On my first day of graduate school at Princeton University in 1979, an older student warned that I would encounter a lot of strange characters in the mathematics department. The strangest of all, he said, was a phantomlike figure known as Nash. In the weeks that followed, I occasionally caught sight of Nash shuffling down the hall in a shabby coat and bright red sneakers or sitting in the cafeteria by himself, staring off into space. But I felt his presence most from the nonsensical messages he wrote on blackboards at night. These often took the form of mathematical equations, but they had as much to do with mathematics as a cat walking across the piano has to do with music.
One day, as a group of students gathered to chat with a professor after a class, someone asked about this mystery man. Lowering his voice, the professor told us that Nash had once been John Nash, the brightest light in the greatest group of students that ever studied math at Princeton. In the late 1940s and 1950s, John Nash had made discoveries that his peers still use every day—the Nash equilibrium, the Nash embedding theorem—even as they averted their eyes from the man himself. But then, somewhere along the line, he lost his grip on reality. He began to believe he was receiving messages from outer space and that there were great and hidden conspiracies against him.
Now, four decades after John Nash was lost to mathematics, mathematics itself may hold the key to treating schizophrenia, the mental illness that held his mind hostage. A new way of analyzing shapes, called morphometrics, may allow doctors to tell what changes occur in the brains of schizophrenics before they lose contact with reality. Morphometrics is also providing clues to the development of fetal alcohol syndrome and Alzheimer’s disease and is improving the ability of brain surgeons to map out the routes they will take to perform delicate operations. In the study of the brain, the shape of things to come is, quite literally, shape.
Fred Bookstein, a statistician at the University of Michigan, has spent more than two decades turning morphometrics into a quantitative science. The basic idea, he says, dates back to the sixteenth century and the work of the German artist Albrecht Dürer. Inspired, perhaps, by the recent discovery of perspective geometry, Dürer tried laying grid lines over the faces in his portraits. By moving the lines, while keeping the features of the face in the same position relative to the grid, he could transform the face any way he wanted, turning a bluff forehead into a sloping one, a weak chin into a lantern jaw.
Bookstein’s modern variations on Dürer’s theme—four little fun house faces—peer down from a bulletin board outside his office. One is a photograph of Bookstein looking like a dour version of Billy Crystal; the other three are “not-Freds”—computer-generated caricatures of the first photograph. To make the caricatures, Bookstein first scanned his photograph into his computer. Then he attached a grid to 13 “landmarks” on the face, such as the top of the forehead and the tips of the ears. When he was done, he simply moved a few of the landmark points around, thus forcing the grid to warp and bend as if a thin metal plate were attached to it. Engineers, it turns out, have used such “thin-plate splines” for years. But it was Bookstein who realized that these composite images are a perfect way to represent changes in what morphometricians call “shape space”—and to detect shape differences both large and small.
Shape is a surprisingly elusive concept. True, humans have an extraordinary capacity for picking out subtle differences in shape. Otherwise it would be impossible to identify a friend in a crowded subway station or to quickly distinguish between a fashionable skirt and one that is hopelessly passé. But describing differences in shape invariably turns out to be problematic. Consider the hippocampus, a region of the brain involved in long-term memory formation. To Bookstein, it resembles a “scroll, a partially unwrapped Torah.” To another neuroscientist I spoke with, it “basically looks like a teaspoon.” And to the anatomists who named it, it resembled a seahorse (hippocampus means “sea horse” in Latin).
Modern science needs to describe shapes more precisely than this. “Morphometrics gives you a language for talking about shapes,” says Jim Rohlf of the State University of New York at Stony Brook. Rohlf is the author of the most widely used collection of computer programs for morphometric analysis. “You can say this shape is like that shape, except that it’s expanded here and compressed there.” Even so, Rohlf says, morphometrics is about more than pretty pictures. It also has a core of statistical theory, which explains why thin-plate splines are the best way to represent differences in shape space.
Hidden in Bookstein’s desk is a not-Fred that demonstrates, better than the others, what a difference a subtle change in shape can make. While the not-Freds on the bulletin board outside his office look like big lovable galoots, this one is downright spooky. Instead of a silly grin, he has flat, faintly sneering lips; instead of a bulbous, pear-shaped face, he has a broad nose and disconcertingly wide-set eyes. Bookstein keeps the picture around, perhaps, as a reminder of the slim distance separating the normal from the abnormal. To an outsider, it’s also an eerie echo of the event that has fueled Bookstein’s personal drive to apply morphometrics to schizophrenia.
