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Decomposition of high-frequency electrical conductivity into extracellular and intracellular compartments based on two-compartment model using low-to-high multi-b diffusion MRI

Abstract

Background

As an object’s electrical passive property, the electrical conductivity is proportional to the mobility and concentration of charged carriers that reflect the brain micro-structures. The measured multi-b diffusion-weighted imaging (Mb-DWI) data by controlling the degree of applied diffusion weights can quantify the apparent mobility of water molecules within biological tissues. Without any external electrical stimulation, magnetic resonance electrical properties tomography (MREPT) techniques have successfully recovered the conductivity distribution at a Larmor-frequency.

Methods

This work provides a non-invasive method to decompose the high-frequency conductivity into the extracellular medium conductivity based on a two-compartment model using Mb-DWI. To separate the intra- and extracellular micro-structures from the recovered high-frequency conductivity, we include higher b-values DWI and apply the random decision forests to stably determine the micro-structural diffusion parameters.

Results

To demonstrate the proposed method, we conducted phantom and human experiments by comparing the results of reconstructed conductivity of extracellular medium and the conductivity in the intra-neurite and intra-cell body. The phantom and human experiments verify that the proposed method can recover the extracellular electrical properties from the high-frequency conductivity using a routine protocol sequence of MRI scan.

Conclusion

We have proposed a method to decompose the electrical properties in the extracellular, intra-neurite, and soma compartments from the high-frequency conductivity map, reconstructed by solving the electro-magnetic equation with measured B1 phase signals.

Background

Using a conventional MRI scanner without any external electrical stimulation, magnetic resonance electrical properties tomography (MREPT) techniques have been developed and successfully recover the conductivity distribution at Larmor-frequency (about 128 MHz at 3 T ) [1,2,3,4]. The electrical conductivity of biological tissues is proportional to the apparent concentration and mobility of ions in the intracellular and extracellular compartments. As the separated form of electrical conductivity of biological tissues, the low-frequency conductivity (< 1 kHz) is dominantly influenced by the apparent concentration and mobility of ions in the extracellular compartment.

To decompose the high-frequency conductivity into the extracellular and intracellular compartments using a conventional clinical MRI scanner, it requires micro-structural parameters including the extracellular volume fraction (EVF), distributions of electrical charged molecules, and diffusion coefficient coefficients in the extracellular and intracellular compartments [5, 6]. For the model-based micro-structure imaging based on the tissue micro architecture in the brain, the micro-structural parameters for the intracellular and extracellular compartments have been studied, related with specific tissue micro-structure features from Mb-DWI data [7,8,9]. Although a two compartment model is the simplest form, the determination of micro-structural parameters from measured decay MR DWI signals with respect to b-value is ill-posed because the estimation of parameters from the combination of smooth exponential curves is sensitive to noise in the measured DWI data. To stabilize the ill-posedness, two compartment models typically assume some restrictions: intrinsic diffusivity and/or water diffusing in elongated cellular fibres, based on the ball-and-stick model [10, 11].

Various exponential diffusion models have been proposed to describe signal attenuation with DWIs using more than two b-values [12,13,14,15,16,17]. Using the exponential diffusion models, an electrodeless method providing the low-frequency electrical property imaging without any external hardware was proposed [6, 17,18,19]. The proposed methods mainly focus on separating the ion mobility and concentration in the extracellular space (ECS) from the recovered high-frequency conductivity. Although, without injection of external currents, it is a promising work to separate the high-frequency conductivity (combined electrical properties from the extra- and intra-celluar compartments) into the low-frequency conductivity (electrical properties from the extracellular compartment), the developed methods still have difficulties in distinguishing the hindered diffusion of free water molecules and the diffusion-limited compartment [17, 19]. For separating the low-frequency conductivity from the high-frequency conductivity, we need to quantify the micro-structures of extracellular compartment. However, typical two compartment models based on the ball-and-stick model consider the diffusion signals from soma or other large cellular domains as those from the extracellular compartment. The reconstructed low-frequency conductivity using the ball-and-stick model overestimates EVF and, as a result, causes the biased conductivity values, especially in the gray matter region.

Recently, a biophysical model has been proposed for apparent cell body (soma) and neurite density imaging (SANDI), which tries to recover the soma size and density in addition to neurite density [9]. In particular, in ECS, to decompose the electrical properties of each compartment, it is important to distinguish the micro-structural characteristics of the soma size and density belonging to the intracellular compartment. The SANDI model needs to include the direction-averaged DWI signal at high b-values (\(\ge \) 3000 s/\(\hbox {mm}^2\)) to detect the apparent soma size and density. Combining with the estimated DWI data for the higher b-values, we apply the SANDI model to estimate the micro-structures of extracellular compartment and use it to identify the characteristics of low-frequency conductivity.

To determine the micro-structural parameters of biological tissues using the measured Mb-DWI data, we use the random forest regression, an ensemble machine learning algorithm by constructing a multitude of decision trees at training time [20, 21]. The applied machine learning method builds a forest of uncorrelated trees, combined with randomized node optimization and bootstrap aggregating [22].

To demonstrate the electrical property decomposition from the recovered high-frequency conductivity, we generate the high-frequency conductivity using the convection-reaction partial differential equation with a small regularization parameter [1]. The recovered high-frequency conductivity reflects the combined electrical properties by the intra- and extra-cellular compartments using a routine protocol sequence of MRI scan.

To validate the proposed method, a conductivity phantom including a giant vesicle suspension was conducted as a model for the membranes of biological cells. The giant vesicles were cell-like materials with thin insulating membranes to validate the two compartment model including both extracellular and intracellular spaces. We conducted human experiments by comparing the results of reconstructed conductivity of extracellular medium and the conductivity in the intra-neurite and intra-cell body. We extracted the apparent total ion concentration from the high-frequency conductivity map and the estimated micro-structural parameters. The phantom and human experiments indicate that the total ion concentration and the extracellular diffusion tensor can predict the extracellular electrical properties without externally injected currents. The accuracy and precision of the reconstructed low-frequency conductivity distribution in the extracellular compartment were evaluated.

