The variety of achievable contrasts in Magnetic Resonance Imaging (MRI) makes it a highly flexible modality, well suited for anatomical and functional imaging. Usually, acquisition parameters such as echo time (TE), repetition time (TR), flip angle are tuned to obtain various image weightings. In this article, optimal control theory is used to design excitation pulses to produce, in an optimal way, a user-defined contrast. Furthermore, the integration of such pulses in an imaging sequence is detailed and both simulation and real experiments are performed, which can be seen as a proof-of-concept for further developments.

Optimal control theory, via the application of the Pontryargin Maximum Principle¹ (PMP), is a powerful tool to compute the time dependent control that optimizes the trajectory of a given dynamic system. In MRI, the dynamics are ruled by the Bloch equations, which indicate the time dependent magnetization of a given spin:

$$$ \frac{d}{dt}\left(\begin{array}{c}M_x\\ M_y \\ M_z\end{array}\right) = \left(\begin{array}{c}-\frac{1}{T_2} & \Delta_{B_0} & -\omega_y \\ -\Delta_{B_0} & -\frac{1}{T_2} & \omega_x \\ \omega_y & - \omega_x & -\frac{1}{T_1} \end{array}\right)\left(\begin{array}{c}M_x\\ M_y \\ M_z\end{array}\right) + \left(\begin{array}{c}0\\ 0 \\ \frac{M_0}{T_1}\end{array}\right) $$$

Controlling the transverse magnetic field ($$$\overrightarrow{w} = \left(\begin{array}{c} w_x\\ w_y\end{array}\right)$$$) allows us to bring the spins' magnetization in a state that optimizes the desired contrast^(2-4). In the following experiments, the saturation contrast between 2 spins is optimized. Let us define $$$\overrightarrow{M^a}(t)$$$ and $$$\overrightarrow{M^b}(t)$$$ respectively the magnetization of spin $$$a$$$ and spin $$$b$$$. The saturation contrast is defined such that at the end of the excitation pulse ($$$t_f$$$), one spin is saturated while the magnetization norm of the other one is maximized. Following the PMP formalism, the saturation contrast is expressed as a cost function to be minimized: $$$C(\overrightarrow{w}) = ||\overrightarrow{M^a}(t_f)||^2 - ||\overrightarrow{M^b}(t_f)||^2$$$. The optimal magnetization and costate ($$$\overrightarrow{P}$$$) trajectories must also satisfy the Hamiltonian equations:

$$$\dot{\overrightarrow{M}}(t) = \frac{\partial H}{\partial \overrightarrow{P}} \ \ \ \ ; \ \ \ \ \dot{\overrightarrow{P}}(t) =- \frac{\partial H}{\partial \overrightarrow{M}}$$$

with $$$H = \overrightarrow{P}.\dot{\overrightarrow{M}}$$$ and the following boundary conditions:

$$$\overrightarrow{M^{(a,b)}}(t_0) = \left(\begin{array}{c}0\\ 0\\M_0^{(a,b)}\end{array}\right) \ \ \ \ ; \ \ \ \ \overrightarrow{P^{(a,b)}}(t_f) = - \frac{\partial C}{\partial \overrightarrow{M^{(a,b)}}(t_f)}$$$

These equations are solved numerically using GRAPE⁵, which is a gradient descent algorithm that iteratively reduces the cost function by updating the control field.

The contrast experiment is performed on both numerical and real phantoms with similar physical properties. It is composed of 6 samples which relaxation times given in Table 1. The goal is to maximize the saturation contrast between sample 2 and sample 5. The numerical experiment is performed with the ODIN MRI simulator⁶. A $$$B_0$$$ inhomogeneity map with random spatial distribution ranging from 0 to 20Hz is used. The real phantom's samples are built with various concentrations of agar gel and nickel sulfate. In terms of imaging sequence, a 3D FLASH sequence without slice selection is used since the generated pulses are not yet frequency selective. TE is set to its shortest value so that the acquisition takes place right at the end of the excitation, since the optimal magnetization state is reached right at that time. TR is fixed to 3s which to ensure complete longitudinal magnetization recovery. The pulse duration are fixed to 400ms in simulation experiments and 250ms in MRI experiments. The MRI acquisition is performed on a small-animal 4.7T Bruker MRI using a 40mm quadrature mouse coil and an acquisition matrix of 64x64x8.Figure 1 shows the result of the numerical experiments, with the optimal pulse, the spins' trajectories in the YZ plane of the Bloch sphere, and the simulated images. Note that each magnetization trajectory represents a specific $$$B_0$$$ offset for a given spin. Figure 2 shows the central slice result for the MRI experiments. In this context, pulses are computed for inhomogeneities ranging from 0 to 60Hz.The reasonable match between simulation and MRI experiments shows that optimal control theory can be transferred on real MR imagers. In both experiments, the desired contrast is reached, which illustrates the robustness to $$$B_0$$$ field inhomogeneities and other acquisition variables such as $$$B_1$$$ inhomogeneities. The achieved saturation contrast is also quite selective since only the spin to be saturated has almost no detectable signal. The optimal nature of the designed pulse could also be used to quantitatively estimate the maximum amount of contrast one can obtain between given tissues. The quality of the results presented above clearly depends on a reliable estimation of the tissue relaxation times. However, uncertainties on relaxation times could also be inserted in the model in the same manner as $$$B_0$$$ field inhomogeneities were considered, to further improve the pulse robustness.