Comparison of FASTMAP and B0 Field Map Shimming at 4T: Magnetic Field Mapping Using a Gradient-Echo Pulse Sequence

Local susceptibility variations result in B0 field inhomogeneities, causing distortions and signal losses in MR imaging. Susceptibility variations become stronger with increasing B0 magnetic field strength. Active shimming is used to generate corrective magnetic fields, which can be used to improve B0 field homogeneity. FASTMAP is an effective shimming technique for computing optimal coil currents, which uses data from six projection directions (or columns): this technique is routinely used for shimming cubic volumes of interest (VOIs). In this paper, we propose several improvements to FASTMAP at 4T. For each shim coil, using a modified 3D gradient-echo pulse sequence, we compute B0 inhomogeneity maps and project them onto eight 1 and 2 order spherical harmonic functions. This process is repeated for shim currents between −15,000 to 15,000 with increments of 5000 Digital to Analog Converter (DAC) units, and is used to compute the gradient between spherical harmonic coefficients and DAC values for all 8 shim coils—along with the R values of linear fits. A method is proposed (based on R values) to further refine optimal shim currents in respective coils. We present an analysis that is numerically robust and completely flexible in the selection of the VOIs for shimming. Performance analyses, phantom results, and in vivo results of a human brain are presented, comparing our methods with the FASTMAP method.


Introduction
Automatic shimming for optimizing magnetic field uniformities is highly desirable in MR spectroscopy. Objects are often heterogeneous and contain intrinsically unshimmable field variations due to rapid susceptibility changes, which can lead to distortions of the lineshape obtained from the volume [1] [2] [3]. Several shimming techniques using volumes of interest (VOIs) have been proposed in order to improve the B 0 field homogeneity [3]- [8]. For example, Holtz et al. (1988) used a surface coil [3] and the signal integral of the free induction decay (FID) over a VOI, iteratively, for field optimization [7]- [12]; however, this technique is time-consuming and impractical for many in vivo applications. Moreover, the FID (or the spectral peak amplitudes) is sensitive to changes in shim settings [7] [9] [10] [11] [12] [13].
The use of linear shim coils is highly advantageous in MR imaging [10] [11].
FASTMAP works well over reasonably homogeneous volumes with moderate field inhomogeneity [20] [23]. This technique performs well in applications probing smaller volumes (e.g., single voxel spectroscopy) [24], but not larger ones. For example: during human brain imaging studies at high-fields where VOIs are extended into the frontal and inferior brain regions; where off-resonance may be present, or whenever fields rapidly change.
The FASTMAP technique incorrectly assumes that shim coil fields can be fully characterized by a minimal set of spherical harmonics [25] [26]. Therefore, a shimming technique less susceptible to signal voids than projection based methods, and capable of handling arbitrarily shaped VOIs is highly desirable.
In this paper, we follow the same general principals outlined in FASTMAP but propose several improvements. In brief, we propose combining spherical harmonic functions and linear least squares fitting for estimating field inhomogeneity. This method entails the computation of 3D phase images and the determination of first and second-order spherical harmonic coefficients for specific shim currents, by changing the Digital to Analog Converter (DAC) settings, which control voltages across different shim coils. The spherical harmonic calibration constants are then determined by computing the gradients between spherical harmonic coefficients and the DAC values of each coil-followed by a first order correction [see Equation (5)]. Our analysis is numerically robust and completely flexible when selecting VOIs for shimming. A performance analysis comparing our technique with FASTMAP, on a phantom and a human brain, demonstrates Journal of Electromagnetic Analysis and Applications how our proposed method outperforms the FASTMAP technique in terms of B 0 homogeneity.

Imaging
The imaging protocol employed a modified 3D gradient-echo pulse sequence (see Figure S1 in supplementary material), which was used to obtain B 0 field maps. Frequency distortion correction (along the read-out direction) was performed on B 0 field maps. All acquisitions used a 256 × 256 ×256 mm field of view; a 128 × 64 × 64 acquisition matrix; a 10˚ pulse flip angle; a repetition time (TR) of 16 ms, and echo times of 5.25 ms and 7 ms. The data was acquired in the axial orientation, with a slab-selective pulse used for excitation.
After acquisition, inverse Fourier transformation was performed on the acquired 3D k-space data. Subsequently, 3D phase unwrapping was performed on the resultant phase images as necessary. Frequency maps were then computed from the difference of the two phase images (acquired at different echo times) with the following equation: After calculation of the 3D frequency maps, voxels corresponding to the selected VOI were extracted. All image reconstruction steps were performed in Matlab (Mathworks, Natick, MA).
Images were obtained in both a phantom and in-vivo. The phantom was a water sphere with a diameter of 178 mm. In-vivo images of a human head were obtained from a single subject. Consent was obtained with an IRB protocol approved by the University of Cincinatti School of Medicine. The VOI for shimming was defined as the entire spherical phantom and the brain only, respectively (see supplementary material for details). B 0 field maps were acquired both prior to, and after, the shimming procedure outlined below.

