Phase Laplacian Coil Combination
Rüdiger Stirnberg1, Lino Lemmer1, and Tony Stöcker1,2

1MR Physics, German Center for Neurodegenerative Diseases (DZNE), Bonn, Germany, 2Department of Physics and Astronomy, University of Bonn, Bonn, Germany


We propose a novel, computationally efficient coil combination technique for multi-channel phase data based on the Laplacian of single-channel phase images. This renders explicit knowledge or estimation of the receive sensitivities unnecessary. The combined phase Laplacian can be either be transformed back to unwrapped phase domain (Laplacian-based unwrapping) or directly utilized for further analyses based on the phase Laplacian, e.g. harmonic background-field removal. At 3T we demonstrate similar-to-improved phase reconstruction compared to the vendor-provided state-of-the-art coil combination, which uses the body-coil as a uniform reference, and successfully apply the technique at 7T.


To obtain an accurate phase representation of MR images, one optimally combines phased array data by a weighted sum of all complex single-coil images, essentially using the complex conjugate receive sensitivity (RX) profiles as weights1. In the absence of a homogenous “body coil”, typically used to estimate the RX profiles, virtual reference coil methods tend to fail at ultra-high fields (e.g. open ended fringe lines)2. Many state-of-the-art phase analysis techniques, however, require only the Laplacian of the phase field (LAP), e.g. (V)SHARP3,4, HARPERELLA5, LBV6, Laplacian-based phase unwrapping (LU)7,8 or improved preconditioned conjugate gradient Laplacian unwrapping (pccgLU)2. We propose to perform multi-channel phase combination efficiently in the LAP domain, rendering knowledge of receive sensitivities unnecessary.


Let $$$\phi_c = \alpha + \beta + \gamma_c + \eta_c$$$ be the phase image acquired by coil $$$c$$$ of the phased array. The first two terms are the internal, $$$\alpha$$$, and the background phase, $$$\beta$$$ (incl. RF transmission phase9). Coil-specific contributions are represented by the RX phase, $$$\gamma_c$$$, and phase noise, $$$\eta_c\in(-\pi,\pi]$$$. We assume that $$$\gamma_c$$$ is largely harmonic, i.e. any non-harmonic contributions are negligible or cancel partially by a weighted sum across all single-coil phase Laplacians:

(1) $$$\quad \Delta\Phi = \sum_c w_c\Delta\phi_c = \Delta(\alpha+\beta) + \sum_c w_c \Delta\eta_c + \sum_c w_c \Delta\gamma_c$$$

To minimize total noise, we propose to employ normalized weights according to squared coil magnitudes (Gaussian-smoothed, 1 voxel kernel width): $$$w_c = m_c^2/\sum_c m_c^2$$$. The last term in Eq. (1) can be regarded as a residual (non-harmonic) RX term.


Five healthy, young subjects underwent two consecutive whole-brain 3D-GRE measurements on a Siemens MAGNETOM Skyra 3T scanner. The latest software version (VE11C) provided state-of-the-art complex coil combination (32 channel head coil) utilizing the body coil as a reference. Imaging parameters were: 232x256x160 matrix, 0.9mm iso, GRAPPA R=2x1, TE=20ms, TR=27ms. Additionally, one subject was scanned in a Siemens MAGNETOM 7T research scanner using a comparable 32 channel coil, but lacking a body coil. A motion-robust dual-echo 3D-GRE-EPI sequence was employed10 (240x240x160 matrix, 0.8mm iso, GRAPPA R=3x1, Partial Fourier 7/8x1, TE1=11ms, TRvol=45s, 9 averages).


The Laplace operation was implemented as $$$\Delta\phi = \text{div}(b\ \text{grad}(\phi))$$$, where $$$b$$$ is a brain binary mask2 and $$$\phi$$$ can be the wrapped scanner-combined phase or a wrapped single-channel phase. For 3T(7T) data, $$$b$$$ was obtained by eroding a BET11 mask (from the root-sum-of-squares magnitude image) by 2(6) voxels. At this stage, $$$b$$$ is required only for pccgLU (here with 5 iterations). For simplified background-field removal (SHARP)4, the BET mask was eroded by 4(12) voxels and applied to LAP before inversion (non-regularized Laplace inversion based on discrete cosine transforms7).

For the vendor-provided (“reference”) and the proposed coil combination (“laplacian”) the standard deviation of the LAP and SHARP noise distribution was evaluated in two regions-of-interest (ROI) from the difference12 between two separate LAP and SHARP maps. One peripheral brain ROI (“high SNR”, right occipital lobe) and one central brain ROI (“low SNR”) were defined on comparable, homogeneous white matter regions in all five subjects (5.4mm spherical ROI radius).


Fig. 1 illustrates, on one example subject and slice at 3T, the “reference” and the “laplacian” method and shows resultant LAP and SHARP maps. According to Eq. (1), the LAP difference map reveals noise and residual, non-harmonic RX differences between the “reference” and “laplacian” LAP maps (note different windowing). Group statistics at 3T, summarized in Fig. 2, indicate comparable noise propagation using either method. The “laplacian” method even seems to result in reduced noise, although the group size does not allow for statements on significance. The corresponding example (pccg)LU maps and the background field (“BG=pccgLU-SHARP”), displayed in Fig. 3, show very high agreement as well. Fig. 4 demonstrates that the proposed method can readily be applied at at ultra-high fields.


We have proposed a novel multi-channel phase combination technique based on phase Laplacian images rather than on raw phase images. We have demonstrated similar or improved phase reconstruction (with or without background-field) compared to reference-based complex coil combination at 3T. The proposed method does not require a real or virtual reference coil. We could thus successfully apply it to 7T data. Future investigations will increase the group size and include virtual body coil methods in our comparison at 3T and 7T. Although only simplified background-field removal has been demonstrated here, the proposed method is also perfectly suited for advanced methods such as V-SHARP3 or for solving the Laplacian boundary value problem (LBV)6.


No acknowledgement found.


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Illustration of the "reference" and the proposed "laplacian" coil combination at 3T. LAP = phase Laplacian map; SHARP = simplified background-free internal phase map (single Laplace inversion applied to masked LAP). All figures, except first column and LAP difference, are windowed from -π (black) to π (white). The LAP difference is windowed from -π/10 to π/10.

Group results of LAP and SHARP noise standard deviation drawn from comparable "high SNR" and "low SNR" ROIs in five subjects using the "reference" and "laplacian" coil combination. Smaller colored bars indicate reduced noise propagation. Shorter error bars suggest more consistent results across subjects.

Results of Laplacian unwrapping = LU (single Laplace inversion applied to LAP) and preconditioned conjugate gradient Laplacian unwrapping = pccgLU (5 iterations starting from LAP) at 3T. The background field (BG) corresponds to SHARP (Fig. 1) subtracted from pccgLU. All figures are windowed from -π to π.

Results of the proposed "laplacian" method applied to 3D-GRE-EPI data acquired at 7T. All figures are windowed from -π to π.

Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)