To design a head-size permanent magnet with built-in Spatial Encoding Magnetic fields (SEM) to support a portable brain imaging system without gradient coils.

3D MR imaging has been shown in a portable rotating Halbach cylinder magnet¹, where generalized projection imaging is performed in the y-z plane using the magnet’s “natural” B₀ variation as an encoding field (SEM)² and encoding along x is done with Transmit Array Spatial Encoding (TRASE)³. y-z plane resolution depends on the local gradient of the SEM which ideally contains relatively high 1^st order fields because blurring results from low-gradient regions of higher order fields⁴. TRASE requires a spin-echo train of accurate refocusing pulses, which are difficult to achieve over a wide B0 bandwidth (includes built-in SEM)⁵. Additionally, RF coil quality factors (Q) must be low for wide B0 bandwidths, compromising the coil efficiency. Therefore, the ideal SEM is a tradeoff between resolution and bandwidth. We aim to design a magnet with a large enough bore for human brain imaging, a low B₀ bandwidth, and a 1^st order field term (specifically X⁰Y¹Z⁰) that provides at least 2mm in-plane resolution for our standard readout time (6.4ms with SW = 20KHz).

Possible magnet designs are a subset of a full Halbach cylinder model with 3 layers (diameters = 32cm, 41cm, and 50cm) and 24 rungs made up of 1” NdFeB cubes in 18 rows (Figure 1). The magnet bore is smaller than shoulder-width therefore only 7 magnet rows are possible below isocenter (assuming 18cm between shoulders and brain center), so the model includes a 1-row “neck layer” (layer 1) to boost the inferior field. Magnet designs are represented by a vector, X, containing magnet indicator values (0,1, or 2) for each cube location, where 0 indicates air, 1 indicates a grade N42 magnet (Br = 1.32T), and 2 indicates a grade N48 magnet (Br = 1.42T).

We aim to optimize the field in a 20cm DSV at isocenter. The field contribution from each possible cube locations was simulated with Comsol Multiphysics and fit to 6^th order polynomials. The field from an arbitrary magnet design is calculated by summing across the appropriate fields specified by the design vector X. MATLAB’s Genetic Algorithm was used to create a high quality population of X vectors that satisfy 2 non-linear constraints: 1) average field > 70mT, 2) X⁰Y¹Z⁰ coefficient >3.67mT/m. The “fitness” of candidate magnet designs are evaluated by an objective function that removes the X⁰Y¹Z⁰ field component and outputs the residual resonant frequency range (i.e. minimizes field variation beyond the in-plane first-order term). The algorithm uses the fitness values to determine the next “generation”, and terminates when the best fitness value stops improving in subsequent generations.

Figure 2A shows an example run of the algorithm, and Figure 2B shows properties of resulting solutions from 500 runs of the algorithm. The color indicates the strength of the linear field component. Designs closest to the upper left corner are best because they have high average field strengths (higher SNR) and lower non-linear field variation. Figure 2C shows the same solutions plotted as a function of total field variation (including linear term). The chosen magnet design solution, indicated in Figure 2B and 2C, has an average field of 74mT (3.15 MHz), a range of 2mT (86 KHz) over the target volume, and a X⁰Y¹Z⁰ coefficient of 4.9mT/m. Figure 3A-B shows the SEM of the non-optimal prototype magnet and the simulated SEM of the optimized magnet. Simulated images using these SEMs are shown in Figure 3C-D, and 1D simulated images of point sources are shown in Figure 4A. Figure 4B shows the width of the resulting point spread functions. Figure 5A shows a drawing of the magnet cube positions (N42=grey, N48=blue). Figure 5B shows a rendering of the planned magnet housing, 32-element receiver-coil array, and head-support piece. The estimated weight of the NdFeB material in this design is 72kg. Imperfections in the NdFeB cubes will cause the constructed magnet’s field to differ from the simulation. For this reason, the magnet housing includes trays that can be populated with shim magnets. After construction, the magnetic field will be mapped and a variation of the described method will be used to place shim magnets.We have presented a method and resulting design of a lightweight magnet for brain imaging with a built-in SEM. Compared to our previous prototype, this magnet design promises a larger bore, improved resolution, and a total field range that maintains reasonable coil efficiency and transmit bandwidths.