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We provide a mathematical analysis and a numerical framework formagnetoacoustic tomography with magnetic induction. The imaging problem is toreconstruct the conductivity distribution of biological tissue frommeasurements of the Lorentz force induced tissue vibration. We begin withreconstructing from the acoustic measurements the divergence of the Lorentzforce, which is acting as the source term in the acoustic wave equation. Thenwe recover the electric current density from the divergence of the Lorentzforce. To solve the nonlinear inverse conductivity problem, we introduce anoptimal control method for reconstructing the conductivity from the electriccurrent density. We prove its convergence and stability. We also present apoint fixed approach and prove its convergence to the true solution. A newdirect reconstruction scheme involving a partial differential equation is thenproposed based on viscosity-type regularization to a transport equationsatisfied by the electric current density field. We prove that solving such anequation yields the true conductivity distribution as the regularizationparameter approaches zero. Finally, we test the three schemes numerically inthe presence of measurement noise, quantify their stability and resolution, andcompare their performance.

A mathematical and numerical framework for magnetoacoustic tomography with magnetic induction