Post on 28-Mar-2015
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