MR‐zero meets FLASH – controlling the transient signal decay in gradient‐ and RF‐spoiled gradient echo sequences

Abstract

Purpose

The complex signal decay during the transient FLASH MRI readout can lead to artifacts in magnitude and phase images. We show that target‐driven optimization of individual RF flip angles and phases can realize near‐ideal signal behavior and mitigate artifacts.

Methods

The differentiable end‐to‐end optimization framework MR‐zero is used to optimize RF trains of the FLASH sequence. We focus herein on minimizing deviations from the ideally spoiled signal by using a mono‐exponential Look–Locker target. We first obtain the transient FLASH signal decay substructure, and then minimize the deviation to the Look–Locker decay by optimizing the individual (i) flip angles, (ii) RF phases, and (iii) flip angles and RF phases. Comparison between measurement and simulation is performed using Pulseq in 1D and 2D.

Results

We were able to reproduce the complex substructure of the transient FLASH signal decay. All three optimization objectives can bring the real FLASH signal closer to the ideal case, with best results when both flip angles and RF phases are adjusted jointly. This solution outperformed all tested conventional quadratic RF cyclings in terms of (i) matching the Look–Locker target signal, (ii) phase stability, (iii) point spread functions ideality, (iv) robustness against parameter changes, and (v) magnitude and phase image quality. Other target functions for the signal could as well be realized, yet their response is not as general as for the Look–Locker target and needs to be optimized for a specific context.

Conclusion

Individual flip angle and RF phase optimization improves the transient signal decay of FLASH MRI sequences.

1. INTRODUCTION

The FLASH sequence—or gradient‐ and RF‐spoiled gradient echo sequence—is one of the major sequences in clinical applications and was one breakthrough that made fast MRI clinically feasible. ¹ , ² , ³ , ⁴ , ⁵ In steady‐state mode, a FLASH sequence provides T₁ or susceptibility weighted imaging. In transient‐state mode, FLASH is especially relevant in magnetization‐prepared versions like MPRAGE sequence, which provide unique strong gray matter (GM)/white matter (WM) contrast. ⁶ In addition, magnetization transfer contrast or CEST sequences can be realized with prepared transient FLASH sequences. ⁷ , ⁸ , ⁹ , ¹⁰ Because the prepared initial state provides the contrast, the transient echo trains are limited in their length, which either leads to small volumes ¹ , ⁷ or requires multi‐shot mode to fill larger k‐spaces.

A common problem in transient FLASH is that the decay of magnetization still affects quality of the image because the decay in k‐space acquisition leads to ringing or blurring artifacts; moreover, the prepared magnetization state is decaying during acquisition. The decay to the FLASH steady‐state signal ($S_{\mathrm{SS}}$) is described by the Look–Locker decay rate given by the joint effects of T₁ and excitation and reads ¹¹ , ¹² :

$$S_{\mathrm{LL}}(n·\mathrm{TR})=\left(S_i-S_{\mathrm{SS}}\right)·\exp \left(-\frac{n·\mathrm{TR}}{{\mathrm{T}}_{1,\mathrm{LL}}}\right)+S_{\mathrm{SS}},$$ (1)

with

$$\frac{1}{{\mathrm{T}}_{1,\mathrm{LL}}}=\frac{1}{{\mathrm{T}}_1}-\frac{\ln (\cos \alpha )}{\mathrm{TR}}.$$ (2)

$S_i$ is the initial signal and $n$ is the number of repetitions.

Interestingly, this mono‐exponential reflects an idealized description in the case of ideal spoiling. In real measurements, the signal decay coarsely follows this exponential function but has complex substructure that depends, among other factors, on the specific RF‐cycling. ⁴ The familiar Look–Locker decay expression is only valid for ${\mathrm{T}}_2\to 0.$ ¹² Moreover, this decay is also only observed for constant flip angles, and the decay can effectively be altered by so‐called variable flip angle approaches. ¹³ , ¹⁴ By changing the flip angles and phases of each RF pulse in the train, the intrinsic transient decay substructure can be modified.

In transient FLASH sequences, this possibility was mainly used for two objectives: (i) to realize the best approximation of ideal spoiling, meaning approaching the Look–Locker mono‐exponential decay, or (ii) to realize a specific target signal decay and thereby improved point spread functions (PSF) and image quality. We want to herein revisit objective (i) using the end‐to‐end optimization framework MR‐zero. ¹⁵

MR‐zero consists of a MR simulation, based on phase distribution graphs (PDG), ¹⁶ which accurately describes the signal behavior of arbitrary MR sequences, allowing both encoded and unencoded signal analysis. The simulation considers isotropic diffusion because diffusion can have an influence on the spoiling process. ¹⁷ , ¹⁸ In contrast to extended phase graphs, ¹⁹ , ²⁰ , ²¹ , ²² , ²³ which can only describe amplitude and phase of the echo top, phase distribution graphs can describe the whole echo shape by extending dephased magnetization and arbitrary timing. This allows analysis of how these signal variations affect the MR image in magnitude and phase. The MR‐zero implementation is written in PyTorch and supports version 1.9 and higher, ²⁴ which provides full differentiability via auto‐differentiation in all input parameters, allowing gradient descent optimization of all individual sequence events, for example, here each individual RF pulse angle and RF pulse phase, when providing a target image or signal. Thus, flexible flip angles and RF phases beyond constant trains and quadratic RF cycling can be investigated by simulating MR images.

Specifically, in this paper we want to answer the following question: Can we reproduce the transient decay substructure of the FLASH signal, understand its effect in the image domain, and further improve it using MR‐zero?

2. METHODS

As a first step, this study compares the MR‐zero simulation with measurements and results from Epstein et al. ⁴ These experiments were performed without phase‐encoding gradients.

In a second step, we use MR‐zero to optimize the RF trains with the ideally spoiled unencoded signal as target and then compare the new strategies to existing ones.

In a third step, we investigate how the performance of the different found RF trains translates to 2D imaging in simulation and in vivo. The optimized trains are therefore used in a centric‐reordered 2D FLASH readout scheme. As an outlook, we also show optimizations using other target signals, namely, a constant signal target and a Hanning‐shaped signal target in Figure S3.

2.1. MR sequence

2.1.1. Unencoded measurements

The sequence parameters as described by Epstein et al. ⁴ are employed for both simulation and measurement:  α = 10°, TR = 10 ms, TE = 5 ms, readout bandwidth = 217 Hz/pixel, FOV = 200 mm, slice thickness = 10 mm, matrix = 128 × 128, RF spoiling with a quadratic increment of $\mathrm{\Psi }=117°$ $\mathrm{\Psi }=84°$ $\mathrm{\Psi }=50°$, as well as gradient spoiling. We used a saturation preparation with recovery time of 970 ms, whereas Epstein ⁴ used in his experiments an inversion pulse resulting in similar prepared magnetization. The unencoded signal (k_x = 0, k_y = 0) is given by the center ADC signal at the echo top where k_x = 0 and, because all phase encoding gradients were switched off, also k_y = 0.

