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Keywords:

  • automatic shimming;
  • B0 inhomogeneity;
  • phase mapping

Abstract

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. REFERENCES

Quantitative MRI techniques as well as methods such as blood oxygen level-dependent (BOLD) imaging and in vivo spectroscopy require stringent optimization of magnetic field homogeneity, particularly when using high main magnetic fields. Automated shimming approaches require a method of measuring the main magnetic field, B0, followed by adjusting the currents in resistive shim coils to maximize homogeneity. A robust automated shimming technique using arbitrary mapping acquisition parameters (RASTAMAP) using a 3D multiecho gradient echo sequence that measures B0 with high precision was developed. Inherent compensation and postprocessing methods enable removal of artifacts due to hardware timing errors, gradient propagation delays, gradient amplifier asymmetry, and eddy currents. This allows field maps to be generated for any field of view, bandwidth, resolution, or acquisition orientation without custom tuning of sequence parameters. Field maps of an aqueous phantom show ± 1 Hz variation with altered acquisition orientations and bandwidths. Subsequent fitting of measured shim coil field maps allows calculation of shim currents to produce optimum field homogeneity. Magn Reson Med 51:881–887, 2004. © 2004 Wiley-Liss, Inc.

The homogeneity of the static magnetic field, B0, is critical for many fast, quantitative, and spectroscopic imaging techniques. The trend towards higher static magnetic fields, both clinically and experimentally, is tempered by the increased magnetic field distortions caused by susceptibility differences. One method to compensate for these distortions is using higher-order resistive shim coils. However, the practical optimization of numerous higher-order resistive shim currents is only possible with automated techniques. Manual shimming by monitoring integrated signal magnitude and line shape during repeated adjustment of individual shim currents according to operator judgment or some mathematical algorithm (1) requires prohibitive amounts of time. Iterative readjustment of shim currents is required due to shim coil cross-terms resulting from either imperfect coils or off-center localization. In addition, signal weighting due to variations in longitudinal and transverse relaxation rates can cause inappropriate spatial biasing of these FID envelope methods.

Automated methods of improving homogeneity developed to alleviate these inherent difficulties fall into two classes, projection mapping (2, 3) and volume mapping (4, 5). Projection mapping is advantageous for its short acquisition time. However, it relies on the incorrect assumption that shim coil fields are always fully characterized by a minimal set of spherical harmonics. In general, projection mapping also involves localization techniques that make it ill-suited for disjoint regions such as in multivoxel spectroscopy. Volume field mapping eliminates the reliance on spherical harmonics by using full 3D maps of the magnetic field generated by the shim coils to determine the optimum current settings. Regions of signal void do not affect the shim current calculations, and arbitrary, potentially disjoint regions over which to shim can be specified. This additional flexibility comes at the expense of increased time, since chemical shift imaging (6, 7) and phase mapping (8, 9) techniques that have been proposed to create volume B0 maps require longer acquisition times.

Chemical shift imaging is particularly ill-suited for automated shimming due to the excessively long acquisition times. Although phase mapping sequences are significantly faster, they are exquisitely sensitive to even minimal errors in timing and gradient waveform. This requires that the pulse sequence be specifically tuned and optimized for particular fields of view, resolutions, and orientations. In addition, the use of identical acquisition parameters for shim coil field mapping and in vivo field mapping is often used to minimize potential errors (4, 10). This can severely restrict their application in demanding situations such as zoom imaging and oblique angle echo planar imaging (EPI). Development of a sequence that is self-calibrating for timing and gradient errors would allow field mapping with arbitrary resolution and field of view (FOV). It would also eliminate the requirement to tune field map acquisition parameters after hardware modifications. We demonstrate a fast, accurate, and flexible pulse sequence that can compensate for phase errors and generate absolute field maps regardless of FOV, resolution, acquisition geometry, or bandwidth, making it ideally suited for automated shimming applications or any other technique requiring accurate knowledge of B0.

