Complete Electrode Model (CEM) Overview

• The Complete Electrode Model (CEM) is a rigorous framework that defines current-voltage relationships in EIT by accounting for finite electrode sizes, contact impedances, and precise boundary current distributions.
• It employs a robust variational formulation integrating elliptic PDEs and mixed Robin conditions, enabling accurate forward simulations and optimal numerical approximations.
• CEM underpins inverse problem solutions and regularization strategies in biomedical imaging, crucial for optimizing electrode placement and reconstructing conductivity distributions.

The complete electrode model (CEM) is the mathematically rigorous framework for simulating and interpreting current-to-voltage measurements in electrical impedance tomography (EIT) and related imaging modalities that utilize boundary electrodes. By explicitly incorporating finite-sized electrodes, contact impedances, and the precise distribution of normal current density at the electrode-object interface, the CEM enables accurate forward and inverse modeling essential for robust reconstructions, parameter estimation, and experimental optimization.

1. Governing Equations and Boundary Conditions

The CEM describes a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ (with $n=2,3$), equipped with $M$ disjoint electrodes $E_1,\dots,E_M \subset \partial\Omega$, each characterized by a contact impedance $z_\ell > 0$. The potential $u:\Omega\to\mathbb{R}$ and constant per-electrode potentials $U=(U_1,\dots,U_M)\in\mathbb{R}^M$ are governed by:

• Elliptic PDE in the interior:

$$\operatorname{div}(\sigma \nabla u) = 0 \quad \text{in } \Omega$$

where $\sigma(x)$ is a symmetric uniformly elliptic (possibly tensor-valued) conductivity.

• Insulated (Neumann) boundary on non-electrode portions:

$$\sigma\nabla u \cdot \nu = 0 \quad \text{on } \partial\Omega \setminus \bigcup_\ell E_\ell$$

• Mixed (Robin/impedance) boundary condition on each electrode:

$$u + z_\ell\, (\sigma\nabla u \cdot \nu) = U_\ell \quad \text{on } E_\ell$$

• Integral current constraint on each electrode:

$$\int_{E_\ell} \sigma\nabla u \cdot \nu \, dS = I_\ell$$

• Net current conservation: $\sum_{\ell=1}^M I_\ell = 0$
• Grounding condition for uniqueness: $\sum_{\ell=1}^M U_\ell = 0$

These strong-form equations fully describe the physical behavior of electrode measurements under finite contact impedance and electrode area (Staboulis et al., 2013, Agsten et al., 2016, Garde et al., 2020, Tyni et al., 2023).

2. Variational Formulation and Function Spaces

The CEM admits a robust variational framework essential for numerical approximation and regularity analysis. Define the product space

$$\mathcal{H}^1 = H^1(\Omega) \oplus \mathbb{R}_0^M; \quad \mathbb{R}_0^M = \left\{ V \in \mathbb{R}^M:\, \sum_{\ell=1}^M V_\ell = 0 \right\}.$$

Define the bilinear form

$$B((u,U),(v,V)) = \int_\Omega \nabla u \cdot \sigma \nabla v \, dx + \sum_{\ell=1}^M \frac{1}{z_\ell} \int_{E_\ell} (u-U_\ell)(v-V_\ell)\, dS,$$

and linear functional $\varphi_I((v,V)) = \sum_\ell I_\ell V_\ell$.

The weak CEM problem: Find $(u,U)\in \mathcal{H}^1$ s.t.

$$B((u,U),(v,V)) = \varphi_I((v,V)) \qquad \forall (v,V) \in \mathcal{H}^1.$$

This formulation yields unique solvability and supports both finite element and boundary integral methods (Staboulis et al., 2013, Tyni et al., 2023). The structure is also amenable to Fréchet differentiation with respect to $\sigma$, $z$, electrode geometry, and boundary currents (Hyvönen et al., 2014, Hyvönen et al., 2016).

3. Regularity, Approximation, and Numerical Discretization

The regularity of the forward potential under CEM boundary data is a key determinant of approximation quality:

• For nondegenerate $z_\ell$, $u\in H^{3/2+s}(\Omega)$ for any $s<1/2$; in the limit $z_{\min}\to 0$ (perfect contacts), regularity drops to $H^{1+s}(\Omega)$ (shunt model) (Staboulis et al., 2013).
• The discontinuous Robin boundary introduces edge singularities, limiting $u$ to $H^{2-\varepsilon}$ (Hyvönen et al., 2017).
• Adopting smoothened admittance profiles (smoothened CEM) increases regularity to $H^{k+2}(\Omega)$ for $g\in C^{k+1}(\partial\Omega)$ (Hyvönen et al., 2017, Li et al., 2019).

