Measuring diffusion using the differential form of Fick's law and magnetic resonance imaging

Abstract

Diffusion is an important process in many biological and industrial processes. Diffusion coefficients are traditionally measured using integrated solutions of Fick's law for systems with well-defined boundary conditions. We report a simple method for measuring diffusion coefficients in processes without well-defined boundary conditions or without a simple integrated solution using the differential form of Fick's law. Magnetic resonance imaging (MRI) was used to obtain spatially and time-resolved profiles of the diffusion of H2O from an agarose gel to a neighboring D2O reservoir. The differential form of Fick's second law was used to solve for the diffusion coefficient, D=1.3×10−9 m2 s−1. MRI is well suited to this type of analysis as it naturally generates time- and space-resolved images. This analytical method allows for the determination of diffusion coefficients in systems that lack an integral solution to the diffusion equation.