NOTE Magnetic Resonance in Medicine 75:2055–2063 (2016)
PSF Mapping-Based Correction of Eddy-Current-Induced Distortions in Diffusion-Weighted Echo-Planar Imaging Myung-Ho In,1* Oleg Posnansky,1 and Oliver Speck1,2,3,4 Purpose: To accurately correct diffusion-encoding directiondependent eddy-current-induced geometric distortions in diffusion-weighted echo-planar imaging (DW-EPI) and to minimize the calibration time at 7 Tesla (T). Methods: A point spread function (PSF) mapping based eddycurrent calibration method is newly presented to determine eddy-current-induced geometric distortions even including nonlinear eddy-current effects within the readout acquisition window. To evaluate the temporal stability of eddy-current maps, calibration was performed four times within 3 months. Furthermore, spatial variations of measured eddy-current maps versus their linear superposition were investigated to enable correction in DW-EPIs with arbitrary diffusion directions without direct calibration. For comparison, an image-based eddy-current correction method was additionally applied. Finally, this method was combined with a PSF-based susceptibility-induced distortion correction approach proposed previously to correct both susceptibility and eddy-current-induced distortions in DW-EPIs. Results: Very fast eddy-current calibration in a threedimensional volume is possible with the proposed method. The measured eddy-current maps are very stable over time and very similar maps can be obtained by linear superposition of principal-axes eddy-current maps. High resolution in vivo brain results demonstrate that the proposed method allows more efficient eddy-current correction than the image-based method. Conclusion: The combination of both PSF-based approaches allows distortion-free images, which permit reliable analysis in diffusion tensor imaging applications at 7T. Magn Reson C 2015 Wiley Periodicals, Inc. Med 75:2055–2063, 2016. V Key words: PSF; geometric distortion; eddy-current; EPI; DTI
INTRODUCTION Single-shot spin-echo echo-planar imaging (EPI) is generally used as a method of choice for diffusion tensor 1 Department of Biomedical Magnetic Resonance, Institute for Experimental Physics, Otto-von-Guericke University Magdeburg, Germany. 2 German Centre for Neurodegenerative Diseases (DZNE), Site Magdeburg, Germany. 3 Leibniz Institute for Neurobiology, Magdeburg, Germany. 4 Center for Behavioral Brain Sciences, Magdeburg, Germany. Grant sponsor: German Research Foundation (DFG); Grant number: SP632-4. *Correspondence to: Myung-Ho In, Ph.D., Department of Biomedical Magnetic Resonance, Institute for Experimental Physics, Otto-von-Guericke University Magdeburg, Leipziger Straße 44, 39120 Magdeburg, Germany. E-mail: [email protected]
Received 3 December 2014; revised 27 March 2015; accepted 31 March 2015 DOI 10.1002/mrm.25746 Published online 22 June 2015 in Wiley Online Library (wileyonlinelibrary. com). C 2015 Wiley Periodicals, Inc. V
imaging (DTI) (1) studies. EPI at ultrahigh field (UHF) enables fast brain imaging with high spatial resolution due to the increased signal to noise ratio (SNR). However, increasing spatial resolution or use of stronger diffusion-gradients leads to longer echo times (TE) and consequently undesirable T2 signal loss (2), which can be more harmful at UHF due to relatively shorter T2. To compensate or correct for eddy-current-induced geometric distortions, the benefit of the Stejskal-Tanner diffusion scheme allowing the use of a minimized TE reinitiates the need for alternative techniques of the twice-refocused spin-echo method (3,4). To take eddy-current-induced distortions into account, several correction approaches involving image-based correction (5–7) and calibration methods (8–10) have been proposed previously. However, they often require time consuming post-processing procedures (5–7). Although calibration methods (8–10) can measure eddy-currentinduced distortions, only simplified distortion models, including shearing, scaling, and translation, were determined from the calibration data and used for correction. All these methods may be insufficient to rectify nonlinear eddy-current effects within the long readout acquisition window of EPI (11,12). Although a recent approach (12) considered even the nonlinear effects, it still can be a time consuming procedure, especially in highresolution diffusion-weighted (DW) imaging with many diffusion encoding directions. Point spread function (PSF) methods (13–16) have been adapted for susceptibility-induced distortion correction in EPI because they potentially allow very reliable distortion mapping. Due to an additional PSF (or spin-warp) phase-encoding (PE) added to the original EPI sequence, the reconstructed PSFs are represented along the correlation (or diagonal) line in the PSF- (or nondistorted) and EPI-PE (or distorted) coordinates after three dimensional (3D) inverse Fourier transformation (13–16). Mapping of the PSFs’ deviations from the diagonal line in both nondistorted and distorted coordinates results in shift (or distortion) maps in both coordinates. Several approaches (13–16) have been proposed to extract the distortion information from the measured PSF data more accurately leading to a higher correction fidelity. In this work, a fast PSF-based eddy-current calibration method is newly proposed to determine eddy-currentinduced distortions including nonlinear eddy-current effects. With the calibration data, additional investigations were carried out to demonstrate that (I-i) an eddycurrent map with arbitrary diffusion gradient amplitude at each axis can be estimated using interpolation chosen from the dependence of eddy-current-induced distortion strength on the diffusion gradient amplitude, (I-ii)