$\mathrm{\Psi }=50°$ RF increment was added because some vendors use it and papers recommend it, ²⁵ , ²⁶ , ²⁷ and Preibisch et al. suggest it, at least for robust T₁ mapping. ²⁸ In order to enable the comparison between measurements and simulations, an additional normalization measurement without preparation was performed with adequate relaxation time and used for signal normalization purposes. As excitation pulse, a slice‐selective sinc‐shaped pulse with apodization of 0.5 and time bandwidth product of 4 was used. The duration of the pulse was 1 ms.

Repeating the sequence of these unencoded experiments with phase encoding gradients and image reconstruction can be found in the in Figure S1.

2.1.2. Encoded measurements

To make the existing signal fluctuations more visible in an image, we decreased TE and increased the flip angle. To achieve shorter TE and TR of TE = 3.2 ms and TR = 6 ms, the readout bandwidth was increased to 500 Hz/pixel. The excitation flip angle was increased to α = 19.5°, and no magnetization preparation before image acquisition was used. All other parameters were chosen equally to the unencoded experiment, but phase encoding was switched on and all ADC samples were used in the reconstruction. A centric reordering for acquisition was used. Images with higher resolution of 256 were measured with the same sequence parameter using two shots per image acquisition and the identical RF train for each shot.

2.2. Simulation

All described sequences were simulated and also optimized in the MR‐zero framework, which consists of a phase distribution graph simulation ¹⁶ written in PyTorch ²⁴ that provides full differentiability in all input parameters, allowing gradient descent optimization via auto‐differentiation. The end‐to‐end training procedure used the ADAM optimizer. ²⁹ The approach extracts additionally useful information about the magnetization evolution such as the latent signal evolution within a sequence. Encoded and unencoded simulation can be performed. Using the Pulseq standard, ³⁰ , ³¹ , ³² sequences are directly transferable to a real scanner.

The simulation requires a 3D object defining the MR parameters (proton density, T₁, T₂, T₂′, relative B₁ (rB₁), ΔB₀, and the isotropic diffusion constant D) to simulate the MR signal. Two different in silico phantoms are used within this paper: a single voxel phantom and a whole brain phantom. The brain phantom is built from the data provided by the BrainWeb database. ³³ Segmented maps for GM, WM, and CSF are filled with values for the various physical properties. ³⁴ , ³⁵ , ³⁶ Then they are summed together, weighted by their contribution to every voxel as given by the BrainWeb data, resulting in the input maps used by the simulation. ΔB₀ (−10.5 Hz—44.4 Hz) and rB₁ (0.84—1.14) inhomogeneities were added by using Normal and Cauchy distributions coarsely matching in vivo data. The single voxel phantom is a 3D phantom with one non‐zero voxel in the center with one set of MR parameters. Exclusively, this single voxel phantom is used for all optimization processes in this paper with proton density set to 1 and T₂′ to 30 ms. For optimization T₁, T₂ and D of the single voxel phantom were adjusted for the two kind of sequences (α = 10°/19.5°) described more in detail in the unencoded and encoded measurement section above. For optimization tasks of encoded experiments (α = 19.5°), one voxel was used with parameters matching the median of the brain phantom (T₁ ≈ 1.5 s, T₂ ≈ 0.11 s, D ≈ 0.84·10⁻³ mm²/s). For the optimization tasks of unencoded in vitro experiments (α = 10°), the optimization phantom consists of a filled voxel, where phantom parameters were gained by a signal fit of quantitative measurements of the phantom measurement (T₁ ≈ 1.0 s, T₂ ≈ 0.17 s, D ≈ 1.6·10⁻³ mm²/s) The used in vitro phantom is described in the measurement section below.

2.3. Experiments/Optimization Objectives

The ideally spoiled signal is for constant flip angles identical to the Look–Locker decay, which we can explicitly generate with phase distribution graph simulation. In the following, $S_{\mathrm{LL}}$ is the ideal Look‐Locker signal for a constant excitation flip angle α = 10° for comparison with Epstein and 19.5° for the image quality experiment. For $S_{\mathrm{LL}}$, the phase cycling is irrelevant.

Task 1: ($\overrightarrow{\alpha }\mid S_{\mathit{LL}};\mathrm{\Psi }=117°$). Optimizing the flip angles $\overrightarrow{\alpha }$ only, using $S_{\mathrm{LL}}$ as a target, while quadratic phase cycling with $\mathrm{\Psi }=117°$ is preserved. The objective function of this is

$${\overrightarrow{\alpha }}_{\mathrm{opt}}={\text{argmin}}_{(\overrightarrow{\alpha })}\left(‖S(\overrightarrow{\alpha })-S_{\mathrm{LL}}‖\right)\mathrm{s}.\mathrm{t}.\mathrm{\Psi }=117°.$$ (3)

Task 2: ($\overrightarrow{\phi }|S_{\mathrm{LL}};\alpha =10°⁄19.5°$). Optimizing the RF phases $\overrightarrow{\phi }$ only, using $S_{\mathrm{LL}}$ as a target, while the flip angle α = 10°/19.5° is preserved. The objective function of this is

$${\overrightarrow{\phi }}_{\mathrm{opt}}={\text{argmin}}_{(\overrightarrow{\phi })}\left(‖S(\overrightarrow{\phi })-S_{\mathrm{LL}}‖\right)\mathrm{s}.\mathrm{t}.\alpha =10°⁄19.5°.$$ (4)

Task 3: ($\overrightarrow{\alpha },\overrightarrow{\phi }|S_{\mathrm{LL}}$). Optimizing both flip angles $\overrightarrow{\alpha }$ and RF phases $\overrightarrow{\phi }$, using $S_{\mathrm{LL}}$ as a target. The objective function of this is

$${\overrightarrow{\alpha }}_{\mathrm{opt}},{\overrightarrow{\phi }}_{\mathrm{opt}}={\text{argmin}}_{(\overrightarrow{\alpha },\overrightarrow{\phi })}\left(‖S(\overrightarrow{\alpha },\overrightarrow{\phi })-S_{\mathrm{LL}}‖\right).$$ (5)

Thus, each task aims for the smallest deviation from the ideally spoiled signal $S_{\mathrm{LL}}$. All other parameters of the sequence, such as timing and bandwidth, are kept fixed; thus, if $\overrightarrow{\phi }$ is not optimized, the quadratically growing RF phases of each pulse is used with $\mathrm{\Psi }=117°$.

