The Trees for the Forest

1. 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

2. 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

3. 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)

4. 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)

5. Structure of a tumor (image from http://www.wisc.edu/wolberg/Insitu/in_situ.html)

6. Genetics & Evolution in Cancer (image from http://www.fhcrc.org/science/education/courses/cancer_course/basic/molecular/accumulation.html)

7. 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)

8. 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 • Perfect spherical symmetry • Necrosis caused by nutrient deficiency only • Toxin leads to decreased growth rate

9. 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

10. ri re 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

11. 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

12. A Typical MCS Simulation

13. Challenges with the SEM • 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

14. Concluding Thoughts • Current models of avascular tumor development, while mathematically useful, do not capture the extremely heterogeneous nature of the disease structure. • An agent based model system, the SEM, can be constructed to fully explore within tumor processes, tumor-host interactions, and adaptative and evolutionary paths. • The advent of massively parallel supercomputers makes this model computationally tractable and able to offer insight and predictive power

15. 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., 52:317 (1972) Hanahan & Weinberg. “The Hallmarks of Cancer” Cell100: 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)