Modeling Human Drug Trials — Without the Human

Hologram body

Dr. David M. Eddy is one of the rare people for whom the “Dr.” refers to both a medical and academic degree. Dr. Eddy received his Ph.D. in applied mathematics in 1978, ten years after becoming a medical doctor, and he’s been a pioneer in applications of mathematics to medicine ever since. His latest project sounds like a massive undertaking: a computer program called Archimedes, which is intended to represent the various aspects of human physiology well enough so that new drugs, tests, or procedures could be tried out on virtual human subjects before any actual humans went under treatment. A description of the project’s genesis and development appears in this story from Wired magazine.

The program was a kind of SimHealth: a vast compendium of medical knowledge drawn from epidemiological data, clinical trials, and physician interviews, which Eddy had laboriously translated into differential equations over the past decade. Those equations, Eddy hoped, would successfully reproduce the complex workings of human biology — down to the individual chambers of a simulated person’s virtual heart…. a soup-to-nuts model that would capture everything known by modern medicine, from the evolution of disease in different people — as shaped by factors like race, genetic risk, and number of hours spent doing yoga — to specific physiological details, such as the amount of heart muscle that dies in the hours after a heart attack and the degree to which medications like aspirin can limit that damage. Tests could be run in hours instead of years, and the model could be constantly updated with the latest research.

According to the article, the model could already do a very good job of predicting the results of previous clinical trials, and the next step—coming soon no doubt—was to have it ‘virtually’ conduct some drug trials, trials that might otherwise be too difficult, costly, or dangerous for drug companies to do themselves.

The 100 Top Science Stories of 2010

Every year Discover magazine lists its 100 Top Science Stories, and a number of these stories, particularly those involving physics and engineering, require a lot of math in their execution. Beyond that, however, four of the stories feature mathematics centrally. In numerical order:

  • In #51 A Computer Rosetta Stone we find a computer program that deciphers ancient heiroglyphics statistically. MIT computer scientist Regina Barzilay has developed the program, which compares unknown letters and words to letters and words of known languages in order to find parallels. When she tested it by seeing how much of ancient Ugaritic the program could decipher using the related language Hebrew as the ‘parallel’, the program correctly matched 29 of the 30 Ugaritic letters to their Hebrew equivalent, and 60% of the Ugaratic words that had Hebrew cognates. More importantly, it did the work in a matter of hours, whereas human translators needed decades (and the chance find of an ancient Ugaritic axe that had the word “axe” carved on it) to accomplish similar feats. While the program certainly cannot replace the intuition and feel for language that human scientists possess, “it is a powerful tool that can aid the human decipherment process,” and could already be of use in expanding the number of languages that machine translators can handle.
  • #60 Fighting Crime with Mathematics details the work of UCLA mathematicians Martin Short and Andrea Bertozzi who, along with UCLA anthropologist Jeff Brantingham, developed a mathematical model of the formation and behavior of crime ‘hotspots.’ After calibrating the model with real-world data, it appears that hotspots come in two varieties: “One type forms when an area experiences a large-scale crime increase, such as when a park is overrun by drug dealers. Another develops when a small number of criminals—say, a pair of burglars—go on a localized crime spree.” According to the work, the typical police reaction of targeting the hotspots appears to work much better on the first type of hotspot, but hotspots of the second type usually just relocate to a less-patrolled area. As the story notes, “By analyzing police reports as they come in, Short hopes to determine which type of hot spot is forming so police can handle it more effectively.”
  • There seems to be a steady stream of stories recently that remark on how some animals instinctively know the best way to do things. One example from this blog is Iain Couzin’s work on animal migration. And here’s another: #92 Sharks Use Math to Hunt. Levy flight is the name given a search pattern which has been long suspected by mathematicians of being one of the most effective hunting strategies when food is scarce. David Sims of the Marine Biological Association of 
the United Kingdom logged the movements of 55 marine animals from 14 different species over 5,700 days, and confirmed that the fish movements closely matched Levy flight. (The marine animals included tuna and marlin, by the way, but sharks always get the headlines.)
  • #95 Rubik’s Cube Decoded covers a story already mentioned on this blog about “God’s Number”, the maximum number of moves that an omniscient being would need in order to solve any starting position of Rubik’s cube. The answer, as you can read in this story or by reading my earlier blog post, is 20.

