A Regularized Electric Flux Volume Integral Equation for Brain Imaging

Abstract

Volume integral equations for modeling biological tissues in the frequency domain typically suffer from illconditioning for high dielectric contrasts and low frequencies. These conditioning breakdowns severely compromise the accuracy and applicability of these models and render them impractical despite their numerous advantages. In this work, we present an electric flux volume integral equation (D-VIE) free from these shortcomings when computed on biologically compatible simply connected objects. This new formulation leverages on careful spectral analysis to obtain volume quasi-Helmholtz projectors capable of curing both sources of ill-conditioning. In particular, the normalization of the projectors by the material permittivity allows for an inhomogeneous re-scaling of the equation which stabilises the high contrast breakdown together with the lowfrequency breakdown. Numerical results show the applicability of this new formulation in realistic brain imaging.

On a Low-Frequency and Contrast Stabilized Full-Wave Volume Integral Equation Solver for Lossy Media

“…The reader should however note that, although in our numerical validations we focus on biomedical applications and brain modeling in particular, the solver we propose is a completely general purpose one and several other application scenarios such as the modeling of lossy interconnects in printed circuit boards [30] could benefit from its use. Very preliminary results of this work have been presented in the conference contribution [37]. This paper is organized as follows: in Section II we set the background and notation, including the definition of the D-VIE along with its discrete Helmholtz decomposition.…”