Improved Shim Method Based on the Minimization of theMaximum Off-Resonance Frequency for Balanced Steady-State Free Precession (bSSFP)

Jongho Lee,1,2* Michael Lustig,2 Dong-hyun Kim,3 and John M. Pauly2

In this work, a shim method that minimizes the maximum off-resonance frequency (min-max shim) in balanced steady-statefree precession (bSSFP) is tested for brain imaging at 3T withconstant and linear shim terms. The method demonstrates im-provement of spatial coverage and banding artifact reduction overstandard least-squares shimming. In addition, a new method(modified min-max shim) is introduced. This method reducesboundary band regions where the artifact is inevitable due to theexcessive off-resonance frequency distribution. In comparison tostandard least-squares shimming, the min-max based shimmethod either eliminates or reduces the size of banding artifacts.The method can be used to increase the signal-to-noise ratio(SNR) in bSSFP imaging or to increase the functional contrast inbSSFP functional MRI (fMRI) by allowing a longer usable repetitiontime (TR). Magn Reson Med 61:1500–1506, 2009. © 2009 Wiley-Liss, Inc.

Key words: min-max; minimax; least-squares; shimming; shim;SSFP

In MRI, magnetic field shimming is a critical step foracquiring high quality images. Almost every scan sessionstarts with a shim protocol to reduce the spatial fieldinhomogeneity in the imaging volume. Most modern scan-ners are equipped with zeroth- and first-order shim coils,and many are also equipped with higher-order shims tocompensate for the complex field inhomogeneity intro-duced by the human body. Several shim algorithms andtechniques have been developed using linear shim meth-ods (1–4), higher-order shim methods (5–9), dynamic orreal-time shim update methods (10–13), and local fieldcorrection shim methods (14,15).

In balanced steady-state free precession (bSSFP; alsoreferred to as FIESTA, TrueFISP, and balanced FFE, (16)).shimming an imaging volume or region of interest (ROI) isparticularly important because the signal is susceptible tosignificant drops in intensity (e.g., the banding artifact) asa function of off-resonance frequency and dark bands ap-pear periodically with a period of 1/(2TR) Hz (Fig. 1b). If

shimming is not properly performed or the off-resonancedistribution is wider than the flat magnitude frequencyrange in the bSSFP profile (the pass-band in Fig. 1b), thebanding artifact is observed (Fig. 1a). To remove this arti-fact, a very short TR (a few milliseconds) is commonlyused to increase the off-resonance frequency coverage. Al-ternatively, multiple acquisitions are obtained, each withshifted center frequencies (17). However, using a short TRor multiple acquisitions is not always favorable. Multipleacquisitions increase the total scan time. A short TR de-creases the data acquisition efficiency and increases thespecific absorption rate (SAR). In bSSFP functional MRI(fMRI) (18–20), a recent fMRI method for high-resolutionfunctional studies, a short TR decreases the level of func-tional contrast since the relative signal change dependslinearly on TR (20). Therefore, it is favorable to use alonger TR with a single acquisition, while keeping theimaging volume or ROI free of banding artifacts.

One way to reduce the field inhomogeneity and increasethe banding-free spatial coverage is to optimize the shimparameters for bSSFP imaging. Most of the previouslymentioned methods try to find the shim values that mini-mize the mean square of the residual field (least-squaresshim) in the ROI or imaging volume (2). However, thisleast squares shimming does not guarantee the largest spa-tial region without banding artifact. In bSSFP the actualoff-resonance frequency of a voxel does not matter as longas it stays within the pass-band of the bSSFP profile,approximately within 80% of �1/(2TR) Hz. Therefore, abetter shim algorithm can be designed to utilize the toler-ance of bSSFP to off-resonance frequency within the pass-band. Based on this observation, adjusting the center fre-quency has been proposed for cardiac applications (21,22).However, this solution is not optimal as it uses only thezeroth-order shim. In this work, we propose a new methodthat utilizes both the zeroth- and first-order shim coils forbetter spatial coverage. The method minimizes the maxi-mum off-resonance frequency (min-max shim (2)) within anROI to improve the off-resonance coverage in bSSFP. Com-bined with this min-max shim, we developed an approachthat ensures reduced artifact regions compared to the least-squares shim when the banding artifact is unavoidable (mod-ified min-max shim). This new shimming method was ap-plied to the bSSFP brain imaging to demonstrate the im-provement over the traditional least-squares method.

