Weimin Han’s article “Mathematical Theory and Numerical Analysis of Bioluminescence Tomography” is not bathtub or beach reading. But it reveals details about how he applies mathematical expertise toward solving medical problems and, ultimately, saving lives. This is the sort of work that has garnered Han, the chair of the University of Iowa’s Applied Mathematical and Computational Sciences Program, a prestigious fellowship from the Simons Foundation.

“It’s great to have this Simons Fellowship,” Han says. “With that support, I can take a semester for exclusively doing research.”

An author or co-author of 11 books, Han is one of 77 outstanding mathematicians and theoretical physicists recognized by the Simons Foundation. He conducts his research at the intersection between numerical analysis and biomedical optics.

“For me, I firmly believe that as applied mathematicians, we should find problems not just from reading papers,” says Han, “but from other disciplines, especially those which deal with real world applications.”

When he began to do research in physics and materials science, plasticity and contact mechanics, Han didn’t have much background in the field. He learned as he went, conducting mathematical research as he gained information in a field previously unfamiliar to him.

For Han, learning by doing, making new connections, and launching collaborations all lead to new knowledge. “I like to do interdisciplinary research,” says Han, “because I like to work with researchers and scholars from all kinds of backgrounds.”

Han believes one of the most efficient ways to learn new material is to research it. Han began to apply his mathematical expertise in the field of biomedical imaging about seven years ago, spurred by conversations with a colleague in biomedical engineering, Professor Ge Wang.

Biomedical imaging is used to develop new tools and techniques to help doctors differentiate between normal and abnormal cells and tissues. Doctors use imaging to examine specific physical processes or chemical reactions in the body; normal and abnormal tissues exhibit different characteristics that can be visually observed and quantified.

“Our job is to establish mathematical models,” says Han. “It’s mostly equations, but it could sometimes also be data collection. So we either need to derive new equations or identify existing equations useful for that kind of physical process.”

After gaining an understanding of the properties of the problem and developing corresponding equations, Han says, the task turns to creating “efficient and effective numerical methods to solve the problem.”

“For real world problems, it’s almost impossible to have closed-form solution formulas,” says Han. “Most formulas exist only in textbooks,” useful mainly as teaching tools.

Numerical methods that solve real world problems require mathematicians to develop algorithms. The strength of these algorithms is tested through further analysis and experiments. The properties of the models are analyzed and the models are improved as needed.

“Everything is dynamic,” Han says. “It’s not like we have something fixed and then we will always use it. Whenever possible we try to improve the model and do a better job in analyzing and simulating the model.”

The next step is to develop programming tools that implement the numerical methods. From this process, researchers finally begin to gain the numerical results necessary to quantify, describe, and understand the biophysical phenomenon they are studying

“It’s very important for the numerical results to be practically meaningful,” says Han. “We are not doing academic exercises.”

At this point, numerical simulation results are compared with experimental results of biomedical engineers with whom Han works.

For his research on biomedical optics, which is supported by the Simons Foundation, Han studies the “propagation of light within the biological medium.”

Biomedical optics researchers usually consider two kinds of light effects: 1) absorption, measuring the decrease of the light intensity as light travels in the same direction; and 2) scattering, the degree to which some part of the light moves away from the original direction.

For certain cancers or other diseases, parts of the body will behave differently with respect to the light absorption or scattering properties. Light absorption analysis is the basis for CT (computer tomography) scans.

“With a CT, you can reconstruct the absorption properties to get a picture of that part of the body,” says Han. “But with a CT, it’s about the location and size of something. It’s not about quantitative properties.”

Where numerical analysis meets biomedical optics, the absorption and scattering properties of light are measured in terms of actual numbers, not just location and size.

“It’s more digital,” says Han. “Precision can be greatly enhanced.”

Han says improvements in x-ray mammography could save many lives. While x-rays have been very useful in decreasing the death rate due to breast cancer, it is too often an inaccurate technology, missing a certain percentage of cancer cases and also giving some false positives.

Han says further study of light scattering would be an improvement on the current imaging technique.

Although Han is currently focused on interdisciplinary applied mathematics, his initial training was more theoretical. In China, his numerical analysis major was already determined before his first day of college.

“In the college, at the beginning I just took courses, studied textbooks, but did not know what kind of important uses were there,” says Han. “Then gradually I felt that it was most important to be able to solve real-world problems with good mathematical tools. We look for both aspects: the beauty, the rigorousness of the mathematical results and, also, the usefulness of the results in relation to applications.”

The beauty?

“Mathematics is very beautiful,” says Han. “There can be some examples where you need to use many, many words to describe something. With mathematics it’s just enough to put down an equation or a formula, and it tells you everything.”