MESS — multi-echo SSFP in Pulseq

Reference implementation of multi-echo steady-state free precession (MESS):

H. Hong, M. N. Hoff, W. Yang, Y. Dong, Z. Zhou, P. Hu. Multi-Echo SSFP: A Method for Synthetic bSSFP MRI Contrast Without Banding Artifacts at High Field. Magn Reson Med 2026;96(4):1696–1708. doi:10.1002/mrm.70472

A bSSFP image is the coherent sum of all SSFP coherence pathways — the dephasing orders $k$ — at one echo time, and their interference produces dark bands wherever the off-resonance is unfavourable. MESS refocuses the orders one after the other inside every TR and samples their echoes $S_k$ separately:

Top: prephaser, readout and rewinder of one TR with their moments in units of $u=N_\text{RO}/(2,\text{FOV}\text{RO})$, and the extended phase graph of the acquired pathways $F_2,\dots,F{-3}$; each forms its echo where it crosses zero inside the readout. Below: the image of every echo, magnitude $|S_k|$ (each on its own grey scale) and phase $\arg S_k$, placed under the echo it comes from (MR-zero simulation of a brain phantom, notebook 3). None of the echoes shows banding.

From one acquisition you can then

• sum the echoes coherently with a virtual phase-cycle increment $\Delta\phi$, $M_\Sigma(\Delta\phi)=\sum_k S_k e^{ik\Delta\phi}$, which reproduces RF phase-cycled bSSFP, bands included (Eq. 6 of the paper), or
• sum their magnitudes, $M_{\Sigma|\cdot|}(\Delta\phi)=\sum_k |S_k|,e^{i{k\Delta\phi+[u(k)-1]\pi}}$, which gives bSSFP-like contrast without banding, tunable after the scan with $\Delta\phi$ (Eq. 7).

Installation

pip install -e .              # the mess package: numpy + pypulseq
pip install -e ".[notebooks]" # + matplotlib, scipy, MRzeroCore, torch, jupyter

The sequences of the paper were built with PyPulseq-Matlab-like (Pulseq v1.5.1), whose make_slr_pulse provides the SLR excitation:

pip install git+https://github.com/m-a-x-i-m-z/pypulseq-matlab-like@master

With a PyPulseq that lacks make_slr_pulse, the protocols fall back to a sinc pulse and say so.

Quick start

import numpy as np
import mess

protocol = mess.MESS3D(orders=(2, -3), flip_angle=50)   # the 3D protocol of the paper
print(protocol)             # moments G1..G3, TR, TE_0, delta TE and every TE_k
protocol.make_sequence(phase_cycle=0.0).write("mess.seq")
protocol.bssfp().make_sequence(phase_cycle=np.pi).write("bssfp.seq")   # matched reference

After splitting every readout into its echoes and Fourier-transforming each of them (mess.split_echoes), the images S (orders along the first axis) are combined with

M_sigma = mess.complex_sum(S, protocol.orders, dphi=np.pi)     # Eq. 6
M_mag = mess.magnitude_sum(S, protocol.orders, dphi=np.pi)     # Eq. 7

Sequences for the scanner

python examples/write_3d_sequences.py    # seq/3d/: the 3D protocols used for scanning
python examples/write_2d_sequences.py    # seq/2d/: the 2D sequences simulated in the notebooks

• seq/3d/paper/ — the protocol of the paper: orders $[2,\dots,-3]$, 150 × 150 × 75 at 1.3 mm isotropic, flip angles 20°, 50°, 80°, $\Delta\psi=0$ and $\pi$.
• seq/3d/small_fov/ — 80 × 80 × 24 over 100 × 100 × 30 mm, 10°, 30°, 50°, with an external trigger on every readout.
• seq/2d/50deg/ — the six-echo MESS, the 24 phase-cycled bSSFP references and the FISP, PSIF, DESS and TESS sequences of the notebooks, byte for byte the files that MR-zero simulated.

Every MESS file comes with a bSSFP reference of identical TR, TE, RF pulse and encoding. The .seq files are generated, not tracked by git.

Arbitrary dephasing orders (Algorithm 1)

mess.readout_moments(k1, kN) returns the zeroth moments $(G_1, G_2, G_3)$ of prephaser, readout and rewinder, in units of $u=N_\text{RO}/(2,\text{FOV}_\text{RO})$, that acquire any contiguous list of orders $[k_1,\dots,k_N]$. mess.plotting.readout_figure draws them as in Fig. 1 of the paper (notebook 1):

Pathway $F_k$ enters the TR with the dephasing $k,(G_1+G_2+G_3)$ and leaves it as $F_{k+1}$. It forms Echo $k$ where it crosses zero inside the readout lobe, i.e. where the moment accumulated since the RF pulse equals $-k,(G_1+G_2+G_3)$, at $\text{TE}_k=\text{TE}_0-k,\Delta\text{TE}$. Reversing the list flips the sign of the net moment and reads the same pathways out in the opposite order.

