Combining phase images from multi-channel RF coils using 3D phase offset maps derived from a dual-echo scan

Authors


  • This article was published online on 19 Jan 2011. Errors were subsequently identified. This notice is included in the online and print versions to indicate that both have been corrected 14 Feb 2011.

Abstract

A method is presented for the combination of phase images from multi-channel RF coils in the absence of a volume reference coil. It is based on the subtraction of 3D phase offset maps from the phase data from each coil. Phase offset maps are weighted combinations of phase measurements at two echo times. Multi-Channel Phase Combination using measured 3D phase offsets (MCPC-3D) offers a conceptually and computationally simple solution to the calculation of combined phase images. The dual-echo data required for the phase maps can be intrinsic to the high-resorlution gradient-echo scan to be reconstructed (MCPC-3D-I). Alternatively, a separate, fast, low-resolution dual-echo scan can be used (MCPC-3D-II). Both variants are shown to give near perfect phase matching, yielding images with high SNR throughout and high GM-WM contrast. MCPC-3D is compared with other reference-free phase image crombination methods; high-pass phase filtering, phase difference imaging, and matching using constant offsets (MCPC-C). Multi-Channel Phase Combination using measured 3D phase offsets method does not need an overlap between the signals from individual coils and can be used with parallel imaging, making it ideally suited to multi-channel coils with a large number of elements, and to high and ultra-high field systems. Magn Reson Med, 2011. © 2011 Wiley-Liss, Inc.

There has been a rapid development of phase imaging methods over the past 5 years and a concurrent development in phased array RF receiver coils for high and very high fields. At present, however, there is no standard method for the direct combination of phase images from multiple coils in the many systems that do not have an integrated body RF coil or other volume reference coil.

Enhanced microscopic and mesoscopic dephasing effects at high field have stimulated the development of methods that are reliant on the phase of the MR signal for contrast, particularly phase imaging (1), susceptibility-weighted imaging (2), and susceptibility mapping (3). These gradient-echo methods are being applied to assessment of the venous anomalies such as developmental venous anomaly and vascular malformations such as cavernomas (4) and the imaging of iron stores in pathologies such as dementia (5), Huntington's, Alzheimer's (6) and Parkinson's disease.

Phased array coils offer advantages over volume RF coils, particularly at very high field. SNR is higher close to small surface coils than it is with volume coils (7). Phased arrays also allow a reduction in the number of phase-encoding steps required for spatial encoding of the MR signal, replacing these by sensitivity encoding, enabling accelerated acquisition (8). As a result, phased arrays with an increasing number of elements are being developed for high field strengths (e.g. Ref.9).

Data from multiple coils are optimally combined by weighting coil values at each voxel by the complex sensitivity of the relevant coil at that voxel position (10). Accurate maps of coil sensitivities are required and are ideally generated by division of complex data acquired with each coil element by complex data acquired with a volume reference coil (8). Some high field systems, and most very high field systems, do not have integrated body RF coils or alternative volume coils available for this purpose.

For magnitude imaging, the magnitude image from each receiver serves as a reasonable approximation to the sensitivity of that element, and a sum of squares reconstruction is a computationally efficient means of creating a combined image (11). Other approaches, such as self-calibrated SENSE (12) and adaptive filtering (13), use sophisticated methods to estimate coil sensitivities without a reference image. Although these lead to improved magnitude images, phase images generated with these reference-free methods do not necessarily represent the true phase (as measured with a volume coil) (14, 15). Self-calibrated SENSE has recently been extended for the optimized reconstruction of phase and complex images (14). Although this approach is promising, it is mathematically complex and computationally demanding, and has yet to be shown to be effective with a large number of coils at very high field.