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temporal stability of eddy-current-induced distortion maps is sufficiently high such that phantom reference acquisitions need not be performed too often, and (I-iii) the linear superposition of measured or estimated eddycurrent maps from the principal gradient axes provides similar performance compared with dedicated acquisitions. (II) For comparison of correction accuracy, an image intensity-based eddy-current correction was additionally applied. (III) Together with a PSF-based susceptibility-induced distortion correction approach suggested previously (16), a PSF-based comprehensive distortion correction was performed to accurately diminish both susceptibility- and eddy-current-induced distortions in DW-EPIs. METHODS
In PSF mapping, the PSF image I(y,s) is obtained by multiplying the proton density q(s) with the PSF H(y,s). A single delta function approximation of the PSF leads to (15): [1]
where d is the Dirac delta-function, and s and y correspond to the nondistorted (or PSF) and distorted (or EPI) PE coordinates, respectively. The distortion term D(s) represents deviations of the PSFs from the diagonal line along the y-direction in the s-axis (15) and yields geometric distortion in the acquired EPI image. If PSF mapping is performed with DW gradients, the obtained shift map D(s) ¼ D(s)diff becomes a combination of susceptibility- D(s)suscep and eddy-current- D(s)EC induced shift maps. Because all distortions cause PSFs’ deviations in the PSF mapping, this map includes even geometric distortions induced by nonlinear eddy-current effects within the readout acquisition window. Therefore, subtraction between the two shift maps calculated from PSF data with and without DW gradient (i.e. D(s)diff and D(s)suscep) yields the eddy-current-induced shift map D(s)EC including nonlinear eddy-current effects in the nondistorted coordinate (s). DðsÞEC ¼ DðsÞdiff DðsÞsuscep :
Temporal Stability of Eddy-Current-Induced Distortions To verify temporal stability of the eddy-current maps, four calibration data sets with identical protocols, acquired at different times over 3 months, were compared. The differences between calibration data with less than 10-min, 1-month, and 2-month calibration intervals were calculated as histograms. Spatial Variations of Direct Calibration versus Linear Superposition of Principal Axes
PSF-Based Eddy-Current Calibration
Iðy; sÞ ¼ rðsÞHðy; sÞ rðsÞdðs þ DðsÞ yÞ
data. In contrast to the linear fitting, a constraint of zero offset was applied to the 2nd order polynomial fitting, because the eddy-currents are assumed to be zero without any diffusion gradient. The differences between the measured and the fitted data were calculated as standard deviation, absolute maximum, and absolute mean values.
[2]
This map can be applicable for correction when the spatial distribution of eddy-current effects is constant across measurements and reproducible, even over long time periods. This assumption was tested as described in the following section. Dependence of Eddy-Current-Induced Distortion Strength on the Amplitude of Diffusion Gradients The linearity of eddy-current-induced distortions as a function of the diffusion gradient amplitude was investigated. For each gradient axis, sixteen eddy-current maps were measured with different b-values from 50 to 800 s/ mm2 and steps of 50 s/mm2 resulting in different gradient amplitudes (up to 92% of the nominal maximum amplitude) with identical duration. To test linearity, 1st and 2nd order polynomial fittings were applied to the
We tested spatial variations of direct calibration of all diffusion encoding directions versus the linear superposition of the three principal gradient directions to investigate whether linear superposition of the eddy-current maps for the x, y, and z diffusion-encoding directions allows calculation of eddy-current maps for arbitrary diffusion-encoding directions without prolonging calibration time or if diffusion-encoding directions are changed. Based on six eddy-current maps (two maps with two different amplitudes of the diffusion gradient for each axis, 27.18 and 65.62 mT/m, to generate the 12 diffusionencoding directions), eddy-current maps for the 12 diffusion-encoding directions were generated by linear superposition and compared with the measured eddycurrent maps. To demonstrate spatial variations over the entire volume including all diffusion directions, in addition to representative values (such as standard deviation, absolute maximum, and absolute mean), standard deviation and absolute mean were calculated along the x, y, and z directions in 3D image space. Calibration data, the differences between calibration data, and differences between calibration data and their linear superposition were used for these calculations. Comparison with FSL-Based Eddy-Current Correction For comparison, the image intensity-based eddy-current correction implemented in FSL (http://fsl.fmrib.ox.ac.uk/ fsl/) was also applied. The eddy-current correction was performed either with the proposed method or based on FSL using the option of “eddy-correct” with splineinterpolation. To verify the improvement quantitatively, standard deviation of DW-EPIs and fractional anisotropy (FA) values in cerebrospinal fluid (CSF) regions of the entire volume were computed as histograms before and after eddy-current correction. PSF-Based Comprehensive Distortion Correction The steps for the eddy-current calibration and comprehensive distortion correction are illustrated in Figure 1. A final eddy-current map was obtained after extrapolation of the shift map D(s)EC to the areas outside of the substantial signal region and 3 3 median filtering. Although the eddy-currents were calibrated in the
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FIG. 1. A schematic description of the proposed calibration and PSF-based comprehensive distortion correction. Note that the calibration is performed as a separate session using a phantom.
