ARTICLE IN PRESS YNIMG-03684; No. of pages: 14; 4C:

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1 YNIMG-03684; No. of pages: 14; 4C: DTD 5 Technical note Application of k-space energy spectrum analysis to susceptibility field mapping and distortion correction in gradient-echo EPI Nan-kuei Chen, a, * Koichi Oshio, a,b and Lawrence P. Panych a a Department of Radiology, Brigham and Women s Hospital, Harvard Medical School, 75 Francis Street, Boston, MA 02115, USA b Department of Diagnostic Radiology, Keio University, Japan Received 27 June 2005; revised 2 December 2005; accepted 16 December NeuroImage xx (2006) xxx xxx Echo-planar imaging (EPI) is widely used in functional MRI studies. It is well known that EPI quality is usually degraded by geometric distortions, when there exist susceptibility field inhomogeneities. EPI distortions may be corrected if the field maps are available. It is possible to estimate the susceptibility field gradients from the phase reconstruction of a single-te EPI image, after a successful phaseunwrapping procedure. However, in regions affected by pronounced field gradients, the phase-unwrapping of a single-te image may fail, and therefore the estimated field maps may be incorrect. It has been reported that the field inhomogeneity may be calculated more reliably from T2*-weighted images corresponding to multiple TEs. However, the multi-te MRI field mapping increases the scan time. Furthermore, the measured field maps may be invalid if the subject s position changes during dynamic scans. To overcome the limitations in conventional field mapping approaches, a novel k-space energy spectrum analysis algorithm is developed, which quantifies the spatially dependent echo-shifting effect and the susceptibility field gradients directly from the k-space data of single-te gradient-echo EPI. Using the k-space energy spectrum analysis, susceptibility field gradients can be reliably measured without phase-unwrapping, and EPI distortions can be corrected without extra field mapping scans or pulse sequence modification. The reported technique can be used to retrospectively improve the image quality of the previously acquired EPI and functional MRI data, provided that the complex-domain k- space data are still available. D 2005 Elsevier Inc. All rights reserved. Introduction * Corresponding author. address: Nan-kuei_Chen@hms.harvard.edu (N. Chen). Available online on ScienceDirect ( Echo-planar imaging (EPI) (Mansfield, 1977), with its high temporal resolution, has become an important technique for various dynamic studies. For example, EPI-based functional MRI (Kwong et al., 1992; Ogawa et al., 1992; Kwong, 1995) is now popularly used in both basic neuroscience research and for presurgical evaluation. EPI is also a potentially valuable tool for realtime monitoring of MR guided interventional procedures. For example, EPI may be used to dynamically measure the temperature changes during ablation procedures based on laser or focused ultrasound treatments (Stafford et al., 2004). It is well known that EPI data are usually geometrically distorted because of the Bo field inhomogeneity and phase error accumulation in single-shot k-space acquisition (Weisskoff and Davis, 1992; Jezzard and Balaban, 1995). In the past few years, various field inhomogeneity mapping and data processing techniques have been developed for EPI distortion correction (Weisskoff and Davis, 1992; Jezzard and Balaban, 1995; Wan et al., 1997; Kadah and Hu, 1997; Reber et al., 1998; Chen and Wyrwicz, 1999, 2001; Zeng and Constable, 2002; Zaitsev et al., 2004; Zeng et al., 2004). It has been shown that, only after geometric distortion correction, EPI-based dynamic or functional data can be accurately co-registered with structural images (usually acquired with other more time-consuming MRI pulse sequences). However, there are several limitations in the previously reported field mapping and EPI distortion correction methods. First, Bo field mapping requires extra scan time, which may not be ideal for certain clinical studies or MR-guided intervention (Stafford et al., 2004). Second, if the subject s position changes between the field mapping scan and the EPI studies, the measured Bo field inhomogeneity information may be invalid for EPI distortion correction (Hutton et al., 2002). It would be desirable therefore to have the capability of measuring the Bo field inhomogeneity at multiple time points in a dynamic EPI scan series. Bo field mapping for every EPI volume acquisition can be achieved with two approaches. Firstly, it has been reported that Bo field inhomogeneity can be measured dynamically using modified EPI sequences, in which a low-resolution multi-te scan is embedded within each dynamic EPI acquisition (Roopchansingh et al., 2003; Ward et al., 2002). However, the Bo field map obtained with this approach has a relatively low spatial resolution and may not provide detailed information on the susceptibility field gradients near the air tissue interfaces. Secondly, it was pointed out by Posse that the in-plane susceptibility field gradient can be estimated directly from gradient-echo MRI (Posse, 1992) through /$ - see front matter D 2005 Elsevier Inc. All rights reserved. doi: /j.neuroimage

2 2 N. Chen et al. / NeuroImage xx (2006) xxx xxx quantifying the echo-shifting effect due to field gradients. Deichmann et al. recently reported that field gradient map may also be calculated directly from gradient-echo EPI data, which can then be applied to further estimate the spatially dependent echo time (TE) and blood oxygenation level dependent (BOLD) sensitivity (Deichmann et al., 2002). Here, we further develop the approach suggested by Posse and Deichmann et al. (Posse, 1992; Deichmann et al., 2002) and investigate its application to gradient-echo EPI distortion correction. A novel algorithm, termed k-space energy spectrum analysis, is presented to map the susceptibility field gradients directly from gradient-echo EPI data. Neither extra field mapping scan nor pulse sequence modification is needed. Based on the calculated susceptibility field gradient maps, the Bo field inhomogeneity map can be derived and used to remove the geometric distortions in EPI data. The proposed method is superior to conventional field map-based EPI correction methods in three ways. First, using the k-space energy spectrum analysis, the field maps can be obtained without performing additional field mapping scan. Second, even in the presence of subject movement during dynamic EPI scans, the valid field maps corresponding to different scan time points can be reliably obtained and used to remove distortions. Third, in the proposed algorithm, the complicated in-plane phase-unwrapping procedure that is required in most conventional field mapping methods can be avoided. It should be noted that the proposed method can be used to retrospectively remove the geometric distortions in previously acquired EPI and fmri data, provided that the complex-domain k-space data are available. Theory In this