### Abstract

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

Echo-planar imaging (EPI) is an ultrafast magnetic resonance (MR) imaging technique prone to geometric distortions. Various correction techniques have been developed to remedy these distortions. Here improvements of the point spread function (PSF) mapping approach are presented, which enable reliable and fully automated distortion correction of echo-planar images at high field strengths. The novel method is fully compatible with EPI acquisitions using parallel imaging. The applicability of parallel imaging to further accelerate PSF acquisition is shown. The possibility of collecting PSF data sets with total acceleration factors higher than the number of coil elements is demonstrated. Additionally, a new approach to visualize and interpret distortions in the context of various imaging and reconstruction methods based on the PSF is proposed. The reliable performance of the PSF mapping technique is demonstrated on phantom and volunteer scans at field strengths of up to 4 T. Magn Reson Med 52:1156–1166, 2004. © 2004 Wiley-Liss, Inc.

Single-shot echo-planar imaging (EPI) allows for the acquisition of a 2D image following a single radiofrequency excitation (1). The extraordinarily high temporal resolution of EPI makes it the method of choice for a variety of magnetic resonance imaging (MRI) applications, such as functional MRI (fMRI), perfusion MRI, or diffusion tensor imaging. However, the prolonged readout time of EPI results in an extremely low readout bandwidth for the phase-encoding direction. The above makes EPI susceptible to various artifacts in the presence of magnetic field inhomogeneities in the sample, e.g., close to tissue/bone or tissue/air interfaces for in vivo imaging. These artifacts typically manifest themselves as geometric and intensity image distortions. Along with time-invariant off-resonance effects, such as *B*_{0} field inhomogeneities, dynamic field variations induced by eddy currents may also contribute significantly to the observed image distortions, e.g., in diffusion-weighted imaging (2). Concomitant field effects arising from fast gradient switching, so-called “Maxwell terms” (3, 4), may also produce severe image distortions. In the most general case, the frequency offsets causing image distortions in EPI vary in space and in time.

The problem of geometric distortions in EPI was recognized soon after its conception and since then numerous correction techniques have been proposed. However, despite the variety of correction methods available in the literature none has gained dominance in routine applications. This motivates the continued search for a fast, robust, and noninteractive correction technique. Among the currently available methods, the most popular is the field map approach (5, 6). For this method the variations of the magnetic field are measured and then used to calculate the local pixel shifts in the image. However, often the need to perform phase unwrapping in image space may cause significant difficulties in the calculation of field maps, especially close to the object boundaries, in regions of high field inhomogeneity, or for disconnected object regions. Additionally, unless special care is taken, the method ignores the time dependency of the magnetic field variations and replaces the true evolution with a value estimated at the echo time. Nevertheless, the technique has proven effective at reducing echo-planar image distortions despite the aforementioned difficulties (5, 6).

Multireference techniques attempt to measure the phase error accumulation in *k*-space directly. For example, in the method proposed by Xin et al. (7), *i*-1 phase-encoding blips are played out prior to the readout gradient during the *i*th reference scan so that all the data acquired following the *i*th excitation are equally phase encoded. Thereafter, a filter is computed based on the acquired reference data and the EPI data are corrected directly in *k*-space. The approach appears to be effective as long as the variation of the magnetic field is smooth and slow, which is often the case. Another multireference technique, proposed by Chen and Wyrwicz (8), uses a multiecho gradient echo (GE) imaging sequence with a number of echoes equal to the EPI echo train length. The phase modulation function calculated from the reference data is applied then to correct echo-planar images. However, the correction operation is extremely calculation-intensive and involves 2*N*_{y} Fourier transformations, where *N*_{y} is the number of phase encoding lines. With regard to the acquisition times, the original method of Chen and Wyrwicz requires *N*_{y} reference scans and the optimized version (9), which enables the mapping of intrinsic eddy current components more accurately, requires 2*N*_{y} scans.

