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Lecture #23: Internal Flows
Body Plan Evolution
1 cell
cellular sheet
cellular bilayer
one way gut
endoderm
ectoderm
bilayered canister
mouth
anus
cephalization
mesoderm
Basic circulatory circuit
lung/gill
heart
body
diffusion
in dedicated
exchangers
intestine
convection
In dedicated
plumbing
Convection vs. Diffusion
x
Fick’s Law:
C1
S
C2
mass
C2  C1
J
  DS
time
x
C= concentration in mass/volume
D = diffusion coefficient
Units = L2/T
Basic strategy of circulatory systems:
Pluming uses bulk flow (convection) to move fluids to capillary beds where diffusion
can take place over short distances.
Relative importance of bulk flow to diffusion given by Peclet number:
ul
Pe 
D
Problems with gas exchange:
consider simple gas exchanger:
convection
C2  C1
J  DS
x
water
DIFFUSION
blood
driving
force
partial
pressure
(02)
equilibrium
distance
Problems with gas exchange:
consider countercurrent gas exchanger:
C2  C1
J  DS
x
water
blood
driving
force
partial
pressure
(02)
distance
What about
lungs?
air
blood
partial
pressure
(02)
distance
C2  C1
J  DS
x
Birds have more efficient system
Birds have more efficient system
What determines flow in pipes?
L
r
P1
x
P2
a
If Re < 2000 (i.e. laminar flow):
P (a  r )
u x (r ) 
L
4
2
2
• flow ~ pressure gradient
• flow ~ 1 / viscosity
• parabolic flow distribution
P (a )
u x max 
L 4
2
What is maximum flow velocity?
At center of pipe, r=0:
What determines flux through pipe?
Flux (Q) = velocity x area:
P a 2 1 2 P a 4
Q
( 2 a ) 
L 4
L 8
= Hagen-Poiseuille equation
Flux through a system:
• proportional to pressure gradient
• inversely proportional to viscosity
• has fourth order dependence on diameter
lung
heart
Pressure
is lost (drops)
across network
of pipes.
10% of our total
metabolic cost!
5% of our total
Weight in blood!
body
intestine
pressure
flow velocity
heart
lung
intestine
distance
body
Problems with blood
From Hagen-Poiseuille Equation:
‘Resistance’
P 8L

 4
Q a
Blood is very viscous due to red blood cells
Blood is not a
‘Newtonian’ fluid,
Mostly because of red
blood cells.
optimum
at 58%
% hematocrit
carrying capacity
viscosity
02 carried/unit cost
Thoughts about plumbing:
Consider simple branch point:
S1
If S1 = 2 S2
then velocity is same in
all branches; flux
is ½ the original value.
S0
S1
Consider change in diameter:
a0
a2
If a0 = 2 a1
then 16 times the
pressure is required
in small pipe for same
flux!
Circulatory systems cannot compensate with large trunks –
Blood volume would become too large.
Murray’s Law: what is geometry of branching network?
1) Cost to pump = Q x pressure gradient, or
P
Q 2 8

Q
4
L
a
 Ma 2
2) Cost to make new pipe
Total cost
Q 2 8
2


M

a
4
a
3) Find optimum as a function of diameter:
2
d Q 8
aopt  da (

M

a
)
4
a
2
16 1/ 6
aopt  Q (
)
M
1/ 3
if
1/ 3
then
aopt ~ Q
Q  ka
3
Mass flux ~ cube of
vessel diameter
But, by law of continuity,
Q0  Q1  Q2
a1
Q1
a0
Q0
thus
a a a
Q2
3
0
a2
3
1
3
2
a.k.a. Murray’s Law
For simple symmetrical branching case:
a1  0.79a0
S1  0.63S0
u1  0.26u0
More generally……
a  a  a  a  ...a
3
0
3
1
3
2
3
3
3
n
How does a growing vascular network ‘know’ to follow Murray’s Law?
r
x
du/dr
Shear stress at wall,
It can be shown that:
4Q
t 3
r
t = du/dr
But by Murray’s Law:
Q  ka
a
3
So with r = a (at wall):
t
4k

Thus, shear stress at wall is constant in network obeying Murray’s Law.
Algorithm could be: ‘Grow vessel until shear stress reaches certain value.’