nondistorted coordinates (s), it is sufficient to use them without transformation from nondistorted to distorted coordinates because eddy-currents generally vary slowly in space and the difference is negligible (12). As described by Zaitsev et al (15), the eddy-current-induced distortion correction in DW-EPIs was performed with the final eddy-current maps obtained in the nondistorted coordinate (s). Individual susceptibility-induced distortion correction followed after eddy-current correction. An improved PSF-based susceptibility-induced distortion correction approach (16) was adopted in this study. In contrast to the calibration, mapping of susceptibility-induced distortions was carried out in the distorted coordinates (y) because they can strongly vary locally. Because a measured PSF profile in the EPI-PE coordinates (EPI-PSF) was derived and directly used as a convolution kernel for distortion correction in DW-EPI (16), the correction fidelity was improved, especially in regions with strong susceptibility-induced distortions. Three FA maps were calculated using FSL from DWEPIs without correction, with eddy-current correction and with comprehensive correction.
Experiments The proposed PSF-based eddy-current calibration was performed on a 7 Tesla (T) scanner (Siemens Healthcare, Erlangen, Germany) equipped with a gradient coil type “SC72AB” (70 mT/m and 200 T/m/s). A spherical silicon-oil phantom (diameter 16 cm) and an eightchannel head array coil (Rapid Biomedical, W€ urzburg, Germany) were used. The PSF scan was performed with an acceleration factor of 18 (corresponding to 10 PSFencoded samples for each direction) (15). The imaging protocols were identical to the in vivo DW-EPI experiment (described below) with these modifications: repetition time (TR) ¼ 2000 ms, 20 axial slices, and slice thickness ¼ 4.8 mm. TR and slice number were reduced to minimize the calibration time and the slice thickness was increased to keep the same field of view (FOV)
along the slice direction, as in the in vivo experiments as well as to increase SNR. Because the eddy-currentinduced distortions vary smoothly along the slice direction, linear interpolation was applied to generate eddycurrent maps covering the entire volume of the in vivo experiments. The acquisition time (TA) for calibration with and without diffusion gradients was 4 min 33 s. For DTI experiments, two healthy subjects after institutional review board (Otto-von-Guericke University Magdeburg, Germany) approved written consent were scanned. A PSF dataset to determine susceptibilityinduced distortions varying with the object and corresponding DW-EPIs were sequentially acquired using a 32-channel head array coil (Nova Medical, Wilmington, MA). The imaging protocols were: 80 axial slices, slice thickness ¼ 1.2 mm, TR/TE ¼ 8200/56 ms (for DW-EPIs) and 6100/39 ms (for the PSF scan), bandwidth ¼ 1544 Hz/pixel, (effective) echospacing ¼ (263) 790 ms, FOV ¼ 224 224 mm2, matrix size ¼ 180 180, partial Fourier ¼ 6/8, GRAPPA factor ¼ 3 with 48 reference lines. The Stejskal-Tanner diffusion-encoding scheme was applied and the number of diffusion directions for DWEPI was 12 (with b-value ¼ 1000 s/mm2) and repeated four times for averaging. For the PSF scan, an acceleration factor of 3 (15) was applied to generate a distortionfree reference image for comparison. To demonstrate further acceleration of PSF mapping, only 25 PSF samples among 60 were used for the calculation using a nonequidistant acquisition scheme (with a further acceleration factor of 3, plus additional five centered PSF samples) (16). The TA for EPI and PSF data were approximately 7 and 6 min (2 min 30 s with nonequidistant sampling), respectively. RESULTS Dependence of Eddy-Current-Induced Distortion Strength on the Amplitude of Diffusion Gradients Overall, the magnitude of eddy-current-induced distortions increased linearly with the diffusion gradient strength (Table 1A). Over the entire volume, the
The values were calculated over the entire volume (20 slices) within substantial signal areas* as well as across all twelve diffusion directions**. The histograms (IIa-e) and spatial distributions (IIIa-f) were demonstrated in Fig. 2II and 2III, respectively. a
1.59 (33.54) 0.37 (7.81) 0.30 (6.33) Absolute maximum difference/pixel (/Hz)** Absolute mean difference/pixel (/Hz)** Standard deviation /pixel (/Hz)** B
Time interval
0.10 (2.21) 0.18 (3.87) 0.17 (3.54) 0.26 (5.78) 0.24 (5.15) 0.26 (5.46) 0.28 (5.82) 0.33 (6.92) 0.32 (6.68) 0.32 (6.71) 0.02 (0.44) 0.03 (0.68) 0.03 (0.63) 0.05 (0.98) 0.05 (1.09) 0.05 (1.13) 0.06 (1.16) 0.06 (1.27) 0.06 (1.26) 0.07 (1.43) 0.03 (0.56) 0.04 (0.85) 0.04 (0.78) 0.06 (1.27) 0.06 (1.37) 0.07 (1.43) 0.07 (1.50) 0.07 (1.57) 0.08 (1.60) 0.09 (1.79)
Twomonth
IIIf
Onemonth Onemonth < tenminute
IIIe
< tenminute Onemonth < tenminute
Difference between direct calibration vs. linear superposition
< tenminute Twomonth Onemonth
IIe, IIId IId, IIIc IIc
Difference between direct calibrations
IIb, IIIb
Eddycurrent-induced distortions
IIa, IIIa
0.08 (1.68) 0.01 (0.19) 0.01 (0.15) 0.09 (1.92) 0.01 (0.20) 0.01 (0.16) 0.09 (1.83) 0.01 (0.16) 0.01 (0.14)
z (head to feet) y (anterior to posterior) x (right to left) z (head to feet)
0.12 (2.54) 0.02 (0.38) 0.01 (0.28) 0.12 (2.55) 0.01 (0.22) 0.01 (0.17) 0.10 (2.19) 0.01 (0.26) 0.01 (0.22)
y (anterior to posterior) x (right to left) Gradient axis