Typically, multireference approaches provide better correction quality and/or are more stable compared to single-reference methods. However, a common shortcoming of most multireference techniques is the prolonged acquisition time. *N*_{y} scans are required for a matrix size of *N*_{x} × *N*_{y} for both approaches described above. This results in acquisition times as long as 3–10 min for typical applications, which appears to be exceedingly large if a separate reference scan is required for each acquisition. Nonetheless, in a number of applications the time overhead of several minutes is perfectly acceptable, e.g., in fMRI where the reference scan is only to be acquired once for the entire examination.

The technique of point spread function (PSF) mapping by constant time imaging was introduced by Robson et al. (10) and recently adopted to the distortion correction problem by Zeng and Constable (11). This multireference method enables mapping of distortions in regions of low and high field inhomogeneity. PSF mapping is realized by the application of an additional gradient encoding to the EPI readout. A schematic sequence timing diagram is depicted in Fig. 1. This additional encoding is performed by repetition of the EPI readout multiple times with varying areas of the phase-encoding prewinder. The point spread function for each image pixel is recovered by a Fourier transformation of the acquired data. Since the method is based on the original EPI readout timing, the inherent eddy currents and concomitant gradients cause identical distortions of the PSF data as they do for EPI and, thus, are also mapped faithfully. Although the reconstruction of the distortion maps in the PSF method may appear less straightforward compared to the field mapping technique, it is computationally cheap and intrinsically stable. This paper presents further developments and optimization of the PSF mapping technique. Special emphasis is given to implementation details in order to encourage the readers to realize the technique on their imaging hardware.

The fully encoded PSF acquisition requires *N*_{y} reference scans. However, the full data set used for the PSF correction is sparsely populated, which makes it possible to apply reduced encoding schemes similar to the reduced FOV technique (12). Thereby, the acquisition time for the reference data can be notably shortened without any loss of precision in the reference data. The sparse character of the PSF reference data set makes it especially suitable for the implementation with parallel imaging techniques with high reduction factors.

In this paper we demonstrate various possibilities for combining the PSF data acquisition with parallel imaging techniques. Indeed, the echo-planar method for which distortions are to be mapped may employ benefits of parallel imaging techniques in order to shorten the echo time and/or readout time. In order to map distortions faithfully in this case, the PSF mapping technique should use exactly the same readout as the EPI scan. This requires a parallel imaging reconstruction technique to be applied to the PSF data during reconstruction. On the other hand, if imaging is performed with a coil array, one may wish to shorten the PSF acquisition and apply a parallel imaging reconstruction procedure to the PSF-encoding dimension. Finally, the parallel imaging reconstruction may be applied to both phase-encoding and PSF-encoding dimensions at the same time. The combination of the parallel imaging acceleration with the reduced FOV technique for the PSF is also possible and is demonstrated below.

In addition, this paper introduces and makes intensive use of a new representation of the PSF space, which allows new insights into the properties of the imaging techniques and reconstruction methods under various conditions.

### THEORY

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

For the sake of completeness and in order to endow the reader with a better understanding of the data acquired by the sequence and the required processing steps we review the theory of the method.

In the presence of off-resonance conditions the resonant frequency can be written as

- (1)

where ω_{0} is the system frequency and *f*(**r**,*t*) is the off-resonance term that varies with spatial location **r** and time *t*. The extra phase accumulated during signal acquisition at time *t* can be expressed as

- (2)

For an EPI *k*-space trajectory the time, *t*, of the acquisition of the respective *k*-space line can be approximated as

- (3)

where *T*_{L} is the duration required to encode and acquire one line in *k*-space (often referred to as “echo spacing”), **k**_{1} is the *k*-space vector corresponding to the EPI encoding with **k** as the initial vector for the EPI encoding scheme, **p** is the vector pointing in phase-encoding direction with a length equal to the inverse of the *k*-space line spacing; vector product (**k**_{1} − **k**) · **p** equals essentially the *k*-space line number. Using the above notation, it is possible to write the standard EPI signal equation in the presence of off-resonance effects:

- (4)

The integral in the exponent expresses the undesired space-varying phase accumulation, which is causing echo-planar image distortions. It is to be noted that due to the approximation given by Eq. [3] phase effects associated with read dimension are neglected.

