This poster was presented as Poster #194 at the Fifth International Conference on Functional Mapping of the Human Mapping (HBM'99) held in Düsseldorf, Germany, June 22-26, 1999.

The abstract for this poster was published in NeuroImage, vol. 9, part 2, pg. S194, 1999.

A QUASI-CONFORMAL FLAT MAP OF THE CEREBELLAR CORTEX 1Monica K. Hurdal, 1De Witt L. Sumners, 2Kelly Rehm, 3Kirt Schaper, 1Philip L. Bowers, 4Ken Stephenson, 2,3David A. Rottenberg 1Department of Mathematics, Florida State University, Tallahassee, FL, U.S.A. 2Department of Radiology, University of Minnesota, Minneapolis, MN, U.S.A. 3PET Imaging Center, VA Medical Center, Minneapolis, MN, U.S.A. 4Department of Mathematics, University of Tennessee, Knoxville, TN, U.S.A. 1,2,3,4Members of the International Neuroimaging Consortium

Abstract mapping the cerebellar cortex is essential for the description and analysis of spatial and functional relationships within and between cortical regions flat maps may assist in the analysis of activated foci buried in interfolial fissures We present cortical flat maps of the cerebellum using our novel method of circle packing. These maps are conformal in character as angular distortion is controlled.

Our Approach to Flat Mapping use math theory to create mathematically unique flat maps of the brain it is impossible to flatten a curved surface in 3D space without linear and areal distortion the Riemann Mapping Theorem (1854) says it is possible to preserve conformal (angular) information our approach attempts to preserve the conformal structure between the original cortical surface and the flattened surface

Flat Mapping the Cerebellum interested in describing activated cerebellar foci in functional neuroimaging high resolution T1-weighted MRI volume obtained [1] cerebellum extracted from volume and a topologically correct triangulated surface representing the cerebellum produced functional data obtained from a target interception experiment (see Figure 1) various lobes and fissures color coded for identification purposes (see Figures 2 & 3)

use circle packing [2] to create quasi-conformal flat maps of the cerebellum in the conventional Euclidean plane, in the hyperbolic plane and on a sphere (see Figures 4, 5, 6 & 7) Figure 1: Axial (left), coronal (middle) and sagittal (right) images showing functional activation of the cerebellum from a target interception task.

Figure 2: The human cerebellum is colored according to the following cortical regions: forest green = lobulus seminlunaris, lobulus semilunaris inferior, lobulus biventer; red = tonsils, flocculus; yellow = lingula, lobulus centralis, lobulus quadrangularis; blue = lobulus simplex, lobulus semilunaris superior; grey = white matter; bright green = fissura prima; cyan = fissure secunda; magenta = fissura horizontalis.

Figure 3: The surface representing the cerebellum is exctracted from the MR images and rendered to show a view from the front and back. Colors correspond to regions shown in Figure 2 and purple corresponds to the boundary used for flat maps.

Figure 4: Flattened map of the cerebellum (in the Euclidean plane). The colors in the left figure correspond to anatomical regions shown in Figure 2. The colors in the right figure correspond to regions of fucntional activation shown in Figure 1.

Figure 5: Cerebellum mapped to a disk (in the hyperbolic plane). The origin (map focus) is marked in black in the center of the maps. The second map has been transformed to a different map focus to display regions in the lower portion of the map. The black circle located near the bottom of the first map is used as the map origin in the second map. Similarly, the black circle located near the top of the second map corresponds to the map focus of the first map.

Figure 6: Same maps as Figure 5 but regions of functional activation are shown from a target interception task. The map foci are marked in black. Fissures are indicated in dark grey and lobes are indicated in light grey.

Figure 7: Cerebellum mapped to a sphere. Two different views are shown. Anatomical information is displayed on the upper maps and functional information is displayed on the lower maps.

Advantages and Benefits conformal mappings presereve angle proportion, are canonical and hence mathematically unique easy to impose a coordinate system when given two anatomical landmarks, which allows comparison of different maps easy to transform and change locations of map distortion no additional surface cuts are required clinical tool for analyzing anatomical and functional differences

For more information... on the mathematics behind this approach, see our poster "Generating Conformal Flat Maps of the Cortical Surface Via Circle Packing" (poster #195) for a demonstration of the software used to create these flat maps, see our poster "CirclePack: Software for Creating Quasi-Conformal Flat Maps of the Brain" (poster #250) contact: Dr. Monica K. Hurdal, Department of Mathematics, Florida State University, Tallahassee, FL., U.S.A., 32306-4510. Phone: (+1-850) 644-7378; Email: mhurdal@math.fsu.edu; WWW: http://www.math.fsu.edu/~mhurdal or visit the International Neuroimaging Consortium page ( http://pet.med.va.gov:8080/incweb/). This work has been supported in part by NIH grants MH57180 and NS33718.

References [1] Holmes, C. J., Hoge, R., Collins, L., Evans, A. C., Neuroimage, 1996, 3:S28. [2] Dubejko, T. Stephenson, K., Experimental Mathematics, 1995, 4:307-348.