Brain Conformal

Discrete conformal methods for cortical brain attening Monica K. Hurdal a, ⁎, Ken Stephenson b a Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA b Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA a b s t r a c t a r t i c l e i n f o Article history: Received 15 October 2008 Accepted 15 October 2008 Available online 11 November 2008 Keywords: Circle packing Conformal map Cortical at map Mapping Riemann Mapping Theorem Surface attening Locations and patterns of functional brain activity in humans are dif cult to compare across subjects because of differences in cortical folding and functional foci are often buried within cortical sulci. Unfolding a cortical surface via at mapping has become a key method for facilitating the recognition of new structural and functional relationships. Mathematical and other issues involved in at mapping are the subject of this paper. It is mathematically impossible to atten curved surfaces without metric and area distortion. Nevertheless, “metric” attening has ourished based on a variety of computational methods that minimize distortion. However, it is mathematically possible to attenwithout any angular distortion — a fact known for 150 years. Computational methods for this “conformal” attening have only recently emerged. Conformal maps are particularly versatile and are backed by a uniquely rich mathematical theory. This paper presents a tutorial level introduction to the mathematics of conformal mapping and provides both conceptual and practical arguments for its use. Discrete conformal mapping computed via circle packing is a method that has provided the rst practical realization of the Riemann Mapping Theorem (RMT). Maps can be displayed in three geometries, manipulated with Möbius transformations to zoom and focus on particular regions of interest, they respect canonical coordinat es useful for inters ubject registrat ion and are locally Euclidean. The versatility and practical advantages of the circle packing approach are shown by producing conformal at maps using MRI data of a human cerebral cortex, cerebellum and a speci c region of interest (ROI). © 2008 Elsevier Inc. All rights reserved. Introduction The cortex of humans is a highly convoluted surface. The folds (gyri) and ssures (sulci) of the brain vary in size and position from person to person, and this variability has made it dif cult for medical researchers to analyze and compare patterns of functional activation wit hin and bet wee n sub je cts. A numbe r of met hod s ha ve bee n impl emen ted that take adva ntage of the two-dimen sion al sheet topology of the cortical surface; the resulting “at mappings” of the cerebral and cerebellar cortex may fac ili tate the rec ogniti on of structural and functional relationships. This paper discusses the mathematics, computations, and applica- tion of a at map pin g approach whi ch uses cir cle pac kin gs to approximate conformal maps of cortical surfaces. Such maps, termed “discrete conformal maps”, are analogues of the conformal mappings of analytic function theory, allowing this rich classical theory to be exploited. Researche rs will nd both concep tual and pr actica l adva ntag es in disc rete conformal mapp ing. The basic s of classica l conf orma l geometry and its discr ete analogue are disc usse d. The computations and mapping tools of circle packing are illustrated by prod ucin g and manipulatingdiscret e conformal at mapsof data from an MRI vol ume of the human cerebral cor te x, cerebellum and a specic region of interest (ROI). Certa in tech nica l prel imi naries are involved in all attening efforts. Firs t is the acq uisit ion of the initial three-d imen siona l (3D) data; anat omical and function al data volumes of the cerebru m can be obtained non-invasively using a variety of neuroimaging modalities, including magnetic resonance imaging (MRI), functional MRI (fMRI), and positron emission tomography (PET) (see Toga and Mazziotta, 1996 for a compilation of various methods). Data preprocessing can incl ude intr a- and inter subj ect regi stra tion , inhomogen eity corr ectio n, segmentation, parcellation and visualization. A common approach to vis ual izi ng fun cti ona l dat a hasbeento pro je ct a foc us of act iv ationon a cr oss -se ction or sli ce of thebrainvolume.This app roach suf fe rs from a number of disadvantages due to the highly folded structure of the cor te x. Ac tiv ated foc i that appearclos e tog eth er ona sli ce or3Dsurface rendering may be quite far apart when visualized on the unfolded cortical surface. In a given individual, foci are often buried within cortical sulci and appear in a number of discrete slices, making it dif cult to compare multiple foci simultaneously. Moreover, inter- subj ect comp ariso ns of focuslocatio n and ext entareaffected, inter alia, by individual differences in folding patterns (Rehm et al., 1998 ). Afte r data acquisition and preprocessing, all attening efforts require that a nite representation of the surface region of interest be extracted from the 3D brain volume. Historically, that surface was reconstructed by tracing contours from histological sections. Wire NeuroImage 45 (2009) S86–S98 ⁎ Corresponding author. Fax: +1 850 644 4053. E-mail address: [email protected] (M.K. Hurdal). 1053-8119/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2008.10.045 Contents lists available at ScienceDirect NeuroImage j ou r nal h o me p a g e: www. e l s ev i e r. c om/ locate/ y ni mg

Documents

Brain Conformal