The signal delivered by the PSF mapping sequence is conveniently described in terms of two wave vectors:

- (5)

where **k**_{1} corresponds to the EPI trajectory, while **k**_{2} describes distortion-free constant-time encoding. For the case of EPI the readout direction may be considered distortion free and the constant-time encoding for the readout direction may be avoided. The Fourier transform of the above equation with respect to both **k**_{1} and **k**_{2} yields the image intensity

- (6)

The reconstructed image is a result of multiplication of the proton density in undistorted coordinates, ρ(**r**_{2}), with a term, which will be referred to as space-varying point spread function, *H*(**r**_{1},**r**_{2}):

- (7)

for the case of stationary off-resonance effects, when *f*(**r**,τ) ≈ *f*_{0}(**r**) the above equation transforms to

- (8)

which is the more familiar form of the point spread function in the presence of off-resonance conditions. The function Δ(**r**_{2}) describes the off-diagonal shift of the PSF peak in (**r**_{1},**r**_{2}) space (see, for example, Fig. 3d). Equation [8] describes the simplified appearance of the PSF in the absence of eddy currents and signal decay. Under realistic conditions the above-mentioned effects cause the measured point spread function to deviate from the δ-function representation and result mostly in spreading of the PSF peaks and phase errors. However, for in vivo applications the PSF broadening caused by the local susceptibility gradients rarely exceeds several pixels. Thus, the PSF peak remains reasonably compact, which makes the δ-function approximation valid. In this case the echo-planar image intensity *I*(**r**_{1}) may be expressed as a convolution product of the true proton density with a shifted δ-function,

- (9)

where Δ(**r**_{2}) denotes the pixel shifts in the undistorted coordinates. With knowledge of the pixel shift map, Δ(**r**_{2}), the corrected echo-planar image may be recovered from the distorted image, *I*(**r**_{1}), by Eq. [9] and an appropriate deconvolution procedure.

It is important to note that integration of the PSF image intensity, *I*(**r**_{1},**r**_{2}), along the distorted dimension, **r**_{1}, yields an undistorted image:

- (10)

The undistorted image is dependent on **r**_{2}, which is the conjugate variable of **k**_{2}. Thus, the resolution of this image is defined by the PSF phase encoding. This resolution should not be lower than that of the corresponding echo-planar image in order to allow for accurate corrections. The resolution in the PSF encoding direction may, in principle, be increased at will. This, however, would not increase the accuracy of the distortion estimation, which is primarily defined by the resolution of the echo-planar image. The undistorted image has an identical echo time and, thus, image contrast as the corresponding echo-planar image and may be used to verify the correction quality or serve as a reference for amplitude correction.

As seen from Eqs. [6–8], the measured PSF data, *I*(**r**_{1},**r**_{2}), differ from zero only in a narrow range of its parameter space, namely when **r**_{1} ≈ **r**_{2} + Δ(**r**_{2}). In the case of moderate distortions the displacement term, Δ(**r**_{2}), is small compared to the image field of view. The few pixels with nonzero intensity are aligned close to the diagonal given by **r**_{1} = **r**_{2}. Here it is of advantage to employ reduced sampling methods, similar to the reduced field-of-view (rFOV) technique (12) widely used in dynamic imaging. The applicability of the rFOV approach to the PSF encoding is demonstrated best upon introduction of a new free parameter **r** = **r**_{2} − **r**_{1}. Then the approximated signal can be written in the following way:

- (11)

In the case of more pronounced distortions, especially when rFOV encoding is applied, the displacement term, Δ(**r**_{2}), may become comparable or even exceed the field of view for the PSF dimension. For the mentioned situation an additional unfolding using parallel imaging methods might be applied. For the parallel imaging methods to be applicable it is required that imaging is performed with a coil array. In this paper we consider self-calibrating parallel reconstruction methods, like the SENSE-based method by McKenzie et al. (13) or GRAPPA (14), which would require several fully encoded reference lines to be collected close to the center of *k*-space. The GRAPPA technique, used throughout this paper, utilizes these additional reference lines to compute *k*-space weighting coefficients. Thereafter, the missing lines for each coil data set are reconstructed in a blockwise way according to the following formula (adopted from (14)):

