Physiological Fluid Dynamics

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• Physiological Fluid Dynamics

Arterial Fluid Dynamics ??? ???????????
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Evolution of Arterial Pressure Away from the heart
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Systemic Arteries

• Conduct blood flow from Left ventricle (LV) to
  peripheral organs
• Aortic valve ? Aortic arch (180 turn)
• Geometry changes Tapering
• Geometry changes Branching
• Mechanical properties changes

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Radius and wall thickness of human arteries
change with age
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Incremental Elastic Modulus of human arteries
change with age
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Stress-Strain relations of rabbits thoracic
aorta
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Fluid Mechanics of Elastic Conduct

• Mass Conservation
• Conservation of momentum
• Conservation of energy

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Background

• Fundamental Variables Pressure? Flow
• Geometrical Variables Size? Thickness ? Length?
  Curvature
• Mechanical Properties Stiffness ?
  Visco-Elasticity

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Equations of Viscous Pipe Flow

• Consider a conduct filled with incompressible
  fluid of density ? and pressure p, let u be the
  only non-zero velocity component

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Poiseuilles Law (1840)

• Assume steady flow, u u(r ) with no body forces,
  the equation of motion

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Laminar Poiseuillean flow

• Rate of flow through the tube
• Mean velocity of flow
• Shear stress at the wall

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Laminar Poiseuillean flow

• Skin friction
• Shear stress in terms of skin friction

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Implication of Poiseuilles Law

• Q is proportional to the fourth power of the
  radius.
• Q is directly proportional to the pressure
  difference.
• Q is inversely proportional to the length of the
  tube.
• If the arteries becomes constricted, the blood
  pressure requires to supply the blood flow
  adequately will rise substantially,leading to the
  state of hypertension.

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Optimum design of Blood Vessel Bifurcation
(Poiseuilles formula)
For a given pressure drop, 1 change in vessel
radius results in a 4 changes in flow
Murray (1926) Rosen (1967)
Work done
Metabolism Energy loss
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Minimum cost function for optimum vessel
configuration
With respect to radius a ? the optimum radius
The optimum vessel radius is proportional flow
to the 1/3 power, and
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Optimum vessel bifurcation that with minimum cost
function
Minimize P at the bifurcation point B
An optimum location B would be
for arbitrary movements of B.
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Let B displaced along A-B direction first
The optimum is obtained when
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Again, let B displaced in the C-B direction
The optimum is obtained when
Similarly, displaced B along D-B direction, we
find
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Similarly, displaced B along D-B direction, we
find
The continuity equation gives
We find
which is often referred to as Murrays Law
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Let ao denotes the radius of the aorta, and
assume equal bifurcation in all generation
If the capillary blood vessel has a radius of 5
um and the radius of the aorta is 1.5 cm.
We find n30.
The total number of blood vessel is about
230?109.
Note in fact arteries rarely bifurcation
symmetrically (a1a2). For human, only one
symmetric bifurcation. For dog, there are none.

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Pulsatile Blood Flow

• Consider pulsatile flow in a circular vessel,
  pp(x, t) and u u(r, t)

• For a sinusoidal flow

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Pulsatile Blood Flow(2)

• The general solution of the ODE in the form
  involves Bessel functions of complex arguments

U(ra)0 (non-slip)
U(r0)finite
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Pulsatile Blood Flow(3)

• Introducing Womersley number ?

• As ??0, the velocity profile becomes parabolic.
• As ???, viscosity is negligible U(r)-i P/??.

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Analysis of Blood Flow using Elastic Theory

• From Poiseuilles Law
• the flux is proportional to the pressure
  difference (p1-p2). However, the blood flow in
  veins are remarkably non-linear.
• The flow in elastic conduct gradually attains a
  maximum value as the pressure difference
  increases and then on longer increases.

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Arterial Flow in Elastic Tube

• Axial velocity, v
• Lumen area, S

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Pressure-Diameter relationship

• Let T denotes the tension of the blood vessel per
  unit thickness, wall thickness h, vessel radius a
• Let ro be the radius of zero tension state, the
  Hookes Law gives elastic constant E as

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Poiseuilles flow in elastic tube

• Consider steady flow in elastic tube of length L,
  assume the tube is long and the pressure is
  function of axial coordinate z, let P1 and P2
  denote the inlet and outlet pressure and the
  external pressure surrounding the tube is P0
• Assume the flow through the tube obey
  Poiseuilles law, the flow becomes

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Transmission of Pulse wave (Velocity) in elastic
tube

