Validation of conductivity tensor imaging against diffusion tensor magnetic resonance electrical impedance tomography

Abstract

Diffusion tensor magnetic resonance electrical impedance tomography (DT-MREIT) and electrodeless conductivity tensor imaging (CTI) are two emerging modalities that can quantify low-frequency tissue anisotropic conductivity properties by assuming similar properties underlie ionic mobility and water diffusion. While both methods have potential applications to estimating neuro-modulation fields or formulating forward models used for electrical source imaging, a direct comparison of the two modalities has not yet been performed in-vitro or in-vivo. 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 DT-MREIT 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 DT-MREIT 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 non-invasive method of measuring low frequency conductivity properties.

Introduction

Direct acquisition of low-frequency (LF) tissue electrical properties is often complicated because of the need to deliver and measure electrical currents via electrodes attached to the body¹. We define LF properties as being those below about 1 MHz.

Magnetic resonance imaging (MRI) approaches provide a unique opportunity to measure electrical properties non-invasively. 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 equations². 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 (DT-MREIT) technique to allow reconstruction of LF conductivity tensor distributions using prior information provided by diffusion tensor data. However, both MREIT and DT-MREIT 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 low-frequency conductivity distributions^9,10 by combining knowledge of high-frequency conductivity $\sigma _H$ obtained from MREPT^11,12 with tissue microstructure parameters estimated from multi-b-value diffusion-weighted images^13,14,15, without administering external current^8,16.

In DT-MREIT, 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, diffusion-weighted and $\sigma _H$ data are combined to reconstruct $\eta$. Both techniques are highly relevant to neurology, neuroimaging and neuromodulation, as subject-specific 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 EEG²⁰ and in determining the likelihood of peripheral neural stimulation in MR imaging²¹.

In both DT-MREIT and CTI,

$$\begin{aligned} \mathbb {C}\approx \eta \mathbb {D}. \end{aligned}$$

(1)

where $\mathbb {C}$ is the conductivity tensor and $\mathbb {D}$ is a representative diffusion tensor measured with a single diffusion-weighting factor b. Both DT-MREIT⁷ and CTI⁸ methods have been used to determine LF anisotropic conductivity distributions in-vivo in canine^8,22 and human brains^16,17. However, the correspondence of the two modalities has not yet been investigated. DT-MREIT 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 range²³, the goal of this study was to determine how well LF electrical tissue conductivities derived from in vivo DT-MREIT 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 DT-MREIT, for example⁷.

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 water-related 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 DT-MREIT experiments made to date have recovered higher gray and white matter conductivities than predicted by the lower-frequency 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 them¹.

Figure 1

Electrical spectra for gray matter, white matter and CSF, showing both conductivity and permittivity ($\varepsilon _r$) for the three tissues.

Factors influencing practical measurement of LF electromagnetic properties using MRI

There are multiple factors that may affect validation of CTI against DT-MREIT. 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 DT-MREIT 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 DT-MREIT imaging when current flow is perpendicular to them^24,25,26, DT-MREIT 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 it²⁷. DT-MREIT-measured 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 square-wave excitations at 5–10 Hz^17,22,28. As noted above, at such low frequencies, little interrogation of intracellular compartments is expected in DT-MREIT due to the conductive barriers posed by cell walls.

A saturation phenomenon may also manifest differences between CTI and MREIT or DT-MREIT 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 unity²⁹. This saturation effect will result in lower than expected reconstructed MREIT or DT-MREIT conductivities in interior areas with high conductivity such as CSF, as the contrast gradient at the CSF-parenchymal boundary is particularly high (see Fig. 1). Regional reconstruction algorithms can be employed to overcome this effect³⁰. We did not use this strategy here, as it requires additional assumptions of interior conductivities to be made.

Reconstructed MREIT or DT-MREIT conductivities may also be dynamically affected by imaging current flows. An extreme difference, observable at very high imaging fields, is that charges displaced by DT-MREIT current flow may be subject to Lorentz forces and magnetohydrodynamic effects that disrupt recordings of the expected magnetic flux patterns caused by the external current³¹.

