Jakob AsslĂ¤nder^{1,2}, Daniel K Sodickson^{1,2}, Riccardo Lattanzi^{1,2}, and Martijn A Cloos^{1,2}

This work analyses relaxation in balanced non-steady-state free precession sequences. Transforming the Bloch equation to polar coordinates gives insights in the spin dynamics and provides the basis for robust numerical optimization of the excitation pattern. The employed optimal control algorithm results in spin trajectories that allow for parameter mapping with considerably reduced noise, as shown in in vivo MR-fingerprinting experiments. The simple shapes of the optimized spin trajectories provide a basis for further analysis of the encoding process of relaxation times for parameter mapping.

The Authors would like to thank Steffen Glaser, Quentin Ansel and Dominique Sugny for the fruitful discussions about optimal control and for giving insights in their implementations.

This work was supported in part by the research grants NIH R21 EB020096 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 dynamics of the on resonant isochromat is depicted for the relaxation times of white matter in the case of free relaxation (a). In the case of an inversion-recovery SSFP sequence, the magnetization is refocused at TE=TR/2, allowing us to neglect dephasing effects. The oscillating RF-pulses force the magnetization onto a cone on which it relaxes until it reaches the steady-state ellipse depicted in blue (b). In an oscillating frame of reference, the same relaxation induced spin dynamics happens purely along the radial direction, motivating a description in polar coordinates (c).

Figure 2: The spin dynamics (red) of optimized excitation pattern are depicted on Bloch-spheres (a,d,g) together with the steady-state ellipse (blue). The excitation angle before and after the optimization are shown in (b,e,h), while (c,f,i) shows the optimized transversal magnetization and its normalized derivatives with respect to the relaxation times. Those three time curves are used to compute the relative Cramer-Rao bound that is minimized by the optimization. All excitation pattern were limited to $$$0\leq\vartheta\leq\pi/2$$$ and serve for theoretical considerations.

Figure 2: The spin dynamics (red) of optimized excitation pattern are depicted on Bloch-spheres (a,d,g) together with the steady-state ellipse (blue). The excitation angle before and after the optimization are shown in (b,e,h), while (c,f,i) shows the optimized transversal magnetization and its normalized derivatives with respect to the relaxation times. Those three time curves are used to compute the relative Cramer-Rao bound that is minimized by the optimization. All excitation pattern were limited to $$$0\leq\vartheta\leq\pi/4$$$ and are used for the in vivo experiments.

Table 1: The relative Cramer-Rao bound is shown for different optimization scenarios (a). The boxes with gray background are the objectives that are minimized by the optimal control algorithm. Table (b) shows the measured relaxation times within a region of interest in the frontal white matter.

Figure 4: The in vivo data was acquired with the excitation pattern depicted in Figure 3. The parameter maps have an in-plane resolution of 1mm and were reconstructed from 840 radial spokes. The acquisition time was 3.8s.