As loss function $\perp (\overrightarrow{x},\overrightarrow{y})$, we used the so‐called perpendicular loss proposed for complex data by Terpstra et al. ³⁷ It shows a symmetric loss landscape, achieving better performance and faster convergence. This loss is defined as

$$\perp (\overrightarrow{x},\overrightarrow{y})=\mathrm{P}(\overrightarrow{x},\overrightarrow{y})+{\ell }^1(\left|\overrightarrow{x}\right|,\left|\overrightarrow{y}\right|),$$ (6)

$$\mathrm{P}(\overrightarrow{x},\overrightarrow{y})=\frac{|\mathrm{Re}(x)\mathrm{Im}(y)-\mathrm{Im}(x)\mathrm{Re}(y)|}{\left|\overrightarrow{y}\right|}.$$ (7)

2.4. Measurement

All measurements were performed at a 3 T Magnetom Prisma scanner (Siemens Healthcare, Erlangen) using a 20‐channel Rx head coil. The used scanner shows a significant B0‐drift from one measurement to another. In order to obtain a match between measurement and simulation, the simulated phase was shifted by the amount of the first measured value so that the first values in simulation and measurement match. This shift differs from one measurement to another.

For in vitro measurements, an agar‐based phantom was created consisting of a 50 mL falcon tube. 13.5 mg copper sulfate was used to control T₁ and T₂ relaxation value, and 281.1 mg agar was added to avoid convection. Measurements of this tube phantom were compared with unencoded simulation where the tube is approximated by a digital phantom consisting of one with phantom parameter–filled voxel. To facilitate a comprehensive comparison between measurement and simulation, phantom parameters were gained by a signal fit of quantitative measurements (T₁ ≈ 1.0 s, T₂ ≈ 0.17 s, D ≈ 1.6·10⁻³ mm²/s). The slice selective pulse efficiency was measured with a standard actual flip angle method resulting in a relative B₁ value of approximately 0.77, which was used to match simulation and experiment flip angles.

Three healthy subjects (m, 23; m, 31; and w, 23) were scanned within this paper after written informed consent and approved by the local ethics committee. In vivo measurements correspond accordingly to the simulation with the data generated with BrainWeb. ³³

3. RESULTS

In a first step, we obtain the finding of Epstein et al., ⁴ who first demonstrated that even with gradient‐ and RF‐spoiling the transient signal decay deviates from an ideally spoiled exponential Look–Locker decay. In Figure 1, we show that we find similar deviations from the ideal Look–Locker decay as Epstein et al., both experimentally as well as by using the phase distribution graph simulation of MR‐zero. The high level of agreement between the experimental data and simulation let us conclude that our simulation framework is able to describe the FLASH signal realistically, and generalization to variable flip angles and altered RF cycling can be expected.

FIGURE 1.

Unencoded FLASH signal for α = 10°, TE = 5 ms, and TR = 10 ms after one 90° preparation pulse with a recovery time of 970 ms. Measured and simulated signal magnitude (A–C) and signal phase (D–F) for $\mathrm{\Psi }=117°$ (A, D), $\mathrm{\Psi }=84°$ (B, E), and $\mathrm{\Psi }=50°$ (C, F). Additionally, the ideally spoiled signal, which is here the Look‐Locker signal $S_{\mathrm{LL}}$, is plotted (gray solid line). All results show a good agreement between measurement and simulation, and the respective plots also agree with the results of Epstein et al., ⁴ where similar deviations from the ideal Look–Locker decay were found. T₁ = 0.987 s and T₂ = 0.167 s were used in simulation.

In the next step, we want to use the MR‐zero framework to optimize individual flip angles and RF phases so that the transient FLASH decay matches the ideal Look‐Locker signal $S_{\mathrm{LL}}$. As described above, we tested three different objectives. Task 1 optimize only the flip angles $\overrightarrow{\alpha }$ for a quadratic phase cycling with increment $\mathrm{\Psi }=117°$. Task 2 optimizes each individual phase of RF phase angle train $\overrightarrow{\phi }$, meaning we explicitly leave the condition of quadratically increasing phases. Task 3 optimizes both ($\overrightarrow{\alpha }$, $\overrightarrow{\phi }$).

Figure 2 depicts the evaluations compared to the ideal mono exponential Look‐Locker decay $S_{\mathrm{LL}}$ (column 2) and to a constant phase (column 3). The first column shows the optimized flip angle and RF phase trains for all three tasks. The overall result of Figure 2 is that for all tasks we get closer to the target signal upon optimization. Task 1 ($\overrightarrow{\alpha }$) leads to signal close to the target, but no constant phase. Task 2 ($\overrightarrow{\phi }$) leads to a constant phase while improving the signal with a RMS error (RMSE) similar to a RF cycling of $\mathrm{\Psi }=84°$. Task 3 ($\overrightarrow{\alpha }$, $\overrightarrow{\phi }$) leads to the best results regarding target signal and constant phase. The last task was initialized with the phases found in the second task; thus, similar phases and only small flip angle variations are observed here. This can also be seen quantitatively in Table 1A, where the RMSE of measurement and simulation was calculated.

FIGURE 2.

The flip angles $\overrightarrow{\alpha }$ and RF phases $\overrightarrow{\phi }$ are shown for the three optimization cases (A) task 1 $(\overrightarrow{\alpha })$, (D) task 2 $(\overrightarrow{\phi })$, and (G) task 3 $(\overrightarrow{\alpha },\overrightarrow{\phi })$. The measured signal magnitude evaluations are shown in (B), (E), and (H), respectively, for the three tasks. Additionally, the ideally spoiled signal $S_{\mathrm{LL}}$ is plotted (gray solid line). The same comparison is done in (C), (F), and (H) for the signal phase. Whereas the flip angle optimization adjusts only the signal amplitude, additional RF phase optimization can also alter the signal phase. The best results are achieved with a combined approach, see Table 1B.

TABLE 1.

(A) RMSE with regard to the ideally spoiled Look‐Locker signal $S_{\mathrm{LL}}$, calculated for Meas and Sim signal magnitude (left value) and phase (right value) for $\mathrm{\Psi }=117°,84°$, and $50°$, and the optimized RF trains of task 1–3. For the RMSE of the PSF, just the absolute signal is considered for calculation. The underlying signals are shown in Figures 1 and 2. The corresponding normalized PSF is displayed in Figure 3. (B) RMSE with regard to $S_{\mathrm{LL}}$, calculated for system parameter changes (T₂, T₁, MI, and rB₁) for typical RF cycling of 117°, 84°, and 50°, and the optimization tasks 1–3 corresponding to the signals shown in Figures 5 and 6. The value in the left column is the error of the signal magnitude; the right column is the error of the signal phase.