The whole Top 100 is worth going through as well. It’s remarkable to realize how much and how quickly science is learning in this day and age.

A Physicist Solves the City

Sometime around 2008 or so a tipping point was reached: for the first time, the number of people worldwide living in cities outnumbered the number of people living in rural areas. The ‘urbanification’ of humanity will likely only continue (you can see the United Nations projections here), and so cities—their structure, their qualities, their creation, maintenance, and growth—are becoming increasingly important objects of study.

I’ve already mentioned the NY Times Magazine’s Annual Year in Ideas issue in a previous post. Included in the same issue is a full-fledged article on the work of Geoffrey West, a former physicist at Stanford and the Los Alamos National Laboratory. West has recently turned his attention away from particle physics and toward biological subjects, and done so with effect; as the article notes, one of West’s first forays was “one of the most contentious and influential papers in modern biology” which has garnered over 1500 citations since published.

The mathematical equations that West and his colleagues devised were inspired by the earlier findings of Max Kleiber. In the early 1930s, when Kleiber was a biologist working in the animal-husbandry department at the University of California, Davis, he noticed that the sprawlingly diverse animal kingdom could be characterized by a simple mathematical relationship, in which the metabolic rate of a creature is equal to its mass taken to the three-fourths power. This ubiquitous principle had some significant implications, because it showed that larger species need less energy per pound of flesh than smaller ones. For instance, while an elephant is 10,000 times the size of a guinea pig, it needs only 1,000 times as much energy. Other scientists soon found more than 70 such related laws, defined by what are known as “sublinear” equations. It doesn’t matter what the animal looks like or where it lives or how it evolved — the math almost always works.

West’s next work went along similar lines, but now the biological subject under the microscope was the city. The first and natural quantity to investigate would be something that played the role of ‘energy’ in the city, and West and his collaborator Luis Bettencourt discovered that indeed a whole host of ‘energy’ measures scaled at a sublinear rate.

In city after city, the indicators of urban “metabolism,” like the number of gas stations or the total surface area of roads, showed that when a city doubles in size, it requires an increase in resources of only 85 percent. This straightforward observation has some surprising implications. It suggests, for instance, that modern cities are the real centers of sustainability…. Small communities might look green, but they consume a disproportionate amount of everything.

Still more surprises arrived when West and Bettencourt looked at measuring not ‘energy’ in terms of infrastructure, but ‘energy’ in terms of people. When people decide to move to a city—and as the United Nations data shows, people are doing so in droves—they often do so not to decrease their expenditures, but to increase their social opportunities. Now it is hard to measure social interactions, but there are related interactions that can be measured, and interestingly enough these seem to scale the same way infrastructure does, but in the opposite direction. Social activity seems to scale in a superlinear way. All sorts of economic activities, from city-wide construction spending to individual bank account deposits, increase by 15 percent per capita when a city doubles in size. Or as West puts it, “[y]ou can take the same person, and if you just move them to a city that’s twice as big, then all of a sudden they’ll do 15 percent more of everything that we can measure.” The bad news is that the ‘everything’ is in fact everything: violent crime, traffic, and AIDS cases for example also see the same type of increase.

West and Bettencourt’s current calculations are controversial and not universally believed. (The author, Jonah Lehrer, seems fairly skeptical himself.) Nevertheless, as with the earlier biological findings, the work described here certainly looks like a very good launching point for some very valuable and much needed future analysis.

The 10th Annual Year in Ideas

NY Times 2010 Year In Ideas

Another year has passed, which means it’s time again for the NY Times Magazine’s annual The Year in Ideas issue, “a high-to-low, silly-to-serious selection of ingenuity and innovation from 2010.” As with the 2009 list, a number of these ideas are based around some bit of mathematics and/or statistical analysis. The ones I’ve listed below are the ones that most prominently feature mathematics ideas, or feature mathematics and/or mathematicians centrally.

