(accepted for publication in Medical & Biological Engineering & Computing)
A Coupled Flow-Acoustic Computational
Study of Bruits from a Modeled Stenosed
Artery
Jung Hee Seo, Rajat Mittal
Department of Mechanical Engineering, Johns Hopkins University,
Baltimore, MD, 21218
Corresponding Author:
Rajat Mittal
e-mail: mittal@jhu.edu
Phone: +1-410-516-4069
Fax: +1-410-516-7254
1
Abstract
The sound generated by blood flow in stenosed arteries is investigated for a model that consists of
a channel with a one-sided constriction. The blood flow-induced arterial “bruits” are computed
directly using a hybrid approach wherein the hemodynamic flow field is solved by an immersed
boundary, incompressible flow solver, and the sound generation is modeled by a first-principles
approach that employs the linearized compressible perturbation equations. The transmission and
propagation of the sound through the surrounding biological tissues is also modeled with a
simplified, linear structural wave equation. The flow field inside the artery and the bruit sound
signal at the epidermal surface are examined to delineate the precise source of the arterial bruit and
the correlation between the bruit and the arterial wall pressure fluctuations. It is found that the
bruits are related primarily to the time-derivative of the integrated pressure force on the poststenotic segment of arterial wall. The current study provides a clear perspective on the generation
of bruits from stenosed arteries and enables an assessment of the conjectures of previous
researchers regarding the source of arterial bruits.
Keywords: Stenosis, Artery, Murmur, Hemoacoustics, Immersed Boundary
Method
2
Introduction
An arterial stenosis is an abnormal narrowing in a artery which is normally caused
by atherosclerosis. It is often found in the coronary, carotid, and femoral arteries,
and presents severe health risks to the patient[4,29]. The diagnosis of arterial
stenoses can be made with ultrasound, MRI (Magnetic Resonance Imaging), or
MRA (Magnetic Resonance Angiogram), but these methods are time-consuming,
expensive, and often invasive. Meanwhile, it is well established that stenosed
arteries produce distinct sounds known as arterial bruits (or murmurs) which can
be heard externally with a stethoscope. This technique of “auscultation” is a noninvasive, inexpensive and safe diagnostic method for arterial stenosis[4,34].
Although it has been generally believed that arterial bruits are associated
with the “disturbances” in the blood flow caused by the stenosis, the precise
source mechanism of the bruit is still poorly understood. Bruns[9] argued that
arterial bruits were generated by the ‘nearly periodic fluctuation in the wake found
downstream of any appropriate obstacle’ and not by post-stenotic turbulence
which is a quadruple sound generation mechanism with exceedingly low strength
at low Mach numbers. Lees and Dewey[21] recorded the spectrum of actual bruit
sounds (a technique called phonoangiography), and suggested a significant
similarity between the bruit sound spectrum and the wall pressure spectrum of a
fully developed turbulent pipe flow which seemed to contradict the postulate of
Bruns. Fredberg[12] derived a theoretical model for the transfer function between
wall pressure spectrum and sensed sound using the Green’s function and a
stochastic analysis of turbulent boundary layer. Wang et al.[39] modeled the
sound generation in a stenosed coronary artery using an electrical network analog
model, and Borisyuk[7] modeled the sound propagation through the tissues
(thorax) theoretically for a simple cylindrical geometry. It should be noted that
both Wang et al. and Borisyuk use an empirical turbulence spectrum as an input to
the sound analysis. Borisyuk[8] also conducted an experimental study for the
sound generation by steady flow through a stenosed duct with a thorax model and
analyzed the sound spectra. Yazicioglu et al.[40] performed an experiment for the
flow inside a constricted viscoelastic tube that was ensconced in a gel phantom
model, and measured the tube wall pressure as well as the surface vibration of the
3
gel phantom Interestingly, Owsley and Hull [31] conducted a similar study but
focused on the propagation of shear waves.