In 1980, driven by “voices in his head,” one of Bookstein’s nephews came home from college and tried to kill his own parents. Fortunately, he was diagnosed as a schizophrenic and medicated in time to avert a tragedy. But his treatment was long and slow, and even today he has no memory of what happened in certain years of his life.
His nephew’s story, Bookstein would later learn, was surprisingly typical. Though schizophrenics are not especially prone to violence, many of them hear voices, hallucinate, or suffer from delusions of grandeur or persecution or both. One recovered schizophrenic calls her experience with mental hospitals a “revolving door,” and many others can’t socialize with other people, even between psychotic episodes. It’s estimated that about 10 percent of schizophrenics commit suicide. And yet, for all its dramatic symptoms, schizophrenia usually sneaks up on its victims. No single cause of schizophrenia has been identified, and most schizophrenics don’t have their first “psychotic break” until their late teens or early twenties. Moreover, the shape of schizophrenic brains doesn’t vary from normal in any consistent way—at least to the naked eye.
Enter morphometrics. About five years ago, prodded by the memory of his nephew’s misfortune, Bookstein studied the brain scans of 14 schizophrenics and 14 non-schizophrenic volunteers, or “controls,” that had been gathered together by his colleague John DeQuardo. He labeled 13 landmark points each of the brains, converted the landmark points into shapes, and averaged them.
At first glance, it’s tough to distinguish between an image of an average brain of normal shape and one of an average brain of schizophrenic shape. But when Bookstein compares the normals with the schizophrenics using a thin-plate spline, the difference is obvious. In the schizophrenics, a little triangular area in the back of the corpus callosum, the central conduit for all communications between the two sides of the brain’s cortex, seems swollen—as if nature had grabbed some landmark points and pushed them apart to create a caricature. Because the shape of the corpus callosum barely varies in healthy people, even this modest swelling turns out to be statistically significant.
“I really did want to see if I could somehow offer help to people who come down with schizophrenia,” Bookstein says, pointing to the swollen corpus callosum. “To the extent that this pattern is correct, it would permit me to figure out who’s going to get it before [they have their first psychotic breaks].” If doctors knew which patients showed signs of developing schizophrenia, they might try prescribing medications in advance. At the very least, the patients could be counseled to avoid alcohol and addictive drugs, which can complicate the disease.
Bookstein’s work on schizophrenia is still a step ahead of mainstream thinking. Then again, his mind has always moved a little bit faster than others. “I was a bit of a prodigy,” he notes matter-of-factly. At the age of 11, he taught himself algebra from library books. At 14, he won a statewide mathematics competition and, at 15, he entered the University of Michigan. He sailed through college in three years and went to graduate school for mathematics at Harvard.
It seemed as if he were prepared to take off into the mathematical stratosphere. But at Harvard, the self-taught math whiz suddenly found he couldn’t just improvise his own solutions anymore. “I lasted about four weeks and realized that this just wasn’t going to work,” Bookstein says. “I was going to be a lousy mathematician.” He switched to sociology, but things didn’t go much better there. All of his ideas for research were either too ambitious or too off-the-wall. With a laugh, he recalls what his dissertation committee thought when he submitted a proposal to use the mathematics of general relativity to measure social change: “We don’t know what this is. We know it’s not sociology. Please find something else to do with your life.”
After a couple years of working various jobs, Bookstein heard about a program at Michigan for oddball scholars with bright ideas. Out of 200 applicants, he was one of seven chosen for the program. Back in Ann Arbor, they still remembered the wunderkind from eight years before. When Bookstein returned there in 1974, he came with a characteristically grand scheme in mind—to work out a mathematically correct theory of shape.
These days, talking with Bookstein can still be an intense experience, though he says he has lightened up since he began running a bed-and-breakfast with his wife, Edith. “When he gives a lecture, the rate of information transmittal is very high,” says Leslie Marcus, a paleontologist at the American Museum of Natural History and a self-described “facilitator” for the new morphometrics. “It’s like having a fire hose put in your mouth.” Indeed, Bookstein talks quickly—in perfectly composed paragraphs, as if quoting from a book—and types even faster. Watching him navigate around a three-dimensional brain image on his computer workstation is enough to induce vertigo.
