# EPTlib: An Open-Source Extensible Collection of Electric Properties Tomography Techniques
## 3 EPT Techniques
### Convection-Reaction-EPT
It reads  
\[\
abla\cdot\left(\gamma\mathbf{\beta}_{\varphi}^{-}\right)+\mathrm{i}\gamma|\
abla \varphi^{-}|^{2}=\mathrm{i}\omega^{2}\mu_{0}\,, \tag{20}\]  
being \(\mathbf{\beta}_{\varphi}^{-}=\
abla\varphi^{-}+\mathrm{i}\
abla\times\left( \varphi^{-}\tilde{z}\right)\). The sum of Equation (19) and (20) leads to  
\[\
abla\cdot\left(\gamma\
abla\varphi^{\pm}\right)-\mathrm{i}\
abla\cdot\left( \gamma\
abla\times\left((\varphi^{+}-\varphi^{-})\tilde{z}\right)\right)+ \mathrm{i}\gamma\left(|\
abla\varphi^{+}|^{2}+|\
abla\varphi^{-}|^{2}\right)=2 \mathrm{i}\omega^{2}\mu_{0}\,, \tag{21}\]  
where the TRx phase \(\varphi^{\pm}=\varphi^{+}+\varphi^{-}\) appears. Finally, assuming that \(\sigma/\omega\gg\epsilon_{\mathrm{F}}\epsilon_{0}\), as it actually is for many biological tissues at 64 MHz (the Larmor frequency of 1.5 T MRI scanners) [18], it is possible to approximate the unknown as \(\gamma\cong\mathrm{i}\omega/\sigma\). Thus, the imaginary part of (21) reads [23; 42]  
\[\boxed{\
abla\cdot\left(\rho\
abla\varphi^{\pm}\right)=2\omega\mu_{0}}\,, \tag{22}\]where the resistivity \(\rho=\sigma^{-1}\) has been introduced. Similarly to Equation (18), also (22) can be approximated in two-dimensions assuming that \(\partial\rho/\partial z\cong 0\). The PDEs on which convection-reaction-EPT is based are of difficult solution because of the lack of a diffusion term. The introduction of an artificial diffusion term in the equations acts like a viscosity-type regularization of the problem [43] both for the complex-valued Equation (17) and its two-dimensional approximation (18) and for the real-valued phase-based approximation (22). The regularized PDE for the complete convection-reaction-EPT is  
Figure 3:0  
where \(\lambda\) denotes the artificial diffusion coefficient. An analogous correction is introduced in the other equations. In _EPTlib_, the class EPTConvReact manages the convection-reaction-EPT technique. The PDEs are solved numerically according to the finite difference method. The complex-valued PDEs are approximated with second-order centred finite differences, whereas the real-valued PDEs derived by the phase-based approximation are discretized with stable first-order upwind finite differences. In all the cases, the artificial diffusion term is discretized with second-order centred finite differences. The obtained linear system is solved using the biconjugate gradient stabilized method implemented in Eigen 3. ```
1:method = 1
2:[input]
3:tx-channels = 1
4:rx-channels = 1
5:[parameter.savitzky-galaxy]
6:size = [2,2,1]
7:shape = 0
8:[parameter.dirichlet]
9:electric-conductivity = 0.1
10:relative-permittivity = 50.0
11:[parameter]
12:volume-tomography = false
13:imaging-slice = 3
14:artificial-diffusion = true
15:artificial-diffusion-coefficient = 0.1
```  
**Configuration File 3** Convection-reaction-EPT specific settings for _EPTlib_app_  
The first- and second-order derivatives of the Tx sensitivity and the TRx phase are estimated by _EPTlib_ applying a Savitzky-Golay filter [36]. Thus, _EPTlib_app_ requires the optional section _parameter.savitzky-galaxy_ in the configuration file, as described above for Helmholtz-EPT. In addition, it requires the settings reported in the example Configuration File 3 for determining the Dirichlet boundary conditions and selecting the proper PDE. Precisely, the optional section _parameter.dirichlet_ set the Dirichlet boundary conditions as constant values in the entire boundary:  
* **electric-conductivity** is the electric conductivity in siemens per meter forced at the domain boundary. By default 0. It is worth noting that this condition is used also by the phase-based approximated convection-reaction-EPT (22), in which case the assigned value must be different from zero in order to allow the correct definition of the resistivity \(\rho\). * **relative-permittivity** is the relative permittivity forced at the domain boundary. By default 1.