NeuroImage 11, Number 5, 2000, Part 2 of 2 Parts
METHODS - AQUISITION
Coordinate Systems for Conformal Cerebellar Flat Maps
Monica K. Hurdal,
De Witt Sumners,
David A. Rottenberg
Department of Mathematics, Florida State University, Tallahassee, U.S.A.
Department of Mathematics, University of Tennessee, Knoxville, U.S.A.
PET Imaging Center, VA Medical Center, Minneapolis, MN, U.S.A.
Departments of Neurology and Radiology, University of Minnesota, Minneapolis, MN, U.S.A.
We have generated quasi-conformal flat maps of the human cerebellum using
circle packings (1,2,3,4). Conformal maps, which preserve angular
proportion and angle direction between curves, are mathematically unique
and carry valuable geometric structure. The quasi-conformal maps we obtain
via circle packing approximate conformal maps and share many of their
mathematical advantages. In particular, they can be created on the
Euclidean and hyperbolic planes as well as on a sphere. The origin of
hyperbolic maps can be interactively transformed so that user-defined
anatomical or functional landmarks can serve as the map focus. One can
impose canonical surface-based coordinate systems on these flat maps by
specifying anatomical or functional landmarks: two for hyperbolic
coordinates, three for spherical coordinates. We describe the use of
anatomical landmarks for imposing canonical coordinate systems on
cerebellar flat maps, and for defining anatomical features and localizing
Quasi-conformal flat maps of the human cerebellum (1,4) were created from
a high-resolution T1-weighted MRI volume (5). A topologically correct
surface was produced from a cerebellar volume defined by a plane parallel
to the posterior commisure-obex line and orthogonal to a plane passing
through the vermal midline. Our quasi-conformal flattening procedure (2,3)
was applied to this surface to produce flat maps; the cortical surface was
parcellated according to (6), and activations produced by a target
interception task were imposed. Eleven readily identifiable anatomical
landmarks, e.g., the apex of the fourth ventricle and the rostral-most tip
of the lingula, were defined. For the hyperbolic coordinate system, one
landmark was used as the map center and a second to specify the map
orientation. We also used a polar coordinate system on the hyperbolic flat
map, with equiangular grid lines radiating from the map focus and
equidistant circles surrounding the focus. For the spherical coordinate
system, one landmark was used as the north pole, one as the south pole,
and a third as a distinguished equatorial point. We used the usual
latitude and longitude coordinates, similar to (7).
Results and Conclusions
A major benefit of (quasi-)conformal flat maps is that they can be used to
generate canonical surface-based coordinate systems for a cortical
surface. Since our flat maps to the hyperbolic plane and the sphere are
1-1 and onto, the coordinate systems produced by these flat maps can be
used to quantitatively localize structure and function. Furthermore, these
flat maps are mathematically unique. Different choices of landmarks and
coordinate systems will increase the utility of flat maps for studying
functional imaging data. We believe that the use of cerebellar flat maps
and canonical coordinate systems based on reproducible anatomical
landmarks will improve our ability to localize functional activity on the
cerebellar cortex and to quantify anatomical and functional differences
between individual subjects and groups of subjects.
1. Hurdal, M.K., Sumners, D.W.L., Rehm, K., Schaper, K., Bowers, P.L.,
Stephenson, K., Rottenberg, D.A., Neuroimage, 1999, 9:S194.
2. Hurdal, M.K., Sumners, D.W.L., Stephenson, K., Bowers, P.L.,
Rottenberg, D.A., Neuroimage, 1999, 9:S195.
3. Hurdal, M.K., Bowers, P.L., Sumners, D.W.L., Stephenson, K.,
Rottenberg, D.A., Lecture Notes in Computer Science, 1999,
4. See http://www.math.fsu.edu/~mhurdal for examples.
5. Holmes, C.J., Hoge, R., Collins, L., Evans, A.C., Neuroimage, 1996,
6. Schmahmann J.D., Doyon J., McDonald D., et al., Neuroimage, 1999,
7. Fischl, B., Sereno, M.I., Tootell, R.B.H., Dale, A.M., Human Brain
Mapping, 1999, 8:272-284.
This work is supported in part by NIH grants MH57180 and NS33718. The
authors would like to gratefully acknowledge Kelly Rehm, Kirt Schaper and
Josh Stern, VA Medical Center, Minneapolis, MN, U.S.A. for providing some
of the data used in this presentation.