Results

Giant vesicle phantom experiment

Giant membrane vesicles dispersed in aqueous solution were prepared as described in [23]. Giant vesicles are a model membrane system for the investigation of lipid membranes. In a 1L round-bottom flask containing 3 mL of chloroform and 400 μm of methanol, 2 mL of phospholipids dissolved with a chloroform solution of 30 mg/mL concentration under argon atmosphere. A 20 mL volume of distilled water or 0.75% NaCl solution was carefully added not to disturb the interface between the aqueous phase and organic solution phase. The flask was installed to a rotary evaporator to remove organic solvent at 47 °C under vacuum for 20 min at 10 rpm and then followed by another 20 min at 60 rpm. During evaporation of organic solvents, phospholipids were assembled to form giant vesicles. The resultant aqueous solution containing giant vesicles were centrifuged at 1500 rpm for 10 min. The volume fraction of the giant vesicles was about 80 to 90% by visual observations. The mean and standard deviation of the diameters of the giant vesicles were 13 ± 4.7 μm.

The conductivity phantom was constructed, which comprised two compartments of electrolyte (Background) and giant vesicle suspension (ROI) (Fig. 1a). The background was an NaCl solution of 3 g/L and ROI was filled with the giant vesicles suspension. The giant vesicle suspension was a mixture of the same NaCl solution of background and giant vesicles. In the giant vesicle suspension, the ratio of NaCl solution and giant vesicle was the same, which means that the volume fraction of the intracellular space was about 0.4–0.45.

A 9.4 T research MRI scanner (Agilent Technologies, USA) was used with a single-channel mouse body coil. The multi-spin-echo pulse sequence was used to acquire B1 phase maps to reconstruct high frequency conductivity images of the phantom. The imaging parameters were as follows: TR/TE = 2200/22 ms, number of echoes (NE) = 6, number of excitation (NEX) = 5, slice thickness = 0.5 mm, field-of-view (FOV) = 65\(\times \)65 \(\hbox {mm}^2\), flip angle = 90\({^\circ }\), image matrix size = 128\(\times \)128\(\times \)6, and scan time = 23 min. Diffusion weighted MR imaging was separately performed using the single-shot spin-echo echo planar imaging sequence. The imaging parameters were as follows: TR/TE = 2000/70 ms, \(\delta /\Delta \) = 6/53.8 ms, NEX = 2, slice thickness = 0.5 mm, FOV = 65\(\times \)65 \(\hbox {mm}^2\), flip angle = 90\(^{\circ }\), and image matrix size = 128 × 128 × 6. The number of directions of the diffusion-weighting gradients was 30 with b-values of 1000, 2200, 3000 and 3600 s/\(\hbox {mm}^2\). The scan time was about 32 min. An additional conventional \(\hbox {T}_1\) weighted scan of 5 min was included for anatomical reference.

Fig. 1
figure1

a Magnitude image. bd Estimated high frequency conductivity,\(\sigma _H\), extracellular volume fraction, \(f_{ec}\), extracellular diffusivity, \(D_{ec}\), and extracellular conductivity, \(\sigma _{ec}\) for different noise levels of (\(\hbox {SNR}_P\), \(\hbox {SNR}_D\)). b (\(\infty ,\infty \)), c (100,50), d (50,10))

The conductivity values of the giant vesicle suspension in ROI (Fig. 1a) were measured by using an impedance analyzer (SI1260A, AMETEK Inc., UK). The conductivity values were 0.57 S/m at 1 kHz and 1.05 S/m at 5 MHz.

Figure 1b shows the estimated high-frequency conductivity, \(\sigma _H\), extracellular volume fraction, \(f_{ec}\), extracellular diffusivity, \(D_{ec}\), and extracellular conductivity, \(\sigma _{ec}\), respectively. Table 1 shows the mean and standard deviation values of the estimated high frequency conductivity, extracellular volume fraction, extracellular diffusivity, extracellular ion concentration and extracellular conductivity within ROI.

Table 1 Mean and standard deviation values of the high frequency conductivity, \(\sigma _H\), extracellular volume fraction, \(f_{ec}\), extracellular diffusivity, \(D_{ec}\), extracellular ion concentration, \({\bar{c}}_{ec}\), and extracellular conductivity, \(\sigma _{ec}\) within ROI

The estimated high-frequency conductivity was \(\sigma _H=1.01\pm 0.02\) S/m in ROI, which is close to the conductivity value 1.05 S/m measured using the impedance analyzer at 5 MHz in ROI. We used the diffusion term \(c=0.05\) to stabilize the convection-reaction partial differential equation in (3). Since the volume fraction of the giant vesicles after the centrifugation was about 80% to 90% and the ratio of NaCl solution and giant vesicles was same in ROI, the controlled extracellular volume fraction was about 0.55–0.60 in ROI. The value of \(f_{ec}\) in ROI was 0.59±0.14 as expected.

To estimated the extracellular ion concentration, \({\bar{c}}_{ec}\), in (12), we assume that the intra-neurite ion concentration and soma ion concentration are same, and further assume that \({\bar{c}}_{in}=\beta {\bar{c}}_{ec}\) for some constant \(\beta \). Since we used the same electrolyte for both inside and outside of the giant vesicle, the ratio of ion concentrations in the intracellular and extracellular compartments, \(\beta \), was 1. The extracellular ion concentration \({{\bar{c}}_{ec}}\) can be estimated as

$$\begin{aligned} {\bar{c}}_{ec}=\frac{\sigma _H}{ \beta (f_{ne} D_{in} + f_{so} D_{is}) + f_{ec} D_{ec} } \end{aligned}$$
(1)

With the estimated high-frequency conductivity and the parameters of the SANDI model for brain micro-structure, we recovered the extracellular ion concentration, \({\bar{c}}_{ec}\), extracellular conductivity, \(\sigma _{ec}\). \(\sigma _{ec}\) = 0.54 ± 0.15 S/m in ROI is close to the measured conductivity value 0.57 S/m at 1 kHz.