In addition to these, there have been many numerical and experimental
studies of the flow field and turbulence characteristics of blood flow in stenosed
arteries including those of Fredberg[13], Kirkeeide et al.[19], Ahmed &
Giddens[1,2], Mittal et al.[24], and Varghese et al.[37,38]. Most of studies on
stenotic flows associated with arterial bruit have focused on the arterial wall
pressure fluctuations as a surrogate for sound[13,18,24] or as a dominant source
for the arterial bruit[7,12,39]. The correlation between the bruit sound and the
arterial wall pressure however remains to be established.
The objective of the present study is to investigate the source mechanism
of arterial bruits and the correlation between bruits and the arterial wall pressure
fluctuation via a physics-based, coupled flow-acoustic computational model. The
direct simulation of blood-flow induced sound is challenging, since the flow Mach
number is very low and wave propagation through different materials is also
involved. In this study, the problem is tackled with an immersed-boundary
method based hybrid approach, and we simulate blood flows as well as the sound
generation and propagation for a canonical model of a stenosed artery.
Methods
At the outset, we point out that the major assumptions made in the current
computational model are that the blood behaves like a Newtonian fluid, the wall
of the blood vessel is not deformed by the blood-flow, the shear waves generated
in the tissue are negligible compares to the acoustic waves and that the viscous
dissipation of the acoustic wave is also negligible. Justifications for these
assumptions is provided in the following section.
Model
A two-dimensional constricted channel is considered as a model of a stenosed
artery and a schematic is shown in Fig. 1. A channel is constricted from one side
(top wall) and the profile of constriction is given by
x x0
b
y ymax 1 cos 2
,
2
D
D ( x x0 ) D
(1)
4
Air
Epidermal Surface
Monitoring Point
Tissue Layer
6D
hT=10D
Constriction
Blood Vessel
hW=0.3D
D
Blood Flow
10D
30D
Figure 1. Schematic of the constricted channel model and acoustic domain; D: arterial diameter,
hw: arterial wall thickness, hT: tissue layer thickness.
where b is the size of constriction, x0 is the center of the stenosis, and D is the
height of the channel. Similar models have been used in past studies of constricted
arteries [1,2,37,38]. Two constriction levels corresponding to b=0.5D and 0.75D
are considered in the present study. The pulsatile pressure drop between the inlet
and exit is assumed to have the following sinusoidal variation in time:
P / U 2 A B sin(2 ft ),
(2)
where constants A and B are chosen to obtain similar maximum flow rate for two
cases (50% and 75% constrictions) and a minimum flow rate close to zero. Thus,
B is fixed to 1.5, while value A is set to 0.225 and 0.75 for the 50% and 75% cases,
respectively. The non-dimensional frequency of pulsation is St=fD/Umax=0.024,
where Umax is the maximum centerline velocity at the inlet, and the Reynolds
number is set to Re=UmaxD/0=2000, where 0 is a kinematic viscosity. The
chosen flow parameters yield a Womersley number[29], ( Re St / 2)1/2 =8.6
which is in the range appropriate for large peripheral arteries[24].
In the current model, the blood flow is assumed to be Newtonian (which is
a good assumption for the larger and medium sized arteries[32]) and the fluidstructure interaction with the arterial wall is neglected. Fluid-structure interaction
with the elastic blood vessel may introduce resonance peaks in the sound
spectrum[23]. However, these resonance peaks are generally diminished due to
5
the damping associated with the surrounding tissue[23] and do not play an
important role in auscultation.
The acoustic domain in the current study includes not only the lumen but
also the arterial wall (blood vessel) and the surrounding tissue (assumed to be
skin). The acoustic material properties are based on Ref. [16]; the density and
speed of sound for the blood, vessel wall and tissue are 1.05 (g/cm3) and 1500
(m/s), 1.1 (g/cm3) and 1580 (m/s), and 1.2 (g/cm3) and 1720 (m/s), respectively.