Absolute maximum difference/pixel (/Hz)* Absolute mean difference/pixel (/Hz)* Standard deviation /pixel (/Hz)* A
Difference between measured and 2nd order polynomial (quadratic) fitted data
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Difference between measured and 1st order polynomial (linear) fitted data
Table 1 Linearity of Eddy-Current-Induced Distortion Strength Depending on the Applied Amplitude of Diffusion Gradient (A) and Eddy-Current Map Differences between Direct Calibrations and between Direct Calibration and the Linear Superposition (B)a
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differences between measured and 1st order fitted data were very small when a nonzero offset was considered (Fig. 2Ia; Table 1A). Interestingly, the errors were further reduced with a 2nd order polynomial fitting with a constraint of zero offset (Fig. 2Ib; Table 1A). However, this reduction, which can be considered as the nonlinear effects within the measurement range, was extremely small (less than 0.01 pixels as an absolute mean value over the entire volume including all axes) and the fitting worked well even with the 1st order model. Nevertheless, because it should be assumed that this offset is equal to zero, the 2nd order model is well-suited to meet the expectation. From the investigation of the dependence on gradient axes as well as position, therefore, a more suitable interpolation may be developed to estimate an eddy-current map with arbitrary diffusion gradient amplitude at each axis. Temporal Stability of Eddy-Current-Induced Distortions Table 1B and Figure 2II demonstrate that the proposed calibration is very stable over time. Due to the use of parallel imaging factor of 3, relatively small eddy-currentinduced distortions between 1.30 and 1.59 pixels were measured (Fig. 2IIa), and reproduced very well in all four calibrations over 3 months. The differences between direct calibrations formed a very narrow Gaussian distribution without any outstanding systematic bias errors in and/or between calibrations (Figs. 2IIb–IIe) and the variance was increased slightly even with 2-month calibration interval (Fig. 2IIe). In addition, because all differences were distributed evenly over space, the spatial variations were very constant along all x, y, and z directions, as shown in histograms (Figs. 2IIb,2IIc,2IIe) and the corresponding spatial variations (Figs. 2IIIb– IIId). When the spatial variations are calculated as representative values over the entire volume including all diffusion directions, the absolute maximum and mean differences between repeated calibration data within less than a 10-min interval were 0.10 and 0.02 pixels (corresponding to 2.21 and 0.44 Hz). The standard deviation was 0.03 pixels (equal to 0.56 Hz). The calibration differences increased to 0.26, 0.05, and 0.06 pixels, respectively, (corresponding to 5.78, 0.98, and 1.27 Hz) within the 2-month interval (Table 1B). Although these are bigger than the experimental errors within a short time, i.e., less than 10-min interval, they still can be considered as negligible differences. Spatial Variations of Direct Calibration versus the Linear Superposition The linear superposition of eddy-current maps generated very similar maps compared with the direct calibration (Fig. 2IIIe). Compared with the differences between direct calibrations over time (Figs. 2IIIb–IIId), the differences between a direct calibration and the linear superposition (Fig. 2IIIe) were slightly larger and further increased when compared with the linear superposition of eddy-current maps acquired with 2-month interval (Fig. 2IIIf). However, the mean and the standard deviation were still small with 0.07 and 0.09 pixels
FIG. 2. Dependence of eddy-current-induced distortion strength on the amplitude of diffusion gradient (I), temporal stability of direct calibrations over time (II), and spatial variation of direct calibration versus the linear superposition (III). In (I), the measured distortion strength is presented in gradient amplitude on the bottom x-axis (in b-value [s/mm2] on the top) together with corresponding 1st (a) and 2nd order polynomial fittings (b) as black dash line at a representative position of 0, -82, and 0 mm, respectively, along x, y, and z directions in image space from the isocenter positon. The histogram of the measured eddy-current-induced distortions from the calibration data (IIa) and differences between calibration data with less than (
PSF Mapping-Based Correction of Eddy-Current-Induced Distortions in Diffusion-Weighted Echo-Planar Imaging Myung-Ho In,1* Oleg Posnansky,1 and Oliver Speck1,2,3,4 Purpose: To accurately correct diffusion-encoding directiondependent eddy-current-induced geometric distortions in diffusion-weighted echo-planar imaging (DW-EPI) and to minimize the calibration time at 7 Tesla (T). Methods: A point spread function (PSF) mapping based eddycurrent calibration method is newly presented to determine eddy-current-induced geometric distortions even including nonlinear eddy-current effects within the readout acquisition window. To evaluate the temporal stability of eddy-current maps, calibration was performed four times within 3 months. Furthermore, spatial variations of measured eddy-current maps versus their linear superposition were investigated to enable correction in DW-EPIs with arbitrary diffusion directions without