Compared with magnitude images, reconstructing phase images poses two additional problems. The first is that phase images are subject to “wraps,” jumps of 2π that arise because phase values of n2π + θ (where n is an integer) are encoded identically. Wraps are phase iso-contours that manifest either as closed loops within the object or begin and terminate at the object boundary. These can be identified using a variety of algorithms (16, 17) and removed to restore the full phase range. The second problem is that individual receivers are subject to different, spatially varying, phase offsets which mean that combining phase information naively, e.g., from summed complex data or by weighting each phase image by the respective magnitude image, leads to interference and regions of signal cancellation (15). Phase images calculated in this way have generally low SNR and frequently show wraps that terminate within the object (corresponding to complete signal cancellation), known as “open-ended fringe lines.” One approach to combining phase images from multiple receivers is to unwrap phase images then apply a high-pass filter (which removes phase offsets, as low frequency features) before calculating a weighted mean over channels (e.g. Ref.18). For ease of reference, we call this method Phase Filtering. Phase Filtering is computationally intensive, as phase images from each channel have to be unwrapped. More importantly, low spatial frequency features are lost, and quantification of combined phase images is heavily dependent on the choice of filter (19).

For dual-echo data, a longstanding approach is to calculate a Phase Difference image, equal to the angle of the sum over channels of the Hermitian inner product of the complex data from the two echo times (from hereon, the Phase Difference method) (20). Channel-dependent phase offsets are removed in the calculation of phase differences, although the subtraction also reduces CNR.

An alternative to the Phase Filtering and Phase Difference Methods has recently been presented, which likewise allows phase images from multiple receivers to be combined without recourse to information from a reference coil (21). It is based on the subtraction of a constant phase value, identified from phase images, from each channel, before calculating the complex sum. We call this Multi-Channel Phase Combination using Constant phase offsets (MCPC-C). A similar phase-sensitive reconstruction method was implemented by de Zwart et al. (22). We show that MCPC-C phase images are prone to phase artifacts and regions of low SNR because phase offsets are not constant throughout the image.

We present a new method, called Multi-Channel Phase Combination using measured 3D phase offsets, or MCPC-3D, in which 3D phase offset maps are calculated for each receiver using a dual-echo acquisition. As in MCPC-C, these are subtracted from respective channels prior to vector addition of the complex signals. This allows phase information from multiple coils to be combined without a reference RF coil, without filtering, and where there is no signal overlap between receivers. Phase images reconstructed with MCPC-3D are compared to those calculated using the Phase Filtering, Phase Difference, and MCPC-C methods.

THEORY

The phase in receiver element l, θl, at echo time TE, is dependent on both the local deviation from the static magnetic field, ΔB0 (the source of image contrast), and the phase offset for that receiver element, θRX,l, as

equation image(1)

neglecting phase wraps.

In the Phase Filtering approach, phase images from each channel are unwrapped, then high-pass filtered. High-pass filtering removes θRX,l, as a low-frequency feature, allowing these images to be combined in a weighted mean over channels.

In the Phase Difference method, a weighted mean phase difference, ΔθHP, is calculated via the sum over channels, l, of the Hermitian inner product of the complex data at the first echo time, S1,l, and that at the second echo time S2,l;

equation image(2)

where denotes ∠ the angle and S1* the complex conjugate of S1 (20). Channel-specific phase offsets are removed in the subtraction.

In the MCPC-C approach, channel offsets, θRX,l, are assumed to be constant throughout the image, and are estimated from the phase at a single voxel location or over a region in which it is presumed that there is signal in all detectors. The scalar θRX,l is subtracted from each channel l, setting the phase of all channels to zero at the point of correction. The weighted mean phase over all channels is then calculated as

equation image(3)

where ∠ denotes the angle of the complex vector and Ml is the magnitude of the signal in channel l from a specific location in the object.

In our method, we assume that at high field θRX,l varies in space due to the fact that the wavelength of RF in tissue is of the order of the object size (7). The phases of two images acquired at TE1 and TE2 are given (in accordance with Eq. 1) by

equation image(4a)

and

equation image(4b)

Combining and rearranging Eqs. 4a and 4b allows the phase offset of each channel l to be determined:

equation image(5)

This formulation has been proposed previously to calculate a phase offset map with a single RF coil in the context of dynamic field mapping (23, 24). The combined phase image is calculated as the angle of the complex sum of phase-corrected signals, according to Eq. 3.

If the optimum echo time for the acquisition of a high-resolution phase image to achieve a particular T2* weighting is TE2, phase information from an additional prior echo acquired at TE1 can be used to calculate and remove channel-specific phase offset dependence (MCPC-3D-I). Alternatively, channel offsets can be calculated from a separate, fast, low-resolution dual-echo scan (MCPC-3D-II). The relative merits of these two variants are investigated, and phase matching and image characteristics compared with the Phase Difference, Phase Filtering, and MCPC-C methods.

