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

Abstract

Magnetic resonance current density imaging (MRCDI) provides a current density image by measuring the induced magnetic flux density within the subject with a magnetic 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 Ω using measured Bz, one component of the magnetic flux density B. In this paper, we analyze the map from current density vector field J to one component of magnetic flux density Bz without any assumption on the conductivity. The map provides an orthogonal decomposition J = JP + JN of the current J where JN belongs to the null space of the map . We explicitly describe the projected current density JP from measured Bz. Based on the decomposition, we prove that Bz 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 JP provides a good approximation of the true current J without accumulating noise effects. Numerical simulations show that JP from measured Bz is quite similar to the target J. Biological tissue phantom experiments compare JP with the reconstructed J via the reconstructed isotropic conductivity using the harmonic Bz algorithm.