BLOOD FLOW:
THE ULTIMATE LIFELINE
M. Zamir
Departments of Applied Mathematics
and of Medical Biophysics
University of Western Ontario
London, CANADA
Workshop IV: Optimal Transport in the
Human Body: Lungs and Blood
May 19 - 23, 2008
What is a “fluid”?
What is a “fluid”?
• It is something that flows ...
What is a “fluid”?
• It is something that flows ...
• It wobbles ...
What is a “fluid”?
• It is something that flows ...
• It wobbles ...
• It fills the space ...
What is “flow”?
What is “flow”?
• It is what fluids do ...
What is “flow”?
• It is what fluids do ...
• It is how fluids move ...
• etc.
The definition (and magic) of
fluids lie in their mechanical
properties ...
The definition (and magic) of
fluids lie in their mechanical
properties ...
We often say that life is not
possible without air and water,
meaning without the chemical
ingredients of air and water ...
The definition (and magic) of
fluids lie in their mechanical
properties ...
We often say that life is not
possible without air and water,
meaning without the chemical
ingredients of air and water ...
But would life be possible
without the mechanical
properties of air and water?
without their fluidity?
• A fluid body cannot support a
nonzero deforming force while
remaining at rest.
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
• A fluid body cannot support a
nonzero deforming force while
remaining at rest.
• It yields outright. It deforms
and continues to deform even
after the force has been
removed.
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
• Flow is a state of continuous
deformation.
• Flow is a state of continuous
deformation.
• Only fluids are able to be in
that state without breaking
apart (sand does not flow).
• Flow is a state of continuous
deformation.
• Only fluids are able to be in
that state without breaking
apart (sand does not flow).
• It is because of this property
that fluids provide the best
means of transport within the
body.
Imagine if you can ...
How it would be possible to
reach every one of many
billions of cells within the
body?
Without fluids as a medium?
And without flow as a
mechanism?
• But, because of this property,
a fluid body does not respond
as a body to the application of
a force ...
• The problem of dealing with
its mechanics becomes more
complicated mathematically ...
Continuum Mechanics
• How to apply laws of
motion to a fluid body?
• Micro/macro scales..
• Continuum concept...
• Concept of fluid
“element”..
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
Fluid Dynamics
• Velocity “at-a-point”..
• Lagrangian/Eularian
velocities..
• Pressure at a point..
• “No-slip” boundary
condition..
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
Fluid Dynamics
• Conservation of mass
“at a point”??
• Equation of
continuity..
Fluid Dynamics
• Navier-Stokes
equations …
force = mass x acceleration
Zamir M: The Physics of Coronary Blood Flow. Springer NY 2005
Classical problem of
(Poiseuille) flow in a
tube, based on a
solution of a simplified
form of the NavierStokes equations.
pressure is directly
related to flow rate
Zamir M: The Physics of Coronary Blood Flow. Springer NY 2005
• Flow in a tube is the cornerstone of
transport within the body …
Zamir M, 2005. The Physics of Coronary Blood Flow. Springer NY.
• Flow in a tube is the cornerstone of
transport within the body …
• A most efficient form of transport …
Zamir M, 2005. The Physics of Coronary Blood Flow. Springer NY.
• Flow in a tube is the cornerstone of
transport within the body …
• A most efficient form of transport …
• The power required to lift 5 L/min of
fluid against gravity to a height of only
1m is sufficient to drive the same rate of
flow in a tube of 3 cm in diameter and
almost 2 km long!
Zamir M, 2005. The Physics of Coronary Blood Flow. Springer NY.
• Flow in a tube is the cornerstone of
transport within the body …
• A most efficient form of transport …
• The power required to lift 5 L/min of
fluid against gravity is sufficient to drive
the same rate of flow in a tube of 3 cm in
diameter and almost 2 km long!
• Did evolution “find” this wonderful tool?
Or, more likely, did it invent it?
Zamir M, 2005. The Physics of Coronary Blood Flow. Springer NY.
Zamir M, 2005. The Physics of Coronary Blood Flow. Springer NY.
- In a tree structure, the calculation is repeated at each bifurcation.
- For this purpose the tree must be mapped in a way that the calculations can
be programmed.
Zamir M, 2005. The Physics of Coronary Blood Flow. Springer NY.
