To retrospectively correct for geometrical distortions, a new dynamic field mapping method suitable for dynamic single-shot gradient-echo type echo-planar imaging (GRE-EPI) is proposed.
To retrospectively correct for geometrical distortions, a new dynamic field mapping method suitable for dynamic single-shot gradient-echo type echo-planar imaging (GRE-EPI) is proposed.
The method requires a single volume additional acquisition and allows the extraction of a field map from each phase volume, assuming invariance across time of the echo time-independent phase component. Performances of the method are assessed using three sets of experiments: the first tests the prerequisite and the modeling; the second tests the method with time-dependent geometrical distortions; and the third presents a comparison with two other methods.
Our results legitimize the modeling procedure and demonstrate that the dynamic method is less sensitive to noise than the other methods. A theoretical explanation for this is proposed in the discussion section.
Given the minor increase in the acquisition time, this method is well suited for functional magnetic resonance imaging; prospective direction. J. Magn. Reson. Imaging 2007;26:747–755. © 2007 Wiley-Liss, Inc.
BECAUSE OF ITS VERY SHORT ACQUISITION TIME, single-shot gradient-echo type echo-planar imaging (GRE-EPI) (1) has become the reference technique for dynamic ultra-fast imaging such as functional MRI (fMRI). However, EPI's exquisite sensitivity to susceptibility artifacts induces both geometric distortions (2–4) and signal losses (5–7). Several methods have been developed to correct for geometrical distortions that can be classified as one-dimensional (1-D) or two-dimensional (2-D), depending on the number of reconstructed directions used in the correction procedure.
In the 1-D methods, a retrospective correction is usually computed using navigator echoes (8). The additional signal required by these methods only slightly increases the acquisition time. However, only simple and reproducible artifacts can be corrected by such methods. For example, navigator echoes have been proposed to correct for dynamic phase discontinuities in interleaved EPI acquisition (9, 10), movement effects in diffusion-weighted EPI (11–13), magnetic field drift during long scan (14–16), or breathing motion artifact (17–19). In order to correct for more complex geometrical distortions, methods based on the acquisition of specific phase-encoded echo trains were proposed. Based on the hypothesis of time invariance of the artifacts during the image acquisitions, such methods were able to correct eddy currents (20–22) and geometrical distortions (23). Recently, a prospective method was proposed, to correct linear shims during dynamic EPI acquisition (24). Combined with a prospective motion correction, this method allows for real-time correction of different artifacts such as distortion correction or signal losses.
In the 2-D methods, three retrospective approaches can be distinguished. In the first, the principles of acquisition of specific phase-encoded echo trains is extended to reconstruct a series of phase images, each one containing the phase accumulation between successive gradient echoes (25, 26). While this method does allow for correction of all off-resonance artifacts, it does so at the detriment of the acquisition time, which is multiplied by the number of phase encoding steps (25, 26) or half the number of phase encoding steps (27). The second approach requires the measurement of the point spread function in the phase-encoded direction in each voxel of the image (28). While the correction provided by this method is optimal, it requires an additional phase encoding with a least 32 steps (29, 30). The time penalty is on the same order of that noted in the first method. The third approach requires the acquisition of a spatial map of the magnetic field variations in order to correct for geometrical distortions. Field maps are classically estimated using two images successively acquired at two different echo times; this method will be referred as the “classical” method in the following discussion. This approach has a major drawback for dynamic imaging, since the acquisition time is doubled. Moreover, successively switching the echo time between two values introduces contrast variations that could be more prominent than those sought in the application itself (for example, functional MRI). Such a drawback led to a strategy that was based on the acquisition of data for the estimation of only one field map before each dynamic series and on the assumption of the field map invariance across time (3, 4, 31, 32). With such a strategy, there is no modification of the dynamic acquisition time resolution; however, subject movement or drifts of the magnetic field invalidate the time-invariance hypothesis and lead to incorrect correction of the distortions (33–35).
In this article, we propose a method for the estimation of a dynamic three-dimensional (3-D) field map without any time resolution loss compared to the classical method; the method is based on a standard 2-D multislice single-shot GRE-EPI. In the following section, after presenting both the theory and implementation of the method, we present two experiments that validate our dynamic field mapping method. We then compare it to the previously published classical and “regression” (31) methods.
The simplest way to compute a field map requires two gradient echo acquisitions at different echo times (TE1 at time t1, TE2 at time t2). This method, referred to as the classical method, leads to an estimate of ΔB0 defined by:
where ΔB̆ is the estimated magnetic field map, ΔTE the positive difference TE2−TE1, and Φ and Φ the two spatially unwrapped phase images acquired with echo times TE2 and TE1, respectively.