section, the impact of the echo-shifting effect on gradient-echo EPI is briefly discussed (Deichmann et al., 2002; Hennig et al., 2003; Posse et al., 2003). A k-space energy spectrum analysis algorithm for measuring EPI s spatially dependent echoshifting effect is presented. Based on the measured echo-shifting effect, the susceptibility field gradients can be calculated and used to correct for geometric distortions. The impact of echo-shifting effect on EPI The gradient waveforms of single-shot gradient-echo EPI are usually designed in such a way that the full k-space data can be sampled using fast-switching readout gradients and phase-blipped gradients after an RF pulse excitation. The k-space energy peak is located at the center of the acquisition window (for full k-space acquisition). However, in the presence of in-plane susceptibility field gradients, the k-space echo peak of gradient-echo EPI may deviate from the center of the targeted k-space coverage. For example, Figs. 1a and b schematically compare the echo formations (displayed along with a gradient-echo EPI sequence) in the absence of, and in the presence of, a background susceptibility field gradient along the phase-encoding direction. The local-field-gradient-induced k-space data distortion has three consequences. First, the distortions in k-space scan trajectories result in geometric distortions in the reconstructed images (Jezzard and Balaban, 1995). Second, the effective echo time (i.e. the time duration between RF excitation and zero k y line acquisition) differs from the desired TE value, when there exists a susceptibility induced echo-shifting effect along the phaseencoding direction (Deichmann et al., 2002). This variation in effective TE causes a non-uniformity of the BOLD sensitivity (Deichmann et al., 2002). Third, because of the echo-shifting effect, the targeted k-space coverage may be under-sampled, which may induce the Gibb s ripple artifact or even signal loss in the reconstructed images. The relationship between EPI s echo-shifting effect (along both readout and phase-encoding directions) and the local susceptibility field gradient values is presented in detail in Appendix A. Eq. (1) summarizes the dependence of zero k y line displacement D zky Ri (due to phase-encoding echo-shifting effect) on Fig. 1. (a) EPI with an ideal echo formation. (b) EPI with the echo-shifting effect due to a background susceptibility field gradient along the phase-encoding direction.

3 N. Chen et al. / NeuroImage xx (2006) xxx xxx 3 EPI scan parameters and local linear susceptibility field gradients (in an image-domain region ). D zky ¼ G ;y TE des ð1þ 1 cfov y þ G Ri;y T esp The zero k x point displacement D zkx Ri (0) (due to readout echoshifting effect) at the effective zero k y line can be calculated from EPI scan parameters and local susceptibility field gradients using Eq. (2). G Ri;x TE des þ D zky D zkx T esp ð0þ ¼ ð2þ 1 cfov x þ G Ri;x T dw where G Ri, y and G Ri, x are the local susceptibility field gradients along the phase-encoding and readout directions respectively, TE des is the desired echo time, T esp is the inter k y line echo-spacing time, and T dw is the readout acquisition dwell time. Mapping the echo-shifting effect with k-space energy spectrum analysis Here, a k-space energy spectrum analysis is proposed to quantify EPI s spatially dependent echo-shifting effect along both readout and phase-encoding directions. The concept of the k-space energy spectrum is first illustrated with a mathematical simulation. k-space energy spectrum along the phase-encoding direction Figs. 2a and b show the mathematical phantom and field inhomogeneity map constructed for simulation. The mathematical phantom consisted of four components with different field inhomogeneity patterns (R1 to R4 in Fig. 2b). There exist significant positive and negative field gradients in regions R2 and R3, respectively, along the phase-encoding direction (i.e. the vertical direction), and there exists a significant field gradient in R4 along the readout direction (i.e. the horizontal direction). Based on Eq. (8) in Appendix A, a single-shot gradient-echo EPI simulation program was implemented in Matlab (assuming matrix size , bandwidth 200 khz, desired effective echo time 60 ms, and k y echo spacing time 0.92 ms). The magnitude representation of the acquired k-space data is presented in Fig. 2c, which shows that the echo signals corresponding to four image-domain components are located at different k-space locations. For example, the echo signals corresponding to R2 and R3 deviate from the k-space center along the phase-encoding direction, due to the significant local field gradients. Therefore, the effective TE values deviate from the desired value in R2 and R3 regions. The Fourier transformed image is shown in Fig. 2d, illustrating field distortions of different levels in R2, R3, and R4 regions. Before the k-space energy spectrum is introduced, we need to first briefly discuss the impact of k-space data truncation on the Fourier transformed image. Basically, truncation of a selected number of k y lines may result in either a severe signal loss or Gibb s ripple artifacts in the Fourier transformed image, depending on whether or not the peaks of the corresponding k-space echo signals are truncated. To illustrate this, 54 negative k y lines of the simulated EPI data are truncated and zero-filled, as illustrated in Fig. 2e. As shown in Fig. 2f, the Fourier transformed image is degraded by severe signal loss in region R2 because the corresponding k-space echo peak is truncated. Note that the R1, R3, and R4 signals in Fig. 2f are affected by Gibb s ripple artifacts because the corresponding high k-space signals are truncated and zero-filled. The ripple artifacts in regions R1, R3, and R4 can be reduced if the zero-filled k y lines are replaced by the values calculated from un-truncated k-space data using the iterative Cuppen s algorithm (Cuppen et al., 1986; McGibney et al., 1993). It has been shown previously that the iterative Cuppen s algorithm is superior to other partial Fourier reconstruction methods and can converge to the ground truth (i.e. the original full k-space data) when (1) accurate image background phase information is available and (2) the echo signals are not truncated (Cuppen et al., 1986; Haacke et al., 1991; McGibney et al., 1993). The magnitude representation of k-space data calculated with Cuppen s algorithm (using background phase information obtained from full k-space data and 4 iterations) and the Fourier transformed image are shown in Figs. 2g and h, respectively. It can be seen that the ripple artifacts in R1, R3, and R4 are reduced. On the other hand, the signal loss in R2 remains since the truncated R2 echo signals cannot be recovered with Cuppen s method (or any other partial Fourier reconstruction method). If a larger portion of k-space data is truncated and zero-filled (Fig. 2i), then the echo peaks corresponding to R1 and R4 may be removed, which in turn results in R1 and R4 