A new imaging modality is introduced to image electrical conductivity of biological tissues via contactless measurements. This modality uses magnetic excitation to induce currents inside the body and measures the magnetic fields of the induced currents. In this study, the mathematical basis of the methodology is analyzed and numerical models are developed to simulate the imaging system. The induced currents are expressed using the A-phi formulation of the electric field where A is the magnetic vector potential and phi is the scalar potential function. It is assumed that A describes the primary magnetic vector potential that exists in the absence of the body. This assumption considerably simplifies the solution of the secondary magnetic fields caused by induced currents. In order to solve phi for objects of arbitrary conductivity distribution a three-dimensional (3-D) finite-element method (FEM) formulation is employed. A specific 7 x 7-coil system is assumed nearby the upper surface of a 10 x 10 x 5-cm conductive body. A sensitivity matrix, which relates the perturbation in measurements to the conductivity perturbations, is calculated. Singular-value decomposition of the sensitivity matrix shows various characteristics of the imaging system. Images are reconstructed using 500 voxels in the image domain, with truncated pseudoinverse. The noise level is assumed to produce a representative signal-to-noise ratio (SNR) of 80 dB. It is observed that it is possible to identify voxel perturbations (of volume 1 cm3) at 2 cm depth. However, resolution gradually decreases for deeper conductivity perturbations.

In three dimensions, there exist modifications of Einstein's gravity akin to the topologically massive gravity that describe massive gravitons about maximally symmetric backgrounds. These theories are built on the three-dimensional version of the Bach tensor (a curl of the Cotton-York tensor) and its higher derivative generalizations; and they are on-shell consistent without a Lagrangian description based on the metric tensor alone. We give a generic construction of these models, find the spectra and compute the conserved quantities for the Banados-Teitelboim-Zanelli black hole.

The forward problem of a new medical imaging system is analysed in this study. This system uses magnetic excitation to induce currents inside a conductive body and measures the magnetic fields of the induced currents. The forward problem, that is determining induced currents in the conductive body and their magnetic fields, is formulated. For a general solution of the forward problem, the finite element method (FEM) is employed to evaluate the scalar potential distribution. Thus, inhomogeneity and anisotropy of conductivity is taken into account for the FEM solutions. An analytical solution for the scalar potential is derived for homogeneous conductive spherical objects in order to test FEM solutions. It is observed that the peak error in FEM solutions is less than 2%. The numerical system is used to reveal the characteristics of the measurement system via simulations. Currents are induced in a 9x9x5 cm body of conductivity 0.2 S m(-1) by circular coils driven sinusoidally. It is found that a 1 cm shift in the perturbation depth reduces the field magnitudes to approximately one-tenth. In addition, the distance between extrema increases. Further simulations carried out using different coil configurations revealed the performance of the method and provided a design perspective for a possible data acquisition system.

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