Abstract
Diffusion tensor magnetic resonance electrical impedance tomography (DTMREIT) and electrodeless conductivity tensor imaging (CTI) are two emerging modalities that can quantify lowfrequency tissue anisotropic conductivity properties by assuming similar properties underlie ionic mobility and water diffusion. While both methods have potential applications to estimating neuromodulation fields or formulating forward models used for electrical source imaging, a direct comparison of the two modalities has not yet been performed invitro or invivo. Therefore, the aim of this study was to test the equivalence of these two modalities. We scanned a tissue phantom and the head of human subject using DTMREIT and CTI protocols and reconstructed conductivity tensor and effective low frequency conductivities. We found both gray and white matter conductivities recovered by each technique were equivalent within 0.05 S/m. Both DTMREIT and CTI require multiple processing steps, and we further assess the effects of each factor on reconstructions and evaluate the extent to which different measurement mechanisms potentially cause discrepancies between the two methods. Finally, we discuss the implications for spectral models of measuring conductivity using these techniques. The study further establishes the credibility of CTI as an electrodeless noninvasive method of measuring low frequency conductivity properties.
Similar content being viewed by others
Introduction
Direct acquisition of lowfrequency (LF) tissue electrical properties is often complicated because of the need to deliver and measure electrical currents via electrodes attached to the body^{1}. We define LF properties as being those below about 1 MHz.
Magnetic resonance imaging (MRI) approaches provide a unique opportunity to measure electrical properties noninvasively. Electrical properties have recently been measured using MRI methods at high (Larmor frequency) and low (ca. 10 Hz) frequencies. In Magnetic Resonance Electric Properties Tomography (MREPT), distortions of scanner coil radiofrequency fields caused by tissue electromagnetic properties have been used to determine isotropic conductivity distributions via solution of Maxwell’s equations^{2}. Magnetic Resonance Electrical Impedance Tomography (MREIT)^{3,4}, uses a single component of magnetic flux density (\(B_z\)) created by externally applied LF currents, encoded in MRI phase data, to reconstruct isotropic conductivity properties. Tissue conductivities are closely related to ionic mobilities and as a consequence also depend strongly on water diffusion properties. Basser et al.^{5} and Tuch et al.^{6} proposed that conductivity and water diffusion eigenvectors could be related via an effective medium assumption. Kwon et al.^{7} developed the diffusion tensor MREIT (DTMREIT) technique to allow reconstruction of LF conductivity tensor distributions using prior information provided by diffusion tensor data. However, both MREIT and DTMREIT methods require external current applications, which may involve risk of injury or peripheral neural stimulation.
Sajib et al.^{8} subsequently extended the association posited between diffusion and electrical properties to propose the conductivity tensor imaging (CTI) method. CTI estimates both isotropic and anisotropic lowfrequency conductivity distributions^{9,10} by combining knowledge of highfrequency conductivity \(\sigma _H\) obtained from MREPT^{11,12} with tissue microstructure parameters estimated from multibvalue diffusionweighted images^{13,14,15}, without administering external current^{8,16}.
In DTMREIT, low frequency conductivity tensor properties are determined by reconstructing images of a scalar quantity \(\eta\) that relates conductivity and diffusivity using both diffusion tensor and MREIT current density data^{7,17}. In CTI, diffusionweighted and \(\sigma _H\) data are combined to reconstruct \(\eta\). Both techniques are highly relevant to neurology, neuroimaging and neuromodulation, as subjectspecific LF electromagnetic models of the head are frequently used to perform dosimetry in transcranial electrical stimulation^{18,19}, in determining dipole source locations in inverse EEG^{20} and in determining the likelihood of peripheral neural stimulation in MR imaging^{21}.
In both DTMREIT and CTI,
where \(\mathbb {C}\) is the conductivity tensor and \(\mathbb {D}\) is a representative diffusion tensor measured with a single diffusionweighting factor b. Both DTMREIT^{7} and CTI^{8} methods have been used to determine LF anisotropic conductivity distributions invivo in canine^{8,22} and human brains^{16,17}. However, the correspondence of the two modalities has not yet been investigated. DTMREIT conductivities are considered representative of the frequency of the externally applied current, while CTI may be considered to represent DC conductivity properties. Because both frequencies are extremely low, and spectral models predict an effectively flat electrical property spectrum in this range^{23}, the goal of this study was to determine how well LF electrical tissue conductivities derived from in vivo DTMREIT and CTI data compared. Both phantom data and data from a scan session from a human subject were compared. The phantom study was performed to determine results from anisotropic and isotropic tissue generated by each approach, as has been done in several previous studies in CTI and DTMREIT, for example^{7}.
Tissue electromagnetic properties
Tissue electromagnetic properties are characteristic of the arrangement and composition of the molecules and electrolytes they contain. Their electrical properties can be summarized by conductivity and permittivity values and spectra. Tissue properties result from multiple influences, including tissue heterogeneity, interfacial effects and anisotropy, and are characteristic of multiple overlapping relaxation properties or dispersions.
Several characteristic relaxations are observed in tissue, with ‘alpha’, ‘beta’ and ‘gamma’ dispersions typically appearing in the Hz, kHz and MHz ranges respectively. At frequencies below around 100 kHz, phase shifts resulting from capacitive tissue characteristics are very small. Almost no current penetrates cell walls and most current flows in extracellular space. The conductive properties of tissues in this range include influences of membrane composition and geometry as well as electrolyte mobility and concentration. Because membranes are effectively ‘shorted out’ at high frequencies, anisotropy only manifests in lower frequency ranges. At higher frequencies, conductivity mostly depends on ionic concentrations. In addition, all tissues show strong waterrelated dispersion in the GHz range.
Gabriel et al.^{23} employed parametric fitting to characterize electrical dispersions over a wide frequency range. Each dispersion, characterized by its parameters, provides an indication of the distribution of relaxation times and the scales of physical processes involved in each range. Examples of gray matter, white matter and cerebrospinal fluid (CSF) conductivity spectra expected using the parametric model developed by Gabriel et al.^{23} are shown in Fig. 1. Gabriel et al.’s models match MREPT measurements well, but CTI and DTMREIT experiments made to date have recovered higher gray and white matter conductivities than predicted by the lowerfrequency ranges of the models. This is likely because of the lack of in vivo experimental impedance measurements in this range, and the difficulty of obtaining them^{1}.
Factors influencing practical measurement of LF electromagnetic properties using MRI
There are multiple factors that may affect validation of CTI against DTMREIT. While both measure LF tissue properties, there are multiple potential differences between them because of the different mechanisms used to extract data. For example, since DTMREIT reconstructions depend on measured electrical current densities \(\textbf{J}\) caused by an applied external current, relative errors in reconstructed conductivities and \(\eta\) values increase where current densities are lower. In addition, since conductivity boundaries are only visible to DTMREIT imaging when current flow is perpendicular to them^{24,25,26}, DTMREIT can be biased by the choice of electrode locations relative to structures in the transverse plane. This may be also be the case if electrode locations result in large component of current flow out of the transverse plane, as the \(B_z\) data used for MREIT does not capture these details, although attempts are made to compensate for it^{27}. DTMREITmeasured conductivities depend on the frequency of the external current, which is typically very low because of restrictions imposed by imaging sequences. Examples in the literature have used squarewave excitations at 5–10 Hz^{17,22,28}. As noted above, at such low frequencies, little interrogation of intracellular compartments is expected in DTMREIT due to the conductive barriers posed by cell walls.
A saturation phenomenon may also manifest differences between CTI and MREIT or DTMREIT reconstructions. As conductivity gradients become larger, the effects on current densities decrease, in accordance with the Laplace equation governing current flow. This is because current sensitivity to conductivity contrasts is greatest when they are closest to unity^{29}. This saturation effect will result in lower than expected reconstructed MREIT or DTMREIT conductivities in interior areas with high conductivity such as CSF, as the contrast gradient at the CSFparenchymal boundary is particularly high (see Fig. 1). Regional reconstruction algorithms can be employed to overcome this effect^{30}. We did not use this strategy here, as it requires additional assumptions of interior conductivities to be made.
Reconstructed MREIT or DTMREIT conductivities may also be dynamically affected by imaging current flows. An extreme difference, observable at very high imaging fields, is that charges displaced by DTMREIT current flow may be subject to Lorentz forces and magnetohydrodynamic effects that disrupt recordings of the expected magnetic flux patterns caused by the external current^{31}.
Concurrently, CTI methods are based on asserting models relating intra and extracellular diffusivity and water content to conductivity^{8,32,33}, and are less directly related to conventionallydetermined conductivity. We will assign the effective frequency of CTI methods to be 0 Hz. However, we expect that values found using either DTMREIT at 10 Hz and CTI results should be statistically equivalent.
Finally, the exact methods employed in reconstruction differ between each approach also affect results. The principal difference occurs because DTMREIT and MREIT require current density data from several different current directions to be acquired to provide a stable solution. This results in smoothing of DTMREIT and MREIT results compared to CTI.
In the sections below we provide a brief overview of the theory behind the approaches used here to reconstruct low frequency DTMREIT and CTI conductivities, and the relationships between them.
CTI reconstructions
Tuch et al.^{6} proposed that conductivity and water diffusion eigenvalues \(\sigma _v\) and \(d_v\) are linked by the approximation
via an effective medium assumption, where \(d_i\) and \(d_e\), and \(\sigma _e\) are the diffusion coefficients and conductivities in intra (i) or extracellular (e) spaces. This leads to the inference that lowfrequency conductivity tensors should be recoverable from measurements of diffusion tensor distribution obtained via MRI, and assumptions or measurements of intra and extracellular diffusion coefficients.
In CTI theory^{15}, the LF equivalent isotropic conductivity
and anisotropic conductivity tensor,
are determined from the reconstructed extracellular water selfdiffusivity, \(d_e^w\) and diffusion tensor \(\mathbb {D}_e^w\). The CTI scalefactor \(\eta\) is determined as^{8}
where (\(1\alpha\)) and \(d_i^w\) denote the estimated intracellular space volume fraction and water diffusivity, respectively and \(\beta\) is the intratoextracellular ion concentration ratio. This last quantity is estimated from independent studies^{34,35} or fitted^{36}. In this work, a convection–reaction based partialdifferentialequation (PDE) was solved to obtain \(\sigma _H\) from the measured \(B_1\)phase data (crEPT)^{12}. The L\(^2\)norm of a multicompartment model was fitted to the observed multibvalueDW data and minimized at each pixel position to find the microstructure parameters (\(\alpha\), \(d_e^w,d_i^w\))^{9}. The \(d_e^w\) and the properties of \(\mathbb {D}\) found at \(b = 1000\) s/mm\(^2\) using a high angular resolution diffusion imaging (HARDI) sequence were used to estimate \(\mathbb {D}_e^w\)^{9}.
DTMREIT reconstructions
At low frequency (ca. 10 Hz) we assumed that the current density \(\textbf{J}\) and voltage distributions u are related by \(\textbf{J}=\mathbb {C}\nabla u\approx \eta \mathbb {D}_b\nabla u\), where \(b = 1000\) s/mm\(^2\). Ma et al.^{37} and Kwon et al.^{7} derived the relation between \(\textbf{J}\), \(\mathbb {D}_b\) and \(\eta\) as
If data from two independent current administrations are measured, Kwon et al.^{7} proposed reconstructing \(\eta\) as
where \(\textbf{J}^P\) is the projected current density^{27} and \(\textbf{e} = [\frac{\partial \ln (\eta )}{\partial x},\frac{\partial \ln (\eta )}{\partial y}]^T\). Logarithmic quantities were calculated relative to 1 \(\text{ S }\cdot \text{ s/mm}^3\). Projected current density computation involves estimating the current density distribution in a uniformconductivity domain having the same external shape as the imaged region, modified by the gradient of measured \(B_z\) data to account for current density variations caused by conductivity inhomogeneity. The projected current density is the best estimate of the true current density \(\textbf{J}\) that can be obtained using a single component of flux density^{38}.
To solve the linear system in (7) for \(\textbf{e}\), an optimum regularization parameter was determined by minimizing the generalized crossvalidation (GCV) function^{39}. The pixelbypixel GCV function was estimated using a \(3 \times 3\) neighbourhood window. A weighting factor that depends on \(B_z\) noise level was also applied. More information on DTMREIT reconstructions are contained in the “Methods” section.
Equation (7) was solved as,
MREIT reconstructions
Using the estimated projected current density^{38} \(\textbf{J}^{P,\mathscr {E}}\), \(\mathscr {E}=1, 2\), from two independent current applications it is also possible to determine the effective isotropic conductivity^{27} \(\left( \textbf{C}_L^{MREIT}\right)\) and conductivity tensor^{7} \(\left( \mathbb {C}^{DTMREIT}=\eta ^{DTMREIT}\mathbb {D}_e^w\right)\). To reconstruct the \(\textbf{C}_L^{MREIT}\), or \(\eta ^{DTMREIT}\) we must solve a twodimensional Poisson’s equation with a known boundary value^{40}. Without prior knowledge of the phantom boundary conductivity or scalefactor values, both MREIT and DTMREIT methods provide only ‘apparent’ conductivity contrasts. Since the CTI method is expected to provide the absolute conductivity, in this study we assigned the DTMREIT and MREIT boundary conductivities to be those obtained from CTI reconstructions. Further details of MREIT reconstructions may be found in the Methods section.
Results
CTI, DTMREIT and MREIT comparisons
CTI scale factors \(\eta ^{CTI}\) calculated for all four slices of phantom data and all three slices of human data respectively are displayed in Figs. 2 and 3A. Corresponding LF conductivity data \(\textbf{C}^{CTI}_L=\eta ^{CTI} d_e\) and isotropic MREIT conductivity \(\textbf{C}_L^{MREIT}\) are shown in Figs. 4 and 5A for phantom and human data slices respectively. We use \(\textbf{C}_L^{MREIT}\) in preference to \(\textbf{C}_L^{DTMREIT}\) which could be calculated as \(\textbf{C}_L^{DTMREIT}=\eta ^{DTMREIT}d_e^w\) as \(\textbf{C}_L^{MREIT}\) is not dependent on diffusion data.
DTMREIT and CTI eigenvalue comparisons
To compare anisotropic properties, we decomposed the reconstructed \(\mathbb {C}\) \(\left( \mathbb {C}^{CTI} or \mathbb {C}^{DTMREIT}\right)\) into longitudinal (\(\lambda _\parallel ^\mathbb {C}=\lambda _1^\mathbb {C}\)) and transversal \(\left( \lambda _\bot ^\mathbb {C}=\frac{\lambda _2^\mathbb {C}+\lambda _3^\mathbb {C}}{2}\right)\) components, where \(\lambda _1^\mathbb {C}\ge \lambda _2^\mathbb {C}\ge \lambda _3^\mathbb {C}\) were the three eigenvalues of \(\mathbb {C}\). Orthotropic components of CTI tensors for the human subject are displayed in Fig. 7A for a representative slice (slice 2). The same components for DTMREIT reconstructions are presented in Fig. 7B.
ROI comparisons
Scale factor and LF conductivity values in entire chicken, potato and background regions of the phantoms were averaged for comparison of CTI and DTMREIT reconstructions. Values in these regions of interest are shown (with standard deviations) in Table 1.
Three regions of interest (ROIs) were selected in each of white matter, gray matter and cerebrospinal fluid compartments of the human subject, within a single focus slice. Values of scale factor, LF isotropic conductivity, and orthotropic conductivity are tabulated in Table 2 in addition to EPT conductivity values \(\sigma _H\). The ROIs in the focus slice are illustrated in Fig. 6. Overall averages of each quantity in white matter, gray matter and CSF compartments are also displayed in Table 2.
Discussion
Overall findings
Comparing both \(\eta\) and conductivity values over DTMREIT and CTI imaging results, DTMREIT results were considerably smoothed, leading to lower \(\eta ^{DTMREIT}\) and \(\textbf{C}_L^{MREIT}\) values in these images. However, values for white and gray matter \(\mathbb {C}\) within ROIs overlapped considerably (Fig. 6). Equivalence tests on entire gray and white matter compartments performed using a two onesided method^{41,42} found that CTI and DTMREIT \(\eta\) values were equivalent within \(\pm ~50\) S ms/mm\(^3\) and that \(\textbf{C}_L^{CTI}\) and \(\textbf{C}_L^{MREIT}\) conductivities were equivalent within ± 0.05 S/m. The mean reconstructed conductivity \(\sigma _H\) in phantom compartment overlapped with \(C_L^{CTI}\) and \(C_L^{MREIT}\), which we attribute to noise in EPT and diffusion microstructure estimations. We also note that orthotropic components of CSF conductivity were somewhat anisotropic both in CTI and DTMREIT reconstructions. This has frequently been observed in similar studies (^{17,24,43}), and is a result of noise in diffusion tensor reconstructions.
Results shown in Figs. 3 and 5 illustrate that CSF conductivity estimations were smoothed and much lower for DTMREIT and MREIT than for CTI. This was due to the influence of \(\sigma _H\) on CTI reconstructions. CSF conductivity estimations were also affected by the relatively large slice thicknesses and voxel sizes used. MREIT and DTMREIT data were also difficult to interpret within CSF because of the large conductivity gradient at brain and CSF boundaries and consequent current density saturation effects. In the sections below we explore the influence of specific parameters on reconstruction results.
Influence of boundary conductivity choice
The same boundary conductivity found for CTI data was also used as the boundary conductivity for both DTIMREIT and MREIT data. This should enable appropriate relative comparison of the reconstruction methods. However, errors in CTI reconstructions may have affected their overall accuracy.
Influences on CTI reconstruction
CTI reconstructions involve optimization over multiple parameters and applying an effective medium approximation via microstructural diffusion modeling. Several methods have been proposed to estimate cellular microstructure parameters for CTI^{8,9,10,32}. In^{8}, model parameters were optimized for volume fraction and water diffusion coefficients of intracellular space, extracellular matrix space, volume fraction of free water component and an offset value. Conversely, the NODDI model^{32,44} optimizes for the volume fraction of intracellular, free water and restricted diffusion compartments in extracellular space, and therefore requires optimization of fewer parameters. It has been demonstrated that cellular microstructure parameters estimated from the NODDI model produce more stable results than that of^{8} and may require a smaller number of gradient direction or multi bvalue diffusion weighted images^{9}. One of the important contributions of Marino et al.^{32} was that they estimated cellular microstructure parameters multiple ways: n times from \(C^{n}_{n1}\) measurements, where n is the number of available gradient directions, and combined these estimated parameters using a maximum intensity projection method. Assuming that the isotropic microstructure parameters are independent of the measured diffusion gradient directions, and since the fitting procedure used in this paper is sensitive to measurement noise, we used the combinatorial method of^{32} to fit the parameters in Sajib et al.’s model^{8} here using a leastsquare method, which improved our method’s stability. Jahng et al.^{10} proposed using the spherical mean technique (MCSMT)^{45,46} to estimate cellular microstructure parameters. However, the imaging protocol used in the SMT method is different than the acquisition protocol used in this paper, requiring data from two bshells (800, 2000 s/mm\(^2\)) with 12 and 24gradientdirections respectively^{10} to estimate microstructural parameters. In future tests we plan to evaluate SMT CTI methods against MREIT reconstructions. The dependence of CTI conductivity on the microstructure model chosen is still an open question, with different studies producing somewhat different \(\eta\) values^{10,16}. Finally, one other dependence not yet explored is any possible dependence on diffusion time.
Influences on EPT reconstructions
Equation (5) shows that the overall scaling factor \(\eta\) directly depends on EPT reconstruction of \(\sigma _H\). Therefore, any inconsistency in this estimation will have a large effect on CTI images. In the sections below we outline factors influencing EPT reconstructions and an estimation of their potential effects on CTI images.
The pulse sequence used may have an effect on the reconstructed \(\sigma _H\). The sequence employed to map \(B_1\) data was a ‘gold standard’ spinecho method, which is likely the most accurate (though lengthy) way of determining \(B_1\) phase. Six echoes were acquired, but only echoes 2–5 were used in optimizing phase images for human data, which may still have resulted in some errors in echo combination^{47,46,49}. Finally, the largest probable influence on the reconstruction of \(\sigma _H\) is likely the choice of regularization parameter c used in solving (10). In Fig. 8 we illustrate the effect of varying c on \(\sigma _H\) in a single profile line of human image data. At lower c values, reconstructed conductivities, particularly that of CSF, were strongly perturbed and exhibited large deviations at each conductivity boundary. However, there was relatively little effect observed on gray or white matter conductivity. We therefore concluded that values reconstructed for these tissues at \(c = 0.1\) provided a stable representation of these tissues’ properties.
Influence of \(\beta\) value
For the human subject we used a fixed value of \(\beta =0.41\) for all CTI reconstructions^{40}. In phantom reconstructions we fixed \(\beta =1\) for all CTI reconstructions^{43}, as it was assumed that postmortem and osmotic processes resulted in equalization of intra and extracellular ionic concentrations. Recently, Kwon et al.^{36} proposed that a \(\beta\)free method may improve the estimation accuracy, although this is yet to be comprehensively tested.
Influences on DTMREIT and MREIT conductivity
As noted above, the error in reconstructed MREIT conductivity^{50} depends on the proportion of \(z\)directional current flow (\(J_z\)). In the phantom images the twodimensional assumption held well in practice as the phantom was cylindrical and inplane surface electrodes were used. However, for the human case \(J_z\approx 0\) does not hold because of the relatively high anisotropic conductivity, nonplanar electrodes and high conductivities in CSF regions causing out of plane flow.
It has been shown in^{28} that the reconstructed equivalent isotropic conductivity value of the white matter region follows the higher conductivity value in the direction of its fibers. Therefore, the conductivity for the white matter region sometimes appears higher compared to gray matter depending on electrode location. This was particularly evident in the orthotropic reconstructions shown in Fig. 7 where the apparent longitudinal conductivity of the forceps major was much brighter (approximately 2 S/m) in DTMREIT reconstructions than in the corresponding region of the CTI reconstruction of the same slice.
Comparisons to other MRbased results, impedance measurements and spectral models
Measuring absolute conductivity at low frequencies via electrodes alone is challenging because of the need to attach them, and unknown effects of electrode polarization that may result in apparent lower conductivity^{1}. MRbased methods are therefore attractive, as while MREIT may require current to be administered via electrodes, measured data are obtained from MR images, therefore removing polarization effects. Reconstructed brain tissue conductivities using different MRbased reconstruction approaches with and without current administration are compared in Fig. 9. Here, MRmeasured white and gray matter conductivities are overlaid on model predictions^{23}. MRbased measurements of brain conductivities at Larmor frequencies of 3 T and above, reconstructed using multiple different approaches, have agreed closely with model predictions. Meanwhile, lowfrequency brain conductivities found using MRbased methods have consistently been approximately 30 times model predictions. Again, the reconstructed conductivities did not appear to depend on the method used. However, values found using lowfrequency MRI conductivity imaging are also close to values obtained using excised animal tissues at similar frequencies to studies cited in the survey of Geddes and Baker^{17,51}.
The disparity between model and lowfrequency MRbased conductivity results might suggest that electrode polarization effects have caused large underestimations of modelpredicted conductivities, and that empirical measurements support values reconstructed by MR methods. However, another likely possibility is that there are unanticipated influences from MR measurement mechanisms, pulse sequences or reconstruction assumptions or processing have affected these findings. At present, we believe there is not enough evidence from the relatively few studies performed so far to indicate a need to redefine existing models.
While the main goal of this study was to measure the correspondence of DTMREIT and CTI reconstructions, the study also provided an opportunity to compare reconstructed conductivities within the phantom materials with standard fourterminal measurements. We found that the conductivity of the potato sample was higher than previously measured for this tissue, while that for chicken was very similar to the phantom background conductivity of around 0.4 S/m. As the phantom was prepared over 6 h before imaging commenced. It is likely the case that salt from the background material was absorbed into the potato starch, and there was also osmotic equalisation of ionic concentrations between the chicken and background material. The correspondence of DTMREIT and CTI reconstructions in the phantom was otherwise reasonable.
Conclusions
Using both biological tissue phantom data and human data we have demonstrated that CTI, DTMREIT and MREIT methods provide tissue conductivities within 0.05 S/m and scaling factors equivalent within 50 s/mm\(^3\) We anticipate that properties of tissues during conduction using MREIT are likely different than those found from diffusion methods but are not presently observable. From our survey of recent findings of brain tissue conductivity using MRbased techniques, we observed that while recent findings at Larmor frequencies closely align with theoretical predictions, those obtained at lower frequencies are roughly thirty times higher than predicted values. However, some experimental measurements at lower frequencies exhibit a closer correspondence with theoretical models. We anticipate that future benchmarking studies using both direct impedance measurements and MRIbased reconstructions will further illuminate the true nature of brain and other tissue conductivities in the lowfrequency range.
Methods
Imaging experiments
All MR data were measured using a 32channel RF head coil in a 3 T Phillips scanner (Phillips, Ingenia, Netherlands) located at the Barrow Neurological Institute (Phoenix, Arizona, USA). The human subject study was approved by the Arizona State University Institutional Review Board under protocol STUDY 00006012. Identical imaging parameters were used for both phantom and human experiments, except as noted below. Briefly, volumes obtained in experiments comprised a highresolution structural study using a MagnetizationPrepared Rapid GradientEcho (MPRAGE) sequence, susceptibilitycorrected high angular resolution diffusion images (HARDI), multispinecho \(B_1\) mapping, and a multigradient echobased sequence used to obtain MREIT data. A transcranial electrical stimulator (DCSTIMULATOR MR, neuroConn, Ilmenau, Germany) was used to deliver 1.5 mAcurrents at approximately 10 Hz through two pairs of opposing electrodes used in either human or phantom imaging studies (\(\mathcal {E}=1,\ 2\)). Finally, a multibvalue EPIbased diffusion volume (fifteen bvalues, six directions) was obtained for determination of tissue microstructure parameters. Table 3 summarizes imaging parameters used in each experiment and sequence.
Phantom construction
A cylindrical acrylic container with a diameter of 150 mm and a height of 140 mm was used to build the biological tissue phantom. Two opposing pairs of 50 \(\times\) 50 mm\(^2\) carbon electrodes (HUREV, South Korea) were attached to the container perimeter. Three sections of chicken muscle, oriented in x (top), y (left), and diagonal xydirections (right) were then placed inside the chamber to provide anisotropic tissue properties (Fig. 10A–C. From previous fourelectrode impedance measurements made on similar cubic tissue samples, we estimated the approximate conductivity of the chicken along muscle fibers was 0.86 S/m and across muscle fibers 0.73 S/m^{24}. A cubic conductive anomaly made of potato (height 40 mm, assumed conductivity of approximately 0.2 S/m at 10 Hz^{24}) was also placed at the bottom of the container. The phantom background was then filled with a 0.4 S/m (10 Hz) agarose gel. Two MREIT runs at each current orientation were performed (total MREIT imaging time of 32 min). Four 5mm thick slices of MREIT data were acquired.
