Relation between water diffusion and conductivity tensors

The effective water diffusion tensor D measured within a voxel using pulsed-gradient-spin echo sequences [30] can be written as a 3 × 3 positive definite symmetric matrix: (1) where the column vectors forming S_D = {s₁, s₂, s₃} are the eigenvectors of D, the superscript T denotes transpose and λ_D,k for k = 1,2,3 are the corresponding eigenvalues. There is no known relationship between water diffusion and electrical conductivity in a free electrolyte solution. However, since the water, ions and other charged particles coexist in the same microscopic environment, Tuch et al. [31] suggested the eigenvalues λ_C,k at low frequency (<1 kHz) conductivity tensor C were related to diffusion coefficients as (2)

Here, C _e is the extra-cellular conductivity, d_i and d_e are the intra- and extracellular diffusion coefficients, respectively, and is bounded as tends to infinity. By neglecting intracellular diffusion in relation (2), as in [8–10, 19, 32, 33] the effective conductivity tensor may be expressed as (3) where the position dependent scale factor η can be determined by measuring magnetic flux density induced due to an externally injected current [9, 10].

Reconstruction of scale factor η using discretization of Faraday’s law

We represent the imaged domain Ω as a stack of axial slices Ω_t arranged perpendicular to the z-axis such that and the entire slice package covers a region of total thickness 2H₀. The divergence-free property of electrical current density, J = −C∇u = −ηD∇u, combined with an externally-injected current density g applied to the domain boundary leads to the following elliptic partial differential equation (4) where u is a voltage distribution, and n = (n_x, n_y, n_z) is the outward-normal unit vector defined at the boundary ∂Ω of the domain Ω.

The Kirchhoff voltage law (KVL) is a low-frequency corollary of Faraday’s law which states that sum of the voltage drops or electric field components around a closed path is zero. In this work, we used the KVL to reconstruct scale factors, η, within each slice of a three-dimensional domain using estimated current densities and diffusion tensor data. To use the KVL we discretized each slice Ω_t as a rectangular grid (Fig 1). Since J = −C∇u = −ηD∇u, the quantity has the dimensions of the electric field, and via the KVL we have that (5) where p_ij,1, p_ij,2, p_ij,3, and p_ij,4 are points at the center of the closed loop defining the rectangular region Ω_ij located at coordinates (x_i−1, y_j−1), (x_i, y_j−1), (x_i, y_j), (x_i−1, y_j) respectively, as shown in Fig 1. It is also possible to design another loop, , such that (6) where , , and are the loop vertices (Fig 1).

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Fig 1. Schematic of the dual-loop network.

The primary loop is shaded in blue and the secondary (primed) loop is shaded in red.

https://doi.org/10.1371/journal.pone.0254690.g001

Note that the dual loop network is designed such a way that x, y-components of (D⁻¹ J) vectors at the points are used simultaneously to determine η values at that position [34].

The representative scale factor in overlapping loops can than be found by combining the identities (5) and (6) to form a dual-loop network i.e. (7a) (7b) With the further assumption that η values are known on the boundary, the dual-loop network defines an overdetermined system containing a total of 2(N_x − 2)(N_y − 2) equations over 2(N_x − 2)(N_y − 2) loops or cells and (N_x − 2)(N_y − 2) internal nodes, for each Ω_t.

Using linear interpolation of (D⁻¹ J) vectors at the center of the nodes shown in Fig 1, the right hand side of Eq (8) contains known boundary voltage differences, or estimated from known boundary η and (D⁻¹ J) values around the loop perimeter.

An expression describing the combined primary and secondary loops may be written (8) where the vector contains inverse scale factor η values at each node, and elements of the stiff matrices (primary) and (secondary) on the left hand side of Eq (8) contain the numerator terms from Eqs (7a) and (7b) respectively.

Solutions for x were found using regularization via a least squares method (9)

Here, λ represents a regularization parameter, I is an identity matrix, and T denotes matrix transpose. The matrix and .

Correction of dual-loop η reconstructions using artificial neural network methods

The reliability of reconstructed η values using the dual-loop method with one current injection is limited, due to noise propagating along equipotential lines [34, 35]. Therefore, an artificial neural network (ANN) approach was used to overcome these limitations. ANN methods use a number of sample input-output pairs to create a regressor model of a non-linear artifact function f mapping the input to the output. In our application, the function mapped dual-loop output to artifact-free images as (10) where X and Y are vectorized images of the uncorrected input and the corrected output η, respectively (Fig 2) and denotes the number of non-zero voxels in each image. The data set in is the result of dual-loop-network calculations, each of which is denoted η* or , and consists of artifact-free η images, each denoted or . Training data are identified using the notation ()*, thus a row of may also be identified as the image .

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Fig 2. Flow diagram of the proposed method demonstrated for a head-shaped domain.