MATERIALS AND METHODS

The study was designed to test the robustness of the proposed phase combination method with two coils (with 8 and 24 receivers) at very high field, and compare all five reconstruction methods; Phase Difference, Phase Filtering, MCPC-C, MCPC-3D-I, and MCPC-3D-II. To this end, high-resolution single-echo, high-resolution dual-echo, and low-resolution dual-echo data were acquired with both 8- and 24-channel coils. These tests were carried out with one 28-year-old healthy male subject. To assess all methods in a range of patients and pathologies, high-resolution dual-echo data were acquired for five tumor patients, aged 27- to 65-years-old (average 43), with an 8 channel coil only (for tolerance reasons).

Measurements were made with a 7 T MR whole body system (Magnetom, Siemens Healthcare, Erlangen, Germany) using both an 8-channel RF coil (RAPID Biomedical, Würzburg, Germany) and a 24-channel RF coil (Nova Medical, Wilmington, USA). The study was approved by the Ethics Committee of the Medical University of Vienna. All participants in the study provided written, informed consent.

The following measurements were made for the healthy subject, with both coils: Scan 1: an axial 3D, high-resolution, single-echo gradient-echo scan with TE/TR = 15/28 ms, GRAPPA factor 2, with a 704 × 572 × 96 matrix and a 16 × 19.8 × 11.5 cm3 FoV (giving 0.3 mm in-plane resolution and 1.2 mm thick slices) and an acquisition bandwidth of 140 Hz/pixel, acquisition time (TA) = 13 min 20 sec, Scan 2: an axial 3D, high-resolution, dual-echo gradient-echo scan with TE/TR = (8, 15)/28 ms, otherwise the same parameters as Scan 1, and Scan 3: a 2D low-resolution dual-echo gradient-echo scan with TE/TR = (4.6, 9.3)/606 ms, GRAPPA factor 4, TA = 27 sec, with a 128 × 128 matrix and 32 slices of 3.0 mm thickness and a 230 × 230 FoV (giving 1.7 mm in-plane resolution). The TE of 15 ms in the high resolution acquisitions was selected to provide near-optimum T2* weighting at 7 T (21). Phase and magnitude data for each channel were stored for all acquisitions.

For the patients, only the high-resolution dual-echo measurements (as in Scan 1, above) were made, with the 8-channel coil.

The Phase Filtering, MCPC-C, and MCPC-3D-II methods need only a single-echo high-resolution scan (here, Scan 1), whereas the Phase Difference and MCPC-3D-I methods require a dual-echo high-resolution scan (Scan 2). To allow reconstruction methods to be compared (based on the same data), Scan 2 was used for all approaches. Scan 1 was used to assess reconstruction times and look for image quality differences which relate to the use of a single-echo rather than a dual-echo acquisition.

Phase Filtering

The Phase Filtering approach was implemented as follows: single-channel phase data from Scan 1 were unwrapped in 2D using PHUN (17). High-pass filtering was achieved by smoothing a copy of the unwrapped phase images with a 3D Gaussian filter with FWHM of 7 mm then subtracting this from the unsmoothed data. The weighted mean of unwrapped, high-pass filtered images was calculated. No further filtering was applied to the combined image in this method.

Phase Difference

Phase Difference images were calculated from Scan 2 according to Eq. 2.

Multi-Channel Phase Combination With Constant Offsets

MCPC-C was applied to echo 2 of Scan 2 (steps indicated in Fig. 1). Phase-matching of channels using constant channel phase offsets was performed according to Hammond et al. (21). The mean phase value in the central 3 × 3 × 3 cube of voxels in each raw phase image (Fig. 1a) was subtracted from the respective channel (Fig. 1b). The combined phase image was calculated as the angle of the sum of the complex, phase-corrected images over all channels (Fig. 1c), and unwrapped (Fig. 1d).

Figure 1.

An overview of steps in the MCPC-C method (illustrated for 8 channels). The value at the centre of each raw phase image (a) was subtracted from that image (b). The angle of the sum of phase-corrected complex data over all channels was calculated (c), and this phase image was unwrapped (d).