Human imaging
All procedures were approved by the Institutional Review Board of Arizona State University under STUDY00006012, and all interventions were performed in compliance with relevant institutional guidelines and regulations. Informed consent was obtained prior to commencing all procedures involving human subjects. The human subject used in this study was aged 60 years. The region of the subject head surveyed along with placement of electrodes for MREIT current delivery is outlined in Fig. 11A and a representative magnitude slice, as well as regions of interest (gray matter, white matter and cerebrospinal fluid) is showed in Fig. 11B,C. For human subjects only, MREIT data were gathered on three slices. This was to satisfy protocol requirements that subjects not undergo more than 6minutes’ continuous AC stimulation at 1.5 mA. Two MREIT runs for each current orientation were performed (total MREIT imaging time of 24 min).
Data processing
Both DTMREIT, CTI and MREIT images are obtained after multiple, sometimes interdependent processing steps. The important factors in each processing pipeline are described below.
Diffusion tensor images
Both human and phantom diffusion EPIbased HARDI diffusion weighted imaging data were processed to tensor information using three image volumes: two sixdirection scans: one gathered with phase encoding steps collected in the anteriorposterior direction and the other with the direction reversed. A third 64direction diffusion scan was then collected using the same field of view and matrix size. All diffusion tensor data were reconstructed using b = 1000 s/mm\(^2\), and diffusion characteristics at this b value were considered representative of extracellular properties. The two sixdirection diffusion volumes were used to correct the 64direction volume for eddy current and susceptibility artifact data using the FSL eddy tool^{54} implemented in the FMRIB Software Library v6.0 (https://fsl.fmrib.ox.ac.uk). To match the dimension of the MREIT/MREPT data, the acquired DW and DT datasets were interpolated to 128 \(\times\) 128 matrix size in subsequent data processing steps. Principal eigenvector data for a representative slice of each volume, modulated by fractional anisotropy are shown in Figs. 10 and 11G respectively. Figure 10 confirms the orientation of the chicken muscles in the leftright (top, red), updown (left, green) and diagonal (right, orange) directions respectively. Some anisotropy was also apparent in the potato sample.
Estimation of \(\mathbb {D}\)
To estimate \(\mathbb {D}=\mathbb {D}^w_{e}\) in this paper, we adopted the method proposed by Lee et al.^{9}. Specifically, eigenvalues (\(d_{b,i}~\text{ where, }~i = 1, 2, 3\)) of DTI tensors (\(\mathbb {D}_{b}\)) obtained from any bvalue were scaled to match the extracellular diffusion coefficient using
For both human and phantom experiments. \(\mathbb {D}\) was estimated using the diffusion tensor estimated at \(b=1000~\text{ s }/\text{mm}^2\) and 64directions. Figures 10 and 11D–G illustrate normalized DWMR images for the phantom and human subject respectively at b values of 500, 2600 and 4500 s/mm\(^2\) as well as a colorcoded fractional anisotropy map of a focus slice.
EPT data processing
Regions of interest used in EPT data processing, a representative T1weighted magnitude slice and a comparison magnitude image of the focal slice (slice 2) of the multispinecho data used to obtain \(B_1\) phase data are shown in Fig. 13A–C respectively. Phase in the representative slice is illustrated in Fig. 13D. \(B_1\) images were reconstructed from raw kspace data. Individual channel data were combined to form phase images using Generic Referenceless Phase Combination (GRPC)^{55,56}. No phase unwrapping was required. Data from all echoes were combined in analysis of phantom data. However, including the first echo of human subject data resulted in SNR degradation. Therefore firstecho data were removed to avoid including inaccurate \(T_2\) weighting factors in the echo combination step^{47}. A 3 \(\times\) 3 median filtering window was also applied to human phase data to remove artifacts in the posterior brain caused by blood flow. Highfrequency conductivity distributions were performed using convection reactionregularized EPT (crEPT) reconstruction^{12,57} implemented in the MRCI toolbox.
Briefly, the crEPT method involves combining Ampere’s and Faraday’s laws^{12,58} to obtain an equation of the form
where \(\phi ^{tr}\) is the \(B_1\) transceive phase, \(\rho\) is resistivity, \(\omega\) is frequency and \(\mu _0\) is the permeability of vacuum. To obtain a stable numerical solution in crEPT to obtain \(\sigma _H\), an artificial diffusion term scaled by a regularizing parameter (c) is added^{12}, therefore producing
In practice, c values around 0.005–0.1 are often employed, but larger values may be required if signal to noise ratios are low.
Figures 12H and 13H show \(\sigma _H\) images reconstructed with diffusion term coefficients c of 0.1 and 0.1 respectively, although a range of c between 0.02 and 0.1 was tested. A boundary conductivity of 0.4 S/m was assumed for phantom data and boundary conductivity of 0.6 S/m was used for human data for EPT reconstructions.
Multibvalue diffusion
Diffusion data at 15 different bvalues were collected and used to estimate \(\alpha , d_i\) and \(d_e\) values for CTI reconstruction.
The extracellular volume fraction, \(\alpha\), extra and intracellular diffusion coefficients, \(d_e\) \(d_i\) were estimated as^{9,32},
The model parameters i.e. isotropic and intracellular volume fraction \(f_{iso}\) \(f_{ic}\) respectively, and the extracellular mean diffusivity \(d_e^{*}\) were estimated as^{9,32}
where \(S_b\) and \(S_0\) are the signal with and without diffusion gradient, \(d_{iso} = 3 \times 10^{3}~\text{ mm}^2/\text{s }\) is the isotropic water diffusion and \(d_{ic}\) = \(1.7 \times 10^{3}~\text{ mm}^2/\text{s }\) is the intracellular diffusion coefficient^{9,32}.
Under the assumption that the CTIparameters are independent of the measured gradient directions, in this paper we estimated \(\hat{\alpha }\), \(\hat{d_e}\) and \(\hat{d_i}\) as
where \(K = \sum _{k=1}^{N_d1} C^{N_d}_{k}\) is the total number of unique subsets of gradient directions and \(N_d\) is the number of acquired gradient directions. For human data, 6diffusion gradient direction data was measured, therefore, \(K=57\), For phantom \(K=57\) was also used.
To estimate the CTIparameters in Eq. (15), the each bvalue signal was averaged over the diffusion gradients as
where \(N_{d, k} < N_d\) is the number of diffusion gradient signals in each subset applied to measure the diffusion tensor at particular bvalue. Note that averaging over gradient directions maintains the exponential property of diffusion signal with respect to bvalues^{8}. Each subset of \(\bar{S}_{b, k}\) was then fitted to the signal model in equation (15) to obtain the microstructure parameters (\(f_{iso, k}\), \(f_{ic, k}\), \(d_{e,k}^{*}\))
The minimization was carried out by employing a variant of the MATLAB (The Mathworks Inc., Natick, MA, USA) fminsearch function, which allowed the definition of boundary conditions of the search space^{32}. For human data, in each individual run \(f_{ic, k}\) and \(f_{iso, k}\) were constrained to be within the range [0, 1], and \(d_{e,k}^{*}\) was limited to be nonnegative^{32}. The initial conditions were set to \(f_{ic, k} =0\), \(f_{iso, k}=0.5\), \(d_{e, k}^{*} = 0~\text{ mm}^2/\text{s }\). For the phantom, the initial value, lower and upper bound of each individual run was set to \(\left( f_{ic, k}, f_{iso, k}, d_{e, k}^{*}\right) =\left( 0, 0.2, 1.5 \times 10^{3}~\text{ mm}^2/\text{s }\right)\), \(\left( 0, 0, 0~\text{ mm}^2/\text{s }\right)\), and \(\left( 1, 1, 3 \times 10^{3}~\text{ mm}^2/\text{s } \right)\), respectively. Note that for phantom these values were determined empirically for faster convergence. The final estimates of \(d_e\), \(d_i\) and \(\alpha\) for phantom and human data are shown in a focus slice for phantom and human data in Figs. 12 and 13E–G respectively.