  • Perfect Parallel Parking by Jascha Hoffman mentions Simon Blackburn’s geometric analysis of parallel parking, which we covered on the blog previously. Updating that earlier story, Hoffman’s entry notes that Jerome White and some fellow teachers at Lusher Charter School in New Orleans subsequently improved the model. (White and company built in allowances for the driver to do a bit more maneuvering.)
  • Aftercrimes visits a topic seen already here in this blog: just as earthquakes typically beget aftershocks, some types of crime beget copycat crimes. Mathematician George Mohler has been able to show that “the timing and location of the crimes can be statistically predicted with a high degree of accuracy.” For more info, check out the entry and the earlier blog post.
  • The entry Social Media as Social Index describes some of the ways that researchers—academic, government, and corporate—are mining social networks like Twitter and Facebook for valuable information. For instance, algorithms analyzing millions of Twitter posts were able to predict how certain movies would perform at the box office and how the Dow Jones Industrial Average would perform in the near future. More social media data mining is undoubtedly in store, as the story ends with one Facebook officer quoted as saying that this is the future of opinion research.
  • Finally, two entries which illustrate the public appetite for data analysis. Do-It-Yourself Macroeconomics describes the growing legion of “ordinary citizens” who are making it their business to “pull apart the [economic] data and come to their own conclusions.” All this is possible, of course, due to the explosion in publicly available economic data, one example of which is described in The Real-Time Inflation Calculator. As the story concludes, thanks to this (freely available) software, “Data on prices, once monopolized by government gatekeepers, are now up for grabs.”

The Aftershocks of Crime

It’s fairly common knowledge that large earthquakes are frequently followed by aftershocks, smaller earthquakes that occur in the same locale or relatively nearby. It is less well-known that some types of crime also have a similar aftershock phenomenon. For instance, burglaries are frequently followed by burglaries in the same neighborhood or a neighborhood nearby. This story in The Economist describes how mathematicians like George Mohler at the University of Santa Clara are using this phenomenon to devise methods of predicting where these “aftercrimes” are most likely to occur. The technique literally adapts the same equations used to describe earthquake aftershocks, and appears to hold some promise.

In one test the program accurately identified a high-risk portion of the city in which, had it been adequately patrolled, police could have prevented a quarter of the burglaries that took place in the whole area that day.

Together with researchers at UCLA, Mohler is extending the work to explore another type of crime in which there are often aftershocks: gang violence. Some of that work, and some additional projects involving ‘predictive policing’, are also detailed in the recent LA Times story “Stopping Crime Before It Starts”, by Joel Rubin.

Popular Science’s Brilliant 10

Each year the magazine Popular Science dubs 10 young scientists their “Brilliant 10”, highlighting the scientists’ work and its implications. In the 2010 edition, more than a few of the profiled rely on mathematics. The work of two, Iain Couzin and Paul Rabadan, are especially mathematical and I’ll mention them here.

Iain Couzin, “the Pattern Maker”, works in ecology and biology, and specializes in identifying the rules that underlie the movements of groups of animals.

The shuffle of life—the wheeling of birds, the silver flash of escaping fish—looks mystically organized. Iain Couzin, who models collective behavior in nature, identifies those patterns mathematically. And he’s finding that certain patterns extend across otherwise unrelated units of life, whether bugs or cancer cells.

This is, of course, one of the great strengths of mathematics: once abstracted, it is easy to recognize a pattern that occurs in different places. Some of Couzin’s earlier work—featured in articles in National Geographic and the NY Times, for instance—involved divining the rules that army ant colonies use to direct their devastating raids. His most recent work, mentioned in Discover, provides an explanation for the large migrations seen in so many animal species. The model, if correct, also provides a warning: tampering with the migrating herds, through hunting or habitat alteration, could devastate the migration instinct itself.

Migration could disappear in a few generations, and take many more to come back, if at all. Indeed, bison in North America no longer seem able to migrate, a fate that may soon be shared by wildebeest in the Serengeti. Migration may vanish at a scale measured in human years, and recover at time scales measured in planetary cycles.