MATERIALS AND METHODS

The conceptual illustration of the min-max shimmingmethod is shown in Fig. 1. Least-squares shimming re-duces the average squared off-resonance frequency.

1Advanced MRI, Laboratory for Functional and Molecular Imaging, NationalInstitute of Neurological Disorders and Stroke, National Institutes of Health,Bethesda, Maryland, USA.2Magnetic Resonance Systems Research Laboratory, Department of Electri-cal Engineering, Stanford University, Stanford, California, USA.3School of Electrical and Electronic Engineering, Yonsei University, Seoul,Republic of Korea.Grant sponsor: National Institutes of Health (NIH); Grant numbers: R01-EB006471, R21-EB002969; Grant sponsor: GE Medical Systems.*Correspondence to: Jongho Lee, 9000 Rockville Pike, Building 10, RoomB1D723A, Bethesda, MD 20892-1065. E-mail: jonghoyi@mail.nih.govReceived 9 April 2008; revised 1 July 2008; accepted 4 August 2008.DOI 10.1002/mrm.21800Published online 24 March 2009 in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 61:1500–1506 (2009)

© 2009 Wiley-Liss, Inc. 1500

Therefore, the method is likely to penalize a small re-gion (e.g., the frontal lobe area) while reducing the over-all off-resonance frequency in other regions. The result-ing field map may exhibit a large peak off-resonanceregion as shown in Fig. 1e and f. In bSSFP, this largepeak off-resonance can result in banding artifacts (theyellow arrow in Fig. 1a, and the red strip in Fig. 1f). Insome cases, one can manipulate the shim parameterssuch that the maximum off-resonance frequency is re-duced to avoid the artifacts (Fig. 1g and h). In thisexample, the center frequency and y-shim were modi-fied to decrease the peak off-resonance frequency. Theresulting field map has a larger mean square error com-pared to the least-squares shim result but the image hasno artifact (Fig. 1d). Hence, one can achieve a widerspatial coverage by reducing the peak off-resonance fre-quency.

This approach can be restated as follows: in bSSFP,better spatial coverage can be obtained by minimizing themaximum absolute off-resonance frequency of the fieldmap (min-max shim). It can be formulated as follows (2):

s0,sx,sy. . .argmin max[|F(x,y,z) � F0s0 � Fxsx � Fysy � Fzsz

� Fxysxy�Fxzsxz – ���|], [1]

where F(x,y,z) is the field map; F0, Fx, Fy, Fz, Fxy, and Fxz

represent the center frequency, x-shim, y-shim, z-shim,xy-shim, and xz-shim induced field basis vectors (or ma-trices); and s0, sx, sy, sz, sxy, and sxz represent the variableshim values. Equation [1] can be reformulated as a linear

program (2) and solved very efficiently (23). The resultingfield map has the minimum peak off-resonance frequencywith both positive and negative peaks the same absolutevalues. Based on the shim result, one can choose a TR thatensures artifact-free images if the TR is longer than theminimum TR of the sequence.

If the choice of TR is constrained (for example, inbSSFP fMRI studies) and the off-resonance frequency ofthe volume is larger than the coverage obtained by theminimum TR, the resulting bSSFP image will inevitablycontain banding artifacts. Moreover, the min-max shimcan potentially create more banding artifacts than least-squares shim because both positive and negative off-resonance frequency peaks may create the artifact, andtheir spatial location may appear in central portion ofthe image. For example, both orange and blue coloredareas in Fig. 1h (min-max shim) can show the bandingartifact for a certain TR, whereas only the red- to orange-colored areas will have the banding artifact in the least-squares shim (Fig. 1e).