direct calibration. For comparison, an image-based eddy-current correction method was additionally applied. Finally, this method was combined with a PSF-based susceptibility-induced distortion correction approach proposed previously to correct both susceptibility and eddy-current-induced distortions in DW-EPIs. Results: Very fast eddy-current calibration in a threedimensional volume is possible with the proposed method. The measured eddy-current maps are very stable over time and very similar maps can be obtained by linear superposition of principal-axes eddy-current maps. High resolution in vivo brain results demonstrate that the proposed method allows more efficient eddy-current correction than the image-based method. Conclusion: The combination of both PSF-based approaches allows distortion-free images, which permit reliable analysis in diffusion tensor imaging applications at 7T. Magn Reson C 2015 Wiley Periodicals, Inc. Med 75:2055–2063, 2016. V Key words: PSF; geometric distortion; eddy-current; EPI; DTI
INTRODUCTION Single-shot spin-echo echo-planar imaging (EPI) is generally used as a method of choice for diffusion tensor 1 Department of Biomedical Magnetic Resonance, Institute for Experimental Physics, Otto-von-Guericke University Magdeburg, Germany. 2 German Centre for Neurodegenerative Diseases (DZNE), Site Magdeburg, Germany. 3 Leibniz Institute for Neurobiology, Magdeburg, Germany. 4 Center for Behavioral Brain Sciences, Magdeburg, Germany. Grant sponsor: German Research Foundation (DFG); Grant number: SP632-4. *Correspondence to: Myung-Ho In, Ph.D., Department of Biomedical Magnetic Resonance, Institute for Experimental Physics, Otto-von-Guericke University Magdeburg, Leipziger Straße 44, 39120 Magdeburg, Germany. E-mail: [email protected]
Received 3 December 2014; revised 27 March 2015; accepted 31 March 2015 DOI 10.1002/mrm.25746 Published online 22 June 2015 in Wiley Online Library (wileyonlinelibrary. com). C 2015 Wiley Periodicals, Inc. V
imaging (DTI) (1) studies. EPI at ultrahigh field (UHF) enables fast brain imaging with high spatial resolution due to the increased signal to noise ratio (SNR). However, increasing spatial resolution or use of stronger diffusion-gradients leads to longer echo times (TE) and consequently undesirable T2 signal loss (2), which can be more harmful at UHF due to relatively shorter T2. To compensate or correct for eddy-current-induced geometric distortions, the benefit of the Stejskal-Tanner diffusion scheme allowing the use of a minimized TE reinitiates the need for alternative techniques of the twice-refocused spin-echo method (3,4). To take eddy-current-induced distortions into account, several correction approaches involving image-based correction (5–7) and calibration methods (8–10) have been proposed previously. However, they often require time consuming post-processing procedures (5–7). Although calibration methods (8–10) can measure eddy-currentinduced distortions, only simplified distortion models, including shearing, scaling, and translation, were determined from the calibration data and used for correction. All these methods may be insufficient to rectify nonlinear eddy-current effects within the long readout acquisition window of EPI (11,12). Although a recent approach (12) considered even the nonlinear effects, it still can be a time consuming procedure, especially in highresolution diffusion-weighted (DW) imaging with many diffusion encoding directions. Point spread function (PSF) methods (13–16) have been adapted for susceptibility-induced distortion correction in EPI because they potentially allow very reliable distortion mapping. Due to an additional PSF (or spin-warp) phase-encoding (PE) added to the original EPI sequence, the reconstructed PSFs are represented along the correlation (or diagonal) line in the PSF- (or nondistorted) and EPI-PE (or distorted) coordinates after three dimensional (3D) inverse Fourier transformation (13–16). Mapping of the PSFs’ deviations from the diagonal line in both nondistorted and distorted coordinates results in shift (or distortion) maps in both coordinates. Several approaches (13–16) have been proposed to extract the distortion information from the measured PSF data more accurately leading to a higher correction fidelity. In this work, a fast PSF-based eddy-current calibration method is newly proposed to determine eddy-currentinduced distortions including nonlinear eddy-current effects. With the calibration data, additional investigations were carried out to demonstrate that (I-i) an eddycurrent map with arbitrary diffusion gradient amplitude at each axis can be estimated using interpolation chosen from the dependence of eddy-current-induced distortion strength on the diffusion gradient amplitude, (I-ii)
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temporal stability of eddy-current-induced distortion maps is sufficiently high such that phantom reference acquisitions need not be performed too often, and (I-iii) the linear superposition of measured or estimated eddycurrent maps from the principal gradient axes provides similar performance compared with dedicated acquisitions. (II) For comparison of correction accuracy, an image intensity-based eddy-current correction was additionally applied. (III) Together with a PSF-based susceptibility-induced distortion correction approach suggested previously (16), a PSF-based comprehensive distortion correction was performed to accurately diminish both susceptibility- and eddy-current-induced distortions in DW-EPIs. METHODS
