Analysis of recoverable current from one component of magnetic flux density in MREIT and MRCDI

Magnetic resonance current density imaging (MRCDI) provides a current density image by measuring the induced magnetic flux density within the subject with amagnetic resonance imaging (MRI) scanner. Magnetic resonance electrical impedance tomography (MREIT) has been focused on extracting some useful information of the current density and conductivity distribution in the subject Omega using measured B-z, one component of the magnetic flux density B. In this paper, we analyze the map T from current density vector field J to one component of magnetic flux density B-z without any assumption on the conductivity. The map T provides an orthogonal decomposition J = J(P) + J(N) of the current J where J(N) belongs to the null space of the map T. We explicitly describe the projected current density J(P) from measured B-z. Based on the decomposition, we prove that B-z data due to one injection current guarantee a unique determination of the isotropic conductivity under assumptions that the current is two-dimensional and the conductivity value on the surface is known. For a two-dimensional dominating current case, the projected current density J(P) provides a good approximation of the true current J without accumulating noise effects. Numerical simulations show that J(P) from measured B-z is quite similar to the target J. Biological tissue phantom experiments compare J(P) with the reconstructed J via the reconstructed isotropic conductivity using the harmonic B-z algorithm.