Generalized Phase based Electrical Conductivity Imaging

Necip Gurler^{1} and Yusuf Ziya Ider^{1}

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.

[1] Voigt T et al. Magn. Reson. Med. 2011;66(2):456-466

[2] Van Lier et al. Magn. Reson. Med. 2012;67:552–561

[3] Katscher U et al. Comput. Math. Methods Med. 2013;2013:546562

[4] Hafalir et al. IEEE Trans. Med Imaging, 2014;33(3) 777-793

[5] Gurler et al. ISMRM22(2014):3247

[6] John V et al. Comput Method Appl M 2007;196:2197-2215

[7] John V et al. Comput Method Appl M 2008;197:1997-2014

[8] Gurler et al. Concepts Magn Reson Part B 2015;45:13-32.

Fig. 1. Birdcage coil model loaded with the head phantom
used in the simulations

Fig. 2. Reconstructed conductivity maps of the human
head simulations under different SNR values. (a) actual conductivity map (b) using
conventional phase based EPT $$$(\sigma=(\nabla^2\phi^tr)/(2\omega\mu_0)$$$ plus 3x3x3 median filter) (c) using generalized phase based EPT (c=0.05)

Fig. 3. Experimental phantom results: (a) selection of
the ROI on the combined magnitude image, (b) magnitude image in the ROI for one
channel. Reconstructed conductivity maps for the same channel (c) using conventional phase based EPT $$$(\sigma=(\nabla^2\phi^tr)/(2\omega\mu_0)$$$ plus 3x3x3 median filter) (d)
using conventional phase based EPT (c=0.05)

Fig. 4. In vivo results: (a) selection of the ROI on the combined magnitude
image (b) magnitude image in the ROI for one channel. Reconstructed
conductivity maps for the same channel (c) using conventional phase based EPT $$$(\sigma=(\nabla^2\phi^tr)/(2\omega\mu_0)$$$ plus 3x3x3 median filter) (d) using conventional phase based EPT (c=0.05)

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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