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 conventionally-determined conductivity. We will assign the effective frequency of CTI methods to be 0 Hz. However, we expect that values found using either DT-MREIT 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 DT-MREIT and MREIT require current density data from several different current directions to be acquired to provide a stable solution. This results in smoothing of DT-MREIT 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 DT-MREIT 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

$$\begin{aligned} \sigma _v=\frac{\sigma _e}{d_e}\left[ d_v\left( \frac{d_i}{3d_e}+1\right) +\frac{d_v^2d_i}{3d_e^2}-\frac{2}{3}d_i\right] +O(d_i^2) \end{aligned}$$

(2)

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 low-frequency 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 theory¹⁵, the LF equivalent isotropic conductivity

$$\begin{aligned} \textbf{C}_L^{CTI}=\eta ^{CTI}d_e^w \end{aligned}$$

(3)

and anisotropic conductivity tensor,

$$\begin{aligned} \mathbb {C}^{CTI}=\eta ^{CTI}\mathbb {D}=\eta ^{CTI}\mathbb {D}_e^w \end{aligned}$$

(4)

are determined from the reconstructed extra-cellular water self-diffusivity, $d_e^w$ and diffusion tensor $\mathbb {D}_e^w$. The CTI scale-factor $\eta$ is determined as⁸

$$\begin{aligned} \eta ^{CTI}=\ \frac{\alpha \sigma _H}{\alpha d_e^w+\beta \left( 1-\alpha \right) d_i^w} \end{aligned}$$

(5)

where ($1-\alpha$) and $d_i^w$ denote the estimated intra-cellular space volume fraction and water diffusivity, respectively and $\beta$ is the intra-to-extra-cellular ion concentration ratio. This last quantity is estimated from independent studies^34,35 or fitted³⁶. In this work, a convection–reaction based partial-differential-equation (PDE) was solved to obtain $\sigma _H$ from the measured $B_1$-phase data (cr-EPT)¹². The L$^2$-norm of a multi-compartment model was fitted to the observed multi-b-value-DW 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.

DT-MREIT 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

$$\begin{aligned} \nabla \times \left( \mathbb {D}_b^{-1}\textbf{J}\right) =-\frac{\nabla \eta }{\eta }\times \left( \eta \nabla u\right) =\nabla \log \eta \times \left( \mathbb {D}_b^{-1}\textbf{J}\right) . \end{aligned}$$

(6)

If data from two independent current administrations are measured, Kwon et al.⁷ proposed reconstructing $\eta$ as

$$\begin{aligned} \begin{bmatrix}\left( \mathbb {D}_b^{-1}\textbf{J}^P_1\right) _y &{} -\left( \mathbb {D}_b^{-1}\textbf{J}^P_1\right) _x \\ \left( \mathbb {D}_b^{-1}\textbf{J}^P_2\right) _y &{} -\left( \mathbb {D}_b^{-1}\textbf{J}^P_2\right) _x \end{bmatrix} \begin{bmatrix}\frac{\partial \log \textbf{e}}{\partial x}\\ \frac{\partial \log \textbf{e}}{\partial y}\end{bmatrix}=\begin{bmatrix}\frac{\partial \left( \mathbb {D}_b^{-1}\textbf{J}^P_1\right) _y}{\partial x} -\frac{\partial \left( \mathbb {D}_b^{-1}\textbf{J}^P_1\right) _x}{\partial y} \\ \frac{\partial \left( \mathbb {D}_b^{-1}\textbf{J}^P_2\right) _y}{\partial x} -\frac{\partial \left( \mathbb {D}_b^{-1}\textbf{J}^P_2\right) _x}{\partial y}\end{bmatrix}. \end{aligned}$$

(7)

where $\textbf{J}^P$ is the projected current density²⁷ 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 uniform-conductivity 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 density³⁸.

To solve the linear system in (7) for $\textbf{e}$, an optimum regularization parameter was determined by minimizing the generalized cross-validation (GCV) function³⁹. The pixel-by-pixel 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 DT-MREIT reconstructions are contained in the “Methods” section.

