Pulsatile Pressure and Flow
Wei-Ting Yeh
July 14, 2014
Outline
• Goals
• History
• Windkessel Model
• Womersley Model
• Improvements
• Summary
• References
Goals (1/1)
• To understand the importance of elasticity of artery system.
• How to indirect measure the stroke volume of heart?
• Model of circulation system.
http://steadystrength.com/glossary/stroke-volume/
History (1/3)
• At 1628, William Harvey observed that blood gushes intermittently from a severed artery (動脈)
but flows slowly and steadily from a laceration of small vessels in the skin.
• At 1733, Stephan Hales explained this phenomenon by comparing the heart and arteries to the
apparatus of an old-fashioned fire engine.
William Harvey (1578-1657)
Stephan Hales (1677-1761)
http://www.studyblue.com/notes/note/n/ap-lab-/deck/5783413
History (2/3)
• How can an occurring intermittency flow (in artery) be converted into a more steady and
continuous stream (in skin vessel)?
• The Windkessel effect helps in damping the fluctuation in blood pressure over the cardiac cycle
and assists in the maintenance of organ perfusion during diastole when cardiac ejection ceases.
Windkessel
http://en.wikipedia.org/wiki/Windkessel_effect
Super Soaker CPS 1200
History (3/3)
• The Windkessel effect also decreases the pulse pressure. So the
reduction of this effect (blood vessel becomes stiffer) will cause
some disease, such as myocardial infraction (心肌梗塞), stroke (中
風), and heart failure (心臟衰竭).
• Early physiologists pictured the arterial system as an elastic chamber
that stretched as it received blood from the heart in systole and then
recoiled to its original volume during diastole.
• Mathematical model is still needed to give a quantitatively study of
this effect.
http://www.tooloop.com/picture-of-blood-circulation-in-thehuman-body/picture-of-blood-circulation-in-the-human-body-2/
Windkessel Model (1/3)
• At 1899, Otto Frank puts Hales’s concept into mathematical form.
• “The basic shape of the arterial pulse. First treatise: mathematical analysis”, translated from
Germany (Z. Biol. 37: 483-526).
Volumetric inflow rate I
Heart
(pump)
Volume V
Pressure P
Artery system
(reservoir)
Volumetric outflow rate Q
Capillary bed
Tube
(hose)
Otto Frank (1865-1944)
Windkessel Model (2/3)
• During the systole process:
𝑑𝑃
=𝐶 𝑃
𝑑𝑉
Volume compliance
of artery
𝑄=
𝑃
𝑅 𝑃
Resistance of tube
𝐼𝑑𝑡 − 𝑄𝑑𝑡 = 𝑑𝑉
Mass conservation
𝐼=
1 𝑑𝑃
𝑃
+
𝐶 𝑃 𝑑𝑡 𝑅 𝑃
We can use this equation to (rough) estimate
the stroke volume of heart from the P-t curve.
Here both C and R are positive function of P. The larger C means stiffer the artery.
• During the diastole process, we set I = 0, so
𝑑𝑃
= −𝑅 𝑃 𝐶 𝑃
𝑑𝑡
If further we assume that the linear compliance and Poiseuille’s law holds, then
𝑃 = 𝑃0 𝑒 −𝐶𝑡/𝑅
Windkessel Model (3/3)
• The Windkessel helps in (1) decreasing the pulse pressure and (2) smoothing out the stream:
Elastic artery
Stiff artery
Volumetric inflow
Systole
Diastole
Systole
Elastic artery
Stiff artery
Systole
Diastole
• Drawbacks of Windkessel model:
1. The pressures generated by the heart are assumed to be transmitted instantaneously through the artery.
2. The effect of each cardiac contraction die away before the next beat.
3. No reflection wave.
Womersley Model (1/5)
• During the period 1955-1958, John R. Womersley published a series of paper of this topic, which
mark a new epoch of the hemodynamics.
• “Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure
gradient is known” (1955, J. Physiol. 127: 553-563).
• Starting point: Linearized Navior-Stokes equation in rigid tube under time-varying pressure
gradient.
