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

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.© H. Ammari et al. / J. Math. Pures Appl. 103 (2015) 1390-1409 1391 de l'onde ultrasonore et dépend non linéairement de la conductivité électrique. Le problème est de reconstruire cette conductivité à partir des mesures de courant. En premier lieu, on utilise une fonction test (potentiel virtuel) pour quantifier le lien entre le signal et la conductivité. Ensuite, à l'aide d'une déconvolution et d'un filtrage, il est possible de ramener le problème à la reconstruction d'une carte de conductivité à partir de la donnée d'un courant électrique interne sur l'ensemble du domaine. On donne d'abord une méthode d'optimisation pour résoudre ce problème. Une seconde méthode de reconstruction directe, utilisant une méthode de viscosité et la résolution d'une équation de transport à coefficients discontinus, est ensuite proposée. On démontre que la résolution de ce problème donne une reconstruction exacte de la conductivité lorsque le paramètre de régularisation tend vers zéro. On illustre les deux méthodes numériquement et on compare leur performances (résolution et stabilité en présence de bruit de mesure).

Rotational magneto-acousto-electric tomography (MAET): theory and experimental validation

“…As in the latter work, only one pair of wide flat electrodes was considered in Ammari et al (2015), i.e. smaller amount of measured information is assumed, comparing to Kunyansky (2012).…”

“…However, similarly to Kunyansky (2012), the assumption was made that the transducer could generate all frequencies and illuminate the object from all the directions within the plane where the object was supported. (Since the experimental setup of Grasland-Mongrain et al (2013) does not deliver multi-directional ultrasound excitation, methods of Ammari et al (2015) cannot be combined directly with the data of Grasland-Mongrain et al (2013).) Since in Ammari et al (2015) only one pair of electrodes was assumed, an explicit reconstruction technique (similar to Kunyansky (2012)) could not be applied, and a more sophisticated minimization procedure was developed.…”

“…(Since the experimental setup of Grasland-Mongrain et al (2013) does not deliver multi-directional ultrasound excitation, methods of Ammari et al (2015) cannot be combined directly with the data of Grasland-Mongrain et al (2013).) Since in Ammari et al (2015) only one pair of electrodes was assumed, an explicit reconstruction technique (similar to Kunyansky (2012)) could not be applied, and a more sophisticated minimization procedure was developed. It has been shown theoretically that this procedure converges to the sought conductivity; the corresponding algorithm was validated in numerical simulations.…”

A Phase-Locking Analysis of Neuronal Firing Rhythms with Transcranial Magneto-Acoustical Stimulation Based on the Hodgkin-Huxley Neuron Model

“…The motion of ions in the presence of a static magnetic field will generate a Lorentz force on the ions. The Lorentz force gives rise to an electric current I ext that oscillates at the ultrasonic frequency including fundamental frequency and modulation frequency (Montalibet et al, 2001; Grasland-Mongrain et al, 2013; Ammari et al, 2015). …”

A mathematical and numerical framework for magnetoacoustic tomography with magnetic induction

“…In order to solve numerically (4.19), we use a method of vanishing viscosity [16]. The field A 1 is known and we can solve uniquely the following problem: Proof.…”