Multi-Channel Phase Combination Using Measured 3D Phase Offsets

The steps in MCPC-3D-I and MCPC-3D-II are illustrated in Figs. 2 and 3, respectively. The MCPC-3D-I method used Scan 2, with both echoes used for the calculation of phase offsets, and echo 2 being reconstructed. For MCPC-3D-II, the low-resolution Scan 3 was used for the calculation of phase offsets. To allow direct comparison with MCPC-3D-I, the second echo from Scan 2 was reconstructed. In both cases, acquired data required for the method are indicated by grayed boxes.

Figure 2.

An overview of steps in the proposed MCPC-3D-I method (for 8 channels). Phase data for both echoes and all channels (a) were unwrapped and corrected for n2π jumps between slices and echoes (b). These dual-echo data were combined according to Eq. 5 to yield 3D maps of phase offsets for each channel (c). These phase offsets were subtracted from raw phase data (a) at the second echo time to yield matched phase data (d). The angle of the sum of phase-corrected complex data over all channels was calculated (e), and this phase image was unwrapped (f).

In the MCPC-3D-I method, phase data for each echo (Fig. 2a) was unwrapped in 2D using PHUN (17). Phase jumps of n2π, where n is an integer, occur between slices when there are n wraps between seed voxels (the position at which unwrapping begins). The occurrence of n2π jumps between slices was identified by comparing the mean in-brain phase value in each slice to that in the adjacent slice, assuming that the true difference between the mean phases in the two slices was closer to 0 than 2π. Phase jumps of n2π also occur between echoes when there are n wraps between seed voxels at the two echo times. The following procedure was used to identify and remove these. A phase difference map was calculated using the Hermitian inner product method (Eq. 2) (20). This phase difference image was unwrapped and added to the unwrapped phase maps for each channel at the first echo time to yield an estimate of the phase at the second echo time, θ2_est,l:

equation image(6)

The mean difference between θ2_est,l and the unwrapped phase maps at the second echo time, θ2,l, was used to identify the occurrence of n2π phase differences between the first and second echoes, as

equation image(7)

where the double bars denote a rounding to the nearest integer. The value of n2π was subtracted from θ2,l to yield unwrapped phase images at TE1 and TE2 that had no interslice or interecho n2π jumps (Fig. 2b). Maps of receiver offsets were calculated according to Eq. 5, and median smoothed with a 5 × 5 × 5 voxel kernel to reduce noise (Fig. 2c). Three dimensional phase offsets were subtracted from the raw phases at the second echo time, phase-matching the channels (Fig. 2d). The combined phase image was calculated as the angle of the complex sum of such phase-matched signals (Fig. 2e), as in the MCPC-C method, and unwrapped (Fig. 2f).

In the MCPC-3D-II method, single-echo phase data (Fig. 3a) are corrected using 3D phase offset maps calculated from a separate, low-resolution dual-echo scan (Fig. 3b). To allow comparison with other methods, the second echo from the dual-echo Scan 2 was reconstructed instead of the single-echo acquisition Scan 1, which is illustrated in the figure. The low-resolution, dual-echo data (Fig. 3b) were unwrapped in 2D using PHUN, and n2π jumps between slices and echo times removed described for the MCPC-3D-I method (Fig. 3c). Low-resolution maps of receiver offsets (Fig. 3d) were calculated from the unwrapped phases at the two echo times for each channel using Eq. 5 and median smoothed with a 5 × 5 × 5 voxel kernel. Low-resolution maps of receiver offsets could be simply enlarged to match high-resolution data. To compensate for any motion between the low- and high-resolution acquisitions, we chose to co-register phase offset maps to the high-resolution data using FSL's FLIRT (www.fmrib.ox.ac.uk/fsl/, 25). To this end, a transformation matrix was applied that was derived from co-registration of the low-resolution magnitude images at the second echo time to the corresponding high-resolution magnitude images. Co-registered, 3D phase offset maps were subtracted from the raw phases, phase-matching the channels (Fig. 3e). The combined phase image was calculated as the angle of the complex sum of such phase-matched signals (Fig. 3f) and unwrapped (Fig. 3g).

