It has been shown that electrical conductivity ($$$\sigma$$$) can be imaged using only MR phase^1-3, without B1-mapping. This approach (called “phase based EPT”) also does not use the transceive phase assumption (TPA), i.e. approximating the transmit phase as half of the transceive phase, and therefore, it can be used for any transmit-receive coil configuration. With these advantages, phase based EPT seems to be the most suitable EPT method for clinical applications. However, in its current form, boundary artifact and low SNR issues precludes the practical application of this method. In this study, we have proposed a new formulation for phase based EPT (called “generalized phase based EPT”) to eliminate the boundary artifact and to provide robustness against noise. We have demonstrated the feasibility of the proposed method using simulation, phantom, and in vivo experiments.

Convection-reaction equation based MREPT (cr-MREPT) formula^4-5 can be written in its logarithm form as

$$$\beta^{\pm }\cdot\nabla\ln(\gamma)-\nabla^{2}B^{\pm}_{1}+i\omega\mu_{0}\gamma{B^{\pm}_{1}}=0$$$

where $$$γ=σ+iωϵ$$$,$$$∇ln(γ)=\begin{bmatrix}∂ln(γ)/∂x\\∂ln(γ)/∂y\\∂ln(γ)/∂z\end{bmatrix}$$$,$$$\beta^{\pm}=\begin{bmatrix}\frac{\partial{B^{\pm }_{1}}}{\partial{x}}\mp{i}\frac{\partial{B^{\pm}_{1}}}{\partial{y}}+\frac{1}{2} \frac{\partial{B_{z}}}{\partial{z}}\\\pm{i}\frac{\partial{B^{\pm}_{1}}}{\partial{x}}+\frac{\partial{B^{\pm}_{1}}}{\partial{y}}\pm{i}\frac{1}{2}\frac{\partial{B_{z}}}{\partial{z}}\\-\frac{1}{2}\frac{\partial{B_{z}}}{\partial{x}}\mp{i}\frac{1}{2}\frac{\partial B_{z}}{\partial{y}}+\frac{\partial{B^{\pm }_{1}}}{\partial{z}}\end{bmatrix}$$$

Substituting $$$B^{\pm}_{1}=\left|B^{\pm}_{1}\right|e^{i\phi^{\pm}}$$$ in the above equation, and assuming $$$\nabla\left|B^{\pm}_{1}\right|=0 $$$ yield transmit (or receive) phase based EPT formula

$$$\Omega^{\pm }\cdot\nabla\ln(\gamma)+(((\frac{\partial{\phi^{\pm}}}{\partial{x}})^2+(\frac{\partial{\phi^{\pm}}}{\partial{y}})^2+(\frac{\partial{\phi^{\pm}}}{\partial{z}})^2)-i\nabla^{2}\phi^{\pm})+i\omega\mu_{0}\gamma=0$$$

where $$$\Omega^{\pm}=\begin{bmatrix}{i}\frac{\partial{{\phi^{\pm}}}}{\partial{x}}\pm\frac{\partial{\phi^{\pm}}}{\partial{y}}\\\mp\frac{\partial{\phi^{\pm}}}{\partial{x}}+i\frac{\partial{\phi^{\pm}}}{\partial{y}}\\+i\frac{\partial{\phi^{\pm}}}{\partial{z}}\end{bmatrix}+\frac{1}{B^{\pm}_{1}}\begin{bmatrix}\frac{1}{2} \frac{\partial{B_{z}}}{\partial{z}}\\\pm{i}\frac{1}{2}\frac{\partial{B_{z}}}{\partial{z}}\\-\frac{1}{2}\frac{\partial{B_{z}}}{\partial{x}}\mp{i}\frac{1}{2}\frac{\partial B_{z}}{\partial{y}}\end{bmatrix}$$$

