How Mathematical Modeling Can Advance Understanding of Diseases

Can math help map out what's next?

1, 1, 2, 3, 5, 8. 

At first glance, it’s a sequence of numbers. If you look closer, you’ll see each number is the sum of the previous two. For centuries, mathematics has been used to uncover hidden patterns and underlying order in the world. Leonardo of Pisa, better known as Fibonacci, used this pattern in the early 13th century to describe a hypothetical population of rabbits, translating biological growth into numbers.

Today, mathematical modeling is still helping us understand the human body better. Physicians are increasingly partnering with mathematicians to analyze complex biological systems, model disease progression and even develop personalized treatment.

Melanie Cree, MD, PhD, a pediatric endocrinologist at Children’s Hospital Colorado, studies how insulin works in children using approaches closely integrated with mathematics. More than a decade ago, Dr. Cree began working on something that had never been done before. She aimed to transform a method used to understand how adult livers handle sugar after a sugary drink into one that could do the same for teenagers.

But the solution wasn’t as simple as scaling down the numbers. Teenagers aren’t smaller adults. Their hormones, metabolism and insulin sensitivity are constantly changing.

“Because we’re in pediatrics, we can’t just take published adult methods and apply them to kids,” Dr. Cree says. “Puberty affects a lot of what I’m looking at.” 

To solve this problem, she needed help from outside the medical field. She needed a biological mathematical modeler who could translate complex physiology into equations that evolve over time. Dr. Cree spent six months searching across the country. And then, she reconnected with former undergraduate classmate and friend Cecilia Diniz Behn, PhD, who had recently moved to Colorado to work in the Department of Applied Mathematics and Statistics at the Colorado School of Mines. Dr. Diniz Behn, now a professor at the Colorado School of Mines and an adjoint professor at the University of Colorado School of Medicine, is a leader in the field of mathematical biology and holds the Joe and Jane Gray Distinguished University Chair. 

“I saw a lot of potential in Dr. Cree’s work and a lot of potential for math contribution,” Dr. Diniz Behn says. “My work was primarily focused on mathematical neuroscience, so it wasn’t what I was doing at all. Jumping into a new field is a big step, especially for assistant professors. It was essential that we already had such trust in each other, and it allowed us to take that risk.” 

That trust allowed them to step outside traditional disciplinary boundaries, translating lab measurements and clinical observations into mathematical language and back into real insights for physicians.

Improving a glucose test through mathematical modeling 

At its core, mathematical modeling uses equations to describe how a system behaves and how it changes. For models to be meaningful, they must be grounded in high-quality biological data.

Take a glucose tolerance test as an example. To get an accurate test, each person must fast for eight to 12 hours beforehand to identify a person’s baseline glucose level. Then, they drink a glucose solution loaded with 75 grams of sugar. With frequent blood sampling, clinicians can see the patient’s metabolic system respond and return to baseline in real time. Oftentimes, diagnoses are based on a snapshot of metabolic health: fasting glucose and two-hour glucose. While this information is clinically useful, it doesn’t capture much about how the glucose and insulin are changing and how the metabolic system is actually functioning.

Mathematical modelers like Dr. Diniz Behn offer tools to analyze the interactions between glucose and insulin across the entire glucose test. Dr. Diniz Behn translates these dynamics to insights about insulin sensitivity — the basis of their original study.

“I use mathematical equations to describe how glucose is changing in time and how it’s being influenced by insulin. By describing that interaction in equations, you can get a sense of that whole trajectory. That’s what modeling allows us to do,” Dr. Diniz Behn says. 

Developing a deeper understanding of the relationship of glucose and insulin 

Dr. Cree and Dr. Diniz Behn have teamed up on several projects to learn more about the relationship between glucose and insulin. One of their recent projects explores the impacts of simplifying the research method for measuring insulin sensitivity. Right now, the insulin clamp procedure is the gold standard. This traditional test takes up to 10 hours. It’s a tedious process, requiring glucose and insulin injections and blood draws every five minutes.

“Our current project is working out the math to get the same insights using a sugary drink,” Dr. Cree says.

By developing a protocol that uses the glucose drink, the team hopes to replace the insulin clamp – shrinking the amount of time spent of the procedure, decreasing blood draws for the participant and decreasing nursing and medical provider time. This approach could allow insulin research to move to the outpatient facility, making studies faster, less expensive and more comfortable for patients. 

“If we can get away from expensive insulin clamp testing, our hope is that we’ll be able to enroll more people in studies,” Dr. Cree says. “It’s going to change how we’re going to do research with Type 1 diabetes.”

Dr. Cree is also applying mathematical models to her current clinical trials. She says one day those models will help identify which medicine will work best in which patients.

In addition to working with Dr. Cree, Dr. Diniz Behn has collaborated with several other clinicians at Children’s Colorado, including insulin sensitivity studies with Kristen Nadeau, MD, circadian rhythm studies with Stacey Simon, PhD, and insulin secretion studies in patients with cystic fibrosis with Christine Chan, MD. 

By bringing mathematics and medicine together, researchers are showing how collaboration across disciplines can reveal insights neither field could uncover alone. The original project that started it all is nearing completion, with one final push to the finish.

“We’re trying to wrap up this question. We’ve answered a lot of questions. We’ve published multiple papers. And we’re still at about 13,000 feet of a fourteener,” Dr. Cree says.