The phase image can be further expressed as:
where Tn is the acquisition time and Φ0 is the echo time–independent phase term. The expression of the phase in Eq.  emphasizes that the classical method implicitly assumes that both Φ0 and ΔB0 are constant at times t1 and t2. The phase image Φ0 contains phase values that are independent of both the echo time and the spatial magnetic field inhomogeneity (36, 37). It is related to the spatial heterogeneity of the transverse magnetic field B1 in the imaged object, which in turn depends on the coil and the Larmor frequency. Other effects can also contribute to the observed Φ0, artifacts in the receiver or transmitter chain; these include drift of the frequency synthesizer, inadequate bandpass filters, or non centered k-space (e.g., pulse timing errors, eddy currents, or imperfect encoding gradients). These latter effects are independent of the imaged object and typically induce independent linear phase variations in the three encoded directions (36).
where ΔB̆ is the estimated magnetic field map, 0 is an estimation of the theoretical echo time-independent phase Φ0 and Φ is the unwrapped phase volume. Compared to the classical method, only Φ0 is now assumed to be constant over the whole acquisition time.
Our method requires a first estimation of Φ0 (see gray area in Fig. 1) that can be computed from the first phase image, according to:
where the initial guess ΔB̆0(x,y,z,t1,t2) is provided by the classical method. Since this 0 map will be used in the computation of each of the field map (see Eq. ), a movement of the subject could invalidate its constancy. In fact, some voxels at the border of the imaged object could toggle from the outside to the inside of the object. Since for such voxel no reliable 0 values can be measured, we replace in Eq. 0 by a model value of this signal, called , that extrapolates the signal beyond the border object. The chosen model, which depends on the MRI scanner, will be presented in the implementation section and its validity heuristically shown in the first experiment (see below). Finally based on Eq.  our method is described by the following equation:
In both Eq.  and Eq. , estimation of Φ requires an unwrapping procedure, in both the spatial and temporal dimensions. The spatial phase unwrapping (noted as SPU in Fig. 1) eliminates all phase jumps of 2π in the volume and results in a smooth map with a global phase offset in reference to the phase value at the seed point (see below, “Implementation”). The purpose of the temporal phase unwrapping (noted TPU in Fig. 1) is to take into account any variations of phase offsets between successive acquisitions. Without such correction, potential phase offset between two successive phase maps would induce a spurious magnetic offset between corresponding field map estimations.
All experiments were performed on a General Electric Signa Echo Speed 1.5-T magnet, running the LX8.2mns software version. This instrument has a 120 mT m–1 second–1 gradient slew rate with a 22 mT m–1 maximum amplitude. A standard birdcage-type quadrature head coil and the EPI blood oxygenation level dependent (BOLD) sequence were used with the following parameters: flip angle = 90°, field of view (FOV) = 24 cm, slice thickness = 3.8 mm, full symmetric k-space coverage, and interleaved contiguous slice acquisition with spatial and spectral radio frequency (RF) excitation. Nyquist ghosts were corrected with an automated version of the Buonocore's method (38).
We first discuss two key points of our dynamic field mapping method: phase unwrapping and echo time–independent phase modeling; we then describe the geometrical distortion correction procedure.
As described above, our method requires both spatial and temporal phase unwrapping (see SPU and TPU in Fig. 1). The former is performed first on each phase map using the method proposed by Cusack and Papadakis (39). Using this method, in each phase map, a seed point is automatically defined and a flood-fill procedure is initiated from this seed point to unwrap all phase values in the object. The temporal phase unwrapping is implemented as a two step procedure. In the first step, taking in account only the phase values at each seed points, the number of 2π phase jumps (jn) in reference to the seed point phase value of the first volume is computed for each successive volume. In the second step, each of the phase map voxel value is offset from the corresponding number of 2π jumps, ΦTEn(x,y,z,tn)−jn2π.
As discussed above, beside the linear variation due to the artifacts in the receiver or transmitter chain, the echo time-independent phase map is related to the transverse magnetic field B1, which induced nonlinear variation. However, since we used a birdcage-type coil at 1.5 T, we expect this field to show only smooth spatial variation across the entire FOV (37). We chose to implement a 3-D spatial global fitting procedure based on a third order polynomial model. Altogether, the polynomial model contains 20 coefficients (10 uncrossed and 10 crossed-product terms) estimated by a linear least-squares regression algorithm.
