Jakob Assländer^{1,2}, Daniel K Sodickson^{1,2,3}, Riccardo Lattanzi^{1,2,3}, and Martijn Cloos^{1,2}

This work analyzes the spin physics in steady-state free precession sequences modified to smoothly vary sequence parameters, as suggested in MR Fingerprinting. We arrive at the conclusion that a transient state develops only in one direction, while the magnetization in the other two dimensions transitions adiabatically between steady states. We provide solutions of the Bloch Equations for this hybrid state and demonstrate the superior T_{1}- and T_{2}-encoding performance of the hybrid state compared to the steady state.

The original MR-Fingerprinting^{1} (MRF) implementation was based on a balanced-SSFP sequence modified to employ varying flip angles. Recent work^{2,3} indicates that spherical coordinates allow for a compact description of relaxation processes in such sequences. Here, we provide a solution of the Bloch Equations in spherical coordinates. A comprehensive analysis of the solution reveals that the spin dynamics can be separated into a transient-state component and two components that adiabatically transition between steady states, which inspires the term "balanced hybrid-state free precession" (bHSFP). Numerical optimizations demonstrate the superior $$$T_{1,2}$$$-encoding power of the hybrid state compared to the steady state.

By applying a first order Taylor expansion with respect to sequence parameters (such as flip-angle, RF-phase, and $$$TR$$$) to the eigen-decomposition of the spin evolution matrix^{4,5}, one can show that only the transient eigen-state parallel to the steady-state magnetization gets populated by smooth parameter variations. Spherical coordinates separate this transient component along the radial dimension $$$r$$$ from the adiabatic components along the direction of the phase $$$\varphi$$$ and the polar angle $$$\vartheta$$$. The latter two are given by the steady-state solution of the Bloch Equations^{6} and are approximated by$$\tan(\varphi-\frac{\phi}{2})=\frac{\cos\phi-\exp(-TR/T_2)}{\sin\phi}$$and$$\sin^2\vartheta=\frac{\sin^2\frac{\alpha}{2}}{\sin^2\frac{\phi}{2}\cdot\cos^2\frac{\alpha}{2}+\sin^2\frac{\alpha}{2}}$$anywhere except near the stop band. Here, $$$\alpha$$$ denotes the flip angle, $$$\phi$$$ the phase accumulated during one $$$TR$$$, and we define $$$\phi=\pi$$$ as on-resonance. Given these adiabatic transitions, the Bloch Equation along the radial direction is uncoupled and is solved by$$r(t)=a(t)\cdot\left(r_0+\frac{1}{T_1}\int_{0}^{t}\frac{\cos\vartheta(\tau)}{a(\tau)}d\tau\right)$$with$$a(\tau)=\exp\left(-\int_{0}^{\tau}\frac{\sin^2\vartheta(\kappa)}{T_2}+\frac{\cos^2\vartheta(\kappa)}{T_1}d\kappa\right),$$in which we assumed departure from $$$r_0$$$, and $$$t$$$ denotes time. Note that neither $$$\vartheta$$$, nor $$$r$$$ depend on $$$TR$$$, given $$$TR\ll\{T_1,T_2\}$$$.

Instead of an initial magnetization, we can also assume periodic or anti-periodic boundary conditions defined by $$$r(0)=\pm{r}(TC)$$$, where $$$TC$$$ denotes the duration of one cycle of the experiment. In this case, the radial Bloch Equation is solved by $$r(t)=\frac{a(t)}{T_1}\cdot\left(\int_{0}^{t}\frac{\cos\vartheta(\tau)}{a(\tau)}d\tau-\frac{a(TC)}{a(TC)\mp1}\int_{0}^{TC}\frac{\cos\vartheta(\tau)}{a(\tau)}d\tau\right).$$

On the search for the most efficient bHSFP sequence, we optimize the $$$\vartheta$$$-pattern. We employ the Broyden-Fletcher–Goldfarb-Shanno algorithm, and use the relative Cramer-Rao bound^{7-11} ($$$rCRB$$$), normalized by $$$TC$$$, as figure of merit. For comparison, optimizations were also performed for an inversion-recovery bSSFP sequence with a constant flip-angle^{12} and for a combination of spoiled gradient echo (SPGR) and bSSFP sequences in the steady state^{13}.

In vivo measurements were performed with the anti-periodic bHSFP sequences, as well as with the optimized steady-state sequence, both with $$$TC = 3.8~\text{s}$$$. All experiments were performed on a 3T Prisma scanner (Siemens, Erlangen, Germany). Spatial encoding was performed with a sagittally oriented 3D stack-of-stars trajectory with golden angle increment^{14}. The spatial resolution is $$$1~\text{mm}\times1~\text{mm}\times 2~\text{mm}$$$ at a FOV of $$$256~\text{mm}\times256~\text{mm}\times192~\text{mm}$$$. The total scan time was approximately $$$12.24~\text{min}$$$. Image reconstruction was performed with a low rank alternating direction method of multipliers approach^{15}. $$$B_0$$$- and $$$B_1$$$-correction were performed on the basis of separate scans.

The optimized bHSFP sequences exhibit smooth $$$\vartheta$$$-patterns (Figure$$$~$$$1), and the spin dynamics, as described by the derived approximate solution of the Bloch Equations, shows good accordance with Bloch simulations anywhere apart from the vicinity of the stop-band (Figure$$$~$$$2).

