This course will attempt to describe general features of nonlinear gradient encoding, as well as provide an overview of the methods and applications reported in the literature.
This lecture will focus on the use of nonlinear gradients in spatial encoding, beginning with basic properties of nonlinear gradients and advancing to the broad categories of sequences that employ them. Topics include:
1) Introduction to nonlinear gradients, including hardware and general properties
2) Phase encoding methods, including image windowing methods
3) Projection imaging methods
4) K-space filling methods
Like standard gradients, nonlinear gradients modulate the magnitude of Bz as a function of position; the difference is that the magnitude as a function of position is generally not linear or unidirectional. The magnitude of Bz as a function of space can either follow some higher order function, follow an arbitrary shape, or be a linear combination of arbitrary shapes designed to approximate a higher order function. These fields can be implemented with a wide range of hardware of varying scale and architecture (1), and combining nonlinear and linear gradients provides still further flexibility to shape the applied field.
One important consequence of nonlinearity in the gradient is that the encoding superimposed on the spins is no longer a uniform sine wave. So nonlinear gradient encoding does not superimpose a Fourier basis and is not directly related to k-space. Furthermore the relationship between frequency and spatial position is not always one to one; it is a non-bijective mapping. However, nonlinear gradients possess a number of intriguing properties[N1] , including spatially varying resolution, reduced peripheral nerve stimulation, and potential to better complement other nonlinear encodings like receive profiles. Each of these features have played a role in the many imaging methods that have been developed and studied. [N1]I assume all the things you mention will be discussed
Frequency and phase encoding methods
In 2008, two important papers were published applying nonlinear gradients to MRI spatial encoding. Ito showed that phase scrambling, i.e. fixed phase encoding, by a quadratic field allowed for unaliased zooming of the field of view (2). This idea was later refined and combined with a modified k-space trajectory, with potential applications in cardiac imaging (3).
In addition, Hennig proposed an intriguing method that encoded an image purely with nonlinear gradients (4). The PatLoc method applied 2nd order gradients in phase and frequency encode patterns identical to those used in standard Cartesian imaging, so Fourier transform mapped out spin density in a warped and non-bijective image space. Applying a Jacobian and using multichannel receivers to resolve the non-bijective ambiguity resulted in an alias free image, with potential for reduced peripheral nerve stimulation and anatomy-adapted field of view. However, since 2nd order fields are flat in the center of the field of view, spatial encoding vanishes there, and images bear a conspicuous void there.
Projection imaging methods
Following this, Constable proposed a projection imaging approach that focused on complementing the geometry of receiver coil profiles with appropriately shaped nonlinear gradients (5). To achieve complementary field shapes, this work was the first to combine linear and nonlinear gradients simultaneously during the readout and was later generalized to arbitrary field shapes (6). Experiments with these methods showed improved image quality in highly undersampled images from the addition of the nonlinear gradient field, particularly for extended readouts.
In addition to these two prototypes for phase/frequency encoding and projection encoding, a number of hybrid methods employing nonlinear and linear gradients both prior and during readout have been studied. These include methods designed to optimize incoherence for compressed sensing, SNR, or uniformity of spatial resolution. Nonlinear gradients have also been used to excite and encode curved slices for a kind of panoramic view MRI, and also to facilitate simultaneous multislice imaging (7-13).
K-Space filling methods
More recently, we have shown that nonlinear gradients can be understood in k-space (14). This perspective has the advantage that it restores an intuitive connection between arbitrary gradient trajectories and image reconstruction. Furthermore, that analysis guided the design of a nonlinear gradient encoding strategy known as FRONSAC, where small perturbations are added to the linear encoding method to better sample gaps in k-space, regardless of the underlying linear trajectory (15). Because nonlinear gradient encodings act as a booster to the linear encoding, the properties of the underlying imaging method are preserved but better undersampled image quality can be achieved. For example, Cartesian-FRONSAC retains features like relative insensitivity to off-resonance spins and timing delays, ease of changing FOV, resolution, and orientation, and relatively simple contrast behavior, while still allowing for higher undersampling factors (16).
1) Littin, S. et al, Development and implementation of an 84-channel matrix gradient coil, MRM 79(2): 1181-1191. (2018)
2) Ito, S. et al, Alias-free image reconstruction using Fresnel transform in the phase-scrambling Fourier imaging technique, MRM 60(2): 422-30. (2008)
3) Witschey, WRT et al, Localization by nonlinear phase preparation and k-space trajectory design, MRM 67(6): 1620-1632. (2012)
4) Hennig, J. et al, Parallel imaging in non-bijective, curvilinear magnetic field gradients: a concept study, MAGMA, 21(2): 5-14. (2008)
5) Stockmann, J. et al, O-space imaging: Highly efficient parallel imaging using second order nonlinear fields as encoding gradients with no phase encoding, MRM 64(2): 447-456. (2010)
6) Tam, L.K. et al, Null space imaging: Nonlinear magnetic encoding fields designed complementary to receiver coil sensitivities for improved acceleration in parallel imaging, MRM 68(4):1166-1175. (2012)
7) Tam, L.K. et al, Pseudo-random center placement O-space imaging for improved incoherence compressed sensing parallel MRI, MRM 73(6): 2212-2224. (2015)
8) Layton, K. et al, Trajectory optimization based on the signal-to-noise ratio for spatial encoding with nonlinear encoding fields, MRM 76(1): 104-117. (2016)
9) Gallichan, D. et al, Simultaneously driven linear and nonlinear spatial encoding fields in MRI, MRM 65(3): 702-714. (2011)
10) Weber, H. et al, Local shape adaptation for curved slice selection, MRM 72(1):112-123. (2013)
11) Kopanoglu, E. et al, Radiofrequency pulse design using nonlinear gradient magnetic fields, MRM 74(3): 826-839. (2014)
12) Littin, S. et al, Simultaneous Multislice techniques using Matrix Gradient Coils, Proc. of 25th ISMRM, p517. (2017)
13) Ertan, K. et al, A z-gradient array for simultaneous multi-slice excitation with a single-band RF pulse, MRM 80(1): 400-412. (2017)
14) Galiana, G. et al, The role of nonlinear gradients in parallel imaging: A k-space based analysis, Concepts in MR, 40A(5): 253-267. (2012)
15) Wang, H., Fast rotary nonlinear spatial acquisition (FRONSAC) imaging, MRM 75(3): 1154-1165. (2016)
16) Dispenza, N.L., Clinical Potential of a New Approach to MRI Acceleration, Scientific Reports, 9:1912. (2019)