# Mathematical Modeling of Cancer Progression: the Impact of Invasion and the Immune System

Faculty Mentor: Jana Gevertz

Students: Warren Jagger

Cancer is a complex disease characterized by several major “hallmarks” that disrupt the normal state of a cell.  My project focused on developing a mathematical model of tumor progression while concentrating on one cancer hallmark: the interaction between a tumor and the immune system.

I developed a 2D system of differential equations that models the rate of change in the number of cancer and immune cells (combination of innate and active). I analyzed the steady state values of the equations, and performed a linear stability analysis to identify how the immune system affects the size of a tumor: does the tumor die out, remain at a benign size, or become malignant?  Since there were a large number of unknown parameters in the model, I used MATLAB (a mathematics computer software package) to help explore these and other features of the system of differential equations.

In MATLAB, I wrote scripts to automate the processes of exploring many parameter scenarios that correspond to different characteristics of a patient’s tumor and immune system.  Data retrieved from the scripts allowed me to categorize the number of equilibria and their stability. Next, graphs were made using pplane8, a MATLAB function that makes direction fields, to visualize how the number of tumor and immune cells change as a function of one another.  Additionally, I used ode45, a numerical ordinary differential equations solver, to study how tumor and immune cells grew over time in order to determine how the behavior of a tumor is dependent on the parameter values of an individual “patient”.

Expanding our research, we have developed a 4D system of differential equations that incorporates more biological details than our original system.  In the future, we hope to analyze this system and implement an immunotherapy term to further our knowledge of tumor-immune interactions.

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