signal loss in the Fourier transformed image (Fig. 2j). Even after the missing k-space data are filled with values calculated with the Cuppen s algorithm (Fig. 2k), the lost signals in R1, R2, and R4 regions cannot be recovered (Fig. 2l). Comparison of Figs. 2d, h, and l demonstrates that the pattern of pixel-wise signal intensity variation in a series of partial Fourier images (with different numbers of truncated k y lines) depends on the peak location of the corresponding k-space echo signals. Actually, the k-space location of echo signals corresponding to a specific image-domain region can be determined by analyzing the pixel-wise signal intensities of those partial Fourier images. The solid curve in Fig. 2m shows the signal intensities of a selected R2 pixel, reconstructed from partial Fourier data with 4 Cuppen s iterations when the number of truncated k y lines varies from 1 to 127. A sudden change of signal intensity is observed when the peak of the k-space echo signals is being truncated. For R2 signals, an abrupt signal change occurs when 49 k y lines are truncated and replaced, indicating that the zero k y line is displaced by 16 lines (i.e. 49 ((128 / 2) + 1)). The solid line is termed the k y space energy spectrum for the signals originating from image-domain region R2. The k-space energy spectrum for R1 and R3 signals is presented by dotted and dashed lines in Fig. 2m, respectively. It shows that (1) R1 echo signals locate very close to the k-space center, (2) the R3 echo signals deviate from the k-space center by 39 k y lines (i.e. 104 ((128 / 2) + 1)). The identified echo signal locations through k-space energy spectrum evaluation agree with the input of our simulation. Application of k-space energy spectrum analysis to human brain EPI (3 T) is illustrated in Fig. 3. Data were acquired from a healthy subject using a single-shot gradient-echo EPI sequence with the following scan parameters: matrix size 96 96, FOV 240 mm 240 mm, readout bandwidth 100 khz, slice thickness 4 mm, desired TE 60 ms, and echo spacing time ms. Three selected axial slices of the acquired EPI images are shown in Fig. 3a. The zero k y line displacement maps were calculated directly from the single-shot EPI data using the proposed k-space energy

4 4 N. Chen et al. / NeuroImage xx (2006) xxx xxx Fig. 2. (a, b) The mathematical phantom and field inhomogeneity map used in our simulations. (c, d) The EPI k-space data and the reconstructed image (matrix size: ). (e, f) The k-space data and reconstructed image after 49 negative k y lines are truncated and zero-filled. (g, h) The k-space data and reconstructed image after the 49 zero-filled k y lines are replaced with values calculated from the un-truncated data using Cuppen s partial Fourier algorithm. (i, j) The k-space data and reconstructed image after 84 k y lines are truncated and zero-filled. (k, l) The k-space data and reconstructed image after the 84 zerofilled k y lines are replaced with values calculated from the un-truncated data using Cuppen s partial Fourier algorithm. (m) The k y space energy spectrum of three pixels in R1, R2, and R3 regions. spectrum analysis procedure, and the results are shown in Fig. 3b. The k-space energy spectrum of three selected pixels is presented in Fig. 3c. As shown, for the chosen blue pixel (p1), an abrupt signal intensity change occurs when the number of the truncated k y lines is approximately 49, indicating that the effective zero k y line for this pixel does not deviate from the desired zero k y line (i.e. (96 / 2) + 1). On the other hand, the k- space energy spectrum of the green pixel (p2) has an abrupt signal change when the number of the truncated k y lines is approximately 25, indicating that the effective zero k y line

5 N. Chen et al. / NeuroImage xx (2006) xxx xxx 5 deviates from the center of the k-space along the phase-encoding direction by 24 lines (i.e. 25 ((96 / 2) + 1)). The k-space energy spectrum of the red pixel (p3) indicates that the corresponding zero k y line deviates from the center of the k- space by 31 lines (i.e. 80 ((96 / 2) + 1)). k-space energy spectrum along the readout direction As shown in the previous section, the zero k y line displacement can be measured directly from EPI data using the k- space energy spectrum analysis along the phase-encoding direction. Similarly, the zero k x point displacement can be measured using the k-space energy spectrum analysis along the readout direction. First, a series of images can be reconstructed with different numbers of truncated and partial Fourier replaced k x lines. Second, the pixel-wise signal intensities of the obtained images are analyzed, and the effective zero k x point for different image-domain regions can be identified (similar to the k y energy spectrum illustrated in Fig. 2e). As indicated by Eq. (24) in Appendix B, the zero k x point displacement changes between different k y lines. The zero k x point displacement identified with the proposed k x energy spectrum analysis is the zero k x point displacement corresponding to the effective zero k y line (Eq. (26)). Calculation of susceptibility field gradients Once the zero k y line displacement and the zero k x point displacement are measured from the k y and k x energy spectrum analysis described in the previous sections, the susceptibility field gradients along both phase-encoding and readout directions can be calculated with Eqs. (3) and (4) (which are derived directly from Eqs. (1) and (2)). D zky G Ri;y ¼ ð3þ cfov y D zky T esp þ TE des D zkx G Ri;x ¼ ð4þ cfov x T esp þ TE des þ D zkx D zky T dw

6 6 N. Chen et al. / NeuroImage xx (2006) xxx xxx In the simulated k-space data shown in Fig. 2c, the energy peaks corresponding to four image-domain regions are well separated. In this case, it is possible to quantify the echo-shifting effect without using the developed k-space energy spectrum analysis algorithm. However, when the local field gradients are strong or nonlinear, the patterns of the corresponding k-space energy peaks may be very complicated or even partially shifted outside the k-space sampling window. In these cases, direct quantification of the k-space echoshifting effect will be difficult, unless the developed k-space energy spectrum analysis method is applied, as illustrated by the simulation data (Fig. 4) and human brain EPI data (Fig. 5). Figs. 4a and b show the proton density map and field map, respectively, used as the input of our simulation study. The simulated field is nonlinear along the phase-encoding (vertical) direction, and this field nonlinearity can be clearly visualized from its 1D plot, presented by the blue solid line in Fig. 4g. The simulated EPI k-space data are shown in Fig. 4c, in which the k-space energy peaks corresponding to different image-domain regions cannot be easily recognized. The magnitude and phase reconstruction of the EPI image are shown in Figs. 4d and e, respectively. In addition to the geometric distortion and phase wrap-around effect, the image data are degraded by local signal