- (12)

where *A* is the acceleration factor, *j* is the current coil number, *L* is the total number of coils, *m* is the index of the missing *k*-space line in the current block, *N*_{b} is the total number of blocks used in the reconstruction, where one block is defined as a single acquired line and A − 1 missing lines. The term *n*(*j*,*b*,*l*,*m*) represents the weights used for the linear combination, where the index *l* counts through the individual coils and the index *b* counts through the individual reconstruction blocks. Following the combination of the *k*-space data *L* single-coil images may be reconstructed by applying the inverse Fourier transform. These images may then be combined into a resulting image using any arbitrary coil combination algorithm.

### METHODS

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

The schematic timing diagram of the PSF mapping sequence is depicted in Fig. 1. The sequence is based on the usual blipped EPI readout with a modified phase-encoding prewinder gradient. The sequence is repeated several times with a stepwise incremented phase-encoding gradient to fill the additional *k*-space dimension, referred to here as PSF dimension. The phase-encoding dimension formed by the EPI readout itself is termed EPI dimension. For each slice the PSF mapping sequence delivers a 3D *k*-space data set with a single frequency-encoding dimension and two independent phase-encoding dimensions.

The data processing steps required for reconstruction of the distortion parameters are visualized as flowchart diagrams in Fig. 2. Fig. 2a presents the initial part of the reconstruction, which transforms the *k*-space PSF data into image space. Additionally, the distorted echo-planar images may be reconstructed from the PSF data with *ky*_{PSF} = 0. For the case when no coil arrays are used for imaging, this initial part of the reconstruction reduces to a trivial 3D Fourier transform. If, however, multiple receiver coils are employed in the experiment, it is of importance to ensure that the channel combination algorithm used to combine the PSF data is able to preserve the phase information (15). In order to use the benefits of accelerated parallel acquisitions along either of the two phase encoding dimensions it is necessary to place the corresponding reconstruction procedures for the missing data at the respective positions in the reconstruction flowchart. These are marked “Grappa EPI” and “Grappa PSF” on the flowchart, respectively.

In this paper we demonstrate the possibility of exploiting the GRAPPA (14) parallel imaging reconstruction technique. GRAPPA was chosen mainly because of its stability and performance. Although it is not demonstrated here, it is in principle possible to employ other parallel imaging methods as long as they satisfy two conditions: (i) the phase information is to be preserved and (ii) the method must allow the restoration of separate coil images. The second requirement may, however, be violated if the accelerated parallel acquisition is applied only along one of the two phase-encoding dimensions.

Figure 2b presents the flowchart for the reconstruction of the pixel shift maps from the 3D PSF data. Initially two distortion-free images are derived from the 3D PSF data. Two different operations are employed to “collapse” the distorted phase-encoding dimension. Integrating the PSF data along the EPI dimension produces the gradient-echo image, as given by Eq. [10], with the image contrast similar to that of the corresponding echo-planar image. Calculation of the SD along the EPI dimension produces another distortion-free image with some useful properties: the signal-to-noise ratio (SNR) in this image appears to be significantly higher and the image contrast appears to be significantly reduced. The above makes this image, termed hereafter StdDev image, perfectly suited for the segmentation of areas with sufficient signal. In our implementation segmentation masks are derived from StdDev images by simple thresholding, followed by the application of a basic cluster filter in order to remove single pixel elements.

