Not every undergraduate has their independent research funded by the National Science Foundation (NSF), but Caleb Mayer ’18 is one of the select few. The mathematics major and psychology minor spent a summer studying mathematical models for how different types of cancer cells respond to various treatments, thanks to an NSF grant as part of their Research Experiences for Undergraduates (REU) program. So when it came time to design a senior thesis project, he knew he wanted to do something similarly interdisciplinary, building on his cancer cell models.

In choosing his thesis topic, Mayer was inspired by his own previous research on mathematical modeling how different types of cancer cells respond to various treatments. His thesis research was unique from his REU study because his thesis modeling focused on how the two categories of cancer cells (stem cancer cells and non-stem cancer cells) move throughout the body and invade the extracellular matrix, rather than on treatment responses.

Mayer’s thesis, “Traveling Wave Solutions for a Cancer Stem Cell Invasion Model,” was advised by Visiting Assistant Professor of Mathematics Eric Stachura. Stachura’s focus is on mathematical physics, but, Mayer says, “he was kind enough to help me explore this mathematical biology topic for my thesis.”

“We discussed many of the main sources in order to refine the topic and develop a relevant model, and he helped greatly throughout the process,” said Mayer. “Much of the mathematics involved was very difficult for me to understand on my own, so he was very instrumental in helping with this and working through the math itself.”

This fall, Mayer is off to the Applied and Interdisciplinary Mathematics Ph.D. program at the University of Michigan, where he hopes to continue his research in mathematical biology and cancer modeling.

**What are the implications for your thesis research?
**We synthesized a variety of mathematical and biological sources in order to developed a model for how cancer stem cells and non-stem cancer cells invade the extracellular matrix. We then performed lots of mathematical analyses of this model— basically, we found a certain possible way in which these different cell populations can invade the extracellular matrix and, hence, lead to cancer spreading. So I’d say the this project could be important in that it provides a model for this cancer invasion, and could help people understand generally how this invasion process occurs. It isn’t super biologically relevant right now, but there hasn’t been much research that combines the cancer stem cell hypothesis and cancer invasion, so I think the thesis has the potential to be valuable.

**What did you learn working on your thesis?
**I obviously learned a lot about both mathematics and biology while working on this project, but I think the biggest takeaway is about the mathematical research process itself. It’s obviously very different than the usual coursework people do in math courses, where professors assign doable problems and then the students work to the required answer. I found it very novel and exciting to be working on problems that don’t necessarily have nice solutions, and having more control over the direction of the research itself. I think I also learned about how to write about math in a more comprehensible way, since making the thesis understandable to a math major is a goal of the project.

*“What They Learned”* *is a blog series exploring the thesis work of recent graduates.*