Information for : Opening a new window on MR-based Electrical Properties Tomography with deep learning

In the radiofrequency (RF) range, the electrical properties of tissues (EPs: conductivity and permittivity) are modulated by the ionic and water content, which change for pathological conditions. Information on tissues EPs can be used e.g. in oncology as a biomarker. The inability of MR-Electrical Properties Tomography techniques (MR-EPT) to accurately reconstruct tissue EPs by relating MR measurements of the transmit RF field to the EPs limits their clinical applicability. Instead of employing electromagnetic models posing strict requirements on the measured MRI quantities, we propose a data driven approach where the electrical properties reconstruction problem can be casted as a supervised deep learning task (DL-EPT). DL-EPT reconstructions for simulations and MR measurements at 3 Tesla on phantoms and human brains using a conditional generative adversarial network demonstrate high quality EPs reconstructions and greatly improved precision compared to conventional MR-EPT. The supervised learning approach leverages the strength of electromagnetic simulations, allowing circumvention of inaccessible MR electromagnetic quantities. Since DL-EPT is more noise-robust than MR-EPT, the requirements for MR acquisitions can be relaxed. This could be a major step forward to turn electrical properties tomography into a reliable biomarker where pathological conditions can be revealed and characterized by abnormalities in tissue electrical properties.


Supplementary Materials and Methods
Phantom and Head Models. 42 cylindrical phantom models and the 20 head models were created in Sim4Life (ZMT AG, Zurich, CH). The ground truth EPs values of these models are reported in the Supplementary Tables S1 and S2, respectively. In order to introduce more variability between the adopted head models, not only the conductivity and permittivity values of WM, GM and CSF were changed between models, but also geometrical transformations were applied with respect to the original models (Duke M0 and Ella M0) 1 . These transformations include compression/dilatation of the head models, as well as rotation and translation, thus mimicking different possible head orientations inside the MR bore. For each head model, ground truth EPs maps are shown for one slice (red plane, Supplementary Figs S1 and S2). This slice was taken on the same plane for all the head models with respect to the considered volume of interest (yellow box). Therefore, the observed variability between subfigures is due to the performed geometrical transformations and variations in the EPs for the simulated head models.

Database Construction.
Two simulations were performed in Sim4Life for each phantom and head model ( Supplementary Fig. S3): one in quadrature mode (QA), and one in anti-quadrature mode (AQ). Contrary to conventional MR-EPT approaches, which reconstruction models require the non-measurable RF transmit phase φ + (approximated with φ ± 2 : the so-called transceive phase assumption) 2 , here the transceive phase (φ ± ) was used, i.e. the phase measurable in an MR experiment. From these simulations, the electromagnetic quantity B 1 + was obtained ( Supplementary Fig. S4). B 1 + consists of the transmit B 1 + field magnitude and the phase φ + proportional to the transceive phase: φ + = ( where φ + ≠ φ + since φ + ≠ φ − . Then, Gaussian noise was independently added to the real and imaginary parts of the computed complex B 1 + field. Finally, the magnitude and the phase of the obtained noise-corrupted B 1 + fields were used as inputs for the cGANs (Supplementary Fig. S4). The SNR of |B 1 + | maps and the precision of φ + maps obtained from the simulations were defined as: To reduce the complexity of the reconstruction problem, cGANs were independently trained for permittivity and conductivity reconstructions, but the same values were used for the network weights λ GAN , λ L1 , and λ L2 . The inputs were the magnitude of the noise-corrupted B 1 + field, the phase φ + (proportional to the transceive phase φ ± measurable in an MR experiment) and a binary mask (1 for tissue and 0 for air). We define this network as cGAN mask . To investigate the impact of tissue information on the accuracy of the reconstructed EPs values, pseudo Spin Echo images were used instead of the binary mask as third input. We define this network as cGAN tissue (Supplementary Fig. S5). These pseudo Spin Echo images were created for each brain model as it follows. First, reference magnitude values were computed for each brain tissue from MRI measurements on a healthy subject performed using a Spin Echo sequence (see Supplementary Materials and Methods -MR Sequences). In particular, these reference values are mean magnitude values computed for each tissue type inside regions with a homogeneous B 1 + magnitude field distribution. These values were applied to the corresponding tissue type of each brain model. Then, the obtained maps were scaled using the simulated B 1 + magnitude field distribution for each head model. Finally, Gaussian noise was added using the same SNR level adopted for the phase maps φ + .  Table S3). where represents {|B 1 + |, φ + , } or {|B 1 + |, φ + , } in the training set, are the corresponding ground truth EPs maps and is a vector drawn from the probability distribution p z 3 . As reported in Isola 3 , p z is ignored by the network and therefore it is left out in the network implementation. Different weights (λ GAN , λ L1 , and λ L2 ) were used during training. The phantom models 12 and 24, which were excluded from the training set, were used in the validation step to choose which combination of λ-weights had the lowest average normalized-root-mean-square error (NRMSE) computed over the reconstructed EPs values of both phantoms. This combination of λ-weights was: λ GAN = 2, λ L1 = 100, and λ L2 = 200 (Supplementary Table S4). This combination was therefore used for testing using the phantom models 38, and 42, the phantom MRI measurements, the head model Duke M0 and the in-vivo MRI measurements. Of course, the phantom and head models, as well as the phantom and in-vivo MRI measurements used for the validation and the testing steps were excluded from the training dataset.