Constructing Calibration Tables for Active Shimming
A one-time procedure was performed to construct shim calibration tables for active shimming. B 0 field maps were acquired upon each of the system's 8 shim coils at different shim current levels. Specifically, the shim current was varied from −15,000 to 15,000 by increments of 5000 per acquisition. Thus, 7 field maps were acquired per shim coil. A spherical phantom (d = 178 mm) was used Journal of Electromagnetic Analysis and Applications as the reference object for this calibration procedure. After reconstruction of the 3D phase images for each shim coil, and shim current setting, frequency distribution maps were computed. The matrix representation of ( ) , , f x y z is given by: where η nm are the coefficients of spherical harmonics, and F n,m is the Cartesian spherical harmonic spatial dependence function (see Figure S4). Using the linear least-squares method, the optimized spherical harmonic coefficients of the first-and second-order shim coils over the selected VOI can be estimated. The frequency distributions of all shim coils (at each DAC step) can be projected onto the spherical harmonics by using Equation (2). We assume that the , , nm g l η of each shim coil is linearly varying with the DAC values. Here, , nm g C is the calibration constant for each spherical harmonic. These values can be estimated using the following expression: The

Correction Procedure for 1 st Order Shims
Generally, first order coils should produce orthogonal fields that correspond to first order spherical harmonics. The second order coils could potentially produce fields that correspond to first and second-order spherical harmonics.
Therefore, we propose the following correction when computing optimal DAC settings of first-order shims, in order to counter the contributions of second-order shims: Here,

Results
Spherical harmonic calibration constants and corresponding R 2 values of linear fits (for all shims) are tabulated in Table 1 and Table 2, which are used to compute optimal DAC settings for shimming an object. Second-order shims seem to exhibit higher R 2 values in spherical harmonic calibration constants for first-order shims ( Table 2). R 2 values that are ≥0.9 are highlighted in light blue in Table 2.
For example, changes in DAC values of the xy coil influence coefficients of some    Table 2 suggest that DAC changes in first-order shims are relatively independent and only influence the first three spherical harmonic calibration constants (e.g., A 11 , A 10 , A 1-1 ). Figure 1(A) and Figure 1(B) show B 0 field distribution in the phantom before and after active shimming. The histograms of magnetic field distributions (over the entire phantom), before and after active shimming, are shown in Figure 2(A) and Figure 2(B). The full width at the half maximum (FWHM) value of the field distribution after active shimming is reduced by approximately 94.8% ( Figure  2(B)). Note: Figure 1(B) and Figure 2(B) are similar to what can be achieved with the proposed shimming method, i.e., using Table 1. These results show that our method improves B 0 homogeneity significantly within the phantom.      (5) to optimal first and second-order corrections. Figure 4. Histograms of the magnetic field variations in the brain corresponding to field maps shown in Figure 3. A narrower histogram indicates a more uniform magnetic field over the entire brain.

Discussion and Conclusion
Performance analyses of phantom results and in vivo results of a human brain showed that our proposed method can significantly outperform FASTMAP. When field maps are derived using all data points within a VOI, B 0 homogeneity can be improved by countering the contributions, or effects, of higher-order shims on first-order shims. First order shims play significant roles in B 0 homogeneity within small VOIs. Accordingly, taking into account the contributions of higher order shims within small VOIs can be important for many MR spectroscopy applications. Specifically, our method highlights the advantage of using spherical harmonic expansion corrections for shimming spherical volumes.
Our method, however, could not improve the magnetic field homogeneity near regions of the nasal sinus to a satisfying degree: these regions are known for significant susceptibility variations. Future research, focusing on combining active and passive shimming, must be pursued in order to further improve field homogeneity in the frontal brain [27]. Combining these two shimming techniques could be very important for high field MR setups which inherently require higher second-order shim fields [8] [28] [29].
Magnetic field gradient pulses can produce eddy-currents in conductive brain regions [17] [30] [31], affecting the accuracy of field map calculations. These effects can be mitigated by fixing the relative timing of gradient pulses immediately preceding excitation pulses or acquisition windows during δ1 and δ2 (see Figure S1 for details).
There may be instances where simultaneous shimming of arbitrary volumes (with differing levels of field uniformity) becomes necessary. For example: to establish a shim over a particular organ, with a tight B 0 range, while maintaining a coarser uniformity over the entire abdominal slice to prevent frequency-based fat-suppression techniques from failing. Thus, our method provides greater flexibility and can be advantageous for shimming arbitrary volumes over FASTMAP.
Here, we followed the method of projecting shim maps onto spherical harmonics: an a priori basis set to represent field maps. Due to some arguments suggesting that the use of spherical harmonics may be sub-optimal [22], Webb et al. (1991) used shim maps themselves as basis sets to produce highly uniform B 0 fields over large volumes [31] [32]. A performance analysis comparing our techniques with theirs should be the focus of future research. Journal of Electromagnetic Analysis and Applications

Supplementary Materials
Inhomogeneous magnetic fields in the MRI scanner can be corrected by adjusting shim coils to produce additional magnetic fields. These shim coils generate unique magnetic field distributions which are modelled using orthogonal spherical harmonic functions [5] [14] [15] [16] [17] [19].
Below, we present the theory and methods to: 1) numerically estimate inhomogeneous magnetic fields by varying shim settings; 2) derive calibration tables, and (3) determine appropriate shim currents for the first and second-order shim coils.

S.1. Modeling the B0 Static Magnetic Field
Assuming a current density of zero ( 0 J = ), the static inhomogeneous magnetic field 0 B ∆ in a region of interest is given by Laplace's equation (S1).
The solution to this equation

, , ∆
We performed a phantom study using the pulse sequence shown in Figure S1 to compute the B 0 field maps.
Here γ is the gyromagnetic ratio in radian/s/T for proton ( )

S.3. Phantom Study
This procedure was repeated on a water phantom to compute frequency distribution maps. Figure S2 and Figure Figure S5 and used in Table 1 and Table 2. Figure S5. The gradient [slope: spherical harmonic calibration constant (C nm,g )] estimates for Y and Z coils. The linear fits of regression (R 2 ) are computed using figures similar to Figure S4. The same procedure described in Figure S4 was followed. Journal of Electromagnetic Analysis and Applications Figure S6. The gradient [slope: spherical harmonic calibration constant (C nm,g )] are estimates for X 2 -Y 2 , XZ, Z 2 C, ZY and XY coils. The linear fits of regression (R 2 ) are computed using figures similar to Figure S4. The same procedure described in Figure  S4 was followed.