Complete Electrode Model

The Complete Electrode Model (CEM) is the standard forward model for real EIT measurements (see Electrode Models). With $L$ electrodes $e_{\ell }\subset \partial \Omega$, contact impedances $z_{\ell }>0$, injected currents $I\in {\mathbb{R}}^L$ and unknown electrode voltages $U\in {\mathbb{R}}^L$, find $(u,U)$ with

$$\begin{array}{rlrl}\nabla \cdot (\gamma \nabla u)& =0& & \text{in }\Omega ,\\ u+z_{\ell }\gamma {\partial }_{\nu }u& =U_{\ell }& & \text{on }{e}_{\ell },\ell =1,\dots ,L,\\ {\int }_{e_{\ell }}\gamma {\partial }_{\nu }u\mathrm{d}s& =I_{\ell }& & \ell =1,\dots ,L,\\ \gamma {\partial }_{\nu }u& =0& & \text{on }\partial \Omega \setminus {\bigcup }_{\ell }{e}_{\ell }.\end{array}$$

Well-posedness. Under charge conservation ${\sum }_{\ell }{I}_{\ell }=0$ and a grounding condition ${\sum }_{\ell }{U}_{\ell }=0$, the problem has a unique solution $(u,U)\in H^1(\Omega )\oplus {\mathbb{R}}^L$. The weak form uses the coercive bilinear form

$$B((u,U),(v,V))={\int }_{\Omega }\gamma \nabla u\cdot \nabla v\mathrm{d}x+\sum _{\ell =1}^L\frac{1}{z_{\ell }}{\int }_{e_{\ell }}(u-U_{\ell })(v-V_{\ell })\mathrm{d}s=\sum _{\ell }{I}_{\ell }{V}_{\ell }.$$

The resulting electrode NtD matrix $I\mapsto U$ is symmetric positive definite on $\{\sum I_{\ell }=0\}$. It is the finite-dimensional analogue of the Neumann-to-Dirichlet Map. As electrodes become small and numerous, the CEM approaches the continuum model. As $z_{\ell }\to 0$ it becomes the Shunt Model; for large $z_{\ell }$ the current density under the electrode becomes uniform, as in the Gap Model.

Integrating the second condition over $e_{\ell }$ gives $U_{\ell }=|e_{\ell }{|}^{-1}{\int }_{e_{\ell }}u\mathrm{d}s+z_{\ell }{I}_{\ell }/|e_{\ell }|$: the voltage of a current-carrying electrode includes the contact voltage drop (see Measurement Protocols). The finite element system is derived in Discrete Electrode Models.

In ModularEIT.jl: CompleteElectrodeModel.

References

1. 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
2. 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
3. N. Hyvönen (2004). Complete Electrode Model of Electrical Impedance Tomography: Approximation Properties and Characterization of Inclusions. SIAM J. Appl. Math. 64(3), 902–931. doi:10.1137/S0036139903423303