The top boundary of the acoustic domain represents the epidermal surface and
given that a stethoscope actually senses transmitted sound via the velocity (or
acceleration) of the epidermis[7], we monitor these quantities in our simulations.
It is assumed that the acoustic waves radiate through all other boundaries.
Hemodynamics
The hemodynamic flow field inside the artery is modeled with an immersed
boundary solver[25] which solves the following incompressible Navier-Stokes
equations,
P
U
(U )U
0 2U ,
0
t
U 0
,
(3)
where U is velocity vector, P is pressure, and 0 is the density of blood. In this
study, the equations are solved by a projection method with a second-order central
finite-difference scheme and a ghost-cell based sharp-interface method is used for
the immersed boundary treatment. The details of the flow solver and the
immersed boundary formulation can be found in Ref. [25].
The blood flow domain is resolved by a 768128 non-uniform Cartesian
grid with the minimum grid spacing x=0.01D. The flow is driven by the pulsatile
pressure gradient and Dirichlet pressure boundary conditions are applied at the
inlet and exit. A Neumann type boundary condition is applied for the velocity at
the inlet and exit, and a no-slip boundary condition is used for the top and bottom
walls. The flow computations are carried out for about 4 pulsation cycles after it
reaches a stationary state.
6
Acoustics
The flow-induced sound in the blood flow region is computed by the linearized
perturbed compressible equations (LPCE)[35] which are given by
u '
1
(u 'U ) p ' 0,
t
0
(4)
p '
DP
2
(U ) p ' 0c0 ( u ') (u ') P
,
t
Dt
where the () represents the compressible (acoustic) perturbation, c0 is the speed of
sound, and D/Dt is the total derivative. The capital letters indicates the
hydrodynamic incompressible variables and they are obtained from the
incompressible flow simulations. The details of the derivation and the validation
of the above procedure can be found in Ref.[35]. The incompressible NavierStokes/LPCE hybrid method is a two-step, one-way coupled approach for the
prediction of flow induced sound at low Mach numbers[26,35,36].
The auscultated sound is in fact the sound signal monitored on the skin
(epidermal) surface. The propagation of the sound through the tissues between the
artery and the epidermal surface is therefore an important aspect of modeling
arterial bruits[5,7,12]. In the present study, the sound propagation through the
arterial wall and surrounding tissue is modeled via a linear structural wave
equation based on the bulk modulus of the tissue material as follows:
u ' 1
p ' 0,
t s
p '
K ( u ') 0,
t
(5)
where u ' is the velocity fluctuation vector (time derivative of displacement) and
p represents the average normal stress (pressure), and s and K=scs2 are the
density and the bulk modulus of the material, respectively. In this model the
propagation of shear waves is not considered; this approach is valid since the
shear modulus of the tissue materials is much smaller than the bulk modulus[30].
Also shear wave length is much shorter than the compression wave and thus it
decays rapidly. This fluid-like assumption of the tissue material for the purpose of
resolving acoustic wave propagation has been widely used for the simulation of
ultrasound[6,30] and acoustic[28] wave radiation in biological materials. Previous
analytical studies[7,12] on arterial bruits also focused on the propagation of
compression waves. The dissipation of the acoustic wave is also neglected in the
7
present study since the frequency range of the bruits is typically on the lower end
of the spectrum (<1000 Hz) and the dissipation of acoustic wave at these low
frequencies is expected to be very small[12]. Specifically, the attenuation loss
coefficient for tissue is about 0.1(neperMHz/cm)[15] and this yields only about a
0.01% loss at 1000 Hz.
