  In this article we propose and investigate a hierarchy of mathematical models
based on partial differential equations (PDE) and ordinary differential
equations (ODE) for the simulation of the biophysical phenomena occurring in
the electrolyte fluid that connects a biological component (a single cell or a
system of cells) and a solid-state device (a single silicon transistor or an
array of transistors). The three members of the hierarchy, ordered by
decreasing complexity, are: (i) a 3D Poisson-Nernst-Planck (PNP) PDE system for
ion concentrations and electric potential; (ii) a 2D reduced PNP system for the
same dependent variables as in (i); (iii) a 2D area-contact PDE system for
electric potential coupled with a system of ODEs for ion concentrations. The
backward Euler method is adopted for temporal semi-discretization and a
fixed-point iteration based on Gummel's map is used to decouple system
equations. Spatial discretization is performed using piecewise linear
triangular finite elements stabilized via edge-based exponential fitting.
Extensively conducted simulation results are in excellent agreement with
existing analytical solutions of the PNP problem in radial coordinates and
experimental and simulated data using simplified lumped parameter models.
