Anisotropic conductivity tensor imaging in MREIT using directional diffusion rate of water molecules

The pulsed gradient spin echo (PGSE) in figure 1(a) excites the spin system by applying time-constant magnetic field gradients before and after the π-pulse and acquires the k-space data. Due to the dephasing of magnetization because of diffusion in the period Δ, the acquired signal intensity reflects the amount of diffusion that has occurred during the diffusion gradient pulses.

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The effective water diffusion tensor D can be written as a positive definite symmetric matrix:

$$\begin{equation} {\bf D}= {\bf S}_D \tilde{{\bf D}}{\bf S}^T_D \quad \mbox{with}\quad \tilde{{\bf D}}= \left( \begin{array}{@{}lll@{}}d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{array} \right) \end{equation} \tag{ 1 }$$

where the column vectors of S_D are the orthogonal eigenvectors of D, the superscript T denotes the transpose and d_i for i = 1, 2, 3 are the corresponding eigenvalues. The signal intensity ρ_D of diffusion MRI is given by

$$\begin{equation} \rho _D=\rho _0\exp (-b{\bf g}^T{\bf D}{\bf g}) \end{equation} \tag{ 2 }$$

where ρ₀ is the signal obtained without diffusion-sensitizing gradient, $\bf g$ is the normalized diffusion-sensitizing gradient vector and b denotes the diffusion-weighting factor depending on the gradient pulse used in the DT-MRI sequence. From figure 1(a),

$$\begin{equation} b=\gamma ^2\delta ^2G^2\left(\Delta -\frac{\delta }{3}\right) \end{equation} \tag{ 3 }$$

where γ = 26.75 × 10⁷ rad Ts⁻¹ is the gyromagnetic ratio of hydrogen and δ and G are the duration and amplitude, respectively, of the diffusion-sensitizing gradient pulse along a given direction. Diffusion in the x-, y- and z-directions can be measured by applying the gradient in the x-, y- and z-directions, respectively.

The eigenvalue c_i of the conductivity tensor C can be represented as

$$\begin{equation} c_i=\frac{\sigma _{{\rm ext}}}{d_{{\rm ext}}}\left[d_i\left(\frac{d_{{\rm int}}}{3d_{{\rm ext}}}+1\right) +\frac{d_i^2d_{{\rm int}}}{3d_{{\rm ext}}^2}+\frac{2}{3}d_{{\rm int}}\right]+\mathcal {O}(d_{{\rm int}}^2), \quad i=1,2,3 \end{equation} \tag{ 4 }$$

where σ_ext is the extra-cellular conductivity, d_int and d_ext are the intra- and extra-cellular diffusion coefficients, respectively, and $\mathcal {O}(d_{{\rm int}}^2)$ is bounded as $d_{{\rm int}}^2$ tends to infinity (Tuck et al 2001, Haueisen et al 2002). For small intra-cellular diffusion d_int ≈ 0, the eigenvalue c_i satisfies

$$\begin{equation} c_i=\frac{\sigma _{{\rm ext}}}{d_{{\rm ext}}}d_i, \quad i=1,2,3. \end{equation} \tag{ 5 }$$

The eigenvalues c_i of C and the eigenvectors of D in S_D can determine the conductivity tensor as

$$\begin{equation} {\bf C}={\bf S}_D \tilde{{\bf C}}{\bf S}^T_D \quad \mbox{where}\quad \tilde{{\bf C}}= \left( \begin{array}{@{}lll@{}}c_1 & 0 & 0 \\ 0 &c_2 & 0 \\ 0 & 0 & c_3 \end{array} \right). \end{equation} \tag{ 6 }$$