The pixel shift reconstruction branch contains the following steps: first the data are phase-corrected by multiplying each PSF “voxel” by the complex conjugate of the phase of the corresponding GE image pixel. This operation is required to avoid the appearance of step-like patterns in the pixel shift maps and will be examined in more detail in the Results. Following phase correction the segmentation mask is applied so that the subsequent peak fitting is only performed for areas of sufficient signal. Peak fitting is the subsequent step. It only considers the real part of the complex PSF data and includes a search for the maximum and interpolation of the peak position. Two interpolation approaches were tested in our implementation: three-point Lorenz and three-point parabolic interpolation, with the latter producing smoother shift maps. Peak fitting is followed by extrapolation of the pixel shift maps to the areas outside of the segmentation mask. This procedure is extremely important for the final image appearance. First, it compensates for inaccuracies in the segmentation mask, correcting for small holes and incorrectly excluded pixels at the boundaries. Second, it improves the appearance of the outer image areas. For example, in head imaging the barely visible signal from the skin is typically excluded from the segmentation mask, but would appear in the corrected image closer to the appropriate place if an adequate extrapolation procedure had been used. Unfortunately, it is not possible to calculate the field distribution to the outside areas accurately, because the spatial distribution of the magnetic susceptibility is generally unknown. Given the circumstances the following simple extrapolation approach was chosen. First, for each image pixel outside of the segmentation mask several closest pixels lying on the boundary of the mask are located. Thereafter the shift at each point of no support is calculated as a weighted sum of shifts in the few boundary pixels selected in the first step. The weights were chosen to be inversely proportional to the square of the distance between the considered pixel and the respective boundary pixel. The extrapolation is followed by the application of a 2D five-pixel median filter in order to remove single pixel “spikes” from the pixel shift maps. Shift maps, thus produced, are smooth, noise free, and extended to areas of no support.

The distortion correction is applied in image space using 1D *b*-spline interpolations of third degree. In our implementation an efficient filter design proposed by Unser et al. (16, 17) was employed, leading to minimal computational overhead for interpolated images of extremely high quality. It was of importance to select an adequate interpolation method because the proposed technique performs corrections in image space. Linear interpolation, widely used for its simplicity and speed, does not appear to fulfil the requirements imposed to the image quality because of the severe blurring it might cause. Sinc interpolation preserves sharp edges in the images; however, it might produce ringing. The other factor that limits applicability of sinc interpolation is the enormously high computational overhead of the method. Maximal-order interpolation methods of minimal support appear to fulfill both requirements of interpolation accuracy and acceptable computation times (18, 19). In the mentioned class of interpolation methods *b*-spline of third degree appears to be most advantageous. In our experience, the images unwrapped using *b*-spline interpolation did not suffer from any noticeable blurring when compared to those produced by sinc interpolation and the computational efforts have only increased by a factor of 2 compared those of the linear interpolation.

The PSF mapping technique was implemented and tested on three different whole-body systems: Siemens Magnetom Sonata 1.5 T, Siemens Magnetom Trio 3 T (Siemens Medical Systems GmbH, Erlangen, Germany), and Bruker MedSpec 4 T (Bruker BioSpin MRI GmbH, Ettlingen, Germany). All imagers were equipped with similar gradient systems, capable of 40 mT/m gradient strength per axis with a minimal rise time of 200 μsec. All imagers were capable of multichannel reception. Eight-channel receive-only coils for head imaging (MRI Device Corp., Waukesha, WI) were available for both Siemens systems.

Phantom measurements were performed in a spherical phantom filled with water doped with Ni(NO_{3})_{2}. In vivo imaging experiments were performed in healthy volunteers. All experiments with human subjects were performed in accordance with local IRB regulations and informed consents were obtained prior to measurements. Typical imaging was 128^{2} 2D image matrix, readout bandwidth 1.4 kHz/pixel, and echo spacing of 800 μsec.

The measurement sequence and the reconstruction procedures were implemented on the scanner and fully integrated into the scanner software platform. Both the reconstruction of the distortion maps and the correction of the echo-planar images acquired later were performed online on the scanner in a fully automated operator-independent manner.