- Pressure distribution along the tree structure depends on the power law
relating flow and vessel diameter.
pulsatile flow
(rigid tube)
pressure-flow relation now
depends on time and on
frequency
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
- flow lags behind pressure
- peak flow is lower than in Poiseuille flow
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
- peak flow depends strongly on frequency
pulsatile flow (elastic tube)
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
3 equations
for p,u,v
coupled with
equations for
wall motion
space
time
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
- wave length / tube length = 1
space
time
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
- wave length / tube length = 10
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
one-dimensional
analysis
(1) forward wave
(2) combined with backward (reflected) wave
(3) reflection coefficient
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
- In the absence of wave reflections, pressure
amplitude is constant throughout.
… forward
- - - reflected
total
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
- In the presence of wave reflections, pressure amplitude depends on
how the forward and backward waves combine, which in turn depends
on the ratio of wave length to tube length.
- wave length / tube length = 1
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
- wave length / tube length = 2
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
- wave length / tube length = 4
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
• Wave reflections change the pressure distribution within the tube,
and in so doing they change everything.
pressure-flow relation now depends not only on time and frequency
but also on wave length to tube length ratio
effects of wave reflections in a
tree structure depend on
reflection coefficient at each
bifurcation
Zamir M, 2005: The Physics of Coronary Blood Flow. Springer NY.
reflection coefficient depends on
characteristic admittance of
branches if there are no reflections
downstream
reflection coefficient depends on
effective admittance of branches if
there are reflections downstream
Zamir M, 2005. The Physics of Coronary Blood Flow. Springer NY.
right coronary artery
from human heart
(vessel segments,
volume in mm^3)
Zamir M, 1998. Mechanics of blood supply to the heart:
wave reflection effects in a right coronary artery. Proceedings
of the Royal Society B265:439-444.
Zamir M, 1996. Tree structure and branching characteristics
of the right coronary artery in a right-dominant heart.
Canadian Journal of Cardiology 12(6):593-599.
approximate branching architecture
calculations based on actual
geometrical data obtained from a
cast of human coronary artery
Zamir M, 1996. Tree structure and branching characteristics of the right
coronary artery in a right-dominant heart. Canadian Journal of Cardiology
12(6):593-599.
effective,
characteristic
- effective admittance is higher than
characteristic admittance
- wave reflections are helping flow
Zamir M, 1998. Mechanics of blood supply to the heart:
wave reflection effects in a right coronary artery. Proceedings
of the Royal Society B265:439-444.
FRACTAL PATTERNS IN
VASCULAR BRANCHING:
FACT OR FICTION?
Zamir M, 2000. The Physics of Pulsatile Flow. Springer-Verlag NY.
Zamir M, 1976. Optimality principles in arterial branching. Journal of Theoretical Biology 62:227-251.
Zamir M, 1988: The branching structure of arterial trees. Comments on Theoretical Biology 1:15-37.
Zamir M, Phipps S, 1987. Morphometric analysis of the distributing vessels of the kidney. Canadian Journal of Physiology and
Pharmacology 65:2433-2440.
Zamir M, 1981. Three-dimensional aspects of arterial branching. Journal of Theoretical Biology 90:457-476.
bifurcation index

d2
d1
Zamir M, 1978. Nonsymmetrical bifurcations in
arterial branching. Journal of General Physiology
72:837-845.
Mandelbrot 1977
• It is desirable therefore to interrupt the
study of rivers and to study instead some
other natural phenomenon …
Mandelbrot BB, 1977. Fractals: Form, Chance, and Dimensions. Freeman, San Francisco.
Mandelbrot 1977
• It is desirable therefore to interrupt the
study of rivers and to study instead some
other natural phenomenon …
• The example of the vascular system is so
suitable to our present purpose that it would
be too painful for us to be prevented from
exploring it here …
Mandelbrot BB, 1977. Fractals: Form, Chance, and Dimensions. Freeman, San Francisco.
Mandelbrot 1977
• The arterial and venous systems involve no
self contact.
Mandelbrot BB, 1977. Fractals: Form, Chance, and Dimensions. Freeman, San Francisco.
Mandelbrot 1977
• The arterial and venous systems involve no
self contact.
• Every point which is not within an artery or
a vein is a point of tissue. There is an artery
and a vein infinitely near every point of
tissue.
