Convection-reaction equation based magnetic resonance electrical properties tomography (cr-MREPT)

Magnetic Resonance Electrical Properties Tomography Based on Modified Physics- Informed Neural Network and Multiconstraints

This paper presents a novel method based on leveraging physics-informed neural networks for magnetic resonance electrical property tomography (MREPT). MREPT is a noninvasive technique that can retrieve the spatial distribution of electrical properties (EPs) of scanned tissues from measured transmit radiofrequency (RF) in magnetic resonance imaging (MRI) systems. The reconstruction of EP values in MREPT is achieved by solving a partial differential equation derived from Maxwell's equations that lacks a direct solution. Most conventional MREPT methods suffer from artifacts caused by the invalidation of the assumption applied for simplification of the problem and numerical errors caused by numerical differentiation. Existing deep learning-based (DL-based) MREPT methods comprise data-driven methods that need to collect massive datasets for training or model-driven methods that are only validated in trivial cases. Hence we proposed a model-driven method that learns mapping from a measured RF, its spatial gradient and Laplacian to EPs using fully connected networks (FCNNs). The spatial gradient of EP can be computed through the automatic differentiation of FCNNs and the chain rule. FCNNs are optimized using the residual of the central physical equation of convection-reaction MREPT as the loss function ( L) . To alleviate the ill condition of the problem, we added multiconstraints, including the similarity constraint between permittivity and conductivity and the l1 norm of spatial gradients of permittivity and conductivity, to the L . We demonstrate the proposed method with a three-dimensional realistic head model, a digital phantom simulation, and a practical phantom experiment at a 9.4T animal MRI system.

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

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.

MR-based electrical property tomography using a physics-informed network at 3 and 7 T

Magnetic resonance electrical propert tomography promises to retrieve electrical properties (EPs) quantitatively and non-invasively in vivo, providing valuable information for tissue characterization and pathology diagnosis. However, its clinical implementation has been hindered by, for example, B1 measurement accuracy, reconstruction artifacts resulting from inaccuracies in underlying models, and stringent hardware/software requirements. To address these challenges, we present a novel approach aimed at accurate and high-resolution EPs reconstruction based on water content maps by using a physics-informed network (PIN-wEPT). The proposed method utilizes standard clinical protocols and conventional multi-channel receive arrays that have been routinely equipped in clinical settings, thus eliminating the need for specialized RF sequence/coil configurations. Compared with the original wEPT method, the network generates accurate water content maps that effectively eliminate the influence ofB → 1 + andB → 1 - by incorporating data mismatch with electrodynamic constraints derived from the Helmholtz equation. Subsequent regression analysis develops a broad relationship between water content and EPs across various types of brain tissue. A series of numerical simulations was conducted at 7 T to assess the feasibility and performance of the method, which encompassed four normal head models and models with tumorous tissues incorporated, and the results showed normalized mean square error below 1.0% in water content, below 11.7% in conductivity, and below 1.1% in permittivity reconstructions for normal brain tissues. Moreover, in vivo validations conducted over five healthy subjects at both 3 and 7 T showed reasonably good consistency with empirical EPs values across the white matter, gray matter, and cerebrospinal fluid. The PIN-wEPT method, with its demonstrated efficacy, flexibility, and compatibility with current MRI scanners, holds promising potential for future clinical application.

Changes of in vivo electrical conductivity in the brain and torso related to age, fat fraction and sex using MRI

This work was inspired by the observation that a majority of MR-electrical properties tomography studies are based on direct comparisons with ex vivo measurements carried out on post-mortem samples in the 90's. As a result, the in vivo conductivity values obtained from MRI in the megahertz range in different types of tissues (brain, liver, tumors, muscles, etc.) found in the literature may not correspond to their ex vivo equivalent, which still serves as a reference for electromagnetic modelling. This study aims to pave the way for improving current databases since the definition of personalized electromagnetic models (e.g. for Specific Absorption Rate estimation) would benefit from better estimation. Seventeen healthy volunteers underwent MRI of both brain and thorax/abdomen using a three-dimensional ultrashort echo-time (UTE) sequence. We estimated conductivity (S/m) in several classes of macroscopic tissue using a customized reconstruction method from complex UTE images, and give general statistics for each of these regions (mean-median-standard deviation). These values are used to find possible correlations with biological parameters such as age, sex, body mass index and/or fat volume fraction, using linear regression analysis. In short, the collected in vivo values show significant deviations from the ex vivo values in conventional databases, and we show significant relationships with the latter parameters in certain organs for the first time, e.g. a decrease in brain conductivity with age.

Bias correction for phase-based cr-MREPT using low resolution B1+ magnitude

ObjectiveFull-form Magnetic Resonance Electrical Properties Tomography (MREPT) requires bothB1+magnitude and phase information. SinceB1+phase can be obtained faster and with higher SNR compared toB1+magnitude, several phase-based methods have been developed for conductivity imaging. However, phase-based methods suffer from a concave bias due to the assumption that∇|B1+|is negligible in the ROI.ApproachIn this paper, we re-derive the central equation of phase-based cr-MREPT without assuming that∇|B1+|is negligible and thus propose a correction method directly integrated into the equation system.Main resultsProposed method successfully corrects the concave bias on both simulated and experimental data and significantly increases image quality.SignificanceThe proposed correction method depends on a very low-resolution|B1+|map, and therefore the imaging time does not increase significantly for obtainingB1+magnitude. Moreover, correction can be achieved using simulatedB1+magnitude, hence completely removing the additional imaging requirement.

Phantom evaluation of electrical conductivity mapping by MRI: Comparison to vector network analyzer measurements and spatial resolution assessment

PURPOSE

To evaluate the performance of various MR electrical properties tomography (MR-EPT) methods at 3 T in terms of absolute quantification and spatial resolution limit for electrical conductivity.

METHODS

Absolute quantification as well as spatial resolution performance were evaluated on homogeneous phantoms and a phantom with holes of different sizes, respectively. Ground-truth conductivities were measured with an open-ended coaxial probe connected to a vector network analyzer (VNA). Four widely used MR-EPT reconstruction methods were investigated: phase-based Helmholtz (PB), phase-based convection-reaction (PB-cr), image-based (IB), and generalized-image-based (GIB). These methods were compared using the same complex images from a 1 mm-isotropic UTE sequence. Alternative transceive phase acquisition sequences were also compared in PB and PB-cr.

RESULTS

In large homogeneous phantoms, all methods showed a strong correlation with ground truth conductivities (r > 0.99); however, GIB was the best in terms of accuracy, spatial uniformity, and robustness to boundary artifacts. In the resolution phantom, the normalized root-mean-squared error of all methods grew rapidly (>0.40) when the hole size was below 10 mm, with simplified methods (PB and IB), or below 5 mm, with generalized methods (PB-cr and GIB).

CONCLUSION

VNA measurements are essential to assess the accuracy of MR-EPT. In this study, all tested MR-EPT methods correlated strongly with the VNA measurements. The UTE sequence is recommended for MR-EPT, with the GIB method providing good accuracy for structures down to 5 mm. Structures below 5 mm may still be detected in the conductivity maps, but with significantly lower accuracy.