Equation (5) is solved in a fully coupled manner with the LPCE. In fact, in the
present study, we combine those two into a single set of equations and the
different material domains are treated by prescribing appropriate material
properties. The following unified single set of acoustic equations result from this
combination:
u '
1
H ( x )(u 'U ) p ' 0,
( x)
t
DP
p '
H ( x ),
H ( x ) (U ) p ' (u ') P K ( x )( u ')
Dt
t
(6)
where H is a Heaviside function of which values is 1 for the blood flow region
and 0 for elsewhere, and the density () and bulk modulus (K=c2) are now
functions of space. By solving Eqs. (6), the wave transmission and reflection at
the interface between the blood and tissue are automatically resolved based on the
difference of acoustic impedance Z=K/c. The same approach has been used in the
simulations of sound wave propagation through heterogeneous materials[6,28].
Equations (6) are spatially discretized with a sixth-order compact finite difference
scheme[22] and integrated in time using a four-stage Runge-Kutta method.
The actual Mach number of blood flows in arteries is M=U/c~O(10-3),
where c is the speed of sound. This extremely low Mach number significantly
increases the computational cost of the acoustic field simulation, because the
time-step size is restricted by the speed of sound, which is much faster than the
flow speed. In order to mitigate this computational expense, we employ a Mach
number of 0.01. This may result in an increase in the absolute sound intensity,
however, the source mechanism and the scaling between sound and pressure are
unaffected, and comparisons between different cases and different source
locations can still be made. It should be noted that even with this increased Mach
number, the acoustic (compression) wave length of the bruit remains much larger
than any other length scale in the problem. A similar, O(10) increase of Mach
number (decrease of speed of sound) was also used in the previous study of
Eienstein et al.[11] for the computation of mitral-valve sound in the heart.
8
The acoustic domain is covered by a 400200 Cartesian grid with a
minimum grid spacing 0.02D. The acoustic wave length is about 20D for the
frequency of St=5, and this wave length is resolved by about 200 grid points. At
the epidermal surface, a zero-stress boundary condition (p=0) is applied [5,7]. A
buffer-zone type radiation boundary condition is applied via grid stretching and
low-pass spatial filtering[14] at all the other boundaries. The flow simulation
results are interpolated onto the acoustic grid in the lumen using a bi-linear
interpolation. The time-step for the acoustic field simulation is 20 times smaller
than the time-step size used for the incompressible flow simulation due to the
acoustic CFL condition, and a second-order Lagrangian interpolation[36] is used
for the temporal interpolation of flow variables.
Analytical Evaluation of Sound Source
The source of the bruit is evaluated analytically for the present model
configuration to aid the investigation of the source mechanism. The wave
equation for the acoustic velocity fluctuation in the absence of shear waves can be
written as
2v '
1 f
2 2
c
v
'
,
s
t 2
s t
(7)
where cs is the speed of sound obtained from the bulk modulus as
cs K / s
, and
f is the external body force per unit volume. In the analytical model, the
inhomogeneity of material properties is not taken into account, since the
differences in the values are quite small. The general solution of Eq. (7) can be
obtained using the Green’s function[17] as
1
v'
s cs 2
1 f
4 | r | t dV ,
(8)
where r is the vector from the source to the observer point, and the squarebracket indicates the value evaluated at the retarded time,
t | r | / cs .
For the
present configuration, the external force is exerted by the blood flow and is
associated with the fluid pressure. The force term in Eq. (8), therefore, can be
replaced by the pressure gradient;
9
v'
1
4 s cs
2
1
( PH ) dV ,
| r | t
(9)
where H is a Heaviside function for which the value is 1 inside the blood flow
domain and 0 otherwise. Integration by parts of Eq. (9) leads to
v'
r P
dV .