To avoid this problem, the min-max shim methodneeds to be improved. When the banding artifact isunavoidable due to excessive off-resonance, our goal isto reduce the size of banding-artifact regions. Moreover,it is favorable to reduce the size of the banding artifactfrom the least-squares shim result without creating newbanding artifacts at arbitrary locations. To ensure re-duced banding regions compared to the least-squaresresult, we modified the min-max shim method as fol-lows. First, an ROI where shimming is performed ischosen. The least-squares shim is performed for the ROI

FIG. 1. Conceptual illustration of the advantage of the min-max shim method in bSSFP imaging. a,d: The bSSFP image. b,c: The bSSFPmagnitude profile. The flat magnitude off-resonance frequency range is referred to as the pass-band. e,h: The color-coded field map of theslice (in Hz). f,g: The off-resonance distribution of the central line in the field map (the red dotted line in e and h). When the min-max shimis used, the peak off-resonance frequency is reduced allowing an artifact-free image.

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to generate a mask where the banding artifact is ex-pected in the least-square shim results. The off-reso-nance frequency range for the pass-band, approximately80% of �1/(2TR), is set for a given TR. Then the regionswhere the off-resonance frequency is larger than thepass-band frequency coverage are calculated from thefield map of the least-squares method. These regions areset to 1 in the mask. Since the off-resonance frequency islarger than the pass-band, some voxels within theseregions, especially the boundaries of the regions, willhave the artifact. Note that the least-squares shim is onlyused to create this mask. After setting the mask, themin-max method is iteratively performed. In each iter-ation, the ROI where the min-max shim is performed isupdated by removing the peak off-resonance frequencyvoxel within the mask. The iteration is terminated whenthe peak off-resonance frequency of the ROI is smallerthan the pass-band frequency coverage. This procedureis referred to as the modified min-max shim and issummarized in Fig. 2. The resulting field map will havelower peak off-resonance frequency than the pass-bandfrequency range for the given TR and the banding arti-fact will only be observed in some of those voxels thatwere removed during the procedure. Since these voxelswere within the mask that was generated by the least-squares shim, the resulting banding areas from the mod-ified min-max shim will have reduced boundary band-ing regions compared to the least-squares shim result. Ifthe peak off-resonance frequency is less than the pass-band frequency coverage after the first min-max shim,the modified min-max shim method becomes the origi-nal min-max shim method.

To demonstrate the efficacy of the proposed min-maxshim method, experiments were performed using a 3T GEEXCITE scanner (40 mT/m and 150 mT/m/ms). A GE stan-dard head coil (transmit/receive) was used to acquire data.Five subjects who provided informed written consentwere scanned under an Institutional Review Board (IRB)-approved protocol. A field map sequence and a bSSFP

imaging sequence were developed for the experiments. Forthe field map sequence, an interleaved 2D gradient re-called echo (GRE) spiral sequence with two echoes wasused (TR � 800 ms, TE1 � 1.5 ms, TE2 � 5.5 ms, flipangle � 70°, slice thickness � 4 mm with 1-mm gaps,FOV � 24 � 24 cm2, number of slices � 8, number ofinterleaves � 4, matrix size � 40 � 40, and bandwidth ��125 kHz). The phase images at the two echo times wereunwrapped individually before generating the field map.The total scan time for the field map sequence was 5 s. Forthe bSSFP imaging, a Cartesian trajectory three-dimen-sional (3D) bSSFP sequence (TR � 5–10 ms, TE � TR/2,FOV � 24 � 24 cm2, matrix size � 128 � 128, slab size �40 mm, and slice thickness � 2.5 mm, BW � �31.25 kHz)was used with a 180° RF cycling.