The distortion correction techniques are dependent on the undistorted or distorted nature of the field map acquisition space; since our method is EPI-based, we chose the distorted space based simulated phase evolution rewinding (SPHERE) method of correction (32). Because of the importance of sensitivity of the correction to noise in the field map, this map first needed to be spatially smooth. Usually, this step is also required to accommodate the toggling of voxels from the outside to the inside of the imaged object. Because of the dynamic nature of our method, we did not need such a large smoothing; thus, we used a conditional median filter to remove only spike noise. This filter consists in replacing the value in each voxel at the position (i, j, k) by the median value of its neighborhoods if the following relation is verified:
where ΔB0(i,j,k) is the value of interest, Ω defines the list of the six nearest neighborhood voxels (face connected) and S is the threshold, empirically fixed at 10–8 T. This threshold corresponds to a 0.02 pixel shift in the case of a typical EPI acquisition with 64 echoes and an echo spacing τesp of 896 μs.
To prove the efficiency of the 3-D fitting procedure and verify the hypothesis of time invariance of Φ0, we acquired 52 consecutive volumes, of either a 20-cm diameter spherical phantom (Exp. Ia) or a human brain (Exp. Ib) (TR = 6 seconds, axial slices = 42, matrix size = 64 × 64). Note that for this latter experiment no specific constraint was imposed on the subject's head. Thus, during the 10 minutes of acquisition, small movements were observed and quantitatively estimated using SPM99 (http://www.fil.ion.ucl.ac.uk) to 0.4 mm translational and 0.2 degree rotational displacements. During acquisition, echo time was switched at each repetition between 40 msec and 50 msec. Each successive pair of repetitions was used to compute a field map with the classical method (Eq. ). The estimated 0 was computed across time (Eq. ) and each one was independently fitted with the 3rd order 3-D polynomial. First, the model adequacy was assessed by computing the coefficient of determination R2 associated with the linear least-squares regression. Second, each parameter value was tested at 0.05 corrected for its significance difference to zero (Student's t-test, degrees of freedom [df] = 25). Finally, each parameter was tested using a linear regression through time, the significance of the regression was assessed using the Fisher test (df1 = 24, df2 = 2, P < 0.05 corrected for multiple comparisons) on the R2 coefficient. These evaluations were performed both on phantom and brain acquisitions.
In the two experiments described below, both the efficiency and accuracy of our field mapping method were assessed by applying estimated dynamic field maps to correct for geometrical distortions in presence of either small, global (Exp. IIa) or large (Exp. 11b), localized magnetic field variations through time. The aim of the first experiment was to test our method in the presence of a magnetic field drift (14); whereas the aim of the second experiment was to test the method when both geometrical distortion and signal losses occur.
In Experiment IIa, a series of 100 volumes in coronal incidence were acquired in 10 minutes (TR = 6 seconds, slices = 42, matrix size = 64 × 64). The echo spacing τesp and EPI readout were 896 μsec and 57.34 msec, respectively. The initial guess of magnetic field map was computed with the classical method, with a delay ΔTE equal to 10 msec. In order to highlight the small drift effect, the first volume of the series was subtracted from each volume of both the uncorrected and corrected time series. To qualitatively assess the distortion correction in each of the series, we estimated the number of voxels that were significantly displaced at each time point. Such voxels were those with absolute intensities greater than four times the standard deviation of the noise (measured at the center of the phantom).
In Experiment IIb, a vial containing gadolinium was placed next to the spherical homogenous phantom. Global shimming was performed in this configuration and a volume encompassing the whole setup was repeatedly imaged twelve consecutive times (TR =12 seconds, axial slices = 42, matrix size = 128 × 128). At the mid-point of the acquisition, the vial was removed, leading to important variations of the magnetic field through time. High-resolution acquisition was used to emphasize susceptibility artifacts (in plane sampling 1.875 × 1.875 mm, 1287 μsec echo spacing τesp, 164.74 msec EPI readout, 110 msec minimum echo time). Circular shapes of the phantom, before and after the vial removal, were graphically appraised for geometrical distortions on both uncorrected and corrected data.
The aim of the third experiment was to compare our dynamic field mapping method to two methods of the literature: the classical method (described above in the theory section) and the regression method (31). In the regression method, a series of volumes with an increasing echo time is acquired. Each voxel field value is estimated by a linear least-squares regression through time on the unwrap phase values (31).
