About "Axial" and "Radial" Diffusivities
C. A. Wheeler-Kingshott1, and M. Cercignani1,2
1Department of Neuroinflammation, Institute of Neurology, UCL, London, London, United Kingdom, 2Neuroimaging Laboratory, Fondazione Santa Lucia, Rome,
Diffusion Tensor Imaging (DTI) allows the quantitative assessment of diffusion anisotropy in tissue. The DT can be diagonalised to determine three
eigenvectors, V1, V2 and V3 and their corresponding eigenvalues, λ1, λ2 and λ3. It is well known that these eigenvalues depend on the underlying tissue
structure and that their direction is affected by uncertainty1,2. Recently the attention has shifted from comparing rotationally invariant anisotropy
measurements, such as Fractional Anisotropy (FA), which are insensitive to the eigenvectors sorting, towards comparing the individual eigenvalues.
The introduction of the terms “axial” and “radial” diffusivities associated to λ1 and the average of λ2 and λ3 respectively, and the results of post-mortem
studies have started a debate on the interpretation of the biophysics of these indices in terms of myelin and axons density3. The problem that arises is
that potentially encouraging correlations between the eigenvalues and histological measures of myelin and axonal degeneration are shadowing the fact
that the eigenvectors V1, V2 and V3 associated with the eigenvalues λ1, λ2 and λ3 may not be aligned with the underlying structure in the same way in
different subjects and therefore comparing “axial” and “radial” diffusivity indices across different subjects/samples can be misleading. We strongly
recommend that current and future studies that deal with “axial” and “radial” diffusivities are accompanied by a thorough investigation of the associated
directions of the eigenvectors, with particular emphasis on areas characterised by low anisotropy, partial volume or an oblate diffusion ellipsoid. To
support our pledge, we have shown a practical example of what sort of errors could occur when analysing the eigenvalues, neglecting the eigenvectors.
Two healthy controls (females, 35 (HCref) and 37 (HC) years old) and two patients with relapsing remitting Multiple Sclerosis (MS) (a female, aged 34
(MSp1), disease duration = 1.5 years, EDSS (Expanded Disability Status Scale) = 2.5 and a male, aged 55 (MSp2), disease duration = 7 years, EDSS =
5.5)) were scanned on a 1.5T MRI scanner using a dual echo fast spin echo (FSE; TR/TE1/TE2=2300/17/103ms) and a pulsed-gradient single shot spin
echo EPI sequence (cardiac gated with TR=20RR≅20s, TE=85ms, 61 distributed directions4 interleaved with 7 non-diffusion weighted b≅0 acquisitions,
maximum b factor=1200smm-2, voxel size=2.3mm3). The DTI data were first realigned and corrected for eddy currents using a 3D affine transformation5;
the tensor was fitted to the data, and FA was calculated. Using HCref as the reference, FA images of all subjects were co-registered and the
transformation was applied to the components of the tensor using the preservation of principal direction algorithm6. The eigenvectors and eigenvalues
of the rotated tensor were then derived. In every voxel we computed the dot product of the principal eigenvectors of two subjects, yielding maps of
cos(θ), where θ is the angle subtended between them. To discern areas where the principle eigenvector is not aligned with the underlying tissue
structure in the same way as the principle eigenvector of the reference data, we thresholded the maps to highlight voxels where θ>45º.
Fig 1 shows the voxels where θ>45º between HCref and HC (left), between HCref and MSp1 (centre) and between HCref and MSp2 (right). There is good
agreement between the direction of the principal eigenvector in the major white matter tracts of HCref and HC, while grey matter areas, voxels affected
by partial volume and a few sparse voxels in white matter areas of low FA show misalignment > 45 º. Areas of misalignment are more widespread in
MSp1, and these areas do not coincide necessarily with MS lesions. Many areas of white matter that are characterised by a change in the direction of
the principal eigenvector are evident in MSp2 and are involving lesion sites too.
This study confirms that the principal eigenvalue of the DT, λ1, and therefore the second and third ones, λ2 and λ3, can represent different underlying
structures in different datasets because of a different orientation of the corresponding principal eigenvector, V1. This different directionality of the
diagonalised DT eigenvectors in different subjects may be due to a real inter-subjects anatomical difference, but it may also be caused by a change of
the main underlying structure or by the presence of a sorting bias introduced by noise or by the shape of the ellipsoid in that particular area. Whatever
the reason, the point is that it underpins the rationale behind the definition of “axial” and “radial” diffusivities and their interpretation in relation to
histology results of myelin content and axonal density measures. In view of this well known problem, we cannot stress enough that analysis which are
using the eigenvalues themselves must include the eigenvectors as well.
HC MSp1 MSp2 HCref MSp2
Fig 1.Voxels in blue are those were the angle between the principal
eigenvector of a given subject and HCref differ by more than 45º.
Olga Ciccarelli for supplying the data, the MS Society of Great Britain and Northern Ireland for support.
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Fig 2. Areas of V1 changes between MSp2 and HCref.The colour coding
represents the direction of V1. The T2-weighted image shows lesions in MSp2.
Proc. Intl. Soc. Mag. Reson. Med. 16 (2008)3271