Signal-domain optimization metrics for MPRAGE RF pulse design in parallel transmission at 7 Tesla

Vincent Gras^{1}, Alexandre Vignaud^{1}, Franck Mauconduit^{2}, Michel Luong^{3}, Alexis Amadon^{1}, Denis Le Bihan^{1}, and Nicolas Boulant^{1}

In a joint optimization of the excitation and inversion pulses of the MPRAGE sequence, the standard FA homogenizing approach typically consists in minimizing the quantity $$$U_{\alpha}=\|f_{\alpha}\|_2^2$$$ where $$$\|\cdot\|_2$$$ refers to the voxel-by-voxel summation of the local FA errors using the $$$\mathcal{L}_2$$$ norm and where $$$f_{\alpha}$$$ is the voxel-wise measure of the FA deviation:

$$f_{\alpha} = \left(1-\frac{\alpha_{\text{Exc}}}{\hat{\alpha}_{\text{Exc}}}\right)^2+\left(1-\frac{\alpha_{\text{Inv}}}{\hat{\alpha}_{\text{Inv}}}\right)^2,$$

$$$\alpha_{\text{Exc}}$$$/$$$\hat{\alpha}_{\text{Exc}}$$$ and $$$\alpha_{\text{Inv}}$$$/$$$\hat{\alpha}_{\text{Inv}}$$$ representing the actual/nominal FA for the excitation and
inversion pulses respectively. The so-called *signal fidelity* is now proposed as an alternative to the FA deviation, which relies on the actual, $$$s(\text{T}_1)$$$, and nominal, $$$\hat{s}(\text{T}_1)$$$, MPRAGE signal^{2} of the central echo, scaled by $$$\text{M}_0$$$. It is defined as:

$$f_s^2 = \frac{\int_I (s-\hat{s})^2 d\text{T}_1}{\int_I \hat{s}^2 d\text{T}_1},$$

where $$$I$$$ denotes a $$$\text{T}_1$$$ interval covering the values found in brain tissue. A method minimizing $$$f_s$$$ may however penalize the local contrast, defined here as the relative signal difference between two voxels exhibiting two different $$$\text{T}_1$$$ values, i.e.:

$$c_s(\text{T}_1,\text{T}'_1)=2 \frac{s(\text{T}_1)-s(\text{T}'_1)}{s(\text{T}_1)+s(\text{T}'_1)}.$$

We thus equally define the contrast fidelity as the distance between the actual ($$$c_s$$$) and nominal contrast ($$$c_{\hat{s}}$$$):

$$f_c^2 = \frac{\iint_{I \times I} (c_s-c_{\hat{s}})^2 d^2\text{T}_1}{\iint_{I \times I} c_{\hat{s}}^2 d^2\text{T}_1},$$

thus vanishing if the actual signal matches the
nominal signal up to a multiplicative constant. We now replace the
objective $$$U_{\alpha}$$$ by the weighted sum of a signal and contrast
fidelity terms, to form the new objective function:

$$U_{f,\lambda}=\|F_s\|_2^2+\lambda \|F_c\|_2^2,$$

whereby $$$F_s$$$ and $$$F_c$$$ are discrete approximations of the continuous integrals $$$f_s$$$ and $$$f_c$$$ and where $$$\lambda$$$ is a weighting factor to be adjusted. The metrics $$$f_{\alpha}$$$, $$$f_s$$$ and $$$f_c$$$ are displayed as functions of $$$\alpha_{\text{Exc}}$$$ (x-axis) and $$$\alpha_{\text{Inv}}$$$ (y-axis) in Fig. 1 for TR/TI=2.6/1.1s, $$$\hat{\alpha}_{\text{Exc}}$$$/$$$\hat{\alpha}_{\text{Inv}}$$$ =9/180° and $$$I = [1,2.3]$$$ s. Interestingly, the signal domain metrics allow compensating a FA < 180° for the inversion pulse by reducing concomitantly the excitation FA by the proper amount. In Fig. 1.d., the ideal excitation FA is always equal to 9° for $$$f_{\alpha}$$$, while for the other metrics, it depends on $$$\alpha_{\text{Inv}}$$$. The proposed new optimization metric was thus investigated first in simulation to check if it indeed could return further RF pulse performance or lower the SAR, and was finally tested in vivo with MPRAGE brain acquisitions. Experiments were performed on a Magnetom 7T Siemens scanner (Siemens Healthcare, Erlangen, Germany) equipped with a home-made 8 channel pTx coil and under real-time local SAR supervision^{6,7}.

[1] Mugler and Brookeman. MRM 1990;15:152-157. [2] Deichmann et al. NeuroImage 2000;12:112-127. [3] Cloos et al. MRM 2012;67:72–80. [4] Cloos et al. NeuroImage 2012;62:2140-2150. [5] Hoyos-Idrobo et al. IEEE TMI 2014;33:739-748. [6] Eichfelder and Gebhardt. MRM 2011;66:1468-2594. [7] Graesslin et al. MRM 2012;68:1664-1674.

Figure 1. Plot
of a) the FA deviation, b) the signal and c) contrast fidelity measures and d)
the interplay between the excitation and inversion pulses in the minimization of *f*_{α}, *f*_{s} or *f*_{c}.

Figure 2. Comparison
of the FA domain (standard approach) and signal domain optimization (proposed
approach) in MPRAGE images acquired at 7T.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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