MR Sequences.
In the Supplementary Tables S5 and S6 are reported the MR sequence parameters used for the Actual Flip Angle Imaging (AFI) sequence and for the two Spin Echo sequences acquired with opposite readout gradient polarities. From the AFI sequence, B 1 + magnitude maps were obtained 4 . From the Spin Echo sequences, φ + maps were computed 5 .

Supplementary Results
EPs Reconstructions. In the Supplementary Fig. S7, the profiles of the reconstructed conductivity and permittivity maps for the phantom model 42 using H-EPT (blue) and cGAN mask (red) are shown. These profiles were taken in direction left/right, as shown in the subfigures on the right (black lines). In these subfigures, the gray circles indicate the region of interest (ROI) used to compute the mean and SD of the reconstructed EPs values for the phantom models used for validation (phantom models 12, and 24) and for testing (phantom models 38, 42, and phantom MR measurements). The same ROI was used for all the other slices of the phantoms. In this way, errors arising from boundary regions in H-EPT reconstructions were excluded. In the Supplementary Fig. S8, the absolute error maps of conductivity and permittivity reconstructions are shown for the phantom model 42 and for the phantom MR measurements, which were used for testing of the selected cGAN mask . The absolute error for conductivity reconstructions is below 0.05 S/m (less than 5% relative error), for both the simulation and the MR measurement. The absolute error for permittivity reconstructions is below 5 for the simulated data, while it is a bit higher (about 8) for the reconstruction from the MR measurement. The higher error in permittivity reconstructions from MR measurements can be explained by intrinsic inaccuracies in the adopted B 1 + magnitude mapping technique. The absolute error for H-EPT reconstructions from simulated data is instead one order of magnitude higher than the error observed for the cGAN mask reconstructions. In the Supplementary Fig. S9, the reconstructed EPs maps for the phantom model 38, which was also used for testing, and the mean ± SD of the reconstructed EPs values are reported. The relative errors for these reconstructions are in line with the relative errors previously observed for the phantom model 42.
In the Supplementary Fig. S10, absolute error maps for conductivity and permittivity reconstructions for the head model Duke M0 are presented. From these maps, it can be observed that the absolute error at tissue boundaries can be reduced if tissue information is given in input to the cGAN. In contrast, the absolute error for H-EPT reconstructions is at least one order of magnitude higher than the errors reported for the adopted cGANs. In the Supplementary Fig. S11, in-vivo DL-EPT reconstructions for the second and the third subject are shown. The mean and SD values of the reconstructed EPs in the WM, GM, and CSF are reported in Supplementary Table S7. These results confirm what was previously observed in the main manuscript for the first subject, thus showing the feasibility of reconstructing tissue EPs in-vivo using DL-EPT.
Impact of SNR. The impact of different SNR levels (no noise, 50, 20, and 5) on EPs reconstructions was investigated for the selected cGAN mask using the head model Duke M0. From the Supplementary Fig. S12 and Table S8, it is visible that only for low SNR levels (less than 20) EPs reconstructions are not accurate anymore. Typical SNR levels in MR experiments are higher than this value, thus suggesting that deep learning approaches would be sufficiently noise-robust for EPs reconstructions from MR measurements. Still, adequate knowledge on the SNR limits for DL-EPT reconstructions would be fundamental to allow for faster MR sequences with higher spatial resolutions (voxel size in the order of 1 mm) than typically employed MR sequences for EPs reconstructions.  Table S9, it can be observed that the combination of λ-weights giving the lowest average NRMSE for cGAN mask is: λ GAN = 2, λ L1 = 1000, and λ L2 = 2000. This cGAN mask was used for DL-EPT reconstructions on the Duke model M0 with a tumor inclusion. It can also be observed that setting λ GAN = 0, thus using a U-Net instead of a cGAN, could in principle lead to accurate results. For sake of completeness, a comparison between EPs reconstructions for Duke M0 using the U-net, and the cGANs adopted in the manuscript is presented in Supplementary Fig. S13. From the Supplementary Fig. S13, it appears that EPs reconstructions using a U-Net are more blurred than cGANs reconstructions. However, we do not exclude that different training parameters and more exhaustive training sets could allow more accurate reconstructions at tissue boundaries. This will be focus of future works.

