Mathematical Model Suggests Link Between Chaotic Oscillations in Kidneys and Hypertension

BUFFALO, N.Y. -- Practitioners of applied mathematics use a lexicon of numerical equations, instead of letters and sentences, to illuminate the secrets of the physical world. They use this "foreign language," which seems so abstract to the rest of the world, to help solve gritty, real-world problems by developing computational models of everything ranging from large industrial systems to tiny biological ones.

Over the past decade, UB mathematics professor Bruce Pitman has done both, applying applied techniques he learned while modeling large industrial systems to the study of the primary functional unit in the kidneys -- the nephron, which measures a mere 20 microns across.

The aim of the National Institutes of Health-funded work, a collaboration with researchers at two other institutions, is to enhance the understanding of how kidneys work.

In the healthy human body, Pitman explained, there are about 1 million nephrons in each kidney, performing the critical job of filtering from blood the water, salts and potassium that the body needs and leaving waste products and urea to be excreted.

"If we understand how the nephron works in healthy animals, it might give us an indication as to the causes of renal disease," he said.

Pitman, also UB vice provost for educational technology, got involved with modeling this piece of the human anatomy quite by accident. He found himself sharing an office at the Courant Institute at New York University with Harold Layton, now a professor of mathematics at Duke University, who had done his graduate work on modeling kidneys.

"Layton wanted to extend an earlier model of blood flow in the nephron, but was unsure how to do some of the computations involved," remembered Pitman.

One afternoon in 1987 over tea, they started talking.

Pitman had done his graduate work on modeling granular behavior, such as how and why large particles such as corn flakes fall out of industrial hoppers in precisely the way that they do -- a far cry from the anatomical obsessions of his officemate.

"But by using the language of mathematics, we were able to communicate," said Pitman. Layton had immersed himself thoroughly in the language of physiology, Pitman recalled, and so could introduce Pitman skillfully to the field.

"What he didn't have was the background to view his model and its potential in the larger context of applied-mathematics work, such as how you use math and computational models to describe fluid flows," he said.

"We spent our first week translating," Pitman recalled.

"He would tell me an idea about the kidneys' function and I would translate that into the language of fluid dynamics. We would work out each idea so that we both understood each other's approach.

"By the end of that week," he added, "we had two formulations side by side, which didn't look anything alike, but were actually the same: one written in the language of physiology and one written in the language of fluid dynamics."

Since then, the research team, which now also includes physiologist Leon Moore from the State University at New York at Stony Brook, has published about a dozen papers developing computational models of the nephron in greater and greater detail.

Experiments have shown that in animals with normal blood pressure, the fluid pressure in the nephron remains rather constant, and oscillations, if they occur, are regular, explained Pitman.

"But in animals with high blood pressure, you see complicated oscillations and sometimes they are even chaotic," he said. "The question is, 'What causes these chaotic behaviors?'"

According to Pitman, experiments by physiologists have shown that as many as half of the nephrons near the surface of the kidneys appear in pairs, triplets or in fours, all of which share a common origin on an artery.

"This coupling is very prevalent," he said, "and in animals with high blood pressure, the coupling of nephrons may contribute to the very complicated behavior in the way fluid flows through the nephron."

Recent work by Pitman and his colleagues suggests that oscillations in nephron flow may coincide with a higher output of salt from the body, a possible link among high blood pressure, salt intake and oscillations in renal blood flow.

"It makes us wonder whether or not these complicated oscillations may occur in order to get rid of excess salt," said Pitman.

Questions like that send the researchers back to their equations, devising formulas that may be able to account for the new behavior. They then adapt the formulas into computer codes that ultimately will be used to develop a model that captures the new information.

"We push the model until it breaks," said Pitman. "In other words, until it doesn't explain something that has been seen experimentally. Then we go back and try to find ways to incorporate the new behavior.

"It's a continual cycle of experiments, analysis of models and computing -- each component relying on the others," he said.