Phase information is used in various magnetic resonance imaging methods, such as Susceptibility-Weighted Imaging (SWI)¹ and Quantitative Susceptibility Mapping.² When acquired with a phased array coil, the phase map of each channel is composed of echo-time-dependent field inhomogeneity and chemical shift terms and an echo-time-independent inherent phase offset term. The latter term originates from B1 field inhomogeneity, coil geometry, and different receiver chain delay and is different in each channel and voxel. Hence, to prevent phase distortion from such inconsistency among channels, the phase offset term needs to be removed before combining phase data from multiple channels. Several strategies such as Multi-Channel Phase Combination using Constant offsets (MCPC-C)³ and Multi-Channel Phase Combination using measured 3D phase offsets (MCPC-3D)⁴ have been suggested to estimate the phase offsets. In this work, we present a method of estimating the phase offsets in multi-echo GRE data, Multi-Channel Phase Combination using all N echoes (MCPC-N).In MCPC-C, phase offsets are assumed to be spatially invariant. The phase data are corrected for a constant phase to make the phase of a center ROI zero.³ In MCPC-3D, voxel-specific phase offset is estimated. The phase of a voxel can be modeled with time dependent term ($$$2πγ∆B_{0_{(x,y)}} TE$$$) and time independent term (i.e. phase offset; $$$θ_{rx,n_{(x,y)}}$$$). Then the phase offset $$$θ_{rx,n_{(x,y)}}$$$ is calculated as follows: $$$\frac{TE_1 θ_{2,n_{(x,y)} }-TE_2 θ_{1,n_{(x,y) }}}{TE_1-TE_2}$$$.⁴ When multi-echo GRE data is acquired, one can use collected echoes to improve estimation of phase offsets. Given m echoes, a model for the phase in a voxel located in (x,y) and in the nth coil at an echo time (TE_m) becomes $$θ_{m,n_{(x,y) } }=2πγ∆B_{0_{(x,y)}} TE_{m}+θ_{rx,n_{(x,y)}}$$ To generate an optimal estimation of the phase offset, a weighted least squares error minimization is performed with the weighting factors that are inverse of variations of noise in order to calculate the phase offset as best linear unbiased estimator. Since the variation of phase noise⁵ is proportional to 1/SNR², SNR² is chosen as a weighting factor.

Data acquisition

2D multi-echo GRE data were acquired at 3T MRI using a 32 channel phased array head coil. Scan parameters were: TR = 2000 ms, TE = 2.7, 5.88, 9.06, 12.24, and 15.42 ms, flip angle = 82°, resolution = 0.9 × 0.9 × 1.0 mm³, and total scan time = 8:34.

Data processing

Phase data were corrected by MCPC-C, MCPC-3D, and MCPC-N. First and second echoes were chosen to generate phase offset in MCPC-3D. In MCPC-3D and MCPC-N, phase data were temporally unwrapped before the estimation of phase offset. After evaluating of the phase offsets, the results were filtered using a 5x5 median filter. The input data for the median filter (i.e. local 5x5 matrix) were columnized and phase maps were unwrapped. MCPC-3D and MCPC-N without the median filter were also tested. In order to quantify the quality of the phase coherence of each result, a Q factor ($$$\frac{|∑Me^{iθ_{corrected} } |}{∑|Me^{iθ} | }$$$)⁶, was calculated for every voxel. SNR was calculated using the magnitude of the complex combined results. To test the effects of SNR in each method, Gaussian noise (3 different levels) was added to uncombined channel data. The second echo was chosen for the evaluation of SNR and Q factor.

Phase offset maps and corrected phase maps calculated by MCPC-3D and MCPC-N, both without and with median filtering are compared in Fig. 1. Without median filtering, a map with MCPC-3D is noisier and the resulting magnitude images show lower SNR in the corrected image compared to MCPC-N. The Q factor maps of MCPC-3D, MCPC-N (both without and with median filtering), and MCPC-C are presented in Fig. 2. MCPC-C was less effective in areas away from the center. Q factor histograms of in-brain voxels are plotted in Fig. 3. The average Q factor for MCPC-C was 0.643 and MCPC-3D and MCPC-N without filtering were 0.954 and 0.979, respectively. After median filtering, these values changed to 0.981 and 0.986, respectively. Hence the MCPC-N shows higher Q values than MCPC-3D although the improvement is relatively small. The effect becomes larger when the SNR of the images becomes low (Fig. 4). For example, the SNR in MCPC-N larger by 56.3% (without the median filter; 19.6% with median filter) than MCPC-3D when the SNR is under 100 (Fig. 4).MCPC-N, which calculates the phase offsets from all echoes, provides more accurate estimation of voxel-wise phase offsets particularly in low SNR.