Forecast uncertain: Chaos theory, weather prediction, and brain cancer
A diagnosis of brain cancer is basically a death sentence. It's a terrible thing for anyone to deal with, and it's only made worse by all the uncertainty. Doctors don't really understand how brain cancer works very well. Beyond death, there's often not a lot that they can tell patients about what to expect—how the cancer will affect the brain, how fast it will spread, where it will spread to.
Eric Kostelich is one of the researchers who is trying to change that, by approaching the problem of brain cancer from a new angle. Kostelich is a mathematician. In particular, he's interested in how we can use math to better predict the behavior of complex and chaotic systems. Right now, this mostly means that he studies the weather. In fact, he's part of a team that developed a new algorithm for weather prediction, called the Local Ensemble Transform Kalman Filter. But Kostelich thinks that the LETKF could have applications outside the nightly news.
In a recent study, published December 21 in Biology Direct, he joined forces with cancer researchers, to see whether the statistical methods that make chaotic weather patterns more predictable could do the same thing for chaotic behavior in cancer cells. The results are promising. A couple of weeks ago, I spoke to Kostelich to find out more about the history of forecasting uncertainty, how algorithms like LETKF work, and what we might learn if we apply these systems to cancer.
Maggie Koerth-Baker: When you set out to apply the methods used to forecast the weather to cancer, why did you choose brain cancer?
Eric Kostelich: Partly it's because I had a family member with a brain tumor. The scientist in me got interested because these really are terrible tumors. Getting to know a number of clinicians, it seemed to me that new ideas, anything that could help people live a little better would be welcome.
From a math standpoint these tumors are interesting in that they don't metastisize on the central nervous system. For instance, melanoma is a process that starts in a mole on surface of skin but spreads to the liver and other vital organs. You have to look at it throughout the entire body, and that's a daunting task. But a brain tumor tends to stay in the brain. That keeps it more mathematically attractive for our initial studies. And the brain also has some interesting geometry, with folds and so on, and with different functions in different places. The liver and kidneys are relatively heterogenous, but where a tumor is in the brain really affects what symptoms you see. My partner in this thought it would be really useful to be able to say, for an individual patient, based on studies of patients with similar tumors, there's a 60% chance that the tumor is more likely to grow in one direction, rather than another. They might be able to give a dose of radiation therapy in that one direction.
If you could help someone live a couple extra months, that would be a big advance for this type of cancer. Patients usually only live 12-14 months. Ted Kennedy suffered from this and his experience was typical. He died about 14 months from diagnosis. We've made a lot of strides in treating other kinds of cancer, but our approach to brain cancer hasn't changed much.
MKB: Why are brain cancers so difficult to treat?
EK: The brain isn't easily accessible. Between the brain and the outside, you've got the skull and accessing it involves drilling a hole. Plus, it's your brain. With a breast cancer you can remove a breast and a margin of tissue and still live. You can live without an arm or a breast. But you can't just remove a big chunk of the brain without crippling the patient. One of the objectives of treatment is to not make the patient's neurological condition any worse than tumor already has made it.
MKB: In this study, you used an algorithm called the Local Ensemble Transform Kalman Filter. What does all that mean? What does this algorithm do when applied to weather?
EK: Meteorologists never make just one prediction. They make many. In any of these models, you always have a grid of points on which you're trying to approximate behavior. To run that model on the computer you put in models for all the grid points, but you can't actually measure them all. What I mean is, you can measure the temperature and barimetric pressure and so forth at one point on the ground and plug it in. But the model also has another grid point above that one, high up in the air, and more points on up into the atmosphere. You can't measure the data at all of those. Another example, if you look at high impact weather like a major hurricane, they get started over remote tropical ocean where it's hard to get any measurements at all. Satellites are a big help, but there's always uncertainty. And because there's uncertainty, you get the idea of an ensemble forecast.
Ensemble forecasts go back to Edward Lorentz. [He's a pioneer in chaos theory—MKB] You run a number of forecasts with slightly different realizations of the numbers at all the different grid points. Forecast uncertainties depend not only on time, but also on space. You might have quite a bit of uncertainty about where a hurricane will end up in five days. But quite a bit of certainty about Phoenix being sunny for five days in June. If you look at many forecasts, you can get a handle of both where your uncertainties are and what magnitude they are.
MKB: What makes it different from other algorithms used for making predictions about uncertain systems?
EK: You're trying to get mathematical combinations of forecasts that best fit observed conditions. Weather models are updated every six hours. Your uncertainties tend to grow with time. You can forecast tomorrow, but a week from now is more iffy. So the models have to be updated on a regular basis. By doing things locally you get better computational efficiency. We thought, "Well, if we can do this for the weather ..."
Our system outperforms the Weather Service's system by quite a bit. The Brazilian government is actually using our approach for their next gen weather prediction systems. What we try to exploit in our approach is, in a mathematical sense, the uncertainties in a forecast. Uncertainties tend to lie in certain directions. All these mathematical models are partial differential equations in dynamical systems. In math, we have a concept we call a phase space. That's "phase" as in phases of the moon. It's a math abstraction and it's where we think of all the math action taking place. In the case of the weather, it appears that the uncertainties in weather models lie primarily in certain directions of the abstract phase space. Our approach takes advantage of that in a clever way. Because we know where the uncertainties are more likely to be, we can pay more attention to those places.
MKB: The ideas here—combining new observations with prior forecasts, and paying the most attention to where you know the most uncertainty is likely to be—these are things that can seem like a bit of an obvious thing to laypeople. How new are these ideas really? Is there something that makes this more special and surprising than it seems on the surface?