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We used a generalized regression neural network (GRNN) [36] to construct the functional relationship f in (10). The probabilistic GRNN architecture consisted of four layers: an input unit, a pattern unit consisting of Gaussian neurons, a summation unit and an output layer (Fig 2) [36]. For an uncorrected input X, the output of the GRNN Y (the conditional mean) may be expressed as [36] (11)

In (11), the spread-constant of the Gaussian neurons is denoted by α, is the k-th artifact-free vectorized image and the total number of neurons (samples) in the pattern unit is [36]. The function [36] (12) measures the distance between the input X and the k-th training vector, . Eq (11) can be interpreted as relating an expression for the expected output Y to a given input data vector , which is the weighted sum of the artifact-free training data sets , where for a chosen α the weighting-factors are determined by the Euclidian distance between input and the training datasets (Eq (12)). Therefore, a particular neuron will produce more weight in the output if it is close to the input data-vector, and vice versa. The spread constant α controls the degree of smoothness of the GRNN output.

As noted in [36], the predicted output of the GRNN-neural network in Eq (11) is associated with the underlying joint probability distribution function (pdf). When the distribution function is not known, Specht [36] suggested using a nonparametric estimate of the joint-pdf which depends on the spread constant and the number of samples in the data sets (as well as the number of neurons in the pattern unit). While there is no general rule that predicts the number of samples required in the pattern unit with a specified degree of accuracy, good prediction results are often obtained with a modest number of samples [37]. Additionally, it has been shown that for a fixed number of data-points the spread-constant of the joint-pdf gradually decreases as sample size increases [36, 37]. However, with large training data sets, Specht [36] suggests a simple clustering technique to reduce the number of neurons in the pattern unit and avoid unnecessary computational burden in the GRNN network. Cluster centers are estimated by averaging the data samples present in the cluster, which eventually improves the underlying noise characteristics of the center-vector of the Gaussian neurons (Eq 12). In this study all training data sets were used in the pattern unit (Fig 2) to predict η.

In the GRNN, the spread-constant α is the only parameter which needs to be determined by the training process. In this paper, we determined an optimum α by minimizing the mean-squared error (MSE) error cost function. For a given α, the mean-squared error for the k-th sample data, MSE_k, is defined as (13) where is the predicted output of the GRNN for the input .

In [36] an optimum spread-constant, was determined from the training data sets using a holdout method. In the holdout method, for a chosen α, a GRNN network is constructed using the data sets, and the mean-squared error (13) is calculated by comparing the network prediction using true data . This process was then repeated for other samples in the data sets to determine the mean MSE value for each α. In the final network, the optimum spread-constant value was set to the value producing the smallest MSE. In this work, we used the constrained optimization function ‘fminbnd’, implemented in the MatLab optimization toolbox (The MathWorks, Inc., Natick, Massachusetts, United States) to determine the value in the range 10⁻³ < α < 10³. This value was used in the final network to predict an artifact-free Y at a given X in (11).

Formation of from MR phase images

In MREIT, the z-component of the current induced magnetic flux density B_z is measured using an MRI scanner. The accumulated phase due to the external current injection depends on the induced B_z and current injection duration T_c. The complex MR signal density developed using a spin-echo MR pulse sequence can be represented in the spatial domain as (14) where ρ(r) is the MR magnitude image, δ(r) represents a systematic phase artifact at position r, γ is the proton gyromagnetic ratio, and the superscript on B_z represents the experimentally measured data. From Eq (14), the magnetic flux densities generated by positive and negative current injections I^± can be recovered using (15)

From the analysis of [3, 38], the noise standard deviation in depends on the signal-to-noise ratio (SNR) of the MR magnitude image Υ_ρ and the current injection duration T_c as (16)

Phantom data for this work were gathered using spin echo MREIT imaging sequences. However, these sequences involve long acquisition times [39] and are inconvenient for human subjects. A faster multi-gradient-echo imaging sequence [40] was used to obtain MREIT data in the human subject. To minimize the noise standard deviation in multi-echo MR images, l = 1, 2, …, N_E measured B_z data were calculated as a weighted sum of individual echoes , where N_E represents the total number of echoes, via (17)

The weighting factors were computed using [40] (18) where ρ_l is the MR magnitude image of the l^th echo.

Reconstruction of current density from measured magnetic flux density

Inside Ω, the current density J = (J_x, J_y, J_z) and magnetic flux density B = (B_x, B_y, B_z) are related by Ampere’s law (19)

Direct calculation of J using Ampere’s law requires knowledge of all three components of B. Instead of measuring the full magnetic flux density vector, we instead estimated the projected current density J^P from alone, with reference to a computational model of the whole domain, Ω constructed using structural T1-weighted images. The model was assigned a uniform conductivity and used to generate a predicted current density distribution J₀, including an estimate of the z-component of the current density. The projected current density was then calculated via [15] (20) where ψ was estimated from the measured B_z via [15] (21)

In (21), is the two-dimensional Laplacian operator. Uniform-conductivity numerical models used to compute J₀ data were constructed using the shape and dimensions of the object found from structural MRI images. These models were then used to solve the Laplace equation, subject to the same boundary conditions g as in experiments, using the COMSOL-MATLAB interface (MLI, COMSOL Inc, Burlington, MA, USA) using (22) where u₀ is the voltage distribution caused by the external current injection g, and C₀ represents an homogeneous and isotropic conductivity distribution.

In the human body, the measured magnetic flux density data suffers because of weak MR signals recovered from bone and skin regions, mainly due to their rapid T₂ decay. Inclusion of such regions cause severe artifacts in reconstructed scale factor images [41]. In the human head studies used in this paper, we therefore reconstructed the regional projected current density, only inside the brain region, (Fig 4 (b)) using the expression (23) where denotes the solution of the Laplace equation in (22), restricted to , and the quantity was estimated from the measured and simulated B_z,0 via [41] (24)

Note that comparing the equation in (21) with (24) only differs in boundary which stems from the fact that for a simply connected local region the normal component of the current density is continuous [41].