### RESULTS

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

Figure 3 Fig. 3 displays images through the brain of a healthy volunteer acquired on the 3-T system with the PSF mapping sequence. Imaging parameters were FOV = 224 mm, TE = 32 msec, TR = 3 sec, A–P phase encoding direction, 128^{3} matrix with 6/8 partial Fourier acquisition for EPI and the full PSF encoding (128 steps). Figure 3a contains the distorted echo-planar image, Fig. 3b shows the corresponding distortion-free GE image, and Fig. 3c presents the measured point spread function in *X*–*Y*_{EPI} coordinates (*X* is horizontal and *Y*_{EPI} is vertical) with *Y*_{GE} fixed to 80. In the absence of distortions the bright line would appear as horizontal line centered at *Y*_{EPI} = 80. Figure 3d presents the same point spread function data in *Y*_{GE}–*Y*_{EPI} coordinates (*Y*_{GE} is horizontal and *Y*_{EPI} is vertical) with *X* fixed to 65. In the absence of distortions the bright line would be aligned to the diagonal of the square. Strong image distortions for the frontal area (corresponds to the top-right part of Fig. 3d) and less apparent distortions in the center of the brain, revealed by Fig. 3d, can be identified in Figs. 3a and b. Figure 3e visualizes the relations among the distorted image, the nondistorted image, and the measured PSF and is provided to facilitate an understanding of the material presented in the paper.

The effect of the phase correction procedure described under Methods on the peak shape of the measured PSF is demonstrated in Fig. 4. The images and graphs presented were derived from the same raw data as shown in Fig. 3. Phase correction effectively removes the shoulder from the profile displayed in Fig. 4b, suppresses checkerboard-like interference patterns in the PSF view seen close to the middle of Fig. 4a, and removes other irregularities visible in Fig. 4a.

The application of the reduced field-of-view technique to the PSF-encoding dimension with fourfold acceleration is demonstrated in Fig. 5. The figure presents the PSF views in *Y*_{GE}–*Y*_{EPI} coordinates acquired in the spherical water-filled phantom on the 3-T system. The *X* coordinate was chosen at the center of the phantom. Imaging parameters were FOV = 192 mm, TE = 52 msec, TR = 2 s, 128 × 128 2D image matrix with full acquisition for EPI, and either 128 or 32 PSF encoding steps.

The entire set of images produced by the PSF mapping method along with the corrected echo-planar image is presented in Fig. 6. The figure presents the images of a single slice extracted from the 32-slice volume image of the brain of the normal volunteer acquired on the 4-T system. Imaging parameters were FOV = 224 mm, TE = 32 msec, TR = 3 sec, L–R phase encoding direction, 128 × 128 2D image matrix with 6/8 partial Fourier acquisition for EPI and 64 PSF encoding steps.

Figure 7 demonstrates the possibility of using the PSF technique to map distortions when the EPI readout is accelerated by means of parallel imaging techniques. Here images of a single slice through the spherical water-filled phantom, acquired on the 1.5-T system with the eight-channel head array coil and a parallel imaging acceleration factor of 2 for the EPI dimension, are presented. Imaging parameters were FOV = 192 mm, TE = 52 msec, TR = 2 sec, 128 (3) matrix, and full PSF encoding.

Figure 8 shows the feasibility of applying the parallel imaging acquisition schemes to the point spread function encoding process with remarkably high acceleration factors. The images of a single slice through the spherical water-filled phantom, acquired on the 1.5-T system with the eight-channel head array coil and a parallel imaging acceleration factor of 4 for the PSF dimension, are presented. Imaging parameters were FOV = 192 mm, TE = 52 msec, TR = 2 sec, 128^{3} matrix with full acquisition for EPI. PSF encoding was performed in accordance with requirements of the GRAPPA technique with 16 reference phase encoding steps, which resulted in acquisition of a total of 44 phase encoding steps. Figures 8a, b, and c were reconstructed by zero-filling the missing lines and are provided for reference to visualize the expected artifacts. It is to be noted that the StdDev image appears to be the most susceptible to artifacts. The GE image in Fig. 8d shows some intensity modulations, which, however, are not of importance for the reconstruction of the correct pixel shift maps.

