Xiang Gao^{1}, V.G. Kiselev^{1}, Thomas Lange^{1}, Jürgen Hennig^{1}, and Maxim Zaitsev^{1}

^{1}Medical Center University of Freiburg, Faculty of Medicine, University of Freiburg, Freiburg im Breisgau, Germany

A new open source recursive magnetization evolution calculation algorithm is proposed for simulating arbitrary pulse sequences efficiently and intuitively. It lifts the sequence symmetry requirements of the Extended Phase Graph and avoids intensive computations associated with direct Bloch equation simulations. The method further allows for tracking the evolution of the MR signal and corresponding k-vectors in presence of time-variant gradients with arbitrary orientations in 3D domain.

To illustrate the developed technique, two simple examples are presented: spoiler design for the PRESS-based magnetic spectroscopic imaging (MRSI) and fast off-resonance calculation for dictionary building in Magnetic Resonance Fingerprinting (MRF).

In this work, we propose an open-source 3D Spatially-Resolved Phase Graph method (3D-SRPG

Under the assumption of piece-wise constant 3D gradients, each isochromat entering Bloch equation is treated as a delta function, and the magnetization vector in Fourier space is modelled as the sum of a finite number of delta functions. Rather than counting the phase paths explicitly, Green function is employed to model the signal evolution. A novel grid-based merging approach is introduced to effectively calculate the signal evolution of non-symmetrical pulse sequences. Isotropic diffusion, magnetization relaxation and global transport, can be considered. Besides, by utilizing the Discrete Fourier Transform (DFT), the relevant k-vector coherences can be transformed into the 3D spatial domain at any time point to represent the resulting signal distribution intuitively.

The schematic diagram of a typical PRESS sequence along with the conventional spoiler and the cutting-edge DOTCOPS

For analysis and illustration of the signal from VOI edges, α1 = 45°, α2 = 90°, and α3 = 90° with volume size (48 mm)

Since increment for k vectors in each phase-encoding step is equivalent to a shift of sampled k-space center, the whole set of phase-encoding steps could be converted into a matrix of the sampled k-space region (receive zone). Both all k-vectors and receive zone are presented in the 3D k-space maps, with projections onto the three orthogonal planes for k-vectors lying inside the corresponding receive zone. Besides, the images transformed from the receive-zone k-vectors via the DFT are also depicted.

Off-resonance simulations can be used to improve dictionary matching accuracy in MRF. In situations when the conventional EPG is not approptiate

Three MRF sequences were tested as reported by Assländer et al.

In the off-resonance MRF example, the magnetization response spectrum is shown for bSSFP, MRF and pSSFP patterns at TE (Fig.4). Similar as the conclusion drawn by Assländer et al., the pSSFP presents a better separation ability for different tissues and fewer magnetization oscillations around the central band for consecutive pulses (d and e) than MRF (b and c).

The proposed algorithm provides a pictorial representation of the signal evolution for a more flexible 3D sequence design framework. Beyond our simple examples, the proposed approach can be helpful in more advanced applications such as spoiler design for sophisticated NMR sequences or k-space trajectory optimization for suppression of eddy-current-induced artifacts in compressed sensing sequences.

1. Hennig J. Echoes—how to generate, recognize, use or avoid them in MR‐imaging sequences. Part I: Fundamental and not so fundamental properties of spin echoes. Concepts in Magnetic Resonance. 1991;3(3):125-43.

2. Kiselev V G. Calculation of diffusion effect for arbitrary pulse sequences. Journal of magnetic resonance. 2003;164(2):205-211.

3. Gao X, Kiselev V G, Zaitsev M. 3D Spatially-Resolved Phase Graph, https://github.com/euler990/3D-Spatially-Resolved-Phase-Graph.git

4. Ma D, Gulani V, Seiberlich N, et al. Magnetic resonance fingerprinting. Nature. 2013;495(7440):187.

5. Landheer K, Juchem C. Dephasing optimization through coherence order pathway selection (DOTCOPS) for improved crusher schemes in MR spectroscopy. Magnetic resonance in medicine. 2019;81(4):2209-22.

6. Assländer J, Glaser SJ, Hennig J. Pseudo steady‐state free precession for MR‐fingerprinting. Magnetic resonance in medicine. 2017;77(3):1151-61.

Fig.1. The
model PRESS sequence with conventional and DOTCOPS spoiler designs. Green
trapezoid present three orthogonal spatial selection gradients, while blue
present spoiler gradients. Static background gradient (G_{stat}) corresponds to a local gradient within the object for example due to shim imperfections.

Fig.2. The
original flip angle pattern of MRF^{4} is displayed on the top left, the pSSFP pattern
on top right. Constant TR and TE were used for original MRF
(bottom left) while TR and TE were varied for pSSFP (bottom right).

Fig.3. 3D
k-space and its corresponding images at Time 1 (TE1), Time 2 (TE1+TE2/2), Time 3
(TE=30ms), Time 4 (TE+20ms) and Time 5 (TE+40ms) created by PRESS-based MRSI with a static gradient (slice-selection gradient with a scaling of 1%, 2% and 3% on the three axes) under conventional (top) and DOTCOPS (bottom)
spoiler designs. The receive zone is indicated by black ‘·’ and whole k-vectors
by color-coded ‘*’. K-vectors are also projected onto three orthogonal
planes (signs ‘□’, ‘△’
and ‘☆’), and transformed into the image domain in
case the k-vectors lie within the receive zone.

Fig.4. The
signal magnitude plotted for a range of off-resonant frequencies. In a,
magnetization spectra are depicted for CSF, WM, GM, and fat in response to a
bSSFP sequence with 1000 RF pulses. Spectra resulting from the original MRF
sequence with 999 and 1000 pulses are depicted in (b) and (c), respectively,
while the response to the pSSFP sequence with 849 and 850 pulses is displayed
in (d) and (e).