Spectral Decomposition of a 4-rank Tensor and Applications to Generalised Diffusion Tensor Imaging

Introduction: In 2003 Ozarslan et al introduced Generalised DTI, and showed how the data from a diffusion MRI experiment can be described by a higher rank tensor [1]. In 2005 the same authors proposed two generalised anisotropy metrics that can be calculated from any n-rank tensor [2]. The normalisation of these metrics is achieved by using a scaling function, with parameters that can be changed to modify the contrast of the anisotropy maps. In this work we show how spectral decomposition [3] of the 4-rank generalised diffusion tensor can be used to characterise brain structure, including the definition of two metrics of anisotropy that do not depend on the arbitrary choice of a normalising function and its parameters. Theory: Any 3D rank-4 tensor, D, such that Dijmn=Dmnij, Dijmn=Djimn, and Dijmn=Dijnm can be mapped into a 6D rank-2 tensor, S, represented by a 6x6 symmetric matrix [3]. The rank-4 generalised diffusion tensor [1], D, is a totally symmetric tensor, and therefore satisfies the required symmetry properties. This correspondence between the 4-rank tensor and the 6x6 matrix can be used to calculate the six eigenvalues (σ1>σ2>σ3>σ4>σ5>σ6) of the generalised diffusion tensor. The six eigenvalues for a 4-rank tensor corresponding to isotropic diffusion can be calculated analytically and are given by: σ1=5/3, σ2=σ3=σ4=σ5=σ6=2/3, where is the mean diffusivity; the six eigenvalues corresponding to unidirectional diffusion are given by: 2 / ) 11 3 ( 1 > < + = D σ ,