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.

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

Schematic diagram of the intestinal wall.

Numerous techniques involving both in vivo and in vitro preparations have been employed in the study of intestinal transport. But because the cells are too small to provide continuous sections large enough for steady-state determinations of their transmission properties in actual physical situations, the distribution of the ions in the cellular and extracellular phases cannot be determined experimentally. Therefore, the idea of analysing such physiological problems using a theoretical approach has arisen. Fadali et al. [

Recently, fractional calculus has been a subject of worldwide attention due to its surprisingly broad range of applications in physics, chemistry, engineering, economics, biology, and so forth [

The fractional anomalous diffusion model considers a two-phase structure of the intestinal wall in which the epithelium is treated as a thin layer. The apical plasma membrane is adjacent to the lumen of the intestine and at the origin of a one-dimensional coordinate system. The rest of the cellular elements form a uniformly distributed array of identical cells. In between are the intercellular spaces which correspond to interstitial phase (Figure

Fick’s law is extensively adopted as a model for standard diffusion processes. For example, the simplest reaction diffusion model in spherical coordinates can be expressed as

where

However, requiring separation of scales, it is not suitable for describing nonlocal transport process. In order to study the anomalous diffusion, the fractional Fick’s law has been proposed [

where

The diffusion of sodium ions is complicated because its flux is determined by both the concentration gradient and the electrical gradient. Considering the motion of sodium ions under all forces, Macey [

where

Here, according to Magin’s idea [

where

We consider a two-phase model consisting of the interstitial phase and the intracellular phase. The mass balance equation in the interstitial phase, which accounts for the molecular diffusion flux and a uniformly distributed continuum of point sinks whose strength is proportional to the local concentration differences between the two phases [

And based on the assumption that diffusion does not contribute significantly to the total molecular transport inside the cell [

which is justified by the fact that the dimensions of the cells are small compared to the thickness of the intestinal wall; therefore, the flux through them is independent of distance.

Meanwhile, we assume that

where

The value

where

Then, we introduce dimensionless parameters

to reduce (

where

Based on the assumption that the concentration of sodium ions in the intestinal lumina surface is equal to the concentration at its abluminal surface for the epithelial is specially thin, the boundary conditions are given by

which mean that the concentration of sodium ions in the lumen is set to 1, whereas at the serosa it is set to 0 at all times. Further, the initial concentration is taken to be 0, a condition justified in the case of in vitro experiments. Both

For the numerical solution of the problem above, we introduce a uniform grid of mesh points

Then, we start to introduce the discretization of the differential operators. The first-order derivatives with respect to the temporal variable

and the first-order derivative with respect to the spatial variable

As for the Riemann-Liouville fractional derivative, using the relationship between the Grünwald-Letnikov formula and Riemann-Liouville fractional derivative, we can approximate the fractional derivative by [

where

Finally, the finite difference method for the above problem is given as follows:

The boundary and initial conditions can be discretized by

The concentrations of the sodium ions in the intercellular phase and intracellular phase are determined at different steps of time and space, and their weighted mean concentration at the blood capillaries can also be obtained.

The thickness of the intestinal wall

According to the Stokes-Einstein formula

Figure

(a) is

Here, based on the above analysis, we take

Figure

(a) is

Figure

(a) is

Figure

(a) is

Figure

(a) is

In summary, in this paper we have derived a fractional anomalous diffusion model for sodium ion transport in the intestinal wall using space fractional Fick’s law. Appropriate partial differential equations describing the variation with time of concentrations of sodium ions in both the interstitial phase and the intracellular phase across the intestinal wall are obtained using Riemann-Liouville space-fractional derivatives and are solved by finite difference methods. The numerical simulations have been discussed, and numerical results are presented graphically for various values of different parameters. It demonstrates that fractional anomalous diffusion model is appropriate for describing the uptake of sodium ions across the epithelium of gastrointestinal mucosa. This research also provides some new points for studying ions transferring processes in biological systems.

The project was supported by the National Natural Science Foundation of China (Grants nos. 11072134, 11102102, and 91130017).