MRI is the most powerful diagnostic imaging modality available. The major drawbacks are cost and complexity, limiting its use mainly to industrialized countries and larger hospitals¹. Multiple studies showed that there are several clinical applications where the diagnostic value gathered from low (B₀=0.2T) field MRI is equivalent to the costly “high” field (B₀≥1.0T) counterparts^2-6. This work investigates permanent magnet Halbach arrangements for low cost low field imaging applications with a focus on absolute B₀, field homogeneity, magnet mass and cost. The concept of Halbach quadrupoles is introduced to form constant gradient fields that can be rotated for spatial encoding removing the need for high power gradient amplifiers⁷. Technical documentation of the presented cost effective open source imaging (COSI) magnet will be made available based upon the principles of open source hardware at www.opensourceimaging.org.Magnetostatic simulations (CST) were performed to investigate a spectrum of Halbach magnet arrangements tailored for low field MRI. To validate the simulations, a dedicated simulation setup was compared to experimental measurements of a Halbach magnet with an inner diameter of d_i=130mm (Fig.1a-b). After validation, B₀, magnet radius, magnet mass as well as number of magnet units (n=16, n=20, n=24) were calculated for different Halbach arrangements with NdFeB magnets (ρ=7.4g/cm³, B_r=1.2T) of varying size (Fig.2a-c). Two ring configurations (Fig.3a) were implemented that afford a uniform B₀ volume in between the rings⁸. Halbach quadrupoles are introduced to enable constant gradient fields for distinct spatial encoding. Finally a COSI magnet was designed and characterized for extremity MRI with an inner diameter of di=180mm.

Simulation results (Fig.1c) are in good agreement with measurements (Fig.1d). The absolute difference along an axial B0 profile through the center of the magnet was <10% (Fig.1e) with a relative difference <1% (Fig.1f).

Using fewer magnet units or a bigger magnet radius increases the magnet mass under the condition of B₀=const (Fig.1d-e). A higher number of magnet units leads to improved homogeneity, since the theoretical periodic Halbach magnetization pattern can be recreated more accurately. Fitting the results to a single equation (for n=16 with the proportionality factor a(m) (Fig.1f)): $$B_0(r,m)=(23.3\cdot m+0.1311)/r^3$$ allows for a quick estimate on expected B₀, magnet radius and mass.

Varying the distance of two ring Halbach configurations allows B₀ homogenization for the central region along z-dimension (Fig.3b). For two Halbach rings (d_i=180mm, n=16) the optimal simulated distance is around 87mm, which is in close agreement with the theoretical value of $$$r\cdot \sqrt{2/3}=90mm$$$ (Fig.3c)⁸.

The arrangement of 16 octagonal magnets (B_r=1.0T) to a Halbach quadrupole with radius r_i1≈18cm is illustrated in Fig.4a. The field profile By (Fig.4b) shows a rapid exponential decay in close vicinity to the magnets (Fig.4c) and a linear decay of 50mT/m in the center (Fig.4d). Reducing the radius of the arrangement to ri2≈13cm allows to increase the gradient strength to dBy/dy=157mT/m (Fig.4d). With this achievement spatial encoding can be performed by means of rotating the quadrupole instead of the whole imaging magnet¹ while the gradient strength can be adjusted changing the quadrupole radius mechanically without the need for costly gradient amplifiers. Rotation and repositioning is feasible with moderate efforts, since the main Halbach magnet has no net magnetic field at the outside.

Combining all these efforts a possible Halbach magnet (d_i=180mm) for extremity MRI together with Halbach gradients (dB_y/dy=157mT/m, d_i=270mm) is exemplified in Fig.5a. The Halbach magnet weighs <45kg and provides an absolute B₀>0.3T with a net magnetic field of B₀≈0 at the outside (Fig.5b). The homogeneity of the Halbach magnet can be deduced from Fig.5c-f. Without any additional shimming the 1000ppm area is (62x50)mm² as shown in Fig.5c. Along 4cm in x- and y-direction the field deviates by 192ppm (Fig.5d) and 574ppm (Fig.5e) respectively.

The combination of uniform Halbach (dipole) magnets with superimposed Halbach quadrupoles is encouraging for low cost low field imaging applications. Utilizing quadrupoles with their excellent gradient field linearity for efficient spatial encoding reduces rotation to the smaller quadrupole magnets only⁷ with the gradient strength being modified by simple mechanical displacement. Significant improvements in B0 homogeneity could be achieved under the constraints of small and light construction for “mobile” applications. Low static magnetic fields inside B₀≈0.3T and B₀≈0 outside the imaging volume improve safety by reducing attraction forces of ferromagnetic objects such as medical equipment or implants. At the same time permanent magnets don’t require a constant/stable power supply and don’t need risk management concerning a quench. Mechanically adjustable imaging gradients furthermore allow for significant noise reduction versus traditional pulsed field gradients used in today’s clinical practice.