First, we acquired two series of 10 volumes on both a spherical phantom (Exp. IIIa) and a human brain (Exp. IIIb). During the acquisition, the echo time was increased by step of 2 msec, starting from 37 msec up to 55 msec (TR = 6 seconds, axial slices = 42, matrix size = 64 × 64). Second, the three different field mapping methods were applied. For the classical method, the 37 msec and 47 msec echo time volumes were used to build the field map. For the regression method, all 10 acquisitions were taken into account. Finally, the dynamic method was applied with an initial guess provided by the classical method and the phase volume was acquired at 37 msec echo time. Differences in the field map estimations were assessed graphically and quantitatively using correlation coefficients.
Figure 2 illustrates one representative sagittal projection of the human brain data (Exp. Ib), with the raw calculated 0 (Fig. 2a), the 3-D model (Fig. 2b), and the residues 0− (Fig. 2c). The phase component 0 exhibits a strong linear phase variation in the slice direction, starting from positive values (around 15 rad) at the top of the brain and ending at negative values (around –15 rad) at the cerebral trunk level. Such a pattern was observed in all 26 phase volumes. As shown in Fig. 2b, this 0 pattern is adequately modeled by a 3rd order 3-D polynomial, the average coefficient of determination between data and model values beingR2 = 0.971 ± 0.009 (N = 26, range = [0.952, 0.990]) and R2 = 0.967 ± 0.032 (N = 26, range = [0.861, 0.999]) in Exp. Ia and Exp. Ib, respectively. As shown for the human data (Exp. Ib) in Table 1 (column 5) only eight parameters were significantly different from 0, linear regression across time for each parameter (Table 1, column 6) being found nonsignificant. As an example, Fig. 2d shows the estimated values through time of three significant parameters of the polynomial model. Figure 3 shows a central axial view of the same volumes shown in Fig. 2: the raw calculated 0 (Fig. 3a) and the 3-D model (Fig. 3b) decomposed in the signal modeled by both the first order linear parameters (Fig. 3c), and the other parameters (Fig. 3d). As stated in the theory, the former phase image is probably related to artifacts in the receiver or transmitter chain (these include drift of the frequency synthesizer, inadequate bandpass filters, or noncenteredk-space). In the latter phase image, the regular B1-related effects (37) can be observed. As shown above, neither of the linear and nonlinear entering terms of the model exhibited significant variations across time. Thus, phase modeling could be considered as being unaffected by both the characteristics of the antenna and the electronic chain.
|Parametera||Mean (×10−4)||SD (×10−4)||Variation coefficient (%)||P-value (H0: mean = 0)b||P-value (linear regression)|
The same results were found for the phantom data (Exp. Ia). This confirmed the constancy of the phase independent model estimation.
Figure 4 shows the results from Exp. IIa. Figure 4a shows an innermost sagittal view of the difference between the last and the first volume (phase and slice encoding directions are horizontal and vertical, respectively). Left and right images show this difference without and with geometrical distortion corrections, respectively. We observe that only lower slices (bottom of the left image in Fig. 4a) are shifted towards the left part of the image. This particular geometrical distortion pattern implies a spatially heterogeneous drift of the magnetic field B0 with an obvious dependence in the slice-selection direction. Our dynamic field mapping methods correctly accounts for these time-dependent variations, as shown by the difference of corrected images (right image in Fig. 4a). For two voxels at the border of the object, Fig. 4c and d shows the time courses of the dynamic magnetic field estimation and both the intensities in uncorrected (continuous line) and corrected (dashed line) images. In both cases, the field maps estimated by our method were able to correct for the geometrical distortion, as shown by the constancy of the intensities after correction (dashed lines). Furthermore, these examples demonstrate, even in the simple phantom case, that the magnetic field drift can exhibit both a monotonic (Fig. 4c) and more complex pattern within the same acquisition (Fig. 4d). These latter types of distortions can only be taken in account with a fully dynamic method such as our dynamic method. Figure 4b shows the time evolution of the number pixels detected as shifted pixels in the plane shown in Fig. 4a. The distortion correction based on our dynamic estimated field maps decrease this number up to a factor of four.
Figure 5 shows the results from Exp. IIb. Figure 5a shows, without distortion correction, an axial slice before (left) and after (middle) suppression of the vial containing gadolinium and the difference (right) between these two images. In the left image, the magnetic field heterogeneities (due to the gadolinium) induced both a signal loss and a stretching of the circular shape of the phantom toward the vial. In the middle image, after vial removal, neither of the artifacts can be seen. The difference image highlights the stretching effect that occurs at the boundary of the phantom. The same images are shown after distortion correction in Fig. 5b. It can be seen in the difference image that the stretching was almost completely corrected and that only the signal loss remains.
