According to ICES Director J. Tinsley Oden, mathematical models of the invasion and growth of tumors in living tissue have been "smoldering in the literature for a decade," and in the last few years, significant advances have been made.

"We're making genuine progress to predict the growth and decline of cancer and reactions to various therapies," said Oden, a member of the National Academy of Engineering.


MODEL SELECTION AND TESTING

Over the years, many different mathematical models of tumor growth have been proposed, but determining which is most accurate at predicting cancer progression is a challenge.

In October 2016, writing in Mathematical Models and Methods in Applied Sciences, the team used a study of cancer in rats to test 13 leading tumor growth models to determine which could predict key quantities of interest relevant to survival, and the effects of various therapies.

They applied the principle of Occam's razor, which says that where two explanations for an occurrence exist, the simpler one is usually better. They implemented this principle through the development and application of something they call the "Occam Plausibility Algorithm," which selects the most plausible model for a given dataset and determines if the model is a valid tool for predicting tumor growth and morphology.

The method was able to predict how large the rat tumors would grow within 5 to 10 percent of their final mass.

"We have examples where we can gather data from lab animals or human subjects and make startlingly accurate depictions about the growth of cancer and the reaction to various therapies, like radiation and chemotherapy," Oden said.

The team analyzes patient-specific data from magnetic resonance imaging (MRI), positron emission tomography (PET), x-ray computed tomography (CT), biopsies and other factors, in order to develop their computational model.

Each factor involved in the tumor response -- whether it is the speed with which chemotherapeutic drugs reach the tissue or the degree to which cells signal each other to grow -- is characterized by a mathematical equation that captures its essence.

"You put mathematical models on a computer and tune them and adapt them and learn more," Oden said. "It is, in a way, an approach that goes back to Aristotle, but it accesses the most modern levels of computing and computational science."

The group tries to model biological behavior at the tissue, cellular and cell signaling levels. Some of their models involve 10 species of tumor cells and include elements like cell connective tissue, nutrients and factors related to the development of new blood vessels. They have to solve partial differential equations for each of these elements and then intelligently couple them to all the other equations.

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