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.

Magnetic resonance electrical properties tomography (MR‐EPT) is a technique used to estimate the conductivity and permittivity of tissues from MR measurements of the transmit magnetic field. Different reconstruction methods are available; however, all these methods present several limitations, which hamper the clinical applicability. Standard Helmholtz‐based MR‐EPT methods are severely affected by noise. Iterative reconstruction methods such as contrast source inversion electrical properties tomography (CSI‐EPT) are typically time‐consuming and are dependent on their initialization. Deep learning (DL) based methods require a large amount of training data before sufficient generalization can be achieved. Here, we investigate the benefits achievable using a hybrid approach, that is, using MR‐EPT or DL‐EPT as initialization guesses for standard 3D CSI‐EPT. Using realistic electromagnetic simulations at 3 and 7 T, the accuracy and precision of hybrid CSI reconstructions are compared with those of standard 3D CSI‐EPT reconstructions. Our results indicate that a hybrid method consisting of an initial DL‐EPT reconstruction followed by a 3D CSI‐EPT reconstruction would be beneficial. DL‐EPT combined with standard 3D CSI‐EPT exploits the power of data‐driven DL‐based EPT reconstructions, while the subsequent CSI‐EPT facilitates a better generalization by providing data consistency.

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

The main objective of electrical-property tomography (EPT) is to retrieve dielectric tissue parameters from ${\widehat{B}}_1^{+}$ 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.

Three-dimensional contrast source inversion-electrical properties tomography (3-D CSI-EPT) is an iterative reconstruction method that estimates the electrical properties of tissue from transmit field magnetic resonance data. However, in order to bring 3-D CSI-EPT into practice for complex tissue structures and to understand the origin and effect of errors, insight in the sensitivities of reconstruction accuracy to the major error-sources is necessary. In this paper, different strategies for implementing 3-D CSI-EPT, including their iterative structure, are presented, of which the regularized implementation shows the most potential to be used in practice. Moreover, the influence of initialization, noise, stopping criteria, incident fields, B1-maps, transceive phase and domain truncation are discussed. We show that of all these different error-sources, initialization, accurate coil models and domain truncation have the most dramatic effect on electrical properties reconstructions in practice.

In this paper, we present an Electrical Properties Tomography (EPT) methodology based on integral (Green's tensor) representations for the electromagnetic field. Inhomogeneous tissue profiles can easily be incorporated in such an approach and the reconstruction method is less sensitive to noise compared with more standard differential based EPT methods since smoothing integral operators act on measured field data. To keep the computational bookkeeping to a minimum, we derive our basic EPT field representations for two-dimensional E-polarized radio frequency fields, which is a valid assumption in the midplane of a loaded birdcage coil. The extension to the fully three-dimensional case is briefly discussed and some initial 3D reconstructions are presented as well. Possible extensions and practical implementation issues are also discussed.