1. Introduction
Electrical conductivity of biological tissues is determined by the cell density, extracellular volume fraction, composition and amount of extracellular matrix materials, and membrane characteristics as well as concentrations and mobility of ions in the extracellular and intracellular fluids [
1]. The apparent macroscopic conductivity of such a composite material has been studied since the early 1900s and can be expressed as a weighted sum of conductivity values of its components based on the volume fractions and other factors [
2,
3]. The extracellular and intracellular fluids are conductors through which conductivity values are determined by concentrations and mobilities of ions and other mobile charge carriers. Cells with thin membranes behave like an insulator and lossy dielectric at low and high frequencies, respectively. Extracellular matrix materials are lossy dielectrics. Therefore, the macroscopic tissue conductivity exhibits frequency dependency [
4,
5].
When elongated cells are aligned towards a certain direction, movements of ions in the extracellular fluid are consequently hindered. Under a low-frequency electric field, the ions in the extracellular space are forced to move along the longitudinal direction, thereby making their mobilities direction-dependent. Therefore, in the white matter and muscle, the conductivity exhibits anisotropic properties at low frequencies. However, at high frequencies, the insulating cell membranes behave like a capacitor, and the anisotropic properties disappear above 1 MHz, for example [
1,
6]. In this paper, we approximately express the low-frequency conductivity of biological tissue as a tensor that is a symmetric positive definite 3 × 3 matrix [
7,
8]. At a high-frequency above 1 MHz, the conductivity is expressed as a scalar quantity [
9].
Electrical conductivity is a passive material property whose measurement requires a probing current to generate a signal affected by the conductivity. In impedance imaging area, there are two different approaches in conductivity imaging using MRI. Magnetic resonance electrical impedance tomography (MREIT) reconstructs an image of low-frequency isotropic conductivity (
) distribution by injecting low-frequency currents into a subject and measuring the induced magnetic flux density distributions using an MRI scanner [
10,
11,
12]. Magnetic resonance electrical properties tomography (MREPT) produces high-frequency isotropic conductivity (
) and permittivity (
) images by generating a radio-frequency (RF) eddy current that is affected by
and
and measuring an induced RF magnetic field using a B1 mapping method [
13,
14,
15].
In the diffusion tensor imaging (DTI) area, conductivity tensor image reconstructions have been investigated based on a physical relationship that conductivity and water diffusion tensors, denoted as
and
, respectively, have the same eigenvectors [
16]. Based on this, Tuch et al. derived the first linear model between
and
, which enabled transformation of a diffusion tensor image into a conductivity tensor image [
17,
18]. Based on the idea of cross-property relation [
17], three more conductivity tensor models and image reconstruction methods using DTI were developed [
19,
20,
21,
22]. These conductivity tensor models are based on a linear relation of
and the corresponding image reconstruction methods determine the scale factor
.
Although MREIT and MREPT reconstruct images of
and
, respectively, through measurement of the effects of both ion concentrations and mobilities, neither of these two methods can produce an image of
. On the other hand, the four methods using DTI produce an image of
, but the effects of ion concentrations are not adequately demonstrated. To overcome these limitations, Sajib et al. developed a novel low-frequency conductivity tensor imaging (CTI) method [
23]. In the first step of the CTI method, MREPT is used to reconstruct an image of
including the effects of both ion concentrations and mobilities at the Larmor frequency. In the second step, they apply a multi-b diffusion weighted imaging method to obtain an image of
and pixel-by-pixel information about extracellular and intracellular spaces. Combining all of these, the scale factor
between
and
is determined for every pixel using the following: (1) a physical relationship between the water diffusivity and ion mobility and (2) a model-based relation between
and
[
6,
24].
Recently, Wu et al. reviewed the DTI-based reconstruction models [
25]. They reported that the conversion coefficient of water diffusion tensor to electrical conductivity tensor should require the information on ion concentration and extracellular volume fraction [
25]. In addition, knowledge of accurate anisotropic conductivity can achieve reliable volume conduction models of electrical brain stimulation and EEG dipole reconstructions [
26,
27]. Over the years, considerable variation in the conductivity reconstruction models used for such approaches has been observed [
27,
28,
29]. Consequently, an in-depth investigation of conductivity tensor models was required.