In a conventional spin-echo MREIT pulse sequence, both positive and negative currents of the same amplitude and duration are injected. These injection currents with the pulse width of T_c accumulate extra phases. The corresponding k-space MR signals can be described as

$$\begin{equation} S^\pm (k_x,k_y)=\int _{\Omega }\rho (x,y)\,{\rm e}^{{\rm i}\phi (x,y)}\,{\rm e}^{\pm {\rm i}\gamma B_z(x,y)T_c}\, {\rm e}^{{\rm i}2\pi (k_xx+k_yy)}\,{\rm d}x\, {\rm d}y \end{equation} \tag{ 7 }$$

where ρ is the T₂ weighted spin density, ϕ is any systematic phase artifact and Ω is a field-of-view (FOV). Here, the superscript of S^± denotes a brief notation for S⁺ and S⁻. For the standard coverage of the k-space, we set

$$\begin{equation} \left\lbrace \begin{array}{@{}ll@{}} k_x = \displaystyle\frac{\gamma }{2\pi } G_x(t-T_E) & \mbox{ for }\quad |t-T_E|<T_s/2 \\ k_y=\displaystyle\frac{\gamma }{2\pi } m\Delta G_yT_{pe} & \mbox{ for }\quad m=-N_y/2,\ldots ,N_y/2-1 \end{array}\right. \end{equation} \tag{ 8 }$$

where G_x is the frequency encoding gradient strength, T_E is the echo time, ΔG_y is the phase encoding step and T_pe is the phase encoding time. The induced magnetic flux densities generated by the positive and negative injection currents I^± are denoted as ±B_z, respectively.

The standard two-dimensional inverse Fourier transform provides the complex MR images ${\mathcal {M}}^\pm$ as

$$\begin{equation} {\mathcal {M}}^\pm (x,y)=\rho (x,y)\,{\rm e}^{{\rm i}\phi (x,y)}\, {\rm e}^{\pm {\rm i}\gamma B_z(x,y)T_c }. \end{equation} \tag{ 9 }$$

From (9), the magnetic flux density B_z can be recovered as

$$\begin{equation} B_z(x,y)=\frac{1}{2\gamma T_c}\tan ^{-1}\left(\frac{\alpha (x,y)}{\beta (x,y)}\right) \end{equation} \tag{ 10 }$$

where α and β are the imaginary and real part of ${\mathcal {M}}^+/{\mathcal {M}}^-$, respectively. From the analysis by Scott et al (1992) and Sadleir et al (2005), the noise standard deviation $sd_{B_z}$ of the measured B_z is given by

$$\begin{equation} sd_{B_z}= \frac{1}{2\gamma T_c \Upsilon _M} \end{equation} \tag{ 11 }$$

where T_c is the current injection duration and ϒ_M is the signal-to-noise ratio (SNR) of the MR magnitude image.

Using the property that the noise standard deviation is inversely proportional to T_c and |ρ|, a multi-echo imaging sequence which acquires a series of echoes has been developed and optimized to reduce the noise level in the measured B_z data (Nam and Kwon 2010a). Figure 1(b) shows a schematic diagram for the injection current nonlinear encoding (ICNE) gradient-multi-echo pulse sequence.

Let Ω be a cylindrical domain with its boundary ∂Ω. We may express Ω as a union of the slices perpendicular to the z-axis:

$$\begin{equation} \Omega =\cup _{t\in (-H,H)}\Omega _t \quad \mbox{where}\quad \Omega _t=\Omega \cap \lbrace (x,y,z)\in \mathbb {R}^3 | z=t\in (-H,H)\rbrace . \end{equation} \tag{ 12 }$$

Here, Ω₀ denotes the center slice. In this paper, we use the following two-dimensional estimations:

$$\begin{equation} \tilde{\nabla }f:=\left(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},0\right), \qquad \tilde{\nabla }^\perp f:=\left(\frac{\partial f}{\partial y},-\frac{\partial f}{\partial x},0\right). \end{equation} \tag{ 13 }$$

Since B_z is the only measurable quantity without rotating the object inside the MRI scanner, we inject current in the orthogonal direction to the main magnetic field through a pair of attached electrodes to maximize B_z. According to the Helmholtz decomposition, Park et al (2007) decomposed the internal current density J into curl-free and divergence-free components as

$$\begin{equation} \mathbf {J}=\mathbf {J}_0+\nabla \times \boldsymbol{\Psi }. \end{equation} \tag{ 14 }$$

