Resolution of complex tissue microarchitecture using the diffusion orientation transform (DOT) Evren O ¨ zarslan, a, * Timothy M. Shepherd, b Baba C. Vemuri, a Stephen J. Blackband, b,c and Thomas H. Mareci d a Department of Computer and Information Science and Engineering, University of Florida, P.O. Box 116120, Gainesville, FL 32611, USA b Department of Neuroscience, University of Florida, P.O. Box 100244, Gainesville, FL 32610, USA c National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA d Department of Biochemistry and Molecular Biology, University of Florida, P.O. Box 100245, Gainesville, FL 32610, USA Received 7 July 2005; revised 19 December 2005; accepted 24 January 2006 Available online 20 March 2006 This article describes an accurate and fast method for fiber orientation mapping using multidirectional diffusion-weighted magnetic resonance (MR) data. This novel approach utilizes the Fourier transform relationship between the water displacement probabilities and diffu- sion-attenuated MR signal expressed in spherical coordinates. The radial part of the Fourier integral is evaluated analytically under the assumption that MR signal attenuates exponentially. The values of the resulting functions are evaluated at a fixed distance away from the origin. The spherical harmonic transform of these functions yields the Laplace series coefficients of the probabilities on a sphere of fixed radius. Alternatively, probability values can be computed nonparametrically using Legendre polynomials. Orientation maps calculated from excised rat nervous tissue data demonstrate this technique’s ability to accurately resolve crossing fibers in anatomical regions such as the optic chiasm. This proposed methodology has a trivial extension to multiexponential diffusion-weighted signal decay. The developed methods will improve the reliability of tractography schemes and may make it possible to correctly identify the neural connections between functionally connected regions of the nervous system. D 2006 Elsevier Inc. All rights reserved. Keywords: MRI; Tensor; Anisotropy; HARDI; Fourier; Spherical harmonics Introduction The diffusional attenuation of MR signal in pulsed field gradient experiments (Stejskal and Tanner, 1965) has been exploited to characterize diffusional anisotropy in fibrous tissues like muscle (e.g. Cleveland et al., 1976) and white matter in animal (e.g. Moseley et al., 1990) and human (e.g. Chenevert et al., 1990) nervous tissue. When the narrow pulse condition is met, i.e. the duration of the applied diffusion sensitizing gradients (d ) is much smaller than the time between the two pulses (D), the fundamental relationship between the MR signal attenuation and average displacement probabilities P(R) is given by a Fourier integral (Callaghan, 1991): P R ð Þ¼ Z E q ðÞ exp 2piq I R ð Þdq; ð1Þ where R is the displacement vector and q is the reciprocal space vector defined by q = gd G/2p, where g is the gyromagnetic ratio and G is the gradient vector. In the above expression E(q )= S (q )/ S 0 , where S (q ) is the signal value associated with the reciprocal space vector q and S 0 is the signal when no diffusion gradient is applied, i.e. when q = 0. Diffusional anisotropy is well-reflected in the water displace- ment probabilities, and it is expected that, in fibrous tissues, the orientations specified by large displacement probabilities will coincide with the fiber orientations. One could in principle estimate these displacement probability functions by using Eq. (1) and the fast Fourier transform (FFT), however, this would require data points all across the space spanned by the diffusion gradients (or q vectors). This q -space approach would require very high gradient strengths and long acquisition times that are difficult to achieve in clinical settings (Basser, 2002). Although attempts have been made to acquire such data sets in vivo (Wedeen et al., 2000), the results typically suffer from undersampled q -space and sacrificed spatial resolution. More than a decade ago, Basser et al. (1994a,b) introduced an imaging method called diffusion tensor imaging (DTI) that replaced the apparent diffusion coefficients that had been calculated in diffusion-weighted imaging studies with a symmetric, positive- 1053-8119/$ - see front matter D 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2006.01.024 * Corresponding author. Present address: National Institutes of Health, 13 South Dr. Rm 3W16, Bethesda, MD 20892-5772, USA. Fax: +1 301 435 5035. E-mail address: evren@helix.nih.gov (E. O ¨ zarslan). Available online on ScienceDirect (www.sciencedirect.com). www.elsevier.com/locate/ynimg YNIMG-03744; No. of pages: 18; 4C: 10, 11, 12 NeuroImage 31 (2006) 1086 – 1103