The New Language of Anatomy

Summer 2012

That was how Miller found his way into computational anatomy. But he found that the field was beset by technical challenges. Some of those had to do with the limitations of that era’s computers; for some of his early projects, Miller had to plead and negotiate for the use of colleagues’ large-scale parallel processors. But some of the problems had to do with the conceptual difficulty of mathematically mapping a space as dense and variable as the human body.

“In map making, you’re building relations between labeled structures in two different places,” Miller says. “The GPS in your phone is constantly computing a map. It knows where you are relative to world coordinates, so it can tell you where you are. In computational anatomy, that’s fundamentally the technology we use: building a global positioning system.”

But anatomical mapping carries two profound challenges that GPS engineers have been spared. First, human anatomy has no single set of “world coordinates.” Instead of mapping a single planet, computational anatomists must build models that can effectively deal with the diversity of 6 billion living human bodies. Second, to a much greater degree than GPS mapping, anatomical mapping must be concerned with three-dimensional structures.

“In lots of map making that we’re familiar with, like Google Maps, the correspondence that they build is very simple,” Miller says. “In the global positioning system that Google uses, you only need to understand the position where you are in space, so that’s three dimensions, and maybe also the way you’re oriented, which is another three dimensions. And then there’s a seventh dimension, which is scale: You can look at smaller or larger fractions of the Google Map atlas.”

Until the late 1990s, most work in computational anatomy used a similar seven-dimensional system. But then Miller had an insight that has profoundly changed the field’s methods. In order to create stylized analyses of diverse human bodies, he realized, scholars should switch to infinite-dimensional mapping, called diffeomorphisms, a special kind of structure-preserving mapping.

“Imagine that you take this object here”-he picks up a watch-“and you took this object here”-he picks up a sheet of paper-“and you said, well, I’m going to understand this map by just using rotation and translation. But these two objects are not necessarily connected. If I rotate one of them, the other might stay in the same position. So you need to attach some number of dimensions to every structure in the brain, or whatever other area you’re mapping. At every place on the continuum, you’d like to be able to deform it smoothly-you know, rotate it, translate it, scale it. And that’s what continuum mechanics is about. Continuum mechanics is about understanding tissue on a continuum, and diffeomorphic deformations are not thinking of it as a rigid body but rather being able to deform it smoothly and continuously.”

To learn how to apply equations from continuum mechanics to the map of the human body, Miller went on a series of pilgrimages to Brown University and to Ecole Normale Supérieure outside Paris, where he has visited annually over the past 10 years. Brown is the home of Ulf Grenander, one of the world’s best-known applied mathematicians. Miller persuaded him to collaborate on a series of theoretical papers in computational anatomy, which later evolved into a book the two men co-authored in 2007. Ecole Normale Supérieure is the home of Alain Trouvé and originally Laurent Younes, now a professor in Miller’s Center for Imaging Science at Hopkins. With Younes, Miller has formalized the field of computational anatomy and the theory of diffeomorphic shape.

“I’ve had the privilege of working with a lot of excellent scientists, but this-this was a lot of fun,” Brown’s Grenander says. “Mike was very knowledgeable, very inventive. He does theoretical mathematics, but he has a very practical point of view. He’s remarkable at getting things done, and at getting his graduate students interested in their work.”

One of Miller’s current graduate assistants at Hopkins, a third-year doctoral student in biomedical engineering named Daniel Tward, says he’s grateful that Miller trusts young researchers to structure their own work. “We have one good solid conversation every week or two,” Tward says. “Other than that, he’s hands-off.”

Tward served as a teaching assistant this spring in Miller’s undergraduate introductory biomedical engineering course, which has roughly 130 students. “He told the TAs that more than anything else, he wanted the students to feel supported and looked after,” Tward says. And that devotion was reciprocated: When Miller expressed surprise at how well the class had done on an early test, Tward pointed out that only one student out of 130 (excluding three who dropped the course) had failed to turn in a weekly homework assignment during the first half of the semester.