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

TL;DR

This paper develops a mathematical and numerical approach for reconstructing tissue conductivity using Lorentz force electrical impedance tomography, leveraging ultrasonic vibrations, magnetic fields, and advanced inverse problem techniques.

Contribution

It introduces a novel framework combining a virtual potential, Wiener filtering, and PDE-based regularization for nonlinear conductivity imaging from Lorentz force measurements.

Findings

The PDE-based reconstruction accurately recovers conductivity with low noise.

The Wiener filter improves measurement stability and resolution.

Numerical tests demonstrate the method's robustness and effectiveness.

Abstract

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 field 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 first 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 filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the current density field. 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.