EK: The notion of data assimilation, combining observations and prior forecasts, has been an integral part of weather forecasting for several decades now. Computers were used to do this on a regular basis, starting around the mid 1960s. By that point they were powerful enough that you could build a realistic enough grid to say something about the weather. You and I take weather forecasting for granted. But the reality is that this is one of the great triumphs of modern science.
Think about Hurricane Katrina. It was known three days in advance that this hurricane would come close to New Orleans. Thirty years ago, we couldn't have been able to tell you that. Today we can evacuate a few hundred miles of coastline instead of telling the entire Gulf region, "This is coming and we don't know where." That's a huge advance.
There's some really interesting history on this. The term forecast goes back to Robert Fitzroy, he was the captain of the Beagle, the ship that Darwin traveled on to the Americas in the 1830s and 40s. He was very intersted in questions of weather and storms because he'd lost several crewmen in a gale. The ship basically tipped over. He saved the ship, but lost several lives. When he was back in England he was appointed meterological statist. He was appointed to keep all the records for the crown. So, by the late 1850s, the telegraph had come in and Fitzroy had people at all the ports telegraph in the weather information to him. He looked at patterns of temperature and pressure rising and falling and was able to combine those patterns with new observations and say, "There's a storm coming in. Stay at port." This was the beginning of forecasting. Shipwrecks fell by half in a few years.
MKB: So where do you make the connection between all of this, and something like brain cancer?
EK: No forecast is perfect. In Fitzroy's day, if the storm didn't materialize, then there were complaints. People lost money by staying in port and Fitzroy actually got into political trouble. They stopped doing forecasting for a while until a fisherman's lobby brought him and his methods back.
Now fast forward 150 years. Say you have a brain tumor. What you'd like to know is, "What is going to happen to me?" But doctors are really very much where Fitzroy was 150 years ago before he invented forecasting. It's like, stick your finger up in the air and make a guess. Nowadays, for weather, we have mathematical models that, combined with satellites, can predict hurricanes before they materialize. We can tell several days in advance where the storm will go. Doctors can't do anything near that with cancer. All they can do is look at a scan and some bloodwork and say we'll see you in a couple months. But we're on the verge of being able to gather lots more information about what's going on in the human body.
What we'd like to do with this oncoming data deluge, what we're trying to do, is devise math tools that will help clinicians make sense of all the new data and associate probabilties with that data. Then they can tell people something more useful. Not perfect. But useful.
MKB: I think most lay people operate under the assumption that weather prediction like this isn't very accurate, beyond a day or so ahead of time. You're wanting to do cancer prediction over 60-day cycles. Why would that kind of time frame be reliably accurate enough to matter?
EK: Weather service looks hard and long at that question. One way in which you can assess the goodness or lack thereof of the forecast is to say, "I'm going to predict the weather two, three, four days out. Then you go out and measure on those days and compare the reality to the forecast. The bigger the difference, the worse the forecast. By that measure, forecasts today made 3-4 days out are as accurate as a 36-hour forecast was 30 years ago. So the weather forecasts are more accurate than people give them credit for. It's just that when you blow it that's what people remember.
A famous case was Veterans' Day a few years ago in Washington DC. The Weather Service forecasted a dusting of snow, but there ended up being something like 14 inches. Basically, the snow was 100 miles off from where they thought it would be. A 100-mile error, 24 hours out, that isn't so bad really in a global perspective, but the local impact is very great. People remember that. On the flip side, though, in 1900 a hurricane hit Galveston, Texas. The Cubans had telegraphed DC to tell us that there was a storm heading West. But we blew the Cubans off and there were no warnings for Galveston. 8000 people drowned. We still have bad hurricanes today, but we don't have 8000 drowning because they don't know it's coming. In that respect, we're doing pretty good. But it's still not perfect.
Our basic approach here, in thinking about cancer in general and brain cancer specifically, is could we adapt the accuracy of a 3-4 day weather forecast for a month or two or three. So that the models show what is likely to happen to an individual patient's tumor. It's a fair question about whether we can move that timeline up. But it works because of differences in what you're forecasting. For weather, the uncertainties tend to double every couple of days. As far as we understand cancer, the uncertainties you have in the state of a tumor, they don't double every two days. They double over a month or two. It's a different mathematical beast than the weather. A couple of months for cancer is like a couple of days for the weather. Now, they can vary quite a bit in how aggressive they are. There are cases in the literature where in some patients the tumors double in size in a couple weeks. But more typically the doubling times are on the order of a month or two.
On the other hand, forecasting cancer can be harder than forecasting the weather. In the case of the weather, air is a fluid, and there's a couple hundred years of physics that go into understanding how fluids move in laboratory conditions. If you're going to write a weather model there's no doubt what equations you start with. In the case of cancer, we don't know how glioblastoma cells really work very well. It's a much greater challenge to write that mathematical model because our understanding is much less complete. We're trying to take into account that whatever model we write down is likely to be off. Possibly by quite a bit. But the question is, "Can data assimiliation system make a clinically useful forecast?" It doesn't have to be perfect to be useful.
MKB: The brain cancers you're looking aren't really treatable. Like you say, most people die from them within 14 months. What's the benefit, then, of having a more accurate prediction of how they will spread? If you still can't treat the cancer, what does it help to know how it will behave?
EK: My understanding, and from personal experience with a family member, is that you're right, this isn't going to cure cancer in general or glioblastoma specifically. But one of the real goals of treatment is to help patients live as well as possible for as long as possible. The age of highest incident for the type of brain cancer we studied is between 40 and 65. If this result allows you to live two months longer than you otherwise would maybe that makes the difference between seeing your daughter get married or not. We can't prevent the inevitable, but we might help them live better or longer. If we can develop good enough mathematical models and be able to tell patients that going through another round of chemo isn't likely to help, then they can decide to spend that time with family instead of in the hospital. That's beneficial in it's own way.
Eric Kostelich's research is available to read, for free, online.
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