Equation (7) was solved as,

$$\begin{aligned} {} \left\{ \begin{array}{lll} \nabla ^2 \log \eta ^{DT-MREIT} = \nabla \cdot \left( \nabla \textbf{e} \right) &{} \text{ in } \quad \Omega _t\\ \log \eta ^{DT-MREIT} = \log \eta ^{CTI} &{} \text{ in } \quad \partial \Omega _t \\ \end{array} \right. \end{aligned}$$

(8)

MREIT reconstructions

Using the estimated projected current density³⁸ $\textbf{J}^{P,\mathscr {E}}$, $\mathscr {E}=1, 2$, from two independent current applications it is also possible to determine the effective isotropic conductivity²⁷ $\left( \textbf{C}_L^{MREIT}\right)$ and conductivity tensor⁷ $\left( \mathbb {C}^{DT-MREIT}=\eta ^{DT-MREIT}\mathbb {D}_e^w\right)$. To reconstruct the $\textbf{C}_L^{MREIT}$, or $\eta ^{DT-MREIT}$ we must solve a two-dimensional Poisson’s equation with a known boundary value⁴⁰. Without prior knowledge of the phantom boundary conductivity or scale-factor values, both MREIT and DT-MREIT methods provide only ‘apparent’ conductivity contrasts. Since the CTI method is expected to provide the absolute conductivity, in this study we assigned the DT-MREIT 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, DT-MREIT 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 $\textbf{C}_L^{DT-MREIT}$ which could be calculated as $\textbf{C}_L^{DT-MREIT}=\eta ^{DT-MREIT}d_e^w$ as $\textbf{C}_L^{MREIT}$ is not dependent on diffusion data.

Figure 2

Reconstructed phantom $\eta$ data for all four slices for (top) CTI and (bottom) DT-MREIT reconstructions.

Figure 3

Reconstructed human $\eta$ data for all three slices for (top) CTI and (bottom) DT-MREIT reconstructions. CTI data were resampled onto DT-MREIT matrix for comparisons.

Figure 4

Reconstructed phantom $\mathbf{C}_L^{CTI}~\mathrm{and}~\mathbf{C}_L^{MREIT}$ data for all four slices for (top) CTI and (bottom) MREIT reconstructions.

Figure 5

Reconstructed human $\mathbf{C}_L^{CTI}~\mathrm{and}~\mathbf{C}_L^{MREIT}$ data for all three slices for (top) CTI (middle) MREIT and (bottom) $\sigma _H$ EPT reconstructions. CTI and EPT data were resampled onto MREIT matrix for comparisons.

Figure 6

ROIs used to compare $\eta$ with $\mathbf{C}_L^{CTI}~\mathrm{and}~\mathbf{C}_L^{MREIT}$ values in human subject. Three CSF, three white matter and three gray matter ROIs were chosen.

DT-MREIT and CTI eigenvalue comparisons

To compare anisotropic properties, we decomposed the reconstructed $\mathbb {C}$ $\left( \mathbb {C}^{CTI} or \mathbb {C}^{DT-MREIT}\right)$ 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 DT-MREIT reconstructions are presented in Fig. 7B.

Figure 7

Orthotropic tensor components in focus slice (slice 2) of (left) CTI and (right) DT-MREIT conductivity reconstructions. Longitudinal components of each tensor are shown at top, and transverse components are shown at bottom.

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 DT-MREIT 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.

Table 1 Comparison of scaling factors, orthotropic tensor components ($\mathbb {C}_p$, $\mathbb {C}_t$), $\mathbf {\sigma }_L$ and $\mathbf {\sigma }_H$ in each phantom compartment over all slices. Standard deviations within each compartment are shown.

Table 2 Comparison of scaling factors, orthotropic tensor components ($\mathbb {C}_p$, $\mathbb {C}_t$), $\mathbf {\sigma }_L$ and $\mathbf {\sigma }_H$ in three ROIs for each tissue chosen from central slice (slice 2). Standard deviations within each compartment or ROI are shown.