	$\mathrm{\Psi }=117°$		$\mathrm{\Psi }=84°$		$\mathrm{\Psi }=50°$		Task 1		Task 2		Task 3
(A)	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase
Meas	12.0·1e‐3	2.23°	5.4·1e‐3	0.68°	7.0·1e‐3	0.93°	2.6·1e‐3	2.62°	5.8·1e‐3	0.35°	3.0·1e‐3	0.53°
Sim	11.5·1e‐3	2.02°	5.5·1e‐3	0.28°	6.6·1e‐3	0.49°	2.2·1e‐3	1.99°	5.2·1e‐3	0.21°	1·1e‐3	0.12°
PSF Meas	15.0·1e‐4		6.3·1e‐4		7.6·1e‐4		14.0·1e‐4		5.7·1e‐4		4.8·1e‐4
PSF Sim	15.0·1e‐4		5.8·1e‐4		6.7·1e‐4		14.0·1e‐4		5.4·1e‐4		1.6·1e‐4
	$\mathrm{\Psi }=117°$		$\mathrm{\Psi }=84°$		$\mathrm{\Psi }=50°$		Task 1		Task 2		Task 3
(B)	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase
T2	17.9·1e‐4	2.13°	9.2·1e‐4	0.34°	11.0·1e‐4	0.68°	6.3·1e‐4	2.11°	9.1·1e‐4	0.41°	4.8·1e‐4	0.33°
T1	16.2·1e‐4	2.06°	7.9·1e‐4	0.31°	9.2·1e‐4	0.50°	3.9·1e‐4	2.05°	7.2·1e‐4	0.22°	1.4·1e‐4	0.14°
MI	15.0·1e‐4	13.03°	6.9·1e‐4	2.66°	8.2·1e‐4	4.04°	9.2·1e‐4	12.03°	5.9·1e‐4	8.14°	2.3·1e‐4	5.57°
rB1	58.7·1e‐4	6.17°	28.2·1e‐4	0.89°	31.6·1e‐4	1.65°	45.6·1e‐4	5.98°	27.2·1e‐4	0.76°	21.4·1e‐4	0.58°
Total	27.0·1e‐4	5.85°	13.1·1e‐4	1.05°	15.0·1e‐4	1.72°	16.3·1e‐4	5.54°	12.4·1e‐4	2.38°	7.5·1e‐4	1.66°

Meas, measured; MI, initial magnetization; PSF, point spread function; RMSE, RMS error; Sim, simulated; rB₁, relative B₁.

The PSFs of the different RF increments and optimization tasks, including the difference to the ideal PSF, are shown in Figure 3. The PSFs of typical quadratic phase cycling approaches show deviations from the ideal PSF. Of the three tested $\mathrm{\Psi }$, a RF cycling increment of $\mathrm{\Psi }=84°$ performs best with regard to RMSE. Yet, the optimization tasks can further improve the signal. Especially with phase optimization, we get closer to an ideal PSF upon optimization. Task 1 slightly improves the PSF. However, the nonconstant signal phase leads to deviations in the PSF. In contrast, tasks 2 and 3 improve signal phase and PSF. Overall, task 3 leads to the best results regarding target signal, constant phase, and therefore also PSF.

FIGURE 3.

Magnitude of PSFs for typical quadratic phase cycling of $\mathrm{\Psi }=117°$ (A–C), $84°$ (D–F), and $50°$ (G–I). The PSFs of the optimization tasks are visualized in (J–L) for task 1, in (M–O) for task 2, and in (P–R) for the last optimization task 3. The measurement data is shown in the top row, the simulation data in the middle row, and the difference between an ideal PSF (gray solid line) to the measured PSF is shown in the bottom row. A closer agreement to the PSF of an ideally spoiled FID can be seen for the optimization tasks, especially when the phases of the pulses are optimized. The best result regarding PSF are achieved by optimization task 3 according to the quantitative results in Table 1A. PSF, point spread function.

The found strategies in tasks 2 and 3 outperform typical quadratic phase‐cycling approaches, which underline that an individual pulse optimization approach can be beneficial even for FLASH.

Before we test the stability of the found solutions against system changes, let us first investigate the underlying strategies. For this, the phase graph analysis provided by the τ‐dephasing latent signal plot of phase distribution graphs is informative (Figure 4). ¹⁶ The latent signal provides insight, which echoes can contribute to the signal in the current or any later repetition. For a gradient‐spoiled FLASH, this would be rephased echoes restored back into the z‐magnetization. The ideal FLASH/Look–Locker signal, consisting only of FIDs, would correspond to only one line at $\tau =0$. Thus, every intensity at $\tau \ne 0$ reflects unwanted signals that can interfere with each other, especially with the FID signal. These originate from higher and higher echoes given by $\tau$, which indicate the dephasing time of transverse magnetization.

FIGURE 4.

Latent signal for FLASH for different RF pulse trains. (A) RF increment of $\mathrm{\Psi }=117°$, (B) $\mathrm{\Psi }=84°$ (B), and (C) $\mathrm{\Psi }=50°$. Optimization task 1–3 are shown in (D, E, F), respectively. Signals below 10⁻⁶ are not considered because their impact to the final signal is negligible. The flip angle optimization task 1 does not affect the structure of the diagram, whereas tasks 2 and 3 do. More and higher dephased and restored echoes contribute to the signal and are the origin of altered signal magnitude and phase.

Small differences already can be observed in the latent signal plots for different quadratic RF spoiling of 117°, 84°, and 50° in Figure 4A–C, with 50° and 84° appearing smoother along the repetition dimension. The latent signal plots of the optimized sequence are shown Figure 4D–F. The flip angle optimization task 1 does not change the structure of the diagram; thus, the higher echo contribution is similar—most probably, the large FID signal is modulated by the flip angle change to counteract restored z‐magnetization. For the latter two tasks, including RF spoiling optimization, the latent signal plots change quite significantly. More and stronger dephased paths contribute to the final signal. Thus, the underlying strategy is most probably a tailored destructive and constructive interference of a higher number of different signals, which leads to the target signal.

Let us now analyze the stability of the observed behavior when changing system parameters, namely, the relaxation times T₁ and T₂, the initial magnetization before the readout MI, and the B1 inhomogeneity rB1. The conventional quadratic RF cycling approaches show no surprises against these system changes (Figure 5). In principle, the same deviations as above are observed, only their magnitude is affected, mostly by T₂ and rB₁. As previously, the $\mathrm{\Psi }=117°$ shows the strongest deviations with regard to the exponential decay of $S_{\mathrm{LL}}$.

FIGURE 5.

Simulation for different T₂ values (a–d), T₁ values (e–h), MI values (i–l), and rB₁ values (m–p) of the signal magnitude (A) and signal phase (B) for $S_{\mathrm{LL}}$, representing a perfect spoiled sequence, and RF cycling of 117°, 84°, and 50°. Only one parameter was varied at a time, whereas the other parameters were kept constant. The yellow curve represents the same curve for the parameters of the tube phantom (T₂ ≈ 0.16 s, T₁ ≈ 1.0 s, MI = 1, rB₁ ≈ 0.77) in all plots. RMSE between system parameters and perfect spoiled FID sequence are shown in Table 1B. RMSE, RMS error.

The stability‐against‐system parameter changes are more diverse for the optimized FLASH signal (Figure 6). Most problematic is the behavior for task 1 ($\overrightarrow{\alpha }|S_{\mathrm{LL}};\mathrm{\Psi }=117°$); the signal patterns appear smooth, but deviations grow when leaving the system parameters of the optimization. More severely, the signal modulation appears inverted for T₁, T₂, MI, or rB₁ below or above the initial value (yellow line in all plots). This fits to our previous interpretation that the flip angle alters the FID signal to counteract the restored signals. But, if the restored signal composition now changes due to the system parameters, this compensation is unbalanced and turns out to be not robust.

