Rick Durrett, James B. Duke professor of mathematics, has developed mathematical models to describe the growth of cancer cells. The Chronicle spoke with Durrett about his findings and potential applications to medical treatment.
The Chronicle: How do you model cancer’s behavior using mathematical equations?
Rick Durrrett: Well, there are a number of approaches to modeling biological systems mathematically. People use partial differential equations to model blood flow or clotting, and in the same way you can model how cancer cells and normal cells compete for oxygen.
When a tumor grows and it doesn’t have blood vessels on the inside, the cells undergo hypoxia—they don’t have enough oxygen—so they switch over to glycolysis, which is an anaerobic method of breaking down glucose that creates acid. Using partial differential equations to model oxygen and blood flow, one can model what cancer cells look like in the bloodstream as well as the interactions between normal cells and a tumor, then make observations from that.
Cancer is the end result of the accumulation of mutations. Some, of course, are simpler than others. Retinoblastoma, for example, is a cancer of the retina that occurs in children, and there’s a gene called RB1 where if both of them get knocked out by a mutation, then tumors develop. In other situations though, it’s a more prolonged sort of cascade. In colon cancer, for instance, some genes get knocked out that cause polyps to grow in the colon, and then other mutations come along and follow the road to cancer.
We also use stochastic models [which incorporate random variables] to examine biology. Differential equations are good at measuring predictable stuff, like fluid flow. But there’s a lot more randomness in some other systems.
In cancer, you might have a kid whose eye is growing, and some cell is going to develop a mutation that causes a retinoblastoma, but a differential equation can’t predict which cell of the many that are “playing the lottery” for cancer will become cancerous. A stochastic model will predict roughly how long it takes for the probabilities of some cells developing mutations at the right times to all combine and cause a cancer to develop, so we can get some data from that to tell us where and when to look as well as what to look for.
TC: When you develop such a mathematical model, does that inform you about how cancer spreads throughout the body as well?
RD: Something like that—I did a collaboration with an undergraduate and two people in obstetrics and gynecology at Duke. They were modeling ovarian cancer—this is a cancer that’s hard to treat because the first time it’s detected, it’s metastasized. If you can detect it before it’s left the female reproductive system, it has a cure rate of about 90 percent, but if you let it go somewhere else, the cure rate, which is really a five-year survival rate, drops to 30 percent.
What happens is that the tumor grows, then when it gets large enough, some cells break off from its surface. Some of these cells then attach within the peritoneal cavity, and new tumors start to grow, starting the metastasis process. It’s sort of this two-stage thing which we use a branching process model to figure out. A branching model predicts the likelihood of one generation of individuals, in this case cells, producing descendants of differing kinds, such as cancer cells. The tumor grows sort of exponentially in time, and cells leave at a certain rate.
There are three types of cells in the model—the primary tumor cells, the cells floating free in the abdomen and the ones that set down and start metastasizing. A lot of these ovarian tumors are too small to detect with ultrasound until they’re a certain size, so there’s a sort of window of opportunity in which to detect these tumors through screening, between the time that it’s half a centimeter in diameter to the time when we have more than a gram of metastasizing cells.
We used our models to work out how long a tumor stays in that window, and it turns out to be around 2.6 years. In order to sort of “catch” this window, then, you’d need to screen women every two years or so. With these studies, we can work out how best to apply treatments, the costs of optimal screening procedures and things like that.
TC: What have you found especially interesting about your work?
RD: People are just learning new things about cancer all the time. Just a few months ago, there was this new discovery that there might be such a thing as cancer stem cells, which had been debated for years, and now there’s some experimental evidence. The mathematics doesn’t change all that quickly, but it’s interesting to see what’s being figured out about cancer.
TC: You are teaching a seminar this semester on mathematical cancer modeling—could you describe that briefly?
RD: There’s a research training grant in mathematical biology, and each semester the seminar has a certain theme. In order to encourage collaboration, they have half of the seminars in the physics building, where the math department is, and the other half in the medical center. Last year it was neurobiology, and now this year the topic is cancer modeling.
We’re trying to make contact with some people over in the Cancer Center, so we thought one way of doing this would be to publicize the seminar series. I’ve been doing cancer modeling now for three or four years, but we’re going to have some people coming in who are really experienced to sort of catalyze interaction between people from the math department and people from the medical school.
TC: How would you say your mathematical models apply to cancer research and treatment?
RD: I don’t work all that closely with doctors or medical researchers, but I’m looking forward to collaborating with people who have more of that medical connection through the upcoming seminar.
One of the most exciting applications of this stuff is how to use mathematics to improve treatment—what is the most effective way to apply radiation therapy or chemotherapy, or how often should surgeries be done?
You can also use these models to look at breast cancer and the kinds of breast cancer that people have, then optimize a treatment based on the behavior of the cancer system that you’re dealing with.
TC: What are you working on right now that ties into all this?
RD: Well, right now we’ve been looking at some spatial models. Again, as in the work on ovarian cancer, you generally assume that there’s this growing population of cancer cells. But a question we were asked was, what does the spatial variability of the mutations look like, and how are these different kinds of cells distributed?
For example, if you have a tumor growing within a breast, you might have different kinds of cancer cells present but only get some of them in a biopsy, and if the others aren’t responsive to the treatment you devise, you won’t fix the problem.
To deal with that, we’re trying to understand the spatial variability of heterogeneity. I work on this from a very mathematical end of things, and it’s probably going to be a while before this stuff really translates into something in the clinic, but I find it interesting to work on things that at least have something to do with the real world.
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