*Feb, 2004*

*Moving beyond the qualitative conclusions of
earlier models in biology,mathematical models are beginning to make
quantitative, testable predictions about real patients.*

When Renee Fister was three years old, she lost her younger brother, then 18 months old, to cancer. Growing up, she hoped to go to medical school so that she could fight the disease that claimed her brother. But along the way, she discovered that she didn’t have the stomach for medicine—and that she liked mathematics a lot. She changed her plans, but reluctantly. “I thought I wouldn’t be able to work on cancer any more,” she says. A generation ago, that might have been true. To mathematicians, biology seemed too nebulous, too hard to pin down with the precise laws and calculations that work so well in physics and engineering. Though mathematicians did have some success in genetics and population biology, a serious mathematical theory of cancer seemed like a figment of the imagination. Now, though, mathematical models are popping up everywhere in cancer research. The approaches are as diverse as the disease they are grappling with. Fister, a mathematician at Murray State University in Kentucky, studies optimal-control models that promise to provide physicians the best timetables for drug treatments. Her colleague Carl Panetta, of the St. Jude Children’s Research Hospital in Tennessee, uses systems of elementary differential equations—typically 30 or 40 at a time—to predict a patient’s response to a given drug regimen. At the University of Washington, applied mathematicians James Murray and Kristin Swanson have developed a brain tumor model that is simple by comparison but uses complex three-dimensional brain anatomy to improve on the predictions of physicians about the spread of the tumors. Meanwhile, in Israel, Zvia Agur of the Institute for Medical Biomathematics is working on the ultimate biomathematical model: a “virtual cancer patient” for non-Hodgkin’s lymphoma, which will take into account all the stages of a cancer cell’s life cycle. If these mathematical descriptions of cancer have any common denominator, it is a scrupulous attention to biological correctness. They are moving past the vague qualitative conclusions of older mathematical models in biology and making quantitative, testable predictions about real patients (or at least laboratory animals). “I have worked in applications of mathematics to biology for nearly 30 years,” says Murray, who was the founding director (in 1983) of the Centre for Mathematical Biology at Oxford University. “The whole tenor has changed in the last 10 years. At the Dundee meeting [of the Society for Mathematical Biology, held last summer], I was delighted to see that almost all of the talks were biologically oriented. The speakers were solving their equations quantitatively and saying what the answers predicted biologically. Having been brought up in Scotland, under the gloom of Calvinism, one shouldn’t be optimistic about anything, but I am particularly optimistic about the future of mathematical biology.”

**Controlling the Uncontrollable **

Fister and Panetta are both
members of the younger generation of mathematical biologists, having
received their doctorates in the 1990s. Panetta, like Fister, has been
intrigued by cancer for as long as he has been a mathematician, and four
years ago he jumped from an academic position at Pennsylvania State
University in Erie to his present job at St. Jude. “It gives me a lot more
chances to work with direct applications of math, plus I have access to
all the data I need,” he says. Panetta’s day-to-day work involves
pharmacodynamics—a mouthful that he defines as the study of “how much of
the drug is getting into the patient’s cells and how fast it is getting
cleared.” To answer these questions, he uses a giant system of linear
differential equations that model cell-to-cell and drug-to-cell
interactions. Because they are linear, the systems can be solved exactly.
But solving them is only half the job. The other half is
parameter-fitting. Every patient reacts a little bit differently to a drug
regimen, and the differences show up in the constant coefficients—the
parameters—in the differential equations. The right parameters for each
patient have to be teased out statistically from the data. Panetta’s
differential equations model very well-established drugs like
metha-trexate, which has been used to treat leukemia for 30 years. But he
has also done more speculative work, such as an optimal-control approach
to chemotherapy that he and Fister described in a 2003 paper in SIAM
Journal on Applied Mathematics (Vol. 63, No. 6). Fister and Panetta
started with one simple differential equation:

**Simplicity Is a Virtue**

“Glioblastoma” is a word that you do
not want to hear your doctor pronounce. One of the most aggressive brain
tumors, it has a puzzling tendency to come back even after surgeons have
removed the entire visible tumor. According to Kristin Swanson, “the
mortality rate is essentially 100 percent. These tumors are considered
uniformly fatal,” with death usually coming within a year of diagnosis.
Given that neither surgery nor chemotherapy nor radiation can stop this
disease, it would be foolish to expect mathematics to change the outlook.
But what math can do—and what Murray’s model, refined by several of his
students and postdocs, has done— is give oncologists a much better
understanding of the foe they are up against. The model is dynamically
quite simple, with only one equation, which describes the diffusion of
cancer cells through brain tissue. “We’d make it more complicated, but so
far we haven’t had to,” says Swanson. “We’re still waiting for the simple
model to be disproven.” The model is plenty complex, though, when it comes
to simulating the brain’s anatomy. Swanson and Murray represent the brain
as a 181 ª 217 ª 181 grid, with each cube in the grid representing either
gray or white matter; gliomas progress at different rates through the two
types of tissue. The results, when presented as a computer animation, are
sobering in the extreme. See illustration on facing page; readers can
visit http://www.amath.washington.edu/ ~swanson/research.html for other
examples of simulations. The typical glioma grows unsuspected for nearly
500 days before it becomes detectable by magnetic resonance imaging.
(Until that time the cancer cells are not dense enough.) At 500 days, when
a surgeon would typically find and remove a small tumor, the cancer cells
have already spread through one whole temporal lobe of the patient’s
brain. No wonder surgery has little effect. “I like to use the iceberg
analogy,” Swanson says. “You see 10 percent above the surface, and there’s
90 percent below the surface that you don’t see.” Swanson and Murray can
plug a patient’s MRI scans into the model and predict where the cancer
will invade next. This allows the doctors to plan their radiation therapy
accordingly. The model also provides them with temporal information— how
long the patient has to live. “What amazed me was how accurate the
predictions of survival time were,” Murray says. In Murray’s opinion, the
model could benefit patients by convincing doctors not to attempt risky
and hopeless surgery. “If we do surgery, the patient might lose mobility,
or it might affect their sight or speech.” says Murray. “The surgeons
already know that, but most were not aware how far the infiltration [of a
patient’s brain by the tumor] has gone.” Sometimes, he says, the best
treatment may be no treatment at all. He cites the example of a patient
who chose not to undergo surgery, and enjoyed nine more months of normal
life. “I completely sympathize with her,” Murray says.

**. . . But So Is Complexity**

At the other end of the spectrum,
mathematically, lie the simulations developed by Zvia Agur. A biologist by
training, she extols the value of simple models for providing insight, but
argues that simplicity is not the way to go when you want clinically
useful information. “Physicians are very busy people,” she says. “They pay
attention to models that give them something to use, not intellectual
understanding.” One of Agur’s models could help mitigate a side effect of
chemotherapy, the destruction of blood platelets. Some doctors have had
the idea of stimulating the body to produce its own platelets by injecting
a naturally occurring protein called thrombopoietin (TPO). However,
recombinant or synthetic TPO can provoke an immune reaction, because the
body recognizes it as a foreign substance. Administering TPO is thus a
delicate balancing act—injecting just enough to stimulate platelet
production, without overdoing it and causing an immune reaction. Unlike
most cells in the body, the ones that manufacture platelets, called
megakaryocytes, do not divide after reproducing their DNA. Some
megakaryocytes, in effect, are more “mega” than others. Some are only
little workshops, with twice the normal amount of DNA, while others are
veritable factories, with 32 or 64 sets of DNA or even more. Not
surprisingly, the workshops and the factories differ in their response to
TPO. Accordingly, Agur’s model includes ten variables representing the
number of cells of each size, plus an eleventh variable representing the
amount of TPO. She had to estimate more than 100 parameters, some of which
had never been estimated before. She did this in two steps—first finding
averages that work for the entire species, and then fine-tuning them for
individual mice or monkeys. Her results, published in the November issue
of the British Journal of Haematology, were for the most part spectacular.
In the mice, the predicted platelet levels almost perfectly matched the
observed values for 11 days after injection. In four rhesus monkeys, the
models gave good predictions for more than 30 days. But in the fifth
monkey, who had an immune reaction against the TPO, the computer’s output
bore no resemblance to reality. “There are no miracles,” Agur says.
“Sometimes your model doesn’t fit. Then you have to do good scientific
thinking to find out why not.” Still, the results were promising enough
that Agur is planning to start human trials with 10 to 100 patients. Best
of all, she believes that she can improve on the currently recommended
dosages of TPO. Her model predicted that she could get an equally strong
platelet-building effect by giving one-tenth of the normal dose, but
giving it four times as often. This would lessen the chance of an immune
reaction. Experiments with mice and monkeys bore out the prediction—in her
words, an “unprecedented” case of a new protocol suggested by theory being
validated in the lab. Although the medical community as a whole may still
be suspicious of the value of mathematics, most of the cancer modelers
have nothing but positive things to say about the support they have
received from individual biologists. Murray says that the glioma model
would never have gotten started without the instigation of Ellsworth
Alvord, a pathologist at the University of Washington. Swanson, who has a
joint appointment in pathology, adds, “When I started here as a postdoc,
there was definitely some skepticism at first. But the transition to, ‘Why
don’t you help me next?’ was very exciting. The fact is that for gliomas,
doctors are at wit’s end when it comes to treatment options. They’re
willing to try anything that helps.”

- Dana Mackenzie writes from Santa Cruz, California.

SIAM News, Volume 37, Number 1, February 2004.