Mandelbrot BB, 1977. Fractals: Form, Chance, and Dimensions. Freeman, San Francisco.
Mandelbrot 1977
• The arterial and venous systems involve no self
contact.
• Every point which is not within an artery or a vein
is a point of tissue. There is an artery and a vein
infinitely near every point of tissue.
• The volume of all the arteries and veins must be
only a small percentage of the body volume,
leaving the bulk to tissue.
Mandelbrot BB, 1977. Fractals: Form, Chance, and Dimensions. Freeman, San Francisco.
Mandelbrot 1977
•… blood vessels
crisscross organs so
tightly that tissue is
a counterpart of the
Lebesgue-Osgood
monster.
Mandelbrot BB, 1977. Fractals: Form, Chance, and Dimensions.
Freeman, San Francisco.
Fractal?
Space filling?
Zamir M, 1988: The branching structure of arterial trees. Comments on Theoretical Biology 1:15-37.
Zamir M, 1999. On fractal properties of arterial trees. Journal of Theoretical Biology 197:517-526..
Zamir M, 1999. On fractal properties of arterial trees. Journal of Theoretical Biology 197:517-526..
bifurcation index

d2
d1
Zamir M, 1999. On fractal properties of arterial trees. Journal of
Theoretical Biology 197:517-526..
area ratio
d12  d 22

d 02
Zamir M, 1999. On fractal properties of arterial trees. Journal of
Theoretical Biology 197:517-526..
power law index
d1x  d 2x  d 0x
Zamir M, 1999. On fractal properties of arterial trees. Journal of
Theoretical Biology 197:517-526..
Zamir M, 1999. On fractal properties of arterial trees. Journal of
Theoretical Biology 197:517-526..
• How does one deal with this degree of variability?
• How does one deal with this degree of variability?
• We should look at vasculature as a functional rather than a
purely geometrical object …
Mandelbrot BB, 1977. Fractals: Form, Chance, and
Dimensions. Freeman, San Francisco.
Zamir M, Wrigley SM, & Langille BL (1983) Arterial
bifurcations in the cardiovascular system of a rat.
Journal of General Physiology 81:325-335.
Burton AC, 1965. Physiology and Biophysics of the Circulation. Year Book Medical Publishers, Chicago.
Zamir M, 2001. Fractal dimensions and multifractility in vascular
branching. Journal of Theoretical Biology 212:183-190.
Zamir M, 2001. Fractal dimensions and multifractility in vascular
branching. Journal of Theoretical Biology 212:183-190.
Zamir M, 2001. Fractal dimensions and multifractility in vascular branching. Journal of Theoretical Biology 212:183-190.
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Zamir M, 2001. Fractal dimensions and
multifractility in vascular branching. Journal of
Theoretical Biology 212:183-190.
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Zamir M, 2001. Fractal dimensions and
multifractility in vascular branching. Journal of
Theoretical Biology 212:183-190.
Zamir M, 2001. Fractal dimensions and multifractility in vascular branching. Journal of Theoretical Biology 212:183-190.
Zamir M, 2001. Fractal dimensions and multifractility in vascular branching. Journal of Theoretical Biology 212:183-190.
Sernetz et al 1985
Sernetz M, Gelleri B, Hofman F, 1985. The organism as a bioreactor, Interpretation of the reduction law of metabolism in terms
of heterogeneous catalysis and fractal structure. Journal of Theoretical Biology 117:209-230.
log M  0.75 log W  log A
M  AW
M W
M r
0.75
0.75
2.25
W r
3
=> Body mass is not a three dimensional solid but
a fractal surface of fractal dimension 2.25.
container
volume  V  r 3
sheet
area  S
• If V is doubled, does S double ?
• In other words, how does S scale with V or with r ?
S  r2 ?
S  r3 ?
It turns out that
S  rD
where D is a fractal dimension, with value
in the range
2 D3
• The sheet does not quite fill the space of the container
as a solid.
• The sheet does not quite fill the space of the container
as a solid.
• The crumpled sheet is an object of topological
dimension 2 and fractal dimension D.
• The sheet does not quite fill the space of the container
as a solid.
• The crumpled sheet is an object of topological
dimension 2 and fractal dimension D.
