Highly accelerated Bloch-Siegert B1+ mapping using variational modeling

Andreas Lesch^{1}, Matthias SchlĂ¶gl^{1}, Martin Holler^{2}, and Rudolf Stollberger^{1,3}

Fast and accurate B_{1+}-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 B_{0}-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 B_{1+}-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
B_{1+}-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 B_{1+}-field is spatially
smooth, we enforce this property by employing H_{1}-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=230^{2}x64mm, matrix-size=128^{2}x32, 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 B_{1}_{+}-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. B_{1+}-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-B_{1+}-mapping which allows for vast under-sampling and therefore drastic scan-time reduction.

[1] Sacolick, L. I., Wiesinger, F., Hancu, I., & Vogel, M. W. (2010). B1 mapping by Bloch-Siegert shift. Magnetic Resonance in Medicine, 63(5), 1315-1322.

[2]
Saranathan, M., Khalighi,
M. M., Glover, G. H., Pandit, P., & Rutt, B. K. (2013). Efficient
bloch-siegert B1+ mapping using spiral and echo-planar readouts. Magnetic Resonance in Medicine, 70(6), 1669-1673.

[3]
Khalighi,
M. M., Glover, G. H., Pandit, P., Hinks, S., Kerr, A. B., Saranathan, M., & Rutt, B. K. (2011) Single-Shot Spiral Based Bloch-Siegert B1+ Mapping. Proc. Intl. Soc. Mag. Reson. Med. 19, 578

[4] Sharma, A., Tadanki, S.,
Jankiewicz, M., & Grissom, W. A. (2014). Highly-accelerated
Bloch-Siegert| B1+| mapping using joint autocalibrated parallel image
reconstruction. Magnetic Resonance in Medicine, 71(4), 1470-1477.

[5] Knoll, F., Bredies, K., Pock, T., & Stollberger, R. (2011). Second order total generalized variation (TGV) for MRI. Magnetic resonance in medicine, 65(2), 480-491.

[6] Bredies, K., Kunisch, K., & Pock, T. (2010). Total
generalized variation. SIAM Journal on Imaging Sciences, 3(3), 492-526.

[7]
Chambolle, A., & Pock, T. (2011). A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision, 40(1), 120-145.

[8] Walsh, D. O., Gmitro, A. F., & Marcellin, M. W. (2000). Adaptive reconstruction of phased array MR imagery. Magnetic Resonance in Medicine, 43(5), 682-690.

[9] Lesch, A., Petrovic, A., Stollberger, R., (2015). Robust Implementation of 3D Bloch Siegert B1 Mapping. Proc. ISMRM 23rd p.2381

Figure 1: Fully-sampled reference, proposed reconstruction-method and its difference to the reference for 12^{2} and 6^{2} ACL ($$$acc_{eff}=28.4$$$ and $$$acc_{eff}=113.8$$$) in each
phase-encoding direction. The B_{1+}-field is shown as relative magnitudes in % of the nominal flip-angle.

Figure 2: Median, maximum and both percentiles (25th and 75th)
of the absolute difference between the proposed method and the
fully-sampled reference inside a VOI shown in Fig.1 for different sizes of the ACL-region $$$n_{ACL}$$$.

Figure 3:
Magnitude profile of rel. B_{1+}-field in the transverse plane
comparing reference, proposed reconstruction and the low-resolution estimate.
The B1+-field is shown in relative magnitudes in % of the nominal
flip-angle.

Table 1:
Number of ACL $$$n_{ACL}$$$ in relation to the achieved
effectice-acceleration-factor $$$acc_{eff}$$$ and the root-mean-squared-error
(RMSE) of the proposed method and the low-resolution estimate in comparison to
the full-sampled reference inside the VOI.

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

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