Nonlinear Inverse Problems in Imaging

Chapter 9

Magnetic Resonance Electrical Impedance Tomography

For high-resolution static imaging of a conductivity distribution inside the human body, there have been strong needs for supplementary data to make the inverse problem well-posed and to overcome the fundamental limitations of the electrical impedance tomography (EIT) imaging methods. To bypass the ill-posed nature of EIT, magnetic resonance electrical impedance tomography (MREIT) was proposed in the early 1990s to take advantage of an MRI scanner as a tool to capture internal magnetic flux density data induced by externally injected currents (Birgul and Ider 1995; Birgul and Ider 1996; Woo et al. 1994; Zhang 1992).

MREIT aims to visualize conductivity images of an electrically conducting object using the current injection MRI technique (Joy et al. 1989; Scott et al. 1991, 1992). To probe the passive material property of the conductivity, low-frequency electrical current is injected into the imaging object through surface electrodes. This induces internal distributions of voltage u, current density J = (J_x, J_y, J_z) and magnetic flux density B = (B_x, B_y, B_z) dictated by Maxwell's equations. At a low frequency of less than a few kilohertz, we can ignore the effects of permittivity and consider only conductivity. For conductivity image reconstructions, MREIT relies on a set of internal magnetic flux density data, since it is dictated by the conductivity distribution σ according to Ampère's law

where μ₀ is the magnetic permeability ...