METHODS, PHYSIOLOGY, COGNITION

CirclePack: Software for Creating Quasi-Conformal Flat Maps of the Brain

Monica K. Hurdal, De Witt L. Sumners, Ken Stephenson, Philip L. Bowers,
David A. Rottenberg
Department of Mathematics, Florida State Univeristy, Tallahassee, U.S.A.
Department of Mathematics, University of Tennessee, Knoxville, U.S.A.
PET Imaging Center, VA Medical Center, Minneapolis, U.S.A.
Department of Radiology, University of Minnesota, Minneapolis, U.S.A.

Abstract
Flat maps of the cortical surface serve as a visualization tool that can enhance the informational content of anatomical and functional neuroimages by revealing spatial relationships that were not previously apparent and by facilitating comparisons between individuals and groups of subjects. We will demonstrate software which can be used to create cortical flat maps that exhibit conformal behavior.

Methods
Although cortical flat maps can be created in many different ways, it is usually desirable to reduce areal or angular distortion. We use the analytic technique of circle packing (1, 2) to create a flat map of the cerebral cortical surface which is conformal in character; by definition, angular proportion and direction are preserved in a conformal map. The approach is demonstrated using triangulated macaque (3) and human cortical surfaces and CirclePack software, which runs on Linux, Sun or Solaris workstations.

Results and Conclusions
Our approach for creating a quasi-conformal flat map of the brain offers several advantages over other flat mapping techniques: computation time is minimal (minutes), the mapping is conformal in character (angular distortion is controlled), cuts in the cortical surface are not required, and the resultant maps can be displayed in the hyperbolic plane as well as in the conventional Euclidean plane. The origin of flat maps in the hyperbolic plane can be interactively transformed, so that different anatomical landmarks can be brought to the map focus and distortion relegated to the periphery; moreover, map transformations can be accomplished in real time. As with other flat maps, color can be used to identify anatomical features. Although CirclePack software is still under development, it promises to be a valuable new tool for the visualization and analysis of PET and fMRI functional activation data.

Reconstruction of Macaque Cortex (3)	Euclidean Flat Map	Hyperbolic Flat Map

1. Dubejko, T., Stephenson, K., Experimental Mathematics, 1995, 4:307-348.
2. 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.
3. Drury, H. A., Van Essen, D. C., Anderson, C. H., Lee, C. W., Coogan, T. A., Lewis, J. W., 1996, J. Cog. Neurosci., 8:1-28.

Acknowledgments
The authors would like to gratefully acknowledge Heather Drury and David Van Essen from Washington University Medical School, St. Louis, for their kind assistance in providing some of the data used in this demonstration.
This work is supported in part by NIH grants MH57180 and NS33718.

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