THEORY

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. REFERENCES

Even small eddy currents, sequence timing errors, gradient propagation delays, and gradient waveform imperfections can cause large errors in phase images. These errors result because the signal echo is shifted relative to the acquisition window, which leads to a linear phase gradient in the reconstructed images as described by the Fourier shift theorem (11). Because these errors are dependant on timing and gradient history, they can produce different effects on each echo within a multiecho sequence typically used for phase-based field mapping. Therefore, they must be corrected by careful measurement and subsequent compensation by tuning the pulse sequence and/or reconstruction. Since any change in FOV will alter gradient strengths and timings, this tuning procedure is required for every set of acquisition parameters, thereby limiting the potential utility of these volume-mapping sequences.

Although sequence timing may not be predictable due to variable hardware and software overhead, it is typically extremely precise. Any timing errors are consistently reproduced in subsequent acquisitions. By collecting two echo trains with identical timing and reversed gradient polarity, the effects of the timing can be removed because the echo shift, Δk, is in the opposite k-space direction for positive vs. negative readout gradients. This results in phase images acquired with positive, Φ+, and negative, Φ, polarity that contain phase gradients in the readout direction, x, relative to the true phase image, Φ, given by:

  • equation image(1)
  • equation image(2)

The true phase is therefore just the average of the two opposite polarity phase images. Short time-constant eddy currents, with time constants on the order of gradient rise times, effectively act as broadband low pass filters of the gradient waveform and hence only introduce a waveform delay and corresponding k-space shift which is corrected by this compensation method.

Unequal area for positive and negative gradient lobes because of gradient amplifier asymmetry will cause the signal echo to move farther from the center of the acquisition window with each echo in an echo train. This causes an increasing linear phase gradient in subsequent phase images. Averaging of phase images acquired with opposite polarity actually compounds this artificial phase gradient, because the echo shift occurs in the same k-space direction for echo trains of both polarities. However, this shift is necessarily linear in both echo number and readout position. By collecting echoes that are unevenly spaced in time, the phase due to gradient amplifier asymmetry effects, g, can be separated from the phase accrual due to the frequency offset caused by B0 inhomogeneity. The phase is given by:

  • equation image(3)

where F and x are vectors of frequency offset and readout position for each voxel, t and n are vectors of time and echo number for each phase image in the echo train, and Φ is a voxels by echoes matrix of phase.

Long time-constant eddy currents, with time constants on the order of repetition time, also present a unique challenge. The high load and frequent switching of the readout gradient required for multiecho gradient echo imaging can cause a buildup of eddy currents that results in a magnetic field gradient that is typically in the readout direction. This magnetic field gradient should not be corrected with the shim coils, as it is a result of the pulse sequence and not a true variation in the static polarizing magnetic field. Using readout profiles acquired before any steady-state state pulse and immediately after acquisition, the magnitude of the steady-state magnetic field gradient along the readout direction caused by the imaging sequence can be determined.

It should be emphasized that this is a precision technique with complete flexibility in acquisition parameters. The residual eddy currents on our system are typically less than 0.01%, comparable to or exceeding clinical scanners. Thus, the shimming approach described here is not specific to our system, but in fact overcomes typical imperfections found on all MRI systems. Instrument-specific tuning for these imperfections is the reason that common clinical shimming algorithms have limited flexibility.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. REFERENCES

Pulse sequences were developed in VNMR and evaluated on a Varian UNITYINOVA 4 T magnet equipped with a Siemens Sonata gradient coil and Cascade amplifier operating at SR120, and a Resonance Research resistive shim power supply providing up to ±10 A on 5 second order shim coils and first order gradient offsets. Postprocessing and calculation of shim current is done using compiled software written in MATLAB 6 (MathWorks, Natick, MA).

Pulse Sequence

A multiecho, 3D gradient echo sequence is used for field mapping (Fig. 1). In particular, a train of eight echoes with linearly increasing echo spacing is used. At least three echoes are required; however, the accuracy of the field map improves with echo number. An entire 3D volume (Fig. 1a) is acquired with one gradient polarity and then repeated (Fig. 1b) with the opposite gradient polarity. Reference profiles consisting of a single pulse repetition without phase encoding are acquired immediately before and after each volume. Three hundred fifty steady-state pulses without data acquisition are used to drive the eddy currents and magnetization into a steady state after collecting the first reference profile in each volume. The acquisition of the two volumes is separated by 4 sec to allow eddy currents to dissipate prior to beginning acquisition of the second volume. To minimize any possible magnetic field gradients along phase encode directions, all gradient spoiling is limited to the readout gradient, and RF spoiling using 117° increments is used to offset this deliberate reduction of gradient spoiling (12).

thumbnail image

Figure 1. Three-dimensional gradient echo pulse sequence for mapping magnetic field. Serial acquisitions a and b use opposite gradient polarity and echo trains have increasing echo spacing.

