Turbo spin-echo (TSE) sequences represent a challenge in high-field MRI due to inhomogeneity of the transmit RF field. By directly acting on the signal, Malik et al [1] have shown how ad-hoc computed dynamical shimming settings can be improve the contrast over the whole FOV. In this paper, we formalize the design of variable refocusing angles (and phases) in the TSE sequences and we cast it in the form of a discrete-time optimal control problem. This problem is quickly solved by applying the adjoint states method (ASM) for the extended phase graph (EPG). For pTx systems, design of optimal dynamic shimming settings for TSE refocusing trains can be done in a patient-specific, on-line fashion.

The EPG can be formulated as a discrete time-variant dynamical system:

\begin{equation}\left\{\begin{array}{lcl} f_{n+1} & = &P_{n}f_{n}\\ f_0 &=& b \end{array}\right.\label{def}\end{equation}

where $$$f_n$$$ contains all configuration states at the $$$n$$$-th echo (the $$$n=0$$$ state is the equilibrium while $$$n=1$$$ denotes the states right after the first excitation).$$$P_{n}$$$ represents the transition matrix, which depends on the sequence and tissue parameters $$$\alpha_n,\phi_n,\tau,T_1$$$ and $$$T_2$$$. $$$\tau$$$ is the echo spacing.

For example, $$$P_n = R(\alpha_n,\phi_n)E(\tau,T_1,T_2)S$$$ represents dephasing ($$$S$$$) followed by decay ($$$E$$$) and refocusing ($$$R$$$). For fixed $$$\tau$$$, $$$T_1$$$ and $$$T_2$$$, the design parameters are $$$\alpha=(\alpha_0,\dots,\alpha_{N-1})$$$ and $$$\phi=(\phi_0,\dots,\phi_{N-1})$$$. The signal at echo-time $$$n-1$$$ is given by the second component of $$$f_n$$$.

For the pTx setup, the dynamic shimming settings are given in terms of the real, $$$x_{n,\ell}$$$, and imaginary, $$$y_{n,\ell}$$$, parts of the $$$\ell$$$-th channel. The effective amplitude, $$$\alpha_n$$$ and phase $$$\phi_n$$$ are the sum over all channels weighted by the spatially dependent transmit fields.

The sequence design is cast as a standard minimization problem:

\begin{equation}\left\{\begin{array}{ll} \mathrm{Minimize} & s(\alpha,\phi)\\\mathrm{s.t.} & p_i(\alpha,\phi) = 0\quad(i = 1,\dots,I) \\ & q_j(\alpha,\phi) \leq 0\quad(j = 1,\dots,J)\end{array} \right.\label{minimization}\end{equation}

where $$$s$$$, $$$p$$$ and $$$q$$$ represent, respectively, the objective, the equality and inequality constraints functions. $$$I$$$ and $$$J$$$ denote, respectively, the number of equality and inequality constraints. Examples of these functions are reported in table 1.

To solve the design problem, the derivative of the functions with respect to the design variables are needed. They can be quickly and exactly determined by the Adjoint States Method (ASM), where the adjoint states are calculated by the backward recursion formulas:

\begin{equation}\left\{ \begin{array}{lcl}\lambda_{N-1} & = & (f_N-t_N)^HC_N\\\lambda_{N-2} & = & \lambda_{N-1}P_{N-1}+(f_{N-1}-t_{N-1})^HC_{N-1}\\ & \vdots & \\\lambda_{n} & = & \lambda_{n+1}P_{n+1}+(f_{n+1}-t_{n+1})^HC_{n+1}\\ & \vdots & \\\lambda_{0} & = & \lambda_{1}P_{1}+(f_{1}-t_{1})^HC_1.\end{array}\right.\label{adjoint}\end{equation}

Afterwards, the derivatives can be easily calculated:

\begin{equation}\frac{\partial s}{\partial \alpha_{n}} = \lambda_n\frac{\partial P_n}{\partial \alpha_{n}}f_n,\quad\quad\frac{\partial s}{\partial \phi_{n}} = \lambda_n\frac{\partial P_n}{\partial \phi_{n}}f_n,\quad\quad\frac{\partial s}{\partial x_{n,\ell}} = \lambda_n\frac{\partial P_n}{\partial x_{n,\ell}}f_n\quad\mathrm{and}\quad\frac{\partial s}{\partial y_{n,\ell}} = \lambda_n\frac{\partial P_n}{\partial y_{n,\ell}}f_n. \label{derivatives1}\end{equation}

where the derivatives of the transition matrix $$$P_n$$$ can be analytically obtained.

First, we test our method for a problem whose solution is analytically known. Afterwards, we address the pTx, spatially resolved EPG in experimental settings.

Test 1: We calculate the minimum power refocusing angle train for constant signal intensity, $$$I_c$$$. The exact solution is given in [2].

Test 2: A 8 channels transceive headcoil and a 7T scanner (Philips) are employed. A spherical phantom of 10 cm diameter is used. The $$$B_1^+$$$ maps are plotted in Fig. 2a. We design amplitude and phase settings for a 3D TSE scan, TSE factor = 113 with constraints on the RF total and peak power. $$$(T_E,T_R)=(188,3200)$$$ ms; echo-spacing: 2.9 ms; resolution: 1 mm$$$^3$$$; scan duration: 10'11".

Test 3: We design a 2D TSE sequence for a volunteer's brain scan with constraints on peak and total RF power. The $$$B_1^+$$$ maps are displayed in Fig. 4a. $$$(T_E,T_R)=(60,3500)$$$ ms; echo-spacing: 12 ms; resolution: 0.3 mm$$$^2$$$; scan duration: 7'45".

Test 1 The refocusing angles for the constant signal intensity are shown in Fig.1. The analytical and algorithmic values coincide.

Test 2 The computation time for the 3D sequence is 45 seconds. The obtained amplitude and phase trains are shown in Fig. 2b. Figure 3a shows the simulated signal intensities in four slices. The obtained images are shown in Fig. 3b-c. The signal homogeneity is clearly improved.

Test 3 The computation time is 8 seconds. The obtained trains and invivo images are shown in Fig. 4b-c. The contrast and signal are more homogeneous over the whole FOV (see arrows).

We presented a generalized and efficient numerical approach to the design of pulse amplitudes and phases in TSE sequences. The flexibility and speed of the proposed approach are exploited to homogenize the response of the TSE over the whole FOV in an 8 channels pTx system at 7T. Contrast and signal are recovered. The computation time is compatible with patient-specific online implementation.

The code is available at [3]