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

A software tool for fabricating phantoms mimicking human tissues with designated dielectric properties and frequency

OBJECTIVES

Dielectric materials play a crucial role in assessing and refining the measurement performance of dielectric properties for specific tasks. The availability of viable and standardized dielectric materials could greatly enhance medical applications related to dielectric properties. However, obtaining reliable phantoms with designated dielectric properties across a specified frequency range remains challenging. In this study, we propose software to easily determine the components of dielectric materials in the frequency range of 16 MHz to 3 GHz.

METHODS

A total of 184 phantoms were fabricated and measured using open-ended coaxial probe method. The relationship among dielectric properties, frequency, and the components of dielectric materials was fitted through feedforward neural networks. Software was developed to quickly calculate the composition of dielectric materials.

RESULTS

We performed validation experiments including blood, muscle, skin, and lung tissue phantoms at 128 MHz, 298 MHz, 915 MHz, and 2.45 GHz. Compared with literature values, the relative errors of dielectric properties are less than 15 %.

CONCLUSIONS

This study establishes a reliable method for fabricating dielectric materials with designated dielectric properties and frequency through the development of the software. This research holds significant importance in enhancing medical research and applications that rely on tissue simulation using dielectric phantoms.

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.

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.

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.

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.