# Texas Tech Mathematicians Use Time Scale Approach To Cancer Treatment

By: Karen Michael

How much of a medicine does a person need? How long should a cast be worn? How long should a person wait between vaccine doses? Mathematics is hardly the first concern a patient may have when diagnosed with an illness, but numbers are critical in treatment of many illnesses.

Texas Tech University mathematicians, including Casey Mills, a doctoral candidate, and Raegan Higgins, associate professor of mathematics, are using a mathematical approach to modeling the treatment of prostate cancer, a disease that will likely affect one in six men in the United States.

Patients who have been diagnosed with prostate cancer typically undergo intermittent androgen deprivation therapy prior to starting radiation or when surgery or radiation are not an option. Mills said these patients undergo cancer treatment for a few months, then go without treatment for a few months. Traditionally, continuous ordinary differential equations are used to model treatment.

“They alternate between periods of on-treatment and off-treatment, and those periods last several months,” Mills said. “We've created a new approach using something called time scales, which are able to combine discrete and continuous time very well.”

A discrete variable is countable in a finite amount of time, but a continuous variable will continue on forever.

Most clinical models currently use strictly continuous time frames, Mills said. Because of the change between on- and off-treatment intervals, she and her advisor wanted to see if it was possible to combine the continuous and discrete to create a new model.

“We were able to do that, and we've been able to make some improvements to our novel model,” Mills said. “Currently, we're introducing this very fresh thing called fractional calculus, which allows non-whole-number derivatives to be computed.”

Higgins said previous approaches to the prostate cancer treatment the researchers are aware of were made on continuous time with no breaks.

“What we're doing introduces the fact that you can model both the continuous time and discrete time. We want to more realistically reflect the break of switching from on-treatment to off-treatment to on-treatment to off-treatment, without just assuming that it's always continuous time,” Higgins said. “The benefit of this intermittent therapy is it gives patients a better quality of life. They're not getting so much treatment at once. The doctors can see if the treatment is working and adjust accordingly.”

Previous models consider either discrete time or continuous time exclusively, but do not combine the two. This is where time scales come in, Higgins said.

“You want to allow something to happen (continuous time), pause (discrete time), and then pick it (continuous time) back up,” Higgins said.

In using data from six patients whose treatment intervals best fit their mathematical formulas, the Texas Tech mathematicians said they accomplished their goal of building a solid foundational model.

“Right now, we have a very, very novel model, and it will require more building and collaboration with biologists to increase its chances of being used in a clinical setting,” Mills said.

But the Texas Tech mathematicians are also interested in other uses for the time scales. Mills said a co-advisor on her dissertation thought it could be applied to ecology and algae growth.

“Any intermittent data set could use time scales. There are countless potential applications,” Mills said.