Finite element modeling of coupled diffusion with partitioning in transdermal drug delivery

The finite element method is employed to simulate two-dimensional (axisymmetric) drug diffusion from a finite drug reservoir into the skin. The numerical formulation is based on a general mathematical model for multicomponent nonlinear diffusion that takes into account the coupling effects between the different components. The presence of several diffusing components is crucial, as many transdermal drug delivery formulations contain one or more permeation enhancers in addition to the drug. The coupling between the drug and permeation enhancer(s) results in nonlinear diffusion with concentration-dependent diffusivities of the various components. The framework is suitable for modeling both linear and nonlinear, single- and multicomponent diffusions, however, as it reduces to the correct formulation simply by setting the relevant parameters to zero. In addition, we show that partitioning of the penetrants from the reservoir into the skin can be treated in a straightforward manner in this framework using the mixed method. Partitioning at interface boundaries poses some difficulty with the standard finite element method as it creates a discontinuity in the concentration variable at the interface. To our knowledge, nonlinear (concentration-dependent) partitioning in diffusion problems has not been treated numerically before, and we demonstrate that nonlinear partitioning may have an important role in the effect of permeation enhancers. The mixed method that we adopt includes the flux at the interface explicitly in the formulation, allowing the modeling of concentration-dependent partitioning of the permeants between the reservoir and the skin as well as constant (linear) partitioning. The result is a versatile finite element framework suitable for modeling both linear and nonlinear diffusions in heterogeneous media where the diffusivities and partition coefficients may vary in each subregion.