Simulation of current density and magnetic flux density data

In calculating training data from computational models of segmented MR magnitude data, it is necessary to determine both the predicted current density and the z-component of the magnetic flux density, B_z, caused by the current flow in Ω at any position r in three dimensional space. Current density data were calculated using COMSOL via the methods described for determining J above, but with the conductivity distribution determined using standard segmentation of brain tissues and literature-estimated conductivity values. The magnetic flux density distribution was then computed from simulated J data using the Biot-Savart law (25) where μ₀ is the magnetic permeability of free space. A fast Fourier transform implementation of (25) was used to improve calculation speeds [42]. The Biot-Savart law was used to calculate and images required for training data sets (Fig 2). Magnetic flux density data derived from uniform models (B_z,0) were also employed for reconstructions of current density in (24) used in estimations of projected current densities used for generating both training and input data for the GRNN.

Reconstruction of diffusion tensor data

Diffusion tensor volumes were reconstructed following Basser et al. [30], by comparing the diffusion weighted signal, with the non-diffusion weighted signal S₀ to estimate (26) where, , j = 1, 2, …, N_d are the normalized magnetic field gradient vectors of diffusion gradients applied in the sequence. Before estimating the diffusion tensor, any susceptibility-related geometric distortion caused by the EPI diffusion weighted imaging sequence was corrected [8, 43]. Six gradient directions were used in measuring phantom diffusion data, and 15 directions were employed in measuring diffusion properties of the human subject.

Generation of training data sets

To generate the training data sets and , we constructed models containing piecewise-constant η values for each distinct tissue, and solved a total of forward models with different discrete combinations of η values using the experimental montage . This process produced sets of simulated B_z data, denoted , using the methods described above.

We additionally simulated results from a complementary montage, (shown in Fig 2) to generate a second set of B_z data sets, , using the same current injection amplitude as in experiments. The complementary montage model in each case differed only in the position of the two electrodes. These complementary positions were chosen to ensure current flow was not collinear with that caused by , so that a unique conductivity reconstruction could be produced with the two-current DT-MREIT algorithm [6]. Noise was added independently to both and , depending on the noise levels measured in experimental data for each imaging experiment. A common isotropic η distribution, designated , was reconstructed from noisy and to obtain artifact free data using the two-current injection DT-MREIT algorithm [9, 10, 19, 33].

Two independent current densities J* and were calculated using Eqs (20) (or (23)) for the experimental and complementary electrode montages, respectively.

The curl-free property of the electric field leads to the following equation [9, 10, 19, 33] (27)

This equation can be re-expressed in matrix form as (28) where , ln quantities were calculated relative to 1 S ⋅ sec/mm³, and the matrices and are (29)

The weights w_i in Eq (29) relate to the MR-magnitude ρ and the noise level h associated with the measured noise (Fig 2) [33] as (30)

In all reconstructions, a 3 × 3 window was used to determine the stiff matrix and load vector (29). Singular value decomposition was then used to decompose the 2N × 2 stiff matrix into UΣV^T, where the two singular values of are entries in Σ = diag(s₁, s₂) and .

The distribution e was solved for by minimizing the generalized cross-validation (GCV) function [32] (31)

Here, ζ is a regularization constant estimated from the region around the voxel position (x, y).

Finally, the images were obtained from e values by solving the two-dimensional Poisson Eq (32) assuming a known boundary value [10, 20, 33] (32)

Next, we reconstructed η* from simulated noisy data alone, using the dual-loop network, to create an input data set . The two data sets and were then used to build the network using the ‘newgrnn’ function implemented in the MATLAB Neural Network Toolbox. After generating the training data sets for each projection, one thousand hidden neurons were used to perform ANN correction, both for phantom and human experiments. We found that the training time for data used in this paper averaged around 30 minutes ysing a MacPro computer with an Intel(R) Xenon(R) (E5–2697 v2, 2.70 GHz) 12 core CPU and 64.0 GB RAM.

Phantom image training data.

To calculate and distributions used to reconstruct phantom experiment data, we segmented tissue masks into three different ROIs, shown in Fig 3(c) for the center slice. Each of the three distinct tissues (agar, chicken, potato) were simulated at ten linearly spaced η values, resulting in a total of models. Agar was simulated with η between 0.1 and 1 S ⋅s/mm³, chicken between 0.2 and 1.25 S ⋅ s/mm³ and potato between 0.01 and 0.3 S ⋅ s/mm³. Noise was added to and data independently, based on the SNR determined in the agar region of phantom magnitude images. From 16 we estimated the standard deviation of measured B_z data in phantom experiments was about 0.18 nT.

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Fig 3. Phantom experiment setup.

(a) Phantom design, (b) MR magnitude image from spin-echo pulse sequence during MREIT experiment, (c) Segmented ROIs for volume conductor model. The ROIs were segmented using the unsupervised data partitioning method implemented in the MATLAB command kmeans. (d) images induced due to the 10 mA current injection, (e) Mean diffusivity map obtained from six direction diffusion data sets with b-value 1000 sec/mm². Images in (b)-(e) cropped to 100 × 100 pixels² to show detail.

https://doi.org/10.1371/journal.pone.0254690.g003

Human image training data.

