# EPTlib: An Open-Source Extensible Collection of Electric Properties Tomography Techniques
###### Abstract  
Electric properties tomography (EPT) is a novel magnetic resonance imaging-based method to estimate non-invasively the distribution of the electric properties in the human body. In this paper, _EPTlib_, an open-source extensible C++ library collecting ready-to-use algorithms for electric properties tomography, is presented. Currently, _EPTlib_ implements three techniques, named Helmholtz-EPT, convection-reaction-EPT and gradient-EPT, whose derivation and implementation is deeply discussed. Moreover, the configuration files needed by the terminal application included in _EPTlib_ to apply the implemented techniques are outlined. The three techniques are applied to a couple of model problems in order to highlight their main features and the effects of the tunable parameters.  
electric properties tomography; open-source software; magnetic resonance imaging; quantitative imaging +
Footnote †: journal: Applied Sciences  
1
Footnote 1: email: {[[https://doi.org/10.3390/app11073237](https://doi.org/10.3390/app11073237)]([https://doi.org/10.3390/app11073237](https://doi.org/10.3390/app11073237))}  
and the phase \(\varphi^{+}\), namely the Tx phase, of \(B_{1}^{+}\) are usually acquired separately. The former by applying a B1-mapping sequence, like the actual flip-angle [15] or the Bloch-Siegert shift [16]; the latter under special symmetry conditions as half of the so-called transceive (TRx) phase \(\varphi^{\pm}\), which is obtained as the phase of complex images acquired with common MRI sequences [5].  
Amongst the EPT techniques based on the elaboration of \(B_{1}^{+}\), many mathematical approaches can be distinguished. For example, there are direct techniques [17; 18; 19; 20; 21; 22; 23], which provide the estimate of the EPs after a direct filtering of \(B_{1}^{+}\), and inverse techniques [24; 25; 26; 27; 28; 29], which retrieve the EPs by minimizing a discrepancy function. The same methods can be classified in local techniques [17; 18; 21], which estimate the EPs in a point by relying only on the \(B_{1}^{+}\) measured in nearby pixels, and global techniques [19; 20; 22; 23; 24; 25; 26; 27; 28; 29], which retrieve the EPs distribution in the entire domain at once. Moreover, data-driven approaches based on deep-learning have been proposed recently [30; 31].  
Tools implementing other quantitative imaging methods for the EPs have been developed, like the EIDORS software [32] for the electric impedance tomography (EIT) [33] and the MRCI toolbox [34] for the magnetic resonance-based electric impedance tomography (MREIT) [35]. However, despite the large number of techniques proposed in the literature, up to the author's knowledge, there are no publicly available algorithms implementing EPT, limiting its diffusion among the scientific community. In this paper, _EPTlib_, an open-source extensible C++ library collecting ready-to-use algorithms for EPT that has recently been made available ([[https://eptilb.github.io](https://eptilb.github.io)]([https://eptilb.github.io](https://eptilb.github.io))), is presented. This library represents a new common platform addressed both to scientists interested in applying the EPT techniques and to those aiming at developing new techniques. Currently, _EPTlib_ implements three EPT techniques--Helmholtz-EPT [17; 18], convection-reaction-EPT [19; 23] and gradient-EPT [20]--but its object-oriented programming design allows to easily extend it with other implementations. The library and the implemented techniques are described thoroughly in the next section. Then, some examples of applications of _EPTlib_ in realistic model problems of head imaging are provided and discussed.  
All the websites linked by this paper have been accessed in date 3 April 2021.