Professors Find Mathematics Can Optimize Cancer Treatments

Posted: February 14, 2008 at 1:00 am, Last Updated: November 30, -0001 at 12:00 am

By Robin Herron

Better, faster, cheaper — these are the bywords of modern industry and the goal of almost any enterprise today. How do businesses continually improve their products? The answer is rooted in one of the oldest and most classic of academic disciplines: mathematics.

The recent research efforts of Mason professors Roman Polyak and Ariela Sofer demonstrate that mathematics is essential to today’s most innovative technology.

Both scientists work in the field of mathematical optimization. Optimization, as the name implies, involves making something as effective as possible; mathematical optimization uses math models to achieve this.

As Polyak, who holds joint appointments in the Departments of Systems Engineering and Operations Research (SEOR) and Mathematical Sciences, says, “People have been working on optimization since the ancient Greeks learned that a string encloses the most area when it is formed into the shape of a circle.”

Roman Polyak and Igor Griva
Roman Polyak, left, and Igor Griva, Mason alum and now assistant professor, work together.

Nearly 25 years ago, Polyak developed a theory and methods for solving constrained optimization problems called nonlinear rescaling (NR) that has become essential in solving complicated, real-world technological problems with thousands of variables. NR has been applied to image processing, designing building structures, and finding optimal distribution of electricity across a power grid. It has even been applied to medical diagnostics.

Take a goal such as improving radiation therapy for cancer treatment. Polyak’s NR concept has been adapted and modified by others, notably German researchers Rembert Reemtsen and Markus Alber, to improve the efficiency of a treatment for cancerous tumors known as intensity modulated radiotherapy treatment (IMRT). The linear accelerator deployed in IMRT depends on optimization to determine the angles, intensity, and duration of radiation beams that most effectively destroy a cancerous tumor — but without damaging nearby healthy tissue. Software fundamentally based on NR is now built into the radiotherapy systems used in various hospitals.

“I never dreamed, 25 years ago, that rather abstract mathematics could be used for cancer treatment. Of course, it makes me very happy that my NR methods have been used for such important applications. Mathematics, when it develops, you have a tool, and it can be applied in any field,” says Polyak.

Polyak has continued to refine his theories. In 2006, he and Igor Griva, his former graduate student who is now an assistant professor at Mason, published a paper that outlined their new NR-based method for achieving faster and more accurate solutions to large-scale constrained optimization problems. The pair also earned a patent for Mason in 2007 for their NR optimization tools.

Ariela Sofer
Ariela Sofer
Creative Services photos

Sofer, who chairs SEOR in the Volgenau School of Information Technology and Engineering, was one of the first researchers to focus on the application of operations research to medical treatment and diagnosis, and she has become a leader in the effort to make operations research in medicine into a field.

She became interested in the medical applications of mathematical optimization nearly 10 years ago when she collaborated with Calvin Johnson, a doctoral student who worked at the National Institutes of Health (NIH) on the problem of image reconstruction in positron emission tomography, a medical imaging technique for investigating the level of metabolism and blood flow in an organ.

Sofer and Johnson developed reconstruction methodology and software that outperformed leading methods both in the quality of image and in the solution time. Through this research she says she discovered how rewarding it was to work on applications that could help save or extend human lives.

In her current work, Sofer has been using optimization to help fine-tune the application of a relatively new medical procedure called radiofrequency ablation. The procedure is used to kill liver tumors in patients who are not candidates for surgery. She is collaborating in this research with NIH physician Bradford Wood, an interventional radiologist.

“Ablation kills the tumor by applying heat,” Sofer explains. “The physician inserts a needle, and electrical current in the range of radiofrequency is applied. That cooks the tumor; however, it also kills adjacent healthy tissue. The key is to apply the heat in exactly the right region. You don’t want to damage vital tissue or structures.”

Computed tomography scans and ultrasound imaging are used to determine the tumor’s location and monitor the needle’s path.

Sofer has been working on the problem of determining how to place one or more needles to maximize the effectiveness of the procedure. It’s not as easy as it sounds. As in the radiation example, there are many variables. Not only must she consider how many needles should be used and where the needle or needles should be inserted, but also the angle and depth of insertion. Other questions that come into play are how to minimize the number of insertion points when multiple needles must be used. And in a real-life situation, this optimization must often be determined in a matter of minutes.

Sofer, who spent a sabbatical at Georgetown University Medical Center, also has worked on a project to improve a prostate biopsy procedure.

“These are very large problems,” Sofer says, “but I get a sense of satisfaction knowing that this work may actually do some good for human beings.”

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