In simulating scaling factors for the human experiment, we segmented tissue masks into five different ROIs, as shown in Fig 4(e) for the center slice. We assumed skin and bone regions had isotropic conductivity values of 0.43 S/m [29] and 0.015 S/m [44] respectively. The brain region, consisting of gray matter (GM), white matter (WM) and CSF was modeled by scaling the measured diffusion tensor [31, 45]. A total of sets of numerical model data were generated by independently varying model scale factor values for GM between 0.210 and 0.736 S ⋅ s/mm³ [46], for WM between 0.210 and 0.736 S ⋅ s/mm³ [46], and for CSF between 0.700 and 0.844 S ⋅ s/mm³ [31, 45] respectively. Table 1 summarizes the range of effective isotropic conductivity values calculated from the conductivity tensor used to generate numerical model data. As for phantom data noise was added to and data. The SNR determined in the white matter region of magnitude images was used to estimate noise levels for human data. Noise standard deviations of 0.21 nT were added to B_z data for the human imaging experiment.

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Fig 4. Human experiment set-up.

(a) Sagittal view of computer model of human subject with attached electrodes (Fpz, Oz, T7, T8). The corresponding lead wire trajectories are marked as , , , . (b) High resolution T₁-weighted image corresponding to the MREIT slice shown in (e). Locations of six ROIs used for evaluation of reconstructed conductivity values shown in Table 5. (c) Segmented tissues in volume conductor model. Tissue segmentation was performed using the method described in Huang et al. [50]. (d) Colour-coded fractional anisotropy map obtained from fifteen-direction diffusion data sets with b-value 1000 sec/mm². (e) MR magnitude image from multi-gradient multi-echo pulse sequence during MREIT acquisition. (f) Brain ROI (), mask used in electromagnetic field reconstruction. (c) Echo-combined images induced due to the 1.5 mA current injection. Calculated stray magnetic fields were subtracted from individual echo images before further processing. Images in (b)-(c) are cropped to 175 × 210 and those in (d), (f)-(g) are cropped to 85 × 100.

https://doi.org/10.1371/journal.pone.0254690.g004

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Table 1. Range of isotropic conductivity values used to generate numerical model data.

Isotropic conductivities were estimated using [29], where C_l = λ_C,1 and C_t = (λ_C,2 + λ_C,3)/2 are the longitudinal and tranversal components of the conductivity, respectively and λ_C,1 ≥ λ_C,2 ≥ λ_C,3.

https://doi.org/10.1371/journal.pone.0254690.t001

Electric field reconstruction

In some cases, electric field images were constructed from reconstructed current density and conductivity tensor data. Using the reconstructed projected current density, or and estimated conductivity tensor C, electric field vectors at locations r within images were reconstructed as (33a) (33b)

Verification of reconstruction performance

To evaluate the proposed one-current reconstruction method, a common isotropic scale factor distribution, designated , was also reconstructed from experimentally measured projected current density () data, by solving Eqs (31) and (32) using the known boundary scaling factor [20].

Dual-loop results corrected by the ANN were verified against DT-MREIT reconstructions and also against known conductivity values, where available. Dual-loop and DT-MREIT reconstructions were compared using a relative L² measure, where, for any reference or true data and reconstructed data , the relative L²-difference or error RE defined as (34)

Here, ‖⋅‖ represents the Euclidean distance.

We also measured the structural similarity (SSIM) of reconstructions [47]. The structural similarity measures the similarity between a true or reference image and an input or reconstructed image by comparing the luminance, contrast and their structures (35)

Here, , , , and are the local mean and standard deviations of the image and respectively and C₁ and C₂ are two regularization constants chosen to avoid instability when the denominator of the expression is close to zero [47]. A 5 × 5 window was used to calculate means and standard deviations and C₁ and C₂ values were set to be 1 × 10⁻⁴ and 9 × 10⁻⁴ respectively. The mean SSIM value was then computed as (36) where is the total number of non-zero voxels in the image slice.

Phantom experiment description

A phantom experiment was conducted to test the proposed method. The cylindrical phantom had a 55 mm diameter and was 50 mm high. Two opposing pairs of 10 × 10 mm² carbon electrodes (Hurev Co. Ltd., South Korea) were attached to the perimeter as shown in Fig 3(a). Three cubes of chicken muscle (∼15 × 15 × 15 mm³) oriented in x (left), y (right), z-directions (top) shown in were placed inside the phantom, centered on the electrode plane. A small cube of potato (∼8 × 8 × 15 mm³), assumed isotropic, was also placed inside the phantom. The background of the phantom was filled with a 1.0 S/m conductive gel made with agarose.

