heterogeneous nature of the disease
   structure.

1. An agent based model system, the SEM, can be
   constructed to fully explore within tumor
   processes, tumor-host interactions, and
   adaptative and evolutionary paths.

1. The advent of massively parallel supercomputers
   makes this model computationally tractable and
   able to offer insight and predictive power

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Acknowledgements
Dr. Yang Kuang (advisor)
In the Math Department
Dr. Timothy Newman (co-advisor)
Abdessamad Tridane
Dr. John Nagy
Dr. Steven Baer
In the Physics Department
Dr. Hal Smith
Erik DeSimone Erick Smith
References
Greenspan H.P. Models for the growth of a solid
tumor by diffusion Stud. Appl. Math., 52317
(1972)
Hanahan Weinberg. The Hallmarks of Cancer
Cell 100 57 (2000)
Nagy, J. D. The Ecology Evolutionary Biology
of Cancer A Review of Mathematical Models of
Necrosis and Tumor Cell Diversity  MBE 2 (2)
381 (2005)
Newman T. J. Modeling Multicellular Systems
Using Subcellular Elements MBE 2 (3) 613 (2005)