We provide a mathematical analysis and a numerical framework for Lorentzforce electrical conductivity imaging. Ultrasonic vibration of a tissue in thepresence of a static magnetic field induces an electrical current by theLorentz force. This current can be detected by electrodes placed around thetissue; it is proportional to the velocity of the ultrasonic pulse, but dependsnonlinearly on the conductivity distribution. The imaging problem is toreconstruct the conductivity distribution from measurements of the inducedcurrent. To solve this nonlinear inverse problem, we first make use of avirtual potential to relate explicitly the current measurements to theconductivity distribution and the velocity of the ultrasonic pulse. Then, byapplying a Wiener filter to the measured data, we reduce the problem to imagingthe conductivity from an internal electric current density. We first introducean optimal control method for solving such a problem. A new directreconstruction scheme involving a partial differential equation is thenproposed based on viscosity-type regularization to a transport equationsatisfied by the current density field. We prove that solving such an equationyields the true conductivity distribution as the regularization parameterapproaches zero. We also test both schemes numerically in the presence ofmeasurement noise, quantify their stability and resolution, and compare theirperformance.
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