Elevating tensor rank increases anisotropy in brain areas associated with intra‐voxel orientational heterogeneity (IVOH): a generalised DTI (GDTI) study

Abstract

Rank-2 tensors are unable to represent multi-modal diffusion associated with intra-voxel orientational heterogeneity (IVOH), which occurs where axons are incoherently oriented, such as where bundles intersect or diverge. Under this condition, they are oblate or spheroidally shaped, resulting in artefactually low anisotropy, potentially masking reduced axonal density, myelinisation and integrity. Higher rank tensors can represent multi-modal diffusion, and suitable metrics such as generalised anisotropy (GA) and scaled entropy (SE) have been introduced. The effect of tensor rank was studied through simulations, and analysing high angular resolution diffusion imaging (HARDI) data from two volunteers, fit with rank-2, rank-4 and rank-6 tensors. The variation of GA and SE as a function of rank was investigated through difference maps and region of interest (ROI)-based comparisons. Results were correlated with orientation distribution functions (ODF) reconstructed with q-ball, and with colour-maps of the principal and second eigenvectors. Simulations revealed that rank-4 tensors are able to represent multi-modal diffusion, and that increasing rank further has a minor effect on measurements. IVOH was detected in subcortical regions of the corona radiata, along the superior longitudinal fasciculus, in the radiations of the genu of the corpus callosum, in peritrigonal white matter and along the inferior fronto-occipital and longitudinal fascicula. In these regions, elevating tensor rank increased anisotropy. This was also true for the corpus callosum, cingulum and anterior limb of the internal capsule, where increasing tensor rank resulted in patterns that, although mono-modal, were more anisotropic. In these regions the second eigenvector was coherently oriented. As rank-4 tensors have only 15 distinct elements, they can be determined without acquiring a large number of directions. By removing artefactual underestimation of anisotropy, their use may increase the sensitivity to pathological change.

Physical foundations, models, and methods of diffusion magnetic resonance imaging of the brain: A review

“…For l ¼ 2, GA is akin to FA. GA and SE are not rankinvariant, and it has been shown both in simulations and on the basis of data acquired in vivo that they increase with rank in regions where IVOH is present (50,74). Their relative merits remain to be characterized: while from a theoretical viewpoint SE may be preferable since it is not clear why the square of the difference and not any other power should be considered in the definition of GA, from a computational viewpoint GA is preferable since the logarithm in the entropy function forces to resort to numerical integration (74).…”

“…Since rank-2 tensors can represent only a single directional diffusivity maximum, least-squares fitting results in oblate-shaped ellipsoids when more than one directional orientation is present: in such a case, the direction of the principal eigenvector e 1 conveys no useful information, and anisotropy indexes from rank-2 tensors are no longer valid indicators of axonal density (50,(53)(54)(55).…”

“…Such a situation may originate from a number of topologies: an interface between adjacent axonal bundles occupying the same voxel due to partial voluming, an actual crossing of the axonal matrices of multiple bundles, or a single bundle spreading or narrowing with a fan-like structure (50,63).…”

“…To the extent to which denser bundles are assumed to imply stronger anatomical connectivity, anisotropy may be considered as an indicator of connection strength (14,48,50).…”

Biexponential and diffusional kurtosis imaging, and generalised diffusion-tensor imaging (GDTI) with rank-4 tensors: a study in a group of healthy subjects

“…As a consequence, the apparent diffusion coefficient (ADC) that one measures depends on the b-value [1][2][3]. It has also been shown that, at the millimetre-scale resolution which is typical of current in vivo imaging studies, intersecting axonal bundles and partial volume effects introduce intra-voxel orientational heterogeneity; as a consequence, rank-2 tensors may take an oblate 123 shape, biasing measurements of anisotropy to artificially low values which are often interpreted as indicating low axonal density [4][5][6].…”