In PSF mapping, the PSF image I(y,s) is obtained by multiplying the proton density q(s) with the PSF H(y,s). A single delta function approximation of the PSF leads to (15): [1]
where d is the Dirac delta-function, and s and y correspond to the nondistorted (or PSF) and distorted (or EPI) PE coordinates, respectively. The distortion term D(s) represents deviations of the PSFs from the diagonal line along the y-direction in the s-axis (15) and yields geometric distortion in the acquired EPI image. If PSF mapping is performed with DW gradients, the obtained shift map D(s) ¼ D(s)diff becomes a combination of susceptibility- D(s)suscep and eddy-current- D(s)EC induced shift maps. Because all distortions cause PSFs’ deviations in the PSF mapping, this map includes even geometric distortions induced by nonlinear eddy-current effects within the readout acquisition window. Therefore, subtraction between the two shift maps calculated from PSF data with and without DW gradient (i.e. D(s)diff and D(s)suscep) yields the eddy-current-induced shift map D(s)EC including nonlinear eddy-current effects in the nondistorted coordinate (s). DðsÞEC ¼ DðsÞdiff DðsÞsuscep :
Temporal Stability of Eddy-Current-Induced Distortions To verify temporal stability of the eddy-current maps, four calibration data sets with identical protocols, acquired at different times over 3 months, were compared. The differences between calibration data with less than 10-min, 1-month, and 2-month calibration intervals were calculated as histograms. Spatial Variations of Direct Calibration versus Linear Superposition of Principal Axes
PSF-Based Eddy-Current Calibration
Iðy; sÞ ¼ rðsÞHðy; sÞ rðsÞdðs þ DðsÞ yÞ
data. In contrast to the linear fitting, a constraint of zero offset was applied to the 2nd order polynomial fitting, because the eddy-currents are assumed to be zero without any diffusion gradient. The differences between the measured and the fitted data were calculated as standard deviation, absolute maximum, and absolute mean values.
[2]
This map can be applicable for correction when the spatial distribution of eddy-current effects is constant across measurements and reproducible, even over long time periods. This assumption was tested as described in the following section. Dependence of Eddy-Current-Induced Distortion Strength on the Amplitude of Diffusion Gradients The linearity of eddy-current-induced distortions as a function of the diffusion gradient amplitude was investigated. For each gradient axis, sixteen eddy-current maps were measured with different b-values from 50 to 800 s/ mm2 and steps of 50 s/mm2 resulting in different gradient amplitudes (up to 92% of the nominal maximum amplitude) with identical duration. To test linearity, 1st and 2nd order polynomial fittings were applied to the
We tested spatial variations of direct calibration of all diffusion encoding directions versus the linear superposition of the three principal gradient directions to investigate whether linear superposition of the eddy-current maps for the x, y, and z diffusion-encoding directions allows calculation of eddy-current maps for arbitrary diffusion-encoding directions without prolonging calibration time or if diffusion-encoding directions are changed. Based on six eddy-current maps (two maps with two different amplitudes of the diffusion gradient for each axis, 27.18 and 65.62 mT/m, to generate the 12 diffusionencoding directions), eddy-current maps for the 12 diffusion-encoding directions were generated by linear superposition and compared with the measured eddycurrent maps. To demonstrate spatial variations over the entire volume including all diffusion directions, in addition to representative values (such as standard deviation, absolute maximum, and absolute mean), standard deviation and absolute mean were calculated along the x, y, and z directions in 3D image space. Calibration data, the differences between calibration data, and differences between calibration data and their linear superposition were used for these calculations. Comparison with FSL-Based Eddy-Current Correction For comparison, the image intensity-based eddy-current correction implemented in FSL (http://fsl.fmrib.ox.ac.uk/ fsl/) was also applied. The eddy-current correction was performed either with the proposed method or based on FSL using the option of “eddy-correct” with splineinterpolation. To verify the improvement quantitatively, standard deviation of DW-EPIs and fractional anisotropy (FA) values in cerebrospinal fluid (CSF) regions of the entire volume were computed as histograms before and after eddy-current correction. PSF-Based Comprehensive Distortion Correction The steps for the eddy-current calibration and comprehensive distortion correction are illustrated in Figure 1. A final eddy-current map was obtained after extrapolation of the shift map D(s)EC to the areas outside of the substantial signal region and 3 3 median filtering. Although the eddy-currents were calibrated in the
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FIG. 1. A schematic description of the proposed calibration and PSF-based comprehensive distortion correction. Note that the calibration is performed as a separate session using a phantom.