loss (in the top of the magnitude image) since the corresponding echo energy peaks are partially shifted outside the acquisition window. Fig. 4e shows a very significant phase wrap-around effect in the area indicated by the arrow. As a result, it is difficult to quantify the field gradient by directly analyzing the image-domain phase values (Deichmann et al., 2002). The proposed k-space energy spectrum analysis algorithm quantifies the k-space patterns by analyzing the voxel signal intensities corresponding to a series of partial Fourier reconstruction. Therefore, unlike the previously reported phase analysis procedure, the performance of the k-space energy spectrum analysis algorithm is not degraded even there exists a significant phase wrap-around effect. The field map measured directly from EPI data using the k-space energy spectrum analysis algorithm is shown in Fig. 4f, which is represented in the distorted coordinates. The measured 1D field profile along the phase-encoding direction is shown by the red dashed line of Fig. 4g, which has different spatial extension compared with the simulation input (i.e. blue solid line) because of the different coordinate systems. Fig. 4. (a, b) The mathematical phantom and field map used as the input of our simulation. (c) The EPI k-space data. (d, e) The magnitude and phase reconstruction of EPI image. (f) The field inhomogeneity map calculated from EPI data using the k-space energy spectrum analysis algorithm. (g) The 1D field profiles along the phase-encoding direction of input field map (blue solid line; in non-distorted coordinates) and the calculated field map (red dashed line; in distorted coordinates). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

7 N. Chen et al. / NeuroImage xx (2006) xxx xxx 7 Fig. 5. (a, b) The magnitude and phase reconstruction of human brain EPI. (c) The zero k y line displacement map calculated directly from EPI data using the k-space energy spectrum analysis algorithm. Figs. 5a and b show the magnitude and phase reconstruction of an axial slice human brain EPI image acquired at 3 T. The magnitude image signal loss and the phase image wrap-around effect are induced by strong local field gradients in the yellow and red regions. The phase wrap-around effect in the yellow region is not very significant, and it is feasible to quantify the local phase values and the field gradients after applying appropriate unwrapping procedures. On the other hand, the conventional phase-unwrapping procedures may fail in the red region because of the significant wrap-around effect and the low signal-to-noise ratio. It is difficult to determine the actual phase values in the red region with conventional phase analysis procedures. Using the proposed k-space energy spectrum analysis, the zero k y line displacement map can be reliably calculated even in critical brain regions (Fig. 5c), which can be directly converted to field gradient map using Eq. (3). Our simulation and human brain EPI data indicate that the k-space energy spectrum analysis method is capable of mapping the strong field gradients and nonlinear fields as long as the corresponding k-space energy peaks are not completely shifted outside the k-space sampling window. Bo field mapping and EPI distortion correction A Bo field inhomogeneity map can be estimated from the susceptibility field gradient values obtained with Eqs. (3) and (4) if the Bo off-resonance value of any pixel is known and used as a reference. Assuming the Bo offset in the reference pixel is DB o (x o, y o ), then Bo offset values of the neighboring pixels can be estimated using Eqs. (5) and (6). DB o ðx o þ yx; y o Þ DB o ðx o ; y o Þ þ yx G xðx o ; y o ÞþG x ðx o þ yx; y o Þ 2 DB o ðx o ; y o þ yyþ DB o ðx o ; y o Þ þ yy G yðx o ; y o ÞþG y ðx o ; y o þ yyþ ð6þ 2 It should be noted that the yy value (in Eq. (6)) of two neighboring EPI pixels depends on the local field gradient. As described by Deichmann et al. (2002), this scaling effect ð5þ may be accounted for by a spatially dependent Q factor in Eq. (7). yy ¼ FOVy N y where Q Ri Q Ri ð7þ ¼ 1 T esp FOV y G Ri;y In a single EPI data set acquired at a fixed TE, the reference value of Bo off-resonance is not available, and therefore the estimated Bo field map may have a constant offset from the actual values. This constant offset in estimated Bo field map may be minimized with two approaches. First, assuming the RF excitation and reception are set to on-resonance mode in data acquisition, then the constant offset value in estimated Bo field map can be adjusted in such a way that the averaged Bo values in brain regions are zero. Second, the EPI pulse sequence may be modified so that a low-resolution Bo map can be calculated and used as a reference to the Bo offset (Roopchansingh et al., 2003). The estimated Bo field inhomogeneity map can be used to convert EPI data from distorted to non-distorted coordinates using previously reported methods, such as phase-modulation (Weisskoff and Davis, 1992; Chen and Wyrwicz, 1999, 2001), pixel-shifting (Jezzard and Balaban, 1995), and point-spread-function-based algorithm (Zeng and Constable, 2002; Zaitsev et al., 2004). It should be noted that the Bo field map estimated from EPI data is represented in distorted coordinates and can be directly applied to EPI distortion corrections (Reber et al., 1998; Zeng et al., 2004). As stated in the previous paragraph, the estimated Bo field map may have a constant offset value (i.e. same offset for every pixels). As a result, there may exist residual global image translation along the phase-encoding direction, even after field-map-based distortion correction. Fortunately, this global translation can be easily detected and corrected with any rigid body image alignment algorithm when EPI-based data need to be co-registered with structural data. Nonlinear EPI distortions, which have been the main challenge to existing rigid body and non-rigid body image alignment algorithms, can be reliably removed using the proposed EPI distortion correction technique. Methods A phantom study was performed at 1.5 T to test the proposed EPI distortion correction technique based on k-space energy spectrum