Raul Rabadan, “the Outbreak Sleuth” has a background in string theory, but his numerical experience is serving him well now in his hunt for the agents behind various biological diseases.

Raul Rabadan hunts deadly viruses, but he has no need for biohazard suits. His work does not bring him to far-flung jungles. He’s neither medical doctor nor epidemiologist. He’s a theoretical physicist with expertise in string theory and black holes, and he cracks microbial mysteries in much the same way he once tried to decode the secrets of the universe: He follows the numbers.

Rabadan has been a pioneer of a data analysis technique called Frequency Analysis of Sequence Data that has been able to pinpoint previously unknown viruses as the cause of major disease outbreaks in various animal (and human) populations. Some of his work focused on tracing the origins of the H1N1 swine flu virus, with articles about the work appearing in Wired and online at CNN and USA Today.

Tying Shoes: Math May Make Case for How We Lace

This is an older story but I couldn’t resist mentioning it here once I’d heard about it: Burkard Polster has determined the best ways to lace a shoe. Now if you’re like me you may have gone through much of your life blind to the endless shoelace arrangements out there. The diagram above provides you (and me) some clue about the 43,200 possibilities there are. (The 43,200 number is for six eyelets on each side; as the number of eyelets goes up the possibilities increase exponentially.) Which arrangements are cooler or more attractive than the others is a matter of taste, but in terms of things quantifiable it’s apparent that some lacings require more shoelace than others, and that some lacings are stronger than others. Thinking in those terms, which shoelace arrangements give you the strongest fit for your buck? Dr. Polster, a mathematician from the University of Monash in Australia, has done the work to figure it out. His findings help confirm why I and so many others have been so ignorant all these years.

The shoe-tying world is dominated by two lacings. The most popular is a simple crisscross, each end of the shoelace pulled through the next eyelet on the opposite flap. The second common way is to pass one end of the lace from the last eyelet across to the top eyelet on the opposite flap and then zigzag the other end in an N pattern through the eyelets….When the eyelets are narrowly spaced and the flaps relatively far apart, the crisscross lacing is strongest. If the eyelets are farther apart, then the zigzagging N pattern provides the tightest tie.

In short, Dr. Polster has demonstrated that sometime in the past the human brain had already “evolutionarily” figured out the optimal ways to lace shoes, and we’ve been largely doing so ever since. The original scientific article detailing the findings appeared in Nature, and the story was then picked up by National Geographic and the NY Times (where the quote above is taken). If you Google Polster and shoelaces, you’ll see that his foot work has garnered Polster mentions on literally hundreds of websites devoted to shoes, and even a mention of sorts in the May 2007 Women’s Health magazine….where they unfortunately decided he was “nutso” for bothering with the calculations. On a more positive note, Polster was inspired to write a short book about his explorations entitled The Shoelace Book: A Mathematical Guide to the Best (And Worst) Ways to Lace Your Shoes, which appears from the online reviews to be a little more appreciated.

Sizing Up Consciousness By Its Bits

This NY Times article details the efforts of Dr. Giulio Tononi to develop a means to measure a person’s level of consciousness as easily as a blood pressure sleeve measures a person’s blood pressure. Dr. Tononi is one of the world’s experts on consciousness, especially that peculiar form of half-consciousness known as sleep. While most people, researchers included, have long thought of consciousness as a kind of synchronization of brain waves, Dr. Tononi noticed that in particular kinds of unconsciousness, like during epileptic seizures, brain waves were even more synchronized than during wakeful periods. It seemed a new paradigm for consciousness was required. And for that paradigm, Dr. Tononi turned to information theory.

While in medical school, Dr. Tononi began to think of consciousness in a different way, as a particularly rich form of information. He took his inspiration from the American engineer Claude Shannon, who built a scientific theory of information in the mid-1900s. Mr. Shannon measured information in a signal by how much uncertainty it reduced. There is very little information in a photodiode that switches on when it detects light, because it reduces only a little uncertainty…. Our neurons are basically fancy photodiodes, producing electric bursts in response to incoming signals. But the conscious experiences they produce contain far more information than in a single diode. In other words, they reduce much more uncertainty.

