Phase based MR-EPT methods mainly use the fast balanced SSFP sequence to obtain B1 transceive phase distribution. The contrast of CSF in SSFP magnitude images may be 6-10 times the contrast of brain tissue.¹ This high contrast ratio causes Gibbs ringing artifact, especially at CSF-tissue boundaries. This artifact manifests itself as phase spikes and oscillations at and near tissue boundaries. These phase artifacts severely obstruct phase based conductivity reconstruction methods that use any order of derivatives of the phase image. This study proposes a k-space apodization method to remove phase artifacts and elucidates its use in MR-EPT.

Defining three regions (A), (B) and (C) whose conductivities are 1.5S/m, 1S/m and 0.5S/m respectively inside a simulation phantom, as shown in Figure1, and simulating it in a birdcage coil model --using Comsol Multiphysics--, we retrieved the transceive phase values of the B1 field inside the phantom with a space increment of 2 mm in each direction. This phase is taken as the phase of the spatial MR image to simulate the phase obtained by an SSFP sequence. To simulate the magnitude of the SSFP image, the contrast ratios of the regions (A)/(C) and (B)/(C) are assigned as 10 and 6, respectively. Gibbs artifact was created by the truncation of the higher frequency Fourier components of the simulated image with frequencies above 63.5% of half the sampling frequency. Apodization is achieved by multiplying the k-space data by a 3-D Kaiser window whose bandwidth is determined by the β parameter.² For the phantom MRI experiments, we used the Siemens Tim Trio 3T MR scanner with a quadrature body coil and an 12-channel receive only phased array head coil. The phantom background is prepared using an agar/saline solution (20gr/L Agar, 2gr/L NaCl, 0.2gr/L CuSO₄ ) and four higher conductivity regions are prepared using only saline solution (8.8gr/L NaCl; 0.5gr/L CuSO₄ for left two regions in Figure4(a) and 1gr/L CuSO₄ for right two regions in Figure4(a)). CuSO₄ is used for increasing the magnitude contrast in the higher conductivity regions. Left two of these regions, shown in Figure4(a), have approximately 4.5 and the right two regions have approximately 7.5 times the background magnitude contrast. MRI scan parameters are: FOV=256mm, 2x2x2mm, FlipAngle=65°, TE/TR=2.21ms/4.42ms. Results of two different MR-EPT methods are evaluated. The first method “Basic Phase Based MR-EPT”, which is described in ^3,4,5, is incapable of reconstructing the conductivity at regions where conductivity varies (internal conductivity boundaries) and is based on the following equation:$$\sigma=\nabla^{2}\phi^{tr}/(\omega\mu_0)$$

The second method, “Generalized Phase Based EPT”, is capable of reconstructing the conductivity at internal conductivity boundaries. This method is based on the solution of the following equation:

$$-c\nabla^2\rho+\nabla\phi^{tr}\cdot\nabla\rho)+\nabla^{2}\phi^{tr}\rho-2\omega\mu_0=0$$In the above equations, $$$\rho=1/\sigma$$$(resistivity), $$$c$$$ is the constant diffusion coefficient, $$$\phi^{tr}$$$ is the measured MR transceive phase, $$$w$$$ is the Larmor frequency, and $$$\mu_0$$$ is the free space permeability. Details of the second method can be found in^6.

Gibbs artifacts in simulated data, caused by truncation, are observed in Figure2(a,b,c). Especially on the (A-C) boundary, random phase spikes are observed. The effect of arbitrary jumps in the phase image, as shown in Figure(2c) yields perturbed conductivity reconstructions by both algorithms, as shown in Figure3(a,b). Figure2(d,e,f) depicts the suppression of the Gibbs artifacts in phase and magnitude images when apodization is used with Kaiser window, β=3. Much more accurate conductivity reconstructions are obtained with the proposed apodization operation, as shown in Figure3(c,d). Table1 depicts the error values, calculated using the L² - norm, on conductivity and phase reconstructions of the simulation phantom for different β values. It is observed that with increasing β, the phase error rapidly decreases and it starts levelling at a certain β value. The conductivity error attains its minimum at that particular β value. This specific β value, being the optimal one for the cases covered in this study, is found as 3. Non-apodized and apodized phantom SSFP magnitude and phase images, are demonstrated in Figure 4(a,b,e,f). The effects of ringing artifacts, shown in Figure4(b,c,d), are reduced with the apodization operation, as shown in Figure4(f,g,h). Being compatible with the simulation results, highly corrupted regions encircled in Figure4(d) are properly reconstructed in Figure4(h), when the apodization is used.Results of both conductivity reconstruction algorithms are enhanced with the use of the proposed phase artifact elimination method. However, this method has a low pass filter effect, which reduces the original resolution of the image. It is suggested that researchers using MR-EPT algorithms for conductivity reconstruction find the optimal Kaiser window bandwidth parameter β for their specific data, using the procedure stated in the “results” section.