Download The mathematics of digestion

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
L1   xx  rr   r  r  
r
L1  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
 2ˆ 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