A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography☆

We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic eld induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we rst make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener lter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We rst introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial dierential equation is then proposed based on viscositytype regularization to a transport equation satised by the current density eld. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance.