The Trees for the ForestThe Trees for the Forest

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

Ph.D. student in Mathematics/Computational BioscienceDept. of Mathematics & Statistics

Arizona State University

Workshop on Mathematical Models in Biology & Medicine

Craig J. Thalhauser

OutlineOutline Biological Review of CancerBiological Review of Cancer

Structure, Genetics, and Evolution Structure, Genetics, and Evolution Model Systems Model Systems in vitro in vitro

Review of Mathematical Models of CancerReview of Mathematical Models of Cancer Models of the Multicellular Spheroid (MCS) Tumor: The Models of the Multicellular Spheroid (MCS) Tumor: The

Greenspan Model and BeyondGreenspan Model and Beyond Continuous and Hybrid models; Cellular AutomataContinuous and Hybrid models; Cellular Automata

The Subcellular Element Model ApproachThe Subcellular Element Model Approach Derivation of the MCS systemDerivation of the MCS system Tumor-environment interactionsTumor-environment interactions

What is Cancer?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 cellWhat makes a transformed cellCancer 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 tumorStructure of a tumor

(image from http://www.wisc.edu/wolberg/Insitu/in_situ.html)

Genetics & Evolution in CancerGenetics & Evolution in Cancer

(image from http://www.fhcrc.org/science/education/courses/cancer_course/basic/molecular/accumulation.html)

The Multicellular SpheroidThe Multicellular SpheroidThe 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 MCSGreenspan’s Model of the MCS

R0(t): Outer radius of MCSRg(t): Inner radius of growthRi(t): Radius of Necrotic Core

(r,t): Diffusible nutrient from media(r,t): Diffusible toxin from tumor

Assumptions

1. Perfect spherical symmetry2. Necrosis caused by nutrient deficiency only3. Toxin leads to decreased growth rate

(Nagy 2005) and (Greenspan 1972)

Moving Beyond the Greenspan Moving Beyond the Greenspan SystemSystem

Cellular Automata: Hybrid of Nutrient Reaction-Diffusion Equations + Cellular Automata cell densities (Mallet & Pillis. J. Theo. Bio. 2005)

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

Model predicts tumor-host interface structure is strongly dependent upon tumor growth rate

The Subcellular Element Model The Subcellular Element Model (SEM)(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

ri

re

The SEM and the MCSThe SEM and the MCS

Agent Actions

1. Reacts to external chemical fields

2. Responds to nearest neighbor actions

3. Attempts to grow at all costs

0

3

),(

)1,0(

)(),(0

NtN

NftrN ii

Ni(x,y,z,t) = concentration of nutrient I

(x,y,z,t) = interpolated density of tumor cellsf(N) = absorption/utilization rate of nutrient

Growth and/or movement of neighbors leads to changes in local density, which leads to interactions via contact laws

Assemble sufficient nutrients to allow for growthStochastic mutations to growth parameters allow cells to adapt to a changing environment

A Typical MCS SimulationA Typical MCS Simulation

Challenges with the SEMChallenges 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

Time Cost of SEM-MCSy = 1E-06x2 + 0.004x

R2 = 0.9998

0

200

400

600

800

1000

1200

1400

1600

0 10000 20000 30000 40000

number of elements

itera

tion

time

(s)

SEM, 1 element/cell

Poly. (SEM, 1 element/cell)

Solution: Optimize for massively parallel computers

Concluding ThoughtsConcluding 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

AcknowledgementsAcknowledgements

ReferencesReferences

Dr. Yang Kuang (advisor)Dr. Timothy Newman (co-advisor)Dr. John NagyDr. Steven BaerDr. Hal Smith

In the Math Department:

Abdessamad Tridane

In the Physics Department:

Erik DeSimoneErick Smith

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)

Greenspan H.P. “Models for the growth of a solid tumor by diffusion” Stud. Appl. Math.,52:317 (1972)