• Let w be the longitudinal fluid velocity at a distance r from the center, the steady state equation is
𝑑 2 𝑤 1 𝑑𝑤 𝐺0
+
+
=0
𝑑𝑟 2 𝑟 𝑑𝑟
𝜇
where 𝐺0 is the constant pressure gradient. The solution is (usual Poiseuillel flow)
𝐺0 2
𝑤=
𝑅 − 𝑟2
4𝜇
Womersley Model (2/5)
• Now if we let the pressure gradient is not a constant, then w becomes function of r and t, and the
equation of motion have an additional term to include the inertial force:
𝜕 2 𝑤 1 𝜕𝑤 𝐺 1 𝜕𝑤
+
+ +
=0
𝜕𝑟 2 𝑟 𝜕𝑟 𝜇 𝜈 𝜕𝑡
• Due to the linear nature of the equation, we can search the analytical solution for each mode, and
then sum of them to fit the required initial/boundary conditions.
• Assuming 𝐺 = 𝐺0 cos 𝜔𝑡 − 𝜙 , then it is simple (but tedious) to show that
Poiseuille flow:
𝐺0 2
𝑤=
𝑅 − 𝑟2
4𝜇
2
where 𝛼 ≝
𝑅2𝜔
𝜈
𝐺0 𝑅2 𝑀′ 0
𝑤=−
cos 𝜔𝑡 − 𝜙 − 90° − 𝜖′0
2
𝜇 𝛼
𝐺0 𝜋𝑅4 𝑀′10
𝑄=
cos 𝜔𝑡 − 𝜙 − 90° − 𝜖′10
2
𝜇
𝛼
is called Womersley number. Both 𝑀′ 0 and 𝜖′0 are function of 𝛼 and r, 𝑀′10
and 𝜖′10 are function of 𝛼.
Womersley Model (3/5)
measured
𝛼2
calculated
𝑅2 𝜔
≝
𝜈
As 𝛼 tends to zero, we recover the
steady state solution. In general,
there is a phase lag and smaller
magnitude than steady one.
Femoral artery of a dog
𝐺0 𝜋𝑅4 𝑀′10
𝑄=
cos 𝜔𝑡 − 𝜙 − 90° − 𝜖′10
2
𝜇
𝛼
Womersley Model (4/5)
• An example: Let R = 0.25 cm, 𝜌 = 1.06 gm/ml, 𝜇 = 0.035 poise, 𝐺0 = 1 mmHg/cm.
Steady flow
(𝜶 = 𝟎)
Osc. with f = 0.02 Hz
(𝜶 = 𝟎. 𝟎𝟓)
Osc. with f = 2.1 Hz
(𝜶 = 𝟓)
Phase lag of Q
0°
≅ 0°
70°
Amplitude of Q
58 ml/sec
≅ 58 ml/sec
14 ml/sec
𝑅2 𝜔
𝛼 ≝
𝜈
2
• In any one species, the fundamental frequency and the properties of blood are the same, and 𝛼
varies in different vessels by virtue of their radii. For different species:
Womersley Model (5/5)
• The velocity profile can also be calculated:
Systole
Diastole
Femoral artery of a dog
Theoretical calculation
Experiment
Improvements (1/1)
• Thin elastic walled tube helps in increasing average velocity, which can be 10% greater than in a
rigid tube under the same pressure gradient.
• Wave propagation is considered. Increasing the wall stiffness implies that the larger phase velocity
and the smaller transmission length.
• If the viscous behavior of tube is considered, the phase velocity would be slightly increasing and a
more prominent decrease in transmission.
Elastic (thin wall)
Elastic (thick wall)
Transmission length
Phase velocity
Rigid
Elastic (thin wall)
Rigid
Elastic (thick wall)
2𝜔
𝑅
𝛼2 ≝
𝜈
Summary
• Some models of circulation system is presented. We can calculate the stroke volume and velocity
profile if the pressure gradient as a function of time is known.
• Elasticity of artery system is essential, which helps in decreasing the pulse pressure and smoothing
out the stream of blood.
• The hemodynamics is quantified by the dimensionless number – Womersley number.
References
• Milnor WR (1982), Hemodynamics, Williams & Wilkins.
• Frank O (1899), The basic shape of the arterial pulse. First treatise: mathematical analysis (
translated from Germany), Z. Biol. 37: 483-526.
• Womersley JR (1955), Method for the calculation of velocity, rate of flow and viscous drag in
arteries when the pressure gradient is known, J. Physiol. 127: 553-563.
• Hale JF, McDonald DA, and Womersley JR (1955), “Velocity profile of oscillating arterial flow,
with some calculations of viscous drag and the Reynolds number”, J. Physiol. 128: 629-640.
END