Two experiments, one for the min-max shim and theother for the modified min-max shim, were performed oneach subject. First, the original min-max shim that wastargeted for the band-free images was compared with theleast-squares shim method at an upper brain area withmoderate off-resonance. The scan session started with a 3Dlocalization and the field map sequence followed. An ROIthat excludes the non-brain area was selected manuallybased either on the magnitude or field map images. Toavoid low-intensity voxels that can induce large off-reso-nance frequency variations, a slightly tighter area waschosen for the ROI. Then, the shim values were calculatedusing the least-squares shim method and the scanner shimsettings were adjusted accordingly. The field map se-quence was repeated up to twice using the same ROI tomeasure and correct for the off-resonance field distribu-tion. The bSSFP images (TR � 5–10 ms with a 1-ms step)were then acquired to see the results of the shim method.After the bSSFP imaging, the shim values were adjustedbased on the min-max shim result. The field map sequencewas scanned again, once or twice, to confirm the min-maxshim result and the bSSFP images were obtained with thesame TRs as in the least-squares experiments.

FIG. 2. a: Flow chart for the modified min-max shim method. The Rk(x,y,z) represents the kth ROI, M(x,y,z) represents the mask from theleast-square shim, FLS(x,y,z) the field map result of the least-squares shim, FmM,k(x,y,z) the kth field map result of the min-max shim, and fthresh

for the one side of the frequency range of the pass-band (here, 80% of 1/(2TR)). The ROI was gradually refined by removing the peakoff-resonance frequency voxel within the mask. b: Initial field map result of the min-max shim. c: Final field map result after the modifiedmin-max shim. The frontal lobe area was removed from the ROI during the procedure. [Color figure can be viewed in the online issue, whichis available at www.interscience.wiley.com.]

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For the second experiment, a lower slab of the brain thathad severe off-resonance was scanned to further evaluatethe performance of the modified min-max shim. A fixedTR (5 ms) was used for the bSSFP imaging and the maxi-mum allowable off-resonance frequency range was set tobe 80% of 1/TR (i.e., –80 Hz to 80 Hz). After selecting aslab for the imaging, the field map was acquired and anROI was drawn manually. Then, the ROI was refined byexcluding the excessive peak off-resonance frequency vox-els as described in Fig. 2 (modified min-max shimmethod). The shim parameters were adjusted based on themodified min-max shim result. The off-resonance fre-quency distribution was measured and corrected by up totwo more runs of the field map sequence. bSSFP imageswith a 5-ms TR were acquired. After that the shim param-eters were adjusted to the least-squares shim with the sameROI. The field maps were acquired and corrected (up totwo times) and the bSSFP sequence was run.

In both experiments, the order of the least-squares shimand the min-max shim was randomly reversed. For theshim parameters, the linear shim terms (x-, y-, and z-shim)and the resonance frequency (f0) were corrected. The shimterms were orthogonalized based on the measured fieldbasis vector (9). The step size for the resonance frequencycorrection was 1 Hz, whereas it was 0.0015 mT/m for thelinear shim parameters.

RESULTS

Figure 3 illustrates the improvement of the spatial cover-age in the bSSFP imaging using the min-max shim method.When the least-squares shim was used, the banding arti-

fact was observed in the frontal lobe of the brain (arrows inFig. 3a). It disappeared when the shim was reconfigured tothe min-max shim for the same TR (10 ms; Fig. 3b). Ashorter TR (7 ms) was necessary to avoid the artifact in theleast-squares shim (Fig. 3c). Hence, the min-max shimenabled to increase TR from 7 ms to 10 ms. The off-resonance frequency range measured by the field map waschanged from [–33.6, �54.5] Hz (least-squares shim; Fig.4a) to [–36.6, �36.2] Hz (min-max shim; Fig. 4b), showinga reduced peak off-resonance in the min-max shimmethod. When averaged over the five subjects, the peakoff-resonance frequency was reduced by 39% (100*[fpk_LS

– fpk_mM]/fpk_mM, where fpk_LS represents the peak off-reso-nance frequency of the least-squares result and fpk_mM rep-resents that of the min-max result) and the TRs were ableto be increased by 1–3 ms when the min-max shim wasused (Table 1). Regional banding artifacts at the edges ofthe frontal lobe of the brain were observed in both least-squares and min-max shim results (Fig. 3). These artifactsarose most likely due to the tight ROI that did not properlycover the edge voxels and the artifacts might have disap-peared if the ROI had been refined. Note that in foursubjects, the min-max shim still allowed us to use a longerTR by 1 to 2 ms compared to the least-squares shim, whenthe TR was further reduced to make these artifacts unno-ticeable (the numbers in the parentheses in Table 1).