Feasibility of undersampled spiral trajectories in MREPT for fast conductivity imaging

PURPOSE

To investigate spiral-based imaging including trajectories with undersampling as a fast and robust alternative for phase-based magnetic resonance electrical properties tomography (MREPT) techniques.

METHODS

Spiral trajectories with various undersampling ratios were prescribed to acquire images from an experimental phantom and a healthy volunteer at 3T. The non-Cartesian acquisitions were reconstructed using SPIRiT, and conductivity maps were derived using phase-based cr-MREPT. The resulting maps were compared between different sampling trajectories. Additionally, a conductivity map was obtained using a Cartesian balanced SSFP acquisition from the volunteer to comparatively demonstrate the robustness of the proposed method.

RESULTS

The phantom and volunteer results illustrate the benefits of the spiral acquisitions. Specifically, undersampled spiral acquisitions display improved robustness against field inhomogeneity artifacts and lowered SD values with shortened readout times. Furthermore, average of conductivity values measured for the cerebrospinal fluid with the spiral acquisitions were 1.703 S/m, indicating a close agreement with the theoretical values of 1.794 S/m.

CONCLUSION

A spiral-based acquisition framework for conductivity imaging with and without undersampling is presented. Overall, spiral-based acquisitions improved robustness against field inhomogeneity artifacts, while achieving whole head coverage with multiple averages in less than a minute.

A database for MR-based electrical properties tomography with in silico brain data-ADEPT

PURPOSE

Several reconstruction methods for MR-based electrical properties tomography (EPT) have been developed. However, the lack of common data makes it difficult to objectively compare their performances. This is, however, a necessary precursor for standardizing and introducing this technique in the clinical setting. To enable objective comparison of the performances of reconstruction methods and provide common data for their training and testing, we created ADEPT, a database of simulated data for brain MR-EPT reconstructions.

METHODS

ADEPT is a database containing in silico data for brain EPT reconstructions. This database was created from 25 different brain models, with and without tumors. Rigid geometric augmentations were applied, and different electrical properties were assigned to white matter, gray matter, CSF, and tumors to generate 120 different brain models. These models were used as input for finite-difference time-domain simulations in Sim4Life, used to compute the electromagnetic fields needed for MR-EPT reconstructions.

RESULTS

Electromagnetic fields from 84 healthy and 36 tumor brain models were simulated. The simulated fields relevant for MR-EPT reconstructions (transmit and receive RF fields and transceive phase) and their ground-truth electrical properties are made publicly available through ADEPT. Additionally, nonattainable fields such as the total magnetic field and the electric field are available upon request.

CONCLUSION

ADEPT will serve as reference database for objective comparisons of reconstruction methods and will be a first step toward standardization of MR-EPT reconstructions. Furthermore, it provides a large amount of data that can be exploited to train data-driven methods. It can be accessed from  https://doi.org/10.34894/V0HBJ8.

Iterative Reconstruction Algorithms in Magneto-Acousto-Electrical Computed Tomography (MAE-CT) for Image Quality Improvement

Magneto-acousto-electrical computed tomography (MAE-CT) is a recently developed rotational magneto-acousto-electrical tomography (MAET) method, which can map the conductivity parameter of tissues with high spatial resolution. Since the imaging mode of MAE-CT is similar to that of CT, the reconstruction algorithms for CT are possible to be adopted for MAE-CT. Previous studies have demonstrated that the filtered back-projection (FBP) algorithm, which is one of the most common CT reconstruction algorithms, can be used for MAE-CT reconstruction. However, FBP has some inherent shortcomings of being sensitive to noise and non-uniform distribution of views. In this study, we introduced iterative reconstruction (IR) method in MAE-CT reconstruction and compared its performance with that of the FBP. The numerical simulation, the phantom, and in vitro experiments were performed, and several IR algorithms (ART, SART, SIRT) were used for reconstruction. The results show that the images reconstructed by the FBP and IR are similar when the data is noise-free in the simulation. As the noise level increases, the images reconstructed by SART and SIRT are more robust to the noise than FBP. In the phantom experiment, noise and some stripe artifacts caused by the FBP are removed by SART and SIRT algorithms. In conclusion, the IR method used in CT is applicable in MAE-CT, and it performs better than FBP, which indicates that the state-of-the-art achievements in the CT algorithm can also be adopted for the MAE-CT reconstruction in the future.

PIFON-EPT: MR-Based Electrical Property Tomography Using Physics-Informed Fourier Networks

We propose Physics-Informed Fourier Networks for Electrical Properties (EP) Tomography (PIFON-EPT), a novel deep learning-based method for EP reconstruction using noisy and/or incomplete magnetic resonance (MR) measurements. Our approach leverages the Helmholtz equation to constrain two networks, responsible for the denoising and completion of the transmit fields, and the estimation of the object's EP, respectively. We embed a random Fourier features mapping into our networks to enable efficient learning of high-frequency details encoded in the transmit fields. We demonstrated the efficacy of PIFON-EPT through several simulated experiments at 3 and 7 tesla(T) MR imaging, and showed that our method can reconstruct physically consistent EP and transmit fields. Specifically, when only 20% of the noisy measured fields were used as inputs, PIFON-EPT reconstructed the EP of a phantom with ≤ 5% error, and denoised and completed the measurements with ≤ 1% error. Additionally, we adapted PIFON-EPT to solve the generalized Helmholtz equation that accounts for gradients of EP between inhomogeneities. This yielded improved results at interfaces between different materials without explicit knowledge of boundary conditions. PIFON-EPT is the first method that can simultaneously reconstruct EP and transmit fields from incomplete noisy MR measurements, providing new opportunities for EPT research.

Data-driven electrical conductivity brain imaging using 3 T MRI

Magnetic resonance electrical properties tomography (MR-EPT) is a non-invasive measurement technique that derives the electrical properties (EPs, e.g., conductivity or permittivity) of tissues in the radiofrequency range (64 MHz for 1.5 T and 128 MHz for 3 T MR systems). Clinical studies have shown the potential of tissue conductivity as a biomarker. To date, model-based conductivity reconstructions rely on numerical assumptions and approximations, leading to inaccuracies in the reconstructed maps. To address such limitations, we propose an artificial neural network (ANN)-based non-linear conductivity estimator trained on simulated data for conductivity brain imaging. Network training was performed on 201 synthesized T2-weighted spin-echo (SE) data obtained from the finite-difference time-domain (FDTD) electromagnetic (EM) simulation. The dataset was composed of an approximated T2-w SE magnitude and transceive phase information. The proposed method was tested three in-silico and in-vivo on two volunteers and three patients' data. For comparison purposes, various conventional phase-based EPT reconstruction methods were used that ignoreB 1 + magnitude information, such as Savitzky-Golay kernel combined with Gaussian filter (S-G Kernel), phase-based convection-reaction EPT (cr-EPT), magnitude-weighted polynomial-fitting phase-based EPT (Poly-Fit), and integral-based phase-based EPT (Integral-based). From the in-silico experiments, quantitative analysis showed that the proposed method provides more accurate and improved quality (e.g., high structural preservation) conductivity maps compared to conventional reconstruction methods. Representatively, in the healthy brain in-silico phantom experiment, the proposed method yielded mean conductivity values of 1.97 ± 0.20 S/m for CSF, 0.33 ± 0.04 S/m for WM, and 0.52 ± 0.08 S/m for GM, which were closer to the ground-truth conductivity (2.00, 0.30, 0.50 S/m) than the integral-based method (2.56 ± 2.31, 0.39 ± 0.12, 0.68 ± 0.33 S/m). In-vivo ANN-based conductivity reconstructions were also of improved quality compared to conventional reconstructions and demonstrated network generalizability and robustness to in-vivo data and pathologies. The reported in-vivo brain conductivity values were in agreement with literatures. In addition, the proposed method was observed for various SNR levels (SNR levels = 10, 20, 40, and 58) and repeatability conditions (the eight acquisitions with the number of signal averages = 1). The preliminary investigations on brain tumor patient datasets suggest that the network trained on simulated dataset can generalize to unforeseen in-vivo pathologies, thus demonstrating its potential for clinical applications.

