A boundary integral equation method for the complete electrode model in electrical impedance tomography with tests on experimental data

Abstract

We develop a boundary integral equation-based numerical method to solve for the electrostatic potential in two dimensions, inside a medium with piecewise constant conductivity, where the boundary condition is given by the complete electrode model (CEM). The CEM is seen as the most accurate model of the physical setting where electrodes are placed on the surface of an electrically conductive body, and currents are injected through the electrodes and the resulting voltages are measured again on these same electrodes. The integral equation formulation is based on expressing the electrostatic potential as the solution to a finite number of Laplace equations which are coupled through boundary matching conditions. This allows us to re-express the solution in terms of single layer potentials; the problem is thus re-cast as a system of integral equations on a finite number of smooth curves. We discuss an adaptive method for the solution of the resulting system of mildly singular integral equations. This solver is both fast and accurate. We then present a numerical inverse solver for electrical impedance tomography (EIT) which uses our forward solver at its core. To demonstrate the applicability of our results we test our numerical methods on an open electrical impedance tomography data set provided by the Finnish Inverse Problems Society.

Figures (11)

• Figure 1: An example configuration of a domain $\Omega_0$ with three bodies $\Omega_1,\Omega_2$ and $\Omega_3$ with their associated conductivities $\sigma_0,\ldots,\sigma_3$, respectively. On the intersection $\Omega_2\cap\Omega_3$ we define the conductivity to be $\sigma_2+\sigma_3$. In this example there are four electrodes $e_i$, $i=1,2,3,4$, whose locations on $\partial\Omega_0$ are marked with gray color.
• Figure 2: Left: The adaptively selected quadrature grid. The quadrature nodes on interfaces are denoted by dots colored by the conductivity inside the interface on which they are located. The electrode locations are marked with solid red and green curves at the top and bottom of the outer interface. The red crosses mark the panel breaks. Note how our adaptive method refined locations where the interfaces are close to each other. Right: The computed solution $u$ of \ref{['eq:complete-electrode-model']}. We observe that the potential $u$ is approximately constant near the electrodes. Recall that on the overlap region of two bodies the conductivity is defined to be the sum of the conductivities of the bodies, which can be seen in the left figure on the overlap of the two bottom-left bodies. The top-right body has a lower conductivity than the two bodies on the bottom-left.
• Figure 3: Numerical validation of the BIEM by comparison of the potentials $u$ obtained using FEM and BIEM in a simple configuration. The $16$ electrode locations are marked with black lines: active electrodes are located at the right side of the domain. The largest relative difference $\vert u_\mathrm{BIEM}-u_\mathrm{FEM}\vert/\Vert u_\mathrm{FEM}\Vert_{L^\infty(\Omega_0)}$ between the two solutions is attained at the edges of the active electrodes.
• Figure 4: Left: the numerical solution $u$ to \ref{['eq:complete-electrode-model']} with $1158$ nodes in total. The point $(x_0,y_0)$ is marked by a red cross. Middle: the refinement study of the absolute differences $\vert u_{1158}(x_0,y_0) - u_{i}(x_0,y_0)\vert$ as a function of the total number of nodes at a point $(x_0,y_0)\in\Omega_0$. Right: the difference $|{\rm V}_{1158}-{\rm V}_{i}|$ between voltages as a function of the total number of nodes.
• Figure 5: The initial guess on the left; the found solution from raw data in the middle; the found solution from corrected data on the right. The score values of the cost functions ${\rm Cost}_{\rm raw}$ and ${\rm Cost}_{\rm corrected}$ at the found solutions are displayed above each photo. The visually estimated and the numerically found centers of masses for the bodies are marked by a blue cross and a circle, respectively.