All data were measured using a single-channel RF volume coil in a 7.0 T Bruker scanner (Bruker Biospin MRI, Billerica, MA, USA) located at the Barrow Neurological Institute (Phoenix, Arizona, USA). A custom-designed constant current source [48] was used to deliver 10 mA current synchronously with spin-echo MREIT acquisitions with a total current injection time T_c, of 20 ms in each TR. Two sets of data were collected over five slices, with an image matrix size of 128 × 128. Other imaging parameters were, TR/TE = 1000/20 ms, field-of-view, FOV = 80 × 80 mm², slice thickness, 3 mm (no slice gap), number of excitation, NEX = 8, and number of echoes, N_E = 1. B_z images were calculated using Eq (15). Fig 3(b) and 3(d) shows the acquired MR magnitude and images respectively for the center slice.

Diffusion tensor data of the same five slices were then collected using a single-shot spin-echo echo planar imaging (SSSE-EPI) pulse sequence with b of 1000 sec/mm² and a total of six diffusion directions. Data were acquired with TR/TE = 2500/32.67 ms. Other parameters were the same as for MREIT image sequences. One reference data set with b = 0 sec/mm² was also collected. The diffusion tensor distribution was reconstructed using (26). The condition number of the DTI scheme was 4.7. Therefore, prior to reconstruction the diffusion weighted data set was denoised by taking local features of the diffusion weighted images into account, using an adaptive Wiener filter [49] implemented in MATLAB (https://www.mathworks.com/matlabcentral/fileexchange/43992).

Human experiment description

Human experimental protocols were approved by the Arizona State University institutional review board and all procedures were carried out in accordance with these protocols after obtaining informed consent from participants. Data obtained from a 58-year old volunteer male human subject were used to demonstrate the performance of the proposed method. Data were acquired using a 32-channel RF head coil and a 3.0T Phillips scanner (Phillips, Ingenia, Netherlands) located at the Barrow Neurological Institute (Phoenix, Arizona, USA). A transcranial electrical stimulator (DC-STIMULATOR MR, neuroConn, Ilmenau, Germany) was used to deliver 1.5 mA currents using both Fpz-Oz and T8-T7 electrode montages. Sets of data were measured over three axial slices using a multi-echo-gradient-echo pulse sequence and an image matrix size of 128 × 128. Other imaging parameters were, TR/TE/ES = 50/7/3 ms, field-of-view, FOV = 224 × 224 mm², slice thickness, 5 mm (no slice gap), number of excitations NEX = 24, and number of echoes, N_E = 10. The current injection time T_c in each TR was 32 ms. B_z images of each echo were calculated using Eq (15). Individual echo images were then combined using (17) to improve the SNR of the acquired B_z signals. Prior to echo combination, stray magnetic field corrections were applied to each echo image [51]. Fig 4(e) and 4(g) show acquired MR magnitude and images respectively for the center slice.

Diffusion-weighted images of the three MREIT slices were also acquired, using a single-shot, spin-echo-echo-planar (SE-EPI) imaging sequence, with b = 1000 sec/mm² and a total of fifteen diffusion directions. The condition number of the DTI scheme was 1.3. Data were acquired with TR/TE = 2000/139 ms and NEX = 2. The acquisition matrix size was 64 × 64, with the same FOV as in MREIT sequences. The data were then interpolated to 128 × 128 matrix to match the MREIT data. One reference data set with b = 0 sec/mm² was also collected.

Parameters used in both phantom and human experiments are summarized in Table 2. For each experiment, diffusion data were co-registered with T₁ weighted MR-images, and diffusion tensors were reconstructed using Eq (26). ure Fig 3(e) shows the mean diffusivity map () estimated from the denoised diffusion weighted images for the phantom, and Fig 4(d) shows the colour-coded fractional anisotropy map estimated from the diffusion weighted images for the human subject. In the human experiment, B_z data were corrected to take into account any stray magnetic field [51, 52] created by lead wires that may have affected phase images. For the phantom experiment, we estimated J^P (Eq (20)) using the Laplacian (∇²) of the measured data (Eq (21)). Therefore, it was not necessary to consider wire-related stray magnetic fields in the phantom data because [6]. However, for the human experiment, the regional projected current density in Eq (23) [41] was estimated using the local boundary condition (24). In this case the stray magnetic field must be accounted for. Therefore, a numerical model of the wires in the human experiment was constructed from 1 mm³-resolution T₁-weighted images in order to estimate these stray magnetic fields via the Biot-Savart law (see S1 File).

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Table 2. Experimental image parameters.

https://doi.org/10.1371/journal.pone.0254690.t002

Phantom experiment results

ure Fig 5(a) and 5(b) shows reconstructed projected current density magnitudes in the biological tissue phantom. Images were calculated using Eq (20).

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Fig 5. Model-predicted current density images for the central slice (slice 3) of the phantom obtained from a homogeneous (I, III) numerical model and estimated current density images found using Eq (20) (II, IV) for (a) vertical and (b) horizontal current injection.

Normalized arrow plots overlaid on images show current flow directions. Parts (c) and (d) illustrate reconstructed scale factor images of the three central slices (2, 3, 4) of the phantom object for vertical (c) and horizontal (d) currents. images in parts (c) and (d) are solutions of the dual-loop matrix system (9). η_v,h images show ANN-corrected scale factors. Part (e) () shows scale factor images recovered from data measured using two current injections. Images cropped to 95 × 95.