In this paper, we compared the accuracy of the five methods in reconstructed conductivity images of two phantoms using a 9.4 T research MRI scanner. The data acquired from five human subjects using a clinical 3 T MRI scanner were used to reconstruct conductivity tensor images of the brains using the five methods. The reconstructed conductivity tensor images of the human brains were analyzed and compared with each other and also with previous literature values.
5. Discussion
In this paper, we compared five conductivity tensor models and image reconstruction methods using two phantoms and five human subjects. The data from the phantom experiments were used to evaluate the accuracy of the reconstructed conductivity images against the measured conductivity values obtained using the impedance analyzer. In vivo human experiment data were used to compare the values of the reconstructed conductivity tensors with existing literature values and also with each other using the CTI method as a reference.
In the phantom experiments, the LEM and FEM methods could not distinguish the electrolytes with different NaCl concentrations. The errors in the reconstructed conductivity images using the LEM and FEM were relatively large. The errors of the VCM method were the smallest since we used the conductivity values measured using the impedance analyzer in place of
in (
9). In most clinical applications where the value of
is unknown, the VCM method may fail to recover the inhomogeneous conductivity values. In addition, living tissues are heterogeneous, and their conductivity varies with the pathophysiology condition. Thus, a single global value of conductivity to the entire tissue type may not sufficient. The VFM method could not be applied to the phantoms since it was explicitly designed for anisotropic brain tissues. Without relying on
prior information of mean isotropic conductivity values, the CTI method recovered conductivity values with an error ranging from 1.10% to 5.26% for all six different compartments in the two phantoms having different ion concentrations and mobilities. Furthermore, the CTI method could properly handle the effects of different cell densities in two giant vesicle suspensions with different extracellular volume fractions.
Using the data from in vivo human brain imaging experiments, we produced conductivity tensor images of the brains as shown in
Figure 3 using the five methods. Compared to the conductivity values of the WM, GM, and CSF from existing literature, the values using the LEM and FEM methods had large amounts of bias in the WM and GM regions. Since the global scale factor was used for all pixels in the FEM method, the method’s ability to handle intra-subject and inter-subject variability appeared to be primarily limited. Although the scale factor
in the LEM method contains inter-subject variability, but still lacks intra-subject tissue heterogeneity. Furthermore, the global scale factor does not account for position dependence, and its deviation in the tissues only explains the water diffusion alterations. The conductivity values in CSF from both LEM and FEM methods were significantly underestimated compared to the literature values. The calculated conductivity tensor in both these methods was simply linearly scaled water diffusion tensor, and hence may explain their considerable low conductivity measurements [
17,
19]. The conductivity values measured in our study were matched with previous studies by Rullman et al. and Sekino et al. [
18,
20].
In the volume constrained method (VCM), conductivity values of GM and CSF were matched with the literature and also with the CTI method in our study. In contrast, conductivity values of WM were underestimated at longitudinal directions. This might be due to the use of similar literature values of
in all three fiber directions. This method still lacks the intra-voxel heterogenic property of the tissues, and segmentation inaccuracies may end up assigning incorrect
values. An introduction of the isotropic extracellular electrical conductivity from MREIT could be helpful to overcome this limitation [
12]. In the volume fraction method (VFM), conductivity values in WM and GM were highly overestimated. This stems from the fact that the eigenvalues of conductivity tensor were computed as a weighted sum of conductivity values associated with four compartments of individual fiber bundle direction. VFM method is computationally complicated, and the ill-posed problem may lead to failed fit in noisy voxels [
22]. In addition, the assumption of constant
and
, which were then set to be literature conductivity value, somehow neglected the intra-voxel heterogeneity, and reduced the anisotropic ratio. Instead of such limitations, VFM method effectively handled the partial volume effect.
