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

Stomach

Intestinal wall

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)

x

r=f(x,t)

r

Wavelength:8cm

h = 1.25cm

Decouple the system:

1. Calculate the flow u of the Chyme assuming Stokes flow and long wavelength

2. Calculate the Nutrient transport and absorption

oncDcuct2.

oncKcnD a).(

Approximations to the flow: I

7 Compartmental and Transit (CAT) Model

nnnnnnnnn cDcKcUcUdt

dc 11

INTESTINE

Stomach

Absorption K1 Absorption K7

Degradation D1 Degradation D7

cnInflow

Inflow

Outflow

Outflow

Absorption

Degradation

Approximations to the flow: II Macro-transport

,*

D

aDD

),0[*** onCKCDCUC xxxt

cDcruc xt2)( on cKDc ar

1D:

)100(* DD

Stoll Stoll et al et al (Chem Eng Sci 2000)(Chem Eng Sci 2000) ‘A Theory of Molecular ‘A Theory of Molecular Absorption from the Small Intestine’Absorption from the Small Intestine’

Approximate flow u by 2D Approximate flow u by 2D Poiseuille flowPoiseuille flow and consider a 1D and consider a 1D equation for the average concentration C equation for the average concentration C (Taylor,Moffatt)(Taylor,Moffatt)

Consider Consider peristalsis as enhanced diffusionperistalsis as enhanced diffusion

2D:

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

• Velocity & Stokes Streamfunction

upuut

u 2).( 0. u

p ˆ

e

ru

ere )/( 0)( e

,ˆ eu

rr

L rrrxx

11

01 L

)/)(2cos(),( txhtxfr

),,( trx ),( rtxz FIXED FRAME WAVE FRAME

No slip on boundary

Change from

Impose periodicity

z

z

h

rr

0,0 ˆˆˆ rrr

ˆˆˆ1

ˆˆ ˆˆˆˆˆ2 rrrzz r

rrw ˆˆˆˆ ˆ

0ˆˆ1ˆˆ

ˆˆˆˆˆ2 rrrzz r

Axisymmetry

)ˆ2cos(1)ˆ( zzf

h

h

• Amplitude:

• Wave Number:

Small parameters

Flow depends on:

w ˆˆ

,6.0h

16.08

25.1

cm

cmh

Flow rate Proportional to pressure drop

Amplitude

Wave number

gives Poiseuille flow0

2Develop asymptotic series in powers of

• Reflux Pressure Rise Particles undergo net retrograde motion

• Trapping

Regions of Pressure Rise & Pressure DropStreamlines encompass a bolus of fluid particles

Trapped Fluid recirculates

0ˆ p

Distinct flow types

A

B

C D E

FG

0ˆ p

0ˆ p4/)1( 2

Flow regions

w

Poiseuille

A: Copumping, Detached TrappingA: Copumping, Detached TrappingB: Copumping, Centreline TrappingB: Copumping, Centreline TrappingC: Copumping, No TrappingC: Copumping, No Trapping

Illeocecal sphincter openIlleocecal sphincter open

D: Pumping, No TrappingD: Pumping, No TrappingE: Pumping, Centreline TrappingE: Pumping, Centreline Trapping

Illeocecal sphincter closedIlleocecal sphincter closed

4/)1( 2

Case A: Copumping, Detached Trapping

RecirculationParticle paths

x

Case B: Copumping, Centreline Trapping

RecirculationParticle paths

x

Case C: Copumping, No Trapping

Poiseuille FlowParticle paths

x

Case D: Pumping, No Trapping

Poiseuille Flow

Reflux

Particle paths

x

Case E: Pumping, Centreline Trapping

Recirculation

Reflux

Particle paths

Calculate the concentration c(x,r,t)

oncDcuct2).(

oncKcnD a).(

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

Poiseuille flow Peristaltic flow

Type C flow: no trapping

Poiseuille flow Peristaltic flow

Type E flow: trapping and reflux

x

x

x x

t

Nutrient absorped

Cross sectional average of nutrient

Location of absorped mass at final time

Peristaltic flow

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