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Reconstructed scale factor images solving Eq (9) estimated from measured current density and diffusion tensor data are displayed as and in Fig 5(c) and 5(d) for vertical and horizontal currents, respectively. Images labeled η_v and η_h in Fig 5(a) and 5(b) show ANN-corrected scale factor images obtained from the trained neural network. The optimum spread-constant values were found to be 0.75 for both vertical- () and horizontal- () current networks. ure Fig 5(e) shows the reconstructed scale factor images () using the two-current injection DT-MREIT algorithm. We also computed the relative L²-differences and measured the similarity index [47] of these reconstructed scale factor images. For the vertical projection, relative L²-differences were found to be 0.11, 0.12 and 0.12 for the second, third and fourth slices, respectively, after ANN correction. Similarly, using the single-current dual loop estimation for the horizontal current injection, relative L²-differences were calculated to be 0.11, 0.12 and 0.12. Similarity indices for vertical current injections were found to be 0.94 in each of the second, third and fourth slice positions after ANN correction. For the single-current dual loop estimation from horizontal current injection, similarity indices for these three slices were also found to be 0.94.

Reconstructed scale factors were multiplied with diffusion tensors to produce conductivity tensors as in (3). Results for the center (third) slice are shown in Fig 6(a). Fig 6(a) shows reconstructed diagonal components of the conductivity tensor found using the ANN-corrected single-current injection for vertical (C_v) and horizontal (C_h) projections, respectively, and Fig 6(a) also displays the conductivity tensor found using the two-current DT-MREIT for the center slice. Actual reconstructed conductivity values and anisotropic ratio, , where λ_C,k k = 1, 2, 3 are the three eigenvalues of the reconstructed conductivity tensor, are summarized in Table 3. Note that due to the scaled relationship of the conductivity and water diffusion tensors, anisotropic ratios in reconstructed conductivity tensors were the same as in water diffusion tensors. Therefore, these values are identical for all three reconstructions compared in Table 3. The proposed one-current injection method showed good agreement with the two-current DT-MREIT reconstructions. However, the anisotropic ratio, which was directly derived from diffusion weighted data sets, was small for all five ROIs, especially for the top (z-oriented) chicken muscle tissue sample, where the C_zz conductivity component was expected to be largest, along with the anisotropic ratio.

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Table 3. Reconstructed diagonal components of the conductivity tensor (S/m) and anisotropic ratio AR for biological tissue phantom images in Fig 6.

Underlined entries show tensor components along tissue orientation directions.

https://doi.org/10.1371/journal.pone.0254690.t003

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Fig 6. Reconstructed conductivity tensor and electric field comparisons for central slice.

Part (a) compares of reconstructed conductivity tensor using proposed one-current injection method with the two-current injection DT-MREIT algorithm [33] using vertical injection (C_v) and horizontal injection (C_h) only or () using the two-current injection method. (b) Electric field maps derived from reconstructed conductivity tensors of the central phantom slice. Images labeled E_v and E_h show electric field magnitudes estimated from reconstructed conductivity tensors C_v and C_h respectively. Images and show estimated electric field distributions for horizontal and vertical current injection, respectively, from conductivity tensor reconstructed from two-injection current injection . The normalized arrow plots overlaid on (b) show E field directions. Images cropped to 95 × 95.

https://doi.org/10.1371/journal.pone.0254690.g006

For comparison purposes, we also measured biological tissue conductivities at 100 Hz using an impedance analyzer and a four-probe method. These values showed good agreement with the reconstructed conductivity (Table 4). For example, the conductivity of the potato sample at 100 Hz was found to 0.18±0.006 S/m, an approximate 0.73% relative L² error. Similarly, for the chicken breast muscle the measured impedances indicated a conductivity of 0.86±0.01, and 0.73±0.008 S/m along and across the fiber directions, respectively. For all three tissue samples the average conductivity value along the fibers was found to be (Table 4) 0.82 S/m, while those across fibers were 0.71 S/m, an approximate 2–4% error. For the agar background the relative L² error was found to be 1.60% with respect to impedance analyzer conductivity measurements.

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Table 4. Comparison of tissue conductivities measured (S/m) by an impedance analyzer using a four-probe method and conductivity values averaged over phantom regions of interest for chicken, potato or agar materials from the image shown in Fig 6, using the proposed method and the two-current injection DT-MREIT algorithm [33].

The longitudinal and transverse components of the conductivity, C_l = λ_C,1 and C_t = (λ_C,2+λ_C,3)/2 were calculated from the eigenvalues (λ_C,1≥λ_C,2≥λ_C,3) of the reconstructed conductivity tensor. Effective isotropic conductivities of potato and agar backgrounds were calculated using the relation, [29] for potato and agar regions for comparison with impedance analyzer measurements.

https://doi.org/10.1371/journal.pone.0254690.t004

Fig 6(b) compares magnitudes of estimated vertical or horizontal electric fields found using Eq (33). The relative L²-differences between single-current injection and DT-MREIT reconstructed electric fields for vertical projection data were found to be 0.16, and 0.12 for the horizontal projection in the center slice shown in Fig 6(b).

Human experiment results

Fig 7(a) and 7(b) shows the magnitude of the reconstructed current density for Fpz-Oz and T8-T7 current injections respectively. Reconstructed scale factor images solving Eq (9) estimated from measured current density and diffusion tensor data are displayed in Fig 7(c) and 7(d) for Fpz-Oz and T8-T7 currents respectively. Images labeled η in Fig 7 show ANN-corrected scale factor images obtained from the trained neural network. The values were found 1.85 for Fpz-Oz and 1.03 for the T7-T8 electrode montage, respectively.

