Computational Anatomy: An Emerging Discipline

BRIEF HISTORY OF A NEW FIELD
APPLICATIONS OF COMPUTATIONAL ANATOMY
NEW DIRECTIONS

he new discipline of computational anatomy uses mathematical analysis to learn how tissues grow, assume new shapes, and "morph" into mature structures that, at first glance, might look like the action in popular music videos. "Our mathematical maneuvers are distinct from the crude morphing seen in a Michael Jackson video," said Michael I. Miller, director of the Center for Imaging Science at Johns Hopkins University, an NPACI partner in the Neuroscience thrust area. "The video is done without any coordinate system, by simple photometric transformation. Our mathematical formulation deforms structures in a coordinate space, and thus the original structures are entirely recoverable computationally. That is the difference between morphing and morphometrics." Miller and his colleagues study, quantitatively, how one brain differs from another, and they’re extending their techniques to predict anatomical anomalies, such as tiny tumors.

"Neuroscientists have long attended to questions like these," Miller said, "and if not for their sorties into this field, computational anatomy would not have become the rapidly emerging discipline that it is today."

Figure 1: Growth in the Corpus Callosum of Girls With the most rapid growth shown in red, these images were produced by Paul Thompson and colleagues in the Laboratory of Neuro Imaging directed by Arthur Toga at UCLA. They are members of the NPACI Neuroscience thrust area.

ESTABLISHING NORMS

For the past decade, Miller’s group has been developing computational methods to analyze gross anatomical structures in the human brain, with the objective of creating tools to help neuroscientists and diagnosticians learn from changes in brain substructures. The underlying mathematics are supplied by metric pattern theory, a formalism developed by Ulf Grenander in the Division of Applied Mathematics at Brown University.

"Computational anatomy could be described as a digital textbook of anatomy, with all its variability in healthy humans, adjusted for things like gender, age, and ethnicity, and also in pathological situations that affect anatomy," said Grenander. "The main difficulty is that anatomical substructures form highly complex systems, with variation being the rule," Miller said. As he said in an interview published earlier this year1, "If machines can compute structures that are equivalent to the structures we see in the world, then we can begin to understand them. In computational anatomy, we now have equations that describe how tissues can grow and bend and morph and change. These equations seem to generate very realistic structures."

Figure 2: Horn Shapes The tubular horns of sheep, goats, oxen and other herbivores can be circular, triangular, or more complicated polygons, and the variable rates of horn growth cause the angles or edges to trace an ensemble of homologous spirals. In some, such as the male muskox, the horn is not developed in a continuous spiral, changing shape as the individual grows, showing that it does not constitute a logarithmic spiral.2

Johns Hopkins graduate student Mirza Faisal Beg compared the problem to the much simpler problem of determining the "normal" human heart rate. "When we take the pulse of many thousands of people over many years, we see that the norm is a large cluster around 72 beats per minute, plus or minus three to five beats," said Beg. "We’re just in the beginning stages of norm establishment with brains, because the notion of normed shape is only now being defined mathematically. But it is at the core of the metric basis for shape in computational anatomy."

Miller has pioneered an approach to volume metric mapping that considers a starting anatomical description as a deformable template. The template anatomy is varied via transformations applied to subvolumes, contours, and surfaces. The equations governing the transformations are generalizations of the Euler equations of fluid mechanics. The rules of growth and transformation, while programmed through genes, obey the physical laws of nature represented in the equations. The group of Grenander, Miller, and French mathematicians Alain Trouvé and Laurent Younes, applies this work to constructing metrics on shape and form, with an emphasis on variation in brains.

BRIEF HISTORY OF A NEW FIELD

One of the first attempts to bring mathematical and physical insight to bear on nature’s limitless variety was made by D’Arcy Wentworth Thompson (1860—1948), a zoologist, mathematician, and classicist who worked at the Scottish universities of Dundee and St. Andrew’s. It was Thompson who found it significant that surface-to-volume ratios of species decline as those species get larger. Small creatures are governed more by surface forces, larger creatures by gravitational (volumetric) forces. In 1917, Thompson wrote:

Miller noted that Thompson’s work pioneered the principles of growth and form at a time when there were few, if any, computational methods for actualizing them (Figure 2). The theory was juxtaposed to more genetically based theories of variation, until Grenander appreciated the fundamental basis of the transformational approach as the foundation of a metric pattern theory. Other pioneers include Fred L. Bookstein of the University of Michigan and Ruzena Bajcsy of the University of Pennsylvania. "Building the digital textbook along general lines for humans and others is a major undertaking that will require years of hard work," Grenander said, adding, "if anyone can do it, it is Michael Miller."

Figure 3: Flattening the Macaque Cerebral Cortex In color, the cortex as it appears in the normal macaque (top) and in the flattened version used for comparisons (bottom). In black and white, the grid or source map used for the deformation of one comparison brain (A), the deformed source map (B), and the vector field of displacements from one to the other (C). The folds (shading) and target registration lines (black lines) on the reference (atlas) map are shown in (D), while (E) shows the difference between the comparison brain and the atlas map with respect to the same folds.