When a lower brain area was scanned with a fixed TR (�5 ms), the modified min-max shim results provided re-duced banding regions compared to those of the least-squares shim (Fig. 5). As a result, the method providedmore slices without the banding artifact (slices 4 and 5)and reduced banding regions (slices 1–3) in the frontal

FIG. 3. Min-max shim result vs.least-squares shim result. a: Whenthe TR was 10 ms, the bSSFP im-ages from the least-squares shimshow the banding artifact (arrows)near the sinus area. b: The artifactdisappeared when the shim wasreconfigured to the min-max shimfor the same TR. c: In the least-squares shim, the TR needed tobe shortened to 7 ms to yield arti-fact-free images.

Min-Max Shimming for bSSFP 1503

lobe area. When the modified min-max shim method wasused, the peak off-resonance frequency was reduced, onaverage, by 48% compared to that of the least-squares shimresults. All the individual results are summarized in Table1.

DISCUSSION AND CONCLUSIONS

In this work, we proposed a new shim method based onthe minimization of the maximum absolute off-reso-nance frequency. The min-max shim algorithm was orig-inally suggested by Prammer et al. (2) but, to the best ofour knowledge, it has not been popularly used nor dem-onstrated in vivo in MRI. Moreover, the modification ofthe min-max shim to adapt to bSSFP imaging has notbeen proposed. Here, we verified that the min-max shimand the modified version of the method can reduce thepeak off-resonance frequency in the brain and can beapplicable to bSSFP imaging. The results showed im-proved spatial coverage with a longer usable TR. Themethod will be helpful to increase the signal-to-noiseratio (SNR) by allowing a higher readout duty cycle andalso to increase the functional contrast in bSSFP fMRI.

FIG. 4. Field map measurements of (a) the least-squares shim and(b) the min-max shim methods. The min-max shim field map showsa reduced maximum off-resonance frequency in the frontal lobearea.

Table 1Summary of the Results

Experiment 1 (TR � 5-10 ms) Experiment 2 (TR � 5 ms)

Least-squares shim method Min-max shim methodLeast-squaresshim method

Modified min-maxshim method

Off-resonance (Hz) TRa,b (ms) Off-resonance (Hz) TRa,b (ms) Off-resonance (Hz)c Off-resonance (Hz)c

1 �42.0 to 49.7 8 (�5) �39.5 to 36.4 9 (�5) �62.4 to 113.2 �86.8 to 76.52 �45.2 to 58.5 6 (�5) �48.4 to 48.6 7 (5) �48.2 to 135.7 �81.1 to 79.13 �42.7 to 78.0 5 (5) �43.6 to 43.7 8 (6) �62.9 to 112.0 �76.0 to 76.0d

4 �33.6 to 54.5 7 (7) �36.6 to 36.2 10 (9) �51.8 to 127.8 �83.7 to 78.25 �45.7 to 62.7 7 (�5) �50.1 to 52.3 9 (6) �53.2 to 115.1 �81.1 to 77.4

aThe maximum TR that showed no significant banding artifact in the bSSFP imaging.bThe number in the parentheses represents a TR that shows no discernable banding artifact even at the edges of the images.cOff-resonance frequency ranges are based on the ROI after removing excessive off-resonant voxels by the modified min-max shimmethod.dThe off-resonance frequency range was calculated from the field map.