8 8 N. Chen et al. / NeuroImage xx (2006) xxx xxx analysis. Data were acquired with single-shot gradient-echo EPI sequence with the following scan parameters: matrix size 64 64, FOV 180 mm 180 mm, TE 40 ms, echo-spacing time ms, readout bandwidth 100 khz, and axial slice thickness 4 mm. Multiple data sets were acquired under different shim settings and RF frequency offsets. Using the developed k-space energy spectrum analysis algorithm, zero k y line displacement and zero k x point displacement were quantified, and in-plane susceptibility field gradients along both phase-encoding and readout directions were calculated using Eqs. (3) and (4). The Bo field maps corresponding to different shim settings were then estimated from the calculated field gradient maps, with an assumption that the averaged Bo frequency offset was zero. The estimated Bo field maps were then used to remove geometric distortions through the previously reported phase modulation algorithm (Weisskoff and Davis, 1992; Chen and Wyrwicz, 1999, 2001). The developed technique was also applied to map the Bo field inhomogeneities and remove EPI distortions in human brain EPI data (3 T). Using the calculated Bo field map, the geometric distortions in EPI data were corrected with the phase modulation algorithm. The corrected EPI data were then compared with the reference images acquired with 16-shot EPI pulse sequence of the same matrix size. Multiple EPI data sets of twelve different TEs were also acquired so that the field inhomogeneity measured with the conventional field mapping method could be compared with the field map calculated with the proposed k-space energy spectrum analysis algorithm. Results Phantom EPI images acquired at 1.5 T under different shim settings and RF frequency offsets are shown in Figs. 6a to c. EPI data obtained with mis-adjusted x-shim value are skewed along the readout (horizontal) direction (Fig. 6a). On the other hand, the image obtained with mis-adjusted x- and y-shim values is skewed along the readout (horizontal) direction and further deformed along the phaseencoding (vertical) direction, as shown in Fig. 6b. The image shown in Fig. 6c was acquired with the same shim setting as that in Fig. 6b, but with RF frequency offset of 60 Hz. It can be seen that this RF frequency offset results in additional image translation along the phase-encoding (vertical) direction, as well as the in-plane signal non-uniformity due to the reduced slice-refocusing efficiency. Using the proposed k-space energy spectrum analysis algorithm, the field inhomogeneities corresponding to these three data sets were measured directly from EPI data and applied to remove the EPI distortions. As shown in Figs. 6d to f, the nonlinear geometric distortions can be effectively removed, without the need of additional field mapping scan. However, it can be seen in Fig. 6f that the linear translation along the phase-encoding direction due to the RF frequency offset cannot be measured or corrected with the proposed method (when the RF frequency offset information is not available). This residual linear translation can be recognized and corrected reliably with rigid body image alignment algorithm when the corrected EPI data are co-registered with structural images. Application of the developed method to geometric distortions in human brain EPI (3 T) is presented in Fig. 7. As shown in Fig. 7a, there exist significant geometric distortions in two selected axial slices of brain EPI images. Using the proposed method, the field inhomogeneity was measured directly from EPI data sets and applied to correct the distortions (Fig. 7b). Comparison of corrected EPI images (Fig. 7b) and the reference images (Fig. 7c: 16-shot EPI of the same matrix size) indicates that the majority of the geometric distortions in human brain single-shot EPI data can be removed with the proposed technique. Bo field inhomogeneities measured with conventional field mapping scans and the proposed k-space energy spectrum analysis algorithm are presented in Figs. 7d and e, respectively. It can be seen that the field maps obtained with the proposed method are very similar to that obtained with conventional field mapping scan, and the difference of the field maps obtained with these two approaches is not significant (Fig. 7f). Discussion The proposed method is superior to conventional field-mapbased EPI correction methods in three ways. First, using k-space Fig. 6. (a) Phantom EPI data acquired with mis-adjusted x-shim. (b) EPI data acquired with mis-adjusted x- and y-shim. (c) EPI data acquired with the same shim setting as that in panel b but with RF frequency offset of 60 Hz. The corresponding EPI images after distortion correction are shown in panels d f.

9 N. Chen et al. / NeuroImage xx (2006) xxx xxx 9 energy spectrum analysis algorithm, the field maps can be obtained without performing additional field mapping scan. The proposed method can also be applied to retrospectively remove the geometric distortions in previously acquired EPI and fmri data, provided that the complex-domain k-space data are available. Second, in the presence of subject movement during dynamic EPI scans, the valid field maps corresponding to different scan times can be reliably obtained with the proposed post-processing method. Third, in the proposed algorithm, complicated in-plane phase-unwrapping procedure that is required in most conventional field mapping methods can be avoided. However, the time-dependent eddy current effects due to fast-switching readout gradients and phase-blipped gradients, which have been shown contribute to EPI distortions (Chen and Wyrwicz, 1999, 2001), may not be measured with the proposed method. In order to measure and remove EPI distortions due to eddy current effects, previously reported multi-echo field mapping protocols need to be applied (Chen and Wyrwicz, 1999, 2001). In comparison to conventional field mapping procedures, the proposed k-space energy-spectrum-analysis-based field measurement requires a higher computation cost. For example, using a Matlab program implemented on a Mac computer (1.5 GHz PowerPC G4 with 512 MB memory), it takes 5.7 s to perform the k-space energy spectrum analysis for measuring the susceptibility field gradient along one dimension (e.g. the phase-encoding direction) of a single-slice EPI image. It therefore takes 11.4 s to calculate a 2D field gradient map. On the other hand, using a conventional field mapping procedure (Reber et al., 1998), it only takes 0.3 s to reconstruct a 2D field map of the same matrix size. The high computation cost of the k-space energy spectrum analysis is mainly related to the iterative partial Fourier reconstruction and the associated Fourier transformation. Using the proposed k-space energy spectrum analysis, the susceptibility field gradient values along both readout and phaseencoding directions for each voxel can be calculated directly from gradient-echo EPI data. As illustrated in previous sections, the Bo field may be estimated from the calculated susceptibility field gradient values and used to remove EPI distortions if the averaged off-resonance value of the scanned region is known (for example, the averaged value is zero when an on-resonant RF pulse is applied). On the other hand, if the averaged off-resonance value is unknown, then the estimated Bo field map will deviate from the actual values by a (spatially invariant) constant value. In this case, there will exist residual translational artifact after correcting EPI distortions based Fig. 7. (a) Human brain EPI images. (b) The EPI images after distortion correction. (c) The reference images obtained with 16-shot segmented EPI sequence. (d) The field maps measured with a conventional multiple-te imaging method. (e) The field maps measured with the proposed k-space energy spectrum analysis method. (f) The difference between panels d and e.

