Franz Patzig^{1}, Bertram Wilm^{1}, Kasper Lars^{1}, Maria Engel^{1}, and Klaas Pruessmann^{1}

A major problem of single-shot acquisition techniques are distortions due to local offsets of the static magnetic B0 field. To avoid relying on separately acquired field maps, the object and the B0 map can be jointly estimated, which usually involves updating object and B0 map in an alternating fashion. A new optimization strategy to solve the non-convex B0 sub-problem is suggested. The number of unknowns is significantly reduced by modelling the B0 maps by a smaller basis and a modified version of the simulated annealing algorithm is implemented to better handle the non-convexity. First in-vivo results are presented.

*Joint Estimation of B0
map and object*

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The aim of JE is
to estimate the object $$$o(x)$$$ and the field
map $$$\omega_0(x)$$$ from the
acquired signal $$$s(t)$$$ by minimizing
the cost function $$$L$$$

$$L(\omega_0(x), o(x)) = \left\lVert s(t) - E(k(t), \omega_0(x)) \cdot o(x) \right\rVert^2$$

$$\hat{\omega}_0(x), \hat{o}(x) = \text{argmin}\;L(\omega_0(x), o(x))$$

where $$$E$$$ is the encoding matrix with entries

$$E_{mn} = \exp(- i \omega_0(x_n)t_m) \exp( - i 2 \pi k(t_m) \cdot x_n)$$.

The linear optimization of updating the object guess (conjugate gradient algorithm [4]) is employed as in reference [3], whereas the B0 optimization is modified.

* *

*Parameter Reduction*

* *

To address the excessive parameter space in the B0 step, the B0 map is modelled as a linear combination of smooth (DCT) basis functions (Fig. 1A). This indirectly regularizes the B0 map to be smooth while significantly reducing the number of unknowns as compared to representing B0 on a voxel basis (NxN-dim). Explicit regularization as implemented in [3] is thus not required. The cost function $$$L$$$ as a function of DCT coefficients remains non-convex (Fig. 2). The B0 maps were modelled such that a maximum deviation of a fully resolved map of 6 Hz is achieved, which was possible by using 437 basis functions (Fig. 1).

*Improved Optimization Algorithm*

* *

A modified version of the SA algorithm [5] was implemented which can escape local minima (Fig. 2). The range of the DCT coefficients is constrained in the optimization according to pre-acquired B0 maps (gray lines Fig. 1A). In addition multiple coefficients (10% of the total amount) were changed in each step of the SA algorithm to increase convergence speed [6].

*MR Experiments and Reference B0 Map Calculation*

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MR Scanning was performed on a 3T MR system (Philips Healthcare, Best, The Netherlands) using an 8-channel head coil array equipped with 16 magnetic field sensors (Skope MR Technologies, Zurich Switzerland) [7] to record the actual k-space trajectory. An EPI up/down and a spiral in/out sequence (two interleaves) with a TE of 35 and 20 ms respectively was played out (FOV: 22 cm, in-plane resolution: 1 mm). An initial B0 guess was fitted from the spiral in and out phase images. As reference, a field map was fitted from a 2-echo gradient echo sequence.

*Spiral In/Out*

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In comparison to the non B0 corrected image reconstruction (Fig. 3(A,E)) significant improvement in the reconstructed object is achieved using the initial B0 map guess (Fig. 3(B,F)). The B0 map found by the JE step further removed blurring artifacts from the image (Fig. 3(C,G)), which showed best agreement with the reference image (Fig. 3(D,H)).

*EPI Up/Down*

* *

For the EPI experiment, the field map found by JE had an overall similar pattern as the field map found by the spiral in/out data and the reference field map (Fig. 4).

[1] Jezzard, Balaban – Correction for Geometric Distortion in Echo Planar Images from B0 Field Variations

[2] Matakos et al. – Estimation of geometrically undistorted B0 inhomogeneity maps

[3] Sutton et al. – Dynamic Field Map Estimation Using a Spiral-In / Spiral-Out Acquisition

[4] Barmet et al. - Sensitivity encoding and B0 inhomogeneity - A simultaneous reconstruction approach, Proceedings of the ISMRM 2005

[5] Kirkpatrick et al. – Optimization by Simulated Annealing

[6] Berthiau et al. – Enhanced Simulated Annealing for Globally Minimizing Functions of Many-Continuous Variables

[7] Kennedy et al. – An industrial design solution for integrating NMR magnetic field sensors into an MRI scanner

Example
of a DCT approximation of an in-vivo B0 map: A maximum deviation of 5.8 Hz from
the original B0 map (C) was achieved in the approximated map (B) using 437
basis functions (yellow line in (A)). The gray lines in (A) show the limits
employed in the SA optimization determined from pre-acquired in-vivo B0 maps.

Examples
of the cost function in the vicinity of the global minimum for a spiral in/out
data as a function of DCT basis functions with n/2 and m/2 cycles in x and y
direction (DCT(n,m)). Both, (A) and (B) exhibit several local minima in
periodical distances, resulting from phase wraps occurring when $$$\hat{\omega}_0$$$ satisfies $$$\varphi_0 = T_E \cdot \omega_0 \approx T_E \cdot \hat{\omega}_0$$$.

Results
of the proposed algorithm for the spiral in/out acquisition. Comparison of different
B0 map estimates and the according reconstructed images: constant B0 map (= 0)
(A,E); distorted B0 map from unwrapped phases of spiral in/out phase images (B,F);
B0 map after JE step using (B) as initial guess (C,G); B0 map fitted from
two GE phase images for comparison (D,H).

B0
map comparison for different trajectories. Reference B0 map from a
multiple-echo GRE sequence (A). B0 map after JE step using a distorted B0 map as initial guess for the spiral in/out trajectory (B) (as shown in Fig. 3)
and for the EPI up/down trajectory (C).