FIGURE 6.

Simulation for different T₂ values (a–d), T₁ values (e–h), MI values (i–l), and rB₁ values (m–p) of the signal magnitude (A) and signal phase (B) for $S_{\mathrm{LL}}$ signal, representing a perfect spoiled sequence and optimization tasks 1–3. Only one parameter was varied at a time, whereas the other parameters were kept constant. The yellow curve represents the same curve for the parameters of the tube phantom (T₂ ≈ 0.16 s, T₁ ≈ 1.0 s, MI = 1, rB₁ ≈ 0.77) in all plots. RMSE between system parameters and perfect spoiled FID sequence are shown in Table 1B. MI, initial magnetization.

Most interestingly, tasks 2 ($\overrightarrow{\phi }|S_{\mathrm{LL}};\alpha =10°$) and task 3 ($\overrightarrow{\alpha }$, $\overrightarrow{\phi }|S_{\mathrm{LL}}$) also show here the best performance and robustness. Similar as for conventional RF cyclings, these optimized cases also behave even more robustly against the introduced changes and lead to signal and phase most similar to ideal Look‐Locker signal in all cases. Interestingly, task 3 outperforms the pure RF phase optimization (especially visible in the smoother signal magnitude in Figure 6A (d, h, l, p)), despite the fluctuations observed for the pure FA optimization. However, this is mostly due to the much smaller flip angle variation in task 3 compared to task 1.

To summarize our results: All three objectives can bring the real FLASH signal closer to the ideal case, but pure flip angle approaches are less robust against parameter changes. Best results are achieved when the phase cycling is optimized first, and then the flip angle pattern is slightly adjusted jointly with the phases. This solution also outperformed all tested conventional RF cyclings in terms of (i) matching the target signal, (ii) phase stability, (iii) PSF ideality, and (iv) robustness against parameter changes, as quantified by the average RMSE given in Table 1B.

In a last step, we investigate the influence of the different RF trains on image quality. For the chosen sequence parameters TE = 5.0 ms, TR = 10 ms, bandwidth of 500 Hz/pixel, and FA α = 10.0°, the effect is less visible. Susceptibility and B0 inhomogeneity effects are in the same order (data shown in Figure S1). Therefore, an adaptation of the sequence was performed by reducing TE and TR and increasing the bandwidth and FA α. The optimization was performed for the same three tasks as in the first part but employing a voxel phantom with median phantom parameters (T₁ ≈ 1.5 s, T₂ ≈ 0.11 s, D ≈ 0.84·10⁻³ mm²/s). In this case, measurements were taken for both the unencoded sequence and a centric reordered encoded sequence. The FLASH magnitude and phase images can be seen in Figure 7 for simulation (Figure 7A,B) and measurement (Figure 7C,D). The RF increment of 117° produces clear artifacts in the magnitude image, as well as in the phase image, clearly visible in the difference images. This is reduced especially by tasks 2 and 3 but not task 1. This is in line with the 1D result that flip angle optimizations are not robust against system parameter changes.

FIGURE 7.

Simulation (A, B) and measurement (C, D) of centric‐encoded FLASH readouts using α = 19.5°, TE = 3.2 ms, and TR = 6 ms for typical RF increments of 117° (b), 84° (c), and 50° (d); and optimization task 1 (e), task 2 (f), and task 3 (g). In simulation, the ideal spoiled Look–Locker image $S_{\mathrm{LL}}$ was additionally calculated. The phantom for simulation was built from data provided by the BrainWeb database. There, differences of signal magnitude S (Ah‐m) and phase arg(S) (Bh‐m) are calculated for each image with respect to the Look–Locker target (a). For the magnitude images (Cb‐g) and phase images (Db‐g), task 3 is used as reference (g) in measurement because it shows the smoothest phase image indicating the best spoiling. This is also validated by the smoothest phase image in the simulation. An overall good agreement between simulation and measurement can be seen. Artifacts in the phase image can be seen for all standard RF increments, especially for 117°. Task 3 can reduce these artifacts coming from the RF cycling and generate the smoothest phase image. In Figure S1, the images corresponding to the unencoded experiment show similar deviations, but less intense, especially in phase images due to overlaying susceptibilities and B₀ inhomogeneities and longer TE. For a B₀ shim of 50 Hz and a TE = 3.2 ms, the measured phase artifact is less than 10% of the accumulated phase.

For the sake of completeness, we also provide the unencoded signals, reflecting the average signal values for the whole slice of the in vivo measurement in Figure S2, which is in line with previous results in simulation and in vitro: Flip angle optimization alone fails when system parameters are changed; task 2 and 3 lead to improved signal decays. $\mathrm{\Psi }=84°$ and $\mathrm{\Psi }=50°$ look the smoothest here, whereas task 2 and 3 introduce noise‐like patterns in higher repetitions, resembling changes toward shorter T₂ (see Figure 6A(d),B(d)). This noise‐like fluctuation seems to have no clear influence on the images in Figure 7g. Task 2 and task 3 especially improve the smoothness of the decay along the first repetitions. There, a very smooth signal profile for task 3 is visible (Figure S2k). In comparison to that, an increment of 117° exhibits a pronounced edge in the unencoded signal and significant phase variations of up to 20°.

Whereas some of the image artifacts look like motion artifacts, motion between the images was not visible during the 3‐min measurement. Furthermore, similar artifacts are visible in the simulation in which no motion is considered.

Another indication that this artifact is not caused by motion comes from the experiment with a simple water bottle. Whereas the artifact is clearly visible for a quadratic increment of 117° (Figure 8A), task 2 can completely eliminate it (Figure 8C). Whether the artifact is visible depends heavily on the sharpness of the edge (Figure 8F–K). A sharp edge produces Gibbs ringing, which can mask the artifact, making it difficult to detect. On the other hand, a very smooth edge generates neither Gibbs ringing nor the artifact. For certain edges, the artifact appears even without Gibbs ringing, but it resembles Gibbs ringing and should not be confused with it.

FIGURE 8.

The magnitude images of the FLASH readout of a simple water bottle using α = 19.5°, TE = 3.2 ms, and TR = 6 ms show artifacts for a RF increment of 117° (A) and 84° (B), whereas task 2 (C) shows no artifact. Line plots (D, E) show the elimination of artifacts of quadratic phase increments of 117° and 84° by the optimization task. Nonuniform intensity of the phantom top is generated by receive coils. Different edges (F, G, H) influence the visibility of the artifact, as shown in simulation. For sharp edges, Gibbs ringing overlap the artifact (I), whereas for beveled edges no effect is visible (K). In (J), the Gibbs ringing is negligibly small, whereas the artifact arising from the quadratic increment is visible. The object was convolved once with a perfect exponential function $S_{\mathrm{LL}}$, and with the decay of a quadratic increment of 117° showing variations.