• It is more than two dimensional but less than three
dimensional.
• The sheet does not quite fill the space of the container
as a solid.
• The crumpled sheet is an object of topological
dimension 2 and fractal dimension D.
• It is more than two dimensional but less than three
dimensional.
• And so it is suspected that metabolic rate scales as
r2.25 because we are not quite three dimensional as a
solid object.
• The sheet does not quite fill the space of the container
as a solid.
• The crumpled sheet is an object of topological
dimension 2 and fractal dimension D.
• It is more than two dimensional but less than three
dimensional.
• And so it is suspected that metabolic rate scales as
r2.25 because we are not quite three dimensional as a
solid object.
• We are packed with vasculature equivalent to a
convoluted surface of fractal dimension 2.25
Coronary Blood Flow
• Coronary blood flow is blood flow to the heart for its own
metabolic needs, the fuel which the heart needs to perform its
pumping action..
Coronary Blood Flow
• Coronary blood flow is blood flow to the heart for its own
metabolic needs, the fuel which the heart needs to perform its
pumping action..
• Failure of this flow to reach its destination is the cause of
the overwhelming majority of heart failures.
Coronary Blood Flow
• Coronary blood flow is blood flow to the heart for its own
metabolic needs, the fuel which the heart needs to perform its
pumping action..
• Failure of this flow to reach its destination is the cause of
the overwhelming majority of heart failures.
• What is generally referred to as heart "disease" is truly not a
disease of the heart at all but of the vessels that carry blood
flow to the heart, failure of the heart to receive the fuel it
needs.
Coronary Blood Flow
• Coronary blood flow is blood flow to the heart for its own
metabolic needs, the fuel which the heart needs to perform its
pumping action..
• Failure of this flow to reach its destination is the cause of
the overwhelming majority of heart failures.
• What is generally referred to as heart "disease" is truly not a
disease of the heart at all but of the vessels that carry blood
flow to the heart, failure of the heart to receive the fuel it
needs.
• How does the heart get its blood supply?
blood must leave the heart
first, then some of it is drawn
back to supply its own
metabolic needs
Zamir M, 1990. Flow strategy and functional design of the coronary network. In: Coronary Circulation, F. Kajiya, G.A. Klassen,
J.A.E. Spaan & J.I.E. Hoffman, eds., Springer Verlag.
left and right main
coronary arteries
Zamir M, 1990. Flow strategy and functional design of the coronary network. In: Coronary Circulation, F. Kajiya, G.A. Klassen,
J.A.E. Spaan & J.I.E. Hoffman, eds., Springer Verlag.
“coronary”:
like a “crown”
Zamir M, 2005. The Physics of Coronary Blood Flow. Springer NY.
heart tissue is totally filled with vasculature
Zamir M, 1990. Flow strategy and functional design of the coronary network. In: Coronary Circulation, F. Kajiya, G.A. Klassen,
J.A.E. Spaan & J.I.E. Hoffman, eds., Springer Verlag.
a massive vascular / fluid dynamic system: how does it work?
Zamir M, 1990. Flow strategy and functional design of the coronary network. In: Coronary Circulation, F. Kajiya, G.A. Klassen,
J.A.E. Spaan & J.I.E. Hoffman, eds., Springer Verlag.
Is it a simple “plumbing” system?
no blockage = flow
blockage = no flow
Obsession with fat and colesterol is based
on this assumption.
- Yet, (as we saw earlier) the “coronary circulation” is a
highly sophisticated dynamical system in which the
dynamics and physics of the flow are as important as the
integrity of the conducting vessels.
- Yet, (as we saw earlier) the “coronary circulation” is a
highly sophisticated dynamical system in which the
dynamics and physics of the flow are as important as the
integrity of the conducting vessels.
- While an obstruction in the conducting vessels is a
fairly obvious and clearly visible cause of disruption in
coronary blood flow, any discord in the complex
dynamics of the system can cause an equally grave,
though less conspicuous, disruption in the flow.
dynamic “norms”?
= Dynamic conditions under which
the system is designed to operate?
dynamic “pathologies”?
= States of disturbed dynamics at
which the system’s operation is
less than optimum?