CTI
After determination of \(d_e\), \(d_i\), \(\alpha\) and \(\mathbb {D}\), Eq. (5) was used to calculate \(\mathbb {C}_{CTI}\). The unknown \(\beta\)value was set to 0.41, based on earlier empirical estimations of this factor^{8,34,35}. For the phantom, we assumed that \(\beta\) was 1.
DTMREIT
Projected current density reconstruction
The measured multigradientecho phase data gathered during application of MREIT currents, in Philips PAR/REC format, was optimized over all 10 echoes using the method of Oh et al.^{59} and converted to \(B_z\) distributions via the relation \(B_z=\phi /\gamma T_c\), where \(\phi\) is the measured phase, \(\gamma\) is the proton gyromagnetic ratio (\(267.5 \times 10^{6}\) rad T\(^{1}\) s\(^{1}\)) and \(T_c\) is the current application time (34 ms). Human \(B_z\) data were corrected for stray magnetic fields created by wiring^{60,61}, but no corrections were applied to phantom data.
The projected current density was estimated as^{38}
For phantom data \(\psi\) was determined via,
For human data regional projected current density^{30}, the projected current density was found as
where
In (24), \(R_t\) is a twodimensional closed domain excluding the skull and skin regions and \(\textbf{J}_0\), and \(B_z^0\) are the homogeneous distribution of current and \(z\)component of magnetic flux density obtained by solving the MREIT forward problem with \(\sigma _0 = 1\) S/m, respectively.
Overall geometry and a representative magnitude slice (slice 3) of the multigradient echo used in reconstructing MREIT data are shown in Fig. 14A,B respectively. The optimized \(B_z\) and reconstructed projected current density magnitude images \({\textbf{J}}^{P}_{\mathcal {E}},\ \mathcal {E}=1,\ 2\) in the phantom are shown in a focus slice in Fig. 14C,D,E,F respectively. Similarly, a representative magnitude slice (slice 2) of human multi gradient echo data and a 3D view of phantom geometry are shown in Fig. 15A,B, and optimized \(B_z\) and reconstructed images for a representative slice are displayed in Fig. 15C,D,E,F respectively.
MREIT
For comparisons to both DTMREIT and CTI, the effective lowfrequency conductivity was also reconstructed using the noniterative Harmonic \(B_z\) algorithm^{50} implemented in the MRCI toolbox^{40}. Using two independent current injections, the noniterative Harmonic \(B_z\) algorithm estimates the parameter \(\textbf{s}(\textbf{r}) = \nabla {\log {\sigma }}(\textbf{r})\) as
Here, the stiffness matrix, \(\textbf{A} = \begin{bmatrix} (\textbf{J}^{P}_1)_y(\textbf{r}) &{}  (\textbf{J}^{P}_1)_x(\textbf{r}) \\ (\textbf{J}^{P}_2)_y(\textbf{r}) &{}  ( \textbf{J}^{P}_2)_x(\textbf{r}) \end{bmatrix}\) and load vector \(\textbf{b} = \frac{1}{\mu _0}\begin{bmatrix} \nabla ^2 B_{z,1}(\textbf{r}) \\ \nabla ^2 B_{z,2}(\textbf{r}) \end{bmatrix}\). The pixelwise regularization parameter was chosen to be \(\lambda (\textbf{r}) \propto \frac{1}{\left \textbf{A}^T \textbf{A}\right }\) and \(\textbf{I}\) is the \(2 \times 2\) identity matrix.
The MREIT conductivity was reconstructed by solving Poisson’s equation
Here, \(\Omega _t\) is the imaging slice and \(\sigma _L^{CTI}=\eta d_e\) is the estimated low frequency conductivity reconstructed using the CTI method. Note in Eq. (26) this conductivity was assigned only on the slice boundary \(\partial \Omega _t\).
Data equivalence tests
Comparisons of \(\eta\) obtained using CTI or DTMREIT, and \(\textbf{C}^{CTI}_L\) and \(\textbf{C}_L^{MREIT}\) were performed using a two onesided test (TOST)^{41,42}. An equivalence tolerance of ± 50 S ms/mm\(^3\) in \(\eta\) values and \(\pm ~0.05\) S/m for \(\textbf{C}^{CTI}_L\) and \(\textbf{C}_L^{MREIT}\) comparisons was specified. The significance level was set at 0.05.
Code availability
Accession codes: The datasets generated during and analyzed during the current study are available in the Arizona State University Research Data Repository by searching for the Authors, title or DOI of this publication.
Abbreviations
 LF:

: Low frequency (< 1 MHz)
 MRI:

: Magnetic resonance imaging
 MREIT:

: Magnetic resonance electrical impedance tomography
 MREPT, EPT:

: Magnetic resonance electric properties tomography
 CTI:

Conductivity tensor imaging
 DTMREIT:

Diffusion tensor magnetic resonance electrical impedance tomography
 \(\mathbf {\sigma }_H\) :

EPT conductivity
 \(\mathbb {C}^{CTI}\) :

CTI conductivity tensor
 \(\mathbb {\eta }^{CTI}\) :

CTI conductivity scale factor
 \(\textbf{C}_L^{CTI}\) :

Isotropic LF CTI conductivity
 \(\mathbb {C}^{DTMREIT}\) :

DTMREIT conductivity tensor
 \(\mathbb {\eta }^{DTMREIT}\) :

DTMREIT conductivity scale factor
 \(\textbf{C}_L^{MREIT}\) :

Isotropic LF MREIT conductivity
References
Gabriel, C., Peyman, A. & Grant, E. H. Electrical conductivity of tissue at frequencies below 1 MHz. Phys. Med. Biol. 54, 4683–4878 (2009).
Katscher, U., Stehning, C. & Tha, K. K. The impact of CSF pulsation on reconstructed brain conductivity. In Proceedings of the 26th Annual Meeting of the ISMRM 546 (ISMRM, 2018).
Seo, J. K. & Woo, E. J. Electrical tissue property imaging at low frequency using MREIT. IEEE Trans. Biomed. Eng. 61, 1390–1399 (2014).
Seo, J. & Woo, E. Magnetic resonance electrical impedance tomography (MREIT). SIAM Rev. 53, 40–68 (2011).