REC-NN: A reconstruction error compensation neural network for Magnetic Resonance Electrical Property Tomography (MREPT)

Electrical properties (EPs) are expected as biomarkers for early cancer detection. Magnetic resonance electrical properties tomography (MREPT) is a technique to non-invasively estimate the EPs of tissues from MRI measurements. While noise sensitivity and artifact problems of MREPT are being solved progressively through recent efforts, the loss of tissue contrast emerges as an obstacle to the clinical applications of MREPT. To solve the problem, we propose a reconstruction error compensation neural network scheme (REC-NN) for a typical analytic MREPT method, Stab-EPT. Two NN structures: one with only ResNet blocks, and the other hybridizing ResNet blocks with an encoder-decoder structure. Results of experiments with digital brain phantoms show that, compared with Stab-EPT, and conventional NN based reconstruction, REC-NN improves both reconstruction accuracy and tissue contrast. It is found that, the encoder-decoder structure could improve the compensation accuracy of EPs in homogeneous region but showed worse reconstruction than only ResNet structure for tumorous tissues unseen in the training samples. Future research is required to address overcompensation problems, optimization of NN structure and application to clinical data.

Physics Informed Neural Networks (PINN) for Low Snr Magnetic Resonance Electrical Properties Tomography (MREPT)

Electrical properties (EPs) of tissues facilitate early detection of cancerous tissues. Magnetic resonance electrical properties tomography (MREPT) is a technique to non-invasively probe the EPs of tissues from MRI measurements. Most MREPT methods rely on numerical differentiation (ND) to solve partial differential Equations (PDEs) to reconstruct the EPs. However, they are not practical for clinical data because ND is noise sensitive and the MRI measurements for MREPT are noisy in nature. Recently, Physics informed neural networks (PINNs) have been introduced to solve PDEs by substituting ND with automatic differentiation (AD). To the best of our knowledge, it has not been applied to MREPT due to the challenges in using PINN on MREPT as (i) a PINN requires part of ground-truth EPs as collocation points to optimize the network's AD, (ii) the noisy input data disrupts the optimization of PINNs despite the noise-filtering nature of NNs and additional denoising processes. In this work, we propose a PINN-MREPT model based on a canonical analytic MREPT model. A reference padding layer with known EPs was added to surround the region of interest for providing additive collocation points. Moreover, an optimizable diffusion coefficient was embedded in the analytic MREPT model used in the PINN-MREPT. The noise robustness of the proposed PINN-MREPT for single-sample reconstruction was tested by using numerical phantoms of human brain with extra tumor-like tissues at different noise levels. The results of numerical experiments show that PINN-MREPT outperforms two typical numerical MREPT methods in terms of reconstruction accuracy, sensitivity to the extra tissues, and the correlations of line profiles in the regions of interest. The advantage of the PINN-MREPT is shown by the results of an experiment on phantom measurement, too. Moreover, it is found that the diffusion term plays an important role to achieve a noise-robust PINN-MREPT. This is an important step moving forward to a clinical application of MREPT.

Correlation analysis between the complex electrical permittivity and relaxation time of tissue mimicking phantoms in 7 T MRI

Dielectric relaxation theory describes the complex permittivity of a material in an alternating field; in particular, Debye theory relates the time it takes for an applied field to achieve the maximum polarization and the electrical properties of the material. Although, Debye’s equations were proposed for electrical polarization, in this study, we investigate the correlation between the magnetic longitudinal relaxation time T1 and the complex electrical permittivity of tissue-mimicking phantoms using a 7 T magnetic resonance scanner. We created phantoms that mimicked several human tissues with specific electrical properties. The electrical properties of the phantoms were measured using bench-test equipment. T1 values were acquired from phantoms using MRI. The measured values were fitted with functions based on dielectric estimations, using relaxation times of electrical polarization, and the mixture theory for dielectrics. The results show that, T1 and the real permittivity are correlated; therefore, the correlation can be approximated with a rational function in the case of water-based phantoms. The correlation between index loss and T1 was determined using a fitting function based on the Debye equation and mixture theory equation, in which the fraction of the materials was taken into account. This phantom study and analysis provide an insight into the application relaxation times used for estimating dielectric properties. Currently, the measurement of electrical properties based on dielectric relaxation theory is based on an antenna, sometimes invasive, that irradiates an electric field into a small sample; thus, it is not possible to create a map of electrical properties for a complex structure such as the human body. This study could be further used to compute the electrical properties maps of tissues by scanning images and measuring T1 maps.

A Method for Electrical Property Tomography Based on a Three-Dimensional Integral Representation of the Electric Field

Magnetic resonance electrical properties tomography (MREPT) noninvasively reconstructs high-resolution electrical property (EP) maps using MRI scanners and is useful for diagnosing cancerous tissues. However, conventional MREPT methods have limitations: sensitivity to noise in the numerical Laplacian operation, difficulty in reconstructing three-dimensional (3D) EPs and convergence not guaranteed in the iterative process. We propose a novel, iterative 3D reconstruction MREPT method without a numerical Laplacian operation. We derive an integral representation of the electric field using its Helmholtz decomposition with Maxwell's equations, under the assumption that the EPs are known on the boundary of the region of interest with the approximation that the unmeasurable magnetic field components are zero. Then, we solve the simultaneous equations composed of the integral representation and Ampere's law using a convex projection algorithm whose convergence is theoretically guaranteed. The efficacy of the proposed method was validated through numerical simulations and a phantom experiment. The results showed that this method is effective in reconstructing 3D EPs and is robust to noise. It was also shown that our proposed method with the unmeasurable component H^- enhances the accuracy of the EPs in a background and that with all the components of the magnetic field reduces the artifacts at the center of the slices except when all the components of the electric field are close to zero.

