Fast and accurate B₁₊-mapping is an important prerequisite for many MRI-techniques at high and very high field-strength. The Bloch-Siegert (BS) method was presented recently as a fast, yet robust and accurate technique[1]. Nevertheless, specific-absorption-rate (SAR) is very high, which limits the minimal TR such that the acquisition-time for a whole 3D-volume is in the order of minutes. One possible acceleration strategy would be a faster data acquisition using spiral-trajectories or echo-planar readouts as proposed in [2] and [3], but spiral-trajectories are not commonly available on clinical scanners and both are prone to artifacts caused by B₀-inhomogeneities or gradient imperfections. Acceleration is also possible by applying compressed-sensing strategies as proposed in[4], where SPIRiT-reconstruction was used. In this work we investigate a highly accelerated reconstruction integrated B₁₊-mapping method based on a problem specific regularization.

As described in [1] the phase-shift $$$\phi_{BS}$$$ due to a BS-pulse depends on the peak-magnitude $$$B_{1,peak}$$$ of the RF-pulse and the pulse-constant $$$K_{BS}$$$[1].

$$\phi_{BS}=B_{1,peak}^2\,\cdot\,K_{BS}$$

The B₁₊-map can easily be calculated from this phase-shift. To remove undesirable background phase, the acquisition is performed twice with positive and negative resonance-offset. The signals of the BS-encoded images $$$I_+$$$ and $$$I_-$$$ are

$$I_+=\left|M\right|e^{j\left(\phi_0+\phi_{BS}\right)}$$

$$I_-=\left|M\right|e^{j\left(\phi_0-\phi_{BS}\right)}$$

with magnetization $$$M$$$, background-phase $$$\phi_0$$$ and the desired BS-phase $$$\phi_{BS}$$$. Conventionally both images are reconstructed independently and $$$\phi_{BS}$$$ is obtained by a complex division of both images. Our approach is a joint-reconstruction of image and BS-phase and consists of two steps: First we reconstruct the morphologic structure out of the positive BS-encoded dataset $$$k_+$$$ by solving the following optimization-problem.

$$\hat{u}=\arg\min_u\parallel\,k_+-\mathcal{F}\left(u\right)\,\parallel_2^2+\lambda\,\cdot\,TGV\left(u\right)$$

$$\hat{u}\,\,\hat{=}\,\left|M\right|e^{j\left(\phi_0+\phi_{BS}\right)}=I_+$$

As it was shown by [5] a piecewise-smooth constraint is suitable for reconstructing morphologic MR-images from under-sampled data, which is enforced by a TGV-regularization[6]. This optimization-problem is solved using the primal-dual-algorithm proposed by [7]. $$$\mathcal{F}$$$ represents the Fourier-operator including under-sampling-pattern and coil-sensitivities, which are estimated using the Walsh-algorithm[8]. The result of this optimization-problem $$$\hat{u}$$$ corresponds to $$$I_+$$$. The whole morphologic information is represented by $$$\hat{u}$$$. The second step is now to reconstruct $$$\phi_{BS}$$$ out of the negative BS-encoded dataset $$$k_-$$$ and $$$\hat{u}$$$. Since we know that the B₁₊-field is spatially smooth, we enforce this property by employing H₁-regularization on $$$\hat{v}$$$. Because of the mathematical formulation of the optimization-problem all morphologic information cancels out of the result and $$$\hat{v}$$$ only contains the double-BS-phase $$$2\phi_{BS}$$$ with magnitude one.

$$\min_v\parallel\,k_--\mathcal{F}\left(\hat{u}\,\cdot\,v\right)\,\parallel_2^2+\mu\parallel\nabla\,v\parallel_2^2$$

$$\hat{u}\cdot\hat{v}\,\,\hat{=}\,\left|M\right|e^{j\left(\phi_0+\phi_{BS}\right)}\cdot\hat{v}=\left|M\right|e^{j\left(\phi_0-\phi_{BS}\right)}\Rightarrow\hat{v}\,\hat{=}\,\,e^{-j2\phi_{BS}}$$

We applied the method to in-vivo 3D-datasets at 3T (Skyra, Siemens, Erlangen Germany) with the following sequence-parameters: FOV=230²x64mm, matrix-size=128²x32, TR_min=95ms (SAR-constraint), TE=13.5ms, $$$\alpha=25°$$$. A Gaussian BS-pulse with on-resonant flip-angle $$$\alpha_{BS}=1000°$$$, resonance-offset $$$\Delta\omega_{RF}=4kHz$$$ leading to $$$K_{BS}=53.4rad/G^2$$$ was used, with an acquisition described in [9], to correct for phase-drifts. The reconstruction-results are compared to a conventional BS-reconstruction of the fully-sampled dataset. The undersampling-pattern only consists of a symmetric auto-calibration-region in the k-space-center with $$$n_{ACL}$$$ auto-calibration-lines (ACL) in both phase-encoding directions.

Fig.1 shows the reconstructed B₁₊-map in the brain with their difference to the fully-sampled reference (in % of the nominal flip-angle) in transversal, sagittal and coronal orientation for different effective-acceleration-factors $$$acc_{eff}$$$. In Fig.2 maximum, median and both percentiles (25th and 75th) of the reconstruction-error (absolute difference in % of the nominal flip-angle) inside a volume-of-interest (VOI) (blue box in Fig.1) are shown for different $$$n_{ACL}$$$. Fig.3 illustrates a profile of the rel. B₁₊-magnitude through the transversal plane to show the reconstruction-quality compared to the fully sampled reference and a conventional low-resolution reconstruction. In Tab.1 you can see the achievable effective-acceleration-factors $$$acc_{eff}$$$ and the corresponding RMSE for the proposed method and the low-resolution estimate. An acceleration of >100 is achievable with a RMSE below 1%. Conventional low-resolution reconstruction of the under-sampled data leads to totally over-smoothed field maps, resulting in much higher RMSE-values (Tab.1).

The acquisition of the full dataset with TR=95ms (min. due to SAR-constraint) leads to an acquisition-time of ~12min. With standard acceleration-strategies like Partial-Fourier or GRAPPA, acquisition-times of 4-5min are possible, but this is far too long for a precalibration step. Using $$$\bf{n_{ACL}=10^2}$$$ we reach an effective-acceleration-factor of $$$\bf{acc_{eff}=40.9}$$$ (Tab.1). For e.g. abdominal-imaging this renders full-liver coverage possible within typical breath-hold times (<20s) with an average error of ~0.5%. Only in low-signal regions and near the object boundary the max-error is ~3,9%. Allowing for maximum error of 1% in average ~110-fold scan-time reduction is achievable using only $$$\bf{n_{ACL}=6^2}$$$ calibration-lines for patients with extreme breath-hold capabilities, leading to an acquisition-time of ~7s. The results were shown in the human brain, to allow the acquisition of a fully-sampled dataset without motion artifacts.

We successfully introduced a novel reconstruction-method for BS-B₁₊-mapping which allows for vast under-sampling and therefore drastic scan-time reduction.