Discussion

Overall findings

Comparing both $\eta$ and conductivity values over DT-MREIT and CTI imaging results, DT-MREIT results were considerably smoothed, leading to lower $\eta ^{DT-MREIT}$ 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 one-sided method^41,42 found that CTI and DT-MREIT $\eta$ 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 DT-MREIT 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 DT-MREIT 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 DT-MREIT 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 DTI-MREIT 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. In⁸, 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 of⁸ and may require a smaller number of gradient direction or multi b-value diffusion weighted images⁹. One of the important contributions of Marino et al.³² was that they estimated cellular microstructure parameters multiple ways: n times from $C^{n}_{n-1}$ 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 of³² to fit the parameters in Sajib et al.’s model⁸ here using a least-square method, which improved our method’s stability. Jahng et al.¹⁰ proposed using the spherical mean technique (MC-SMT)^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 b-shells (800, 2000 s/mm$^2$) with 12- and 24-gradient-directions respectively¹⁰ 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’ spin-echo 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.

Figure 8

Effect of artificial diffusion term value c on EPT reconstructions. (A) Profile line shown on focus slice (slice 2). (B) Plots of reconstructed $\sigma _H$ on profile line as a function of distance showing dependence on c. (C) Mean and standard deviations of reconstructed $\sigma _H$ in each compartment over all slices as a function of c value. Shading in part (B) indicates tissue type as (dark gray) gray matter, (light gray) white matter, and (purple) CSF.

Influence of $\beta$ value

For the human subject we used a fixed value of $\beta =0.41$ for all CTI reconstructions⁴⁰. In phantom reconstructions we fixed $\beta =1$ for all CTI reconstructions⁴³, 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 DT-MREIT and MREIT conductivity

As noted above, the error in reconstructed MREIT conductivity⁵⁰ depends on the proportion of $z-$directional current flow ($J_z$). In the phantom images the two-dimensional assumption held well in practice as the phantom was cylindrical and in-plane surface electrodes were used. However, for the human case $J_z\approx 0$ does not hold because of the relatively high anisotropic conductivity, non-planar electrodes and high conductivities in CSF regions causing out of plane flow.

It has been shown in²⁸ 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 DT-MREIT reconstructions than in the corresponding region of the CTI reconstruction of the same slice.

Comparisons to other MR-based 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 conductivity¹. MR-based 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 MR-based reconstruction approaches with and without current administration are compared in Fig. 9. Here, MR-measured white and gray matter conductivities are overlaid on model predictions²³. MR-based measurements of brain conductivities at Larmor frequencies of 3 T and above, reconstructed using multiple different approaches, have agreed closely with model predictions. Meanwhile, low-frequency brain conductivities found using MR-based 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 low-frequency 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 low-frequency MR-based conductivity results might suggest that electrode polarization effects have caused large underestimations of model-predicted 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.

Figure 9

Electrical conductivity values for gray matter and white matter measured in this work (hoc. op.) using comparable MR-based methods at both Larmor frequency and low (DT-MREIT, CTI) frequencies overlaid on model data. (A) Shows conductivities reported in this work using CTI (plotted at 1 Hz) against Marino et al. and Katoch et al.^16,32 and DT-MREIT at 10 Hz against Chauhan et al. and Sajib et al.^17,43. Part (B) plots EPT conductivities observed in this work at 3 T (128 MHz) against Mandija et al.⁵² and 7 T (298 MHz, van Lier et al.⁵³) respectively.

While the main goal of this study was to measure the correspondence of DT-MREIT and CTI reconstructions, the study also provided an opportunity to compare reconstructed conductivities within the phantom materials with standard four-terminal 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 DT-MREIT and CTI reconstructions in the phantom was otherwise reasonable.

Conclusions

Using both biological tissue phantom data and human data we have demonstrated that CTI, DT-MREIT 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 MR-based 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 MRI-based reconstructions will further illuminate the true nature of brain and other tissue conductivities in the low-frequency range.

Methods

Imaging experiments

All MR data were measured using a 32-channel 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 high-resolution structural study using a Magnetization-Prepared Rapid Gradient-Echo (MPRAGE) sequence, susceptibility-corrected high angular resolution diffusion images (HARDI), multi-spin-echo $B_1$ mapping, and a multigradient echo-based sequence used to obtain MREIT data. A transcranial electrical stimulator (DC-STIMULATOR MR, neuroConn, Ilmenau, Germany) was used to deliver 1.5 mA-currents at approximately 10 Hz through two pairs of opposing electrodes used in either human or phantom imaging studies ($\mathcal {E}=1,\ 2$). Finally, a multi-b-value EPI-based diffusion volume (fifteen b-values, six directions) was obtained for determination of tissue microstructure parameters. Table 3 summarizes imaging parameters used in each experiment and sequence.