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For shimming a human head, typical acquisitions parameters are a 32 × 32 × 32 acquisition matrix, 100 kHz readout bandwidth, 12 ms TR, first echo time at 1.2 ms, initial echo spacing of 0.7 ms, incremental echo spacing increase of 0.2 ms, and eight echoes for a total acquisition time of 37 sec. A slice-selective sinc pulse with a 6° flip angle is used to restrict the FOV in the third dimension. Readout bandwidths of 100 kHz or higher are used to minimize geometric distortion caused by static magnetic field gradients. The optimum FOV size and position is automatically derived from preplanned acquisitions for subsequent experiments.

Computation

Data acquired with the mapping sequences is processed on the spectrometer's host computer, a dual 450 MHz Sun Ultra 80 with 1 GB of RAM. Receiver DC offsets are removed from the k-space data that is subsequently Fermi-filtered to minimize truncation artifacts. Positional offsets in the phase-encode directions are applied as linear phase ramps prior to Fourier transforming into image space. Magnitude images are used to eliminate regions of insufficient signal-to-noise from further computation and shim regions of interest are automatically selected from FOV information of preplanned subsequent acquisitions and can be graphically evaluated and altered.

The phase images acquired with opposite polarities are averaged and temporally phase unwrapped. The increment delay increase ensures that phase has a quadratic dependence on echo number, allowing second derivative minimization to be used. This enables unwrapping of phase jumps of greater than ± π that can occur between later echoes due to the increasing echo spacing and gradient imbalance effects. Typically frequencies between ±2.5 kHz can be unwrapped without requiring spatial unwrapping. For larger frequency offsets, 3D spatial unwrapping algorithm can be used (13, 14). However, these tend to be time-consuming and prone to spatial propagation of unwrap errors, making temporal unwrapping preferable. Equation 3 governs the resulting phase image and a nonlinear minimization (MATLAB 6 Optimization Toolbox 2.2 fmincon using a Sequential Quadratic Programming method) is used to find F and g that minimize:

  • equation image(4)

The four reference profile echo trains are Fourier transformed into image space and converted into reference phase profiles, R. The two initial phase references of opposite polarity are subtracted from the two final phase references:

  • equation image(5)

This difference in phase accrual over time is used to determine the magnetic field gradient established by long time-constant eddy currents generated during acquisition, E, by standard least squares solution of:

  • equation image(6)

A gradient balancing term is not required because phase accrual due to imbalanced readout gradients is cancelled in the subtraction of the final and initial phase reference profiles. Using E and F, the main magnet field map, B, is simply:

  • equation image(7)

Calibration

Effects of shim coils must be empirically determined prior to performing automated shim current optimization. Repeated field maps of a homogenous phantom spanning the imaging volume of the MRI system with known shim current offsets allows mapping of the shim coil field. Shim coil calibration is performed infrequently and has a significant effect on accuracy. Therefore, 10 or more shim current offsets for each shim coil and covering a wide range are typically used. Because only differences in field maps are required, acquisition and processing is greatly simplified. Only one polarity of echo train without reference scans is acquired. Shim current settings, pi, and phase images, Pi, for the ith repetition are referenced to the first acquisition.

