NeuroImage 9, Number 6, 1999, Part 2 of 2 Parts


A Quasi-Conformal Flat Map of the Cerebellar Cortex

Monica K. Hurdal, De Witt L. Sumners, Kelly Rehm, Kirt Schaper, Philip L. Bowers,
Ken Stephenson, 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 behavior.

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 individuals.

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:S28.
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.