APPLICATIONS OF COMPUTATIONAL ANATOMY

The group applies their mathematical methods to original anatomies derived from various modes of brain imagery, including positron emission tomography (PET), computerized tomography (CT), and magnetic resonance imagery (MRI). Miller and J. Tilak Ratnanather, an assistant research professor in biomedical engineering at Johns Hopkins, and their collaborators, Arthur Toga and Paul Thompson, of UCLA’s Laboratory for Neuro Imaging, an NPACI Neuroscience partner, are studying growth and development in young, normal brains. An example (Figure1) shows growth in the corpus callosum area of the juvenile brain. The fastest growth is in the isthmus, which carries fibers to areas of cerebral cortex that support language function. As Toga and Thompson have shown, applying mathematical theories of deformation in computational anatomy to a grid overlying the youngest anatomy results in a "prediction" of the later anatomy that accords well with what was seen in the older subjects.

Key here is placing similar structures in the same metric space and then "deforming" one structure to another, called the target structure. The deformations have mathematical properties that persist through subsequent structures; and a statistical calculation of sameness and difference can be made across an entire group of similar structures. "You’d like to have some quantitative measure that actually says that one anatomical shape is moving closer to or farther from another. The distance between one structure and another can also be thought of as the square root of the energy required to transform the first structure onto the metric of the second," Miller said, "with the assumption that normal transformations follow the least-energy path."

One of Miller’s favorite examples of the uses of computational anatomy comes from work done in the laboratory of NPACI Neuroscience partner David Van Essen at Washington University in St. Louis, MO (Figure 3). Van Essen, one of the earliest pioneers in computational anatomy, has "built a metric space in which monkey brains and human brains can be usefully compared and contrasted," Miller said. The work of cortical "flattening" pursued by the Van Essen group is a challenge to the mathematicians, said Miller, to further develop metric pattern theory and realize its promise in neuroanatomy so the geometries of the cortical folds can be precisely quantified in terms of their metric shape.

"The study of brain geometry is now the study of all submanifolds of anatomical significance: landmarks, curves, surfaces, and subvolumes, all taken together to form a complete volume," Miller said. The more dimensions in which a structure is mapped and the more external physical forces and flows considered, the more degrees of freedom are introduced into any calculation. The complete methodology combines mappings that carry all of the submanifolds of points, curves, surfaces, and subvolumes together. The only constraint, according to Miller, is that the original anatomies must be "topologically identical"–an object in one must occur in the other and must change smoothly from one to the other. "The generalized Euler equations of motion for describing shapes of the same topology are extremely computationally intensive," Miller said.

The image-matching codes developed by Beg to compute these metric distances run on SDSC’s IBM machine, Blue Horizon, and on Zeus (a similar IBM machine) at the Center for Imaging Science at Johns Hopkins. Beg said, "Image-matching codes are naturally amenable to parallelization via decomposition of the image data into smaller pieces, each given to a different processor." The Center for Imaging Science was chosen for an SDSC Strategic Applications Collaboration project, and Beg worked with Tim Kaiser from SDSC on optimization and efficient partition algorithms.

NEW DIRECTIONS

The higher resolution of new brain imagery allows the scientists to consider transformations of a given anatomy not only through mappings of coordinate systems, but also through transformations of the luminance values of the imagery. Since these often correspond to the appearance or creation of new structures or substructures, "our methods actually let us catch a tumor as it grows or a brain region like the hippocampus as it shrinks," Miller said.

"Work like this gives us new insight into the way that brain structures grow and develop," Miller said, "and the comparison of this normal growth with abnormal structure can be more accurate as a result. Computations like these in the hands of diagnosticians may ultimately lead to earlier detection and treatment of brain tumors." –MM

1 G. Taubes (2002): An interview with Dr. Michael I. Miller in ISI’s In-Cytes: www.incites.com/scientists/DrMichaelIMiller.html

2 D.W. Thompson: On Growth and Form, 1917 (abridged edition edited by John Tyler Bonner, Canto Editions, Cambridge University Press, 1992).

Project Leader
Michael I. Miller
Johns Hopkins University

Participants
Ulf Grenander
Brown University

Alain Trouvé
Université Paris XIII, France

Laurent Younes
Ecole Normale Supérieure, Cachan, France

Mirza Faisal Beg,
J. Tilak Ratnanather
Johns Hopkins University

U. Grenander and M.I. Miller (1998): Computational anatomy: An emerging discipline, Quarterly of Applied Mathematics 56, 617-694.

U. Grenander (1993): General Pattern Theory, Oxford University Press.

M.I. Miller, A. Trouvé, and L. Younes (2002): On the metrics and Euler-Lagrange equations of computational anatomy, Annual Reviews of Biomedical Engineering 4: 375-405.