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Fig 7.

Model-predicted current density in a central slice of human head data (I, III) obtained from Eq (22) and estimated current density images found using Eq (23) (II, IV) are shown for (a) Fpz-Oz and (b) T8-T7 electrode montages. Normalized arrow plots overlaid on images show current flow directions. Reconstructed scale factor images of the central slice of in-vivo human subject are shown in parts (c) and (d) for Fpz-Oz or T8-T7 current injections respectively. Images in (c) and (d) labeled show solutions of the dual-loop matrix system in (9). Images labeled η show ANN-corrected scale factors. Part (e) shows a scale factor image recovered from data measured from both current injections. Images cropped to 75 × 100.

https://doi.org/10.1371/journal.pone.0254690.g007

Fig 7(e) shows a reconstructed scale factor image found using the two-current-injection DT-MREIT algorithm. We computed relative L²-differences and similarity indices for scale factor images reconstructed the two methods. For the Fpz-Oz and T7-T8 electrode montages, relative L²-differences and similarity indices were found to be 0.14 and 0.93, respectively after ANN correction.

Reconstructed scale factors were multiplied with diffusion tensors to produce conductivity tensors as in (3). Results for the center slice are shown in Fig 8(a). The top and middle rows of Fig 8(a) show reconstructed diagonal components of the conductivity tensor using ANN-corrected single-current injection for Fpz-Oz and T8-T7 projections, respectively, and the bottom row displays the conductivity tensor components found using the two-current DT-MREIT for the center slice. Fig 8(b) compares the reconstructed conductivity tensors within the selected ROI. Values for diagonal components of reconstructed conductivity tensors are summarized in Table 5. Fig 8(c) compares magnitudes of estimated Fpz-Oz or T8-T7 electric fields found using Eq (20). Relative L² errors and MSSIM results for all phantom and human data slices are summarized in Table 6. The relative L² difference between single-current injection and DT-MREIT reconstructed electric fields for the human subject Fpz-Oz projection data was found to be 0.13 for the center slice shown in Fig 8(c). For the same slice and T7-T8 data, the relative L² difference in electric field was 0.09. Structural similarities to corresponding two-current-measured data for these electric field images were 0.94 and 0.98 respectively.

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Fig 8. Reconstructed conductivity tensor and electric field comparisons for in-vivo human experiment.

(a) Comparison of reconstructed conductivity tensor using proposed one-current injection method with the two-current injection DT-MREIT algorithm [33] using Fpz-Oz injection (C _Fpz−Oz) and T8-T7 injection only (C _T7−T8) or using the two-current injection method (). Part (b) shows tensor plots of the reconstructed conductivity images in (a). The ROI is marked in the inset T₁-weighted image. The size of each ellipsoid in (b) is proportional to the tensor eigenvalues at that location, and its color and orientation represent principal eigenvectors. Part (c) shows electric field maps derived from reconstructed conductivity tensors of the central slice, with the Fpz-Oz image at left and T8-T7 montage at right. Images labeled E denote electric field magnitudes estimated from reconstructed conductivity tensors shown in (a). Images labeled are estimated electric field distributions for were found from conductivity tensors reconstructed from the two-injection current injection shown in (a). Normalized arrow plots overlaid on (c) show E field directions. Reconstructed images cropped to 75 × 100.

https://doi.org/10.1371/journal.pone.0254690.g008

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Table 5. Reconstructed diagonal components of the conductivity tensor (S/m) for in-vivo human experiment shown in Fig 8.

The ROIs are displayed in Fig 4(d).

https://doi.org/10.1371/journal.pone.0254690.t005

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Table 6. Relative L²-diferences (34) and mean structural similarity MSSIM indices (36) of reconstructed scale factor and E fields for proposed single-current and two-current injection DT-MREIT algorithms [33].

https://doi.org/10.1371/journal.pone.0254690.t006

Relationship between MREIT and DT-MREIT

We first note properties of the DT-MREIT reconstruction problem that are distinct from purely electromagnetic approaches. Without loss of generality, consider a three dimensional conductive domain Ω consisting of a background region containing one anomaly . The conductivity of the background is denoted and the anomaly conductivity is defined as C = C _b + δ C. From fundamental electromagnetic principles [53], any point on the anomaly subdomain boundary, satisfies the relations (37) where ν and τ are normal and tangential unit vectors on respectively, and the voltage distributions u⁺ and u⁻ are defined as (38)

Decomposing the current density vector J into normal and tangential part at the subdomain interface, J(ξ) = (J(ξ) ⋅ ν)ν + (J(ξ) ⋅ τ)τ, and using Eq (37) we have [34] (39) where the current density vectors J⁺ and J⁻ are defined in the background and anomaly regions respectively in a similar manner to u⁺ and u⁻. However, in DT-MREIT the relationship D⁻¹ J = −η∇u at the subdomain interface satisfies [18] (40)

Eqs (39) and (40) show the fundamental differences between MREIT and DT-MREIT. Eq (39) implies that it is nearly impossible to have any distinguishable contrast along an anomaly boundary if current flow is orthogonal to its tangent vector. In addition, Park et al. [15] show that it is possible to determine the conductivity uniquely if the conductivity value is known at the boundary. To avoid the ambiguity caused by (39), most MREIT algorithms use two independent currents to reconstruct unique isotropic conductivity distributions. On the other hand, (40) implies that a distinguishable contrast at tissue edges can obtained using measured diffusion tensors D, if diffusion values are different in the two regions. These assumptions are physically distinct, because diffusion coefficients are principally determined by the medium viscosity. Hence, incorporation of D allows the possibility of using only one current injection to reconstruct position dependent conductivity scale factors. However, from (.27) if ∇ln(η) is parallel to (D⁻¹ J) vectors it is not possible to obtain any distinguishable boundary information using DT-MREIT.