  • equation image(8)
  • equation image(9)

The difference in phase is attributed to the difference in magnet field caused by the shim coil. Effects due to eddy currents, timing errors, and gradient imbalance are identical for all acquisitions and therefore cancel in the subtraction. Standard linear least-squares methods are used to solve:

  • equation image(10)

for the magnetic field changes, Hi, for each acquisition. Using the calculated changes in magnetic field and shim currents settings, the effect of all shim coils, Q, is given by the linear least-squares solution to:

  • equation image(11)

Shimming

Optimum shim current corrects, s, are computed using least-squares solution to:

  • equation image(12)

If currents beyond the shim coil power supply are calculated a constrained least-squares solution to Eq. 12 (MATLAB lsqlin) that prevents excessive current requests can be found. In shimming over regions of limited extent that do not contain voxels in at least two positions along one or more shim coil axes, shim coils must be removed from the computation to prevent ill conditioning (4).

RESULTS

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. REFERENCES

Phase images acquired with positive and negative gradient polarity are shown in Fig. 2a,b. The differences in phase caused by gradient propagation delays and short time-constant eddy currents can cause errors up to 330 μT m−1 or 290 Hz across an adult human head. The temporal series of average phase images (Fig. 2c) is composed of phase accrual due to B0 inhomogeneity (Fig. 2d) and gradient asymmetry (Fig. 2e). Phase due to gradient infidelity can cause gradient errors of 10 μT m−1 and offsets of 380 nT or 16 Hz, which is comparable to, if not greater than, the phase accrual caused by B0 inhomogeneity. These two components must be separated since the gradient asymmetry will otherwise suggest an erroneous first-order linear shim in the readout direction. This first-order shim will act to balance the gradient asymmetry, and a second B0 map would suggest improved homogeneity because phase due to the magnetic field and gradient asymmetry would cancel. However, the apparent improvement is actually achieved by degrading the B0 homogeneity. This highlights the inadequacy of repeated measures for shim evaluation if subtle gradient characteristics have not been fully taken into account.

thumbnail image

Figure 2. Phase images for a typical acquisition with positive gradient polarity (a) and negative gradient polarity (b), and average of both acquisitions (c), which has a magnetic field inhomogeneity (d) and gradient asymmetry (e) component. All images are referenced to first average phase image.

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To validate that the field maps are independent of gradient infidelity, field maps of an aqueous phantom were collected using three orthogonal readout directions (Fig. 3). The FOV acquired was 18 cm isotropic on a 32 × 32 × 32 matrix with a readout bandwidth of 100 kHz. The high degree of correlation (R2 > 0.996) and low standard deviation of the difference (1 Hz) between these field maps indicates that the acquisition is largely insensitive to gradient infidelity (Table 1). Histograms of field map differences in Fig. 4 visualize the small standard deviation and global offsets that do not affect shimming. Neglecting the use of reference profiles to correct long time-constant eddy currents produces comparably poor correlations (R2 < 0.959) and large SDs of 5.3–5.8 Hz in the difference between field maps. This results from the 3 μT m−1 gradient that the eddy currents establish at the chosen FOV and bandwidth. Changes in readout bandwidth and corresponding gradient strength produced field maps with a difference SD of 1 Hz.

thumbnail image

Figure 3. Comparison of frequency maps showing central slice in xy plane extracted from image volumes generated using three orthogonal readout directions.

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Table 1. Standard Deviations of the Difference and Cross-Correlation Between Field Maps Acquired With Three Orthogonal Readout Directions With and Without Correcting for Long Time–Constant Eddy Currents
Readout CorrectedUncorrected
YXZYXZ
CorrectedY 0.9960.9970.9480.9340.945
 X4.08 ± 1.05 0.9970.9590.9140.949
 Z1.72 ± 0.91−2.36 ± 0.89 0.9570.9300.953
UncorrectedY0.17 ± 3.94−3.91 ± 3.57−1.55 ± 3.63 0.9000.908
 X3.51 ± 4.40−0.57 ± 4.971.78 ± 4.503.34 ± 5.54 0.885
 Z0.29 ± 3.92−3.79 ± 3.78−1.43 ± 3.630.12 ± 5.26−3.22 ± 5.82 
thumbnail image

Figure 4. Histogram of frequency difference for field maps acquired with three orthogonal readout axis.

