Electrical Impedance Tomography

Electrical Impedance Tomography (EIT) is an imaging method that recovers the electrical conductivity $\gamma$ inside a body $\Omega$ from measurements made only on its boundary $\partial \Omega$.

Electrodes are attached around the boundary. Known currents are driven through some electrodes and the resulting voltages are recorded on all of them. Repeating this for many current patterns gives a set of boundary voltage–current pairs. The aim is to infer the interior conductivity that is consistent with all of them.

Mathematically the problem splits into two parts:

• the forward problem: given $\gamma$, predict the boundary data. It is governed by the Conductivity Equation and encoded in the Dirichlet-to-Neumann Map;
• the inverse problem, the Calderón Problem: given the boundary data, recover $\gamma$. It is severely ill-posed (see Stability of the Calderón Problem), so practical reconstructions need regularization.

Real devices only see finitely many electrodes and have contact impedances, which the Complete Electrode Model accounts for. Many analyses and simulations instead use the idealised continuum model, where every point of $\partial \Omega$ is available for injecting current and measuring voltage (see Electrode Models).

Compared with other Tomographic Imaging Modalities, EIT is cheap, fast, portable and uses no ionising radiation, but its spatial resolution is low. See Applications of EIT.

References

1. M. Cheney, D. Isaacson, J. C. Newell (1999). Electrical Impedance Tomography. SIAM Review 41(1), 85–101. doi:10.1137/S0036144598333613
2. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201
3. D. S. Holder (ed.) (2004). Electrical Impedance Tomography: Methods, History and Applications. CRC Press. doi:10.1201/9781420034462
4. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344