10 10 N. Chen et al. / NeuroImage xx (2006) xxx xxx on the estimated Bo field map. This residual translational artifact can be detected and corrected using a rigid body image alignment algorithm. It should be noted that the proposed field mapping method based on post-processing of k-space data is actually compatible and complementary to a recently reported dynamic field mapping technique (Roopchansingh et al., 2003), in which Bo field inhomogeneity information is obtained from a lowresolution field mapping scan acquired with each EPI volume set. From the embedded low-resolution field mapping scan, a low-resolution Bo field map (e.g. 64 8) can be derived without excessive field mapping time, and the acquired field maps are valid even when the subject moves during dynamic scans. Even though the obtained low-resolution field map itself cannot provide detailed susceptibility field information in critical brain regions, it provides accurate information on the RF frequency offset and can be combined with our k-space energy spectrum analysis method for an accurate high-resolution field inhomogeneity mapping. Posse showed that the echo-shifting effect along the readout direction of gradient-echo MRI can be quantified by applying shifted k-space filters with a small bandwidth to the raw data (Posse, 1992), and Deichmann et al. recently reported that EPI s echo-shifting effect along the phase-encoding direction can be directly obtained from the phases of the complex image data (Deichmann et al., 2002). In the present work, the echo-shifting effect along both readout and phase-encoding directions of gradient-echo EPI is quantified with the k-space energy spectrum analysis algorithm, which is based on a series of partial Fourier image reconstruction with different partial Fourier ratios, as illustrated by Fig. 2. We further demonstrate that the obtained information on the echo-shifting effect can be applied to remove the geometric distortions in gradient-echo EPI. Although the proposed k-space energy spectrum analysis algorithm has been applied to process the gradient-echo EPI data in this paper, it can be applied to analyze the echo-shifting effect and remove the geometric distortions in asymmetric spin-echo EPI as well. On the other hand, since there is no echo-shifting effect in spin-echo EPI, the proposed method cannot be used to quantify the susceptibility field gradients in spin-echo EPI. The proposed technique is designed to measure the in-plane susceptibility field gradient and to correct the resultant EPI geometric distortions. Our data suggest that the developed method is effective even in the presence of local nonlinear field gradient as long as the corresponding k-space energy peaks are still within the k-space sampling window. However, when the k- space energy peaks are shifted completely outside the sampling window due to a very significant in-plane local field gradient, the k-space energy spectrum analysis method cannot generate accurate field map values in those critical regions. Similarly, the intravoxel dephasing artifact due to through-plane susceptibility field gradients cannot be compensated for using the k-space energy spectrum analysis method. The intravoxel dephasing artifact due to in-plane and through-plane field gradients may be reduced using the 3D gradient compensation method recently reported by Posse et al. (2003). The developed k-space energy spectrum analysis algorithm and EPI distortion correction technique are compatible with parallel imaging methods. Even though the EPI distortions can generally be reduced when data are acquired with parallel imaging schemes, the residual distortions may still degrade the spatial accuracy, especially at high field. Using the proposed post-processing algorithm, the residual distortion in parallel EPI can be further reduced. In addition to the susceptibility field mapping, the developed k-space energy spectrum analysis method can also be applied to quantify the k-space echoshifting effect due to in-plane motion, as reported by Wedeen et al. (1994). Acknowledgments This study is supported by NIH grant R03EB003902, R01NS37922, and U41RR Appendix A. Echo-shifting effect along the phase-encoding direction In gradient-echo EPI, two-dimensional (2D) spatial encoding is achieved with fast-switching readout gradients (G x (t)) and phaseblipped gradients (G y (t)) during time-domain data acquisition. The pulse sequence of single-shot gradient-echo EPI is shown in Fig. 8a. The acquired time-domain signals (S R1 (t)), originating from an image-domain region R1, can be represented by Eq. (8) (Jezzard and Balaban, 1995), where D(x,y) is the proton density map of the scanned object, Dx(x,y) represents the off-resonance factors (including B o field inhomogeneity, eddy current effect, chemical shift effect, etc.). ZZ S R1 ðþ¼ t Dx; ð yþe i 2 pfkx ðþx t þ kyðþy t þ Dx ð x;y Þtg e t T24ðx;yÞ dxdy R1 where k x ðþ¼c t k y ðþ¼c t Z Z t t V¼ 0 t tv¼ 0 G x ðtvþdtv G y ðtvþdtv EPI s gradient waveforms are designed in such a way that the echoes are refocused at the center of the acquisition window (Fig. 8a), when the off-resonance factor (Dx in Eq. (8)) is absent. The desired value of the effective echo time TE des (i.e. the duration between RF pulse excitation and zero k y line acquisition) satisfies Eq. (9), where T pre is the pre-acquisition delay, and T ACQW is the time duration of the acquisition window (see Fig. 8a). k y ðte des Þ ¼ c Z TEdes t ¼ 0 G y ðþdt t ¼ 0 where TE des ¼ T pre þ T ACQW 2 As described by Hennel (1997), 2D EPI s scan trajectory can be represented in a three-dimensional (3D) k x k y t space. The projections of an ideal 3D k x k y t trajectory (when Dx is absent) onto a 2D k y t plane and a 2D k x k y plane are schematically shown in Figs. 8b and c, respectively. When there exists a non-zero off-resonance factor (Dx), the 3D k x k y t space trajectories are distorted. We first consider a case that, within an image-domain region R2, the off-resonance factor Dx R2 is a time-independent linear susceptibility field gradient anti-parallel to the phase-blipped gradient (Fig. 8d), as ð8þ ð9þ