As a final processing step for the Phase Difference, MCPC-C, and MCPC-3D methods (but not the Phase Filtering method), phase images were high-pass filtered. To achieve this, a copy of the unwrapped combined phase images was smoothed with a 3D Gaussian filter with FWHM of 7 mm and subtracted from the original. Image processing was carried out with a PC with 24 X7460 2.66 GHz Intel Xeon processors and 97 GB of RAM, running 64-bit server Ubuntu 9.10. Parallel processing was not used.

To assess the quality of phase matching over all voxels, voxel-wise ratios were calculated of the magnitude of the summed complex data to the sum of individual channel magnitudes. If phases are well matched over channels by the correction method, the magnitude of the complex, phase-matched sum is similar to the sum of the individual magnitudes.

Processing times were recorded for all methods, and image noise and grey-white matter contrast to noise were evaluated for the healthy subject. The average standard deviation over six white matter regions of interest was assessed as a noise metric, and grey-white matter contrast to noise ratio was calculated for three pairs of adjacent grey and white matter ROIs.

RESULTS

Maps of channel phase offsets (Figs. 2c and 3d) showed slow variation throughout the image and no susceptibility effects (i.e., no B0 offsets in frontal and ventral regions) as would be expected from Eq. 5. Phase variation was ∼3π across phase offset maps, consistent with the size of the brain and a wavelength in tissue close to 12 cm at 300 MHz (26). In the MCPC-C method, single-channel phase images were matched at the correction position (the image center) but showed variation throughout the image (Fig. 1b). Single-channel phase images corrected with the MCPC-3D method were very well matched, with phase wraps occurring in near-identical positions (Figs. 2d and 3e).

The quality of phase images reconstructed with the Phase Difference, Phase Filtering, MCPC-C, MCPC-3D-I, and MCPC-3D-II methods is illustrated in Fig. 4. Although the Phase Filtering, MCPC-C and MCPC-3D-II methods only require a single-echo high resolution scan, the dual-echo data (Scan 2) were used for all methods, to ensure that differences were attributable to differences in the reconstruction approach, and not differing properties of the underlying acquisitions. Phase images are illustrated with the same dynamic range. Phase images reconstructed with the Phase Difference method had lower GM-WM contrast and vessel visibility than those reconstructed using Phase Filtering and MCPC-3D. Phase Filtering images had high SNR and good GM-WM contrast, and were similar in quality to those reconstructed with MCPC-3D. MCPC-C phase images showed regions of reduced SNR, and the complete loss of image features in some regions (see enlargement). Poor phase matching with MCPC-C was apparent as signal cancellation in the magnitude images extracted from the summed complex data, marked at arrow 1. Phase images reconstructed with MCPC-3D-I and MCPC-3D-II had high SNR throughout and excellent grey-white matter contrast (arrows numbered 2). Uniformity in the magnitude of the complex sums (see magnitude insets in Fig. 4) indicated excellent phase matching throughout (see S1 panel in Fig. 5 for quantification). There were no consistent differences between images reconstructed with MCPC-3D-I and MCPC-3D-II.

Figure 3.

An overview of steps in the proposed MCPC-3D-II method (for 8 channels). Low-resolution dual-echo phase data (b—Scan 3) were unwrapped and corrected for n2π jumps between slices and echoes (c). These data were combined according to Eq. 5 to yield low-resolution 3D maps of phase offsets for each channel (d), which were co-registered to high-resolution data (a) (registration step not shown). These 3D phase offset maps were subtracted from high-resolution raw phase data (a) to yield matched phase data (e). The angle of the sum of phase-corrected complex data over all channels was calculated (f), and this phase image was unwrapped (g).

Figure 4.

Comparison of the quality of phase images reconstructed with the Phase Difference, Phase Filtering, MCPC-C, MCPC-3D-I, and MCPC-3D-II methods (24-channel coil, healthy subject). Data from the dual-echo Scan 2 was used for all methods (the second echo for the Phase Filtering, MCPC-C and MCPC-3D methods and both echoes for the Phase Difference method). The Phase Difference image has low GM-WM contrast. GM-WM contrast is very good and noise low in the Phase Filtered images (see enlargement). The MCPC-C phase image shows regions of reduced SNR and poor GM-WM contrast, and some areas of complete signal cancellation (enlargement). Poor phase matching is reflected in low magnitude in the summed complex data (arrow 1). Phase images reconstructed with the MCPC-3D methods have high SNR throughout, excellent grey-white matter contrast (see arrow 2), and show good uniformity in magnitude images extracted from the summed complex data.

