Magnetic Resonance Electrical Impedance Tomography (MREIT)

Magnetic resonance electrical impedance tomography (MREIT) is a recently developed medical imaging modality visualizing conductivity images of an electrically conducting object. MREIT was motivated by the well-known ill-posedness of the image reconstruction problem of electrical impedance tomography (EIT). Numerous experiences have shown that practically measurable data sets in an EIT system are insufficient for a robust reconstruction of a high-resolution static conductivity image due to its ill-posed nature and the influences of errors in forward modeling. To overcome the inherent ill-posed characteristics of EIT, the MREIT system was proposed in the early 1990s to use the internal data of magnetic flux density ${\bf B}=(B_x,B_y,B_z)$, which is induced by an externally injected current. MREIT uses an MRI scanner as a tool to measure the $z$-component $B_z$ of the magnetic flux density, where $z$ is the axial magnetization direction of the MRI scanner. In 2001, a constructive $B_z$-based MREIT algorithm called the harmonic $B_z$ algorithm was developed and its numerical simulations showed that high-resolution conductivity image reconstructions are possible. This novel algorithm is based on the key observation that the Laplacian $\Delta B_z$ probes changes in the log of the conductivity distribution along any equipotential curve having its tangent to the vector field ${\bf J}\times (0,0,1)$, where ${\bf J}=(J_x,J_y,J_z)$ is the induced current density vector. Since then, imaging techniques in MREIT have advanced rapidly and have now reached the stage of in vivo animal and human experiments. This paper reviews MREIT from its mathematical framework to the most recent human experiment outcomes.