Table 3 Imaging experiment parameters.

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 xy-directions (right) were then placed inside the chamber to provide anisotropic tissue properties (Fig. 10A–C. From previous four-electrode 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/m²⁴. A cubic conductive anomaly made of potato (height 40 mm, assumed conductivity of approximately 0.2 S/m at 10 Hz²⁴) 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 5-mm thick slices of MREIT data were acquired.

Figure 10

(A) Computational model geometry of phantom with attached electrodes and wire tracks, showing focus slice location. (B) MR magnitude image of third slice acquired during MREIT experiment. (C) Regions of interest within phantom. Direction combined normalized DW-MR images at b-values of (D) 500 (E) 2600 and (F) 4500 s/mm$^2$ respectively. (G) Color-coded fractional anisotropic map calculated at b = 1000 s/mm$^2$.

Figure 11

(A) Computational model geometry of the human head showing attached electrodes and wire tracks, showing focus slice location (slice 2). (B) MR magnitude image of third slice acquired during MREIT experiment. (C) Regions of interest within human subject. Direction combined normalized DW-MR images at b-values of (D) 500 (E) 2600 and (F) 4500 s/mm$^2$ respectively. (G) Color-coded fractional anisotropic map calculated at b = 1000 s/mm$^2$.

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 6-minutes’ 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 DT-MREIT, 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 EPI-based HARDI diffusion weighted imaging data were processed to tensor information using three image volumes: two six-direction scans: one gathered with phase encoding steps collected in the anterior-posterior direction and the other with the direction reversed. A third 64-direction 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 six-direction diffusion volumes were used to correct the 64-direction volume for eddy current and susceptibility artifact data using the FSL eddy tool⁵⁴ 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 left-right (top, red), up-down (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 b-value were scaled to match the extra-cellular diffusion coefficient using

$$\begin{aligned} \mathbb {D} = \left( \frac{3d_e}{d_{b, 1}+d_{b, 2}+d_{b, 3}}\right) \mathbb {D}_{b} \end{aligned}$$

(9)

For both human and phantom experiments. $\mathbb {D}$ was estimated using the diffusion tensor estimated at $b=1000~\text{ s }/\text{mm}^2$ and 64-directions. Figures 10 and 11D–G illustrate normalized DW-MR images for the phantom and human subject respectively at b values of 500, 2600 and 4500 s/mm$^2$ as well as a color-coded fractional anisotropy map of a focus slice.

EPT data processing

Regions of interest used in EPT data processing, a representative T1-weighted magnitude slice and a comparison magnitude image of the focal slice (slice 2) of the multi-spin-echo 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 k-space 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 first-echo data were removed to avoid including inaccurate $T_2$ weighting factors in the echo combination step⁴⁷. 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. High-frequency conductivity distributions were performed using convection reaction-regularized EPT (cr-EPT) reconstruction^12,57 implemented in the MRCI toolbox.

Briefly, the cr-EPT method involves combining Ampere’s and Faraday’s laws^12,58 to obtain an equation of the form

$$\begin{aligned} \left( \nabla \phi ^{tr}\cdot \nabla \rho \right) +\nabla ^2\phi ^{tr}\rho =2\omega \mu _0 \end{aligned}$$

(10)

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 added¹², therefore producing

$$\begin{aligned} -c^2\nabla ^2\rho +\left( \nabla \phi ^{tr}\cdot \nabla \rho \right) +\nabla ^2\phi ^{tr}\rho =2\omega \mu _0 \end{aligned}$$

(11)

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.

Multi-b-value diffusion

Diffusion data at 15 different b-values were collected and used to estimate $\alpha , d_i$ and $d_e$ values for CTI reconstruction.

