Transcript The Trees for the Forest - Arizona State University

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
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
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)
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)
Structure of a tumor
(image from http://www.wisc.edu/wolberg/Insitu/in_situ.html)
Genetics & Evolution in Cancer
(image from http://www.fhcrc.org/science/education/courses/cancer_course/basic/molecular/accumulation.html)
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_group/research/tumor.htm)
Greenspan’s 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
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
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
ri
3. Elements within a cell behave as if
connected by a nonlinear spring
4. Elements between cells repel with
a modified inverse-square law
re
The SEM and the MCS
Agent Actions
1. Reacts to external chemical fields

0  N i   (r , t ) f ( N i )
  (0,1)3
N (, t )  N 0
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
A Typical MCS Simulation
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
y = 1E-06x 2 + 0.004x
R2 = 0.9998
Time Cost of SEM-MCS
Solution: Optimize for
massively parallel
computers
iteration time (s)
1600
1400
1200
1000
800
600
SEM, 1 element/cell
Poly. (SEM, 1 element/cell)
400
200
0
0
10000
20000
number of elements
30000
40000
Concluding Thoughts
1. Current models of avascular tumor development, while
mathematically useful, do not capture the extremely heterogeneous
nature of the disease structure.
2. An agent based model system, the SEM, can be constructed to
fully explore within tumor processes, tumor-host interactions,
and adaptative and evolutionary paths.
3. The advent of massively parallel supercomputers makes this
model computationally tractable and able to offer insight and
predictive power
Acknowledgements
Dr. Yang Kuang (advisor)
Dr. Timothy Newman (co-advisor)
Dr. John Nagy
Dr. Steven Baer
Dr. Hal Smith
In the Math Department:
Abdessamad Tridane
In the Physics Department:
Erik DeSimone
Erick Smith
References
Greenspan H.P. “Models for the growth of a solid tumor by diffusion” Stud. Appl. Math.,
52:317 (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)