Fig 1.
Edge weight, w(e), values as seed density increases for a 3D straight edge.
For this calculation, the image resolution, d, is taken to be 1, which results in a voxel volume of 1, and a surface area of 6. The calculation of w2(e) assuming an FA value of 1. These assumptions made the value for w1(e), w2(e), and w3(e) of Table 1 equal. For a different value of FA, w2(e) behaves similarly to w1(e) scaled by the appropriate FA value along the track. As described by Hagmann et. al. [16], w4(e) removes the length dependence. However, w4(e) still displays a divergent asymptote as seed density increases. The edge weight proposed in this paper, w(e) (Eq 1), displays a constant value of 1/6 for all seed densities.
Table 1.
Edge weights, w(e), derived from DWI tractography.
The edge weight, w1(eij), introduced by Li, et.al., is defined by Sij, which is the number of set of streamlines connecting node i to j. The edge weight, w2(eij) (introduced by Lo, et.al.), depends on Sij and FA, which is the average fractional anisotropy value from all voxels making up the edge. The edge weight, w3(eij) (introduced by Buchanan et.al.), depends on Sij and the volume of the connected nodes, Vx, when x = i or j. The edge weight, w4(e)) (introduced by Hagmann et.al.) depends on the inverse of l(f), which is the length of the streamline f (in units of distance) from the set of all streamlines connecting node i to j, and Ai and Ai are the surface areas of node i and j. Finally the edge weight described in this manuscript (w(eij), 5th row) is defined over the set M of voxels making up the edge, m is a voxel from set M, Pvoxel is the number of seed points per voxel, p is a seed point out of all Pvoxel from voxel m, Vvoxel is the image voxel volume, Ai is the surface area of node i and fm,p corresponds to a streamline originating from voxel m and seed point p. Generally, edge weights use in literature are some variation of weighting with the number of streamlines, FA or 1/ l(f).
Fig 2.
Scheme for streamline filtering.
In the left diagram, three nodes (n1, n2 and n3) are shown connected by two edges, e12 and e23. In the center diagram, the nodes of interest, n1 and n2, are connected by M voxels. In the right diagram, region R contains the seed points that contribute to the desired edge weight, e12.
Fig 3.
Two-dimensional WM fibers are contained within the edge, eij (white pixels), connecting nodes (gray pixels), ni and nj, as labeled in Part A.
For straight-line edges, the fiber lies within pixels of the edge. The streamline tracks are shown in pixels of the edge performed with (B) one seed-per-pixel and (C) four seeds-per-pixel in a single-pixel edge, and (D) one seed-per-pixel in a two-pixel edge. (E) A fully connected 2D network with straight edges on each face of a central node, ni (dark gray), connected to four other nodes (light gray), each through a single fiber (white) similar to the fiber shown in part B-D.
Fig 4.
Three-dimensional edge, eij (white pixels), connecting nodes (gray pixels), ni and nj, as shown in Parts A and B.
As shown on the left in A, the voxel has dimension, d by αd by βd. Part B illustrates a fully connected 3D network with a central node, ni, (dark gray) connected to six other nodes (light gray), each through a single straight edge (white) similar to the fiber shown in Part A. The front node and edge are render transparent is this diagram to make the central node visible.
Fig 5.
Diagram of two cubic, single-voxel nodes connected by a fiber in 3D.
(A) Arched streamlines joining two nodes (dark gray) in the same plane of voxels connected at one face of each cubic node. (B) Nodes in the same plane of voxels connected by 45° streamlines at two adjoining orthogonal faces of each node. The nodes are separated by one or more voxels (one voxel separation shown in B). In this diagram, the streamlines joining the nodes at the closest point have a length equal to √2 times the voxel width. (C) Nodes connected at three adjoining orthogonal faces of each node by streamlines slanting at a polar angle of 45° and azimuthal angle of 54.1°. The nodes are in two voxel-planes separated by one or more voxels (one voxel separation shown in C). In this diagram, the streamline joining the nodes at the closet point will have a length equal to √3 times the voxel width.
