Yusuf Ziya Ider^{1}, Gokhan Ariturk^{1}, and Gulsah Yildiz^{1}

Clarification of the contrast resolution (CR) and spatial resolution (SR) limits of phase based MREPT for conductivity imaging is essential for assessing its success success in clinical applications. Noise analysis of conventional phase based MREPT is performed to find the SNR needed for the MR sequence used for measuring B1-phase. It is found that with 1000-2000 SNR values about 0.01 S/m CR can be achieved. For SR evaluation, generalized phase based MREPT, which does not suffer from internal boundary artefacts, is considered. It is found by phantom experiments that 3.5mm spatial resolution is easily obtained with the state-of-art MR methods.

For CR evaluation we have considered conventional phase based MREPT (method1) which has the formula $$$\sigma = \frac{1}{w \mu_0}\triangledown^2 \phi$$$ where $$$\phi$$$ is the transmit B1-phase^{4}. Given the approximation to the Laplacian on a
Cartesian grid, $$$[\triangledown
^2(\phi)]_{i,j,k} = \frac{(\phi_{i,j,k+1} + \phi_{i,j,k-1} + \phi_{i,j+1,k} +
\phi_{i,j-1,k}+ \phi_{i+1,j,k} + \phi_{i-1,j,k}) -6\phi_{i,j,k} }{(\triangle
x)^2}$$$, one can find that $$$s_\sigma
= \frac{1}{w \mu_0} \sqrt{\frac{6s_\phi ^2}{(\triangle x)^4} +\frac{36s_\phi
^2}{(\triangle x)^4}} = \frac{1}{w\mu_0}\frac{6.5s_\phi}{(\triangle x)^2}$$$ where $$$s_\phi$$$ is the stdev of error in the the conductivity estimate and $$$s_\sigma$$$ is the stdev of noise in the phase. It is known that $$$s_\phi=\frac{1}{SNR}$$$ where SNR is the signal-to-noise ratio of MR magnitude image^{3}. Therefore $$$s_\sigma
= \frac{1}{w\mu_0}\frac{6.5}{(\triangle x)^2} \frac{1}{SNR}$$$. Use of low pass filtering is often necessary to obtain reasonable images. If the Laplacian and the filter are cascaded then the coefficients of the overall system are to be used. In general $$$s_\sigma = \frac{1}{w\mu_0}\frac{\alpha}{(\triangle x)^2} \frac{1}{SNR}$$$ where $$$\alpha$$$ is 6.5, 0.14 or 0.21
for the cases of only Laplacian, only filter (Gaussian with N=5) or both
respectively. NxNxN Gaussian filter kernels used in this study are designed to have FWHM=N/2.

For SR evaluation generalized phase based MREPT (method2) is used because it eliminates internal boundary artefacts^{1,2}. This method achieves conductivity reconstruction by solving the equation $$$\triangledown\phi.\triangledown\rho+ (\triangledown^2 \phi)\rho=2\omega\mu_0$$$ where $$$\rho$$$ is resistivity and $$$\phi$$$ is tranceive phase. For regularization purposes a diffusion term, $$$c\triangledown^2\rho$$$, is added to the LHS where the diffusion constant $$$c$$$ is experimentally found.

For experiments two cylindrical agar phantoms are used: SR-phantom (Figure 2) and PSF-phantom (Figure 4). SSFP sequence is used to measure B1-phase. Quadrature QBC body coil is used for transmission. For reception, QBC body coil is used for the PSF-phantom and 4-channel head coil is used for the SR-phantom.

