Discrete Electrode Models

After a Galerkin discretisation with basis functions ${\phi }_i$, every electrode model becomes a linear system with an injection matrix $P$ and a measurement matrix $Q$:

$$A(\sigma )\mathbf{x}=PI,V=Q\mathbf{x},$$

where $I$ collects the injected currents and $V$ the measured voltages. For all models except the CEM, $A(\sigma )=L_{\sigma }$ is the Weighted Stiffness Matrix and $\mathbf{x}=\mathbf{u}$ the nodal potential.

model	$P$ (column $\ell$)	$Q$ (row $m$)
continuum	${\int }_{\partial \Omega }{\phi }_i{\phi }_j\mathrm{d}s$ restricted to boundary nodes ($I$ = density coefficients)	unit row of boundary node $m$
point	unit vector of node $x_{\ell }$	unit row of node $x_m$
gap	${\int }_{e_{\ell }}{\phi }_i\mathrm{d}s/|e_{\ell }|$	${\int }_{e_m}{\phi }_j\mathrm{d}s/|e_m|$
CEM	unit vector of the unknown $U_{\ell }$	unit row of $U_m$

Injection and measurement sites are independent: $P$ has one column per drive electrode, $Q$ one row per measurement electrode (see Measurement Protocols). For the gap model with the same electrodes, $Q=P^{\top }$, which is discrete reciprocity.

Complete electrode model

The CEM adds the electrode voltages $U\in {\mathbb{R}}^L$ as unknowns. With the electrode mass matrices $(M_{\ell }{)}_{ij}={\int }_{e_{\ell }}{\phi }_i{\phi }_j\mathrm{d}s$ and vectors $(d_{\ell }{)}_i={\int }_{e_{\ell }}{\phi }_i\mathrm{d}s$, the weak form of the Complete Electrode Model gives

$$\left[\begin{array}{cc}{L}_{\sigma }+{\sum }_{\ell }{z}_{\ell }^{-1}{M}_{\ell }& -D\\ -D^{\top }& \mathrm{diag}(|e_{\ell }|/z_{\ell })\end{array}\right]\left[\begin{array}{c}\mathbf{u}\\ U\end{array}\right]=\left[\begin{array}{c}0\\ I\end{array}\right],D=[d_1/z_1\cdots d_L/z_L].$$

The matrix is symmetric positive semidefinite. Its null space is spanned by the constant vector on $(\mathbf{u},U)$ jointly, and the grounding ${\sum }_{\ell }{U}_{\ell }=0$ fixes the constant. Only the upper-left block depends on $\sigma$. The contact terms form a constant matrix $A_0$, so $A(\sigma )=A_0+L_{\sigma }$.

As $z_{\ell }\to \infty$, the current density under each electrode becomes uniform, and $U_{\ell }$ approaches the gap-model voltage plus $z_{\ell }{I}_{\ell }/|e_{\ell }|$.

Voltage-driven (Dirichlet) problems

Every model also has a voltage-driven counterpart. Prescribe the unknowns on a set $B$ of Dirichlet degrees of freedom through an expansion matrix $E$, ${\mathbf{x}}_B=EU$, and solve for the free degrees of freedom $F$:

$$A_{FF}{\mathbf{x}}_F=-A_{FB}EU,I=C^{-1}{E}^{\top }(A\mathbf{x}{)}_B,C=E^{\top }{P}_B.$$

• continuum: $B$ = boundary nodes, $E=\mathrm{Id}$, $C$ = boundary mass matrix (the currents come out as densities);
• point: $B$ = electrode nodes, $E=\mathrm{Id}$, $C=\mathrm{Id}$;
• gap: $B$ = all nodes of the drive electrodes, $E$ = electrode indicator. This is the Shunt Model, $C=\mathrm{Id}$;
• CEM: $B$ = the voltages $U$, $E=\mathrm{Id}$, $C=\mathrm{Id}$.

$(A\mathbf{x}{)}_B$ is the residual of the discrete equation on the Dirichlet nodes, the variationally consistent discrete normal current. The matrix $I\mapsto U$ inverts $U\mapsto I$ whenever both problems describe the same physics (continuum, point, CEM). The map $U\mapsto I$ is the Schur complement $A_{BB}-A_{BF}{A}_{FF}^{-1}{A}_{FB}$, a discrete Dirichlet-to-Neumann Map (see Discrete Boundary Operator).

In ModularEIT.jl: ForwardModel, ContinuumModel, PointElectrodeModel, GapModel, 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. P. J. Vauhkonen, M. Vauhkonen, T. Savolainen, J. P. Kaipio (1999). Three-dimensional electrical impedance tomography based on the complete electrode model. IEEE Trans. Biomed. Eng. 46(9), 1150–1160. doi:10.1109/10.784147
3. J. P. Kaipio, V. Kolehmainen, E. Somersalo, M. Vauhkonen (2000). Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography. Inverse Problems 16(5), 1487–1522. doi:10.1088/0266-5611/16/5/321
4. N. Polydorides, W. R. B. Lionheart (2002). A Matlab toolkit for three-dimensional electrical impedance tomography: a contribution to the Electrical Impedance and Diffuse Optical Reconstruction Software project. Meas. Sci. Technol. 13(12), 1871–1883. doi:10.1088/0957-0233/13/12/310