NeuroImage 9, Number 6, 1999, Part 2 of 2 Parts
METHODS, PHYSIOLOGY, COGNITION
A Quasi-Conformal Flat Map of the Cerebellar Cortex
Monica K. Hurdal,
De Witt L. Sumners,
Philip L. Bowers,
David A. Rottenberg
Department of Mathematics, Florida State University, Tallahassee, U.S.A.
Department of Radiology, University of Minneapolis, Minneapolis, U.S.A.
PET Imaging Center, VA Medical Center, Minneapolis, U.S.A.
Department of Mathematics, University of Tennessee, Knoxville, U.S.A.
Mapping of the cerebellar cortex is essential for the
description and analysis of spatial and functional relationships within and
between cortical regions (1). Flat maps of the cortex facilitate the
recognition of individual differences in cortical organization and the
localization of activated foci in functional neuroimages. Given adequate
spatial resolution, flat maps may also aid in the analysis of activated foci
buried within interfolial fissures.
A high-resolution T1-weighted MRI volume was obtained
from Dr. Alan Evans, Coordinator of the McConnell Brain Imaging Centre (2). The
cerebellum, defined by a plane parallel to the posterior commissure-obex line
and orthogonal to a plane passing through the vermal midline, was extracted
from the MRI volume and triangulated using the marching cubes algorithm.
Since this algorithm may produce a topologically incorrect surface, its
output was examined and repairs were made as necessary in order to produce
a topologically correct polygonal representation of the cortical surface.
Cortical regions defined by the fissura prima, fissura horizontalis and
fissura secunda were color coded, and a circle packing algorithm (3, 4) was
used to create a flat map of the cerebellar surface which exhibits conformal
Results and Conclusions
A conformal map preserves angular
proportion and direction between curves. Because our flat map is conformal in
character, it is mathematically unique and requires no cuts in the cerebellar
surface: angular distortion is controlled. Maps can be displayed in the
conventional Euclidean plane and in the hyperbolic plane. The hyperbolic maps
can be transformed interactively by dragging user-selected anatomical
structures to the origin (map focus), forcing areal distortion to the
periphery. Our flat-map approach provides an ideal method for comparing
cerebellar surface anatomy and functional localization within and between
1. Schmahmann, J. D., Doyon, J., Holmes, C., Makris, N., Petrides, M.,
Kennedy, D., Evans, A. C., Neuroimage, 1996, 3:S122.
2. Holmes, C. J., Hoge, R., Collins, L., Evans, A. C., Neuroimage, 1996,
3. Dubejko, T., Stephenson, K., Experimental Mathematics, 1995, 4:307-348.
4. Hurdal, M. K., Sumners, D. W. L., Stephenson, K., Bowers, P. L.,
Rottenberg, D. A, "Generating Conformal Flat Maps of the Cortical Surface via
Circle Packing", Abstract submitted to 5th International Conference on
Functional Mapping of the Human Brain (HBM99), 1999.
This work is supported in part by NIH grants MH57180 and NS33718.