Here, J₀ := −∇α is the background current density determined by

$$\begin{equation} \left\lbrace \begin{array}{@{}ll@{}} \nabla ^2 \alpha = 0 & {\rm in}\quad \Omega \\ \nabla \alpha \cdot \mathbf {n} = g & {\rm on} \quad \partial \Omega \end{array} \right. \end{equation} \tag{ 15 }$$

where n is the outward unit normal vector on ∂Ω and g denotes the Neumann boundary data subject to the injection current. The vector potential $\boldsymbol{\Psi }$ satisfies

$$\begin{equation} \left\lbrace \begin{array}{@{}ll@{}}-\nabla ^2\boldsymbol{\Psi }= \nabla \times \mathbf {J} & \mbox{in}\quad \Omega \\ \nabla \cdot \boldsymbol{\Psi } = 0 & \mbox{in}\quad \Omega \\ \nabla \times \boldsymbol{\Psi } \cdot \mathbf {n}= 0 & \mbox{on}\quad \partial \Omega . \end{array} \right. \end{equation} \tag{ 16 }$$

We define the vector field J_P as the projected current density, which can be directly calculated from any vector field J, with

$$\begin{equation} \mathbf {J}_P:=\mathbf {J}_0+\tilde{\nabla }^\perp \psi \quad \mbox{in } \quad \Omega _t \end{equation} \tag{ 17 }$$

where the potential ψ solves

$$\begin{equation} \left\lbrace \begin{array}{@{}ll@{}} \tilde{\nabla }^2 \psi = \displaystyle\frac{1}{\mu _0}\nabla ^2 B_z &\mbox{in}\quad \Omega _t \\ \tilde{\nabla }^\perp \psi \cdot \mathbf {n} = 0 &\mbox{on}\quad \partial \Omega _t. \end{array} \right. \end{equation} \tag{ 18 }$$

Note that J_P may not be identical to J of which the z-component is not negligible. The projected current density J_P is the quantity observable from the measured B_z data, which satisfies ∇ · J_P = 0 and the following stability condition (Park et al 2007):

$$\begin{equation} \Vert \mathbf {J}-\mathbf {J}_P\Vert _{\Omega _t} \le C\left( \left\Vert \frac{\partial }{\partial z}(J_z-J_{0,z}) \right\Vert _{\Omega _t}+ \Vert J_z-J_{0,z}\Vert_{\Omega _t}\right). \end{equation} \tag{ 19 }$$

Here, the constant C depends only on Ω_t and not on J. The estimated stability between J and J_P implies that J_P is close to the true current density J depending on the z-component of J − J_P.

As described by Ma et al (2013), from (5) and (6), we can get

$$\begin{equation} {\bf C}={\bf S}_D \tilde{{\bf C}}{\bf S}^T_D=\frac{\sigma _{{\rm ext}}}{d_{{\rm ext}}}{\bf S}_D \tilde{{\bf D}}{\bf S}^T_D=\frac{\sigma _{{\rm ext}}}{d_{{\rm ext}}}{\bf D}. \end{equation} \tag{ 20 }$$

The internal current densities J_i for i = 1, 2 subject to two externally injected currents can be represented as

$$\begin{equation} \mathbf {J}_i=-{\bf C}\nabla u_i=-\eta _{{\rm ext}}{\bf D}\nabla u_i \end{equation} \tag{ 21 }$$

where u_i is the voltage potential corresponding to the injection current g_i for i = 1, 2 and

$$\begin{equation} \eta _{{\rm ext}}:=\frac{\sigma _{{\rm ext}}}{d_{{\rm ext}}} \end{equation} \tag{ 22 }$$

is a scale factor called the ECDR.