In summary, 84° delivers a good solution for a smooth transient decay, as already suggested by Epstein et al. ⁴ By optimization of task 2 and especially task 3, improvement of the signal decay can be achieved in vivo, even compared to 84°. The nongeneralization of task 1 is clearly visible in the signal and images.

In Figure 9, the impact of image quality for a high‐resolution image of 256 by using a two‐shot sequence resulting in less blurry images was investigated. The strength of the signal phase artifacts in the unencoded experiment corresponds to the artifacts observed in the centric‐encoded FLASH phase images. Because the phase image has less contrast, such artifacts are easiest to spot in the phase image. An ideal Look–Locker experiment should have a constant phase in the unencoded case, thus the smoothest phase image. Phase artifacts translate to artifacts in the magnitude image. The phase image of $\mathrm{\Psi }=117°$ shows artifacts in both unencoded phase (Figure 9A(a)) and image phase (Figure 9B(a)), which is improved by $\mathrm{\Psi }=84°$ (Figure 9A(b),B(b)) and outperformed by our optimized sequence (Figure 9A(c,d),B(c,d)). The remaining phase artifacts of a quadratic RF increment of 84° vanish for the optimization task, as can be seen in the Figure 9C(a),C(d), where example profile line plots of magnitude and phase images are shown.

FIGURE 9.

Unencoded simulation (A) of the FLASH signal using α = 19.5°, TE = 3.2 ms, and TR = 6 ms for typical RF increments of 117° (a) and 84° (b), and for task 2 (c) and task 3 (d), are shown. By using a two‐shot sequence, a resolution of 256 is achieved using the identical RF train two times, leading to a high‐resolution phase image arg(S) (Ba‐d) and magnitude image |S| (Be‐h) of each FLASH sequence. The amount of the signal phase artifacts (A) corresponds to the artifacts observed in the phase images (B). An ideal Look–Locker experiment should have a constant phase in the unencoded case, thus the smoothest phase image. The phase image of $\mathrm{\Psi }=117°$ shows artifacts in both unencoded signal phase (Aa) and image phase (Ba), which is improved by $\mathrm{\Psi }=84°$ (Ab, Bb), and further improved by our optimized sequence (A(c, d), B (c, d)). Red and blue arrows indicate clearly visible artifacts in the phase and magnitude images, which can be eliminated by the optimization tasks. (C) Line profiles for signal phase (a, b) and signal magnitude (c, d) are chosen as shown in the inlay in (Ca). A clear improvement can be achieved by using a quadratic increment of 84° instead of 117°. The artifact around position 103 can be eliminated in the magnitude signal, as well as in the phase (Ca, Cc). Task 3 even outperforms a quadratic phase increment of 84° where remaining artifacts (position 96) vanish (Cb, Cd).

Finally, instead of the Look–Locker decay we can also use other target functions, for example, a constant signal target or Hanning function target. ³⁸ , ³⁹ For this, we perform an unencoded optimization with no magnetization preparation of another task 4 ($\overrightarrow{\alpha }|S_{\text{const}};\mathrm{\Psi }=84°$) and task 5 ($\overrightarrow{\alpha }|S_{\text{Hanning}};\mathrm{\Psi }=84°$) for a voxel filled with tissue parameters of the median of a brain phantom (GM‐like T₁ value). As seen in Figure S3, this goal can be reached with a flip angle optimization, but just for the tissue parameter of the training. The solution is tissue‐ and problem‐dependent and leads to nonconstant signals for other T₁ values (CSF and WM). Thus, this optimization has to be performed within specific contexts, which is beyond the scope of this work. The Look–Locker target leads to generally applicable solutions.

4. DISCUSSION

4.1. Overview and Interpretation of Findings

In this paper, we revisited the problem of the transient FLASH signal decay using the end‐to‐end MR‐zero framework with a phase distribution graph simulation. The advantage of methodology by using phase distribution graphs instead of an isochromat or spin simulation allows us to achieve a higher level of precision in our simulations compared to Loktyushin, ¹⁵ as shown in the in Figure S4. To describe proper spoiling, a high number of spins is necessary, which requires significant simulation time. In comparison, the PDG simulation is more efficient. The accuracy of our simulation is validated within this paper by the high agreement of simulation and measurement, even for substructures arising from interference of higher echoes, as can be seen in the in Figure S5. The PDG simulation provides information on the composition of the magnetization and the origin of the deviation from the perfect Look–Locker decay. We showed that a general target‐driven signal optimization is possible by optimization of each and every RF pulse angle and phase of the pulse sequence. We further demonstrate in simulation and measurement how artifacts from unencoded experiments transfer to the image domain and differs between optimized RF pulse trains and typical RF cyclings. Despite the general signal optimization potential, we focused herein on the Look–Locker target as discussed in the following.

As an MRI sequence, the FLASH or gradient‐ and RF‐spoiled gradient echo sequence can be seen as one of the simplest sequences attempting to use only the FID signal of fresh z‐magnetization and eliminating the signal contribution of all higher echoes. However, in reality higher echoes play a role because some of them can get rephased and restored into the nonencoded z‐magnetization, which is the origin of the FID signal. In this paper, we showed that the MR‐zero simulation based on phase distribution graphs ¹⁶ is able to accurately simulate these additional signals, leading to deviations from the exponential Look–Locker decay in agreement with the findings of Epstein et al. ⁴ The deviations lead to artifacts in images that resemble Gibbs ringing. To achieve the same agreement also for larger flip angles and/or longer TR, isotropic diffusion is included in the simulation because for these parameter regimes a diffusion dependency is expected. ¹⁷ , ¹⁸ Moreover, the differentiability of our simulation allows to alter these deviations via a gradient descent optimization of the individual RF flip angles and phases. From our results, we can conclude that such an individual optimization can further improve the signal and image quality of FLASH sequences.

Interestingly, our RF phase cycling optimization challenges the often‐used concept of quadratic phase cycling. Thus, we want to point out that quadratic phase cycling was actually derived to generate a signal independent of the repetition n; thus, it is derived for the steady state signal. ⁴⁰ Quadratic phase cycling is still used very generally, also in transient sequences, despite the lack of theoretical foundation in this case. Moreover, the exact value of $\mathrm{\Psi }$, most famous choice of $\mathrm{\Psi }=117°$, is also derived using a steady‐state argumentation by finding the signal that is closest to the dynamic FLASH equation or Look–Locker equation. ³ Despite the work of Epstein et al., ⁴ who already suggested the $\mathrm{\Psi }=84°$ phase increment to be better in transient cases, in many transient state applications, even in vendor implementations of magnetization prepared sequences, quadratic phase increments of $\mathrm{\Psi }=117°$, $\mathrm{\Psi }=50°$, or $\mathrm{\Psi }=123.5°$ still are used. ³ , ²⁵ , ²⁶ , ²⁷ , ⁴¹ We conclude that certain values of $\mathrm{\Psi }$ neither have a fundamental justification for the transient phase of FLASH signals nor the quadratic phase cycling itself. Both are derived for steady‐state. Thus, the optimization we performed regarding the phase cycling $\overrightarrow{\phi }$ in transient state was not yet covered by the literature. Only Epstein et al. ⁴ tested different $\mathrm{\Psi }$ also in the transient case.

