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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) h = 1.25cm r Wavelength:8cm x r=f(x,t) Decouple the system: 1. Calculate the flow u of the Chyme assuming Stokes flow and long wavelength 2. Calculate the Nutrient transport and absorption ct u.c D c on 2 D(n.c) K a c on Approximations to the flow: I 7 Compartmental and Transit (CAT) Model Degradation D1 Inflow Absorption K1 Degradation D7 cn Outflow INTESTINE Absorption K7 Stomach Degradation Outflow dcn U n 1cn 1 U n c n K n cn Dn cn dt 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) Dcr K a c on 2D: ct u (r )cx D c 1D: Ct U C x D C xx K C on [0, ) 2 * * * Consider peristalsis as enhanced diffusion a D D , D * D* (100D ) 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 • Navier Stokes • Slow viscous Axisymmetric flow u (u.)u p 2u t .u 0 ˆ u e , ˆ p • Velocity & Stokes Streamfunction (e / r ) e u e r (e ) 0 1 L1 xx rr r r r L1 0 No slip on boundary r f ( x, t ) h cos( 2 ( x t ) / ) FIXED FRAME Change from Impose periodicity ( x, r , t ) WAVE FRAME ( z x t , r ) • Amplitude: Small parameters • Wave Number: r r h h h f ( zˆ ) 1 cos( 2 zˆ) ˆ ˆ w ˆ rˆ rˆ 1 rˆ 2ˆ zˆzˆ ˆ rˆrˆ ˆ rˆ ˆ 1 ˆ ˆ ˆ zˆzˆ rˆrˆ rˆ 0 rˆ 2 Axisymmetry 0, rˆrˆrˆ 0 z z Flow depends on: ˆ ˆ w Flow rate Amplitude h 0.6, Proportional to pressure drop 0 gives Poiseuille flow h Wave number 1.25cm 0.16 8cm Develop asymptotic series in powers of 2 Distinct flow types • Reflux pˆ 0 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 pˆ 0 (1 ) 2 / 4 A: Copumping, Detached Trapping B: Copumping, Centreline Trapping C: Copumping, No Trapping A B ˆ w Illeocecal sphincter open C E D pˆ 0 F (1 ) 2 / 4 Poiseuille G D: Pumping, No Trapping E: Pumping, Centreline Trapping Illeocecal sphincter closed Case A: Copumping, Detached Trapping Particle paths Recirculation Case B: Copumping, Centreline Trapping Particle paths x Recirculation Case C: Copumping, No Trapping Particle paths x Poiseuille Flow Case D: Pumping, No Trapping Particle paths x Poiseuille Flow 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 ct (u.c) D c on 2 D(n.c) K a c on 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 flow Peristaltic flow Type E flow: trapping and reflux Poiseuille flow Peristaltic flow Cross sectional average of nutrient x x Location of absorped mass at final time x 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