Figure 1 shows the dependence of $$$s_\sigma$$$ on N and SNR. It is observed that for voxel size of 2mmX2mmX2mm, $$$s_\sigma$$$=0.09S/m, and with N=5 we need an SNR of 600. If N=7 is used an SNR of 200 is sufficient. For N=9 and with SNR=200, $$$s_\sigma$$$=0.04S/m can be achieved. If $$$s_\sigma$$$=0.01S/m is required we need to have SNRs of 200, 300, 700, and 1700 for kernel sizes of 13, 11, 9, and 7 respectively. In 3D SSFP experiments (Figure 2) we have observed SNRs of 800 and 1600 in agar and non-agar regions where the surface coil is relatively more sensitive. In 2D SSFP experiments (Figure 4) SNRs of 200 and 400 are achieved in agar and non-agar regions uniformly across the object.

Figures 2 and 3 show the experimental results obtained with the SR-phantom. With method1 it is not possible to evaluate SR because of the internal boundary artefacts. With method2 however the 4.5mm diameter small objects are well reconstructed. The 3-3.5 mm separations between some of the objects are also identifiable. However in this case the conductivity drop in between the objects is less than 50% of the contrast of the objects themselves. With these results we can conclude that spatial resolution is about 3.5 mm.

Results with the PSF-phantom are given in Figure 4. The large homogeneous anomaly in the phantom acts as step input. The profile plot shows a sharp jump in reconstructed conductivity. When the derivative is taken the PSF is obtained. The FWHM value is about 3 pixels corresponding to 3X1.2mm=3.6mm.

We have observed that contrast resolution of 0.01S/m can be achieved with SNR such as 1700 which is experimentally possible. However if lower SNRs are obtained, then larger filter kernels must be used which means a larger homogeneous region is necessary to estimate the conductivity. For example with N=13, Gaussian filter's FWHM value is (13/2)x2mm=13mm. Regarding spatial resolution we have observed that with the experimental conditions we have in this study a spatial resolution of around 3.5 mm is indicated.

This study is supported by TUBITAK 114E522 grant. MR experiments are conducted in UMRAM, Bilkent, Ankara.

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Figure 1: Standard deviation of reconstructed sigma as dependent on MR magnitude SNR and kernel size, N, of a Gaussian filter. Gaussian filter is realized by the "fspecial3.m" function of Matlab (Mathworks, Natick, MA) which adjusts it variance so that the FWHM value becomes N/2. Results are given for a voxel sizes of 2mm X 2mm X 2mm, and 1.5mm 1.5mm X 1.5mm on the left and right respectively.

Figure 2: 3D-SSFP magnitude (top left) and phase (top right) images of the spatial resolution phantom at central slice for one of the receive coils. Sequence parameters are: 128x128x30 image size, TE/TR = 2.35ms/4.7ms, F.A.=40^{o}, 1.56mmx1.56mmx1.56mm voxel size, NEX=32) The phantom has 4.5 mm and 7 mm holes filled with (6 g/L NaCl, 0.2 g/L CuSO4) solution expected to yield 1.5 S/m conductivity. Background is agar with composition (20 g/L agar, 2 g/L NaCl, 0.2 g/L CuSO4) yielding a conductivity of 0.5 S/m. Conventional phase based (lower left) and generalized phase based (lower right) MREPT conductivity images are also shown.

Figure 3:
Profile plots of generalized phase based mrept conductivity reconstruction for the phantom described in Figure 2. Profile lines are shown both on the SSFP magnitude image and also on the conductivity image. Profiles on the 1st, 2nd, 3rd, and 4th lines numbered from left to right, are shown in the inserts middle top, middle bottom, right top, and right bottom respectively.

Figure 4: Point Spread Function analysis using data from an agar phantom with large conductive anomalies. Background is the same as in Figure 2 and anomalies are expected to have a conductivity of 1 S/m. Top left: Reconstructed conductivity using generalized phase based MREPT. Top right: SSFP magnitude image. Lower left: Profile plots drawn for the profile line shown in the upper images. Bottom right: Derivative of the profile plots on the left to get an indication of the PSF of the system. 2D-SSFP sequence parameters are: 128x128 image size, TE/TR = 2.4ms/4.88ms, F.A.=40^{o}, 1.2mmx1.2mmx3mm voxel size, NEX=32.