A Fractional Anomalous Diffusion Model and Numerical Simulation for Sodium Ion Transport in the Intestinal Wall

The authors present a fractional anomalous diffusion model to describe the uptake of sodium ions across the epithelium of gastrointestinal mucosa and their subsequent diffusion in the underlying blood capillaries using fractional Fick’s law. A heterogeneous two-phase model of the gastrointestinal mucosa is considered, consisting of a continuous extracellular phase and a dispersed cellular phase. The main mode of uptake is considered to be a fractional anomalous diffusion under concentration gradient and potential gradient. Appropriate partial differential equations describing the variation with time of concentrations of sodium ions in both the two phases across the intestinal wall are obtained using Riemann-Liouville space-fractional derivative and are solved by finite difference methods. The concentrations of sodium ions in the interstitial space and in the cells have been studied as a function of time, and the mean concentration of sodium ions available for absorption by the blood capillaries has also been studied. Finally, numerical results are presented graphically for various values of different parameters. This study demonstrates that fractional anomalous diffusion model is appropriate for describing the uptake of sodium ions across the epithelium of gastrointestinal mucosa. 1. Introduction The intestinal wall represents a complex system which allows the passage of substances either through the cells or in between the cells. The luminal surface of the intestine is covered with a typically leaky epithelium which enables the passage of ions via the intercellular route. The substance to be absorbed either penetrates into the intercellular space directly through the tight junction or enters the cell cytoplasm through the apical plasma membrane from the lumen of the intestine and then penetrates through the lateral plasma membrane to enter the intercellular space. The latter route leads to the underlying lamina propria, which consists of connective tissue, blood vessels, and lymph capillaries, and thus the substance enters the circulation (Figure 1) [1]. The process in which the ions enter the cell is passive diffusion under concentration gradient and potential gradient. This is mainly because transmural electrical potential differences of 5–12？mV have been reported from a variety of species during recent years [2, 3]. Although the potential differences across the intestinal wall are relatively small, they cannot be ignored in the studies of the intestinal transport of charged species [4]. Figure 1: Schematic diagram of the intestinal wall. Numerous