Diffusion-absorption modeling of drug transport in the gastrointestinal tract: an initial-boundary value problem approach

The absorption of drugs in the gastrointestinal tract is a fundamental process in oral drug delivery involving the complex interplay of diffusion and permeation. This paper develops a mathematical model for drug transport in the gastrointestinal lumen, incorporating Fickian diffusion and a first-order volumetric sink term to represent uptake into the systemic circulation. The model is formulated as an initial-boundary value problem with no-flux boundary conditions at both the gut wall and the luminal center, representing a closed system where drug removal occurs solely through the distributed absorption term. Separation of variables is employed to derive an analytical solution, revealing that the concentration remains spatially uniform throughout the domain and decays exponentially with time as C(x,t) = C0e-kat. The dimensionless absorption number κ = kaL2/D emerges as the key parameter governing the rate of drug elimination, with κ ≫ 1 indicating absorption-dominated transport and κ ≪ 1 indicating diffusion-dominated transport. The analytical solution is verified numerically using a Crank--Nicolson finite difference scheme, demonstrating excellent agreement with errors on the order of machine precision (∼ 10−15). The model provides a rigorous framework for understanding the biophysical interplay between diffusion and absorption in oral drug delivery and has potential applications in the design of controlled release formulations and absorption enhancement strategies.