Figure 9 demonstrates the possibility of combining parallel acquisition in EPI and PSF encoding dimensions. Here, images of a single slice through the spherical water-filled phantom, acquired on the 1.5-T system with the eight-channel head array coil and parallel imaging acceleration factor of 2 for EPI dimension and 4 for the PSF dimension, are presented. Imaging parameters were FOV = 192 mm, TE = 52 msec, TR = 2 sec, 128^{3}, both PSF and EPI encodings used 16 reference steps. As in Fig. 8d the GE image in Fig. 9a shows certain intensity modulations. As seen from Figs. 9c and 9d, the total acceleration factor of 8 does not cause observable degradation of the PSF data.

Figure 10 presents the result of an in vivo imaging experiment where the parallel imaging reconstruction for both EPI and PSF encodings was applied in combination with rFOV encoding for PSF direction. Images of a single slice through the brain of a normal volunteer acquired on the 3-T system with the eight-channel head array coil and a total parallel imaging acceleration factor of 8 are shown. Imaging parameters were FOV = 224 mm, TE = 30 msec, TR = 3 sec, R–L phase encoding direction, 128^{3} matrix. PSF encoding scheme used twofold rFOV acceleration and fourfold parallel imaging acceleration with eight reference encoding steps. EPI encoding used twofold parallel imaging acceleration with 16 reference encoding steps. A certain level of high frequency modulations may be observed in the GE image (Fig. 10b), which, however, does not seem to affect the quality of the pixel shift maps (not shown here). The StdDev image appears to be artifact-free. The successfully corrected echo-planar image (Fig. 10d) demonstrates the feasibility of combining the vastly accelerated PSF acquisition with the accelerated EPI acquisition without sacrificing the quality of correction.

### DISCUSSION

- Top of page
- Abstract
- THEORY
- METHODS
- RESULTS
- DISCUSSION
- CONCLUSIONS
- Acknowledgements
- REFERENCES

The results presented in this paper demonstrate that the PSF mapping technique is capable of measuring and correcting geometric distortions in echo-planar imaging. The method performs reliably at different field strengths and is able to recover correct distortion information even for areas of high magnetic field inhomogeneity. In our institution the PSF distortion mapping technique has been used in routine day-to-day fMRI scanning over several months, providing fully automated online distortion corrections of the functional data with remarkable accuracy and not a single case of introducing additional artifacts has been observed (20).

The increased stability of the PSF approach arises from the fact that it is a multireference technique. The PSF method acquires a larger amount of information and, hereby, affords to omit complicated multidimensional phase unwrapping procedures (21) that are frequently required, for example, by the dual-echo field mapping approach. Any field mapping method is also susceptible to partial volume effects, appearing, for example, due to the presence of fat and water in the same image voxel or differences in precession frequencies across the voxel because of strong field inhomogeneity. Indeed, the presence of severe intravoxel dephasing may cause the phase evolution to deviate from a linear increase with regard to the echo time, introducing errors in the reconstructed field maps. Furthermore, these effects are most pronounced in regions that experience the strongest distortions.

In comparison to other multireference techniques (7, 8) the PSF mapping method affords greater flexibility in decreasing the acquisition time by employing the reduced field-of-view technique and parallel imaging methods. Additionally, for the PSF technique the reconstruction of pixel shift maps and the application of corrections are computationally inexpensive and do not require numerous Fourier transforms or matrix inversions.

The StdDev images used throughout the paper deserve additional remarks. The possibility of using the SD instead of simple summation for collapsing the distorted dimension emerged empirically from the search of a representative parameter for the width of the PSF peak. It turned out that these images had high SNR with reduced image contrast. The increase in SNR arises due to the efficient combination of NMR signals spread along the distorted PSF dimension. Indeed, sum-of-squares summation in the SD calculation avoids complex signal cancellation. The reduction of image contrast is due to the subtraction of the squared mean from the mean square in the SD calculation. Therefore, the StdDev images appear to be well suited for image segmentation.