Basser, P. J., Mattiello, J. & LeBihan, D. MR diffusion tensor spectroscopy and imaging. Biophys. J. 66, 259–267 (1994).
Tuch, D. S., Wedeen, V. J., Dale, A. M., George, J. S. & Belliveau, J. W. Conductivity tensor mapping of the human brain using diffusion tensor MRI. Proc. Natl. Acad. Sci. U. S. A. 98, 11697–11701 (2001).
Kwon, O., Jeong, W. C., Sajib, S. Z. K., Kim, H. J. & Woo, E. J. Anisotropic conductivity tensor imaging in MREIT using directional diffusion rate of water molecules. Phys. Med. Biol. 59, 2955–2974 (2014).
Sajib, S. Z. K., Kwon, O. I., Kim, H. J. & Woo, E. J. Electrodeless conductivity tensor imaging (CTI) using MRI: Basic and animal experiments. Biomed. Eng. Lett. 8, 273–282 (2018).
Lee, M. B., Jahng, G.H., Kim, H. J., Woo, E. J. & Kwon, O. I. Extracellular electrical conductivity property imaging by decomposition of highfrequency conductivity at Larmorfrequency using multibvalue diffusionweighted imaging. PLoS One 15, e0230903 (2020).
Jahng, G.H., Lee, M. B., Kim, H. J., Woo, E. J. & Kwon, O. I. Lowfrequency dominant electrical conductivity imaging of in vivo human brain using highfrequency conductivity at Larmorfrequency and spherical mean diffusivity without external injection current. NeuroImage 225, 117466 (2021).
Seo, J. K. et al. Electrical tissue property imaging using MRI at DC and Larmor frequency. Inverse Prob. 28, 084002 (2012).
Gurler, N. & Ider, Y. Z. Gradientbased electrical conductivity imaging using MR phase. Magn. Reson. Med. 77, 137–150 (2016).
Tanner, J. E. Selfdiffusion of water in frog muscle. Biophys. J. 28, 107–116 (1979).
van Gelderen, P., DesPres, D., van Zijl, P. C. M. & Moonen, C. Evaluation of restricted diffusion in cylinders phosphocreatine in rabbit leg muscle. J. Magn. Reson. Ser. B 103, 255–260 (1994).
Stanisz, G. J., Szafer, A., Wright, G. A. & Henkelman, R. M. An analytical model of restricted diffusion in bovine optic nerve. Magn. Reson. Med. 37, 103–111 (1997).
Katoch, N. et al. Conductivity tensor imaging of in vivo human brain and experimental validation using giant vesicle suspension. IEEE Trans. Med. Imaging 38, 1569–1577 (2019).
Chauhan, M. et al. Lowfrequency conductivity tensor imaging of the human head in vivo using DTMREIT: First study. IEEE Trans. Med. Imaging 37, 966–976. https://doi.org/10.1109/TMI.2017.2783348 (2018).
Indahlastari, A., Chauhan, M., Schwartz, B. & Sadleir, R. J. Changing head model extent affects finite element predictions of transcranial direct current stimulation distributions. J. Neural Eng. 13, 066006. https://doi.org/10.1088/17412560/13/6/066006 (2016).
Antonenko, D. et al. Towards precise brain stimulation: Is electric field stimulation related to neuromodulation?. Brain Stimul. 12, P11591168. https://doi.org/10.1016/j.brs.2019.03.072 (2019).
FernandezCorazza, M. et al. Source localization of epileptic spikes using multiple sparse priors. Clin. Neurophysiol. 132, 586–597. https://doi.org/10.1016/j.clinph.2020.10.030 (2021).
Davids, M., Guerin, B., Malzacher, M., Schad, L. R. & Wald, L. Predicting magnetostimulation thresholds in the peripheral nervous system using realistic body models. Sci. Rep. 7, 5316. https://doi.org/10.1038/s41598017054939 (2017).
Jeong, W. C. et al. Anisotropic conductivity tensor imaging of canine brain using DTMREIT. IEEE Trans. Med. Imaging 36, 124–131. https://doi.org/10.1109/TMI.2016.2598546 (2017).
Gabriel, S., Lau, R. W. & Gabriel, C. The dielectric properties of biological tissues: III Parametric models for the dielectric spectrum of tissues. Phys. Med. Biol. 41, 2271–2293 (1996).
Sajib, S. Z. K., Chauhan, M., Kwon, O. I. & Sadleir, R. J. Magneticresonancebased measurement of electromagnetic fields and conductivity in vivo using single current administration  a machine learning approach. PLoS One 16, e0254690. https://doi.org/10.1371/journal.pone.0254690 (2021).
Song, Y. et al. Low frequency conductivity reconstruction based on a single current injection via MREIT. Phys. Med. Biol. 65, 225016. https://doi.org/10.1088/13616560/abbc4d (2020).
Saurav Z. K. et al. Electrical Properties of Tissues Quantitative Magnetic Resonance Mapping Magnetic Resonance Current Density Imaging (MRCDI) 135–155 (Springer International Publishing Cham, 2022).
Nam, H., Park, C. & Kwon, O. I. Noniterative conductivity reconstruction algorithm using projected current density in MREIT. Phys. Med. Biol. 53, 6947–61 (2008).
Kim, H. J. et al. Conductivity imaging of canine brain using a 3 T MREIT system: Postmortem experiments. Physiol. Meas. 28, 1341 (2007).
Seagar, A. D., Barber, D. C. & Brown, B. H. Theoretical limits to sensitivity and resolution in impedance imaging. Clin. Phys. Physiol. Meas. 8A, 13–31 (1987).
Sajib, S. Z. K., Kim, H. J., Woo, E. J. & Kwon, O. I. Regional absolute conductivity reconstruction using projected current density in MREIT. Phys. Med. Biol. 57, 5841–5859 (2012).
Minhas, A. S., Chauhan, M., Fu, F. & Sadleir, R. J. Evaluation of magnetohydrodynamic effects in magnetic resonance electrical impedance tomography at ultrahigh magnetic fields. Magn. Reson. Med. 81, 2264–2276. https://doi.org/10.1002/mrm.27534 (2018).
Marino, M., CorderoGrande, L., Mantini, D. & Ferrazzi, G. Conductivity tensor imaging of the human brain using water mapping techniques. Front. Neurosci. 15, 694645. https://doi.org/10.3389/fnins.2021.694645 (2021).
Michel, E., Hernandez, D. & Lee, S. Y. Electrical conductivity and permittivity maps of brain tissues derived from water content based on T_{1}weighted acquisition. Magn. Reson. Med. 77, 1094–1103 (2017).
Volkov, A. G., Paula, S. & Deamer, D. W. Two mechanisms of permeation of small neutral molecules and hydrated ions across phospholipid bilayers. Bioelectrochem. Bioenerg. 42, 153–160. https://doi.org/10.1016/S03024598(96)050970 (1997).
Hansen, A. J. Effect of anoxia on ion distribution in the brain. Physiol. Rev. 65, 101–148 (1985).