FIG. 5. Modified min-max shim result vs. least-squares shim result. The least-squares result (a) shows wider banding area (arrows) thanthe modified min-max shim result (b). Hence, the bSSFP images from the modified min-max shim have more banding-free slices (slices 4and 5 counting from the most inferior slice) and smaller boundary banding regions (slices 1–3).

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To check the stability of the shim methods, the field mapmeasurement that was repeated after configuring the shimparameters was analyzed. In the original min-max shimexperiments, both least-squares shim and min-max shimmethods showed an excellent convergence within two it-erations (most of the cases, without iteration) allowing upto 1 shim step size (0.0015 mT/m) variation in the linearshims and less than 2-Hz change in the center frequency.Consequently, the measurement results matched well withthe expected off-resonance frequency range calculatedfrom the field map revealing, on average, a 1.38 � 1.37-Hzdifference (the absolute peak off-resonance frequency dif-ference in one side �SD) in the min-max shim results anda 2.42 � 1.69-Hz difference in the least-squares shim re-sults. The possible sources of the errors include round-offerrors of the finite precision in the shim terms, subjectmotion, and respiration-induced B0 shift (12,24). In themodified min-max shim experiments, slightly larger vari-ability was observed showing a maximum 3-shim step sizevariation in the linear terms and a 3-Hz change in thecenter frequency. However, the resulting field map stillshowed a small field deviation, yielding a 4.88 � 1.64-Hzerror in the min-max shim and a 2.20 � 2.75-Hz error inthe least-squares shim. Some of the possible sources forthe slightly increased instability might be related to thesteep off-resonance change near the sinus area, which canresult in large off-resonance measurement variation due tothe aforementioned sources.

When the modified min-max shim method is per-formed, the mask that encompasses the voxels withexcessive off-resonance frequency, which is beyond the80% width of 1/TR, is set by the least-squares shim fieldmap. Since certain voxels within this mask are removedby the modified min-max shim method, the resultingfield map will have a reduced number of voxels thathave excessive off-resonance frequency. Note that notall the voxels whose off-resonance frequency is largerthan the pass-band frequency coverage have the bandingartifact. However, since the field is continuous, theboundaries of the mask will have close to �1/(2TR) Hzoff-resonance frequency and have the banding artifact. Ifonly a small portion of the voxels within the mask isremoved by the modified min-max shim method, thenthe banding will exists at the boundary of these removedvoxels. Hence, the resulting bSSFP image will have re-duced boundary banding regions, effectively increasingcontinuously banding free regions, as shown in Fig. 4.This modified min-max shim method might not providethe optimum solution since some voxels that were ex-cluded in the earlier iteration could be included in thelater iteration due to the change of the field distribution.However, the method ensures a reduced boundary band-ing region compared to the least-square shim.

One of the important steps in the min-max shim meth-ods is to define a proper ROI. If a voxel that is located atthe outside of the brain region is included in the ROI, itmight produce a large variation in the min-max shimresult since the frequency measurement of the voxel cansignificantly change in each field map result. On theother hand, if some of the edge voxels are excluded fromthe ROI, the resulting bSSFP images might have thebanding artifact in those voxels—that was the case in

our experiments (Fig. 3). Therefore, choosing a properROI is crucial. The current methods may be improved byacquiring the field map multiple times. Then the voxelswith large frequency variation can be avoided by mea-suring their SD. Since the field map sequence takes only5 s for the relatively large area, acquiring it two to threetimes is quite practical. The resulting ROI will improvethe stability of the results.

One major application of the proposed method can befound in spectroscopy. Fat suppression for spectroscopyis crucial. However, remnant lipid signals are oftenpresent due to incomplete suppression originating fromthe field inhomogeneity. For improved fat suppression,we can modify our technique to ensure that the fre-quency in lipid regions is outside of the boundary re-gion. Frequency-selective pulses will also benefit fromour proposed technique. Another possible application isthe EPI distortion reduction in fMRI. One might be ableto reduce the distortion by limiting the maximum allow-able off-resonance range.

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