Using the estimated diffusion tensor D, the current J_i satisfies the following relation:

$$\begin{equation} \nabla \times ({\bf D}^{-1}\mathbf {J}_i)=-\nabla \times (\eta _{{\rm ext}}\nabla u_i)=-\nabla \eta _{{\rm ext}}\times \nabla u_i \end{equation} \tag{ 23 }$$

and it implies

$$\begin{equation} \nabla \times ({\bf D}^{-1}\mathbf {J}_i)=-\frac{\nabla \eta _{{\rm ext}}}{\eta _{{\rm ext}}}\times (\eta _{{\rm ext}}\nabla u_i) =\nabla \log \eta _{{\rm ext}}\times ({\bf D}^{-1}\mathbf {J}_i). \end{equation} \tag{ 24 }$$

The term D⁻¹J_i can be defined as the pseudo-current as suggested by Ma et al (2013). With a current density in place of D⁻¹J_i, the relation (24) is also the basis to recover the isotropic conductivity from two internal current densities in CDII (Hasanov et al 2008).

Assuming that the reconstructed projected current density J_{P, i} ≈ J_i = −η_extD∇u_i by injecting transversal currents, the unknown η_ext can be represented as

$$\begin{equation} \nabla \log \eta _{{\rm ext}}\times ({\bf D}^{-1}\mathbf {J}_{P,i})=\nabla \times ({\bf D}^{-1}\mathbf {J}_{P,i}) \end{equation} \tag{ 25 }$$

where D is known. Setting

$$\begin{equation} {\bf E}_i={\bf D}^{-1}\mathbf {J}_{P,i}, \end{equation} \tag{ 26 }$$

the identity (25) can be written as

$$\begin{equation} \tilde{\nabla }\times (E_{i,x},E_{i,y})= -\tilde{\nabla }\eta _{{\rm ext}}\times \tilde{\nabla }u_i=\tilde{\nabla }\log \eta _{{\rm ext}}\times (E_{i,x},E_{i,y}) \end{equation} \tag{ 27 }$$

where E_{i, x} and E_{i, y} are the x- and y-components of E_i, respectively. Using (27), we have the following matrix equation:

$$\begin{equation} {\mathbf {A}}{\mathbf {x}}={\mathbf {b}} \end{equation} \tag{ 28 }$$

where

$$\begin{equation} {\mathbf {A}}=\left( \begin{array}{@{}cc@{}} E_{1,x}& E_{1,y}\\ E_{2,x}& E_{2,y} \end{array} \right),\qquad {\mathbf {b}}=\left( \begin{array}{@{}c@{}} -\tilde{\nabla }\times (E_{1,x},E_{1,y})\\ -\tilde{\nabla }\times (E_{2,x},E_{2,y}) \end{array} \right) \end{equation} \tag{ 29 }$$

and the unknown variable ${\mathbf {x}}= \big( \frac{\partial \log \eta _{{\rm ext}}}{\partial x}, \frac{\partial \log \eta _{{\rm ext}}}{\partial y} \big)^T$. By solving (28), we get the transversal gradient vector $\tilde{\nabla }\log \eta _{{\rm ext}}$ in the imaging slice Ω_t.

The log-scale log η_ext is recovered by

$$\begin{eqnarray} &&\log \eta _{{\rm ext}}(\mathbf {r})=-\int _{\Omega _{t}}\tilde{\nabla }\Phi _2(\mathbf {r}-\mathbf {r}^{\prime })\cdot \tilde{\nabla }\log \eta _{{\rm ext}}(\mathbf {r}^{\prime })\ {\rm d}\mathbf {r}^{\prime } +\int _{\partial \Omega _{t}}\frac{\partial \Phi _2(\mathbf {r}-\mathbf {r}^{\prime })}{\partial \mathbf {n}}\log \eta _{{\rm ext}}(\mathbf {r}^{\prime })\ {\rm d}l_{\mathbf {r}^{\prime }} \nonumber \\ \end{eqnarray} \tag{ 30 }$$