H-EPT Reconstructions:
These reconstructions demonstrate that H-EPT provides accurate EPs reconstructions only in large homogeneous regions for noiseless cases (Supplementary Fig. S14). However, even if a small kernel is used, severe errors at tissue boundaries are observed. For real cases with the presence of noise, large kernels need to be employed in H-EPT for noise robust reconstructions, however, at the cost of a larger spatial extension of boundary errors. For the SNR level adopted in this manuscript, which is typical for an MRI experiment, H-EPT conductivity reconstructions are of poor quality and permittivity reconstructions are not feasible. This is due to presence of boundary errors as well as errors due to noise amplification introduced by the numerical Laplacian operation 5 . cGAN tissue rescaling: To test whether cGAN tissue would learn a rescaling using only the pseudo Spin Echo image and discarding the transceive phase and the magnitude of the B 1 + , we gave as an input to the cGAN tissue network only the pseudo Spin Echo images of Duke M0. If cGAN tissue output would rely heavily on the pseudo Spin Echo image intensity and learn a simple rescaling for EPs maps generation, we would expect that the cGAN tissue output should still be EPs maps. However, as shown in the Supplementary Fig. S15, this is not the case, indicating that B 1 + magnitude and phase information are needed. Future work should investigate whether other strategies are possible, e.g. providing only boundary information instead of full tissue information.

Supplementary References
Supplementary Figure S1: Conductivity maps of the simulated head models. These maps were taken on the same slice (red plane) inside the considered volume of interest (yellow box). Figure S2: Permittivity maps of the simulated head models. These maps were taken on the same slice (red plane) inside the considered volume of interest (yellow box). Figure S3: The setup adopted in Sim4Life for the electromagnetic simulations on: (a) phantoms, (b) head models. Figure S4: Flowchart of the operations performed to create the input maps for the cGANs. Figure S5: Flowchart of the inputs/outputs of the adopted cGANs (cGAN mask , and cGAN tissue ) for training, validation, and testing. Figure S6: Measured Spin Echo magnitude map (left) and pseudo Spin Echo map (right). The depicted four ROIs are used to compute the mean signal intensity values (see Supplementary Table S3). These maps were normalized between 0 and 1.  conductivity (a, b) and permittivity (d, e) maps using H -EPT (a, d) and cGAN mask (b, e). Phantom MRI measurements: absolute error maps for the reconstructed conductivity (c) and permittivity (f) maps using cGAN mask . Figure S9: Phantom 38 conductivity (a, b,) and permittivity (c, d,) maps reconstructed using H-EPT (a, c,) and cGAN mask (b, d,). The reported numbers are the mean ± SD values computed inside the region of interest indicated in the Supplementary Figure S7. Ground truth EPs values are respectively σ = 1 S/m and ε r = 66 (see Table  S1).  , g), the cGAN mask adopted for the tumor reconstruction from simulations using Duke M0 (c, h), and the cGAN mask (d, i) and cGAN tissue (e, j) adopted for DL-EPT reconstructions in the manuscript, i.e. for the phantom model 42, Duke M0, phantom and in-vivo brain MR measurements. Figure S14: Comparison between H-EPT reconstructions using a small kernel (3×3×3) and a large kernel (7×7×5) for the noiseless case. Notable the errors at tissue boundaries, which spatial extension increases for the large kernel. EPs reconstructions are accurate only inside large homogeneous regions of WM.

Supplementary
Supplementary Figure S15: cGAN tissue reconstructions given only the pseudo Spin Echo images of Duke M0 as input to the network.

Supplementary Table S1: Phantoms EPs values
The models 12 and 24 are used for validation, while the models 38 and 42 are used for testing. The electrical properties values of the 20 head models. T x , T y , and T z are the scaling factors applied to the original models (M0) along the coordinate axis x, y, and z (T x,y,z = 100: no scaling, T x,y,z > 100: dilatation, and T x,y,z < 100: compression). The models Duke M0 and Ella M0 are the reference models.

Supplementary Table S3. Spin Echo Magnitude
Comparison between measured and pseudo Spin Echo magnitude values in the four ROIs depicted in Supplementary Figure S6.

Supplementary Table S5. AFI Sequence Parameters
Sequence parameters used for the AFI sequence. This sequence was adopted to map the magnitude of the transmit MR field.

Supplementary Table S6. Spin Echo Sequence Parameters
For both phantom and in-vivo MR measurements, this sequence was performed twice, i.e. with opposite readout gradient polarities to compensate for eddy-currents related artifacts. This sequence was adopted to map the transceive phase.