Imagine being able to predict the growth of the tumor. How about the vein growth of premature babies’ eyes? All of these applications and more can be determined with the use of a certain mathematical model.

The model for angiogenesis is a stochastic model describing how the veins grow in the body. Angiogenesis is the process by which new blood vessels are created. Veins will grow out from another vein, and from there, the vein can fork or split into different ways. Determining the growth of the tip of the vein can determine the direction and the way it looks.

In order for a tumor to grow, it needs oxygen, and it obtains oxygen through the veins. Without the proper veins and oxygen, tumors would not grow beyond a certain size and not be able to spread through the body. / courtesy of

“If you understand angiogenesis, it has a lot of applications. With babies, the veins in the eyes need to grow adequately or they can become blind,” Björn Birnir, professor of mathematics and director of the Center for Complex and Nonlinear Science, said.

Research initially began with a model biologists had suggested and then developed into a more complicated mathematical model. The model is complex because it shows how tiny veins would grow and what happens over time as they unite into a larger vein. Using a mathematical computer simulation allowed researchers to test out the model.

“We can do it on the tumor growth and give different kinds of medication to prevent vein growth towards the tumor. We test that also on the computer simulations,” Birnir said.

When studying the vein’s tips, the researchers saw a moving pulse formed by the lumps of vessel tips. Birnir describes this pulse as the “most famous soliton of all,” the Korteweg-de Vries (KdV) equation.

A soliton is a wave that maintains its shape while continuing at a constant velocity. In 1834, a soliton was discovered by John Scott Russell, a Scottish engineer and naval architect. He discovered this by witnessing a wave in shallow water maintain its shape when travelling through the Union Canal.

“I spotted the soliton, the most famous soliton, KdV, when doing the mathematical model; a wave that describes a small wave in shallow water,” Birnir said. “It didn’t change its form and continued to stay the same — no shape change.”

Birnir worked with colleagues at the Universidad Carlos III in Madrid, where he held the Chair of Excellence. After about six months of work, the paper “Soliton driven angiogenesis,” was published in Nature Scientific Reports.

On Oct. 19, Birnir will speak about his research at the seminar “The KdV Soliton in Angiogenesis,” hosted by the Center for Interdisciplinary Research in Fluids. This seminar will include a follow-up to the model for angiogenesis and some results from a second research paper.

“If you understand that it’s a soliton, you have a way of controlling it. If you can control it in premature babies, in their eyes, you can give them medication and test it on the mathematical model and see if it works,” Birnir said.