Eric Van Reeth^{1}, Hélène Ratiney^{1}, Kevin Tse Ve Koon^{1}, Michael Tesch^{2}, Denis Grenier^{1}, Olivier Beuf^{1}, Steffen J Glaser^{2}, and Dominique Sugny^{3,4}

This abstract proposes an optimal control strategy for the computation of contrast preparation pulses robust to B1 (+/- 35%) and B0 (+/- 400Hz) inhomogeneities. The problem formulation allows to optimize the compromise between contrast performance and preparation time. An *in vitro* short-T2 enhancing contrast experiment validates the robustness superiority of the proposed preparation compared to a block pulse-based scheme, and shows a good match with simulations.

Contrast
preparation schemes create a significant signal difference on the
longitudinal axis before running the excitation imaging sequence. For
optimum image reliability, it is important to achieve uniform contrast
despite B1 and B0 inhomogeneities, which especially occur at high
magnetic fields. Existing methods typically use rectangular,
composite, or adiabatic pulses, to perform robust T1^{[1]} or robust T2^{[2,3]} weighting. T1-weighted or T2-weighted based on saturation or inversion and echo time respectively limits the achievable contrast performance. Considering both relaxation times significantly complicates the
design of preparation sequences, which is why optimal control has
been proposed as an efficient pulse design strategy that fully
controls the magnetization dynamics^{[4,5]}. This abstract proposes an
optimal control strategy for the computation of B1 and B0-robust
contrast preparation pulses.

The
optimal control pulse computation is performed with GRAPE^{[6]}, as a
two-step procedure. In a first step, only a small number (N=3)
of block pulses and delays are optimized, defined by their respective flip
angle $$$\alpha^{(i)}$$$,
phase $$$\theta^{(i)}$$$
and post-pulse delay $$$\tau^{(i)}$$$.
Note that the duration of each block pulse is neglected with respect to the total preparation duration. This parameterization allows to incorporate the control time $$$(T = \sum \tau^{(i)})$$$
in the cost function,
optimizing both contrast performance and preparation time:

$$C^{a>b} = \Vert \overrightarrow{M^b}(T) \Vert - \Vert \overrightarrow{M^a}(T) \Vert + \gamma \sum_{i<N}\tau^{(i)}$$

with
$$$\overrightarrow{M^a}$$$ and
$$$\overrightarrow{M^b}$$$ the
macroscopic magnetization vectors of samples *a*
and *b*,
whose evolution is described by Bloch equations, and $$$\gamma$$$ a scalar that balances the influence of the control time
penalization term. An example of such a sequence is illustrated in
Figure 1a.

This sequence is subsequently converted into a finely-discretized complex pulse, where each time point becomes an optimization variable, and used as the initialization of the second optimization step. The objective is to make the initial time-efficient B0-robust sequence robust to B1-inhomogeneities by allowing additional degrees of freedom (Figure 1b). The cost function becomes:

$$ C^{a>b}_2 = \sum_i \Vert \overrightarrow{M_i^b}(T) \Vert - \Vert \overrightarrow{M_i^a}(T) \Vert $$

where
the subscript *i*
represents a specific B1-inhomogeneity value, taken inside a
user-defined robustness interval. Note that a similar approach is
used to handle B0-inhomogeneities – not detailed here for synthesis purpose.

The
B1-robustness of the optimized pulse is validated on a small-animal
4.7T Bruker MR system, by performing a non-trivial contrast
experiment that maximizes a short-T2 sample (90 ms), and minimizes a long-T2
sample (117 ms), both samples having similar T1 values (249 ms). The
pulse is optimized for +/-35% around the
theoretical B1 amplitude. The robustness is validated by measuring the
resulting contrast and intensity variations inside the targeted ROIs
for shifted amplitudes of the preparation scheme. Note that the amplitudes of the pulses used for excitation are not shifted, in order to analyze only the robustness of the preparation scheme.

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[2] R. Nezafat, M. Stuber, R. Ouwerkerk, A.M. Gharib, M.Y. Desai, and R.I. Pettigrew. *B1-Insensitive T2 Preparation for Improved Coronary Magnetic Resonance Angiography at 3 T*. Magnetic Resonance in Medicine, 55, 858-864, 2006

[3] E.R. Jenista, W.G. Rehwald, E.L. Chen, H.W. Kim, I. Klem, M.A. Parker, and R.J. Kim. *Motion and flow insensitive adiabatic T2-preparation module for cardiac MR imaging at 3 Tesla*. Magnetic Resonance in Medicine, 70, 1360-1368, 2013

[4] E. Van Reeth, H. Ratiney, M. Tesch, D. Grenier, O. Beuf, S.J. Glaser, and D. Sugny. *Optimal control design of preparation pulses for contrast optimization in MRI. Journal of Magnetic Resonance*, 279:39-50, 2017

[5] M. Lapert, Y. Zhang, M. Janich, S.J. Glaser and D. Sugny. *Exploring the physical limits of saturation contrast in Magnetic Resonance Imaging.* Sci. Rep. 2, 589, 2012

[6] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, and S.J. Glaser. *Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms*. Journal of Magnetic Resonance, 172(2):296-305, 2005

Figure 1: Optimized pulse sequences used in the *in vitro* experiment (a) Step 1: Preparation sequence composed of block pulses (b) Step 2: Complex contrast preparation pulse (blue: B1x - red: B1y) optimized to be robust to variations of +/- 35% around the theoretical B1 value.

Figure 2: Phantom acquisitions of the short-T2 enhancement experiment that respectively maximizes sample A (T2~90ms) and minimizes sample B (T2~117ms). The same quadrature coil and imaging spin-echo sequence are used: TE = 6.7ms – TR = 1.5s – Matrix size = 64x64. Left: Image acquired with the nominal B1 amplitude. Center: Image acquired at B1 + 30% with the B1-robust pulse. Right: Image acquired at B1 + 30% with the block pulse sequence.

Figure 3: Mean intensity values and standard deviation inside samples A and B obtained with the block pulse (top) and the B1-robust (bottom) sequences. Note that the B1-robust pulse also preserves the contrast outside the control range.

Figure 4: Cost function distribution of the B1-robust pulse as a function of B0 and B1 inhomogeneities. Note that the cost function value is directly related to the resulting contrast, which is homogeneous within the control range represented as a white rectangle: +/- 400Hz for B0 (+/-2ppm), and +/-35% for B1.