Kwon, O. I., Lee, M. B. & Jahng, G. H. High frequency conductivity decomposition by solving physically constrained underdetermined inverse problem in human brain. Sci. Rep. 13, 3273 (2023).
Ma, W., DeMonte, T., Nachman, A., Elsaid, N. M. H. & Joy, M. Experimental implementation of a new method of imaging anisotropic electric conductivities. In Proc. 35th Ann. Int. Conf. IEEE EMBS 6437–6440 (2013).
Park, C., Lee, B. & I., K. O. Analysis of recoverable current from one component of magnetic flux density in MREIT. Phys. Med. Biol.52, 11 (2007).
Sajib, S., Lee, M., Kim, H. E. J. W. & Kwon, O. I. Extracellular total electrolyte concentration imaging for electrical brain stimulation (EBS). Sci. Rep. 8, 290 (2018).
Sajib, S. Z. K., Katoch, N., Kim, H. J., Kwon, O. I. & Woo, E. J. Software toolbox for lowfrequency conductivity and current density imaging using MRI. IEEE Trans. Biomed. Eng. 64, 2505–2514 (2017).
Schuirmann, D. J. A comparison of the two onesided tests procedure and the power approach for assessing the equivalence of average bioavailability. J. Pharmacokinet. Biopharm. 15, 657–680. https://doi.org/10.1007/BF01068419 (1987).
Rogers, J. L., Howard, K. I. & Vessey, J. T. Using significance tests to evaluate equivalence between two experimental groups. Psychol. Bull. 113, 553–565. https://doi.org/10.1037/00332909.113.3.553 (1993).
Sajib, S., Chauhan, M., Sahu, S., Boakye, E. & Sadleir, R. J. Comparison of MRbased lowfrequency electrical conductivity tensor using DTMREIT and CTI: A biological tissue phantom study. In Proceedings of 29th Annual Meeting of the International Society of Magnetic Resonance in Medicine, 3786 (2021).
Zhang, H., Schneider, T., WheelerKingshott, C. A. & Alexander, D. C. NODDI: Practical in vivo neurite orientation dispersion and density imaging of the human brain. NeuroImage 61, 1000–1016 (2012).
Kaden, E., Kruggel, F. & Alexander, D. C. Quantitative mapping of the peraxon diffusion coefficients in brain white matter. Magn. Reson. Med. 75, 1752–1763. https://doi.org/10.1002/mrm.25734 (2016).
Li, H., Nikam, R., Kandula, V., Chow, H. M. & Choudhary, A. K. Comparison of NODDI and spherical mean signal for measuring intraneurite volume fraction. Magn. Reson. Imaging 57, 151–155 (2019).
BenEliezer, N., Sodickson, D. K. & Block, K. T. Rapid and accurate T_{2} mapping from multispinecho data using blochsimulationbased reconstruction. Magn. Reson. Med. 73, 809–817 (2015).
Kim, D., Jensen, J. H., Wu, E. X., Sheth, S. S. & Brittenham, G. M. Breathhold multiecho fast spinecho pulse sequence for accurate R_{2} measurement in the heart and liver. Magn. Reson. Med. 62, 300–306 (2009).
Fatemi, Y., Danyali, H., Helfroush, M. S. & Amiri, H. Fast T_{2} mapping using multiecho spinecho MRI: A linear order approach. Magn. Reson. Med. 84, 2815–2830 (2020).
Seo, J. K. & Woo, E. J. Noniterative harmonic B_{z} algorithm in MREIT. Inverse Probl. 27, 1–12 (2011).
Geddes, L. & Baker, L. E. The specific resistance of biological materials: A compendium of data for the biomedical engineer and physiologist. Med. Biol. Eng. Comput. 5, 271–293 (1967).
Mandija, S., Meliadò, E. F., Huttinga, N. R. F., Luijten, P. R. & van den Berg, C. A. T. Opening a new window on MRbased electrical properties tomography with deep learning. Sci. Rep. 9(1), 8895. https://doi.org/10.1038/s4159801945382x (2019).
van Lier, A. L. H. M. W. et al. Electrical properties tomography in the human brain at 1.5, 3, and 7T: A comparison study. Magn. Reson. Med. 71(1) 354–363. https://doi.org/10.1002/mrm.v71.110.1002/mrm.24637 (2014).
Andersson, J. L. R. & Sotiropoulos, S. N. An integrated approach to correction for offresonance effects and subject movement in diffusion MR imaging. Neuroimage 125, 1063–1078 (2016).
Santini, F., Ganter, C., Ehses, P., Scheffler, K. & Bieri, O. A generic referenceless phase combination (GRPC) method: application at high and ultrahigh fields. In Proceedings of the 23rd Annual Meeting of ISMRM, 3724 (2015).
Santini, F., Santin, M. D., de Sousa, P. L. & Bieri, O. Twostep generic referenceless phase combination (GRPC) for accurate phase image reconstruction from multiple receiver coils. In Proceedings of the 24th Annual Meeting of ISMRM, 4314 (2016).
Hafalir, F. S., Oran, O. F., Gurler, N. & Ider, Y. Z. Convectionreaction equation based magnetic resonance electrical properties tomography (crMREPT). IEEE Trans. Med. Imaging 33, 777–793 (2014).
Katscher, U., Minhas, A. S. & Katoch, N. Magnetic Resonance Electrical Properties Tomography (MREPT) 185–202 (Springer International Publishing, 2022).
Oh, T. I. et al. Noise analysis in fast magnetic resonance electrical impedance tomography (MREIT) based on spoiled multi gradient echo (SPMGE) pulse sequence. Phys. Med. Biol. 59, 4723–4738 (2014).
Sajib, S. Z. K. et al. Compensation of leadwire magnetic field contributions in mreit experiment using image segmentation: a phantom study. In Proceedings of the 27th Annual Meeting of the International Society for Magnetic Resonance in Medicine, 5049 (2019).
Göksu, C. et al. Human invivo brain magnetic resonance current density imaging (MRCDI). NeuroImagehttps://doi.org/10.1016/j.neuroimage.2017.12.075 (2018).
Acknowledgements
R.J.S. discloses support for the research of this work from the National Institute of Mental Health under RF1MH114290. R.J.S. and S.Z.K.S. thank Dr. Camelia Gabriel for her advice on the content of the manuscript.
Author information
Authors and Affiliations
Contributions
S.Z.K.S, M.C. and R.J.S. conceived the experiments. M.C., E.B., S.S. and R.J.S. conducted the experiments, S.Z.K.S, R.J.S and M.C. analysed the results. S.Z.K.S. and R.J.S. wrote the manuscript. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher's note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License, which permits any noncommercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/byncnd/4.0/.
About this article
Cite this article
Sajib, S.Z.K., Chauhan, M., Sahu, S. et al. Validation of conductivity tensor imaging against diffusion tensor magnetic resonance electrical impedance tomography. Sci Rep 14, 17995 (2024). https://doi.org/10.1038/s4159802468551z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4159802468551z
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.