• Consider inviscid and incompressible fluid flow
  in elastic tube of lumen area A,
• By linearizing the equations

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Transmission of Pulse wave (Velocity) in elastic
tube

• Combining the continuity and momentum equations,
• The wave equations

Pulse Wave Velocity (PWV)
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Analysis of Aortic Diastolic and Systolic
Pressure Waveforms

• Constitutive relationship between aortic volume
  and pressure where K is the
  volume elasticity of the aorta, and V0 is the
  end-systolic volume.
• If the aorta is very soft (K is very small), let
  I(t) and Q(t) denote the inflow and outflow
  rates, we have

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• During diastole, the aortic valve is closed and
  there is no flow into the aorta. Hence I(t)
  0. where ? is a
  non-invasive measure aortic volume elasticity.
  Let Td be the duration of diastolic phase, the
  aortic pressure (Pd) at the end of this phase or
  just prior to ejection is given by
• The volume elasticity that depicts the
  exponential drop of aortic pressure is given by

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Reynolds Strouhal Womersley

• Reynolds number
• Strouhal number
• Womersley number

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Flows under the action of Oscillating pressure
gradient
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Wave propagation in Blood Vessel

• Pulse wave propagation in arteries
• A(x, t) depends on transmural pressure,

Here c is the wave propagation velocity.
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For thin walled elastic tube

• Consider the elasticity of the tube, arterial
  diameter ? blood pressure

For a thick walled elastic tube
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Balance of Force in Arterial Wall
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Resonant vibration of flow in a circular tube

• When the tube length is equal to the half wave
  length

• This is called the fundamental frequency of the
  natural vibration.
• Hemodynamics Effects of Frequency on the
  Pressure-flow relationship of Arterial tree

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Boundary conditions
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Pressure-Flow
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Mean Velocity Profile (Dog Aorta)
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Velocity waveform at the upper descending aorta
of a dog
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Effect of Womersly number on the velocity
distribution
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Blood PressureEvolution
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Effect of sinusoidal pressure wave speed of
various frequencies on the instantaneous aortic
pressure
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Distribution of Atherosclerotic Sites in Human
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Stress Concentration Conditions
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Atherosclerotic disease at the carotid bifurcation
Stress contours in arterial branching
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Whats the blood flows in reality?

• Unsteady
• Non-uniform geometry
• Bifurcations
• Non-Newtonian
• Viscoelastic wall
• Fluid-solid interactions

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In vitro measurement of artery pressures and flows
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Ultrasonic flowmeters
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Electromagnetic flowmeters

• Electromagnetic flowmeters have existed for
  measurement of blood flow rate outside the body
  during open heart surgery.

This miniature probe is for acute and chronic,
low flow measurements in small animals and
rodents. Sizes from 1 to 10 mm internal
circumference
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Electromagnetic flowmeters
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Electromagnetic flowmeters

• Faraday's principle of electromagnetic induction
  can be applied to any electrical conductor
  (including blood) which moves through a magnetic
  field. The electromagnetic blood flowmeter is
  sometimes used during vascular surgery to measure
  the quantity of blood passing through a vessel or
  graft, before during or after surgery. A circular
  probe with a gap to fit the vessel is fitted
  around the vessel. This probe applies an
  alternating magnetic field across the vessel and
  detects the voltage induced by the flow via small
  electrodes in contact with the vessel.

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Electromagnetic flowmeters

• Alternating magnetic fields (typically at 400 Hz)
  are used since the induced voltages are in the
  microvolt region and d.c. electrode potentials
  may cause significant errors with unchanging
  magnetic fields. A number of probes are required
  to fit the various diameters of blood vessel.
• An alternative design carries the sensing device
  on the tip of a special catheter which passes
  inside the vessel and generates a magnetic field
  in the space around it and has the electrodes on
  its surface.

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SQUARE-WAVE ELECTROMAGNETICBLOOD FLOWMETERS
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Pulse Oximetry

• Takuo Aoyagi(1974) developed the principle of
  pulse oximetry. The next year, Nihon Kohden
  introduced the world's first ear oximeter,
  OLV-5100, which used pulse oximetry to
  noninvasively measure saturated blood oxygen
  without the need to sample blood. All pulse
  oximeters today are based on Dr. Aoyagi's
  original principle of pulse oximetry.

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CCA Root
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Variation of velocity waveforms across the
arterial vessel

• Common Carotid Artery (CCA)

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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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CCA
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Radial Artery
0.23 cm in diameter
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Brachial Artery
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Brachial Vein