Figure 5.

Histograms of the ratio of the magnitude of summed complex data to the sum of individual channel magnitudes for “No phase correction,” Phase Difference, Phase Filtering, MCPC-C and MCPC-3D-I for the healthy subject (S1) and five patients (S2-6) for the 8-channel acquisitions (Scan 2), demonstrating the quality of phase matching (perfectly matched phase yields a complex sum signal with magnitude close to 100% of the sum of the individual magnitude signals). With no phase matching, the distributions of the length ratio metric are broad and peak at low ratios. MCPC-C improves phase matching, but much better phase matching is achieved with Phase Difference, Phase Filtering, and MCPC-3D-I, which all have average values, over all voxels, of 92–94%.

Histograms showing the ratio of the magnitude of the summed complex data to the sum of individual channel magnitude values are illustrated in Fig. 5 for unmatched phase data “No phase correction,” Phase Difference, Phase Filtering, MCPC-C, and MCPC-3D-I for the healthy subject (S1) and five patients (S2-6) [8 channel data, in-brain voxels as defined by masks calculated from magnitude images using FSL's Brain Extraction Technique (27)]. These allow quantification of the relative performance of all approaches. With no phase matching, distributions were broad, and had an average and standard deviation value over subjects (of the average value of the ratio for each subject) of 25.0 ± 3.2%. MCPC-C led to some improvement in phase matching, with the average value over subjects equal to 63.1 ± 10.0%. Phase Filtering gave very good results for S1, S2, S3, and S6 but less satisfactory performance for S4 and S5 (but still better than MCPC-C in all cases). For these patients, a number of unwrapping errors, combined with smoothing, led to a small percentage of poorly matched voxels. Phase difference and MCPC-3D-I gave excellent matching for all subjects. The mean value of the combined length metric was in the range 92–94% for the Phase Filtering, Phase Difference, and MCPC-3D-I methods.

The total acquisition time, data size, and processing times are listed in Table 1 for each method, along with phase image noise and grey-white contrast to noise ratios, assessed for the healthy subject with the 24-channel coil. Data storage requirements were approximately twice those for the Phase Filtering, MCPC-C and MCPC-3D-II methods. In reconstruction, the fastest method was MCPC-C method at 410 s, followed by Phase Difference at 1217 s and MCPC-3D-II at 1778 s. Phase filtering and MCPC-3D-I were both at least three times slower due to the need to unwrap high resolution, separate channel data. MCPC-3D-II was more than five times faster than MCPC-3D-I, the time saving being due to having only to unwrap low resolution single-channel data (Scan 3). Noise was similar in the Phase Difference, Phase Filtering, and MCPC-3D methods (in the range 0.071–0.077 ± circa 0.01 rad), and higher and more variable for MCPC-C (0.169 ± 0.177 rad). The grey-white matter contrast to noise ratio was higher for Phase Filtering, MCPC-3D-I, and MCPC-3D-II (3.95 ± 1.84, 3.90 ± 1.05 and 3.38 ± 1.02, respectively) than for the Phase Difference method (2.04 ± 0.29). MCPC-C had the lowest grey-white matter contrast to noise ratio, at 1.61 ± 0.27.

Table 1. Acquisition Times, Data Storage Requirements, Processing Times, and Phase Image Properties (Noise and Grey-White Matter Contrast to Noise Ratio) for the Methods Being Compared (Healthy Subject, 24 Channel Coil)
 Data acquiredProcessing time (s)Phase noise (rad/s)GW CNR
Acquisition timeSize (GB)
  1. Acquisition times, data sizes, and processing times are given for the data required for each method, which is Scan 1 for Phase Filtering and MCPC-C, Scan 2 for Phase Difference and MCPC-3D-I, and Scans 1 and 3 for MCPC-3D-II. To allow image properties to be compared between reconstructions approaches, the same data—Scan 2—was used for assessment of Phase Noise and GW CNR for all five methods.