Although the scale factor in the LEM, FEM, and VCM methods was computed for each pixel, their ability to handle the intra-subject and inter-subject variability was also limited as a result of fixed literature conductivity values of the WM, GM, and CSF regions for all subjects. The VFM method manages cross-subject variability but failed with intra-subject flexibility. In contrast to the four methods using DTI only, the CTI method produced conductivity tensor images with values that were comparable to existing literature values. Without using prior information about tissue conductivity values, the CTI method appeared to properly handle the intra-subject and inter-subject variability. However, the intra-subject and inter-subject variabilities, may have been affected in all methods by measurement noise, and systematic artifacts.
The recovered conductivity tensors using the LEM and FEM methods in
Figure 5a to
Figure 5b shows similar results as the water diffusion tensor in our study. In spite of the calculating the pixel-dependent scale factor in the VCM method, a higher correlation with water diffusion was observed (
Figure 5c). Considering that the local concentration of a charge carrier can affect the linearity between
and
, the CTI method appeared to properly reflect the effects of position-dependent concentration differences by incorporating the actually measured high-frequency conductivity
into the conductivity tensor image reconstruction. Note that these linear regression analyses should not be considered as a general validity of
. In CTI method, the mutual restriction of both the ionic and the water mobility by the geometry of the brain medium builds the basis for the relationship of
and
. The voxels in conductivity tensor map are expressed as a sum of products of the carrier concentration, extracellular volume fraction, and the mobility tensor. Since all of the five methods utilized the mobility information embedded in the water diffusion tensor
, the anisotropy ratio of
was primarily determined by the anisotropy ratio of
.
Data acquisition and processing steps are more involved in the CTI method than in those other four methods using DTI. Given that there are various sources of error in the data acquisition and processing stages, subsequent studies are necessary to rigorously validate the performance of the CTI method in terms of errors due to measurement noise and artifacts, partial volume effects, and coregistration of different images. In particular, high-frequency conductivity images using MREPT suffer from errors at boundaries of two regions with different conductivity values and also assumes the piecewise constant conductivity [
47], which does not hold in practice. Although the MREPT method we adopted in this paper could reduce such boundary artifacts [
14] at the expense of smooth images, future improvements in high-frequency conductivity image reconstruction algorithms are needed to enhance the accuracy of the CTI method in human subjects. The parameter
in (
15) was estimated from the literature values and assumed to be constant for all pixels. Although the sensitivity of a reconstructed conductivity tensor image to
appeared to be small [
24], clinical studies are needed to find out any practical limitations imposed by this assumption. The CTI method can provide clinically useful information about pathological and physiological changes in cells and cellular structures associated with disease progression. It is one of most promising clinical applications is tumor imaging in terms for early detection, better characterization, and monitoring after a treatment [
48,
49]. The recovered low-frequency conductivity tensor could be also used to predict internal current pathways and electric field distributions subject to externally injected or induced therapeutic currents in electrical stimulation [
28,
29].
This study has several limitations that should be considered in the future works. The distribution of conductivity within the giant vesicle phantom does not reflect an adequate anisotropic environment. Future studies should include phantoms mimicking microstructural properties similar to the brain [
50]. Diffusion MRI suffers from various systematic errors, such as gradient inhomogeneity [
51]. Although we corrected the data using commonly used correction methods, a study of systematic error propagation or noise analysis is needed to enhance the accuracy of the conductivity images. In human brain, conventional DTI measures water diffusion assuming that displacement distribution of water molecules in a given time is a Gaussian function. However, this assumption may not be valid in complex biological tissue where water molecules often show non-Gaussian diffusion [
52]. DTI also has limitations in characterizing the diffusion process in areas of low anisotropy and complex fiber structure in a voxel. Future studies with diffusion kurtosis imaging (DKI) or high angular methods (HARDI) can provide better characterization of human brain architecture [
52,
53]. Cell membranes are assumed to resist low-frequency currents in the CTI approach. As a result, the CTI approach may underestimate the low-frequency conductivity value of tissue including cells with leaky membranes. It would be worthwhile to investigate a more sophisticated CTI model involving such cells.