where $\Phi _2(\mathbf {r}-\mathbf {r}^{\prime })=\frac{1}{2\pi }\log |\mathbf {r}-\mathbf {r}^{\prime }|$ is the two-dimensional fundamental solution of the Laplace equation. From a reference background value of η_ext, the log-scale log η_ext can be iteratively updated based on the equation (30). A similar formula to determine the isotropic conductivity from the measured B_z data was introduced by Oh et al (2003). To guarantee the uniqueness of the solution log η_ext, we set log η_ext := ζ + φ where

$$\begin{eqnarray} \zeta (\mathbf {r}) &=& -\int _{\Omega _{t}}\tilde{\nabla }\Phi _2(\mathbf {r}-\mathbf {r}^{\prime })\cdot \tilde{\nabla }\log \eta _{{\rm ext}}(\mathbf {r}^{\prime })\ {\rm d}\mathbf {r}^{\prime }, \nonumber \\ \varphi (\mathbf {r}) &=& \int _{\partial \Omega _{t}}\frac{\partial \Phi _2(\mathbf {r}-\mathbf {r}^{\prime })}{\partial \mathbf {n}}\log \eta _{{\rm ext}}(\mathbf {r}^{\prime })\ {\rm d}l_{\mathbf {r}^{\prime }}. \end{eqnarray} \tag{ 31 }$$

The following two-dimensional harmonic equation in Ω_t is uniquely determined up to a constant:

$$\begin{equation} \left\lbrace \begin{array}{@{}ll@{}} \tilde{\nabla }^2\varphi (\mathbf {r}) = 0 & \mbox{in } \ \Omega _t \\ \tilde{\nabla }\varphi (\mathbf {r})\cdot \mathbf {n}(\mathbf {r}) = \tilde{\nabla }(\log \eta _{{\rm ext}}(\mathbf {r})-\zeta (\mathbf {r}))\cdot \mathbf {n}(\mathbf {r}) & \mbox{on } \ \partial \Omega _t. \end{array} \right. \end{equation} \tag{ 32 }$$

Therefore, the scale factor log η_ext is uniquely determined up to a constant in Ω_t. The determination of the constant depends on a practical situation including a known background conductivity value or a reference region with a known conductivity value such as the hydrogel under the thin carbon electrode.

We built a three-dimensional numerical model using COMSOL (COMSOL Inc., USA). Figure 2(a) shows the cylindrical model with 100 mm diameter and 140 mm height. We attached two pairs of electrodes on its surface to inject currents. Figure 2(b) shows its finite element mesh with 264 020 tetrahedral and 19 130 triangular elements. As shown in figure 2(c), the model includes two types of anomalies: one is the isotropic agar gel object and the other is the anisotropic muscle tissue with different directional diffusion rates. The orientations of three muscle tissues were $0,\frac{\pi }{2}$ and 0.6 radian as marked in figure 2(c). The background of the model was assumed to be saline.

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To generate simulated data, we set S₀ as the signal obtained without the diffusion-sensitizing gradient:

$$\begin{equation} S_0(\mathbf {r})=\rho (\mathbf {r}) (1-{\rm e}^{-T_R/T_1(\mathbf {r})})\,{\rm e}^{-T_E/T_2(\mathbf {r})}. \end{equation} \tag{ 33 }$$

With the diffusion-sensitizing gradient, we calculated the echo signal intensity as

$$\begin{equation} S_b=S_0\,{\rm e}^{-b{\mathbf {g}}^T{\bf D}_{\mathbf {g}}} \end{equation} \tag{ 34 }$$

where g represents the normalized diffusion-sensitizing gradient vector and

$$\begin{equation} {\bf D}= \left( \begin{array}{@{}ccc@{}} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha &\cos \alpha & 0 \\ 0 & 0 & 1 \end{array} \right) \left( \begin{array}{@{}lll@{}}d_1 & 0 & 0 \\ 0 &d_2 & 0 \\ 0 & 0 & d_3 \end{array} \right) \left( \begin{array}{@{}ccc@{}}\cos \alpha & \sin \alpha & 0 \\ -\sin \alpha &\cos \alpha & 0 \\ 0 & 0 & 1 \end{array} \right)^T \end{equation} \tag{ 35 }$$

where α is the orientation specified in table 1. Figure 2(d) shows the echo signal intensities S_b for the six direction gradients of

$$\begin{equation} {\mathbf {g}}=\left\lbrace (1,0,0),(0,1,0),(0,0,1),\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0\right),\left(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right), \left(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}}\right)\right\rbrace \end{equation} \tag{ 36 }$$

and the used parameters in table 1.

