This abstract discusses the signal behavior when varying the flip angle in a (balanced) steady state free precession sequences (SSFP) sequence, as done in MR-Fingerprinting (MRF) sequences$$$^\text{1}$$$. The flip angle variations prevent a steady state and introduce instabilities regarding magnetic field inhomogeneities and intra-voxel dephasing. We show how a pseudo steady state free precession (pSSFP) results in a more stable MRF reconstruction. Furthermore, we propose an iterative reconstruction algorithm for fingerprinting data.

Disregarding the balanced gradients, a traditional SSFP sequence consists of a train of equidistant RF-pulses with a constant flip angle and a phase increment of $$$\pi$$$ for consecutive pulses. This leads to a steady state in which the magnetization oscillates about the z-axis (Fig.1a,b) and the polar angle $$$\vartheta=\alpha/2$$$ (in spherical coordinates) is approximately half the flip angle. As a consequence, a spin echo$$$^\text{2}$$$ is formed at $$$TE=TR/2$$$. However, when varying $$$\vartheta$$$, the phase of the isochromats is changed (Fig.1c). In contrast, the Euclidean distance between the tips of the magnetization vectors $$$\delta$$$ is not changed when rotating the entire spin ensemble instantaneously$$$^\text{3}$$$. Therefore, we can compare $$$d\delta$$$ for two isochromats that are separated by an infinitesimal phase difference $$$d\omega$$$. Before and after the $$$i^{\text{th}}$$$ RF-pulse it is given by $$d\delta_{i}^+=d\omega\cdot (TR_i-TE_i)\cdot\sin\vartheta_i\\d\delta_{i+1}^-=d\omega\cdot TE_{i+1}\cdot\sin\vartheta_{i+1}\;.$$ The preservation of $$$d\delta$$$ by a hard pulses reveals $$TE_{i+1}=(TR_i-TE_i)\cdot\frac{\sin\vartheta_i}{\sin\vartheta_{i+1}}\;.$$ A similar bandwidth compared to SSFP is achieved with $$\max\left\{ TE_{i+1},\;\;TR_i-TE_i\right\}=TR_{\text{SSFP}}/2\;.$$ The geometrical variables are depicted in Fig.1. With these relations, flip angle dependent values for $$$TR$$$ and $$$TE$$$ can be calculated, which maintain the spin echo-like character of SSFP sequences when varying the flip angle.

Dictionaries were calculated with Bloch simulations in the hard pulse approximation for the original flip angle pattern$$$^\text{1}$$$ with a constant $$$TR=4~\text{ms}$$$ and for the pSSFP approach with $$$TR_{\text{SSFP}}=4~\text{ms}$$$. In the latter sequence, the sections with $$$\alpha=0^\circ$$$ were omitted since they contradict the idea of SSFP. MRF data of a single slice of a volunteer's brain were acquired with a $$$3~\text{T}$$$ PRISMA scanner (Siemens, Erlangen, Germany) employing a radial readout with a golden angle increment. The total acquisition time was $$$4~\text{s}$$$ for the MRF sequence and $$$3.3~\text{s}$$$ for the pSSFP-MRF sequence. The spatial resolution was $$$1\text{mm}\times1\text{mm}\times3\text{mm}$$$, the $$$FOV=256~\text{mm}\times256~\text{mm}$$$ and the bandwidth was $$$1500~\text{Hz/px}$$$. For reconstructing $$$T_1,T_2$$$ and $$$PD$$$ maps we propose to iteratively minimize the cost function $$\tilde{x}=\arg\min_{x~\in~\mathbb{C}^{N \times T}}\sum_{t=0}^{T-1}\left|\left|\mathbf{E}_{t,:,:}\cdot{x_{:,t}}–S_{:,t}\right|\right|_2^2+\lambda_{\mathbf{P}}^2\sum_{n=0}^{N-1}\left|\left|x_{n,:}-\mathbf{P}_{\mathcal{A}}(x_{n,:})\right|\right|_2^2+\lambda_{\mathbf{W}}^2\sum_{n=0}^{N-1}\left|\left|\mathbf{W}(x_{n,:})\right|\right|^1\;.$$ The search variable $$$x$$$ contains the images with all $$$N$$$ voxels of all $$$T$$$ time-frames. The first summand is the data consistency term with the forward operator $$$\mathbf{E}$$$ which incorporates a non-uniform FFT and ESPIRiT$$$^\text{4}$$$ coil sensitivities. The second term compares the time series of each voxel with its projection $$$\mathbf{P}$$$ on the dictionary $$$\mathcal{A}$$$. The third term is an $$$\ell_1$$$-penalty in the wavelet domain$$$^\text{5}$$$. Since this algorithm approaches the Bloch response recovery via iterated projection algorithm$$$^\text{6}$$$ (BLIP) when setting the regularization parameters to $$$\lambda_{\mathbf{P}}\rightarrow\infty$$$ and $$$\lambda_{\mathbf{W}}\rightarrow0$$$, we refer to it as generalized BLIP (gBLIP). The quantitative $$$T_1,T_2$$$ and PD-maps finally result from $$$\mathbf{P}_{\mathcal{A}}(x)$$$ via a look-up table.The spin echo formations known from SSFP (Fig.2b) are lost when varying the flip angle (Fig.2c), but can be restored with the proposed approach (Fig.2d). Comparing the signal evolution of a single isochromat to the signal evolution averaged with a Cauchy distribution corresponding to a decay time of $$$T_2'=40~\text{ms}$$$, significant differences can be observed (Fig.3). The difference becomes obvious in the sections with $$$\alpha=0^\circ$$$, where the single isochromat decays with $$$T_2$$$, while the averaged signal decays with $$$T_2^*$$$. These differences introduce a dependency of the signal evolution and ultimately the quantitative maps to intra-voxel dephasing. This dependency can be undone by the pSSFP approach (Fig.4), as long as the signal contributions originate from the central passband. The maps in Fig.5 reveal the feasibility of acquiring quantitative maps with a matrix size of $$$256\times256$$$ with 1000 and 850 spokes in the case of MRF and pSSFP-MRF, respectively. In the case of MRF, $$$T_1=802\pm79~\text{ms}$$$ and $$$T_2=47.3\pm6.3~\text{ms}$$$ of white matter are reduced compared to literature ($$$T_1=1110\pm45~\text{ms}^\text{7}$$$ and $$$T_2=69\pm3~\text{ms}^\text{8}$$$). Employing the concept of pSSFP, $$$T_1=1052\pm42~\text{ms}$$$ matches literature well, while $$$T_2=96.8\pm6.2~\text{ms}$$$ is slightly overestimated. This behavior translates to other tissues.The proposed adjustments of TR and TE restore the spin-echo-like signal behavior typical for SSFP in fingerprinting sequences, making this approach more robust to intra-voxel dephasing. Along with the proposed gBLIP reconstruction, feasible $$$T_1$$$ and $$$PD$$$-maps can be acquired with a resolution of $$$1~\text{mm}$$$ within $$$3.3~\text{s}$$$. The $$$T_2$$$-information of the acquired data seems to be insufficient. Future work will address this issue by optimizing the flip angle pattern for best discrimination of different tissue employing optimal control.