Assuming the gradients of B_z are negligible compared to $$$B^{\pm}_{1}$$$, the second term of $$$\Omega^{\pm}$$$ can be neglected compared to the first term. The rest are the transmit (or receive) phase components, which cannot be measured directly via MRI. To obtain the equation in terms of measurable transceive phase, i.e. $$$\phi^{tr}=\phi^{+}+\phi^{-}$$$, we can sum the transmit and receive phase based formulae, and this gives us

$$$\begin{bmatrix}{i}\frac{\partial{{\phi^{tr}}}}{\partial{x}}+\frac{\partial{(\phi^{+}-\phi^{-})}}{\partial{y}}\\-\frac{\partial{(\phi^{+}-\phi^{-})}}{\partial{x}}+i\frac{\partial{\phi^{tr}}}{\partial{y}}\\+i\frac{\partial{\phi^{tr}}}{\partial{z}}\end{bmatrix}\cdot\begin{bmatrix}∂ln(γ)/∂x\\∂ln(γ)/∂y\\∂ln(γ)/∂z\end{bmatrix}+(k_{real}-i\nabla^{2}\phi^{tr})+i2\omega\mu_{0}\gamma=0$$$

where $$$k_{real}$$$ has no imaginary components. Writing only the imaginary terms, which are related with the conductivity, and assuming that $$$\sigma^2\gg(\omega\epsilon)^2$$$ yields

$$$(\nabla\phi^{tr}\cdot\nabla\rho)+\nabla^{2}\phi^{tr}\rho-2\omega\mu_0=0$$$

where $$$\rho=\frac{1}{\sigma}$$$ (resistivity). This equation is in the form of convection-reaction-diffusion equation, and the coefficients are solely based on the transceive phase. Since the convection term $$$(\nabla\phi^{tr}\cdot\nabla\rho)$$$ dominates the diffusion term (no diffusion term) in our formulation, the solution will have unwanted spurious oscillations near interior and boundary layers^6-7. Therefore, an artificial diffusion term $$$(-c\nabla^2\rho)$$$ is added for the purpose of stabilization, and the governing equation of the generalized phase based EPT method will be

$$$-c\nabla^2\rho+(\nabla\phi^{tr}\cdot\nabla\rho)+\nabla^{2}\phi^{tr}\rho-2\omega\mu_0=0$$$

where c is the constant diffusion coefficient.

Apart from the previous cr-MREPT studies^4-5, the governing equation is solved for $$$\rho$$$ using finite difference scheme. This is important because since the measured transceive phase is already on Cartesian grid, we do not have to generate a grid for the region of interest (ROI). Partial derivatives in the governing equations are directly represented with the central difference formulae. Final matrix equation ($$$A\rho=b$$$) is solved using MATLAB (Mathworks, Natick, MA) by applying Dirichlet boundary condition. For electromagnetic simulations, RF birdcage coil was modeled and loaded with head phantom (see Fig. 1) using COMSOL Multiphysics⁸. The simulations were made at 128 MHz with a voxel size of 2x2x2 mm³. The conductivity maps were calculated using the simulated transceive phase, which is acquired by the summation of $$$B_1^+$$$ and $$$B_1^-$$$ phases of the coil. For the phantom experiment, background was prepared using agar-saline solution (20 g/L Agar, 2.5 g/L NaCl, 0.2 g/L CuSO4), and anomalies were prepared using a saline solution (8.8 g/L NaCl, 0.2 g/L CuSO4). Finally, a healthy 23-years-old male volunteer has been scanned with the approval of Institutional Review Board of Bilkent University. Experiments were conducted on Siemens Tim Trio 3T MR scanner (Erlangen, Germany) using a quadrature body coil and 12-channel receive only phased array head coil. The transceive phase was acquired using 3D balanced SSFP (FA=40 deg, TE-TR=2.33-4.66 ms, FOV=200x200x50 mm³, RES=1.56x1.56x1.56 mm³, coronal, NEX=32, total scan time ~10min for the experimental phantom, and FA=40 deg, TE-TR=2.23-4.46 ms, FOV=210x210x52.5 mm³, RES=1.64x1.64x1.64 mm³, axial, NEX=7, total scan time ~ 2.5min for volunteer study). For noisy simulations and experiments, isotropic Gaussian filter with 5x5x5 voxels and a standard deviation of 1.06 was applied to the phase data.Fig. 2-4 show the reconstructed conductivity results for the simulated head model, experimental phantom, and human experiments, respectively. Using generalized phase based EPT, boundary artifacts were eliminated completely in the phantom experiment, and were significantly reduced in human head simulations and experiments. The generalized phase based EPT method also provides immunity against noise due to the use of diffusion term without significantly blurring internal layers.With the boundary artifact reduction and noise immunity advantages of generalized phase based EPT, and the inherent advantages of the phase based EPT (fast, TPA free, and being suitable for any transmit-receive coil configuration), the proposed method provides fast and reliable electrical conductivity images for clinical applications.