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 details the implementation and interest of an adapted
parameterization for the computation of contrast preparation
schemes in an optimal control framework. It optimally balances the
effect of T1 and T2 relaxation, penalizes long preparation
sequences in order to improve the compromise between contrast performance and
preparation time, and significantly reduces the computation time. As an example, an *in vitro* experiment validates the contrast benefit over an inversion-recovery scheme. Finally, it offers a huge flexibility in terms of achievable
contrasts, which is demonstrated *in vivo *by a white-matter enhancement experiment on a rat brain.

Numerical
optimal control approaches often consider the complex amplitude of
the pulse at each discrete time point as an optimization variable,
which typically results in several thousands of variables. It is
proposed to only consider a finite number (N=3) of block pulses, defined by
their respective flip angle $$$\alpha^{(i)}$$$,
phase $$$\theta^{(i)}$$$
and post-pulse delay $$$\tau^{(i)}$$$,
resulting in only 9 optimization variables (Figure 1).
The
optimal control implementation is performed with GRAPE^{[5]}, which
iteratively updates the optimization variables to decrease
the following cost function:

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

in
which
$$$ \overrightarrow{M^a}$$$ and
$$$ \overrightarrow{M^b}$$$ are
respectively the macroscopic magnetization vectors of samples *a*
and *b*,
whose evolution is ruled by Bloch equations ;
$$$T$$$
is
the total preparation time (with
$$$T = \sum \tau^{(i)}$$$),
and $$$\gamma$$$
is a scalar that balances the influence of the control time
penalization term. This cost function reaches a minimum when the
contrast between both samples is maximum (with $$$\Vert \overrightarrow{M^a}(T)\Vert > \Vert \overrightarrow{M^b}(T)\Vert $$$). Note that
the proposed parameterization penalizes long pulse sequences via the
control time regularization term, which optimizes the compromise
between contrast and preparation time.
The
method is validated with two experiments carried out on a
small-animal 4.7T Bruker MR system. The first experiment emphasizes
the improvement of the compromise between preparation time and
contrast performance. It consists in contrasting two *in
vitro*
samples: a and b, whose relaxation times are [T1^{a},
T2^{a}]
= [247, 90] ms and [T1^{b},
T2^{b}]
= [381, 131] ms. The objective is to maximize sample *a*
over sample *b*,
which would intuitively be achieved with an inversion recovery
preparation because of the shorter T1 and T2 values of sample *a*.
The second experiment validates the ability of the proposed method to
perform *in
vivo*
short-T2 enhancement, by creating a non-trivial contrast between
white and gray matter in a rat brain.

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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 –
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Figure 1: Comparison of optimal control pulse design strategies.
Left: The standard scheme is based on a fine temporal sampling where
each temporal sample is a complex optimization variable. Right: The
proposed simplified scheme optimizes only N flip angles, phase terms and
post-pulse delays.

Figure 2: Comparison of the 2 preparation
schemes, using the exact same spin-echo excitation scheme for
both images (TE = 6.7ms – TR = 1.5s – Matrix size = 64x64).
Samples *a* and *b* are respectively maximized and minimized, and
displayed on the same intensity window. (a) Result with the proposed
preparation scheme with T = 256ms. Each
pulse is implemented as a 0.2 ms block pulse, whose amplitude is set
according to the flip angle.
(b) Inversion recovery preparation with TI = 256 ms.
(c) Magnetization
trajectories of both samples (on resonance) during the proposed
preparation sequence.

Figure 3: *In vivo* results on a rat brain. Left:
Standard T2 contrast acquired with a fast spin-echo sequence (TE =
56ms). Right: Short-T2 enhanced image acquired with the optimized
preparation scheme, and a spin-echo sequence (TE = 8.5ms – TR = 4s
– Matrix size = 128x128). The corpus callosum (*Cc*: T2~58ms) and
thalamus (*Th*: T2~62ms) appear clearly enhanced compared to the
surrounding longer-T2 gray matter structures such as the hippocampus
(*Hc*: T2~72ms).