nondistorted coordinates (s), it is sufficient to use them without transformation from nondistorted to distorted coordinates because eddy-currents generally vary slowly in space and the difference is negligible (12). As described by Zaitsev et al (15), the eddy-current-induced distortion correction in DW-EPIs was performed with the final eddy-current maps obtained in the nondistorted coordinate (s). Individual susceptibility-induced distortion correction followed after eddy-current correction. An improved PSF-based susceptibility-induced distortion correction approach (16) was adopted in this study. In contrast to the calibration, mapping of susceptibility-induced distortions was carried out in the distorted coordinates (y) because they can strongly vary locally. Because a measured PSF profile in the EPI-PE coordinates (EPI-PSF) was derived and directly used as a convolution kernel for distortion correction in DW-EPI (16), the correction fidelity was improved, especially in regions with strong susceptibility-induced distortions. Three FA maps were calculated using FSL from DWEPIs without correction, with eddy-current correction and with comprehensive correction.
Experiments The proposed PSF-based eddy-current calibration was performed on a 7 Tesla (T) scanner (Siemens Healthcare, Erlangen, Germany) equipped with a gradient coil type “SC72AB” (70 mT/m and 200 T/m/s). A spherical silicon-oil phantom (diameter 16 cm) and an eightchannel head array coil (Rapid Biomedical, W€ urzburg, Germany) were used. The PSF scan was performed with an acceleration factor of 18 (corresponding to 10 PSFencoded samples for each direction) (15). The imaging protocols were identical to the in vivo DW-EPI experiment (described below) with these modifications: repetition time (TR) ¼ 2000 ms, 20 axial slices, and slice thickness ¼ 4.8 mm. TR and slice number were reduced to minimize the calibration time and the slice thickness was increased to keep the same field of view (FOV)
along the slice direction, as in the in vivo experiments as well as to increase SNR. Because the eddy-currentinduced distortions vary smoothly along the slice direction, linear interpolation was applied to generate eddycurrent maps covering the entire volume of the in vivo experiments. The acquisition time (TA) for calibration with and without diffusion gradients was 4 min 33 s. For DTI experiments, two healthy subjects after institutional review board (Otto-von-Guericke University Magdeburg, Germany) approved written consent were scanned. A PSF dataset to determine susceptibilityinduced distortions varying with the object and corresponding DW-EPIs were sequentially acquired using a 32-channel head array coil (Nova Medical, Wilmington, MA). The imaging protocols were: 80 axial slices, slice thickness ¼ 1.2 mm, TR/TE ¼ 8200/56 ms (for DW-EPIs) and 6100/39 ms (for the PSF scan), bandwidth ¼ 1544 Hz/pixel, (effective) echospacing ¼ (263) 790 ms, FOV ¼ 224 224 mm2, matrix size ¼ 180 180, partial Fourier ¼ 6/8, GRAPPA factor ¼ 3 with 48 reference lines. The Stejskal-Tanner diffusion-encoding scheme was applied and the number of diffusion directions for DWEPI was 12 (with b-value ¼ 1000 s/mm2) and repeated four times for averaging. For the PSF scan, an acceleration factor of 3 (15) was applied to generate a distortionfree reference image for comparison. To demonstrate further acceleration of PSF mapping, only 25 PSF samples among 60 were used for the calculation using a nonequidistant acquisition scheme (with a further acceleration factor of 3, plus additional five centered PSF samples) (16). The TA for EPI and PSF data were approximately 7 and 6 min (2 min 30 s with nonequidistant sampling), respectively. RESULTS Dependence of Eddy-Current-Induced Distortion Strength on the Amplitude of Diffusion Gradients Overall, the magnitude of eddy-current-induced distortions increased linearly with the diffusion gradient strength (Table 1A). Over the entire volume, the
The values were calculated over the entire volume (20 slices) within substantial signal areas* as well as across all twelve diffusion directions**. The histograms (IIa-e) and spatial distributions (IIIa-f) were demonstrated in Fig. 2II and 2III, respectively. a
1.59 (33.54) 0.37 (7.81) 0.30 (6.33) Absolute maximum difference/pixel (/Hz)** Absolute mean difference/pixel (/Hz)** Standard deviation /pixel (/Hz)** B
Time interval
0.10 (2.21) 0.18 (3.87) 0.17 (3.54) 0.26 (5.78) 0.24 (5.15) 0.26 (5.46) 0.28 (5.82) 0.33 (6.92) 0.32 (6.68) 0.32 (6.71) 0.02 (0.44) 0.03 (0.68) 0.03 (0.63) 0.05 (0.98) 0.05 (1.09) 0.05 (1.13) 0.06 (1.16) 0.06 (1.27) 0.06 (1.26) 0.07 (1.43) 0.03 (0.56) 0.04 (0.85) 0.04 (0.78) 0.06 (1.27) 0.06 (1.37) 0.07 (1.43) 0.07 (1.50) 0.07 (1.57) 0.08 (1.60) 0.09 (1.79)
Twomonth
IIIf
Onemonth Onemonth < tenminute
IIIe
< tenminute Onemonth < tenminute
Difference between direct calibration vs. linear superposition
< tenminute Twomonth Onemonth
IIe, IIId IId, IIIc IIc
Difference between direct calibrations
IIb, IIIb
Eddycurrent-induced distortions
IIa, IIIa
0.08 (1.68) 0.01 (0.19) 0.01 (0.15) 0.09 (1.92) 0.01 (0.20) 0.01 (0.16) 0.09 (1.83) 0.01 (0.16) 0.01 (0.14)
z (head to feet) y (anterior to posterior) x (right to left) z (head to feet)
0.12 (2.54) 0.02 (0.38) 0.01 (0.28) 0.12 (2.55) 0.01 (0.22) 0.01 (0.17) 0.10 (2.19) 0.01 (0.26) 0.01 (0.22)
y (anterior to posterior) x (right to left) Gradient axis
Absolute maximum difference/pixel (/Hz)* Absolute mean difference/pixel (/Hz)* Standard deviation /pixel (/Hz)* A
Difference between measured and 2nd order polynomial (quadratic) fitted data
In et al.