To verify the proposed method, we conducted two cases. We artificially destroyed the measured signals by adding random noise and motion artifacts. For noisy data, we added the zero-mean Gaussian random noise to both the phase data and diffusion weighted signal. The noise standard deviation were calculated by \(\langle S \rangle \)/(\(\sqrt{2}\) SNR), where \(\langle S \rangle \) is the average signal (phase signal and the signal obtained without diffusion gradient) amplitude and SNR is the signal-to-noise ratio in MR magnitude images. For the phase data, \(\hbox {SNR}_P\) values 200, 100, and 50 were employed. For the diffusion weighted signal, we assumed that \(\hbox {SNR}_D\) of the signal obtained without diffusion gradient, \(S_0\), were 100, 50, and 10. The random noise added to \(S_0\) was added to each \(S_b\).

Figure 1c, d shows images of the estimated high frequency conductivity, \(\sigma _H\), extracellular volume fraction, \(f_{ec}\), extracellular diffusivity, \(D_{ec}\), and extracellular conductivity, \(\sigma _{ec}\) for (\(\hbox {SNR}_P\), \(\hbox {SNR}_D\)) = (100,50) and (50,10). Table 2 shows the mean and standard deviation values of the estimated high frequency conductivity, extracellular volume fraction, extracellular diffusivity, extracellular ion concentration and extracellular conductivity within ROI for the chosen noise levels. The proposed method stably reconstructed the conductivity values when \(\hbox {SNR}_P\) and \(\hbox {SNR}_D\) were above 100.

Table 2 Mean and standard deviation values of the high frequency conductivity, \(\sigma _H\), extracellular volume fraction, \(f_{ec}\), extracellular diffusivity, \(D_{ec}\), extracellular ion concentration, \({\bar{c}}_{ec}\), and extracellular conductivity, \(\sigma _{ec}\) within ROI for different noise levels of (\(\hbox {SNR}_P\), \(\hbox {SNR}_D\))

To investigate the impact of subject motion, we simulated a sinusoidal motion in the phase-encoding direction with amplitude 1 and a frequency of 2 Hz and obtained a motion-corrupted complex MR signals which were the convolution of the measured complex MR signals with the corresponding point spread function [24]. Figure 2a shows the magnitude image with motion artifacts. Figure 2b–d shows images of the estimated high frequency conductivity, \(\sigma _H\), and extracellular conductivity, \(\sigma _{ec}\) for the different diffusion term c in (3). Since the diffusion term c acts as a low-pass filter in the reconstruction algorithm, it was possible to reconstruct the motion-corrected high-conductivity map using a high c value. For \(c=1\), the values of the estimated high frequency conductivity and extracellular conductivity within ROI were 0.99\(\pm 0.02\) and 0.53 ± 0.14, respectively.

Fig. 2
figure2

a Magnitude image with motion artifacts. bd Estimated high frequency conductivity, \(\sigma _H\), and intracellular conductivity, \(\sigma _{ec}\) for different diffusion term c in (3). ((b): \(c=0.1\), c \(c=0.5\), d \(c=1\))

Human experiment

MRI measurements were performed with three healthy volunteers without a documented history of any disease were recruited. The participants were located inside the bore of a 3T MRI scanner with the head coil in transmit and a 32-channel RF head coil (Achieva TX, Philips Medical Systems, the Netherlands). All experimental protocols were approved by the institutional review board of Kyung Hee University (KHSIRB-16-033). All methods were carried out in accordance with the relevant guidelines and regulations and all participants provided written informed consent.

For MREPT imaging experiments, the multi-spin-echo pulse sequence with multiple refocusing pulses was adopted to minimize the measured noise. Before the data acquisition, we applied a volume shimming method with the volume defined to cover the brain region. Imaging parameters were as follows: TR/TE = 1500/15 ms, NE = 6, NEX = 1, slice thickness = 4 mm, FOV = 260\(\times \)260 \(\hbox {mm}^2\), imaging matrix size = 128\(\times \)128\(\times \)5, and scan time = 16 min. After MREPT scans, we performed DWI scans using the single-shot spin-echo echo planner imaging (SS-SE-EPI) pulse sequence. We applied the diffusion weighting gradients in 15 directions with 4 b-values of 1000, 2200, 3000 and 3600 s/\(\hbox {mm}^{2}\), respectively. Imaging parameters were as follows: TR/TE = 2000/70 ms, \(\delta /\Delta \) = 21/33 ms, NEX = 2, slice thickness = 4 mm, and acquisition matrix size = 64\(\times \)64\(\times \)5. The scan time was about 6.2 min. The matrix size of 64\(\times \)64\(\times \)5 was extended to 128\(\times \)128\(\times \)5 to match the spatial resolution (2.03\(\times \)2.03\(\times \)4  \(\hbox {mm}^3\)) of MREPT experiment. To increase the spatial resolution with the matrix size of \(64\times 64\) to the spatial resolution of \(128\times 128\) by reducing scanning times without much loss in resolution or SNR, we used the zero-filling interpolation (ZIP) by zero filling the high spatial-frequency components of the raw k-space data, which places the acquired data in the central regions on k -space and fills the data of zero in the outer regions. An additional conventional \(\hbox {T}_1\) weighted scan of 2 min was included for anatomical reference.

Figure 3 shows the estimated micro-structural parameter maps of SANDI model (5) for the first human subject: the estimated extracellular volume fraction \(f_{ec}\), intra-neurite volume fraction \(f_{ne}\), soma volume fraction \(f_{so}\), extracellular diffusivity \(D_{ec}\), and intra-neurite diffusivity \(D_{in}\) from the first subject.

Fig. 3
figure3

Estimated extracellular volume fraction, \(f_{ec}\), intra-neurite volume fraction, \(f_{ne}\), soma volume fraction, \(f_{so}\), extracellular diffusivity, \(D_{ec}\), and intra-neurite diffusivity, \(D_{in}\), from the first subject. Imaging slices are shown in the first column

Figure 4a, b shows the MR magnitude and the B1 phase images at the third imaging slice of the first subject. For quantitative analyses, \(\hbox {T}_1\) image was segmented into cerebrospinal fluid (CSF), gray matter (GM), and white matter (WM) using the segmentation tool of Statistical Parametric Mapping (SPM 12) [25]. These regions are shown in Fig. 4c.