Interestingly, if we unwrap the RF phase of quadratic increments, they behave periodically, whereas the found results of task 2 are more random, as can be seen in the in Figure S6. But the overall slope is rather similar. This still needs to be investigated in more detail.

An open question might be here how the transition to a steady‐state with fixed $\mathrm{\Psi }$ is handled; however, this could be included in a task, that is, to optimize only the first 200 events of an otherwise quadratic phase cycling of a fixed $\mathrm{\Psi }$. As shown in Figure S7, this might even not be necessary because for typical transient sequences with low flip angle in vivo, the steady‐state is almost independent of $\mathrm{\Psi }.$ ¹⁷ Strong variations of the steady state for different increments are just visible for high flip angles or high TR/T₁‐ratio. ⁵

4.2. Related Work

In this paper, we focused on using the ideal Look–Locker ¹¹ decay as a target function; however, we also showed in the in Figure S3 that different signal target functions are possible. The general optimization of RF flip angles for this objective was already done by several groups as early as in 1992 by Mugler et al. for SSFP ¹³ and 1992 by Stehling for RF‐cycled FLASH. ¹⁴ Both Stehling, as well as Priatna and Paschal in 1994, ⁴² show similar flip angle trains as visualized in in Figure S3. Stöcker and Shah ⁴³ also included the k‐space trajectory in their considerations for SSFP, as well as an EPG‐based simulation similar to the one we used. Li et al. ⁴⁴ did the same for the FLASH sequence. All the above works aimed for a flat signal response, which leads to a tissue‐dependent result: Flat response in the target tissue leads to a non‐flat signal evolution tissues with different relaxation or B₁, as discussed by Worters and Hargreaves ⁴⁵ and shown here in in Figure S3a–c. Interestingly, none of the above papers optimized the RF cycling; only Epstein et al. ⁴ identified the best quadratic phase‐cycling increment $\mathrm{\Psi }=84°$ for the transient decay. Although different signal targets have the benefit of PSF‐tuning, the Look–Locker target has the benefit of an unchanged Look–Locker decay in all tissues, which is why we focused on this target herein.

This has the benefit that all voxel signals still are interpretable by the Look–Locker equation (1), which would be especially relevant for model‐based reconstructions assuming the ideal spoiled Look–Locker decay. ¹¹ , ⁴⁶ , ⁴⁷ , ⁴⁸ For such approaches, our refined RF trains might lead to more accurate quantification. Moreover, for already existing data with, for example, $\mathrm{\Psi }=117°$, our differentiable MR‐zero simulation could be used directly as improved reconstruction model including the substructure, similar as for Bloch model–based reconstructions. ⁴⁹

The present work uses MR‐zero on unencoded signals, which is different from previous applications of MR‐zero that aimed for end‐to‐end target‐driven optimization in the image space for learning encoding, T₁ mapping, ¹⁵ or sharp TSE MRI. ⁵⁰

4.3. Limitations and future directions

The remaining artifacts in RF‐spoiled images can be mitigated by either larger gradient spoiling ²⁶ or variable gradient spoiling. ⁵¹ Both require higher gradient amplitudes; thus, the minimal TR increases, which limits fast imaging applications. Controlling the deviation by the RF events therefore allows lower gradient spoiling and faster sequences.

An incidental finding for the transient response was that for zero starting magnetization MI = 0, the signals generated with typical quadratic RF increments or from task 2 and 3 were already most similar to $S_{\mathit{LL}}$ (red curve in Figures 5A(i,l) and 6A(i,k,l)); only task 1 deteriorates the progression (Figure 6A(j)). This is plausible because the fluctuations result from restored echoes of early excitations. If the early excitations now have very small magnitude, the fluctuations vanish. This is interesting because all saturation recovery sequences (e.g., saturation recovery single‐shot acquisition) fulfill such an initial condition and implicitly should have more accurate exponential recovery curves. ⁵² Furthermore, for a typical MPRAGE sequence, only slight differences between a quadratic RF increment of 117° and 84° could be observed (data shown in Figure S8). Because CSF has low signal but produces most artifacts, only minor influences of different RF spoiling are expected in MPRAGE sequences. However, in tumor‐ or stroke‐affected areas, there might be liquids with different relaxation times and nulling points that are not fully suppressed but still show the mentioned artifacts.

In contrast, the shown artifact in the magnitude image can lead to errors using a simple Look–Locker model for reconstruction for a series of T₁‐weighted images. The Look–Locker model leads to T₁ map differences for the RF cycling of 117°, which are mitigated when using the data acquired with optimized RF trains. This is shown in the in Figure S9. Nevertheless, there are many more details to be explored in silico and in vivo, not only using optimized sequences for simple models but also using the PDG simulation as reconstruction operator for nonoptimized sequences.

Some RF increments lead to visible artifacts in the phase images (Figures 7 and 9). These artifacts can influence results in phase‐based sequences as QSM ⁵³ or fast B₀ mapping approaches. Our insights might help to improve fast implementations of these sequences.

5. CONCLUSION

It was shown that the transient signal decay in FLASH sequences shows deviations from the Lock–Locker decay with a complex substructure. These structures are visible in measurement and simulation and are distinct for different RF cyclings. In the past, quadratic RF cycling was shown to be able to decrease this substructure, but only for distinct phase increments such as, for example, 84°. Typical phase cyclings even in use in clinical transient FLASH sequences (117° or 50°) show pronounced deviations leading to sidebands in the PSF and image artifacts. By introducing the end‐to‐end optimization MR‐zero, we optimize RF phase and flip angles simultaneously, leading to a close match to a perfectly spoiled Look–Locker decay and better PSF resulting in artifact free magnitude and phase images.

FUNDING INFORMATION

Funding for this work was provided by the German Research Foundation (DFG), project 500888779 / RU5534 MR biosignatures at UHF, 442377885.

Supporting information

FIGURE S1 Simulation (A, B) and measurement (C, D) of centric encoded FLASH readouts using α = 10°, TE = 5.0 ms, TR = 10.0 ms for typical RF increments of 117° (b), 84° (c) and 50° (d) and optimization task 1 (e), task 2 (f) and task 3 (g). In simulation the ideal spoiled Look‐Locker image $S_{\mathrm{LL}}$ was additional calculated. The phantom for simulation was built from data provided by the BrainWeb database. There, differences of signal magnitude S (Ah‐m) and phase arg(S) (Bh‐m) are calculated for each image to the Look‐Locker target (a). For the magnitude images (Cb‐g) and phase images (Db‐g) task 3 is used as reference (g) in measurement. Artifacts in the phase image can be seen for all standard RF increments, especially for 117°. Task 3 can reduce these artifacts coming from the RF cycling and generate the smoothest phase image. In vivo fewer clear artifacts are visible compared to Figure 7 with higher excitation flip angle a longer TE and TR. Susceptibility effects are in the same order of magnitude. The seen offset in the phase differences (Dg‐Di) can be explained by the shifting B₀ field of the used scanner. B₀ field of the used scanner. For a B₀ shim of 50 Hz and a TE = 5.0 ms, the measured phase artifact is less than 5% of the accumulated phase.