As demonstrated by our results, the PSF technique is able to map distortions in EPI experiments when parallel acquisitions are employed to accelerate the EPI readout. This is, however, only compatible with techniques capable of recovering the signal phase like numerous SMASH (22, 23) or GRAPPA (14) implementations. For image-based techniques, such as SENSE (24), respective modifications would be necessary.

The application of parallel imaging techniques to the PSF encoding itself appears to be extremely advantageous. Indeed, our results show the great tolerance of the reconstructed point spread functions to an increase in the parallel acquisition acceleration factor. This is due to the special properties of the acquired data: the measured PSF appears as a narrow (1–2 pixel wide) bright line as seen, for example, in Fig. 8f. The ghosts produced by the data undersampling, as seen from Fig. 8c are well resolved and do not overlap with the correct PSF line. This situation creates no difficulties for the parallel reconstruction as long as the number of ghosts does not exceed the number of coils whose sensitivity varies in the given phase encoding dimension. This limits the maximum useful acceleration factor for the eight-element cylindrical head coil used in our experiments to 4, which, however, performs extremely well. The modulation artifacts seen in the GE images (Figs. 8d, 9a, and 10b) do not affect the accuracy of the derived pixel shift maps. The mentioned artifacts in the GE images disappear when lower acceleration factors are used. Further investigation is required to investigate the sources and possible remedial approaches for these artifacts.

An unusual property of the PSF sequence with regard to parallel acquisition techniques is that it enables combination of extremely high acceleration factors on the PSF encoding dimension with acceleration in the standard EPI dimension. This is a consequence of the fact that PSF mapping is not a true 3D technique. It does not encode the 3D structure of the object by applying phase encoding in two dimensions. Instead, it encodes the same dimension twice using two independent phase encoding schemes. In principle, this allows for high acceleration factors for both encodings. However, the situation is more beneficial for PSF encoding, because for that dimension the parallel imaging reconstruction has to restore a thin line in image space, which does not contain overlaps between the image and the ghosts.

As demonstrated in this article, it is possible to combine parallel acquisition of the GE dimension with the rFOV technique in order to further shorten the measurement times. This combination does not appear to be extremely advantageous for high rFOV factors if self-calibrating parallel imaging techniques are used to restore the missing data, because the later techniques require acquisition of a certain number of reference lines, which may be comparable to the total number of lines to be acquired. In certain cases, however, this combination might come in handy. For example, in our experiments on the 4-T system it was required to acquire 64 PSF encoding steps (rFOV factor of 2) due to the large extent of distortions, which resulted in approximately 3-min acquisition time. For this case parallel acquisition can shorten the time required for the reference measurement. For example, the acceleration factor of 4 with 8 reference encoding steps requires 22 PSF encoding steps and reduces the measurement time to about 1 min. In general, the use of high rFOV factors requires prior knowledge on the extent of distortions and makes the reference measurement prone to errors if the distortions are underestimated. For measurements, which for other reasons are performed using coil arrays, it is more advantageous to combine parallel acquisitions of the PSF encoding dimension with moderate FOV reduction factors.

An important issue with distortion corrections in practical applications is the interaction between subject motion in the course of the scan session and image distortions. First, care should be taken to realign the dynamic EPI data to the position at which the pixel shift map was collected. This is achieved by using the distorted echo-planar image, generated by the PSF reconstruction routine, as a reference for motion correction. Assuming that the main field inhomogeneities move together with the head, this effectively eliminates the gross errors, when corrections would be applied to wrong image locations. In our implementation, real-time motion correction of the dynamic fMRI data is performed prior to distortion correction. However, smaller deviations in distortions may persist because the shim field deviations in the sample are dependent on both the position in the sample and the sample itself relative to the shim coils. As reported by Jezzard and Clare (25), in a typical fMRI experiment these additional time varying distortions may be as large as 0.4–0.6 pixels and manifest themselves as apparent noise in the fMRI time series. A potential remedy for these effects would be the use of a real-time autoshimming technique, e.g., similar to that reported by Ward et al. (26)