Fig. 4
figure4

Magnitude, phase, and segmented (CSF, GM, WM) images from the third imaging slice of the first subject

Figure 5 shows the details of reconstructed conductivity images from the first subject. To estimate the high frequency conductivity, \(\sigma _H\), with the acquired transceiver phases of B1 maps, we solved the convection-reaction partial differential equation (PDE) in (3) with the diffusion term \(c=0.02\).

Fig. 5
figure5

Estimated high-frequency conductivity, \(\sigma _H\), extracellular ion concentration, \({\bar{c}}_{ec}\), extracellular conductivity, \(\sigma _{ec}\), intra-neurite conductivity, \(\sigma _{ne}\), and soma conductivity, \(\sigma _{so}\), from the first subject

To estimate the extracellular ion concentration, \({\bar{c}}_{ec}\), in (12), we set that the ratio \(\beta \) for ion concentrations in the intracellular and extracellular compartments to be 0.41 as suggested in [6] by adopting reference values of intracellular and extracellular ion concentrations of four predominant ions (\(\hbox {Na}^+\), \(\hbox {Cl}^-\), \(\hbox {K}^+\), and \(\hbox {Ca}^{2+}\)). Using the reference ratio value \(\beta =0.41\), the extracellular ion concentration \({{\bar{c}}_{ec}}\) can be estimated by (1).

With the estimated high-frequency conductivity and the parameters of the SANDI model for brain micro-structure, we recovered the extracellular ion concentration, \({\bar{c}}_{ec}\), extracellular conductivity, \(\sigma _{ec}\), intra-neurite conductivity, \(\sigma _{ne}\), and soma conductivity, \(\sigma _{so}\).

Figure 6 shows the 3\(\times \)3 extracellular conductivity tensor, \({\mathbf{C}}_{ec}\), and intra-neurite conductivity tensor, \({\mathbf{C}}_{ne}\), images. To estimate conductivity tensors, we used the water molecule diffusion tensors with the b value of 1000 s/\(\hbox {mm}^2\). We fixed the principal diffusion direction (eigenvector corresponding to the maximum eigenvalue of the diffusion tensor) and solved the equations (16) and (17) to obtain the extracellular diffusion tensor, \({\mathbf{D}}_{ec}\), and intra- neurite diffusion tensor, \({\mathbf{D}}_{ne}\), images. Using these diffusion tensors, we reconstructed the extracellular conductivity tensor, \({\mathbf{C}}_{ec}\), and intra-neurite conductivity tensor, \({\mathbf{C}}_{ne}\), in (18) and (19), respectively.

Fig. 6
figure6

Estimated extracellular conductivity tensor, \({\mathbf{C}}_{ec}\), and intra-neurite conductivity tensor, \({\mathbf{C}}_{ne}\), images from the third imaging slice of the first subject

Table 3 summarizes the estimated values of high frequency conductivity, extracellular ion concentration, extracellular conductivity, and intra-neurite conductivity of each subject. In Table 3, longitudinal (L), transverse (T), and average (A) white matter conductivity values were found by computing the principal eigenvalue, the mean of the other two eigenvalues, and the mean of all eigenvalues of the conductivity tensor over all region, respectively.

Table 3 Estimated values of high-frequency conductivity, \(\sigma _H\), extracellular ion concentration, \({\bar{c}}_{ec}\), extracellular conductivity, and intra-neurite conductivity within CSF, white matter, and gray matter regions

The high-frequency conductivity values in CSF regions were between 1.43 and 1.58 S/m and the extracellular conductivity values were between 1.35 and 1.49 S/m. Note SANDI model does not take into account CSF compartment. For this reason, a slight difference between high-frequency conductivity values and extracellular conductivity values was found in CSF regions.

For GM regions, the high-frequency conductivity values were between 0.51 and 0.55 S/m and the extracellular conductivity values were in the range of 0.29 to 0.31 S/m. The extracellular conductivity values were higher than the neurite conductivity values (0.07–0.08 S/m in GM), as expected.

We found longitudinal WM extracellular conductivity values of 0.25–0.28 S/m. Transverse WM extracellular conductivity values were between 0.13 and 0.15 S/m. The average ratio between longitudinal conductivity values and transverse conductivity values was 1.87–2.00. For WM regions, longitudinal intra-neurite conductivity values were 0.19 –0.22 S/m and transverse intra-neurite conductivity values were 0.10–0.11 S/m.

The extracellular conductivity values in GM regions were higher than the average extracellular conductivity values in WM regions, whereas the average intra-neurite conductivity values in WM regions were always higher than the intra-neurite conductivity values in GM regions.

Figure 7 shows the estimated extracellular conductivity tensor, \({\mathbf{C}}_{ec}\), and intra-neurite conductivity tensor, \({\mathbf{C}}_{ne}\), images represented by tri-axial ellipsoids, respectively, in the rectangular ROIs shown in extracellular conductivity, \(\sigma _{ec}\), and intra-neurite conductivity, \(\sigma _{ne}\), images. Since all of the conductivity tensors shared the same eigenvectors from the diffusion tensor, their orientations were same. The radii of each ellipsoid are proportional to the eigenvalues and their axes are oriented along the directions of eigenvectors. As expected, the volume of intra-neurite conductivity ellipsoids appeared larger in WM regions.

