PULSATILE FLOW Blood flow by heart is divided in two phases: Systole, during which blood is pumped from the heart and diastole, during which no blood is pumped from the heart and the ventricles fill with blood. This results in the variation of applied pressure gradient and flow through the arteries with time over the period of the heart relaxation and contraction (pulse). Heart Beat Cycle:

• The Right Atrium, receives "used blood" from the body. Blood will be pushed through the tricuspid valve to the

• Right Ventricle, the chamber which will pump to the lungs through the pulmonic valve to the

• Pulmonary Arteries, providing blood to both lungs. Blood is circulated through the lungs where carbon dioxide is removed and oxygen added. It returns through the

• Pulmonary Veins, which empty into the • Left Atrium, a chamber which will push the Mitral Valve open. Blood then passes

into the • Left Ventricle. Although it doesn't always look like it in drawings done from this

angle, this is the largest and most important chamber in the heart. It pumps to the rest of the body. As it pumps, the pressure will close the mitral valve and open the aortic valve, with blood passing through to the

• Aorta, where it will be delivered to the rest of the body.

Example: Proximal aorta, femoral artery, coronary arteries, terminal arteries Note that pressure pulses are almost nonexistent in the venous system and the flow is steady. Simplified governing equation with boundary conditions:

0:

0:0

==

=∂∂

=

∂∂

∂∂

+∂∂

−=∂∂

z

z

zz

uRrrur

rur

rrzp

tu µρ

Oscillatory/pulsatile pressure gradient (Pumping due to heart)

( ) ]Re[cos * tieAtLp

zp ωω =∆=∂∂

−

Womersley number

2/Re/ SrR == υωα

Separation of variables

( )

01*

2

2

*

=+−+

−=

+−

=

µυω

µρµω ωωω

ω

AVidrdVrdr

Vd

eAdrdVr

drd

reVei

Veu

tititi

tiz

Solution using Bessel’s functions

)/()/( 2/3022/3

01

*

riYCriJCiAV υωυωωρ

++=

Apply B.C.s:

tiz eiJ

RriJiAu ω

αα

ωρ

−=

)()/(

1 2/30

2/30

*

Bessel Function

Bessel functions are canonical solutions y(x) of Bessel's differential equation:

for an arbitrary real number α (the order). The most common and important special case is where α is an integer, n.

Although α and −α produce the same differential equation, it is conventional to define different Bessel functions for these two orders (e.g., so that the Bessel functions are mostly smooth functions of α).

Bessel's equation arises when finding separable solutions to Laplace's equation and the Helmholtz equation in cylindrical or spherical coordinates, and Bessel functions are therefore especially important for many problems of wave propagation, static potentials, and so on. (For cylindrical problems, one obtains Bessel functions of integer order α = n; for spherical problems, one obtains half integer orders α = n+½.) For example:

• electromagnetic waves in a cylindrical waveguide • heat conduction in a cylindrical object. • modes of vibration of a thin circular (or annular) membrane.

Bessel functions also have useful properties for other problems, such as signal processing (e.g., see FM synthesis or Kaiser window).

Bessel functions of the first kind:

Bessel functions of the first kind, denoted with Jα(x), are solutions of Bessel's differential equation which are finite at x = 0 for α an integer or α non-negative. The specific choice and normalization of Jα are defined by its properties below; another possibility is to define it by its Taylor series expansion around x = 0 (or a more general power series for non-integer α):

Here, Γ(z) is the gamma function, a generalization of the factorial to non-integer values. The graphs of Bessel functions look roughly like oscillating sine or cosine functions that decay proportionally to 1/√x (see also their asymptotic forms, below), although their roots are not generally periodic except asymptotically for large x.

Plot of Jα(x) and Yα(x) for α = 0,1,2:

Orthogonal w.r.t. a weighting function. (Classical Sturm-Liouville Problem)

Flow-rate:

ti

RtiR

z

eiJi

RriJiiARQ

xxJdxxxJ

rdriJ

RriJiAerdruQ

ω

ω

αααα

ωρπ

αα

ωρππ

−=∴

=

−==

∫

∫∫

)()/(21

)()(

)()/(

122

2/30

32

2/31

2/3*2

10

02/3

0

2/30

*

0

Viscous drag on the cylindrical surface:

( )[ ] ( )xJxJdxd

rurF Rrz

10

|2

−=

∂∂

−= =πµ

Explain phase lag between velocity and pressure gradient Local maxima near the walls (inertia effects) Limiting case of small Womersley number:

( )+−+−= 26

6

4

4

2

2

0 !32!2221)( xxxxJ

One can retrieve the parabolic profile for the velocity field by taking the limit aØ0. Homework assignment: N/A References:

1. Womersley, J.R. (1955). Method for the calculation of velocity, rate of the flow and viscous drag in arteries when the pressure gradient is known, J. Physiol., 127, 553-563.

2. Truskey, G.A., Yuan, F., Katz, D.F. (2004). Transport phenomena in biological systems, Pearson Prentice Hall, 213-223.

3. http://en.wikipedia.org/wiki/Bessel_function 4. http://www.heartpoint.com/theheart.html