Figure 6a shows the axial (upper row) and sagittal (lower row) field maps measured on a spherical phantom (Exp. IIIa) using (from left to right) the classical, the regression, and our dynamic method. The same display is used in Fig. 6b for data obtained from a human brain (Exp. IIIb). Globally, the different field mapping methods lead to similar results. The correlation coefficients dynamic/classical and dynamic/regression were 0.967 and 0.971, respectively, in the phantom (N = 52,000), and 0.976 and 0.985, respectively, in the human brain (N = 27,000). The three methods show similar field values in the axial slice; however, in the reconstructed sagittal slices, some spurious variations from slice to slice can be observed with both the classical and the regression methods, but not with ours.
The constant phase modeling is a crucial part of our procedure. In fact, its validity cannot be checked without losing the aim of the method, which is to avoid increasing the scan time duration. Although only two experiments have been presented, many other experiments involving different slice orientations, acquisition orders, and thicknesses have been performed in order to select the reliable model and check its robustness. Furthermore, classical and spatial/spectral RF excitations were tested with both a bird cage and a surface coil. First, the empirical selection of the model was assessed. In all experiments, this model was able to explain more than 90% of the time-independent phase pattern. Second, from those experiments, we observed that the main feature of the constant phase pattern was linear, with a minimal value at the center of the imaged object. Such behavior indicates a slice excitation-dependent phenomenon that could be explained by a small error in the sequence implementation (possibly due to a drift of the frequency synthesizer during excitation pulse). This phenomenon, therefore, likely depends on the scanner type. Beside these specific phase patterns, the echo time-independent phase term Φ0 is related to the spatial heterogeneity of the transverse magnetic field, B1, which mainly depends on the type of coil, the imaged object, and on the main magnetic field B0 intensity. This effect on the phase depends on the scanner type and the higher the field, the higher the B1 related in homogeneities are spatially complex. From the B1 map dependency to the B0 field, as shown by Collins and Smith (37), we believe that up to 3 T, our model would still be effective to fit the time-independent phase. However, prior to the application of our dynamic method, it is recommended that one measures the Φ0 map (using sequences equivalent to those used in Exp. I) and adapts the model in order to fit only the time-independent phase components of Φ0.
Each of the three methods gives comparable results, but the field map estimated with the classical method exhibits some spurious discontinuities between slices. This phenomenon is reduced with the regression method and is absent with the dynamic method. Such an artifact was probably missed in earlier works because the analysis was only performed in the image plane (see, for example, Ref.31). This artifact is related to phase errors in images used to compute the field map. We believe these errors are related to phase interferences between the interleaved acquired slices and probably some linear spatial phase variations attributable to miscentered k-space. To understand why our dynamic magnetic field map method is insensitive to such artifacts, we can rewrite Eq.  with an additional error term δΦ(x,y,z,tn):
In the classical method, described by the Eq. , the variance of the error term δΦ is doubled and, thus, its standard deviation multiplied by , which gives a magnetic field error term of: δΦ/γΔTE. In the dynamic method, described by Eq. , the phase error term appears only once and, thus, the magnetic field error term is expressed as: δΦ/γTE. Applying the values of ΔTE (10 msec) and TE (37 msec), used in our experiments, lead to a reduction of the phase error effect in the estimation of the magnetic field errors by a factor 5.3 in the dynamic compared to the classical method. The same computation with the regression method is not straightforward; however, if N is the number of acquisition used in the analysis, a lower bound for the field error term can be expressed as δΦ/(γΔTE). When N = 10, this leads to a correction factor of 1.2 in the dynamic compared to the regression method.
In this article, we did not investigate the efficiency of our method with large movements of the subject. In fact, since our method estimates the magnetic field at each time in the distorted space, it is suitable to take into account any magnetic field variations, particularly those induced by motion. These magnetic field variations are classically not taken into account by geometrical distortion correction based on a unique field map acquired in the undistorted space (21, 40). Consequently, our method should be of particular interest for dynamic imaging, such as fMRI, where optimally geometrical distortion correction improves analysis in terms of anatomical localization and statistical power (40). However, it should be pointed out that if the transverse magnetic field, B1, significantly varies with movement of the subject in the coil, then our model will no longer be valid for all acquisitions through time. In practice, with regular minimal movement of the subject's head, we did not observed any time variation of the parameters of the model .
In conclusion, a new EPI-based dynamic field mapping method is proposed that requires less acquisition time and leads to more accurate phase maps than both the classical and the regression methods. Only one additional acquisition with different echo time is needed without modifying the repetition time during the following dynamic acquisition. This new dynamic field mapping method will allow better geometrical deformation corrections than a technique based on a unique magnetic field map estimated once in a particular position of the subject.