Fig. 7
figure7

Two chosen rectangular ROIs are shown in extracellular conductivity, \(\sigma _{ec}\), and intra-neurite conductivity, \(\sigma _{ne}\), images. Estimated extracellular conductivity tensor, \({\mathbf{C}}_{ec}\), and intra-neurite conductivity tensor, \({\mathbf{C}}_{ne}\), images represented by tri-axial ellipsoids, respectively. The radii of each ellipsoid are proportional to the eigenvalues and their axes are oriented along the directions of eigenvectors. For comparison, an isotropic tensor image with conductivity of 0.5 S/m is located in the center left. The colors of ellipsoids indicate the orientation of principle eigenvector

Discussion

Literature conductivity values including those obtained from direct impedance measurements (IM), MREPT and diffusion tensor MR electrical impedance tomography (DT-MREIT), are summarized in Table 4. The conductivity values of brain tissues heavily depend on the biological tissue structures, participant’s age and pathology, frequency, and the measurement conditions (in vivo, ex vivo, and in vitro) [26,27,28]. Without external injection currents and using conventional MR pulse sequences minimizing magnetic field inhomogeneity, MREPT is a promising research area for practical clinical medical devices. Comparing to MREPT, by injecting a dc current into the imaging subject, MREIT can reconstruct images of the internal low-frequency conductivity distribution. In this paper, we proposed a new way to decompose electrical properties in each compartment (extracellular, intra-neurite, and soma compartments) from the reconstructed high-frequency conductivity using MREPT technique and the micro-structural parameters using SANDI model.

Table 4 Literature conductivity values

From the intrinsic noise in the MR measurements, MREPT reconstruction techniques have been proposed to improve the quality of the high-frequency conductivity map [33,34,35,36]. The high conductivity values in Table 3 using the reconstruction algorithm in [1] also depend on the first term in (3), which is the diffusion term to stabilize the solution. The diffusion term acts as a low-pass filter, leading to some blurring of the final high-conductivity maps.

Because SANDI model does not take into account CSF compartment, a slight difference between the high-frequency conductivity values and the extracellular conductivity values was found in CSF region as was described in Result section. SANDI model estimates the orientation-independent features of microstructure using the direction-averaged DWI signals. In future studies, it is worth exploring a new model that take into account CSF compartment. In the SANDI model, the direction averaged signals using many uniformly distributed directions are advantageous to estimate the microstructural parameters since the DWI data is noisy at a high-b value. Although the accuracy of estimated parameters can be improved as the diffusion directions are increase, the numbers of diffusion directions were 30 and 15 for the phantom and human experiment, respectively, to recover the low-frequency electrical properties within a feasible MR scanning time.

At the low frequency, the internal electrical current flow caused by external current stimulation occurs only in the extracellular space and CSF, excluding the intracellular space, due to the insulation properties of cell membrane [27, 37, 38]. Despite extensive researches for the electrical properties in ECS, the reported low frequency conductivity values do not match and show considerable standard deviation. Recently, a meta-analysis of reported human head electrical conductivity values at low frequency (< 1 kHz) provides a recommended value estimated under suitable and realistic conditions, in which data acquisition techniques were categorized into five groups including directly applied current and MREIT. In Table 4, the reference values for low-frequency conductivity were 1.71 ± 0.3 (CSF), 0.47 ± 0.24 (GM), and 0.22 ± 0.14 S/m (WM) with broadly similar results to ours [30].

At fairly low frequencies, the conductivity of brain tissue, particularly white matter, is known to be anisotropic [32]. Comparing to the anisotropicity of intra-neurite compartment, the average ratio between longitudinal WM extracellular conductivity values and transverse WM conductivity values of 1.87–2.00 found here were lower than 3 and 5.9 reported in [31, 32], respectively. The extracellular conductivity values in GM and WM regions exhibited much stronger frequency dependencies compared to CSF region, because of their complicated tissue structures. More rigorous analysis including geometric and viscous components of the tortuosity of ECS will be needed in the future work.

In this paper, the acquired DWIs are combined because SANDI model assume an isotropic diffusion to distinguish the intra- and extra-cellular diffusion signals. We focus on separating the microscopic parameters for extracellular space from the multiple DWI data. The recovering procedure for the low-frequency electrical properties is very complex and requires further developments, including measurement techniques, noise artifact reduction, main magnetic field inhomogeneity artifact correction, and more reliable models to separate the extracellular and intracellular compartments.

To compensate the difficulties of measuring DWI data for higher b-values, we adopt the hypothesis that the DWI data corresponding to the b-value range (1000-3600 s/\(\hbox {mm}^2\)) reflects distinguishable diffusion signals between the soma and the extracellular space. To avoid the ill-posedness to determine the six unknowns, \(f_{in}, f_{ec}, D_{in}, D_{is}, D_{ec},\) and \(r_s\), in the two compartment model (5) from the smoothly decayed exponential curves, the parameter of \(D_{is}\) was fixed as 2 \(\times 10^{-3}\) \(\hbox {mm}^2\)/sec. However, the determination of parameters by matching the observed DWI data for Mb-DWI data and SANDI model was still very sensitive to measured noise artifacts.

As described in the above, there are several problems to overcome, especially including the follows

  • SANDI model is not yet suitable for clinical MRI scanner because it requires high b-values (higher than 3000 s/\(\hbox {mm}^2\)) DWI and relatively many diffusion directions to apply direction averaged DWI signals.

  • To estimate the extracellular ion concentration, \({{\bar{c}}}_{ec}\), it is assumed that the intra-neurite and soma ion concentrations are the same. Moreover, the ratio of ion concentrations in the intra- and extra-compartments is assumed as a fixed constant, \(\beta =0.41\) for human experiments. Since the ratio \(\beta \) depends on the apparent ion concentration, \(\beta \) cannot be fixed as a constant in the whole brain region. However, no method to experimentally estimate of the human brain is available.

Nevertheless, the method of extracting low-frequency electrical properties in a non-invasive way from the high-frequency conductivity is critical for clinical usefulness.

Electrical brain stimulation (EBS) techniques, such as transcranial direct current stimulation (tDCS) and deep brain stimulation (DBS), are promising treatments for human disorders [39,40,41,42,43]. Since there is no clear explanation for the mechanism, EBS studies have relied on computational modeling using reference conductivity values in the whole brain region. The proposed electrical property decomposition from the high-frequency conductivity can be a promising work for the EBS techniques.