The extracellular volume fraction, $\alpha$, extra- and intra-cellular diffusion coefficients, $d_e$ $d_i$ were estimated as^9,32,

$$\begin{aligned} \alpha = \left( 1-f_{iso}\right) \left( 1-f_{ic}\right) + f_{iso} \end{aligned}$$

(12)

$$\begin{aligned} d_e = \frac{\left( 1-f_{iso}\right) \left( 1-f_{ic}\right) ^2 d_{e}^{*}}{\alpha } + \frac{f_{iso} d_{iso}}{\alpha } \end{aligned}$$

(13)

$$\begin{aligned} d_i = f_{ic} d_{ic} \end{aligned}$$

(14)

The model parameters i.e. isotropic and intra-cellular volume fraction $f_{iso}$ $f_{ic}$ respectively, and the extra-cellular mean diffusivity $d_e^{*}$ were estimated as^9,32

$$\begin{aligned} \frac{S_b}{S_0} = \left( 1 - f_{iso}\right) \left[ f_{ic}e^{-bf_{ic}d_{ic}} + (1-f_{ic})e^{-v(1-f_{ic})d_e^*}\right] + f_{iso} e^{-bd_{iso}} \end{aligned}$$

(15)

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 intra-cellular diffusion coefficient^9,32.

Under the assumption that the CTI-parameters are independent of the measured gradient directions, in this paper we estimated $\hat{\alpha }$, $\hat{d_e}$ and $\hat{d_i}$ as

$$\begin{aligned} \hat{\alpha } = \frac{1}{K}\sum _{k=1}^K \left( 1-f_{iso, k}\right) \left( 1-f_{ic, k}\right) + f_{iso, k} \end{aligned}$$

(16)

$$\begin{aligned} \hat{d_e} = \frac{\sum _{k=1}^K \left[ \left( \left( 1-f_{iso, k}\right) \left( 1-f_{ic, k}\right) ^2 d_{e, k}^{*} + f_{iso, k} d_{iso} \right) \left( \left( 1-f_{iso, k}\right) \left( 1-f_{ic, k}\right) + f_{iso, k}\right) \right] }{\sum _{k=1}^K \left( \left( 1-f_{iso, k}\right) \left( 1-f_{ic, k}\right) + f_{iso, k} \right) ^2} \end{aligned}$$

(17)

$$\begin{aligned} \hat{d_i} = \frac{1}{K} \sum _{k=1}^K f_{ic, k} d_{ic} \end{aligned}$$

(18)

where $K = \sum _{k=1}^{N_d-1} C^{N_d}_{k}$ is the total number of unique subsets of gradient directions and $N_d$ is the number of acquired gradient directions. For human data, 6-diffusion gradient direction data was measured, therefore, $K=57$, For phantom $K=57$ was also used.

To estimate the CTI-parameters in Eq. (15), the each b-value signal was averaged over the diffusion gradients as

$$\begin{aligned} \bar{S}_{b,k} = \frac{1}{N_{d, k}}\sum _{n=1}^{N_{d, k}} S_{b,n} \end{aligned}$$

(19)

where $N_{d, k} < N_d$ is the number of diffusion gradient signals in each subset applied to measure the diffusion tensor at particular b-value. Note that averaging over gradient directions maintains the exponential property of diffusion signal with respect to b-values⁸. Each subset of $\bar{S}_{b, k}$ was then fitted to the signal model in equation (15) to obtain the microstructure parameters ($f_{iso, k}$, $f_{ic, k}$, $d_{e,k}^{*}$)

$$\begin{aligned} \left( f_{iso, k}, f_{ic, k}, d_{e,k}^{*}\right) = \arg \min _{f_{iso}, f_{ic}, d_{e}^{*}} \Vert S_b - \bar{S}_{b, k}\Vert _{2}^{2} \end{aligned}$$

(20)

Figure 12

Phantom CTI parameters in focus slice (slice 3). Regions of interest, T1-weighted image and Magnitude of multi-spin-echo data used for high-frequency conductivity reconstruction are shown in (A–C) respectively. MSE phase data are in (D). Extra-, intra-cellular water diffusion and extra-cellular volume fraction maps are shown in (E), (F), and (G), respectively. Part (H) displays reconstructed high-frequency conductivity ($\sigma _H$) found using MREPT method at 128 MHz.

