A Regularized Electric Flux Volume Integral Equation for Brain Imaging

Volume integral equations for modeling biological tissues in the frequency domain typically suffer from ill-conditioning 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 low-frequency breakdown. Numerical results show the applicability of this new formulation in realistic brain imaging.