On the Convergence of the Harmonic $B_z$ Algorithm in Magnetic Resonance Electrical Impedance Tomography

Abstract

Magnetic resonance electrical impedance tomography (MREIT) is a new medical imaging technique that aims to provide electrical conductivity images with sufficiently high spatial resolution and accuracy. A new MREIT image reconstruction method called the harmonic $B_z$ algorithm was proposed in 2002, and it is based on the measurement of $B_z$ that is a single component of an induced magnetic flux density $\mathbf{B}=(B_x,B_y,B_z)$ subject to an injection current. Since then, MREIT imaging techniques have made significant progress, and recent published numerical simulations and phantom experiments show that we can produce high-quality conductivity images when the conductivity contrast is not very high. Though numerical simulations can explain why we could successfully distinguish different tissues with small conductivity differences, a rigorous mathematical analysis is required to better understand the underlying physical and mathematical principle. The purpose of this paper is to provide such a mathematical analysis of those numerical simulations and experimental results. By using a uniform a priori estimate for the solution of the elliptic equation in the divergent form and an induction argument, we show that, for a relatively small contrast of the target conductivity, the iterative harmonic $B_z$ algorithm with a good initial guess is stable and exponentially convergent in the continuous norm. Both two- and three-dimensional versions of the algorithm are considered, and the difference in the convergence property of these two cases is analyzed. Some numerical results are also given to show the expected exponential convergence behavior.