Finite element convergence rates are dictated by this regularity; classical CEM yields suboptimal $O(h^{1/2+s})$ in $H^1$-norm, while smoothened models achieve optimal rates $O(h^{k+1})$ for piecewise polynomials of degree $k$ (Staboulis et al., 2013, Hyvönen et al., 2017). Immersed boundary and boundary integral equation solvers are also developed for efficient, geometry-agnostic implementation (Dardé et al., 2023, Tyni et al., 2023).

4. Electrode Model Variants and Approximation Theory

Alternative boundary models are widely used for efficiency or application-specific considerations:

• Gap Model (GAP): Imposes uniform current density across each electrode; retains the mean voltage-drop relation $$U_\ell = (1/|E_\ell|) \int_{E_\ell} u\, dS + z_\ell I_\ell$$, omitting spatial shunting effects (Agsten et al., 2016).
• Point Electrode Model (PEM): Treats electrodes as points; $U_\ell = u(y_\ell) + z_\ell I_\ell$ at site $y_\ell$ (Garde et al., 2020).
• Shunt Model: Zero impedance limit ($z_\ell\to 0$) reducing the Robin condition to Dirichlet on electrodes (Staboulis et al., 2013).

Approximation errors between CEM, PEM, and continuum models have been quantitatively analyzed, with CEM–PEM discrepancies scaling as $O(M d^2)$ in the number and width $d$ of electrodes (Garde et al., 2020). In two dimensions, conformal mapping invariance results allow compensation for boundary mis-modeling via electrode adjustment within CEM (Hyvönen et al., 2016).

5. Inverse Problems, Regularization, and Sensitivity Analysis

The CEM forms the foundation for conductivity reconstruction in EIT and hybrid imaging. Inverse approaches leverage model-based Fréchet derivatives:

• Fréchet differentiability of the CEM forward map with respect to conductivity, electrode positions, contact impedance, and source currents underpins gradient or Hessian-based reconstructions and experimental design (Hyvönen et al., 2014, Hyvönen et al., 2016).
• Bayesian experiment design utilizing CEM linearizations enables optimization of electrode placement via A- or D-optimality, using analytic shape derivatives of the measurement map (Hyvönen et al., 2014).
• Regularization strategies—including variational penalties (e.g., Mumford–Shah/Ambrosio–Tortorelli) and smoothened CEMs—address severe ill-posedness, edge preservation, and phase retrieval, with theoretical guarantees established via $\Gamma$-convergence (Jauhiainen et al., 2021, Nachman et al., 2015).

6. Applications in Biomedical Imaging and Neurotechnology

CEM fidelity is essential in applications where accurate modeling of electrode geometry and interface physics is required:

• In transcranial electrical stimulation (tES), temporal interference stimulation (tTIS), and combined tES/EEG, the CEM accurately predicts shunting effects, focality, and skin heating, and is necessary for modeling dense/high-definition arrays or impedance-mismatch scenarios (Agsten et al., 2016, Söderholm et al., 23 Jun 2025).
• In multi-compartment FEM studies for EEG and stereo-EEG source imaging, CEM enables precise lead-field computation, resolving both scalp and invasive contact configurations, and enhances the accuracy and specificity of source localization, particularly in deep brain regions (Prieto et al., 31 Jan 2026).
• Robustness to electrode misplacement and geometric modeling errors is facilitated by including electrode movement as reconstruction variables, justified theoretically in 2D via conformal equivalence of CEM maps (Hyvönen et al., 2016).

7. Impact and Limitations

The CEM is the reference standard in impedance-based imaging, providing maximal physical fidelity for practical and experimental setups. Its adoption is critical when modeling interfacial physics (impedance, shunting, heating), nonideal electrode geometries, or when seeking reliable uncertainty quantification and optimal design. For applications requiring only brain-internal potential distributions, reduced models (GAP/PEM) often suffice and may offer substantial computational gains (Agsten et al., 2016).

Nevertheless, CEM adoption is computationally more demanding due to the coupling between interior fields and electrode unknowns, the lower regularity from boundary singularities, and the need for explicit electrode geometry tracking. Recent work on smoothened boundary profiles, immersed/meshless methods, and efficient evaluation of Fréchet derivatives has addressed several of these constraints, opening possibilities for high-resolution, fully coupled reconstructions across diverse domains (Hyvönen et al., 2017, Dardé et al., 2023, Hyvönen et al., 2014).