sequence	orders	$(G_1, G_2, G_3)$
bSSFP	all, coherently	$(-1, 2, -1)$
FISP	$[0]$	$(-1, 2, 1)$
PSIF	$[-1]$	$(1, 2, -1)$
DESS	$[0,-1]$ / $[-1,0]$	$(-1, 4, -1)$ / $(-3, 4, -3)$
DESS	$[0,1]$ / $[1,0]$	$(-1, 4, -5)$ / $(-3, 4, 1)$
TESS	$[1,0,-1]$	$(-3, 6, -1)$
QESS	$[1,0,-1,-2]$	$(-3, 8, -3)$
MESS	$[2,1,0,-1,-2,-3]$	$(-5, 12, -5)$

Tutorials

The notebooks simulate the sequences with MR-zero, which reads the same .seq files as the scanner. Everything is written in the notation of the paper.

notebook	content
01_readout_design	Algorithm 1, the readout diagrams of Fig. 1 (bSSFP, DESS, TESS, QESS, MESS), the Pulseq sequence and the 3D protocol of the paper — no simulator needed
02_steady_state	how many dummy TRs the steady state needs, and what a flip-angle ramp changes
03_echo_orders	the six echoes against the signal model (Eqs. 3, 4); one acquisition contains FISP, PSIF, DESS and TESS
04_complex_sum	RF phase-cycled bSSFP versus the virtually phase-cycled complex sum (animation), band spacing, number of orders
05_magnitude_sum	banding-free magnitude sum (animation), retrospective contrast tuning, six orders versus DESS

The simulations take about 45 minutes on a laptop GPU the first time (most of it the 24 phase-cycled bSSFP scans of notebook 4) and are cached in notebooks/.cache/ afterwards. Run the notebooks one at a time: mess.simulation.simulate runs a single MR-zero simulation at a time.

Repository

mess/
  orders.py       Algorithm 1 and the echo condition
  sequence.py     MESS2D / MESS3D protocols, their bSSFP reference, flip-angle ramps
  recon.py        echo splitting, complex sum (Eq. 6), magnitude sum (Eq. 7)
  theory.py       steady-state signal model (Eqs. 1-5, Supporting Information S1-S4)
  simulation.py   MR-zero phantoms and simulation (notebooks)
  plotting.py     readout diagrams (Fig. 1), figures and animations (notebooks)
examples/         scripts that write the .seq files for the scanner
seq/              the written sequences (2d/, 3d/; generated, not tracked)
notebooks/        tutorials; figures/ holds the images of this README, written by the notebooks
tests/            python -m pytest

Notation

paper	code
dephasing order $k$, acquired orders $[k_1,\dots,k_N]$	protocol.orders
readout moments $G_1, G_2, G_3$	mess.readout_moments, protocol.moments
$\text{TE}_k$, $\text{TE}_0$, $\Delta\text{TE}$	protocol.echo_times, protocol.te0, protocol.delta_te
RF phase-cycle increment $\Delta\psi$	make_sequence(phase_cycle=...) [rad]
virtual phase-cycle increment $\Delta\phi$	complex_sum(..., dphi), magnitude_sum(..., dphi) [rad]
$F_k^+$ (Eq. 1), $M_{xy}^+(\theta)$ (S1)	theory.epg_coefficient, theory.bssfp_profile
$S_k$ (Eq. 3), $M_\text{bSSFP}$ (Eq. 5)	theory.echo_signal, theory.m_bssfp
$M_\Sigma$ (Eq. 6), $M_{\Sigma\lvert\cdot\rvert}$ (Eq. 7)	mess.complex_sum, mess.magnitude_sum
$\lambda_\pm, \mu_\pm$ (S3.3), closed forms S4.2–S4.4	theory.exponential_coefficients, theory.m_infinity

Flip angles are in degrees, all phases in radians.

Coming from the first version

before	now
from MESS import MESS_3D	import mess
MESS_3D(FA=50, num_RO=150, num_PE=150, num_SPE=75, fov_RO=..., ...)	mess.MESS3D(flip_angle=50, matrix=(150, 150, 75), fov=(...))
make_sequence(2, -3, phase_cycle=180) (degrees)	MESS3D(orders=(2, -3), ...).make_sequence(phase_cycle=np.pi) (radians)
make_sequence(0, 0, balance=True, delay_pre=..., delay_post=...)	protocol.bssfp().make_sequence(...): TR and TE matched automatically
FLAG_min_TR=True	tr=None (the default); tr=... and te=... set TR and $\text{TE}_0$
arbitarySSFP(start, end) → (a, b, c)	mess.readout_moments(k1, kN) → (G1, G2, G3)

The readout and the timing of the protocols are unchanged. Two details were corrected: the phase-encoding tables are now the standard Cartesian (i - N//2) / FOV (before, linspace made the encoded FOV of the paper protocol 0.7 % too small along phase encoding and 1.3 % along the partitions), and the slab gradient is rewound exactly around the RF centre.

Citation

@article{hong2026mess,
  author  = {Hong, Haotian and Hoff, Michael Nicholas and Yang, Wenchao and Dong, Yiyun and Zhou, Zijian and Hu, Peng},
  title   = {Multi-Echo {SSFP}: A Method for Synthetic {bSSFP} {MRI} Contrast Without Banding Artifacts at High Field},
  journal = {Magnetic Resonance in Medicine},
  year    = {2026},
  volume  = {96},
  number  = {4},
  pages   = {1696--1708},
  doi     = {10.1002/mrm.70472}
}

MIT License.