That intimidating style, and the devilish math behind morphometrics, may account for why physicians have been slow to adopt his techniques. “What’s the holdup of using shape measurement? It’s a harder concept,” says David Kennedy, a neuroscientist at the Harvard Medical School Center for Morphometric Analysis. “If I say that the volume of the hippocampus is 13 cubic centimeters, we all know what I mean. If I talk about spherical harmonics or a thin-plate spline, clinicians don’t have a grasp of what is biologically meaningful about that.”
The field of schizophrenia research is booming these days, thanks to the window on the brain provided by magnetic resonance imaging (MRI). But it doesn’t occur to many researchers that they need a whole new theory of shape to interpret what those MRI scans are telling them. Most researchers, like Kennedy, still prefer studying volumes. “To be honest, volumetric measures have done pretty well,” says Paul Thompson, a neuroscientist at the University of California at Los Angeles. Among other things, researchers have found that the hippocampus is usually smaller in schizophrenics, whereas some of the ventricles (four cavities at the center of the brain that are filled with cerebrospinal fluid) are larger.
MEASURE FOR MEASURE
Some of the uses of shape analysis may sound uncomfortably familiar. After all, in the nineteenth century phrenologists published scores of articles and books claiming to show that the brains of “inferior” races were smaller or differently shaped from the “ideal.” (That ideal, conveniently enough, tended to be of the same race and nationality as the researcher.) Have times changed? Or is morphometrics really nothing more than craniometry with an MRI scanner?
According to Bookstein, the true guarantee of the integrity of modern morphometrics—and of all statistics, for that matter—lies in the strict observance of certain safeguards against bias. That is the fine print that can be found in almost any scientific paper, but seldom gets reported in the mass media.
One routine modern precaution, which was never practiced a century ago, is called “blinding.” For example, in his study comparing the brains of people with fetal alcohol syndrome to normal brains, the images were coded so that Bookstein did not know which patients had the syndrome when he labeled the landmark points. If Bookstein had known, his biases might have led him to the wrong conclusion. “I really wanted the corpus callosum to be narrower in the patients with the syndrome,” he says, because this could have led to a new diagnostic test for the disease. Instead, he found that the brains of patients with the syndrome varied from the norm in different ways. Some had narrower callosa, some had wider, but very few had the normal width. Blinding, in this case, opened Bookstein’s eyes to the truth.
But these findings are also maddeningly inconsistent. Three quarters of the published studies have found enlarged ventricles in schizophrenics; one quarter have not. Three quarters find smaller hippocampi; one quarter do not. “There is a significant overlap between subjects with schizophrenia and controls for every imaging (and neuropathological) parameter,” Oxford psychiatrist Paul Harrison wrote in a review article last year. “For this reason. . . schizophrenia cannot be diagnosed using either a brain scan or a microscope.”
Morphometrics may provide the answer, and it has already won some converts. At Case Western Reserve in Cleveland, Ohio, for instance, psychiatrist Peter Buckley has used Bookstein’s methods to show that the ventricles in male schizophrenic brains differ in shape as well as size from those in brains. But Bookstein still worries that his pilot studies have attracted too little attention. “I find myself, quite to my surprise, as radical at age 51 as I was as a graduate student,” he sighs.
In the meantime, more and more scientists have been finding applications for his techniques in other areas. Marcus runs a discussion list that now has more than 400 subscribers, and Rohlf and Bookstein have given seminars in Vienna, Paris, Tuscany, Taiwan, and elsewhere. Because of its sensitivity to small differences, morphometrics is especially useful in the classification of species. Biologists have used Bookstein’s methods to study a whole bestiary of animals: bats, fishes, midges, mice, coral, shrews, and even pinworms.
More important, perhaps, is that brain surgeons now use the science of shape in the operating room, where they have long fretted over just where to do their cutting. The brain is an exceedingly mysterious, delicate, and malleable organ. Slice the wrong part of it, and your patient might lose her peripheral vision or her ability to do needlepoint or understand English. More and more often, therefore, brain surgeons depend on three-dimensional computer images produced by ct and mri scanners to plan operations and even to see what they are doing during the operation. The volume scans allow them to see the inside structure of the brain with millimeter accuracy and to work through small incisions instead of opening a large piece of the skull.
Still, the new technology poses problems that only sophisticated shape analysis can solve. A part of the brain may appear in stunning detail, but what is its function? That question can be answered by a technique called “brain warping,” in which landmarks in the brain are mapped (by a thin-plate spline or a similar transformation) to corresponding points on a “brain atlas.” This tells the computer exactly how the patient’s brain geometry differs from that of a generic brain and allows the boundaries of functional regions, like the visual cortex, to be identified. When the surgeon enters the operating room, he sees a giant color display on the computer monitor, like a Rand McNally guide to that patient’s brain.
