What They Learned: Leela Krishnan ’21

The mathematics major examined models of lung cell repair after patients are put on a ventilator.

When patients in the hospital get put on a ventilator, it stretches the lung tissue. This lung damage can cause breathing and fluid drainage problems, but our lung cells and immune systems can repair that damage over time. Leela Krishnan’s senior thesis studied how existing research models this repair process.

The mathematics major and chemistry minor analyzed the models in a paper by a team of researchers at Virginia Commonwealth University. First, she examined three equations which looked at the rate of change of health, damaged, and dead cells in the lungs over time. 

“With several methods of mathematical analysis, I learned the different equilibrium points for the healthy cell space, damaged cell space, and dead cell space that our lungs approach over time depending on how much damage is present initially and the values of other relevant parameters,” said Krishnan. “For example, a patient with a lower repair rate of lung cells approached an equilibrium with a significantly smaller amount of healthy cell space compared to a patient with a higher repair rate of lung cells.”

She also simplified a system of 18 equations that model these rates of change and the role of the immune system in repair. With the help of her advisor, Assistant Professor Rebecca Everett, Krishnan developed a model which used only nine equations. 

“I found that this extensive model felt overly complicated for its goal,” she said. “My advisor and I developed an immunity model of nine differential equations with one lung compartment that accounted for the immune system’s involvement in lung reparations … We found that on one parameter set we could achieve results of similar magnitude and trend for our state variables.” 

Krishnan is starting work as an actuarial trainee this summer as she works to pass her actuarial exams. As an actuary, she will work with insurance company’s pricing models. 

“This thesis was also a useful exercise in explaining some complicated concepts in simpler terms,” she said. “Communicating complicated ideas to an audience who may not be familiar with the math and technical terms will play a key role in my job as an actuary.”

What inspired your thesis work?
Initially, I wanted to base my thesis on a published paper that modeled the lung dynamics of a patient with Cystic Fibrosis (CF) because my brother has CF. I wanted to learn a little bit more about the inner workings of the disease and to explore ways to treat it. Unfortunately, the papers that my advisor and I found on CF were too complicated to take on for a senior thesis. Instead, we found a paper that explored the dynamics of the lungs and immune system in repairing lungs that were damaged by mechanical ventilation. This topic felt like a good fit because it involved the analysis of lung dynamics in a patient with a respiratory illness and it was relevant because COVID-19 is a respiratory illness that can sometimes lead to the use of a mechanical ventilator. I was excited to learn more about an issue that was prevalent during my senior year. 

What are the implications for your thesis research? 
My thesis research is helpful for other researchers in the field of healthcare because it encourages the revision of complicated models down to smaller, more efficient models. Fewer parameters can reduce error in a healthcare model’s results since these parameters are often difficult to determine biologically. A smaller parameter set can lead to more certainty during computational analyses. There is also the opportunity to go more in depth with analyses and obtain stronger results. This is especially important in healthcare because models with thorough analyses and certainty in their results aid in the development of treatments for diseases and illness.  

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