Conclusion

We have proposed a method to decompose the electrical properties in the extracellular, intra-neurite, and soma compartments from the high-frequency conductivity map, reconstructed by solving the electro-magnetic equation with measured B1 phase signals. By decomposing the electrical conductivity into the product of mobility and charged carrier concentrations, voxel-wise micro-structures including the extracellular volume fraction and diffusivity were investigated using SANDI model by analyzing the Mb-DWI data based on a two compartment model. In the SANDI model, to distinguish the intra-soma compartment from the mixed diffusivity signals in ECS, the Gaussian phase distribution approximation of the tail was used. To determine the micro-structural parameters in each separated compartment using Mb-DWI data, a machine learning algorithm, random forests, was used by constructing a multitude of decision trees. Combining with the predicted DWI data and SANDI model, we separated the extracellular conductivity from the high-frequency conductivity using the decomposed micro-structural diffusion parameters. To verify the proposed method, we conducted human experiments to verify that the proposed method recovered the low-frequency electrical properties using a routine protocol sequence of MRI scan.

Methods

High-frequency conductivity at Larmor frequency using B1 phase map

The electrical conductivity of biological tissue as a function of frequency is complicated by the anisotropic nature of tissue, non-homogeneous natures in the extracellular and intracellular compartments, and randomly distributed cells sizes. The high-frequency conductivity is dominantly isotropic because the electrical current flow tends to pass through the cell membrane.

For the positive (negative) rotating component of the transmit B1 field \(B_1^+={\left| \,B_1^+\,\right| }e^{i\varphi ^+}\) (\(B_1^-={\left| \,B_1^-\,\right| }e^{i\varphi ^-}\)), by assuming \(\sigma _H \gg \omega \epsilon _H\), a phase-based MREPT formula has been proposed and conducted for numerous clinical studies [44,45,46,47]:

$$\begin{aligned} \nabla \varphi ^{tr}\cdot \nabla \tau _H+\tau _H\nabla ^2\varphi ^{tr}-2\omega \mu _0=0 \end{aligned}$$
(2)

where \(\tau _H\) denotes \(\frac{1}{\sigma _H}\) and \(\varphi ^{tr}=\varphi ^++\varphi ^-\) is the measurable transceive phase using MRI [1].

Since the reaction–diffusion equation is sensitive to the measured noise, to stabilize the formula (2), after adding an artificial diffusion term, the equation (2) leads to

$$\begin{aligned} -c\nabla ^2\tau _H+\nabla \varphi ^{tr}\cdot \nabla \tau _H+\tau _H\nabla ^2\varphi ^{tr}=2\omega \mu _0 \end{aligned}$$
(3)

where c is a constant diffusion coefficient.

Detection of the micro-structures of biological tissues based on intracellular and extracellular compartments

For the model-based micro-structure imaging based on the tissue micro architecture in the brain, the model parameters for the intracellular and extracellular compartments are associated with specific tissue micro-structure characteristics from Mb-DWI data [7,8,9]. The signal intensity \(S_b\) by applying a diffusion encoding gradient is given by

$$\begin{aligned} S_b=S_0\exp (-bD) \end{aligned}$$
(4)

where \(S_0\) is the signal obtained without diffusion gradient and b denotes the diffusion-weighting factor. To distinguish the diffusion signals from ECS and the cell bodies of any brain cell type (collectively named soma) [9], SANDI model proposes the following compartment model of brain tissue micro-structure:

$$\begin{aligned} S_b=S_0\left( f_{ic}\left( f_{in}A_{in}(b)+f_{is}A_{is}(b)\right) +f_{ec}A_{ec}(b)\right) \end{aligned}$$
(5)

where \(f_{ic}\) and \(f_{ec}\) are the intracellular and extracellular volume fractions, \(f_{ic}+f_{ec}=1\); \(f_{in}\) and \(f_{is}\) are the neurite and soma relative volume fractions in the intracellular compartment, \(f_{in}+f_{is}=1\); \(A_{in}\) and \(A_{is}\) are the normalized signals for restricted diffusion within neurites and soma, respectively, and \(A_{ec}\) is the normalized signal of the extracellular compartment. To investigate the complicated micro-structural model (5), some assumptions are applied to each parameter. The diffusion of water molecules associated with the extracellular compartment is modeled as isotropic Gaussian diffusion:

$$\begin{aligned} A_{ec}(b)=e^{-bD_{ec}} \end{aligned}$$
(6)

The diffusion signals \(A_{in}\) from neurites are assumed as a collection of sticks (long thin cylinders). The direction-averaged DWI signal \(A_{in}\) is computed as [11]

$$\begin{aligned} A_{in}(b)=\sqrt{\frac{\pi }{4bD_{in}}} {erf}\left( \sqrt{bD_{in}}\right) \end{aligned}$$
(7)

where erf is the error function. The signal contribution, \(A_{is}\), for the inra-soma compartment is computed from the Gaussian phase distribution (GPD) approximation of the tail:

$$\begin{aligned} A_{is}(b)=\exp \left\{ -\frac{2(\gamma G)^2}{D_{is}}\left( \sum _{m=1}^\infty \frac{\alpha ^{-4}_m}{\alpha ^2_mr^2_s-2}\right) \left( 2\delta -\Psi \right) \right\} \end{aligned}$$
(8)

where

$$\begin{aligned} \Psi =\frac{2+e^{-\alpha ^2_mD_{is}(\Delta -\delta )}-2e^{-\alpha ^2_mD_{is}\delta } -2e^{-\alpha ^2_mD_{is}\Delta }+e^{-\alpha ^2_mD_{is}(\Delta +\delta )}}{\alpha ^2_mD_{is}}, \end{aligned}$$

\(D_{is}\) is the bulk diffusivity of water in soma, \(\alpha _m\) is the m-th root of the Bessel equation \((\alpha r_s)^{-1}J_{\frac{3}{2}}(\alpha r_s)=J_{\frac{5}{2}}(\alpha r_s)\).

The total unknown parameters to be determined are \(f_{in}, f_{ec}, D_{in}, D_{is}, D_{ec},\) and \(r_s\). To avoid the ill-posedness to determine the six unknowns from the smoothly decayed exponential curves, typically the parameter of \(D_{is}\) is fixed as 2 \(\times 10^{-3}\) \(\hbox {mm}^2\)/sec [9].