The discovery that has Bookstein most excited nowadays is a possible test for fetal alcohol syndrome, which in some ways lies at the opposite end of the mental illness spectrum from schizophrenia. Fetal alcohol syndrome starts affecting patients’ lives right away, in infancy. It is about half as common as schizophrenia, affecting nearly 1 million Americans, and every bit as hard to diagnose. Many mothers are reluctant to admit they drank heavily during pregnancy. Others can’t care for their babies and give them up for adoption. Then the first people to realize that something is wrong with the child are the adoptive parents, who have no clue about the birth mother’s drinking history. If there were a way to consistently diagnose fetal alcohol syndrome, even without knowing a child had been exposed to alcohol in the womb, many of these unwitting victims could get the specialized help and advocacy they need.
Children with fetal alcohol syndrome have shortened eyelids, a narrow forehead, and a missing philtrum (the fold between the upper lip and the nose). But those with a milder form of the syndrome, known as fetal alcohol effects, may not bear the telltale facial features. “Many of these kids don’t get diagnosed, and then they start acting weirder and weirder,” says Ann Streissguth, a member of the research group that discovered the syndrome in 1973. “Their parents don’t know what’s wrong with them.”
Although people with fetal alcohol syndrome and fetal alcohol effects are rarely retarded (at least according to iq tests) they have trouble tuning out distractions. Often they can’t cope with new situations or tasks. One of Streissguth’s more successful patients got a steady job as a busboy in a restaurant and was doing well until he was asked to substitute for the cashier. “He ended up throwing furniture and had to be taken to the hospital in restraints,” she says. As they move into adulthood, the problems just get worse. Sixty percent of people with fetal alcohol syndrome and fetal alcohol effects drop out of school, get suspended, or get expelled. More than a third go to jail.
Once again, the shape of the corpus callosum may hold an answer. In people with both forms of the syndrome, the callosum is either much wider than normal or much narrower. While an embryo is exposed to alcohol in the womb, Bookstein says, “there’s a process that’s basically out of control.” It’s as if nature were aiming at the correct shape but didn’t have as good aim as usual.
Testing children for early signs of fetal alcohol syndrome or schizophrenia—or tracking the diseases’ development in the brain—is no simple task. mri scans are elaborate, expensive, and somewhat intimidating, and they require special permission from parents. When used on adults, however, Bookstein’s method is already a powerful tool. In a study now being reviewed for publication, Bookstein and Streissguth looked at the results of behavioral tests and brain scans of 45 adult men, 30 of them afflicted with either fetal alcohol syndrome or fetal alcohol effects, the others not. Though neither Bookstein nor Streissguth had ever met the patients, they guessed the correct diagnosis in all but one instance.
In Bookstein’s field, such concrete results are so rare as to seem almost suspect. Most mathematicians take a perverse pride in abstruseness, in their work’s stubborn irrelevance to daily life. “It will be millions of years before we’ll have any understanding,” Hungarian theorist Paul Erdös said. “And even then it won’t be complete understanding, because we’re up against the infinite.” But shape theory, in Bookstein’s hands, is mathematics made flesh: It not only sheds new light on mental illness, it may change a doctor’s diagnosis or his decision on where to cut a living brain.
“When I look back on it, I see [mathematics’] appeal to me as having been primarily aesthetic,” Bookstein says, “and that is not a justification. The justification is that it occasionally makes sense of the world, and does so in quite unexpected ways.” As for schizophrenia, he admits that the disease is so complex and multifaceted that a true diagnostic test for it may be 20 years away. Still, there is a kind of poetic justice to the fact that such a test, when it comes, may be rooted in Bookstein’s work. Like John Nash, Bookstein is an outsider, an autodidact who dared to think “outside the box,” taking on problems that most experts considered unsolvable.
Was it worth it? Bookstein would clearly say yes. But Nash might have thought twice. Had he been given drugs to forestall his schizophrenia, his life would have been immeasurably easier. But then those same drugs might have taken the reckless edge off his mathematical genius. “You would have lost the Nash embedding theorem,” , Bookstein points out, and someone else might have won the 1994 Nobel prize for economics. Is a Nobel prize worth 30 years of madness? For anyone who saw Nash during his long, bleak battle with his own mind, the answer is obvious.