Fig 9(f) shows an example of the condition described above, where some tissue boundaries are not visible in single current injection MREIT data. However, because of the discontinuity at the tissue interface, DT-MREIT can distinguish the tissue boundary even with the single current injection shown in Fig 9(e). The interested reader may find details of this imaging experiment in S2 File.

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Fig 9. Example illustrating distinct MREIT and DT-MREIT properties.

(a) Phantom design. (b)-(c) MR and corresponding images. (d) Mean diffusivity map showing muscle directions. (e) Scale factor images reconstructed using dual-loop method for (I) horizontal, (II) vertical injection only. Part III shows the scale factor image obtained using two-current DT-MREIT method. (f) IV-V show equivalent isotropic conductivities from one current injection, and VI is isotropic conductivity found using two-current injections and the J-substitution algorithm [4]. Parts (g)-(h) show diagonal components of the reconstructed conductivity tensor using one-(horizontal) and two-current injection methods respectively. Images cropped to 40 × 40.

https://doi.org/10.1371/journal.pone.0254690.g009

Phantom results

Single-current L² differences were around 10% different from two-current DT-MREIT reconstructions. We found that the lowest longitudinal conductivities in the muscle samples were measured in the ‘top’ sample, where the tissue was oriented along the z-direction. This could have been related to the particular sample or limitations in the diffusion data. Overall, the anisotropic ratios of the ‘left’ (x-oriented) and ‘right’ (y-oriented) muscle samples measured in the diffusion tensor images were lower than expected (around 1.2). The top (z-oriented) sample had the lowest anisotropic ratio of 1.05, lower than that reconstructed for agar or potato samples in the same phantom. It is possible that anisotropic properties were not sampled adequately at 0.625 millimeter resolution. It may also be the case that there was little anisotropic property remaining in the muscle tissue samples.

Assumption of boundary scaling factor values

Implementation of the dual-loop method requires an assumption of conductivity at the object boundary. We therefore specified the boundary scale factor as based on the ratio of the known conductivity of the agar gel background (1 S/m) and the apparent diffusion tensor map measured in the agar gel background region. However, for the human data we assigned the scale factor value on the boundary to be η_∂Ω = 0.40 S ⋅ sec/mm³ [33]. This may have been responsible for some of the L² differences observed in the phantom and human experimental results.

Study limitations and future work

As noted in Lee et al. [34] and Sajib et al. [35], noise propagation along equipotential lines limits the performance of the dual-loop method. The degree of noise propagation depends on the condition number of the stiff matrix in Eq (9) [34, 35]. However, application of a total variation denoising technique, as in [35], to input data sets before performing dual-loop calculations may improve the condition number and reconstruction performance, but may not greatly reduce streaking artifacts, or reconstruction fidelity because of oversmoothing. It is also possible to reconstruct the scale factor using a single-loop method. However, one-loop reconstructions are strongly influenced by local changes of current flow, noise and reconstruction path [34].

In these studies, ANN methods were able to reconstruct the scale factors shown in Fig 5(c) and 5(d) for phantom data; or reconstruct human data, as in the η images of Fig 7 with reasonable accuracy using 1000 training data sets (S3 File), by employing the prior information provided by dual-loop estimations. While the agreement between standard two-current and single-current dual-loop reconstructions in these preliminary tests was encouraging, further improvements in machine learning algorithms may produce better overall performance. For example, one of the limitations in this study was that we used a general regression neural network (GRNN) to find the regressor function f in Eq (10). Because the centers and the weights in the pattern and summation unit of a GRNN network are determined from subject-specific training data sets, it cannot be generally extended to predict any artifact-free scale-factor image. To avoid this, in future studies we plan to use more diverse training data in conjunction with a deep learning model that is expected to improve reconstruction performance by taking into account local image features.

In future work, we intend to test these techniques in vivo in human heads for other tES electrode montages. In human tES studies it is rare to use in-plane (transverse) electrode montages, and electrode locations are instead chosen close to a presumed cortical target [11]. As noted previously, the difference between projected current density and true current density estimates depends on the out of plane current component J_z [15, 16]. For transverse currents, nearly all current density information is encoded in B_z data, and J_z data is negligible. The quality of projected current density data is key to both dual-loop (9) and DT-MREIT reconstructions (27). Therefore, we expect that the single-current data reconstructions used here will become less accurate as J_z increases. Deep learning methods are also anticipated to be of potential benefit for these more general situations. Training data for deep learning could be generated using partial dual-loop-reconstructed data from off-plane montages or computed predictions of these data with arbitrary electrode locations.