Complex Conductivity

Biological tissue both conducts and stores charge: cell membranes act as capacitors. Under a time-harmonic excitation with angular frequency $\omega$ the material is described by the complex admittivity

$$\gamma (x,\omega )=\sigma (x)+i\omega \epsilon (x),$$

where $\sigma$ is the conductivity and $\epsilon$ the permittivity. The potential then solves $\nabla \cdot (\gamma \nabla u)=0$ with complex $\gamma$ (see Quasi-Static Approximation).

Consequences:

• Voltages and currents have an amplitude and a phase. The system behaves like a linear time-invariant (LTI) system at each frequency, while $\gamma \mapsto$ data stays nonlinear.
• Multi-frequency and frequency-difference EIT use the fact that $\sigma$ and $\epsilon$ of tissues depend on frequency in characteristic ways (tissue spectroscopy). This makes reconstructions possible without a reference measurement.
• For $\omega \epsilon \ll \sigma$ the imaginary part is negligible and one recovers the real Conductivity Equation.

References

1. M. Cheney, D. Isaacson, J. C. Newell (1999). Electrical Impedance Tomography. SIAM Review 41(1), 85–101. doi:10.1137/S0036144598333613
2. A. Adler, A. Boyle (2017). Electrical Impedance Tomography: Tissue Properties to Image Measures. IEEE Trans. Biomed. Eng. 64(11), 2494–2504. doi:10.1109/TBME.2017.2728323