The most commonly used measure to quantify the anisotropy is the fractional anisotropy (FA). We calculated the FA value as the normalized variance of the three eigenvalues of the water diffusion tensor:

$$\begin{equation} {\mbox{FA}=\sqrt{\frac{3}{2}\frac{(d_1-\hat{d})^2+(d_2-\hat{d})^2+(d_3-\hat{d})^2}{d_1^2+d_2^2+d_3^2}}} \end{equation} \tag{ 37 }$$

where $\hat{d}:=\frac{d_1+d_2+d_3}{3}$. Figure 2(e) shows the color-coded FA map of the numerical model to indicate diffusion along the x-axis (red), y-axis (green) and z-axis (blue). We computed the FA map by taking the projection of the FA value on to the principal eigenvector (Jiang et al 2006). For the case of our numerical simulation, the principal eigenvector was [ − cos α, −sin α, 0]^T and, therefore, the color-coded FA map was computed as

$$\begin{equation*} [R,G,B]=[\mbox{FA}\cos \alpha ,\mbox{FA}\sin \alpha ,0]. \end{equation*}$$

For the muscle anomaly oriented at 0.6 radian, the FA value was found to be 0.2610 using equation (37). The corresponding color was [R, G, B] = [0.2154, 0.1474, 0], which is between the red and green. Since the FA value is zero for the isotropic object, the agar object is not visible in the FA map.

For the MREIT simulation, the isotropic conductivity values of the background saline and circular agar anomaly regions were 0.2 Sm⁻¹ and 0.5 Sm⁻¹, respectively. The orientations of the eigenvectors of the anisotropic muscle region were assumed to be the same as those of the diffusion eigenvectors and the eigenvalues were set to be (c₁, c₂, c₃) = (1.00, 0.710, 0.675) Sm⁻¹.

Figures 3(a) and (b) illustrate the phantom design. Around the cylindrical phantom with 200 mm diameter and 140 mm height, we attached four carbon-hydrogel electrodes (HUREV Co. Ltd, Korea). We filled it with 1 S m⁻¹ saline and placed one cylindrical isotropic gel object of 2 S m⁻¹ conductivity. We also put three pieces of anisotropic biological tissue (chicken breast) with the dimension of 35 × 35 × 35 mm³. We placed the phantom inside the bore of the 3 T MRI scanner (Achieva, Philips) equipped with the 32-channel receiver RF coil (SENSE-Head-32ch, Philips).

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We acquired the diffusion-weighted data using the single-shot spin-echo echo planar imaging sequence. We applied the diffusion-weighting gradients in 15 directions with b-value of 1000 s mm⁻². The imaging parameters were as follows: repetition time T_R = 3000 ms, echo time T_E = 73 ms, slice thickness = 5 mm, number of excitations (NEX) = 2, FOV = 180 × 180 mm² and acquisition matrix size = 64 × 64. The NEX means the number of repeated acquisitions to improve the SNR by averaging. Figure 3(c) shows the T₂ weighted MR magnitude image. Three images in figure 3(d) are the b-value diffusion-weighted MR magnitude images with the applied gradients in the x-, y- and z-directions. Figure 3(e) shows the color-coded FA map. The left chicken breast object shows diffusion along the x-direction (red), the middle-top object shows diffusion along intermediate directions (red and green) and the right one shows diffusion along the y-direction (green).

