Digestion in the small intestine

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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

Intestinal wall
Stomach
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)

r
h 1.25cm
x
rf(x,t)
Wavelength8cm
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• Decouple the system
• Calculate the flow u of the Chyme assuming Stokes
  flow and long wavelength
• Calculate the Nutrient transport and absorption

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Approximations to the flow I

• 7 Compartmental and Transit (CAT) Model

Degradation D1
Degradation D7
cn
Inflow
Outflow
INTESTINE
Absorption K1
Absorption K7
Stomach
Outflow
Degradation
Inflow
Absorption
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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
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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
• 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

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• Navier Stokes
• Slow viscous
• Axisymmetric flow
• Velocity Stokes Streamfunction

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No slip on boundary
WAVE FRAME
FIXED FRAME
Change from
Impose periodicity
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• Amplitude
• Wave Number

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

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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
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Case A Copumping, Detached Trapping
Recirculation
Particle paths
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Case B Copumping, Centreline Trapping
Recirculation
Particle paths
x
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Case C Copumping, No Trapping
Particle paths
Poiseuille Flow
x
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Case D Pumping, No Trapping
Poiseuille Flow
Particle paths
x
Reflux
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Case E Pumping, Centreline Trapping
Recirculation
Particle paths
x
Reflux
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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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Type C flow no trapping
Poiseuille flow
Peristaltic flow
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Type E flow trapping and reflux
Poiseuille flow
Peristaltic flow
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Cross sectional average of nutrient
x
x
x
Nutrient absorped
Location of absorped mass at final time
Peristaltic flow
x
t
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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