- RLC system in series: single harmonic input
- inertial time constant = 0.1s, capacitive time constant = 0.0253s
Zamir M: The Physics of Coronary Blood Flow. Springer NY 2005
Only this particular combination of parameter values produces a unique state whereby
the inertial and capacitive effects precisely cancel each other, leaving the system with
pure resistance- any deviation from these values moves the system away from that norm.
- LM2: {{R1+L},{R2+C}}: cardiac pressure wave
- inertial time constant = capacitive time constant = 0.2s, R2/R1 = 1.0
Zamir M: The Physics of Coronary Blood Flow. Springer NY 2005
Only this particular combination of parameter values produces a unique state whereby the
inertial and capacitive effects precisely cancel each other, leaving the system with pure
resistance- any deviation from these values moves the system away from that norm.
γ = power law index, co = Moen-Korteweg wave speed, c = actual wave speed
Zamir M: The Physics of Coronary Blood Flow. Springer NY 2005
Only a certain combination of parameter values produces “impedance matching”, whereby the
form of the input pressure wave remains unchanged as it progresses along the tree- any
deviation from these values moves the system away from that norm.
Zamir M: The Physics of Coronary Blood Flow. Springer NY 2005
normal: vessel diameters and branching architecture follows cube law, E=10^7 dyn/cm^2
vasoconstriction: diameters decreased by 50% in 6 peripheral levels of tree, E=10^9 dyn/cm^2
vasodilatation: diameters increased by 50% in 6 peripheral levels of tree, E=10^9 dyn/cm^2
• Disease processes produce structural or
functional pathologies that affect coronary
blood supply ...
• but the ultimate effect of these is the dynamic
pathologies which they produce.
Arrhythmia
Arrhythmia, in all its forms, is the most
blatant example of a dynamic pathology.
•
Arrhythmia
Arrhythmia, in all its forms, is the most
blatant example of a dynamic pathology.
•
• The underlying cause may be any
factor affecting the pacemaker or
affecting the free passage of its signals.
Arrhythmia
Arrhythmia, in all its forms, is the most
blatant example of a dynamic pathology.
•
• The underlying cause may be any
factor affecting the pacemaker or
affecting the free passage of its signals.
• But the ultimate insult, the ultimate
factor that brings the system down, is a
disruption in the dyanamics of the
system, in other words it is a dynamic
pathology.
Sudden Cardiac Death
Sudden cardiac death is by
definition “death due to
heart failure in which the
timing of the failure cannot
be attributed etirely to any
prevailing heart disease.”
Current Theory suggests that in
sudden cardiac death the fall in
coronary blood flow is caused by a fall
in cardiac output.
Current Theory suggests that in
sudden cardiac death the fall in
coronary blood flow is caused by a fall
in cardiac output.
The difficulty with this course of
events is that it points to a dynamical
system with positive feedback.
Current Theory suggests that in
sudden cardiac death the fall in
coronary blood flow is caused by a fall
in cardiac output.
The difficulty with this course of
events is that it points to a dynamical
system with positive feedback.
This course of events and this dynamic
feature of the coronary circulation are
actually prevented by the mechanism
of “autoregulation” of coronary blood
flow.
A more likely course of events is one in which the fall in coronary blood supply is the cause
rather than the result of a fall in cardiac output- the fall in coronary blood flow itself is due to a
disruption in the dynamics of the coronary circulation.
“Broken Heart”
Syndrome
The phenomenon of
a broken heart has
very recently made
the passage from the
realm of folklore to
that of medicine.
“Broken Heart”
Syndrome
The phenomenon of
a broken heart has
very recently made
the passage from the
realm of folklore to
that of medicine.
It is perhaps the
most celebrated form
of a dynamic
pathology.
• “Sudden emotional distress
... can sometimes produce
severe transient left
ventricular disfunction.”
• “Sudden emotional distress
... can sometimes produce
severe transient left
ventricular disfunction.”
• “Emotional stress can
precipitate severe, reversible
left ventricular disfunction in
patients without coronary
disease.”
• “Sudden emotional distress
... can sometimes produce
severe transient left
ventricular disfunction.”
• Emotional stress can
precipitate severe, reversible
left ventricular disfunction in
patients without coronary
disease.”
• So, in the absence of
coronary artery disease, it is
the dynamics of coronary
blood flow that are being
disrupted, and it is a dynamic
pathology that finally brings
the heart down.