  In this paper we present a new regularized electric flux volume integral
equation (D-VIE) for modeling high-contrast conductive dielectric objects in a
broad frequency range. This new formulation is particularly suitable for
modeling biological tissues at low frequencies, as it is required by brain
epileptogenic area imaging, but also at higher ones, as it is required by
several applications including, but not limited to, transcranial magnetic and
deep brain stimulation (TMS and DBS, respectively). When modeling inhomogeneous
objects with high complex permittivities at low frequencies, the traditional
D-VIE is ill-conditioned and suffers from numerical instabilities that result
in slower convergence and in less accurate solutions. In this work we address
these shortcomings by leveraging a new set of volume quasi-Helmholtz
projectors. Their scaling by the material permittivity matrix allows for the
re-balancing of the equation when applied to inhomogeneous scatterers and
thereby makes the proposed method accurate and stable even for high complex
permittivity objects until arbitrarily low frequencies. Numerical results,
canonical and realistic, corroborate the theory and confirm the stability and
the accuracy of this new method both in the quasi-static regime and at higher
frequencies.
