Eric Van Reeth^{1}, Hélène Ratiney^{1}, Sophie Gaillard^{1}, Michael Tesch^{2}, Olivier Beuf^{1}, Steffen Glaser^{2}, and Dominique Sugny^{3,4}

Optimal control RF pulse design has recently been proposed to address
the optimization of image contrast in MRI - in order to explore the
theoretical contrast bound of a given imaged system. Their use has
recently been validated on a real MRI scanner to contrast
various *in vitro* samples. This abstract extends these results
to *in vivo* applications, and shows that contrasts obtained
with standard weighting strategies on rat and mouse brains can be
improved or inverted. This
demonstrates
both
the
interest
and flexibility
that one can get
when using optimal contrast pulses for *in
vitro*
and *in
vivo*
applications.

Optimal control
problems, via the resolution of the Pontryagin Maximum Principle^{4}
(PMP), consist of computing the control that optimizes a user-defined
cost function, given the dynamical evolution of the considered
system. When applied to MRI, this comes down to the computation of
the RF pulse that leads to a user-defined magnetization state, whose
evolution is ruled by the Bloch equations.

In the context of
contrast optimization, the magnetization target state depends on the
desired contrast. Let $$$\overrightarrow{M_a}(t)$$$ and $$$\overrightarrow{M_b}(t)$$$ represent the magnetization
temporal evolution of respectively spins $$$a$$$ and $$$b$$$. The saturation
contrast is defined so that one spin
is saturated while the magnetization norm of the other one is
maximized at time ($$$t_f$$$):

$$C(w) = ||\overrightarrow{M_a}(t_f)||^2 - ||\overrightarrow{M_b}(t_f)||^2$$

where $$$\overrightarrow{w} = (w_x, w_y)$$$ is the pulse to be optimized. The pulse duration ($$$t_f$$$) is set long enough so that the pulse can optimally combine the effects of excitation and relaxation (T1 and T2) to produce the desired contrast. Following the PMP formalism, the optimal magnetization ($$$\overrightarrow{M}$$$) and costate ($$$\overrightarrow{P}$$$) trajectories must satisfy the Hamiltonian equations:

$$\dot{\overrightarrow{M}} =\frac{dH}{d\overrightarrow{P}} \quad \text{and} \quad \dot{\overrightarrow{P}} = -\frac{dH}{d\overrightarrow{M}}$$

with the following boundary conditions:

$$\overrightarrow{M}(t_0) = (0,0,M_0)^T \quad \text{and} \quad \overrightarrow{P}(t_f) = -\frac{dC}{d\overrightarrow{M}(t_f)}$$

The numerical
resolution of these equations is performed with a gradient-ascent
algorithm (GRAPE^{5}), which iteratively reduces the cost
function by updating the control field. Magnetic field
inhomogeneities are taken into account in the system dynamics to make
the pulse robust to deviations from the Larmor frequency.

In vivo acquisitions were performed on adult mouse and rat brains in accordance with the rules of our institutional ethic committee on animal experimentation, on a 4.7 T Bruker MR system using quadrature coils. In both experiments, the optimal contrast pulse is used as a preparation pulse that creates longitudinal magnetization difference, i.e. contrast along the $$$M_Z$$$ axis. The magnetization is subsequently flipped onto the transverse plane with a slice-selective 90° pulse and refocused with a 180° pulse. TE is set as short as possible in order to preserve the contrast created at the time of acquisition. TR is set long enough to ensure complete longitudinal magnetization recovery. A RARE acceleration factor of 8 is used, with a centric encoding scheme.

The mouse experiment consists of minimizing the signal coming from the brain ($$$[T1^b , T2^b] = [920, 66]$$$ ms) while maximizing the signal coming from the surrounding muscles. ($$$[T1^m , T2^m] = [1011, 30]$$$ ms). Average relaxation times were estimated by fitting the water peak inside a spectroscopy voxel at different TE and TR values. Figure 1 shows the optimal pulse amplitude together with the magnetization evolutions of the brain and muscle during the application of the pulse. The resulting images are shown in Figure 2. The rat experiment consists of maximizing the hippocampus signal while minimizing the thalamus signal. Average relaxation times are estimated to: $$$[T1^h , T2^h] = [921, 68]$$$ ms and $$$[T1^t , T2^t] = [832, 63]$$$ ms. Figure 3 compares standard T2 weighting, when TE is set to maximize the desired contrast (65.4 ms), with the image obtained with the optimal contrast pulse.

1. Lapert M, Zhang Y, Janich M, Glaser SJ, Sugny D. Exploring the physical limits of saturation contrast in magnetic resonance imaging” Scientific Reports, Nature Publishing Group, 2012, 2

2. Bonnard B, Cots O. Geometric numerical methods and results in the contrast imaging problem in nuclear magnetic resonance, Mathematical Models and Methods in Applied Sciences, World Scientific, 24, 187-21, 2014

3. Van Reeth E, Ratiney H, Tesch M, Glaser SJ, Sugny D. Optimizing MRI Contrast with B1 pulses using optimal control theory, IEEE 12th International Symposium on Biomedical Imaging (ISBI), 2016

4. Pontryagin L S, Mathematical theory of optimal processes, CRC Press, 1987.

5. Khaneja N, Reiss T, Kehlet C, Schulte-Herbrüggen T, Glaser SJ. “Optimal control of coupled spin dynamics: design of nmr pulse sequences by gradient ascent algorithms,” Journal of Magnetic Resonance, vol.172, no. 2, pp. 296 – 305, 2005.

Figure 1: RF pulse magnitude and
magnetization trajectories during the application of the pulse.
Trajectories show that complete saturation is reached for the brain
while significant residual longitudinal magnetization is left for the
muscle. Magnetization trajectories are shown for a range of B0
values of +/- 1 kHz around the Larmor frequency.

Figure 2: Mouse brain experiment
results that aims at **saturating the brain** and
**maximizing the surrounding muscles**
signal. Left: image
obtained without contrast preparation (proton density contrast).
Right: same sequence with optimal contrast preparation. For both
images: TR = 5 s, TE = 9.4 ms, matrix size = 192x192, slice thickness
= 1.25 mm.

Figure 3: Rat brain experiment
results that aims at **maximizing the hippocampus **signal while
**minimizing the thalamus** signal. Same window level is used for
both images. Left: image acquired at optimal TE = 65.4 ms for T2
contrast (TR = 5 s, matrix size = 256x128, slice thickness = 1.5 mm).
Right: image obtained with the optimal contrast pulse (TR = 5 s, TE =
8.1 ms, matrix size = 128x128, slice thickness = 1.5 mm).