Phase difference13′20″14.212170.071 ± 0.0082.04 ± 0.29
Phase filtering13′20″7.161490.077 ± 0.0153.95 ± 1.84
MCPC-C13′20″7.14100.169 ± 0.1771.61 ± 0.27
MCPC-3D-I13′20″14.293390.075 ± 0.0163.90 ± 1.05
MCPC-3D-II13′47″7.217780.074 ± 0.0183.38 ± 1.02

DISCUSSION

We have presented a new method for the combination of phase images from multi-channel RF coils, which does not require information from a volume reference coil. As such it is ideally suited to use with high and very high field systems where there is no integrated body coil. Three dimensional phase offsets for each RF channel are estimated from a dual-echo scan, either integrated into the high-resolution gradient-echo sequences (MCPC-3D-I) or derived from an additional fast, low-resolution dual-echo acquisition (MCPC-3D-II). This allows direct summation of wrapped phase images from individual coils without filtering.

Multiecho GE imaging is increasingly being used for phase imaging and susceptibility-weighted imaging due to the opportunity to improve SNR by combining data from a number of echoes, and because it allows T2* mapping from the same data (28). In most applications, no additional time is required to acquire a second echo for the MCPC-3D-I method, as a long TE is generally used for phase imaging to achieve T2* weighting. SNR in MCPC-3D-reconstructed phase images would be expected to be maximized by acquiring the first echo with the shortest possible TE, to minimize noise in phase offset maps. However, phase offset maps have no high-frequency features (other than the object boundary) and may be heavily smoothed, so do not introduce substantial noise into the images being combined. Given that gradient-echo sequences used for phase imaging generally have low receiver bandwidth, the minimum echo spacing is typically in the range 3–8 ms, even the lower end of which is adequate for creating low-noise phase offset maps. However, the use of very low bandwidth acquisitions at very high field might preclude the acquisition of a prior echo the MCPC-3D-I method. In this case, the MCPC-3D-II method can be applied.

MCPC-3D-II requires an additional 20–30 s for the acquisition of low-resolution dual-echo data, but has the advantage over MCPC-3D-I that it is much more computationally efficient to process. There are also advantages to single-echo acquisitions over multi-echo acquisitions. A single echo can be acquired with a low bandwidth, yielding high SNR compared to a one echo from a multiecho series. Single-echo scans are also less acoustically challenging, which can be a criterion for tolerance in patient studies. Although the data for calculation of phase offset maps are acquired separately, MCPC-3D-II would be expected to be robust to moderate motion between the acquisition of the low-resolution and the high-resolution scans given that the two are co-registered, and the phase offset maps vary slowly in space. Low-resolution data for phase offset maps can be acquired with high SNR in a short space of time using echo times typical in field mapping—e.g., 5 ms and 10 ms. The use of a higher acceleration factor than for the high-resolution data (GRAPPA 4) led to no discernable artifacts or SNR problems in phase offset maps. No consistent differences were apparent between the quality of phase images calculated with the MCPC-3D-I and MPCP-3D-II methods.

The 3D phase offset maps calculated here, in accordance with theory, had no residual B0 variation and were consistent with the estimated RF wavelength in tissue at 300 MHz. Corrected images were well phase-matched both at the image level (Figs. 2d and 3e) and when the magnitude of the combined complex vectors were examined (Fig. 5). Combined phase images had high SNR throughout (Fig. 4, center and right columns).

The constant offset subtraction method (MCPC-C) (21) led to destructive interference between complex signals and poor SNR in some regions (Fig. 4) because the assumption that phase offsets are constant throughout images is flawed. The 3D phase offset maps presented here demonstrate wide variation in phase offset over the object. The paper in which the method was introduced (21) showed results for an 8-channel coil. We find that SNR is low in some areas for 8-channel MCPC-C images, but that the shortcomings of the MCPC-C method became clear for 24 channel data: magnitude images showed regions of complete signal nulling, and loss of all image features at corresponding positions in phase images. The same data reconstructed with the MCPC-3D method (and the Phase Filtering method) showed excellent grey-white matter contrast, high SNR throughout, and allowed small veins to be visualized.

MCPC-C relies on signal overlap between channels, where the other methods examined (including MCPC-3D) do not. MCPC-3D can, therefore, be applied to coils arranged in a line, or small coils in which the signal drop off with depth may be such that, even if they are arranged around an object (e.g. on the surface of a cylinder or sphere), there may be no mutual point of detection.