FIGURE S2 Unencoded measurements of the FLASH signal using α = 19.5°, TE = 3.2 ms, TR = 6 ms for typical RF increments of 117° (a, b), 84° (c, d) and 50° (e, f) and for task 1 (g, h), task 2 (i, j) and task 3 (k, l) are shown. In the first row, the signal magnitude |S| and in the second row the signal phase arg(S) of each FLASH sequence is visualize. Strong deviations are again visible for 117° increment, while 84° and 50° leads to less variations. Task 1 performs worse w.r.t. a close signal for a Look‐Locker decay and constant phase signal. Task 2 and 3 could improve the decay while simultaneously having the most constant phase. For outer k‐space lines it results in a more noise like fluctuation which are not visible in images as can be seen in Figure 7.

FIGURE S3 Signal evolution of a voxel with different T₁ values (CSF—4.0 s, WM—0.83 s and GM—1.48 s) are shown in (a,d). Additionally, in gray the constant signal target (a) and Hanning function target ($\mathrm{h}(\mathrm{n})=\frac{\mathrm{\beta }}{2}\left(1+\cos \frac{2\mathrm{\pi n}∆\mathrm{x}}{{\mathrm{\alpha }}^{\prime }}\right)$ with $\mathrm{\beta }=1,{\mathrm{\alpha }}^{\prime }=2,∆\mathrm{x}=1/128$) (d) is plotted. For task 4 ($\overrightarrow{\alpha }|{\mathrm{S}}_{\text{const}};\mathrm{\Psi }=84°$) and task 5 ($\overrightarrow{\alpha }|{\mathrm{S}}_{\text{Hanning}};\mathrm{\Psi }=84°$) the final optimized flip angle trains are given in (b, e). In (c, f) the signal magnitude evolution is visualize again for different T₁ values. The sequence parameters in the shown optimization tasks are α = 5° for task 4 and α = 10° for task 5, a readout bandwidth of 500 Hz/pixel, TE = 3.2 ms and TR = 6 ms is used. No magnetization preparation is applied before the readout in this case. With flip angle optimization the target signal can be reached, at least for the in training used T₁ value. Here, the training is performed with the median of a BrainWeb phantom, which has a GM‐like T₁ value. However, the optimized flip angle train clearly fails for another T₁ values (WM and CSF). The solution is tissue and problem depended.

FIGURE S4 Comparison between spin simulation (colored line) using different number of spins per voxel and the same PDG simulation (black line).

FIGURE S5 Magnitude (a) and phase (b) of an unencoded simulation of the FLASH signal using α = 19.5°, TE = 3.2 ms, TR = 6 ms for a RF increment of 117° and task 2. Magnetization arrows are plotted in a complex plane (c–f). The black arrow represents the ideal Look‐Locker magnetization, while the green or blue arrows depict the simulated magnetization with 117° or optimized phase cycling (task 2), respectively. These arrows are the sum of many contributing PDG magnetization pathways, which are plotted in red. Comparing to the Look‐Locker magnetization, this green vector deviates in both magnitude and direction. However, for task 2, the blue arrow aligns closely with the black arrow, indicating that the residual magnetization of higher echoes has approximately canceled out, and only the newly excited magnetization in each repetition (longest red arrow) contributes to the total signal.

FIGURE S6 The unwrapped phases of typical RF increments of 117° (a), 84° (b) and 50° (c) shows repetitive and oscillating behavior while for $\mathrm{\Psi }=117°$ the fluctuations are strongest. The unwrapped RF phase of task 2 (d) shows more random fluctuations, but the overall increase is similar to a quadratic RF cycling.

FIGURE S7 Steady‐state signal magnitude of FLASH sequence (TR = 5.6 ms) are shown as function of the spoiling phase increment $\mathrm{\Psi }$ for different excitation flip angles for a white matter pixel (T₁ = 0.85 s, T₂ = 0.076 s, D = 0.66 ⋅ 10⁻³ mm²/s). For typical RF phases the images are shown. The signal magnitude of a perfect spoiled sequence doesn't depend on the phase increment (gray color). The corresponding image is visualized as well. Strong deviations from a perfect spoiled steady state signal can be seen for higher excitation flip angles.

FIGURE S8 Standard clinical MRRAGE magnitude (a, b) and phase (c, d) images using typical quadratic RF increments of $\mathrm{\Psi }=117°$ and $\mathrm{\Psi }=84°$. Sequence parameter of linear‐reordered MPRAGE: inversion time: 0.9 s; resolution: 256 x 256; FOV: 200 x 200 mm²; TE: 4.2 ms; echo distance: 8.2 ms; TR: 1.95 s bandwidth: 280 Hz/pixel; average: 2; shots: 2; dummy shots: 3. No significant difference can be seen in magnitude and phase image produced by different RF increments, just a slightly different contrast is visible in the magnitude.

FIGURE S9 The magnitude images for different inversion times given in the gray box are shown in (a). In this simulation, the ideal Look‐Locker decay is considered. A centric encoded FLASH readout was used with parameters α = 19.5°, readout bandwidth of 500 Hz/pixel, TE = 3.2 ms and TR = 6 ms. Before each inversion we assume to be in a completely relaxed thermal equilibrium state. Subfigures (b) and (c) display the absolute difference between (a) and a full simulation that uses all necessary signal distributions for a quadratic RF increment of $\mathrm{\Psi }=117°$, as well as the optimized sequence from task 3. The optimized RF train reduces clearly the error of the magnitude images. Then we quantify T₁ for these both datasets using a Look‐Locker model for reconstruction. The T₁ map of the BrainWeb phantom is presented in (d). The Look‐Locker model leads to T₁ map differences for the RF cycling of 117° (e), which are mitigated when using the data acquired with the optimized RF train of task 3 (f), which is also indicated by a lower RMSE.

Weinmüller S, Endres J, Dang N, Stollberger R, Zaiss M. MR‐zero meets FLASH – controlling the transient signal decay in gradient‐ and RF‐spoiled gradient echo sequences. Magn Reson Med. 2025;93(3):942‐960. doi: 10.1002/mrm.30318

DATA AVAILABILITY STATEMENT

The MR‐zero simulation is accessible via https://pypi.org/project/mrzerocore/ and different script can be directly run on a playground Playground MR0.

Optimization results are published here: https://github.com/MRsources/FLASHzero.