Category: differential equations

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.

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.

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.

Finally, 140-year-old Boltzmann Equation solved

During the late 1800’s, James Clerk Maxwell and Ludwig Boltzmann developed a differential equation that predicted how gaseous material distributes itself in space and how it reacts to changes in things like temperature, pressure or velocity. The only problem: for 140 years, solutions to the equation could only be found for gases that were perfect equilibrium. Enter Philip Gressman and Robert Strain of the University of Pennsylvania. Using recently developed mathematical techniques, they were able to describe solutions to the equation under any conditions. As this article in the Time of India notes, now it’s possible to “describe the location of gas molecules probabilistically and predict the likelihood that a molecule will reside at any particular location and have a particular momentum at any given time in the future.”

Math Mimics Hard-to-Heal Wounds

This brief article from US News & World Report describes a new mathematical model of ischemic wounds. Ischemic wounds are wounds that do not get as much blood flow as normal wounds, and they affect six and a half million Americans each year. The model, developed by Avner Friedman of Ohio State, includes factors that mimic the actions of healing agents like white blood cells, capillary sprouts, blood-vessel-forming proteins and oxygen concentrations, and is the first to accurately predict healing times for these kind of wounds. The hope is that models like this are “the start of something that could give valuable insight to the wound healing problem in the future.”

Best Science Visualization Videos of 2009

Wired magazine presents the best science visualization videos of 2009:

The Department of Energy honored 10 of this year’s best scientific visualizations with its annual SciDAC Vis Night awards, at the Scientific Discovery through Advanced Computing conference (SciDAC) in June. Researchers submitted visualizations to the contest, and program participants voted on the best of the best. From earthquakes to jet flames, this gallery of videos and images show how beautiful (and descriptive) visual data can be.

All of these videos are of course essentially illustrations of mathematical models, models that are so complex that just making the individual frames of the videos requires heavy-duty mathematics and heavy-duty computational power.

When Zombies Attack! Mathematical Modelling of an Outbreak of Zombie Infection

This mathematical research paper by Philip Munz, Ioan Hudea, Joe Imad, and Robert J. Smith? (yes, Robert spells his last name with a “?”), applies mathematical models of epidemic spread to a hypothetical zombie “epidemic”. Their conclusion:

[A] zombie outbreak is likely to lead to the collapse of civilization, unless it is dealt with quickly. While aggressive quarantine may contain the epidemic, or a cure may lead to coexistence of humans and zombies, the most effective way to contain the rise of the undead is to hit hard and hit often. As seen in the movies, it is imperative that zombies are dealt with quickly, or else we are all in a great deal of trouble.

In other words, those movies where nearly everybody is turned into a zombie except for a small band of survivors, but then the small band of survivors figures out a way to turn the tables and vanquish the zombie horde? In actuality, that could never happen.

The paper was featured in the Wall Street Journal online here and Wired magazine here, as well as many other online sites. Addendum: the paper also appeared in the NY Times “The Year in Ideas” issue, 2009 edition.

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