An efficient interlaced multi-shell sampling scheme for reconstruction of diffusion propagators

Abstract

In this paper, we propose an interlaced multi-shell sampling scheme for the reconstruction of the diffusion propagator from diffusion weighted magnetic resonance imaging (DW-MRI). In standard multi-shell sampling schemes, sample points are uniformly distributed on several spherical shells in q-space. The distribution of sample points is the same for all shells, and is determined by the vertices of a selected polyhedron. We propose a more efficient interlaced scheme where sample points are different on alternating shells and are determined by the vertices of a pair of dual polyhedra. Since it samples more directions than the standard scheme, this method offers increased angular discrimination. Another contribution of this work is the application of optimal sampling lattices to q-space data acquisition and the proposal of a model-free reconstruction algorithm, which uses the lattice dependent sinc interpolation function. It is shown that under this reconstruction framework, the body centered cubic (BCC) lattice provides increased accuracy. The sampling scheme and the reconstruction algorithms were evaluated on simulated data as well as rat brain data collected on a 600 MHz (14.1T) Bruker imaging spectrometer.

Fig. 1

A square and a hexagonal pixel with unit area correspond to Brillouin zones of Cartesian and hexagonal sampling in q-space. The area of inscribing disc for a square is about 14% less than the area of the inscribing disc for the hexagon. In 3-D, this difference is about 30%, comparing a cubic voxel to rhombic-dodecahedron, the voxel of the FCC lattice, with the same volume.

Fig. 2

(a) Standard radial sampling scheme, (b) interlaced sampling scheme, (c) the structure of BCC lattice as 2-D Cartesian layers of samples shifted on alternate Z slices.

Fig. 3

Shape of polyhedra used in this paper. From left to right: Icosahedron, 12 vertices, 30 edges, 20 faces, Dodecahedron, 20 vertices, 30 edges, 12 faces, Rhombic triacontahedron, 32 vertices, 60 edges, 30 faces, Icosidodecahedron, 30 vertices, 60 edges, 32 faces.

Fig. 4

(a) Interlaced multi-shell sampling directions defined on rhombic triacontahedron (purple) and icosidodecahedron (copper) in an alternative manner. (b) Standard multi-shell sampling directions defined on rhombic triacontahedron over all the shells. Rhombic triacontahedron has 32 vertices, 60 edges and 30 faces. Icosidodecahedron has 30 vertices, 60 edges and 32 faces.

Fig. 5

Reconstruction of P(r) in the two-fiber case, evaluated at (a) ∥r∥ = 15 and (b) ∥r∥ = 25, using different sampling scheme and embedding lattices. In each figure, we have: first row: true values of P(r), second row: reconstructions using standard scheme and Cartesian lattice, third row: using standard scheme and BCC lattice, forth row: using interlaced scheme and Cartesian lattice, fifth row: using interlaced scheme and BCC lattice.

Fig. 6

The normalized MSE of the reconstruction for synthetic data of two-fiber crossings. In the legend, SC: standard scheme with Cartesian lattice. SB: standard scheme with BCC lattice. IC: interlaced scheme with Cartesian lattice. IB: interlaced scheme with BCC lattice.

Fig. 7

Reconstruction of P(r) in the three-fiber case, evaluated at (a) ∥r∥ = 10 and (b) ∥r∥ = 15. Two fixed fibers are of 120° crossing and the third fiber is rotating to form different angle to the vertical fiber. The arrangement of the figure is the same as Figure 5.

Fig. 8

The normalized MSE of the reconstruction for synthetic data of three-fiber crossings.

Fig. 9

Reconstruction results on dataset #1 evaluated at ∥r∥ = 8.0μm. (a) S₀ image where the region of interest (ROI) is shown in the blue box. Reconstruction using (b) non-interlaced scheme and Cartesian lattice, (c) the proposed interlaced scheme and BCC lattice. Zoom-in views of reconstructions on several voxels using standard scheme with Cartesian lattice (SC) (d), standard scheme with BCC lattice (SB) (e), interlaced scheme with Cartesian lattice (IC) (f), interlaced scheme with BCC lattice (IB) (g).

Fig. 10

Reconstruction of dataset #2 evaluated at ∥r∥ = 9.8μm. The arrangement of the figure is the same as Figure 9.

Fig. 11

Reconstruction of dataset #3 evaluated at ∥r∥ = 10.0μm. The arrangement of the figure is the same as Figure 9.