Magnetic-resonance-based measurement of electromagnetic fields and conductivity in vivo using single current administration-A machine learning approach

Abstract

Diffusion tensor magnetic resonance electrical impedance tomography (DT-MREIT) is a newly developed technique that combines MR-based measurements of magnetic flux density with diffusion tensor MRI (DT-MRI) data to reconstruct electrical conductivity tensor distributions. DT-MREIT techniques normally require injection of two independent current patterns for unique reconstruction of conductivity characteristics. In this paper, we demonstrate an algorithm that can be used to reconstruct the position dependent scale factor relating conductivity and diffusion tensors, using flux density data measured from only one current injection. We demonstrate how these images can also be used to reconstruct electric field and current density distributions. Reconstructions were performed using a mimetic algorithm and simulations of magnetic flux density from complementary electrode montages, combined with a small-scale machine learning approach. In a biological tissue phantom, we found that the method reduced relative errors between single-current and two-current DT-MREIT results to around 10%. For in vivo human experimental data the error was about 15%. These results suggest that incorporation of machine learning may make it easier to recover electrical conductivity tensors and electric field images during neuromodulation therapy without the need for multiple current administrations.

Fig 1. Schematic of the dual-loop network.

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

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

Fig 3. Phantom experiment setup.

(a) Phantom design, (b) MR magnitude image from spin-echo pulse sequence during MREIT experiment, (c) Segmented ROIs for volume conductor model. The ROIs were segmented using the unsupervised data partitioning method implemented in the MATLAB command kmeans. (d) $B_z^{m,\mathcal{E}},\mathcal{E}=1\text{(vertical)}\text{and}\text{2}\text{(horizontal)}$ images induced due to the 10 mA current injection, (e) Mean diffusivity map obtained from six direction diffusion data sets with b-value 1000 sec/mm². Images in (b)-(e) cropped to 100 × 100 pixels² to show detail.

Fig 4. Human experiment set-up.

(a) Sagittal view of computer model of human subject with attached electrodes (Fpz, Oz, T7, T8). The corresponding lead wire trajectories are marked as ${\mathcal{L}}_{\text{Fpz}}$, ${\mathcal{L}}_{\text{Oz}}$, ${\mathcal{L}}_{\text{T7}}$, ${\mathcal{L}}_{\text{T8}}$. (b) High resolution T₁-weighted image corresponding to the MREIT slice shown in (e). Locations of six ROIs used for evaluation of reconstructed conductivity values shown in Table 5. (c) Segmented tissues in volume conductor model. Tissue segmentation was performed using the method described in Huang et al. [50]. (d) Colour-coded fractional anisotropy map obtained from fifteen-direction diffusion data sets with b-value 1000 sec/mm². (e) MR magnitude image from multi-gradient multi-echo pulse sequence during MREIT acquisition. (f) Brain ROI (${\mathcal{R}}_t$), mask used in electromagnetic field reconstruction. (c) Echo-combined $B_z^{m,\mathcal{E}},\mathcal{E}=1\text{(Fpz-Oz),}\text{and}\text{2}\text{(T7-T8)}$ images induced due to the 1.5 mA current injection. Calculated stray magnetic fields were subtracted from individual echo images before further processing. Images in (b)-(c) are cropped to 175 × 210 and those in (d), (f)-(g) are cropped to 85 × 100.

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

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

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

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

Fig 7

Model-predicted current density in a central slice of human head data ${\mathbf{J}}_0{\mid }_{{\mathcal{R}}_t}$ (I, III) obtained from Eq (22) and estimated current density images found using Eq (23) (II, IV) are shown for (a) Fpz-Oz and (b) T8-T7 electrode montages. Normalized arrow plots overlaid on images show current flow directions. Reconstructed scale factor images of the central slice of in-vivo human subject are shown in parts (c) and (d) for Fpz-Oz or T8-T7 current injections respectively. Images in (c) and (d) labeled $\widehat{\eta }$ show solutions of the dual-loop matrix system in (9). Images labeled η show ANN-corrected scale factors. Part (e) shows a scale factor image $\tilde{\eta }$ recovered from data measured from both current injections. Images cropped to 75 × 100.

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

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

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

(a) Phantom design. (b)-(c) MR and corresponding $B_z^{m,\mathcal{E}}\mathcal{E}=1,2$ images. (d) Mean diffusivity map showing muscle directions. (e) Scale factor images reconstructed using dual-loop method for (I) horizontal, (II) vertical injection only. Part III shows the scale factor image obtained using two-current DT-MREIT method. (f) IV-V show equivalent isotropic conductivities from one current injection, and VI is isotropic conductivity found using two-current injections and the J-substitution algorithm [4]. Parts (g)-(h) show diagonal components of the reconstructed conductivity tensor using one-(horizontal) and two-current injection methods respectively. Images cropped to 40 × 40.