Fig 6.
(A) Coronal view of a cingulum node. (B) Saggital view of corpus callosum (CC) nodes. (C) Saggital view of cingulum network shows disk nodes and streamlines connecting them. (D) Coronal view of CC network shows the disk nodes and fibers connecting them. Temporal lobe (TL) network with tracks connecting nodes (E) at the interpolated resolution of 95 μm, (F) at acquisition resolution of 190 μm, and (G) at the degraded resolution of 380 μm. (H) Sketch of the TL simple graph; hippocampus (HC), thalamus (TH), amygdala (AM), and entorhinal cortex, (EC). The color scheme is maintained in all figures. (I) Coronal slice displaying the TL rat nodes.
Fig 7.
System of 2 nodes connected by a streamline at 45° with 4 equally spaced seed points in each pixel and across pixel boundaries.
(A) Nodes (dark gray) are connected by a slanted fiber (light gray) and dark squares are the seeds points used to perform tractography. (B) After keeping the seed points that lie within the region, R, only 12 contribute to the edge weight out of 36 original seed points.
Fig 8.
Edge weight calculations, for the arched and slanted streamline paths of Fig 4, as a function of seed point density.
The seed point density is P = n3, where n = 1, 3, 5, …, 33. Arched edge weight values plateaus at a value of 0.167, for all radii, r (d is the voxel width). Slanted in-plane edge weight plateaus at a value of 0.235, and 3D slant edge weight plateaus at 0.289, when the nodes are separated by 1, 2, or 3 voxels (v).
Fig 9.
Simulation of edge weight values for an arc streamline pathway in the same plane as single voxel nodes (Fig 5, part A) as random error increases.
The nodes closest points were separated by 1, 2, 3, 5, and 10 voxels.
Table 2.
Edge weight, w(e), for edges, e, in major white matter regions (CC, corpus callosum; Cing, cingulum).
Calculated average w(e) value from the ten human datasets at 1 and 8 mm3 isotropic resolution, along with the associated coefficient of variation, cv, of the edge weight. The results are presented for the complete edge, CC long and Cing long, as well as two subdivisions (short 1 and short 2, see text) of these long edges (see Fig 6). The calculations were performed without restriction to the streamlines connecting the nodes (Without Streamline Restriction), and with the inclusion of streamlines fibers restricted (With Streamline Restriction) to originate from the connecting edge, R.
Fig 10.
Excised rat brain TL network parameter values at isotropic resolutions of 95, 190 and 380 μm.
The edge values are in the left column and node values in the right column: (A) Edge weights values (TH-EC values are too small to appear at this scale), (B) node strength values, (C) edge volume, (D) node surface area, and (E) edge lengths. (F) The highest resolution TL network with the edge widths size scaled by the value of edge weight and the node size scaled by the value of the node connection strength. The TH-EC edge is illustrated at the minimum line width that is still visible.
Fig 11.
(A) 3D edge sideways. (B) 3D edge at an angle to display the face where it connects to the node. Nodes are not shown to simplify the visualization. (C-D) Sketch of one of the portions that make up the fiber. (C) Shows that for every short fiber there are two of the long ones. (D) Shows a short side, which has a length of √3 and the longer one is 2√3. (D) Shows the triangular cross section of the fiber, yielding a higher number of longer fibers compared to the short ones.
Fig 12.
Sketch of the portion of the voxel (dark gray) adjacent to the node that contributes to the edge.
(A) Shows the dark gray corresponding to the voxel above the node voxel on a sideways view. (B) Shows the voxel above the node voxel on a view along the z axis. (C) Shows to the voxel above the node voxel on a top view. (D) Sketch of the isolated piece of the voxel contributing to the edge. This volume is repeated along the fiber length except in middle nodes where the entirety of voxel contributes to the edge weight.