Magnetic Resonance Electrical Impedance Tomography(MREIT) is a new medical imaging technique for the cross-sectional conductivity distribution of a human body using both EIT(Electrical Impedance Tomography) and MRI(Magnetic Resonance Imaging) system. MREIT system was designed to enhance EIT imaging system which has inherent low sensitivity of boundary measurements to any changes of internal tissue conductivity values. MREIT utilizes a recent CDI (Current Density Imaging) technique of measuring the internal current density by means of MRI technique. In this paper, a mathematical modeling for MREIT and image reconstruction method called the alternating J-substitution algorithm are presented. Computer simulations show that the alternating J-substitution algorithm provides accurate high-resolution conductivity images.