11 N. Chen et al. / NeuroImage xx (2006) xxx xxx 11 Fig. 8. (a to c) The pulse sequence of single-shot gradient-echo EPI and the corresponding scan trajectories represented in k y t space and k x k y space. (d to f) The sequence and scan trajectories of EPI when there exists a susceptibility field gradient anti-parallel to the phase-blipped gradients. (g to i) The sequence and scan trajectories of EPI when there exists a susceptibility field gradient parallel to the phase-blipped gradients. In the k y t representations of the scan trajectories (b, e, h), the solid lines and solid circles represent the actual scan trajectories; the dashed lines represent the desired trajectories; the dashed lines with two dots represent the phase accumulation as a result of the susceptibility field gradients. In the k x k y representations of the scan trajectories (c, f, i), the solid lines represent the actual scan trajectories and the gray squares represent the desired k-space coverage. described by Eq. (10) in which c R2 and G R2 are constants and G R2 < 0. In this case, EPI s time-domain signals originating from the image-domain region R2 can be represented by Eq. (11), where k y (t) and k y(t) are the ideal and distorted scan trajectories, respectively. Dx R2 ðx; yþ ¼ C R2 þ cg R2 y: ð10þ ZZ S R2 ðþ¼ t R2 Dx; ð yþe i 2 p ð kxðþx t þ k y ðt;g R2 Þy þ C R2 tþ e t T24ðx;yÞ dx dy; ð11þ where k y ðt; G R2 Þ ¼ k y ðþþcg t R2 t: As indicated by Eq. (11), the resultant scan trajectory k y(t,g R2 ) deviates from the desired trajectory k y (t) due to the negative background field gradient G R2. Because of the reduction in k y spacing, more phase-blipped gradients need to be applied before the scan trajectory reaches the zero k y line. Therefore, the effective echo time for image-domain region R2 increases due to the echo-shifting effect along the phase-encoding direction (Deichmann et al., 2002), as schematically illustrated with Fig. 8d. The achieved effective echo time TE eff (i.e. the time that the zero k y line is acquired) satisfies Eq. (12), which can be transformed to Eq. (13) or equivalently Eq. (14). k y ðte eff ; G R2 Þ ¼ 0 ð12þ k y ðte des Þþc Z TEeff Z TEeff G y ðþdt t þ cg R2 TE eff ¼ 0 t ¼ TE des G y ðþdt t ¼ G R2 TE eff TE des ð13þ ð14þ The echo-shifting effect can be quantified by the number of lines between the desired zero k y line and the achieved zero k y line (i.e. zero k y line displacement). Assuming that there exists D zky R2 phase blips between the desired zero k y line and the achieved zero k y line, Eq. (14) can be transformed to Eq. (15), where G Blip and s Blip are the amplitude and duration of each phase-blipped gradient waveform and T esp is the echo-spacing time. G Blip s Blip D zky R2 ¼ G R2 TE des þ D zky R2 T esp ð15þ Based on Eq. (15), the dependence of the echo-shifting effect (quantified by zero k y line displacement) on local susceptibility

12 12 N. Chen et al. / NeuroImage xx (2006) xxx xxx field gradient ( G R2 ) and scan parameters (TE des, FOV y, and T esp ) satisfies Eq. (16). D zky R2 ¼ G R2 TE des ¼ G R2 TE des ð16þ G Blip s Blip þ G R2 T 1 esp c FOV y þ G R2 T esp Fig. 8e schematically shows the k y t plane representation of (1) the desired scan trajectory (dashed line; i.e. k y (t) in Eq. (11)), (2) the background field gradient induced trajectory deviation (dashed line with two dots; i.e. gg R2 t in Eq. (11)), and (3) the actual scan trajectory (solid line; i.e. k y (t,g R2 ) in Eq. (11)). For image region R2, D zky R2 > 0 (i.e. TE eff >TE des ). The corresponding k x k y plane representation of the actual scan trajectory is schematically illustrated in Fig. 8f, indicating that (1) the negative k y space is over-sampled and (2) the positive k y space is undersampled. Now, we consider another case where, within an image-domain region R3, there exists a time-independent linear field gradient parallel to the phase-blipped gradient direction ( G R3 ). In this case, the k-space echo-shifting effect results in the increase of k y line spacing and therefore a decrease of the effective echo time, as schematically illustrated with Fig. 8g. The k y t plane and k x k y plane representations of the resultant scan trajectories are schematically shown in Figs. 8h and i, respectively. A further mathematical examination (not presented here) shows that the zero k y line displacement represented by Eq. (16) is also valid when the linear susceptibility field gradient is parallel to the phase-blipped direction. In this case, D zky R3 < 0 (i.e. TE eff <TE des ). Appendix B. Echo-shifting effect along the readout direction Considering a case where there exists a susceptibility field gradient along the readout direction in an image-domain region R4 (as represented by Eq. (17) in which c R4 and G R4 are constants), the corresponding EPI time-domain signals are represented by Eq. (18), in which k x (t) and k x(t) are the ideal and distorted k x scan trajectories, respectively. Dx R4 ðx; yþ ¼ C R4 þ cg R4 x ð17þ ZZ S R4 ðþ¼ t Dx; ð yþe i 2 p ð k x ðt;g R4 Þx þ k y ðþy t þ C R4 tþ e t T24ðx;yÞ dxdy R4 ð18þ where k x ðt; G R4 Þ ¼ k x ðþþcg t R4 t An EPI pulse sequence is designed in such a way that the signals in every k y lines are refocused at the centers of the readout gradient lobes, in the absence of a background susceptibility field gradient. Therefore, for EPI data of a matrix size N x N y, the ideal k x (t) trajectory satisfies Eq. (19), in which TE des is the desired echo time, T esp is the echo-spacing time, n y is an integer, and s ny is the time that zero k x point is reached in k y line (n y ) of the ideal scan trajectory. k x s ny ¼ 0 where s ny ¼ TE des þ n y T esp ð19þ and Ny 2 þ 1 V n y V Ny 2 As indicated by Eq. (18), the actual scan trajectory k x(t) deviates from the desired trajectory k x (t) due to the susceptibility field