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

\textit{Objective:} In this paper, we introduce Physics-Informed Fourier Networks for Electrical Properties Tomography (PIFON-EPT), a novel deep learning-based method that solves an inverse scattering problem based on noisy and/or incomplete magnetic resonance (MR) measurements. \textit{Methods:} We used two separate fully-connected neural networks, namely $B_1^{+}$ Net and EP Net, to solve the Helmholtz equation in order to learn a de-noised version of the input $B_1^{+}$ maps and estimate the object's EP. A random Fourier features mapping was embedded into $B_1^{+}$ Net, to learn the high-frequency details of $B_1^{+}$ more efficiently. The two neural networks were trained jointly by minimizing the combination of a physics-informed loss and a data mismatch loss via gradient descent. \textit{Results:} We performed several numerical experiments, showing that PIFON-EPT could provide physically consistent reconstructions of the EP and transmit field. Even when only $50\%$ of the noisy MR measurements were used as inputs, our method could still reconstruct the EP and transmit field with average error $2.49\%$, $4.09\%$ and $0.32\%$ for the relative permittivity, conductivity and $B_{1}^{+}$, respectively, over the entire volume of the phantom. The generalized version of PIFON-EPT that accounts for gradients of EP yielded accurate results at the interface between regions of different EP values without requiring any boundary conditions. \textit{Conclusion:} This work demonstrated the feasibility of PIFON-EPT, suggesting it could be an accurate and effective method for EP estimation. \textit{Significance:} PIFON-EPT can efficiently de-noise $B_1^{+}$ maps, which has the potential to improve other MR-based EPT techniques. Furthermore, PIFON-EPT is the first technique that can reconstruct EP and $B_{1}^{+}$ simultaneously from incomplete noisy MR measurements.