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Digestion in the small intestine Chris Budd, Andre Leger, Alastair Spence EPSRC CASE Award with Unilever

Digestion in the small intestine Chris Budd, Andre Leger, Alastair Spence EPSRC CASE Award with Unilever

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## Digestion in the small intestine Chris Budd, Andre Leger, Alastair Spence EPSRC CASE Award with Unilever

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**Digestion in the small intestine**Chris Budd, Andre Leger, Alastair Spence EPSRC CASE Award with Unilever**What happens when we eat?**Stomach Small intestine: 7m x 1.25cm Intestinal wall: Villi and Microvilli**Process:**• Food enters stomach and leaves as Chyme • Nutrients are absorbed through the intestinal wall • Chyme passes through small intestine in 4.5hrs Intestinal wall Stomach Colon, illeocecal sphincter Peristaltic wave Mixing process**Objectives**• Model the process of food moving through the intestine • Model the process of nutrient mixing and absorption • Conclusions … • Peristalsis is effective at mixing the nutrients • It also acts to retard the mean flow of nutrient, allowing for greater nutrient absorption in the first part of the gut**Basic model: axisymmetric flow pumped by a peristaltic wave**and a pressure gradient • Chyne moves at velocity: u(x,r,t) • Nutrient concentration: c(x,r,t) • Peristaltic wave: r = f(x,t) r h = 1.25cm x r=f(x,t) Wavelength:8cm**Decouple the system:**• Calculate the flow u of the Chyme assuming Stokes flow and long wavelength • Calculate the Nutrient transport and absorption**7 Compartmental and Transit(CAT) Model**Approximations to the flow: I Degradation D1 Degradation D7 cn Inflow Outflow INTESTINE Absorption K1 Absorption K7 Stomach Outflow Degradation Inflow Absorption**Approximations to the flow: II Macro-transport**• Stoll et al (Chem Eng Sci 2000)‘A Theory of Molecular Absorption from the Small Intestine’ • Approximate flow u by 2D Poiseuille flow and consider a 1D equation for the average concentration C (Taylor,Moffatt) • Consider peristalsis as enhanced diffusion 2D: 1D:**Good news: Models are easy to use**Bad news: results are poor fits to the numerically computed concentration profiles for complex peristaltic flow • Better approach: • Use an asymptotic approach to give a good approximation to the peristaltic flow velocity u in the case of a small wave number • Identify different flow regimes • Use this in a numerical calculation of the concentration c**Navier Stokes**• Slow viscous Axisymmetric flow • Velocity & Stokes Streamfunction**No slip on boundary**WAVE FRAME FIXED FRAME Change from Impose periodicity**Amplitude:**• Wave Number: Small parameters Axisymmetry**Flow depends on:**Proportional to pressure drop Flow rate gives Poiseuille flow Amplitude Wave number Develop asymptotic series in powers of**Distinct flow types**• Reflux Pressure Rise Particles undergo net retrograde motion • Trapping Regions of Pressure Rise & Pressure Drop Streamlines encompass a bolus of fluid particles Trapped Fluid recirculates**Flow regions**A: Copumping, Detached Trapping B: Copumping, Centreline Trapping C: Copumping, No Trapping Illeocecal sphincter open D: Pumping, No Trapping E: Pumping, Centreline Trapping Illeocecal sphincter closed A B E C D F G Poiseuille**Case A: Copumping, Detached Trapping**Recirculation Particle paths**Case B: Copumping, Centreline Trapping**Recirculation Particle paths x**Case C: Copumping, No Trapping**Particle paths Poiseuille Flow x**Case D: Pumping, No Trapping**Particle paths Poiseuille Flow x Reflux**Case E: Pumping, Centreline Trapping**Particle paths Recirculation x Reflux**Calculate the concentration c(x,r,t)**1. Substitute asymptotic solution for u into 2. Solve for c(x,r,t) numerically using an upwind scheme on a domain transformed into a computational rectangle. 3. Calculate rate of absorption**Type C flow: no trapping**Poiseuille flowPeristaltic flow**Type E flow: trapping and reflux**Poiseuille flowPeristaltic flow**Cross sectional average of nutrient**x x x Location of absorped mass at final time Nutrient absorped Peristaltic flow x t**Conclusions**• Peristalsis helps both pumping and mixing • Significantly greater absorption with Peristaltic flow than with Poiseuille flow • Next steps • Improve the absorption model • Improve the fluid model (Non-Newtonian flow) • More accurate representation of the intestine geometry • Experiments