Characterizing non‐gaussian diffusion by using generalized diffusion tensors

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Abstract
Diffusion tensor imaging (DTI) is known to have a limited capability of resolving multiple fiber orientations within one voxel. This is mainly because the probability density function (PDF) for random spin displacement is non‐Gaussian in the confining environment of biological tissues and, thus, the modeling of self‐diffusion by a second‐order tensor breaks down. The statistical property of a non‐Gaussian diffusion process is characterized via the higher‐order tensor (HOT) coefficients by reconstructing the PDF of the random spin displacement. Those HOT coefficients can be determined by combining a series of complex diffusion‐weighted measurements. The signal equation for an MR diffusion experiment was investigated theoretically by generalizing Fick's law to a higher‐order partial differential equation (PDE) obtained via Kramers‐Moyal expansion. A relationship has been derived between the HOT coefficients of the PDE and the higher‐order cumulants of the random spin displacement. Monte‐Carlo simulations of diffusion in a restricted environment with different geometrical shapes were performed, and the strengths and weaknesses of both HOT and established diffusion analysis techniques were investigated. The generalized diffusion tensor formalism is capable of accurately resolving the underlying spin displacement for complex geometrical structures, of which neither conventional DTI nor diffusion‐weighted imaging at high angular resolution (HARD) is capable. The HOT method helps illuminate some of the restrictions that are characteristic of these other methods. Furthermore, a direct relationship between HOT and q‐space is also established. Magn Reson Med 51:924–937, 2004.