Physics-Coupled Neural Network Magnetic Resonance Electrical Property Tomography (MREPT) for Conductivity Reconstruction

The electrical property (EP) of human tissues is a quantitative biomarker that facilitates early diagnosis of cancerous tissues. Magnetic resonance electrical properties tomography (MREPT) is an imaging modality that reconstructs EPs by the radio-frequency field in an MRI system. MREPT reconstructs EPs by solving analytic models numerically based on Maxwell's equations. Most MREPT methods suffer from artifacts caused by inaccuracy of the hypotheses behind the models, and/or numerical errors. These artifacts can be mitigated by adding coefficients to stabilize the models, however, the selection of such coefficient has been empirical, which limit its medical application. Alternatively, end-to-end Neural networks-based MREPT (NN-MREPT) learns to reconstruct the EPs from training samples, circumventing Maxwell's equations. However, due to its pattern-matching nature, it is difficult for NN-MREPT to produce accurate reconstructions for new samples. In this work, we proposed a physics-coupled NN for MREPT (PCNN-MREPT), in which an analytic model, cr-MREPT, works with diffusion and convection coefficients, learned by NNs from the difference between the reconstructed and ground-truth EPs to reduce artifacts. With two simulated datasets, three generalization experiments in which test samples deviate gradually from the training samples, and one noise-robustness experiment were conducted. The results show that the proposed PCNN-MREPT achieves higher accuracy than two representative analytic methods. Moreover, compared with an end-to-end NN-MREPT, the proposed method attained higher accuracy in two critical generalization tests. This is an important step to practical MREPT medical diagnoses.

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

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.

Electrical Properties Tomography: A Methodological Review

Electrical properties tomography (EPT) is an imaging method that uses a magnetic resonance (MR) system to non-invasively determine the spatial distribution of the conductivity and permittivity of the imaged object. This manuscript starts by providing clear definitions about the data required for, and acquired in, EPT, followed by comprehensively formulating the physical equations underlying a large number of analytical EPT techniques. This thorough mathematical overview of EPT harmonizes several EPT techniques in a single type of formulation and gives insight into how they act on the data and what their data requirements are. Furthermore, the review describes machine learning-based algorithms. Matlab code of several differential and iterative integral methods is available upon request.

Effect of radiofrequency inhomogeneity on water-content based electrical properties tomography and its correction by flip angle maps

Water-content based electrical properties tomography (wEPT) can retrieve electrical properties (EPs) from water-content maps. B₁⁺ field information is not involved in the traditional magnetic resonance electrical properties tomography approach. wEPT can be performed through conventional MR scanning, such as T₁-weighted spin-echo imaging, which provides convenient access to multiple clinical applications. However, the inhomogeneous radiofrequency (RF) field induced by RF coils would cause inaccuracy in wEPT reconstructions during MR scanning. We conducted a detailed investigation to evaluate the effect of inhomogeneous RF field on wEPT reconstructions to guarantee that EP mapping is desired for clinical practice. Two important considerations are involved, namely, multiple typical coil configurations and various flip angles (FAs). We proposed a correction scheme with actual FA mapping to calibrate the RF inhomogeneity and finally validated it by using human imaging at 3 T. This study illustrates a detailed evaluation for wEPT under imperfect RF homogeneity and further provides a feasible correction procedure to mitigate it. The profound knowledge of wEPT provided in our work will benefit its performance in clinical applications.

Configuration-based electrical properties tomography

PURPOSE

To introduce phase-based conductivity mapping from a configuration space analysis.

METHODS

The frequency response function of balanced SSFP (bSSFP) is used to perform a configuration space analysis. It is shown that the transceive phase for conductivity mapping can be directly obtained by a simple fast Fourier transform of a series of phase-cycled bSSFP scans. For validation, transceive phase and off-resonance mapping with fast Fourier transform is compared with phase estimation using a recently proposed method, termed PLANET. Experiments were performed in phantoms and for in vivo brain imaging at 3 T using a quadrature head coil.

RESULTS

For fast Fourier transform, aliasing can lead to systematic phase errors. This bias, however, decreases rapidly with increasing sampling points. Interestingly, Monte Carlo simulations revealed a lower uncertainty for the transceive phase and the off-resonance using fast Fourier transform as compared with PLANET. Both methods, however, essentially retrieve the same phase information from a set of phase-cycled bSSFP scans. As a result, configuration-based conductivity mapping was successfully performed using eight phase-cycled bSSFP scans in the phantoms and for brain tissues. Overall, the retrieved values were in good agreement with expectations. Conductivity estimation and mapping of the field inhomogeneities can therefore be performed in conjunction with the estimation of other quantitative parameters, such as relaxation, using configuration theory.

CONCLUSIONS

Phase-based conductivity mapping can be estimated directly from a simple Fourier analysis, such as in conjunction with relaxometry, using a series of phase-cycled bSSFP scans.

MR electrical properties imaging using a generalized image-based method

PURPOSE

To develop a fast and easy-to-use electrical properties tomography (EPT) method based on a single MR scan, avoiding both the need of a B₁ -map and transceive phase assumption, and that is robust against noise.

THEORY

Derived from Maxwell's equations, conductivity, and permittivity are reconstructed from a new partial differential equation involving the product of the RF fields and its derivatives. This also allows us to clarify and revisit the relevance of common assumptions of MREPT.

METHODS

Our new governing equation is solved using a 3D finite-difference scheme and compared to previous frameworks. The benefits of our method over selected existing MREPT methods are demonstrated for different simulation models, as well as for both an inhomogeneous agar phantom gel and in vivo brain data at 3T.

RESULTS

Simulation and experimental results are illustrated to highlight the merits of the proposed method over existing methods. We show the validity of our algorithm in versatile configurations, with many transition regions notably. Complex admittivity maps are also provided as a complementary MR contrast.

CONCLUSION

Because it avoids time-consuming RF field mapping and generalizes the use of standard MR image for electrical properties reconstruction, this contribution is promising as a new step forward for clinical applications.

Magnetic-Resonance-Based Electrical Property Mapping Using Global Maxwell Tomography With an 8-Channel Head Coil at 7 Tesla: A Simulation Study

OBJECTIVE

Global Maxwell Tomography (GMT) is a recently introduced volumetric technique for noninvasive estimation of electrical properties (EP) from magnetic resonance measurements. Previous work evaluated GMT using ideal radiofrequency (RF) excitations. The aim of this simulation study was to assess GMT performance with a realistic RF coil.

METHODS

We designed a transmit-receive RF coil with 8 decoupled channels for 7T head imaging. We calculated the RF transmit field ( B₁⁺) inside heterogeneous head models for different RF shimming approaches, and used them as input for GMT to reconstruct EP for all voxels.

RESULTS

Coil tuning/decoupling remained relatively stable when the coil was loaded with different head models. Mean error in EP estimation changed from [Formula: see text] to [Formula: see text] and from [Formula: see text] to [Formula: see text] for relative permittivity and conductivity, respectively, when changing head model without re-tuning the coil. Results slightly improved when an SVD-based RF shimming algorithm was applied, in place of excitation with one coil at a time. Despite errors in EP, RF transmit field ( B₁⁺) and absorbed power could be predicted with less than [Formula: see text] error over the entire head. GMT could accurately detect a numerically inserted tumor.

CONCLUSION

This work demonstrates that GMT can reliably reconstruct EP in realistic simulated scenarios using a tailored 8-channel RF coil design at 7T. Future work will focus on construction of the coil and optimization of GMT's robustness to noise, to enable in-vivo GMT experiments.

SIGNIFICANCE

GMT could provide accurate estimations of tissue EP, which could be used as biomarkers and could enable patient-specific estimation of RF power deposition, which is an unsolved problem for ultra-high-field magnetic resonance imaging.

