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

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What happens when we eat? Stomach Small intestine: 7m x 1.25cm Intestinal wall: Villi and Microvilli

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Process: Food enters stomach and leaves as Chyme Nutrients are absorbed through the intestinal wall Chyme passes through small intestine in 4.5hrs Stomach Intestinal wall Colon, illeocecal sphincter Peristaltic wave Mixing process

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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

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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) x r=f(x,t) r Wavelength:8cm h = 1.25cm

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Decouple the system: 1.Calculate the flow u of the Chyme assuming Stokes flow and long wavelength 2. Calculate the Nutrient transport and absorption

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Approximations to the flow: I 7 Compartmental and Transit (CAT) Model INTESTINE Stomach Absorption K 1 Absorption K 7 Degradation D 1 Degradation D 7 cncn Inflow Outflow Absorption Degradation

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Approximations to the flow: II Macro-transport 1D: 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:

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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: 1.Use an asymptotic approach to give a good approximation to the peristaltic flow velocity u in the case of a small wave number 2. Identify different flow regimes 3. Use this in a numerical calculation of the concentration c

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Navier Stokes Slow viscous Axisymmetric flow Velocity & Stokes Streamfunction

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FIXED FRAME WAVE FRAME No slip on boundary Change from Impose periodicity

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Axisymmetry Amplitude: Wave Number: Small parameters

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Flow depends on: Flow rate Proportional to pressure drop Amplitude Wave number gives Poiseuille flow Develop asymptotic series in powers of

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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 Distinct flow types

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A B C D E F G Flow regions Poiseuille 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 Illeocecal sphincter closed

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Case A: Copumping, Detached Trapping Recirculation Particle paths

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x Case B: Copumping, Centreline Trapping Recirculation Particle paths

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x Case C: Copumping, No Trapping Poiseuille Flow Particle paths

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x Case D: Pumping, No Trapping Poiseuille Flow Reflux Particle paths

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x Case E: Pumping, Centreline Trapping Recirculation Reflux Particle paths

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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

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Poiseuille flow Peristaltic flow Type C flow: no trapping

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Poiseuille flow Peristaltic flow Type E flow: trapping and reflux

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x x x x t Nutrient absorped Cross sectional average of nutrient Location of absorped mass at final time Peristaltic flow

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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

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