# EPTlib: An Open-Source Extensible Collection of Electric Properties Tomography Techniques
## 3 EPT Techniques
### Helmholtz-EPT  
Assuming a constant magnetic permeability equal to the one of vacuum \(\mu_{0}\)[37], the time-harmonic RF magnetic flux density \(B_{1}\) generated by the MRI scanner in the biological tissues satisfies  
\[\
abla\times\left(\xi^{-1}\
abla\times B_{1}\right)=\omega^{2}\mu_{0}B_{1}\,, \tag{3}\]where \(\omega=2\pi f\) is the angular frequency, being \(f\) the Larmor frequency, and \(\tilde{\epsilon}=\epsilon_{\mathrm{r}}\epsilon_{0}-\mathrm{i}\sigma/\omega\) is the complex permittivity, being \(\epsilon_{0}\) the electric permittivity of vacuum. This equation can be written equivalently as  
\[\
abla\tilde{\epsilon}^{-1}\times(\
abla\times B_{1})-\tilde{\epsilon}^{-1} \
abla^{2}B_{1}=\omega^{2}\mu_{0}B_{1}\,, \tag{4}\]  
whose left hand side is deduced from the vector calculus identity \(\
abla\times(\
abla\times B_{1})=\
abla(\
abla\cdot B_{1})\)\(-\
abla^{2}B_{1}\) and from the [PERSON]'s law for magnetism \(\
abla\cdot B_{1}=0\). In the regions where the EPs are almost homogeneous, namely \(\
abla\tilde{\epsilon}^{-1}\cong 0\), the latter coincides with the vector Helmholtz equation. Since the positively rotating component \(B_{1}^{+}\) of the RF coil is defined as [38]  
\[B_{1}^{+}=\frac{B_{1,x}+\mathrm{i}B_{1,y}}{2}\,, \tag{5}\]  
the Cartesian components of the vector Helmholtz equation can be combined linearly to obtain the Helmholtz equation for \(B_{1}^{+}\), which inverted algebraically with respect to \(\tilde{\epsilon}\) leads to the fundamental equation of Helmholtz-EPT [17],  
\[\boxed{\tilde{\epsilon}=-\frac{\
abla^{2}B_{1}^{+}}{\omega^{2}\mu_{0}B_{1}^{+ }}}\,. \tag{6}\]  
Clearly, where the EPs does not satisfy the assumption of homogeneity, like at the tissue boundaries, this equation introduces systematic errors. By denoting \(B_{1}^{+}=|B_{1}^{+}|\mathrm{e}^{\mathrm{i}\sigma^{+}}\), an explicit equation for each EP can be deduced. Precisely,  
\[\epsilon_{\mathrm{r}}=-\frac{\
abla^{2}|B_{1}^{+}|}{\omega^{2}\mu_{0}\epsilon _{0}|B_{1}^{+}|}+\frac{|\
abla\varphi^{+}|^{2}}{\omega^{2}\mu_{0}\epsilon_{0} }\,, \tag{7}\]  
and  
\[\sigma=\frac{\
abla^{2}\varphi^{+}}{\omega\mu_{0}}+2\frac{\
abla|B_{1}^{+}| \cdot\
abla\varphi^{+}}{\omega\mu_{0}|B_{1}^{+}|}\,, \tag{8}\]  
where the Tx phase \(\varphi^{+}\) is estimated as half of the measurable TRx phase \(\varphi^{\pm}\), when the same RF coil is used both in transmission and in reception with a polarization switch [5]. In the case of \(|\
abla\varphi^{+}|^{2}\ll|\
abla^{2}|B_{1}^{+}|/|B_{1}^{+}|\), the relative permittivity can be estimated with the magnitude-based approximation of (7) [18]  
\[\boxed{\epsilon_{\mathrm{r}}\cong-\frac{\
abla^{2}|B_{1}^{+}|}{\omega^{2}\mu_ {0}\epsilon_{0}|B_{1}^{+}|}}\,, \tag{9}\]  
which underestimates systematically the permittivity. Similarly, when the Tx sensitivity \(|B_{1}^{+}|\) is enough homogeneous, so that \(2|\
abla|B_{1}^{+}|\cdot\
abla\varphi^{+}|/|B_{1}^{+}|\ll|\
abla^{2}\varphi^ {+}|\), the electric conductivity can be estimated with the phase-based approximation of (8) [18]  
\[\sigma\cong\frac{\
abla^{2}\varphi^{+}}{\omega\mu_{0}}\,. \tag{10}\]  
It is worth noting that the homogeneity of \(|B_{1}^{+}|\) is often satisfactorily verified by MRI scanners up to 3 T with volume Tx coils [39], since they are designed to achieve it.