Fig 1.
Continua of possible shapes for ellipsoidal, superquadric, and supertoroidal glyphs.
Ellipsoidal (A), superquadric (B), and supertoroidal (C) glyphs are functions of geometric shape metrics CS, CP, and CL. Glyph shapes are rendered at 30 degrees with respect to the viewer. The glyph field is color-coded according to eigenvalue configuration. Supertoroids overcome some of the visual ambiguities associated with ellipsoids and superquadrics due to the increase in shape genus.
Fig 2.
Comparison of the toroidal volume (TV) and mean diffusivity (MD) indices.
Diffusivity indices TV and MD, each normalized to [0, 1], and displayed as a function of eigenvalue configuration. MD is a linear function whereas TV exhibits a nonlinear response to the increase in anisotropy. The inflection point in the TV curve profile from isotropy to anisotropy enhances tissue discrimination as compared to MD.
Fig 3.
Toroidal curvature (TC) as a function of eigenvalue configuration.
Gaussian curvature of the toroidal surface is shown as a function of the polar angle (ϕ) in isotropy (blue), transverse isotropy (green), and anisotropy (red). TC is the maximum of the Gaussian curvature values, as indicated at the positive peaks. The corresponding toroidal surfaces and curvature maxima are shown on the right panel.
Fig 4.
Diffusion tensor representations of a normal brain near the splenium.
FA map near the splenium in a normal brain color-coded by the orientation of the primary eigenvector (A). Rendering of this region using ellipsoidal (B), superquadric (C), and supertoroidal (D) glyphs using the same orientation encoding. Supertoroidal glyphs enhance the visualization of local structure and orientation due to the increase in the shape genus.
Fig 5.
Comparison between toroidal and traditional diffusion tensor maps.
Top row shows TV (A), MD (B), TC (C), and FA (D) maps of a mid-axial slice of the brain of a healthy volunteer. In the TV map of the normal brain, a distinction can be made between white matter (WM) and gray matter. The anisotropy coefficient TC displays the WM structure with detail equivalent to the FA map. Bottom row shows TV (E), MD (F), TC (G), and FA (H) maps of a mid-axial slice of a brain containing a GBM (red arrow). The ROIs in the tumor, edema, grey and white matter regions are indicated. The TV map (E) reveals WM information, which is less apparent in the MD map (F). The anisotropy coefficient TC displays some vestiges of WM in the tumor region also evident in the FA map.
Fig 6.
Supertoroidal field of a brain with a GBM color-coded with TV and eigenvalue configuration.
(A) T2-weighted image of a mid-level axial cross-section of a brain with a GBM. (B) Supertoroidal glyph field color-coded by the orientation of the primary eigenvector and weighted by FA. (C) Supertoroidal glyph field color-coded with normalized TV index using a hot-to-cold color map, revealing the necrotic core and surrounding edema. (D) Supertoroidal glyphs encoded by the eigenvalue configuration with color saturation modulated by TV. These schemes (C, D) provide visual cues that enhance the identification of tissue architectural changes, as seen in the tumor-infiltrated tissue.
Fig 7.
Comparison of MD, TV, FA, and TC within different regions of interest.
Bar plots representing MD (A), TV (B), FA (C), and TC (D) values within tumor (T), edema (E), gray matter (GM), and white matter (WM). Significant differences (p<0.05) are denoted between the T region (*), E region (#), and GM region (~) with respect to other ROIs. Discrimination of more tissue types was possible with TV than with MD owing to the nonlinear response of TV. The greater range of TC improves discrimination between degrees of anisotropy within the same WM structure and provides complementary information to that of FA.
Fig 8.
Mass effect due to a GBM depicted by the integration of tractography with supertoroids.
Tractography (with conventional orientation encoding) of the white matter in the presence of a GBM, augmented with supertoroidal glyphs color-coded by eigenvalue configuration (A, B). This demonstrates the mass effect exerted by the GBM, in which surrounding white matter tracts are pushed away from their normal trajectories as a result of tumor growth.