Fast calculation of the sensitivity matrix in magnetic induction tomography by tetrahedral edge finite elements and the reciprocity theorem

Abstract

Magnetic induction tomography of biological tissue is used to reconstruct the changes in the complex conductivity distribution by measuring the perturbation of an alternating primary magnetic field. To facilitate the sensitivity analysis and the solution of the inverse problem a fast calculation of the sensitivity matrix, i.e. the Jacobian matrix, which maps the changes of the conductivity distribution onto the changes of the voltage induced in a receiver coil, is needed. The use of finite differences to determine the entries of the sensitivity matrix does not represent a feasible solution because of the high computational costs of the basic eddy current problem. Therefore, the reciprocity theorem was exploited. The basic eddy current problem was simulated by the finite element method using symmetric tetrahedral edge elements of second order. To test the method various simulations were carried out and discussed.

Electromagnetic induction tomography

“…When a particular coil is used as a transmitter, it is fed by currents of certain values to produce alternating magnetic fields; other coils are used to measure the secondary field due to the velocity-induced magnetic field and the magnetic field produced by the eddy currents in the objects. Several authors (Norton & Bowler, 1993;Dyck, et al, 1994;Hollaus, Magele, Merwa, & Scharfetter, 2004b;Yin & Peyton, 2010b) have reported derivations for the sensitivity matrix for a range of electromagnetic problems. These typically follow the approach of defining an adjoint problem to which the reciprocity theorem is then applied.…”

Numerical modeling of magnetic induction tomography using the impedance method

“…This procedure requires a large computational effort since for each source position and for each perturbation voxel, the complete calculation in all sensor positions should be performed twice to determine the variation of the field. Hollaus et al [4] used the finite element method for calculating the fields and the sensitivity was obtained by applying the reciprocity theorem to calculate the mutual impedance variation between source and sensor. Thus, the assembly of the sensitivity matrix requires only two steps of field calculation for each sourcesensor pair, significantly reducing the amount of computation required.…”

A Large and Quick Induction Field Scanner for Examining the Interior of Extended Objects or Humans

“…5(f) shows the calculated scanner signals from a single, small, and conductive cube without a conducting background For this scanner with moving test objects, the sensitivity for a local deviation inside an extended 3D object apparently does not depend strongly on the deviation's position. This finding is helpful, as static sensitivity fields for opposing coils tend to show heterogeneous sensitivity and obtain relatively weaker signals from a deviation in the middle of a conducting body (e.g., as shown in [3,25], and [26] and as seen for the sheet Fig. 5(e)).…”