Model-parameter estimation using random forest regression

Random forest is a popular machine learning algorithm using the bootstrap aggregating (bagging) with a tree model as the base model. Random forest regression is an ensemble supervised learning method that combines bagging decision trees with random subset sampling of the predictors for constructing each node split [22]. To perform the random forest regression, each individual decision tree produces a prediction individually and then predictions of all decision trees are combined to generate a prediction of the ensemble. The number of trees can be adapted to find the desired trade-off between accuracy and computational efficiency of the detection process.

The five parameters \(f_{in}, f_{ec}, D_{in}, D_{ec},\) and \(r_s\) in (5) are estimated by random forest regression using the scikit-learn python toolkit [48] as in [9]. We generated \(13^5\) synthetic signals using the model (5) with \(13^5\) combinations of the five parameters chosen uniformly distributed within the intervals: \(f_{in},\,\,f_{ec} \in [0.01,0.99]\), \(D_{in},\,\,D_{ec} \in [0.1, 3]~ \times 10^{-3}\) \(\hbox {mm}^2\)/sec, and \(r_s \in [3, 20]~ \mu \)m. We added rician-distributed noise to the synthetic signals. We split the synthetic signals into random train and test subsets with test size of 20%. Hyperparameter selection in random forest regression was performed using a grid search for the number of decision trees (150, 180, 200, 230, 250, 280, 300 and 330 trees). The remaining hyperparameters were left to default in scikit-learn toolkit. By comparing the mean squared errors obtained from the different number of trees, the final random forest regressor was built with 300 trees. One decision tree in the random forest regressor is displayed in Fig. 8. To produce an understandable image, the depth of the decision tree was limited to three.

Fig. 8
figure8

Sample visualization for a small decision tree

High-frequency conductivity decomposition

The recovered high-frequency conductivity \(\sigma _H\) at Larmor frequency, obtained by solving the equation (3), can be decomposed as the following compartment mode:

$$\begin{aligned} \sigma _H=\left( \sigma _{ne}+\sigma _{so}\right) +\sigma _{ec} \end{aligned}$$
(9)

where \(\sigma _{ne}\) and \(\sigma _{so}\) denote the conductivity in the intra-neurite and soma compartments, respectively, and \(\sigma _{ec}\) is the conductivity in the extracellular compartment. At each compartment, the apparent conductivities \(\sigma _{ne}\), \(\sigma _{so}\), and \(\sigma _{ec}\) are expressed as the sum of products of concentration, charge carrier mobility, and the charger of carrier. For simplicity of notation, we write \(f_{ne}\) and \(f_{so}\) instead of \(f_{ic} f_{in}\) and \(f_{ic}f_{is}\), respectively:

$$\begin{aligned} \sigma _{ne}&=f_{ne}\sum _{j=1}^{N_{n}}z_n^jqc^n_jm^n_j =f_{ne}\sum _{j=1}^{N_{n}}z_n^jqc^n_j\left( \frac{r_wq}{r_jk_BT}\right) D_{in}\nonumber \\&=f_{ne}{\bar{c}}_{in}D_{in} \end{aligned}$$
(10)
$$\begin{aligned} \sigma _{so}&=f_{so}\sum _{j=1}^{N_{s}}z_s^jqc^s_jm^s_j =f_{so}\sum _{j=1}^{N_{s}}z_s^jqc^s_j\left( \frac{r_wq}{r_jk_BT}\right) D_{is}\nonumber \\&=f_{so}{\bar{c}}_{is}D_{is} \end{aligned}$$
(11)
$$\begin{aligned} \sigma _{ec}&=f_{ec}\sum _{j=1}^{N_{ec}}z_e^jqc^e_jm^e_j =f_{ec}\sum _{j=1}^{N_{ec}}z_e^jqc^e_j\left( \frac{r_wq}{r_jk_BT}\right) D_{ec} \nonumber \\&=f_{ec}{\bar{c}}_{ec}D_{ec} \end{aligned}$$
(12)

where \({\bar{c}}_{ec}:=\sum _{j=1}^{N_{ec}}z_e^jqc^e_j\left( \frac{r_wq}{r_j k_BT}\right) \) denote the apparent ion concentrations with respect to the water molecule diffusivity in the extracellular compartments. The other symbols of the physical quantities are as follows: \(r_w\) and \(r_j\) are the Stoke’s radius of a water molecule and an ion, respectively, \(q=1.6\times 10^{-19}\)C is the electric charge carried by a single proton, \(k_B\) is the Boltzmann constant, and T is the absolute temperature. \(c^e_j\), \(m^e_j\) \(z^j_eq\), and \(N_{ec}\) are concentration, the charge carrier mobility, the charge of carrier, and the number of electrical charges in the extracellular space, respectively.

By the same argument, \(\sigma _{ne}=f_{ne}{\bar{c}}_{in}D_{in}\) and \(\sigma _{so}=f_{so}{\bar{c}}_{is}D_{is}\), where \({\bar{c}}_{in}\) and \({\bar{c}}_{is}\) denote the apparent ion concentrations with respect to the water molecule diffusivity in the intra-neurite, and soma compartments, respectively.

Extracellular diffusion tensor

Diffusion process is sensitive to intracellular, extracellular, and cell density. For a fixed b value, the measured non-singular diffusion tensor \(\mathbf {D}_b\) can be diagonalized as

$$\begin{aligned} \mathbf {D}_b= {\mathcal {S}}_D {\tilde{\mathbf {D}}}_b {\mathcal {S}}^T_D \quad \text{ with }\quad {\tilde{\mathbf {D}}}_b= \left( \begin{array}{ccc} d^b_1 &{} 0 &{} 0 \\ 0 &{}d^b_2 &{} 0 \\ 0 &{} 0 &{} d^b_3 \end{array} \right) \end{aligned}$$
(13)

where the column vectors of \({\mathcal {S}}_D\) are the orthonormal eigenvectors of \(\mathbf {D}_b\), the superscript T denotes the transpose and \(d^b_1\ge d^b_2\ge d^b_3\) are the corresponding eigenvalues.

