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The tensor distribution function
Author(s) -
Leow A. D.,
Zhu S.,
Zhan L.,
McMahon K.,
de Zubicaray G. I.,
Meredith M.,
Wright M. J.,
Toga A. W.,
Thompson P. M.
Publication year - 2009
Publication title -
magnetic resonance in medicine
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.696
H-Index - 225
eISSN - 1522-2594
pISSN - 0740-3194
DOI - 10.1002/mrm.21852
Subject(s) - diffusion mri , tensor (intrinsic definition) , eigenvalues and eigenvectors , orientation (vector space) , function (biology) , mathematical analysis , probability density function , distribution function , probability distribution , gaussian , displacement (psychology) , distribution (mathematics) , mathematics , diffusion , geometry , physics , magnetic resonance imaging , statistics , medicine , psychology , quantum mechanics , evolutionary biology , biology , psychotherapist , radiology , thermodynamics
Diffusion weighted magnetic resonance imaging is a powerful tool that can be employed to study white matter microstructure by examining the 3D displacement profile of water molecules in brain tissue. By applying diffusion‐sensitized gradients along a minimum of six directions, second‐order tensors (represented by three‐by‐three positive definite matrices) can be computed to model dominant diffusion processes. However, conventional DTI is not sufficient to resolve more complicated white matter configurations, e.g., crossing fiber tracts. Recently, a number of high‐angular resolution schemes with more than six gradient directions have been employed to address this issue. In this article, we introduce the tensor distribution function (TDF), a probability function defined on the space of symmetric positive definite matrices. Using the calculus of variations, we solve the TDF that optimally describes the observed data. Here, fiber crossing is modeled as an ensemble of Gaussian diffusion processes with weights specified by the TDF. Once this optimal TDF is determined, the orientation distribution function (ODF) can easily be computed by analytic integration of the resulting displacement probability function. Moreover, a tensor orientation distribution function (TOD) may also be derived from the TDF, allowing for the estimation of principal fiber directions and their corresponding eigenvalues. Magn Reson Med 61:205–214, 2009. © 2008 Wiley‐Liss, Inc.