Theory of magnetic resonance in non-uniform, non-static, and non-unidirectional fields: From bending frames to kappa spaces

The theoretical foundations of conventional magnetic resonance imaging (MRI) assume a static and uniform main magnetic field and relatively weak gradient fields, the total precession field being unidirectional. Even unconventional MRI systems that deviate from these assumptions still apply the conventional theoretical framework that is based on a unidirectional and static precession axis. The picture of the magnetization dynamics and image encoding process is therefore incomplete. To address these limitations, this paper introduces an extended theoretical framework with concepts such as the 'bending frame' and 'kappa space' to accommodate new kinds of MRI scanners, including those with spatial and/or temporal variation in the applied magnetic field direction and magnitude. Also the effects of concomitant gradients, significant in low-field MRI, are covered. Kappa spaces can be classified as 'pure' or 'dispersive', indicating whether or not the image can be reconstructed with a combination of Fourier reconstruction and spatial unwarping. We also introduce techniques such as variable-length gradient pulses and dynamic scaling of the main magnetic field during the gradient trajectory. These techniques render a dispersive kappa space into a pure one, thus allowing reconstruction with Fourier methods and unwarping, even if significant spatiotemporal variations in field magnitude and direction are present. The general framework can, however, be applied to situations that abandon the Fourier transform altogether, but it also provides computational efficiency and is designed to have sufficient conceptual simplicity to be practical even when the deviations from the traditional case are small.