gradient G R4, and therefore the signals may not be refocused at the centers of the readout gradient lobes. The zero k x points for different k y lines of the distorted scan trajectory satisfy Eq. (20), in which sñy is the time that the scan trajectory reaches the zero k x point corresponding to k y line n y. Eq. (20) can be transformed to Eq. (21) or equivalently Eq. (22). k x s ny ¼ 0 ð20þ Z s ny k x s ny þ c Z s ny t ¼ s ny t ¼ s ny G x ðþdt t ¼ G R4 s ny G x ðþdt t þ cg R4 s ny ¼ 0 ð21þ ð22þ The displacement of zero k x point corresponding to a specific k y line n y is defined by Eq. (23), in which T dw is the dwell time. Based on Eq. (22), the zero k x point displacement satisfies Eq. (24). D zkx R4 n y ¼ s ny s ny T dw ð23þ G x T dw D zkx R4 n y ¼ GR4 TE des þ n y T esp þ D zkx R4 ðn yþt dw N y 2 þ 1 V n y V N y ð24þ 2 The dependence of zero k x point displacement at zero k y line (D zkx Ri (0)) on the susceptibility field gradient and other scan parameters (FOV, TE des, T dw, and T esp ) can be derived, as represented by Eq. (25). D zkx R4 ð0þ ¼ G R4 TE des ð25þ 1 c FOV x þ G R4 T dw In general, zero k y line displacement (D zky R4 ) and zero k x point displacement (D zkx R4 (n y )) may co-exist, when local susceptibility field gradients have projection components along both readout and phase-encoding directions. A generalized form of zero k x point displacement at effective zero k y line is represented by Eq. (26), in which D zky R4 can be calculated using Eq. (16). D zkx R4 0 ð Þ ¼ G R4 TE des þ D zky R4 T esp ð26þ 1 c FOV x þ G R4 T dw Appendix C. Spatially dependent echo-shifting effect In actual MRI studies, the susceptibility field gradient within a scanned object may not be a linear function, and it is not apparent if the theory described in the previous two sub-sections is generally applicable. To understand the echo-shifting effect in general, the scanned object is mathematically divided into many small subregions so that the susceptibility field gradient in each sub-region may be described by a linear function. As presented in Eq. (27), the time-domain signals S(t) originating from the whole scanned object are a linear combination of signals S Ri (t) originating from all the sub-

13 N. Chen et al. / NeuroImage xx (2006) xxx xxx 13 regions, in which the susceptibility field gradient Dx Ri (x,y) is approximately linear. E will be covered by different number of k y represented by Eq. (31). lines (m), as St ðþ¼ XN i ¼ 1 ¼ XN i ¼ 1 ¼ XN i ¼ 1 S Ri ðþ t 8 < ZZ : Ri Ri Dx; ð yþe i 2 p f kx ðþx t þ ky ðþy t þ DxR ðx;yþt i g e t ð T2 4 x;y 9 = Þ dxdy ; 8 9 < ZZ = ð27þ Dx; ð yþe i 2 p ð k xðt;g ;xþx þ k yðt;g ;yþy þ C tþ e t T2 4 ðx;yþ dxdy : ; where Dx Ri ðx; yþ ¼ C Ri þ cg Ri;y y þ cg Ri;x x k x t; G Ri;x ¼ kx ðþþcg t Ri;x t; and k y t; G Ri;y ¼ ky ðþþcg t Ri;y t Eq. (27) implies that the echo-shifting effect of the whole scanned object varies spatially. The zero k y line displacement number corresponding to different sub-regions can be calculated from the local linear susceptibility field gradients and scan parameters, as presented in Eq. (28). The zero k x point displacement at zero k y line can be calculated using Eq. (29). D zky ¼ G ;y TE des ð28þ 1 c FOV y þ G Ri;y T esp D zkx ð0þ ¼ G Ri;x TE des þ D zky T esp ð29þ 1 c FOV x þ G Ri;x T dw Appendix D. Transition width of the k-space energy spectrum As shown in Figs. 2m and 3c, the transition point of the k- space energy spectrum (i.e. the echo-shifting effect) depends on the local field gradient. Actually, the transition width of the k- space energy spectrum also depends on the local field gradient. For example, in Fig. 3c, the transition width of p3 is larger than that of p2. The dependence of the k-space energy spectrum transition width on the local field gradient is mathematically described below. As stated in Appendix A, the actual k-space trajectory (k y(t)) deviates from the desired trajectory (k y (t)) as a result of the local field gradient. Based on Eq. (11), the phase increment between two ideal k y lines (yk y ) and the phase increment between two actual k y lines (yk y) satisfy Eq. (30), where T esp is the echo-spacing time. y k y ¼ yk y þ cg R2 T esp ¼ 1 FOV y þ cg R2 T esp ð30þ Assuming the width of the k-space energy peak (corresponding to a certain image-domain object) is E, which can be covered by a certain number of k y lines (n), when there is no background susceptibility field gradient. In the presence of a susceptibility field gradient, the phase increment between two sampling k y lines changes, and therefore the same energy width E ¼ nyk y ð31þ ¼ my k y Based on Eqs. (30) and (31), the local field gradient ( G R2 ) changes the k y line number needed for covering a certain energy peak by a factor R, as shown by Eq. (32). R ¼ m n ¼ 1 ð32þ 1 þ cg R2 T esp FOV y The ratio n Eq. (32) represents the ratio of the k-space energy spectrum transition width with and without the background susceptibility field gradient. For example, when the local field gradient is anti-parallel to the phase-blipped gradient (i.e. G R2 < 0), the phase increment between two k y lines decreases (Fig. 8f) and therefore the transition width increases (R > 1; e.g. p3 in Fig. 3). On the other hand, when the local field gradient is parallel to the phaseblipped gradient, the phase increment between two k y lines increases, and therefore the transition width decreases. In our current implementation, the local field gradient is calculated from the k-space energy spectrum transition point. Theoretically, the field gradient can also be calculated from the k-space energy spectrum transition width. It is also possible to use both information of transition point and transition width to further improve the robustness of the k-space energy spectrum analysis based field mapping. References Chen, N.-K., Wyrwicz, A.M., Correction for EPI distortions using multi-echo gradient-echo imaging. Magn. Reson. 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