Statistical analysis of the accuracy of water content-based electrical properties tomography

Water content-based electrical properties tomography (wEPT) can retrieve electrical properties (EPs) from water content maps, thereby eliminating the need for B₁ field measurement in the traditional magnetic resonance electrical properties tomography method. The wEPT is performed by conventional MR scanning, such as T₁ -weighted spin-echo imaging, and thus can be directly applied to clinical settings. However, the random noise propagation involved in wEPT causes inaccuracy in EP mapping. To guarantee the EP estimates desired for clinical practice, this study statically investigates the noise-specific uncertainty of wEPT through probability density function models. We calculated the probability distribution of EP maps with different noise levels and examined the effects of scan parameters on reconstruction accuracy with various flip angles (FAs) and repetition time (TR) settings. The theoretical derivation was validated by Monte Carlo simulations and human imaging experiment at 3 T. Results showed that a serious deviation could occur in tissues with large conductivity value at a low signal-to-noise ratio and quantitatively demonstrate that such deviation could be mitigated by increased FAs or TRs. This study provided useful information for the setup of scan parameters, evaluation of accuracy of the wEPT under specific SNR levels, and promote its clinical applications.

Deep learning-based reconstruction of in vivo pelvis conductivity with a 3D patch-based convolutional neural network trained on simulated MR data

PURPOSE

To demonstrate that mapping pelvis conductivity at 3T with deep learning (DL) is feasible.

METHODS

210 dielectric pelvic models were generated based on CT scans of 42 cervical cancer patients. For all dielectric models, electromagnetic and MR simulations with realistic accuracy and precision were performed to obtain B 1 + and transceive phase (ϕ^± ). Simulated B 1 + and ϕ^± served as input to a 3D patch-based convolutional neural network, which was trained in a supervised fashion to retrieve the conductivity. The same network architecture was retrained using only ϕ^± in input. Both network configurations were tested on simulated MR data and their conductivity reconstruction accuracy and precision were assessed. Furthermore, both network configurations were used to reconstruct conductivity maps from a healthy volunteer and two cervical cancer patients. DL-based conductivity was compared in vivo and in silico to Helmholtz-based (H-EPT) conductivity.

RESULTS

Conductivity maps obtained from both network configurations were comparable. Accuracy was assessed by mean error (ME) with respect to ground truth conductivity. On average, ME < 0.1 Sm^-1 for all tissues. Maximum MEs were 0.2 Sm^-1 for muscle and tumour, and 0.4 Sm^-1 for bladder. Precision was indicated with the difference between 90^th and 10^th conductivity percentiles, and was below 0.1 Sm^-1 for fat, bone and muscle, 0.2 Sm^-1 for tumour and 0.3 Sm^-1 for bladder. In vivo, DL-based conductivity had median values in agreement with H-EPT values, but a higher precision.

CONCLUSION

Anatomically detailed, noise-robust 3D conductivity maps with good sensitivity to tissue conductivity variations were reconstructed in the pelvis with DL.

A robust electrical conductivity imaging method with total variation and wavelet regularization

PURPOSE

This study aims to develop and evaluate a robust conductivity imaging method that combines total variation and wavelet regularization to enhance the accuracy of conductivity maps.

THEORY AND METHODS

The proposed approach is based on a gradient-based method. The central equation is derived from Maxwell's equation and describes the relationship between conductivity and the transceive phase. A linear system equation is obtained via a finite-difference method and solved using a least-squares method. Total variation and wavelet transform regularization terms are added to the minimization problem and solved using the Split Bregman method to improve reconstruction stability. The proposed approach is compared with conventional and gradient-based methods. Numerical simulations are performed to validate the accuracy of the developed method, and the effects of noise are determined. Phantom and in vivo experiments are conducted at 3 T to verify the clinical applicability of the proposed method.

RESULTS

Numerical simulations show that the proposed method is more robust than other methods and can suppress the effects of noise. The quantitative conductivity value of the phantom experiment agrees with the measured value. The in vivo experiment results present a clear structure, and the conductivity value of the tumor region is significantly higher than that around healthy tissues.

CONCLUSION

The proposed electrical conductivity imaging method can improve the quality of conductivity reconstruction, and thus, has future clinical applications.

Magnetic resonance electrical property mapping at 21.1 T: a study of conductivity and permittivity in phantoms, ex vivo tissue and in vivo ischemia

Electrical properties (EP), namely conductivity and permittivity, can provide endogenous contrast for tissue characterization. Using electrical property tomography (EPT), maps of EP can be generated from conventional MRI data. This report investigates the feasibility and accuracy of EPT at 21.1 T for multiple RF coils and modes of operation using phantoms. Additionally, it demonstrates the EP of the in vivo rat brain with and without ischemia. Helmholtz-based EPT was implemented in its Full-form, which demands the complex [Formula: see text] field, and a simplified form requiring either just the [Formula: see text] field phase for conductivity or the [Formula: see text] field magnitude for permittivity. Experiments were conducted at 21.1 T using birdcage and saddle coils operated in linear or quadrature transceive mode, respectively. EPT approaches were evaluated using a phantom, ex and in vivo Sprague-Dawley rats under naïve conditions and ischemic stroke via transient middle cerebral artery occlusion. Different conductivity reconstruction approaches applied to the phantom displayed average errors of 12%-73% to the target acquired from dielectric probe measurements. Permittivity reconstructions showed higher agreement and an average 3%-8% error to the target depending on reconstruction approach. Conductivity and permittivity of ex and in vivo rodent brain were measured. Elevated EP in the ischemia region correlated with the increased sodium content and the influx of water intracellularly following ischemia in the lesion were detected. The Full-form technique generated from the linear birdcage provided the best accuracy for EP of the phantom. Phase-based conductivity and magnitude-based permittivity mapping provided reasonable estimates but also demonstrated the limitations of Helmholtz-based EPT at 21.1 T. Permittivity reconstruction was improved significantly over lower fields, suggesting a novel metric for in vivo brain studies. EPT applied to ischemic rat brain proved sensitivity to physiological changes, motivating the future application of more advanced reconstruction approaches.

Numerical Experiments on the Contrast Capability of Magnetic Resonance Electrical Property Tomography

Purpose:

Magnetic resonance electrical property tomography (MR EPT) is a technique for non-invasively obtaining the electric property (EP) distribution of biological tissues, with a promising potential for application in the early detection of tumors. However, the contrast capability (CC) of this technique has not been fully studied. This work aims to theoretically explore the CC for detecting the variation of EP values and the size of the imaging region.

Methods:

A simulation scheme was specifically designed to evaluate the CC of MR EPT. The simulation study has the advantage that the magnetic field can be accurately obtained. EP maps of the designed phantom embedded with target regions of designated various sizes and EPs were reconstructed using the homogeneous Helmholtz equation based on B1+ with different signal-to-noise ratios (SNRs). The CC was estimated by determining the smallest detectable EP contrast when the target region size was as large as the Laplacian kernel and the smallest detectable target region size when the EP contrast was the same as the difference between healthy and malignant tissues in the brain, based on the reconstructed EP maps.

Results:

Using noise free B1+, the smallest detectable contrast_σ and contrast_εr were 1% and 3%, respectively, and the smallest detectable target region size was 1 mesh unit (the base unit size used in the simulation) for conductivity and relative permittivity. The smallest detectable EP contrast and target region size were decreased as the B1+ SNR increased.

Conclusion:

The CC of MR EPT was related with the SNR of the magnetic field. A small EP contrast and size were necessary for detection at a high-SNR magnetic field. Obtaining a high-SNR magnetic field is important for improving the CC of MR EPT.

