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The Right Equation: Mathematicians Work To Predict Tumor Growth

Michael Smith

Statistics is well accepted as a tool in designing clinical trials and in analyzing clinical data. Similarly, mathematical theory has found applications in oncology; the current use of dose-dense chemotherapy, for example, arose from mathematical considerations. (See News, Vol. 95, No. 4, p. 254.)

But the goal of the new mathematical oncology is to be able to model tumors so well that researchers can begin to use computers to guide treatment on an individual basis. Those working in the field say that recent successes are beginning to excite a new interest among physicians in applying math to cancer.

"For the longest time, you had mathematicians sitting around proving nice theorems and physicians who didn't want to talk to the mathematicians because they didn't see anything useful coming out," said Siv Sivaloganathan, Ph.D., a mathematician at Canada's University of Waterloo, in Ontario.



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Siv Sivaloganathan

 
Sivaloganathan co-chaired a workshop on mathematical oncology last year at Toronto's Fields Institute for Research in Mathematical Sciences, which among its other activities administers the prestigious Fields Medal, regarded as the Nobel prize for mathematics.

That meeting brought together physicians, radiological physicists, mathematicians, and even one researcher trained as an astrophysicist. "There's now a definite bridge between the disciplines," Sivaloganathan said. "And it isn't just mathematical theory ... people actually have experimental data."

One researcher with data in hand is Kristin Swanson, Ph.D., of the University of Washington in Seattle. She and her colleagues have shown that the growth of gliomas can be accurately modeled using only two parameters—net proliferation and tumor cell motility.

Gliomas are "clearly much more complicated than those two parameters," Swanson admits, but surprisingly, her model—derived from experimental data from nearly 400 patients—is sufficiently robust to determine "the mechanistic behavior of the tumor."

One implication is that treatment could in theory be individualized; given the net rate of proliferation and the motility of a given tumor, oncologists can use the model to predict how fast it will grow and in what directions, when it is likely to recur, and where the recurrence is likely to happen.

So, for instance, radiation fields could be directed to areas that appear to be cancer-free, but actually harbor the seeds of recurrence, Swanson explained.

Whether such models will eventually be applied in the clinic remains to be seen, and Swanson's glioma model is a first step in that direction. But, in general, mathematical oncologists are far from being able to match her success, says bioengineer Lance Munn, Ph.D., of the E.L. Steele Laboratory for Tumor Biology in the Department of Radiation Oncology at Harvard Medical School.

"That's the goal—to know enough about cancer to be able to put the rules into a computer, push the button, and let it go," Munn said. "Unfortunately, from the experimental side, we don't know all the rules yet."

Munn and colleagues are attempting to model the tangled web of tumor vasculature using what is called a lattice-Boltzmann approach. As is well known, tumors recruit new blood vessels as they grow, but the resulting vasculature is a mess of loops, leaks, sudden changes in vessel diameter, and stagnant areas.

"It is a very inefficient system for perfusing the tissue," Munn said. The downside of this is that anticancer drugs—reliant on this messy system for delivery—often don't get where they are needed.

The goal of the modeling, he says, is to see if there are ways to alter the tumor vasculature so that it becomes more efficient and—paradoxically for the tumor—more susceptible to chemotherapy.

(One effect of antiangiogenesis drugs, it is now thought, may be to prune the redundant vasculature, making the system more efficient. This is being tested in clinical trials at Harvard and, in parallel, in Munn's computer models.)

Although Munn says the computer models of cancer are not yet ready for prime time, he argues that advances in both computer power and the biological understanding of cancer mean that day is not far off.

Indeed, the advent of fast personal computers is already making a difference in the treatment of prostate cancer, says Marco Zaider, Ph.D., professor of physics in radiology at the Cornell University Weill Medical College, and an attending physicist at Memorial Sloan-Kettering Cancer Center in New York.

"There are two different aspects in which mathematics can and does guide radiation oncology," Zaider said. At Memorial Sloan-Kettering, oncologists are already using a sophisticated software package—based on what's called a "genetic algorithm"—to design placement plans for the radioactive seeds used in brachytherapy.

The design takes place in the operating room, using real-time data from ultrasound images of the prostate, so that "we know exactly the geometry at the time of implantation," Zaider said.

But researchers are also developing a method—using the mathematics of stochastic (or random) processes—to predict what Zaider calls the "tumor control probability (TCP) ... essentially the probability to destroy all the cells in the tumor." At the same time, the model predicts the impact on normal tissue—the normal tissue complication probability (NTCP).

The model allows physicians to guide treatment based on "biological optimization," Zaider says, rather than optimizing by physical dose.

The next step, he says, is to add the mathematics of decision theory to account for the patient's wishes. For example, one patient may want few complications and a lower probability of tumor control, while another may want the tumor destroyed as much as possible—which comes with a higher risk of morbidity.

This will increasingly be the norm in all forms of cancer, Zaider says. "The future will go in the direction of optimization driven by the patient himself or herself," he said.

There's no fundamentally new mathematics involved in oncology today, Waterloo's Sivaloganathan says; what's interesting is the challenge of translating math into new and important areas and gaining acceptance for a new tool. Adds Zaider, the math is "well understood but physicians need to be educated."

Success in guiding and improving treatment may be the most important step in that direction, says Washington's Swanson. "In my institution, they've become very receptive," she said. "People are pretty excited about it here."



             
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