We used a custom-designed MREIT current source to sequentially inject 10 mA currents for the horizontal and vertical directions (Kim et al 2011). The current source was synchronized with the ICNE multi-gradient-echo pulse sequence to optimize the multiple B_{z, l} data for l = 1, ..., N_E by minimizing the noise effect (Nam and Kwon 2010a). The imaging parameters were as follows: repetition time T_R = 35 ms, first echo time $T_{E_1}$ = 1.88 ms, echo space E_s = 2.2 ms, slice thickness = 5 mm, number of echoes N_E = 15, NEX = 50, flip angle = 6.65°, FOV = 260 × 260 mm², imaging matrix = 128 × 128. The imaging time (T_R × 128 × NEX) was 3.7 min. For each current injection direction, we acquired the k-space data twice, subject to the positive and negative injection currents to cancel out the systematic phase artifact. Therefore, to obtain the full data set for the horizontal and vertical directions, the total imaging time was 14.8 min. The multiple echoes corresponding to the different echo times $T_{E_l}=1.88+2.2\times (l-1)$ ms for l = 1, ..., 15 were acquired by the alternating read-out gradients.

Figure 4(a) shows the simulated data of B_{z, 1} and B_{z, 2} subject to the horizontal and vertical current injections, respectively. Figure 4(b) shows the x- and y-components of the recovered projected current densities J_{P, 1} and J_{P, 2} for the horizontal and vertical injection currents, respectively.

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Figure 4(c) shows the water diffusion tensor D. Since it is symmetric with six variables, we estimated them from the six diffusion-sensitizing gradient vectors and one reference S₀ by taking the logarithm on both sides of (34). From the estimated water diffusion tensor, we determined its three eigenvectors by diagonalizing it. Using J_P in figure 4(b) and D in (c), we solved the matrix equation (28) to compute the ECDR factor η_ext using a regularization factor ξ as

$$\begin{equation} \tilde{\nabla }\log \eta _{{\rm ext}} = ({\bf A^{T}A}+\xi {\bf I})^{-1}{\bf A}^{T}{\bf b} \end{equation} \tag{ 38 }$$

where I is the 2 × 2 identity matrix. The value of ξ was set to be inversely proportional to the determinant of A. We, therefore, added a relatively large value of ξ where |A| was small and vice versa. Figures 4(d) and (e) show the computed ECDR η_ext and the reconstructed conductivity tensor C, respectively.

We estimated the L²-error of the reconstructed C using the following formula:

$$\begin{equation} \mathcal {E}({\bf C})=\sqrt{\frac{\sum _{(x_i,y_j)\in \Omega _t}{({\bf C}(x_i,y_j)-{\bf C}^*(x_i,y_j))^2}}{N_{xy}}} \end{equation} \tag{ 39 }$$

where Ω_t is the imaging region, C* denotes the true conductivity tensor and N_xy is the number of pixels in Ω_t. The L²-error values were 0.0900, 0.0885, 0.0895, 0.0066, 0.000 and 0.000 for σ₁₁, σ₂₂, σ₃₃, σ₁₂, σ₁₃ and σ₂₃, respectively.

To evaluate the performance of the proposed method at different noise levels, we added the white Gaussian random noise to both the diffusion weighted signal and B_z data assuming that the DT-MRI and MREIT experiments were conducted independently. For the DT-MRI experiment, we assumed that the SNR of S₀ in the background saline varied from 70 to 80 dB, in steps of 5 dB. Since the SNRs of the different regions depend on their relaxation properties, we used the following relation to compute the SNRs of the gel and tissue regions:

$$\begin{equation} \Upsilon _{M,k}= \Upsilon _{M,0} \frac{(1-{\rm e}^{-T_{1,k}/T_R})\,{\rm e}^{-T_{2,k}/T_E}}{(1-{\rm e}^{-T_{1,0}/T_R})\,{\rm e}^{-T_{2,0}/T_E}}\end{equation} \tag{ 40 }$$