Integral MR-EPT With the Calculation of Coil Current Distributions

Many integral equation (IE)-based magnetic resonance electrical property tomography (MR-EPT) methods use unloaded incident radio-frequency (RF) fields from simulations that may not fully reflect the real situation and thus lead to reconstruction errors. To improve the accuracy of IE-based MR-EPT methods, a novel approach that enables the calculation of loaded coil current distributions and avoids the explicit use of incident RF fields is presented in this paper. In the proposed method, a hybrid source composed of the current source from the coil and the contrast source from the subject are introduced in the integral equations. Because the loaded coil current distributions can be extracted from the reconstructed hybrid source, the simulated incident RF fields are eliminated from the problem formulations. To improve the convergence performance, a modified conjugate gradient (CG) scheme was used where the gradients of the current source and contrast source were balanced through using different weighting parameters. The proposed method was verified through full-wave simulations at 9.4 and 7 T involving a heterogeneous ball and an anatomical head phantom. The numerical results indicated that by using the proposed method, an accurate coil current distributions and EPs profiles can be reconstructed and the desirable robustness against noise can also be achieved.

Transceive phase mapping using the PLANET method and its application for conductivity mapping in the brain

Purpose

To demonstrate feasibility of transceive phase mapping with the PLANET method and its application for conductivity reconstruction in the brain.

Methods

Accuracy and precision of transceive phase (ϕ^±) estimation with PLANET, an ellipse fitting approach to phase‐cycled balanced steady state free precession (bSSFP) data, were assessed with simulations and measurements and compared to standard bSSFP. Measurements were conducted on a homogeneous phantom and in the brain of healthy volunteers at 3 tesla. Conductivity maps were reconstructed with Helmholtz‐based electrical properties tomography. In measurements, PLANET was also compared to a reference technique for transceive phase mapping, i.e., spin echo.

Results

Accuracy and precision of ϕ^± estimated with PLANET depended on the chosen flip angle and TR. PLANET‐based ϕ^± was less sensitive to perturbations induced by off‐resonance effects and partial volume (e.g., white matter + myelin) than bSSFP‐based ϕ^±. For flip angle = 25° and TR = 4.6 ms, PLANET showed an accuracy comparable to that of reference spin echo but a higher precision than bSSFP and spin echo (factor of 2 and 3, respectively). The acquisition time for PLANET was ~5 min; 2 min faster than spin echo and 8 times slower than bSSFP. However, PLANET simultaneously reconstructed T₁, T₂, B₀ maps besides mapping ϕ^±. In the phantom, PLANET‐based conductivity matched the true value and had the smallest spread of the three methods. In vivo, PLANET‐based conductivity was similar to spin echo‐based conductivity.

Conclusion

Provided that appropriate sequence parameters are used, PLANET delivers accurate and precise ϕ^± maps, which can be used to reconstruct brain tissue conductivity while simultaneously recovering T₁, T₂, and B₀ maps.

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.

Accuracy and precision of electrical permittivity mapping at 3T: the impact of three B 1 + mapping techniques

Purpose

To investigate the sequence‐specific impact of B1+ amplitude mapping on the accuracy and precision of permittivity reconstruction at 3T in the pelvic region.

Methods

B1+ maps obtained with actual flip angle imaging (AFI), Bloch–Siegert (BS), and dual refocusing echo acquisition mode (DREAM) sequences, set to a clinically feasible scan time of 5 minutes, were compared in terms of accuracy and precision with electromagnetic and Bloch simulations and MR measurements. Permittivity maps were reconstructed based on these B1+ maps with Helmholtz‐based electrical properties tomography. Accuracy and precision in permittivity were assessed. A 2‐compartment phantom with properties and size similar to the human pelvis was used for both simulations and measurements. Measurements were also performed on a female volunteer’s pelvis.

Results

Accuracy was evaluated with noiseless simulations on the phantom. The maximum B1+ bias relative to the true B1+ distribution was 1% for AFI and BS and 6% to 15% for DREAM. This caused an average permittivity bias relative to the true permittivity of 7% to 20% for AFI and BS and 12% to 35% for DREAM. Precision was assessed in MR experiments. The lowest standard deviation in permittivity, found in the phantom for BS, measured 22.4 relative units and corresponded to a standard deviation in B1+ of 0.2% of the B1+ average value. As regards B1+ precision, in vivo and phantom measurements were comparable.

Conclusions

Our simulation framework quantitatively predicts the different impact of B1+ mapping techniques on permittivity reconstruction and shows high sensitivity of permittivity reconstructions to sequence‐specific bias and noise perturbation in the B1+ map. These findings are supported by the experimental results.

Developments in Electrical-Property Tomography Based on the Contrast-Source Inversion Method

The main objective of electrical-property tomography (EPT) is to retrieve dielectric tissue parameters from B ^ 1 + data as measured by a magnetic-resonance (MR) scanner. This is a so-called hybrid inverse problem in which data are defined inside the reconstruction domain of interest. In this paper, we discuss recent and new developments in EPT based on the contrast-source inversion (CSI) method. After a short review of the basics of this method, two- and three-dimensional implementations of CSI-EPT are presented along with a very efficient variant of 2D CSI-EPT called first-order induced current EPT (foIC-EPT). Practical implementation issues that arise when applying the method to measured data are addressed as well, and the limitations of a two-dimensional approach are extensively discussed. Tissue-parameter reconstructions of an anatomically correct male head model illustrate the performance of two- and three-dimensional CSI-EPT. We show that 2D implementation only produces reliable reconstructions under very special circumstances, while accurate reconstructions can be obtained with 3D CSI-EPT.

Use of dielectric padding to eliminate low convective field artifact in cr-MREPT conductivity images

PURPOSE

Convection-reaction equation-based magnetic resonance electrical properties tomography (cr-MREPT) provides conductivity images that are boundary artifact-free and robust against noise. However, these images suffer from the low convective field (LCF) artifact. We propose to use dielectric pads to alter the transmit magnetic field (B₁ ⁺ ), shift the LCF region, and eliminate the LCF artifact.

METHODS

Computer simulations were conducted to analyze the effects of pad electrical properties, pad thickness, pad height, arc angle, and thickness of the pad-object gap. In 3T MR experiments, water pads and BaTiO₃ pads were used with agar-saline phantoms. Two data sets (e.g., with the pad located on the left or on the right of the object [phantom]) were acquired, and the corresponding linear systems were simultaneously solved to get LCF artifact-free conductivity images.

RESULTS

A pad needed to have 180° arc angle and the same height with the phantom for maximum benefit. Increasing the pad thickness and/or the relative permittivity of the pad increased the LCF shift, whereas excessive amounts of these parameters caused errors in conductivity reconstructions because the effect of neglected B_z terms became noticeable. Conductivity of the pad, on the other hand, had minimal effect on elimination of the LCF artifact. Combining 2 data sets (i.e., with 2 different dielectric pad positions) resulted in more accurate conductivity maps (low L² -errors) as opposed to no pad or single pad cases in experiments and simulations.

CONCLUSIONS

Using the proposed technique, LCF artifact is significantly removed, and the reconstructed conductivity values are improved.

