The small intestine is the foremost site of nutrient absorption in the body. In fact 90% of nutrients are absorbed here. Therefore models that accurately depict what goes on in the intestine would provide numerous applications to the food and drug industry. The complicated structure and mechanisms of the small intestine is the main obstacle to its modelling, with mechanisms such as peristalsis and segmentation mixing and transporting the intestinal contents. The structure of the small intestine, in place to aid absorption of nutrients, consists of three levels: on the macro scale, folds or plicae intrude into the lumen; on the mesoscale villi, small, finger like projections, cover the surface, as shown in the image below; and on the micro scale microvilli cover the villi on a cellular level.
The small intestine is too complicated to model in full, so the researchers concentrated on modelling the effects of the villi on the flow and absorption in the small intestine, a factor which has been ignored in most previous absorption models. On the level of the villi most of the motility patterns can be ignored, with the flow being represented as a simple shear flow over the villi. They created a model of a two-dimensional wavy-walled channel, where the waves are small in comparison with the channel width and represent the villi. The aims of this project were to investigate the effects of the villi on the transport of nutrients as well as to determine effective boundary conditions for the concentration that could be applied on a flat wall, to incorporate the effects of the villi. These could be then used in future models which include more of the complicated intestinal mechanisms or larger-scale geometry.
Technical Summary
To determine the effects of the villi on the transport we used asymptotic methods to split the channel into a large core region and a thin region near the wavy wall. We could then investigate the flow and uptake in the wall region, using a variety of numerical and analytical methods to find solutions and using matching methods to determine far-field boundary conditions and to relate the wall region solutions to effective boundary conditions that could be applied on the core region.
At the wall, the flow circulates in the troughs between the villi, causing a string of eddies to form. These affect the uptake, causing nutrients to circulate between the villi for certain parameter values. Looking at the solutions of the concentration, the effect of villi is very dependent on the uptake parameter (or effective permeability) of the nutrient being absorbed. For nutrients or molecules with a low uptake parameter, such as ibuprofen, which are not readily absorbed, the villi increase the uptake proportional to the surface area, hence ensuring that the molecule is absorbed over a shorter length scale. This is the expected result, and is thought to be the purpose of the villi.
For molecules with a large uptake parameter however, such as glucose, which are easily absorbed, the villi decrease the uptake, relative to that of a flat wall. Therefore high uptake molecules would be more readily absorbed over a flat wall than one with villi. This is due to these nutrients being pulled in strongly at the peaks of the villi, leaving few nutrients to permeate through to the troughs. This effectively decreases the surface area available for absorption, hence decreasing the uptake. These results can be seen in the figures below, where molecules with a low uptake parameter are absorbed slowly across the villi, while those with a high uptake parameter have a high flux at the peaks, dropping to a nearly zero flux in the troughs. As advection starts to dominate over diffusion (i.e. the Peclet number is large) the effects of the flow can be seen more easily, with the nutrients being circulated by the eddies and pulled further into the troughs.
These effects may seem detrimental in the context of the small intestine, however, the villi increase absorption for low uptake molecules whilst the high uptake molecules will always be totally absorbed over the length of the intestine, so a small increase in the absorption length scales will make little difference. Previous experimental work has shown that certain molecules, such as sugars and amino acids which have high uptake parameters, are absorbed mainly on the top third of the villi, and the small intestine has adapted to this factor with, for example, sites of glucose absorption only situated on the upper third of the villi.
This model has shown unexpected effects of villi on the uptake of nutrients, as well as providing boundary conditions for future models that incorporate more detail, hence producing a more accurate portrait of the small intestine. This work could also be applied to other industrial situations such as heat transport in a rough-walled pipe, or other types of transport over surfaces covered with small projections.
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