We separate the apparent diffusion tensor \(\mathbf {D}_b\) into the extracellular and intracellular compartments:

$$\begin{aligned} \mathbf {D}_b=\mathbf {D}_{ec}+\mathbf {D}_{ne}+\mathbf {D}_{so} \end{aligned}$$
(14)

where \(\mathbf {D}_{ec}\) and \(\mathbf {D}_{ne}\) denote the apparent diffusion tensors in ECS and intra-neurite compartment, respectively, and \(\mathbf {D}_{so}\) is the isotropic diffusion. By assuming that the diffusion tensors \(\mathbf {D}_{ec}\), \(\mathbf {D}_{ne}\), and \(\mathbf {D}_b\) share the eigenvectors, the intrinsic diffusion tensors \(\mathbf {D}_{es}\) and \(\mathbf {D}_{ne}\) can be expressed as

$$\begin{aligned} \mathbf {D}_{ec}= {\mathcal {S}}_D {\tilde{\mathbf {D}}}_{ec} {\mathcal {S}}^T_D \quad \text{ and }\quad \mathbf {D}_{ne}= {\mathcal {S}}_D {\tilde{\mathbf {D}}}_{ne} {\mathcal {S}}^T_D \end{aligned}$$
(15)

where \({\tilde{\mathbf {D}}}_{ec}\) and \({\tilde{\mathbf {D}}}_{ne}\) are the diagonal matrices consist of the eigenvalues of \(\mathbf {D}_{ec}\) and \(\mathbf {D}_{ne}\), respectively. To translate the estimated intrinsic diffusivities \(D_{ec}\) and \(D_{in}\) to apparent diffusion tensors in each compartment, we define scale parameters as

$$\begin{aligned} \eta _{ec} = f_{ec}\frac{3 D_{ec}}{tr(\mathbf {D}_b)}, \quad \text{ and }\quad \eta _{ne} = f_{ne}\frac{3 D_{in}}{tr(\mathbf {D}_b)} \end{aligned}$$
(16)

where \(tr(\mathbf {D}_b)=d^b_1+d^b_2+d^b_3\) denotes the trace of \(\mathbf {D}_b\). Under the hypothesis that the extracellular diffusion tensor \(\mathbf {D}_{ec}\), the diffusion tensor \(\mathbf {D}_{ne}\), and the diffusion tensor \(\mathbf {D}_b\) share the eigenvectors, we can determine the decomposed diffusion tensors:

$$\begin{aligned} \mathbf {D}_{ec}=\eta _{ec} \mathbf {D}_b \quad \text{ and }\quad \mathbf {D}_{ne}=\eta _{ne} \mathbf {D}_b \end{aligned}$$
(17)

From the relation (17), the conductivity tensors in ECS and the neurite compartment can be expressed as the following

$$\begin{aligned} {\mathbf{C}}_{ec}= {\bar{c}}_{ec} \mathbf {D}_{ec}= {\bar{c}}_{ec} \eta _{ec} \mathbf {D}_{b} \end{aligned}$$
(18)

and

$$\begin{aligned} {\mathbf{C}}_{ne}= {\bar{c}}_{in} \mathbf {D}_{ne}= {\bar{c}}_{in} \eta _{ne} \mathbf {D}_{b} \end{aligned}$$
(19)

Data processing

The motion, eddy current distortion, and EPI distortion of DWI images were corrected using the DIFFPREP tool of TORTOISE [49, 50]. To reduce the noise artifacts, we used odd echoes of six measured complex MREPT signals to avoid the background phase signal due to the consecutive 180\(^\circ \) RF pulses. Since the accumulated noise artifacts in the phase signal is inversely proportional to MR magnitude intensity, \(\tilde{{\mathcal {S}}}_k,~k=1,3,5\), the measured phase signal was optimized as a weighted averaging using the weight of [51]

$$\begin{aligned} w_k = \frac{|\tilde{{\mathcal {S}}}_k|^2}{|\tilde{{\mathcal {S}}}_1|^2 +|\tilde{{\mathcal {S}}}_2|^2 +|\tilde{{\mathcal {S}}}_3|^2}, ~k=1,3,5 \end{aligned}$$
(20)

The procedure of combining the high frequency conductivity and diffusion parameter (e.g. extracellular volume fraction and extracellular diffusivity) requires an accurate registration. Using the co-registration tool of Statistical Parametric Mapping (SPM 12), the diffusion images and the diffusion parameter maps were transformed to the first echo magnitude of MREPT image.

Availability of data and materials

All data generated or analyzed during this study are included in this published article.

Abbreviations

Mb-DWI:

multi-b diffusion-weighted imaging

MREPT:

Magnetic resonance electrical properties tomography

EVF:

Extracellular volume fraction

SANDI:

Soma and neurite density imaging

ECS:

Extracellular space

SS-SE-EPI:

Single-shot spin-echo echo planner imaging

CSF:

Cerebrospinal fluid

GM:

Gray matter

WM:

White matter

IM:

Impedance measurements

DT-MREIT:

Diffusion tensor magnetic resonance electrical impedance tomography

EBS:

Electrical brain stimulation

tDCS:

Transcranial direct current stimulation

DBS:

Deep brain stimulation

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Acknowledgements

This paper was supported by Konkuk University in 2018.

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OIK designed the research topic and experiments, analyzed and drafted the manuscript. MBL and HJK prepared and performed the experiments and wrote up the experimental and result sections. All authors read and approved the final manuscript.

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Correspondence to Oh In Kwon.

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Lee, M.B., Kim, H.J. & Kwon, O.I. Decomposition of high-frequency electrical conductivity into extracellular and intracellular compartments based on two-compartment model using low-to-high multi-b diffusion MRI. BioMed Eng OnLine 20, 29 (2021). https://doi.org/10.1186/s12938-021-00869-5

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Keywords

  • Magnetic resonance electrical property tomography
  • High-frequency conductivity decomposition
  • Multi-b diffusion weighted imaging
  • Low-frequency conductivity tensor
  • Random forest