Mapping electrical properties heterogeneity of tumor using boundary informed electrical properties tomography (BIEPT) at 7T

PURPOSES

To develop and evaluate a boundary informed electrical properties tomography (BIEPT) technique for high-resolution imaging of tumor electrical properties (EPs) heterogeneity on a rodent tumor xenograft model.

METHODS

Tumor EP distributions were inferred from a reference area external to the tumor, as well as internal EP spatial variations derived from a plurality of relative transmit B1 measurements at 7T. Edge sparsity constraint was enforced to enhance numerical stability. Phantom experiments were performed to determine the imaging accuracy and sensitivity for structures of various EP values, as well as geometrical sizes down to 1.5 mm. Numerical simulation of a realistic rodent model was used to quantify the algorithm performance in the presence of noise. Eleven athymic rats with human breast cancer xenograft were imaged in vivo, and representative pathological samples were acquired for comparison.

RESULTS

Reconstructed EPs of the phantoms correspond well to the ground truth acquired from dielectric probe measurements, with the smallest structure reliably detectable being 3 mm. EPs heterogeneity inside a tumor is successfully retrieved in both simulated and experimental cases. In vivo tumor imaging results demonstrate similar local features and spatial patterns to anatomical MRI and pathological slides. The imaged conductivity of necrotic tissue is higher than that of viable tissues, which agrees with our expectation.

CONCLUSION

BIEPT enables robust detection of tumor EPs heterogeneity with high accuracy and sensitivity to small structures. The retrieved quantitative EPs reflect tumor pathological features (e.g., necrosis). These results provide strong rationale to further expand BIEPT studies toward pathological conditions where EPs may yield valuable, non-invasive biomarkers.

Reference-Based Integral MR-EPT:Simulation and Experiment Studies on the 9.4T MRI

Current integral-equation (IE) based MR electrical properties tomography (EPT) methods utilize simulated incident radio-frequency (RF) fields, which are inaccurate and lead to reconstruction errors. To improve the accuracy and practicability of IE-based MR-EPT methods, a new approach is presented that obtains the incident fields using reference subjects and RF field mapping techniques. The Incident field approximation (IFA) is first demonstrated in this paper. This approximation assumes that two imaged subjects with similar coil/subject interactions will have similar incident RF fields, thus one can feed the estimation of the incident fields within the imaged subject into the calculation of those within a homogeneous subject (reference subject). This is done by measuring the total RF fields ( field) of the reference using field mapping techniques, using the known EPs of the reference subject and by rearranging Ampere's Law and the integral equations. The calculated incident RF fields are then used to reconstruct the EPs' distribution with a three-dimensional (3D) integral-based MR-EPT method. Numerical simulation results indicated that the incident RF fields obtained from the reference subject provide accurate 3D reconstruction of EPs with less than 16% root mean square error (RMSE) in noise-free scenario while the conventional IE method had more than 28% RMSE. The phantom-based experiments at 9.4T MRI system have also been conducted to evaluate the performance of the proposed method and the results indicated that the proposed method achieved desirable robustness against the noise in practical scenario with less than 21% RMSE while the conventional differential equation-based method showed worse than 37% RMSE.

bSSFP phase correction and its use in magnetic resonance electrical properties tomography

PURPOSE

Balanced steady-state free precession (bSSFP) sequence is widely used because of its high SNR and high speed. However, bSSFP images suffer from "banding artifact" caused by B₀ inhomogeneity. In this article, we propose a method to remove this artifact in bSSFP phase images and investigate the usage of the corrected phase images in phase-based magnetic resonance electrical properties tomography (MREPT).

THEORY AND METHODS

Two bSSFP phase images, obtained with different excitation frequencies, are collaged to get rid of the regions containing banding artifacts. Phase of the collaged bSSFP image is the sum of the transceive phase of the RF system and an error term that depends on B₀ and T₂ . By using B₀ and T₂ maps, this error is eliminated from bSSFP phase images by using pixel-wise corrections. Conductivity maps are obtained from the uncorrected and the corrected phase images using the phase-based cr-MREPT method.

RESULTS

Phantom and human experiment results of the proposed method are illustrated for both phase images and conductivity maps. It is shown that uncorrected phase images yield unacceptable conductivity images. When only B₀ information is used for phase correction conductivity, reconstructions are substantially improved, and yet T₂ information is still needed to fully recover accurate and undistorted conductivity images.

CONCLUSIONS

With the proposed technique, B₀ sensitivity of the bSSFP phase images can be removed by using B₀ and T₂ maps. It is also shown that corrected bSSFP phase images are of sufficient quality to be used in conductivity imaging.

Electrical properties tomography: Available contrast and reconstruction capabilities

MR-based electrical properties tomography converts the MRI transmit/receive RF field measurements to tissue electrical property maps through dedicated reconstruction algorithms. Recent reports showed that despite limitations, electrical properties tomography holds promise for generating additional contrast for tumor detection and patient-specific modeling of tissue-RF field interactions. This review summarizes the available tissue electrical property contrasts and compares them with the capabilities of the most commonly used electrical properties tomography reconstruction method. Future directions and prospects of clinical translation are discussed.

Redesign of the Laplacian kernel for improvements in conductivity imaging using MRI

PURPOSE

To develop an electrical property tomography reconstruction method that achieves improvements over standard method by redesigning the Laplacian kernel.

THEORY AND METHODS

A decomposition property of the governing PET equation shows the possibility of redesigning the Laplacian kernel for conductivity reconstruction. Hence, the discrete Laplacian operator used for electrical property tomography reconstruction is redesigned to have a Gaussian-like envelope, which enables manipulation of the spatial and spectral response. The characteristics of the proposed kernel are investigated through numerical simulations and in vivo brain experiments.

RESULTS

The proposed method reduces textured noise, which hampers observing features of the conductivity image. Furthermore, the proposed scheme can mitigate the propagation of local phase error such as flow-induced phase. By doing so, the proposed method can recover feature information in conductivity (or resistivity) images. Lastly, the proposed kernel can be extended to other electrical property tomography reconstructions, improving the quality of images.

CONCLUSION

An alternative design of the Laplacian kernel for conductivity imaging has been developed to mitigate the textured noise and the propagation of local phase artifact.

Dictionary-based electric properties tomography

PURPOSE

To develop and validate a new algorithm called "dictionary-based electric properties tomography" (dbEPT) for deriving tissue electric properties from measured B₁ maps.

METHODS

Inspired by Magnetic Resonance fingerprinting, dbEPT uses a dictionary of local patterns ("atoms") of B₁ maps and corresponding electric properties distributions, derived from electromagnetic field simulations. For reconstruction, a pattern from a measured B₁ map is compared with the B₁ atoms of the dictionary. The B₁ atom showing the best match with the measured B₁ pattern yields the optimum electric properties pattern that is chosen for reconstruction. Matching was performed through machine learning algorithms. Two dictionaries, using transmit and transceive phases, were evaluated. The spatial distribution of local matching distance between optimal atom and measured pattern yielded a reconstruction reliability map. The method was applied to reconstruct conductivity of 4 volunteers' brains. A conventional, Helmholtz-based Electric properties tomography (EPT) reconstruction was performed for reference. Noise performance was studied through phantom simulations.