where T_{1, 0}, T_{2, 0} and ϒ_{M, 0} were the T₁ and T₂ relaxation values and the SNR of the background saline region, respectively. The subscript k denotes either the gel or tissue regions and the T_{1, k} and T_{2, k} values are given in table 1. We chose T_R and T_E as 4000 and 40 ms, respectively, which were similar to those used in the typical experimental study. After computing the SNR in each region, we calculated the noise standard deviation sd_k of each region using the following relation:

$$\begin{equation} sd_k = 0.66\frac{\langle S_{b,k} \rangle }{\Upsilon _{M,k}} \end{equation} \tag{ 41 }$$

where 〈S_{b, k}〉 is the average signal amplitude in each region and the factor 0.66 accounts for Rayleigh statistics. We generated and added the complex white Gaussian noise with zero mean and the standard deviation given by (41).

For the MREIT experiment, we assumed that the SNR of the background saline region ϒ_{M, 0} varied from 70 to 100 dB, in steps of 15 dB. With T_R = 1200 and T_E = 15 ms, we computed the expected noise levels for the gel and tissue regions using (40). For the current pulse width, T_c = 30 ms, the expected noise standard deviation, $sd_{B_z,k}$ of B_z, for each region, was

$$\begin{equation} sd_{B_z,k} = \frac{1}{2\gamma T_c \Upsilon _{M,k}}. \end{equation} \tag{ 42 }$$

We tried four different noise levels of $\big(\Upsilon _{M,0}^D,\Upsilon _{M,0}^{B_z}\big)=(70,\infty ),(70,100),(70,85)$ and (70, 70) dB where $\Upsilon _{M,0}^D$ and $\Upsilon _{M,0}^{B_z}$ are the background region SNRs of the DT-MRI and MREIT experiments, respectively. Figure 5 shows images of the ECDR η_ext and diagonal elements σ₁₁, σ₂₂, σ₃₃ of C for the chosen noise levels. Table 2 summarizes the estimated L²-errors showing that the proposed method stably reconstructed the conductivity tensor C for the chosen noise levels.

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Table 2. L²-errors of the reconstructed conductivity tensor for different noise levels.

$\Upsilon _{M,0}^{B_z}$ (dB)	σ11	σ22	σ33	σ12	σ13	σ23
∞	0.0916	0.0909	0.0907	0.0221	0.0213	0.0220
100	0.1097	0.1078	0.1052	0.0225	0.0216	0.0226
85	0.1132	0.1109	0.1056	0.0235	0.0232	0.0243
70	0.1240	0.1211	0.1186	0.0247	0.0242	0.0256

Figure 6(a) shows the measured B_{z, 1} and B_{z, 2} data from the phantom in figure 3 corresponding to the horizontal and vertical current injections, respectively. Figure 6(b) plots computed J_{P, 1} and J_{P, 2} for the horizontal and vertical injection currents, respectively. Figure 6(c) shows the diffusion tensor map D. In the three chicken breast regions with muscle fibers oriented in the x-, y- and xy-directions as shown in figure 3(e), the diffusion tensor map shows different diffusion anisotropy effects in each region. The cylindrical gel region shows the isotropic diffusion. Figure 6(d) and (e) show images of the ECDR η_ext and the conductivity tensor C, respectively, reconstructed from the measured D and B_z data without using any referred or assumed extra-cellular and diffusivity values.

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Figure 7 shows the color-coded FA and conductivity values in the three chicken breast tissue regions. The reconstructed conductivity values of the tissues exhibit their anisotropic characteristics depending on the muscle fiber orientation. We can see that σ₁₁ is relatively higher in the left tissue and σ₁₂ is higher in the top middle tissue.

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Table 3 shows the conductivity values, eigenvalues and FA values of the reconstructed conductivity tensor C at five different regions. The maximum eigenvalues of the background saline and the agar gel object were 1.05 and 2.22 S m⁻¹, respectively, which were close to the true values of 1 and 2 S m⁻¹. For the anisotropic muscle regions, the ratio between the maximum and minimum eigenvalues was 1.38.