Incorporating Adhesion in a Cellular Automata Model of Melanoma Growth Chris DuBois (’06) , Ami Radunskaya*

1. Sample Diffusion Simulation Incorporating Adhesion in a Cellular Automata Model of Melanoma GrowthChris DuBois (’06), Ami Radunskaya* Dept. of Mathematics, Pomona College, Claremont, CA Research Supported by the HHMI Grant 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 Cij(t). Vijrepresents 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 growth2, as seen in Figure * below. 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(ni)/dn is least; if this amount exceeds neq, then the remaining cells move into the neighbor with the next smallest de(n)/dn value. 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 treatment6. 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 treatment6. 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 glycolytic7. 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. Cellular Automata Cellular automata (CA) are common discrete implementations because of their ability to replicate the complexity of biological systems3. 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. 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 therapies1. 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. • 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 Physiology151:386-394. • 2. Gatenby, R.A. and Gawlinski, E.T. (2003). The Glycolytic Phenotype in Carcinogenesis and Tumor Invasion: Insights through Mathematical Models. Cancer Research63: 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. 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 patients3. 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 tissue3. 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 matrix3. 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. 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).