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A Mathematical Model of Cancer: Tumor Growth & Invasion

Faculty Mentor: Jana Gevertz

Student: Kayla Spector

The focus of this project was cancer cell invasion, the process by which cancerous cells leave the primary tumor site and enter healthy tissue.  Invasion is the first step of metastasis, the formation of secondary cancer colonies.  My goal was to build a mathematical model which accurately describes how the tumor microenvironment impacts cancer cell proliferation and invasion.

Using MATLAB, I developed a computational algorithm based on biological knowledge of how cancer cells grow and invade.  It involves several components: proliferating, invasive, hypoxic (low oxygen), and necrotic (dead) cancer cells as well as the extracellular matrix (ECM) which surrounds them. The algorithm is initialized by introducing a small tumor in a simulated tissue region containing an ECM of uniform density.  A novel feature of our biologically-based algorithm is how cancer cells interact with the microenvironment: the probability of division and invasion depend on pressure imposed by the ECM, and if a cell divides, the location of the daughter cell is determined using physical and geometric constraints of the microenvironment.  Further, physically-motivated functions are introduced for “pushing” the ECM as a tumor grows.

Using these rules we can study the dependence of tumor growth on ECM density. Tumors that develop in higher density environments are characterized by a small necrotic and hypoxic core with large amounts of branching and a higher probability of invasion, while low-density environments foster tumors that are more circular and isotropic with less invasion.  These findings are consistent with biological studies.  While we currently determine the probability that a daughter cell becomes invasive, our simulation of invasion is incomplete.  Once this aspect of the algorithm is developed, we would compare our output with experimental data and possibly calibrate parameters.  Eventually, a more complex microenvironment could be implemented to include blood vessels, non-uniform ECM density, and drug therapy.