Circular Spectrum Mapping for Intravoxel Fiber Structures Based on High Angular Resolution Apparent Diffusion Coefficients Wang Zhan, 1 Hong Gu, 1 Su Xu, 2 David A. Silbersweig, 1 Emily Stern, 1 and Yihong Yang 1 * A method is presented for mapping intravoxel fiber structures using spectral decomposition onto a circular distribution of measured apparent diffusion coefficients (ADCs). The zeroth-, second-, and fourth-order harmonic components of the ADC distribution on the circle spanned by the major and median eigenvectors of the diffusion tensor can be used to provide quantitative indices for isotropic, linear, and fiber-crossing dif- fusion, respectively. A diffusion-weighted MRI technique with 90 encoding orientations was implemented to estimate the cir- cular ADC distribution and calculate the circular spectrum. A digital phantom was used to simulate various diffusion pat- terns. Comparisons were made between the circular spectrum and regular DTI-based index maps. The results indicated that the zeroth- and second-order circular spectrum maps exhibited a strong consistency with the DTI-based mean diffusivity and linear indices, respectively, and the fourth-order circular spec- trum map was able to identify the fiber crossings. MRI experi- ments were performed on seven healthy human brains using a 3T scanner. The in vivo fourth-order maps showed significantly higher densities in several brain regions, including the corpus callosum, cingulum bundle, superior longitudinal fasciculus, corticospinal tract, and middle cerebellar peduncle, which in- dicated the existence of fiber crossings in these regions. Magn Reson Med 49:1077–1088, 2003. © 2003 Wiley-Liss, Inc. Key words: apparent diffusion coefficient; diffusion tensor im- aging; fiber crossing; high angular resolution; spectral decom- position Diffusion-weighted MRI has been widely used to noninva- sively investigate the structural organization of biological tissues. Much attention has been paid to diffusion anisot- ropy, in which a scalar apparent diffusion coefficient (ADC) is insufficient to fully describe the diffusion process (1– 4). The diffusion tensor imaging (DTI) method intro- duced by Basser et al. (5,6) overcame this difficulty to a large extent by interpreting the anisotropic diffusion as a second-order function of the encoding orientation. Much effort has been directed toward optimizing the DTI acqui- sition scheme (7–9), and significant progress has been made in the interpretation of DTI data by various quanti- tative indices based on the eigenstructure of the diffusion tensor (10 –12). As a promising application of DTI, “fiber tracking” techniques have been developed to allow a non- invasive delineation of the neuronal pathways inside the brain (13–16). However, a major challenge for DTI tech- niques is the problem caused by fiber crossings, including fiber “intersection,” “dispersion,” “kissing,” and “branch- ing,” whereby multiple fiber compartments share a single voxel. In this case, the major eigenvector of the tensor can be substantially biased from the real fiber orientation, pre- venting fiber-tracking algorithms from following the actual fiber connection. Moreover, reducing the voxel size does not remedy this problem completely because of the “pow- der averaging” effects of multiple-fiber singularity (15). Some tensor-derived indices, e.g., the fractional anisot- ropy (FA), are vulnerable to the artifacts associated with the partial volume effects of crossing fibers (17). To resolve these problems, Poupon et al. (14,18) proposed an im- proved fiber-tracking algorithm to estimate a regularized fascicle trajectory by taking into account prior knowledge of a local direction map. Wiegell et al. (19) investigated the fiber crossings in several regions of interest (ROIs) in the human brain by employing the full eigenstructure of the diffusion tensor, and found that the planar model spanned by the major and median eigenvectors of the tensor could be used to characterize the intersection and dispersion of fibers. Some geometric measurement (spherical case (CS), (linear case (CL), and (planar case (CP)) indices, describing the tensor shape in regard to spherical, linear, and planar degrees, respectively, were proposed by Westin et al. (20) and investigated extensively by Alexander et al. (21). However, the efforts made within the DTI formulation cannot offer a complete solution to the fiber-crossing prob- lem, mainly because of the limitation of the tensor model itself. The CP index, for example, is unable to distinguish an inherent planar diffusion tensor (e.g., in a thin mem- brane filled with water) from the planar tensor induced by an orthogonal intersection of two linear fibers (17). More importantly, it should be noted that the diffusion tensor is only a second-order approximation (in terms of mean square fitting) to the real 3D diffusion process (22), and thus this model cannot make full use of this intravoxel non-Gaussian information. Therefore, a model beyond conventional DTI is essential to solve the problem of fiber crossing. In fact, a number of studies beyond the DTI framework have been carried out. An important contribu- tion came from q-space imaging, a technique in which the tissue structures are probed by directly calculating the diffusion displacement distribution in each voxel (23–25). This method is theoretically rooted in the Fourier-trans- form relationship between the echo attenuation in q-space and the diffusion displacement probability distribution. 1 Functional Neuroimaging Laboratory, Department of Psychiatry, Weill Med- ical College of Cornell University, New York, New York. 2 Department of Medical Physics, Memorial Sloan-Kettering Cancer Center, New York, New York. *Correspondence to: Yihong Yang, Ph.D., MRI Physics Unit, Neuroimaging Research Branch, National Institute on Drug Abuse, NIH, 5500 Nathan Shock Drive, Rm. 383, Baltimore, MD 21224. E-mail: YiHongYang@intra.nida.nih.gov Received 12 June 2002; revised 13 January 2003; accepted 15 January 2003. DOI 10.1002/mrm.10484 Published online in Wiley InterScience (www.interscience.wiley.com). Magnetic Resonance in Medicine 49:1077–1088 (2003) © 2003 Wiley-Liss, Inc. 1077