Difference between measured and 1st order polynomial (linear) fitted data
Table 1 Linearity of Eddy-Current-Induced Distortion Strength Depending on the Applied Amplitude of Diffusion Gradient (A) and Eddy-Current Map Differences between Direct Calibrations and between Direct Calibration and the Linear Superposition (B)a
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differences between measured and 1st order fitted data were very small when a nonzero offset was considered (Fig. 2Ia; Table 1A). Interestingly, the errors were further reduced with a 2nd order polynomial fitting with a constraint of zero offset (Fig. 2Ib; Table 1A). However, this reduction, which can be considered as the nonlinear effects within the measurement range, was extremely small (less than 0.01 pixels as an absolute mean value over the entire volume including all axes) and the fitting worked well even with the 1st order model. Nevertheless, because it should be assumed that this offset is equal to zero, the 2nd order model is well-suited to meet the expectation. From the investigation of the dependence on gradient axes as well as position, therefore, a more suitable interpolation may be developed to estimate an eddy-current map with arbitrary diffusion gradient amplitude at each axis. Temporal Stability of Eddy-Current-Induced Distortions Table 1B and Figure 2II demonstrate that the proposed calibration is very stable over time. Due to the use of parallel imaging factor of 3, relatively small eddy-currentinduced distortions between 1.30 and 1.59 pixels were measured (Fig. 2IIa), and reproduced very well in all four calibrations over 3 months. The differences between direct calibrations formed a very narrow Gaussian distribution without any outstanding systematic bias errors in and/or between calibrations (Figs. 2IIb–IIe) and the variance was increased slightly even with 2-month calibration interval (Fig. 2IIe). In addition, because all differences were distributed evenly over space, the spatial variations were very constant along all x, y, and z directions, as shown in histograms (Figs. 2IIb,2IIc,2IIe) and the corresponding spatial variations (Figs. 2IIIb– IIId). When the spatial variations are calculated as representative values over the entire volume including all diffusion directions, the absolute maximum and mean differences between repeated calibration data within less than a 10-min interval were 0.10 and 0.02 pixels (corresponding to 2.21 and 0.44 Hz). The standard deviation was 0.03 pixels (equal to 0.56 Hz). The calibration differences increased to 0.26, 0.05, and 0.06 pixels, respectively, (corresponding to 5.78, 0.98, and 1.27 Hz) within the 2-month interval (Table 1B). Although these are bigger than the experimental errors within a short time, i.e., less than 10-min interval, they still can be considered as negligible differences. Spatial Variations of Direct Calibration versus the Linear Superposition The linear superposition of eddy-current maps generated very similar maps compared with the direct calibration (Fig. 2IIIe). Compared with the differences between direct calibrations over time (Figs. 2IIIb–IIId), the differences between a direct calibration and the linear superposition (Fig. 2IIIe) were slightly larger and further increased when compared with the linear superposition of eddy-current maps acquired with 2-month interval (Fig. 2IIIf). However, the mean and the standard deviation were still small with 0.07 and 0.09 pixels
FIG. 2. Dependence of eddy-current-induced distortion strength on the amplitude of diffusion gradient (I), temporal stability of direct calibrations over time (II), and spatial variation of direct calibration versus the linear superposition (III). In (I), the measured distortion strength is presented in gradient amplitude on the bottom x-axis (in b-value [s/mm2] on the top) together with corresponding 1st (a) and 2nd order polynomial fittings (b) as black dash line at a representative position of 0, -82, and 0 mm, respectively, along x, y, and z directions in image space from the isocenter positon. The histogram of the measured eddy-current-induced distortions from the calibration data (IIa) and differences between calibration data with less than (