The Phase Difference method took a similar amount of processing time to MCPC-3D-II and showed excellent, robust phase matching. Noise levels in reconstructed images were similar to those in MCPC-3D images, but grey-white matter contrast to noise was substantially lower, wherein lies the principal disadvantage of this approach. With the sequence used here, contrast in Phase Difference images could be improved slightly by acquiring the first echo earlier. The shortest first echo time possible with the bandwidth used here is 6.2 ms, which would increase the echo spacing from 7.0 to 8.8 ms, which would be expected to modestly increase GM-WM CNR.

The Phase Filtering method, which like MCPC-3D-I requires the unwrapping of separate channel high-resolution data, was slower than MCPC-3D-II by a factor of 3.5, and yielded images with similar noise and contrast properties to the MCPC-3D methods, although it was more affected by phase unwrapping errors (because of high-pass filtering), so phase matching was less satisfactory for some voxels than MCPC-3D or the Phase Difference approach.

Another solution to the calculation of combined phase images in the absence of reference coil data is offered by the PO-SENSE method (14). PO-SENSE is an extension to phase of the regularized in vivo SENSE method of Lin et al. (29). The authors have demonstrated that phase images can be reconstructed with high SNR and excellent grey-white matter contrast, for a single subject, with a 32-channel coil at 3 T and with an 8-channel coil at 7 T. The principal disadvantages of this approach are its mathematical complexity and computational load. It has also not yet been fully tested. Coil sensitivity estimates are based on estimated reference phases, which are the complex sum over uncorrected channel phases. It remains to be seen whether this approach works at 7 T and higher field strengths with a large number of detectors, where we have shown that the complex sum of signals (even matched with a constant correction (MCPC-C)) yield ill-defined phase corresponding to regions of complete signal cancellation (Fig. 4). The authors have demonstrated the method with an 8-channel coil at 7 T (where we find MCPC-C to perform suboptimally, but not lead to complete signal cancellation) but not with a coil with a large number of elements—a regime that would seem likely to confound the PO-SENSE approach. No calculation times are given by Chen et al. (14), but the analogous magnitude method of Lin et al. (29) required of the order of one hour per slice for a modest 256 × 256 imaging matrix. Although computer hardware has improved since that study, it seems likely that regularization optimization approaches will remain too computationally demanding to be realized as on-console methods in the near future.

Data storage requirements are the same for the MCPC-3D-II method as for the Phase Filtering method (neglecting the additional low-resolution scan). MCPC-3D-I, being a dual-echo method, requires twice the storage space needed for Phase Filtering. In the MCPC-3D methods, however, once combined phase images have been calculated, the separate channel data can be dispensed with. High-pass filtering is carried out on the combined image rather than separate channel images, which means that the effect of different spatial filters can be investigated, computationally efficiently, without recourse to the separate channel data.

The most computationally demanding step in both the Phase Filtering and MCPC-3D methods is phase unwrapping. The MCPC-3D-II is very fast in this regard, as only the low resolution separate channel data has to be unwrapped. With the increasingly fast and reliable unwrapping methods that have recently become available (e.g. Refs.17, 30), MCPC-3D-II could be implemented as an on-console reconstruction method. This would make phase imaging more clinically accessible, and eliminate data storage and transfer problems inherent to offline processing of high-resolution multi-channel data.

In summary, we have demonstrated the effectiveness of a new, simple method for the combination of phase images from multi-channel coils without the need for a volume reference coil or prior processing such as spatial filtering. The method is based on the calculation and subtraction of 3D phase offsets for each channel which are derived from a dual-echo scan. This data comes either from the high-resolution scan (MCPC-3D-I) or is acquired in a separate, fast, low-resolution scan (MCPC-3D-II). MCPC-3D-I generally requires no additional acquisition time. MCPC-3D-II is computationally efficient, can be applied to single-echo high-resolution data, and requires only a short (circa 30 s) additional measurement. Both MCPC-3D-I and MCPC-3D-II are compatible with parallel imaging.

Acknowledgements

This method was filed with the Austrian Patent Office (A 986/2010). The authors thank the anonymous reviewers for constructive suggestions.

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