Unifying the mathematical modeling of in vivo and in vitro microdialysis

A unifying approach is presented for developing mathematical models of microdialysis that are applicable to both in vitro and in vivo situations. Previous models for cylindrical probes have been limited by accommodating analyte diffusion through the surrounding medium in the radial direction only, i.e., perpendicular to the probe axis, or by incomplete incorporation of diffusion in the axial direction. Both radial and axial diffusion are included in the present work by employing two-dimensional finite element analysis. As in previous models, the nondimensional clearance modulus (Θ) represents the degree to which analyte clearance from the external medium influences diffusion through the medium for systems exhibiting analyte concentration linearity. Incorporating axial diffusion introduces a second dimensionless group, which is the length-to-radius aspect ratio of the membrane. These two parameter groups uniquely determine the external medium permeability, which is time dependent under transient conditions. At steady-state, the dependence of this permeability on the two groups can be approximated by an algebraic formula for much of the parameter ranges. Explicit steady-state expressions derived for the membrane and fluid permeabilities provide good approximations under transient conditions (quasi-steady-state assumption). The predictive ability of the unifying approach is illustrated for microdialysis of sucrose in vivo (brain) and inert media in vitro, under both well-stirred and quiescent conditions.