RESULTS

Quantitative values of conductivity agree with literature values. Results of the 2 dictionaries exhibit only minor differences. Somewhat larger differences are visible between dbEPT and Helmholtz-based EPT. Quantified by the correlation between conductivity and anatomic images, dbEPT depicts brain details more clearly than Helmholtz-based EPT. Matching distance is minimal in homogeneous brain ventricles and increases with tissue heterogeneity. Central processing unit time was approximately 2 minutes per dictionary training and 3 minutes per brain conductivity reconstruction using standard hardware equipment.

CONCLUSION

A new, dictionary-based approach for reconstructing electric properties is presented. Its conductivity reconstruction is able to overcome the EPT transceive-phase problem.

3-D Contrast Source Inversion-Electrical Properties Tomography

Contrast source inversion-electrical properties tomography (CSI-EPT) is an iterative reconstruction method to retrieve the electrical properties (EPs) of tissues from magnetic resonance data. The method is based on integral representations of the electromagnetic field and has been shown to allow EP reconstructions of small structures as well as tissue boundaries with compelling accuracy. However, to date, the CSI-EPT has been implemented for 2-D configurations only, which limits its applicability. In this paper, a full 3-D extension of the CSI-EPT method is presented, to enable CSI-EPT to be applied to realistic 3-D scenarios. Here, we demonstrate a proof-of-principle of 3-D CSI-EPT and present the reconstructions of a 3-D abdominal body section and a 3-D head model using different settings of the transmit coil. Numerical results show that the full 3-D approach yields accurate reconstructions of the EPs, even at tissue boundaries and is most accurate in regions where the absolute value of the electric field is highest.

MR-based electrical property tomography using a modified finite difference scheme

Magnetic resonance electrical property tomography (MR-EPT) reconstructs electrical properties (EPs) from measured magnetic fields in magnetic resonance imaging (MRI) systems. In this study, an MR-EPT method was proposed that utilized a new finite difference approximation of the involved differential wave equation. Compared with existing MR-EPT approaches, the construction of the system matrix involves applying the first derivative twice based on a larger number of neighbouring finite-difference grids, which is different from a standard Laplacian operator on a regular grid structure, leading to a better conditioned linear inverse problem. With improved noise robustness, more faithful EPs can be obtained by the proposed method, particularly at tissue boundaries and regions with a poorly measured magnetic field (low signal-to-noise ratio). Numerical simulations with a specially designed multi-slice phantom and an anatomically accurate head model (Duke) have demonstrated that the proposed method can provide a more faithful reconstruction of EPs compared to existing methods, which usually offer unreliable solutions associated with traditional finite difference approximation of the central wave equation and unrealistic assumptions. Experiments on a 9.4 T MRI system have been conducted to validate the simulations.

In vivo imaging of electrical properties of an animal tumor model with an 8-channel transceiver array at 7 T using electrical properties tomography

PURPOSE

To develop and evaluate a technique for imaging electrical properties ((EPs), conductivity and permittivity) of an animal tumor model in vivo using MRI.

METHODS

Electrical properties were reconstructed from the calculated EP gradient, which was derived using two sets of measured transmit B1 magnitude and relative phase maps with the sample and radiofrequency (RF) coil oriented in the positive and negative z-directions, respectively. An eight-channel transceiver microstrip array RF coil fitting the size of the animal was developed for generating and mapping B1 fields to reconstruct EPs. The technique was evaluated at 7 tesla using a physical phantom and in vivo on two Copenhagen rats with subcutaneously implanted AT-1 rat prostate cancer on a hind limb.

RESULTS

The reconstructed EPs in the phantom experiment was in good agreement with the target EP map determined by a dielectric probe. Reconstructed conductivity map of the animals revealed the boundary between tumor and healthy tissue consistent with the boundary indicated by T1 -weighted MRI.

CONCLUSION

A technique for imaging EP of an animal tumor model using MRI has been developed with high sensitivity, accuracy, and resolution, as demonstrated in the phantom experiment. Further animal experiments are needed to demonstrate its translational value for tumor diagnosis. Magn Reson Med 78:2157-2169, 2017. © 2016 International Society for Magnetic Resonance in Medicine.

Error analysis of helmholtz-based MR-electrical properties tomography

PURPOSE

MR electrical properties tomography (MR-EPT) aims to measure tissue electrical properties by computing spatial derivatives of measured B1+ data. This computation is very sensitive to spatial fluctuations caused, for example, by noise and Gibbs ringing. In this work, the error arising from the computation of spatial derivatives using finite difference kernels (FD error) has been investigated. In relation to this FD error, it has also been investigated whether mitigation strategies such as Gibbs ringing correction and Gaussian apodization can be beneficial for conductivity reconstructions.

METHODS

Conductivity reconstructions were performed on a phantom (by means of simulations and MR measurements at 3T) and on a human brain model. The accuracy was evaluated as a function of image resolution, FD kernel size, k-space windowing, and signal-to-noise ratio. The impact of mitigation strategies was also investigated.

RESULTS

The adopted small FD kernel is highly sensitive to spatial fluctuations, whereas the large FD kernel is more noise-robust. However, large FD kernels lead to extended numerical boundary error propagation, which severely hampers the MR-EPT reconstruction accuracy for highly spatially convoluted tissue structures such as the human brain. Mitigation strategies slightly improve the accuracy of conductivity reconstructions. For the adopted derivative kernels and the investigated scenario, MR-EPT conductivity reconstructions show low accuracy: less than 37% of the voxels have a relative error lower than 30%.

CONCLUSION

The numerical error introduced by the computation of spatial derivatives using FD kernels is one of the major causes of limited accuracy in Helmholtz-based MR-EPT reconstructions. Magn Reson Med 80:90-100, 2018. © 2017 International Society for Magnetic Resonance in Medicine.

Electrical Properties Tomography Based on $B_{{1}}$ Maps in MRI: Principles, Applications, and Challenges

OBJECTIVE

The purpose is to provide a comprehensive review of the electrical properties tomography (EPT) technique, which was introduced to image the electrical properties (EPs) of tissue noninvasively by exploiting the measured field data of MRI.

METHODS

We reviewed the principle of EPT, reconstruction methods, biomedical applications such as tumor imaging, and existing challenges. As a key application of EPT, the estimation of specific absorption rate (SAR) due to MRI was discussed in the background of elevated risk of tissue heating at high field.

RESULTS AND CONCLUSION

Since the originally proposed local, homogeneous Helmholtz equation-based reconstruction algorithm, advanced EPT algorithms have emerged to address the challenges of EPT, including reconstruction error near tissue boundaries, noise sensitivity, inaccurate phase estimation, and elimination of the unmeasurable component, along with demonstrations of in vivo experiments. EPT techniques have been applied to investigate EPs of both healthy and pathological tissues in vivo and factors contributing to various EP value, including sodium, water content, etc. More studies are anticipated to consolidate the current findings. EPT-based subject-specific SAR estimation has led to in vivo demonstration of its feasibility and prediction of temperature increase of phantom during MRI scans merely using measured data.

SIGNIFICANCE

EPT has the advantage of high resolution and practical feasibility in a clinical setup for imaging the biomedically interesting EPs of tissue in the radiofrequency range. EPT-based SAR estimation is another promising topic for predicting tissue heating of individual subjects during a specific MRI scan.