3D-Spatial encoding with permanent magnets for ultra-low field magnetic resonance imaging

Abstract

We describe with a theoretical and numerical analysis the use of small permanent magnets moving along prescribed helical paths for 3D spatial encoding and imaging without sample adjustment in ultra-low field magnetic resonance imaging (ULF-MRI). With our developed method the optimal magnet path and orientation for a given encoding magnet number and instrument architecture can be determined. As a proof-of-concept, we studied simple helical magnet paths and lengths for one and two encoding magnets to evaluate the imaging efficiency for a mechanically operated ULF-MRI instrument with permanent magnets. We demonstrate that a single encoding magnet moving around the sample in a single revolution suffices for the generation of a 3D image by back projection.

Figure 1

PMA design for the ULF-MRI developed at the Centre for Advanced Imaging (CAI). It comprises a switchable Array A with 12 magnets for sample pre-polarization field B_p, Arrays B and C with 24 magnets and 36 magnets, respectively, and Array D shown with one encoding magnet. With the chosen design parameters, described in the methods section, B_p ≈ 48 mT parallel to the x-axis and B_m ≈ 140 µT aligned with the y-axis. The magnetisation directions are indicated by green (Array A), blue (Array B) and red (Array C) arrows for each magnet. The insets show the simulated fields as surface plots (COMSOL colour scheme Rainbow) on the cubic FOV, located at the centre of the arrays, with a section removed to view the fields within the FOV. (a) Array A with the Halbach magnetization pattern. The B_p distribution illustrated in the inset has the typical field characteristics of a cylindrical dipole Halbach array. (b) Array A with the tangential magnetisation pattern (B_p = off), with B_m shown in the inset. (c) Detail of FOV with the 3D cross-shaped sample used for this study. For illustration purposes the front section of the sample has been removed. Shown are two small encoding magnets Ma1 and Ma2 with position parameters used in Equations 5 and 6. The inset shows the magnetic field B_e generated by Ma1 with magnetisation m at an arbitrary location. The red arrows indicate the magnetic field orientation at discrete locations within the FOV; their length indicates the local field strength.

Figure 2

3D encoding magnet paths and corresponding condition number exemplified for one encoding magnet Ma1. (a) 3D view show the position angles for Ma1 and Ma2. The magnet orientation m is described by the polar angle θ with respect to the x-axis and azimuthal angle ϕ to the xy-plane. (b) For Ma1 three helical paths are shown with linear and non-linear height variation z₁(α). The height varies from z₁(α₁) = −0.15 m to z₁(α₃) = 0.15 m. Each line segment corresponds to one encoding step location and magnet orientation for Ma1 (see inset), shown here for θ = 0° and ϕ = 0°. α varies from α₁ = 0° (initial angle) to α₃ = 360° (final angle), equivalent to one revolution. z₁(α) varies linearly if the intermediate angle α₂ = 180° (red path 2) and quadratically if α₂ = 100° (black path 1) and α₂ = 240° (blue path 3). (c) Condition number vs possible Ma1 orientation for the helical paths shown in (b). (d) Minimum condition vs intermediate angle α₂. (c,d) confirm that the optimal height variation for Ma1 is nearly linear (α₂ ≈ 180°) with optimal orientation θ ≈ 0° and ϕ ≈ 0°.

Figure 3

Minimum condition number vs encoding Ma1 path length for linear height variation. (a) 3D helical paths with different lengths are shown for final angles α₃ = 360° (red path 3), α₃ = 240° (blue path 2) and α₃ = 180° (black path 1) with α₁ = 0°. (b) Minimal condition number vs final angle, or equivalently path length. The condition number varies by less than one order of magnitude for final angles α₃ > 240°, which indicates that a full revolution of Ma1 might not be required.

Figure 4

Image reconstruction with one encoding magnet Ma1. (a) Calculated error for image reconstruction with an iterative Kaczmarz-based method. The images show 5 cross sections (see text) of the 3D sample (see Fig. 1c) after 1, 4 and 16 iterations. Image convergence occur after about 8 iterations. (b) Image quality dependence on path length for α₃ = 180° (black), α₃ = 240° (blue) and α₃ = 360° (red). Images are shown for each path at one cross section through the sample (z = 0, see inset) after 10 iterations with standard deviations calculated for α₃ = 180° (0.0231), α₂ = 240° (0.0221) and α₂ = 360° (0.0200).

Figure 5

Condition number vs magnet orientations and image reconstruction with Ma₁ and Ma₂. (a) The paths and the arrows indicate the magnet motion with configuration 1 (left column) and configuration 2 (right) column. At each encoding step the magnets are opposite to each other (xy-plane projection). The condition number distribution is shown for Ma₁ assuming optimal orientation of Ma₂ and vice versa. Like for one encoding magnet, the optimal orientations are θ ≈ 0° and ϕ ≈ 0° for both encoding magnets. (b) Image reconstruction for Ma₁ (black) and Ma₂ (red) indicated by the arrows for two configurations shown after 10 iterations. The cross section locations correspond to Fig. 4. The standard deviations are 0.0254 (configuration 1) and 0.0287 (configuration 2).