Electrical properties tomography (EPT) is a method to calculate the conductivity and permittivity of a material from measured $B_1^+$ fields using MRI. These calculations result in high noise amplification, warranting the use of high signal-to-noise ratio (SNR) images. Multi-channel receivers can be used to improve SNR of the $B_1^+$ images, but the individual coil data must be combined while leaving the image phase intact. Combining coil data prior to EPT calculations provides the benefit of eliminating phase artifacts in areas of low signal, which would lead to highly inaccurate electrical properties. Current approaches to combining phase data include SENSE-like methods¹ and methods using some sort of reference coil^2,3. We present an adaptation of coil compression^4,5 to optimally combine the $B_1^+$ data such that the phase is preserved for EPT. This method requires no reference images or sensitivity maps. We demonstrate the SNR gains and reconstruction efficacy with phase-based conductivity mapping⁶: $$\sigma = \frac{1}{\omega \mu_0}\nabla^2\phi^+$$ where σ is conductivity, ω is the scanner frequency, µ₀ is vacuum permeability, and φ+ is the phase of $B_1^+$.

Multi-channel array data was compressed into one virtual coil using the singular value decomposition (SVD), which provides the globally optimal combination of k-space data. The procedure for a fully-sampled slice with dimensions [nx, ny] from a N-channel array is as follows:

1. Perform a 2D IFFT for each coil image and place into a matrix X, which is size [N, nx*ny], such that each row is the vectorized k-space data from a different coil.

2. Calculate the SVD: X = UΣV^H. Keep only the first column of U, the first element of Σ, and the first row of V^H. Multiply these truncated matrices to form the virtual coil k-space, X_V1.

A spatially varying combination can be calculated in a similar manner. This requires an alignment of the singular vectors to ensure the compressed image is smooth, as described by Zhang⁵. This local compression technique is called geometric decomposition coil compression with alignment, GCC-Align in this work.

The phase from each compressed coil was divided by two to approximate the transmit phase for conductivity mapping, calculated using a model-based method⁷.

Data was acquired for a saline phantom and a healthy control subject’s brain, each with an 8-channel receiver (GE Healthcare, Waukesha, WI) and a 32-channel receiver (Nova Medical) on a GE Discovery MR750 3.0T scanner. For the phantom, a spin echo (SE) scan was performed with TE/TR = 11/1000 ms, FOV = 24cm x 24cm, 256x256 pixels, 3mm slices. Nominal values of the phantom were measured using the coaxial probe method⁸. For the brain, a SE scan was performed with TE/TR = 14/1000 ms, FOV = 24cm x 24cm, 256x256 pixels, 3mm slices.

Noise levels in the virtual coils were compared in a uniform spherical phantom using the same acquisition parameters as the saline phantom. The noise variance was calculated as the variance of the residuals after fitting a parabolic surface to phase data in a 15x15 pixel sliding window.

Figure 1 shows that while both compression techniques provide substantial noise reduction in the phase images, the GCC-Align method provides a more uniform noise profile across the object. Figures 2 and 3 show representative coils from the multi-channel arrays and the respective coil combinations for a saline phantom. The arrow in Figure 3 shows an open-ended fringe line in a low signal region. This problematic feature is eliminated in the virtual coil. The phantom conductivity maps are shown in Figure 4, along with the measured values. Human subject conductivity maps are shown in Figure 5. Compressing the SE image data to a single virtual coil decreased the noise variance by at least an order of magnitude. For a small increase in computation, we create a more uniform noise profile with the GCC-Align virtual coil. The conductivity maps shown in Figure 4 show values consistent with those measured, but the GCC-Align images have less variation farther from the coil. While true values for the brain are not available, the higher conductivity of the cerebrospinal fluid is easily identified. The GCC-Align image shows more uniformity in the white matter, as evidenced by fewer black patches, due to reduced phase noise.We have applied the idea of multi-channel compression to virtual coils for the purpose of high SNR phase maps for EPT. This increases the accuracy of the reconstruction, eliminates the need for reference coils in phase combination, and can be done for the entire FOV or in a locally-varying manner.