Figure 13

Human CTI parameters in focus slice (slice 2). Regions of interest, T1-weighted image and Magnitude of multi-spin-echo data used for high-frequency conductivity reconstruction are shown in (A–C) respectively. MSE phase data are in (D). Extra-, intra-cellular water diffusion and extra-cellular volume fraction maps are shown in (E), (F), and (G), respectively. Part (H) displays reconstructed high-frequency conductivity ($\sigma _H$) found using MREPT method at 128 MHz.

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 space³². 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 non-negative³². 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.

DT-MREIT

Projected current density reconstruction

The measured multi-gradient-echo 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 as³⁸

$$\begin{aligned} \textbf{J}^P = \textbf{J}_0 + \left( \frac{\partial \psi }{\partial y}, \frac{\partial \psi }{\partial x}\right) . \end{aligned}$$

(21)

For phantom data $\psi$ was determined via,

$$\begin{aligned} {} \left\{ \begin{array}{lll} \nabla ^2 \psi = \frac{1}{\mu _0}\nabla ^2B_z &{} \text{ in } \quad \Omega _t\\ \psi = 0 &{} \text{ in } \quad \partial \Omega _t \\ \end{array} \right. \end{aligned}$$

(22)

Figure 14

Phantom DT-MREIT data at focus slice (slice 3). Parts (A) and (B) show 3D view of phantom geometry and representative slice (slice 3) of multi-gradient echo image respectively. Reconstructed $B_z$ (C, E) and $|J^P|$ (D, F) images for horizontal and vertical electrode pairs.

For human data regional projected current density³⁰, the projected current density was found as

$$\begin{aligned} \textbf{J}^P = \textbf{J}_0|_{R_t} + \left( \frac{\partial \psi |_{R_t}}{\partial y}, \frac{\partial \psi |_{R_t}}{\partial x}\right) . \end{aligned}$$

(23)

where

$$\begin{aligned} \left\{ \begin{array}{lll} \nabla ^2 \psi |_{R_t} = \frac{1}{\mu _0}\nabla ^2B_z &{} \text{ in } \quad R_t\\ \psi |_{R_t} = \frac{1}{\mu _0}\left( B_z-B_z^0\right) &{} \text{ in } \quad \partial R_t \\ \end{array} \right. \end{aligned}$$

(24)

In (24), $R_t$ is a two-dimensional 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 multi-gradient 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.

Figure 15

Human DT-MREIT data at focus slice (slice 2). Parts (A) and (B) show multi-gradient echo magnitude image of representative slice (slice 2) and 3D view of subject volume respectively. Reconstructed $B_z$ (C,E) and $|J^P|$ (D,F) images for T7–T8 and Fpz-Oz electrode pairs.

MREIT

For comparisons to both DT-MREIT and CTI, the effective low-frequency conductivity was also reconstructed using the non-iterative Harmonic $B_z$ algorithm⁵⁰ implemented in the MRCI toolbox⁴⁰. Using two independent current injections, the non-iterative Harmonic $B_z$ algorithm estimates the parameter $\textbf{s}(\textbf{r}) = \nabla {\log {\sigma }}(\textbf{r})$ as

$$\begin{aligned} {} \nabla \textbf{s} \left( \textbf{r}\right) = \left( \textbf{A}^T \textbf{A} + \lambda \textbf{I}\right) ^{-1}\left( \textbf{A}^T b\right) \end{aligned}$$

(25)

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 pixel-wise 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

$$\begin{aligned} {} \left\{ \begin{array}{lll} \nabla ^2 \log \textbf{C} = \nabla \cdot \left( \nabla \textbf{s} \right) &{} \text{ in } \quad \Omega _t\\ \log \textbf{C}_L^{MREIT} = \log \sigma _L^{CTI} &{} \text{ in } \quad \partial \Omega _t \\ \end{array} \right. \end{aligned}$$

(26)

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 DT-MREIT, and $\textbf{C}^{CTI}_L$ and $\textbf{C}_L^{MREIT}$ were performed using a two one-sided 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.