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Figure 5 shows the improvement in EPI that can be achieved using RASTAMAP to compute optimal shim currents. Severe distortions due to the sinus are corrected using RASTAMAP with a whole head shim region as compared with manually optimized shims. RASTAMAP can also shim single voxels with a single acquisition. In a 1.5 cm isotropic voxel, water linewidth was reduced to 10.9 Hz from 19.6 Hz when a globally optimized shim was used (Fig. 6). Repeated use of RASTAMAP, suggested negligible changes in shim current that did not alter linewidth, and further manual adjust of shims was unable to improve linewidth.

thumbnail image

Figure 5. Echo planar imaging images acquired with optimal manual shimming before and after using RASTAMAP show significant improvement in the sinus region.

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Figure 6. Water spectrum acquired with LASER localization of 1.5 cm isotropic voxel before (dashed line) and after automated shimming (solid line).

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DISCUSSION

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. REFERENCES

Other fitting techniques and cost functions, as suggested by Wen and Jaffer (15), can be used with this field mapping technique to compute optimal shim currents. Kim et al. (10) have shown that the matrix inversion involved in solving for required shim correction terms can lead to numerically unstable results. Shimming regions distant to the magnetic iso-center where interactions between shim coils are larger may require regularization or coordinate transformation to increase numerical stability, and their proposed regularization and singular value decomposition techniques, can be applied to solve Eq. 12. However, for typical imaging near iso-center this has not been required. The increased resolution available by using an adaptive field map enables better characterization of shim coil fields and decreases susceptibility to ill conditioning.

Optimum shim currents can be computed for each slice to enable dynamic shimming (16, 17). This technique can also be easily extended to multivoxel imaging and parallel image systems used to image multiple objects simultaneously (18). Optimum shim currents can be found for individual objects or volumes and used with dynamic changing of shim currents, or all objects can be simultaneously optimized.

The field mapping sequence is not limited to shimming applications. It can be applied whenever the field distribution is required, such as techniques using tailored RF pulses to reduce dephasing caused by intrinsically unshimmable B0 inhomogeneity (19) or postprocessing techniques used to correct for residual inhomogeneity (20, 21). Field maps can be used to separate tissue-specific effects on effective transverse relaxation rates, R2*, and RF reversible dephasing relaxation rates, R2′, from effects of macroscopic gradients (22).

The background magnetic field that arises due to the long time-constant eddy currents can be corrected as long as the eddy currents perpendicular to the readout direction are negligible. This is typically not the case if the readout direction is not one of the principle gradient axes. The direct eddy currents for each axis are invariably different, resulting in a background magnetic field gradient that is not in the readout direction and therefore cannot be corrected using the reference scan. The use of 3D reference scans would be required to fully characterize the steady-state field. However, since the steady-state magnetic field begins to decay immediately following completion of acquisition, time constraints make it difficult to obtain reliable 3D measurements.

Successful shimming cannot be demonstrated by repeated acquisitions of a field map, as suggested in the literature (4, 10, 15, 23). Errors produced by steady-state eddy current fields and sequence timing will produce frequency map artifacts that are propagated to the shim currents. Thus, the shim coils are used to produce an offsetting magnetic field so that subsequent field show reduced field inhomogeneity. However, the true magnetic field homogeneity may actually be worse.

A fast, flexible, and self-calibrating acquisition and postprocessing technique has been developed for in vivo mapping of magnetic fields. This approach allows acquisitions to be tailored to sample geometry without any special calibration or tuning, providing optimum field maps for any required purpose. B0 maps generated with different readout directions produced field maps with voxel-by-voxel differences with SD of 1.0 Hz over an 18-cm spherical phantom volume. Small global differences of less than 4 Hz are likely real differences in B0 due to thermal fluctuations between acquisitions or differences in B0 eddy currents generated by the gradient coils. However, they do not affect the shim results since global shifts are incorporated in the transmitter frequency offset. These field maps can be used to improve magnetic field homogeneity for improved signal-to-noise and reduced artifacts.

Acknowledgements

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. REFERENCES

We thank Dr. Chris Bowen and Joe Gati for helpful discussions concerning system hardware and eddy currents.

REFERENCES

  1. Top of page
  2. Abstract
  3. THEORY
  4. MATERIALS AND METHODS
  5. RESULTS
  6. DISCUSSION
  7. Acknowledgements
  8. REFERENCES