When plotting the optimized $$$rCRB$$$ as a function of $$$TC$$$ (Figure$$$~$$$3), one can discriminate different aspects of underlying spin physics: All transient-state flavors result in a lower (i.e. better) $$$rCRB$$$ compared to the steady-state optimizations for scans longer than $$$2~\text{s}$$$. Note that the (anti-)periodic solutions represent transient state equivalents to the steady state in the sense that the RF-train can be repeated without waiting times, allowing for an efficient and flexible implementation of long RF-trains, e.g., for 3D imaging. Contrary to the periodic, the anti-periodic boundary condition allows for repeated visits of the southern hemisphere (Figure$$$~$$$1), which is reflected in a reduced $$$rCRB$$$ (green vs. black marks in Figure$$$~$$$3). The $$$rCRB$$$ difference between anti-periodic boundary condition and fully relaxed inversion-recovery design reflects the increased net-magnetization available at the start of the sequence (black vs. red marks). Further, comparing the IR-bSSFP to the IR-bHSFP optimization (brown vs. red marks) highlights the improvement in SNR due to a variable flip angles. These improvements are particularly pronounced at long $$$TC$$$ where the IR-bSSFP converges to a steady state. Figure$$$~$$$4 verifies that the $$$rCRB$$$ improvements span over large areas in $$$T_1$$$-$$$T_2$$$-space.

In vivo images reconstructed using full Bloch simulations and the proposed solution are in excellent agreement (Figure 5; $$$T_1=965\pm23~\text{ms}$$$ vs. $$$T_1=988\pm23~\text{ms}$$$ and $$$T_2=48.2\pm3.0~\text{ms}$$$ vs. $$$T_2=49.7\pm2.9~\text{ms}$$$). The superior SNR efficiency of the bHSFP patterns compared to an steady-state approach ($$$T_1=1035\pm72~\text{ms}$$$, $$$T_2=37.8\pm2.6~\text{ms}$$$) can also be seen in Figure$$$~$$$5. Systematic deviations can be noted, which we associate mainly with magnetization transfer^{16} and partial volume effects^{17}, which are not yet considered in this work.

The authors would like to thank Steffen Glaser, Quentin Ansel and Dominique Sugny for fruitful discussions, and for giving insights into their optimal control implementation. The authors would also like to acknowledge Jeffrey Fessler and Gopal Nataraj discussions regarding the solution of the simplified Bloch Equation.

This work was supported by the research grants NIH/NIBIB R21 EB020096 and NIH/NIAMS R01 AR070297, and was performed under the rubric of the Center for Advanced Imaging Innovation and Research (CAI2R, www.cai2r.net), a NIBIB Biomedical Technology Resource Center (NIH P41 EB017183).

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Figure 1: The optimized spin trajectories are depicted for periodic (r(0) = r(TC)) and anti-periodic (r(0) = -r(TC)) boundary conditions, as well as with the search space limited to the bSSFP steady-state ellipse (blue line in **g**) and the SPGR steady-state ellipse (red line in **g**). Note that ϑ of the bSSFP segment in (**g**-**i**) was retrospectively sorted in increasing order. The polar angle was limited to 0 ≤ ϑ ≤ π/4 for the transient state optimizations (**a**-**f**) and to 0 ≤ ϑ ≤ π/2 for the steady-state optimization (**g**-**i**).

Figure 2: The derived solution of the Bloch Equations is verified against Bloch simulations at the example of the optimized anti-periodic pattern. In agreement with the approximate nature of the derivation, good accordance can be observed anywhere apart from the vicinity of the stop band. The gray areas indicate time points that were not acquired in the in vivo scan since the polar angle is close to zero.

Figure 3: The depicted relative Cramer-Rao bounds can be understood as a lower bound of the squared inverse SNR efficiency per unit time. The legend is organized in blocks, where the first one (original DESPOT^{13}, MRF^{1} and pSSFP^{2}) are sequences taken from literature without modifications. The second block (steady-state and IR-bSSFP) are traditional sequences with optimized parameters and the last block contains the bHSFP sequences described by the derived approximate solution of the Bloch Equations. Note that all three sub-figures result from separate optimizations with rCRB(T_{1}), rCRB(T_{2}) and rCRB(T_{1}) + rCRB(T_{2}) as figures of merit and the data depicted in (**c**) are not the sum of the data depicted in (**a**) and (**b**).

Figure 4: The performance of the optimized patterns is illustrated through plots of the relative Cramer-Rao bounds, which provide a lower bound for the noise in the retrieved relaxation times. All patterns were optimized for T1 = 781 ms and T2 = 65 ms, as indicated by the red square, and were tested for the entire parameter space in a sample MRF dictionary. Note the logarithmic scale in all three dimensions.

Figure 5: A single sagittal slice of an in vivo 3D data set is depicted. The data were acquired with the excitation patterns depicted in Figure 1e,h. The parameter maps have a resolution of 1 mm x 1 mm x 2 mm. The anti-periodic data set was reconstructed once with a dictionary calculated with Bloch simulations, and once with a dictionary calculated based on the derived approximate solution of the Bloch Equations. The red box indicates a region of interest used for extracting T_{1} and T_{2} values. Note the logarithmic scale of the color maps.