Delineating white matter structure in diffusion tensor MRI with anisotropy creases

Abstract

Geometric models of white matter architecture play an increasing role in neuroscientific applications of diffusion tensor imaging, and the most popular method for building them is fiber tractography. For some analysis tasks, however, a compelling alternative may be found in the first and second derivatives of diffusion anisotropy. We extend to tensor fields the notion from classical computer vision of ridges and valleys, and define anisotropy creases as features of locally extremal tensor anisotropy. Mathematically, these are the loci where the gradient of anisotropy is orthogonal to one or more eigenvectors of its Hessian. We propose that anisotropy creases provide a basis for extracting a skeleton of the major white matter pathways, in that ridges of anisotropy coincide with interiors of fiber tracts, and valleys of anisotropy coincide with the interfaces between adjacent but distinctly oriented tracts. The crease extraction algorithm we present generates high-quality polygonal models of crease surfaces, which are further simplified by connected-component analysis. We demonstrate anisotropy creases on measured diffusion MRI data, and visualize them in combination with tractography to confirm their anatomic relevance.

Fig. 1

FA calculation does not commute with convolution-based reconstruction or differentiation. A slice of sampled tensor data D is shown with the standard RGB colormap (a) (?) or FA map (d). Convolving the discrete tensor data with a continuous reconstruction kernel B produces a continuous tensor field, in which we measure FA (b) and |∇FA| (c). Convolving the discrete FA data FA(D) with B gives a continuous FA field (e) and |∇FA| (f). Note that the shape and boundaries of the finer structures in (b) appear blurred in (e), especially near the center of the image. This is confirmed by the |∇FA| images; interfaces between major tracts visible in (a) remain sharp in (c) but indistinct in (f).

Fig. 2

Demonstrate of FA ridge and valley surfaces in a synthetic dataset. The tractography of the two arcs in (a) is color-coded by the usual RGB(e₁). The lower FA between the arcs is highlighted in (b). In (c), the red and green ridge surfaces correctly model the shape of bands, and the white valley surface captures the interface between them.

Fig. 3

Functional components of crease feature definition. The ridge surface strength s_rs (a), valley surface strength s_vs (b), and ridge line strength s_rl (c) are all defined in terms of the eigenvalues of the FA Hessian. These are used to modulate the display of the ridge surface (d), valley surface (e), and ridge line (f) functions defined in terms of the FA gradient g and Hessian eigenvectors e_i. The crease features are visible as dark lines (in the case of crease surfaces) or dark dots (in the case of ridge lines) in the bright areas.

Fig. 4

Anisotropy Creases Near the Corpus Callosum. CC = corpus callosum; IC = internal capsule; CR = corona radiata; FX = fornix; CB = cingulum bundles; SFO = superior fronto-occipital fasciculus; SLF = superior longitudinal fasciculus.

Fig. 5

Anisotropy Creases in the Brainstem

Fig. 6

Ridge surface extraction results from a single scan, showing different numbers of connected components: all 742 (left), nine largest (middle), and single largest (right).

Fig. 7

Ridge surface extraction results from six scans, viewed from an anterior (and slightly right) viewpoint, showing largest connected component (solid) and next eight largest CCs (translucent).