Electrode Models

The mathematical boundary conditions have to model how current actually enters the body through $L$ metal electrodes $e_1,\dots ,e_L\subset \partial \Omega$. The standard models, in increasing realism:

1. Continuum model. Current density and voltage are known at every boundary point. This idealisation underlies the Dirichlet-to-Neumann Map and the Calderón Problem. It is convenient for theory and for simulations in which every boundary node of the mesh acts as an electrode. Measuring on all boundary nodes is equivalent to choosing a basis of boundary functions, such as Fourier modes.
2. Point model. Each electrode is a single boundary point with a point current source. Justified for small electrodes; voltages can only be measured at points that carry no current.
3. Gap model. Current density $I_{\ell }/|e_{\ell }|$ is constant on each electrode and zero in the gaps. It ignores that the electrode, being a good conductor, forces the voltage (not the current) to be constant, and it overestimates resistivity.
4. Shunt model. The electrode is a perfect conductor: $u=U_{\ell }$ on $e_{\ell }$ and ${\int }_{e_{\ell }}\gamma {\partial }_{\nu }u=I_{\ell }$. It ignores the thin, highly resistive layer at the electrode–skin contact.
5. Complete Electrode Model (CEM). The shunt model plus a contact impedance $z_{\ell }$. It predicts measured voltages to within measurement precision and is the standard for real data.

Cheng et al. (1989) showed experimentally that only the CEM matches measured data to instrument precision.

Injection versus measurement. Current-carrying electrodes and voltage-measuring electrodes need not coincide. On an electrode that carries current, the measured voltage contains the contact voltage drop $z_{\ell }{I}_{\ell }/|e_{\ell }|$, which only the CEM describes. On electrodes without current, all models agree closely. Which model is adequate therefore depends on the measurement protocol. The finite element versions of all models are collected in Discrete Electrode Models.

In ModularEIT.jl: ContinuumModel, PointElectrodeModel, GapModel, CompleteElectrodeModel.

References

1. K.-S. Cheng, D. Isaacson, J. C. Newell, D. G. Gisser (1989). Electrode models for electric current computed tomography. IEEE Trans. Biomed. Eng. 36(9), 918–924. doi:10.1109/10.35300
2. E. Somersalo, M. Cheney, D. Isaacson (1992). Existence and Uniqueness for Electrode Models for Electric Current Computed Tomography. SIAM J. Appl. Math. 52(4), 1023–1040. doi:10.1137/0152060