Tononi has developed a measure called phi that seems to track how rich in information a mental state is, and the article mentions some preliminary medical work that is lending support to his model. The research is in its infancy and much more work is needed, but the same could be said for all science-based inquiries into consciousness. The point here is that the Dr. Tononi’s work “translating the poetry of our conscious experiences into the precise language of mathematics” holds promise—at the very least, enough promise to warrant featuring in this article.

Math Model May Decrease Phantom Traffic Jams

More traffic-related stories here. The ‘phantom’ traffic jams in the title are those where there is no apparent cause. These traffic jams typically occur when there are a lot of cars on the road and some small irregularity—one driver hitting the brakes hard, for instance—starts a chain reaction of driver reactions that blows up back through the line of cars and into a full-fledged stoppage. The “chain reaction” and “blows up” language turns out to be appropriate here. Recently a team of MIT researchers looking at the equations that describe these traffic jams—equations that had been unsolved since the 1950s—made a breakthrough when they realized that the equations were very similar to solved equations describing the detonation of explosives! The story’s combination of cars and explosions was catnip to the media, of course, with the story appearing among other places at MSNBC and Wired and overseas at the Telegraph.

And in another traffic story coming down the pike, another researcher, Morris Flynn at the University of Alberta, has proposed that outfitting cars with some kind of ‘interactive GPS’ could help decrease traffic density and thus decrease the occurrence of these phantom jams. Flynn’s idea was number 39 of Discover’s Top 100 Stories of 2009. Minnesota has used special stoplights at entrance ramps to great effect to do the same thing, as described in Numbers Rule Your World: The Hidden Influence of Probabilities and Statistics on Everything You Do, a recent book by Kaiser Fung.

And finally, to complete this math and traffic pile-up, we have another story featuring Morris Flynn which has an interesting psychological angle. Instead of solving the governing traffic equations, Flynn has been using computers to ‘kick the tires’ of various different mathematical models for traffic flow, and one of the models he investigated looked at those who abide by traffic rules assiduously, the rule-abiders, versus drivers who are willing to bend the rules every now and then, the rule-breakers. Not surprisingly, when rule-breakers rule the road traffic typically doesn’t flow well, since it’s impossible to ‘go with the flow’ if you don’t have a good idea of what the other drivers are going to do. Somewhat surprisingly, rule-abiders don’t do so well by themselves either: when a group of them gets into jams they are in a jam for a good long time. Most surprisingly, traffic works best with a mix of both. In fact, according to Flynn’s calculations a 60%/40% mix of rule-abiders vs. rule-breakers is just about ideal. This story appeared in “A Cure for Traffic Jams: Rule-Breakers” at ABC News.

Short-Circuiting Malaria

Malaria affects hundreds of millions of people throughout the world annually, killing over one million of them. Almost all of these victims live in third-world countries, and many of them are children—one estimate is that a child dies of malaria every 30 seconds. This Newsweek article, by Daniel Lyons, describes Intellectual Ventures, an American start-up that is trying to use first-world know-how to combat the disease. The company got its start when its founder, Nathan Myhrvold, was told by Bill Gates, “Come up with some good ideas and I’ll come up with some money to pursue them.” Some of the inventors mentioned are mathematicians, and one of those ideas is a massive mathematical model of the disease that “lets researchers see the effect of potential vaccines that don’t exist, so they can choose which one to develop.” Other ideas are also mentioned and many of them, as Myhrvold admits, sound farfetched. But really, as Lyons concludes, how can you argue against trying?

Addendum: In another development on the malaria front, researchers at Case Western Reserve University recently developed techniques that can quickly identify drug resistance in strains of malaria. The new technique is expected to “enable the medical community to react quickly to inevitable resistance and thereby save lives while increasing the lifespan of drugs used against the disease,” according to the article “New Methods, New Math Speed Detection of Drug-Resistant Malaria” from ScienceDaily. The key—developed by mathematics Prof. Peter Thomas and a student, Drew Kouri—involved “using a nontraditional mathematical analysis that’s proved more accurate than traditional methods.” (That ‘nontraditional analysis’ mentioned, by the way, was simply switching to polar coordinates.)