4 s cs 2 | r |3 t
1
(10)
The volume integration is therefore reduced to the blood flow domain which is
denoted by in Fig.1. If we now assume that the pressure on the upper and the
lower surfaces of the lumen are not significantly different, Eq. (10) can be
approximated by the trapezoidal rule for the y-component of velocity fluctuation
on the epidermal surface as
v'
D
2 s cs 2
Lz
0
Lx
0
sin P
dx dz
r 2 t
,
(11)
where is the angle between the x-axis and r . Here x and z are the axial and
spanwise directions of the channel, respectively, and y is the direction towards the
epidermal surface. Furthermore, Lx and Lz are the streamwise and spanwise
lengths of the blood flow region, and r | r | . The boundary condition on the
epidermal surface ( v '/ y 0 , with a zero stress boundary condition) is also applied
on the Eq. (11) by means of an anti-symmetric imaginary source. If it is assumed
that
r ra const . ,
where ra is the average distance, and / 2 , the above
expression can be further simplified to;
v'
dFy
D
;
2 2
2 s cs ra dt
Fy
Lz
0
Lx
0
P dx dz , (12)
where Fy is the pressure force integrated on the upper(or lower) boundary surface
of the blood flow domain. Note that the above equation is the result for threedimensional wave radiation. For a two-dimensional case, the equivalent form of
Eq. (12) is written in the frequency domain as
vˆ '( )
dF
iDk
H1(1) (kra ) y ,2 D ( );
2
2 s cs
dt
Lx
Fy ,2 D P dx , (13)
0
where k=/cs is the wave number, H1 is the Hankel function of order 1, and hat
(^) indicates a Fourier transform. The time signal may be given by
10
v '(t ) vˆ '( )e it d
. The analytical expression derived here suggests that the
vertical velocity fluctuation detected by a stethoscope is generated by the timederivative of the integrated pressure force, Fy. This theoretical estimate can be
examined using the current computational model.
a: 50% Constriction
b: 75% Constriction
Figure 2. Time evolution of vorticity field; 0/4T: maximum flow rate, and 2/4T: minimum flow
rate phase.
Results
Hemodynamics
Unless otherwise noted, all the data presented in this paper are nondimensionalized by the velocity scale: Umax, length-scale: D, time-scale: D/Umax,
and pressure scale: Umax2. The instantaneous hemodynamic flow fields are
visualized in Fig. 2 by contours of spanwise vorticity. For the 50% constriction
case (Fig. 2a), it is observed that the vortex roll-up starts from the maximum flow
rate phase (0/4T, where T is the period of pulsation). The detachment of
separation bubble in the wake of the stenosis, and the boundary layer separation at
the bottom surface are clearly visible. The shear layers become unstable during
deceleration and a coherent vortex street is formed as shown at 2/4T with an
overall wavelength of about ~1D. For the 75% case (Fig. 2b), more complex and
stronger (see the contour legend) vortex motions are observed. The separation
bubble in the wake of the stenosis rapidly becomes unstable, and a strong, jet-like
flow through the gap below the constriction induces large-scale vortex roll-up (of
which length scale ~1D) as well as the formation of smaller-scale vortices. At the
minimum flow rate and beyond, a vortex street similar to the 50% constriction
11
case is observed and for both cases, a clear signature of the vortex street persists
into the next cycle. The overall flow patterns are similar to the 3D large-eddy
simulation (LES) results of Mittal et al.[24]. However, in 3D LES, the large
vortex structures break into smaller eddies in the post-stenotic region reducing the
coherence of the vortex street.
a
b
0.2
s= -1D
0.2
0
0
-0.2
-0.2
0.2
s= 1D
0.4
-0.2
s= 4D
0.2
-0.4
2
0
0
-0.2
-2
0.2
s= 1D
0
dP/dt
dP/dt
0
s= -1D
s= 6D
1
0
s= 4D
s= 6D
0
-0.2
-1
1
2
3
4
t/T
1
2
3
4
t/T
Figure 3. Time variations of temporal wall pressure fluctuation represented by the time derivative
of pressure at several locations on the upper lumen, where s=(x-x0) is the distance from the center
of the stenosis and T is the period of pulsation. a) 50% constriction case, b) 75% case. Vertical
dashed lines indicate the maximum flow rate phase.
The temporal wall pressure fluctuations represented by the timederivatives of pressure, dP/dt are monitored at the following locations on the
upper wall; 1D upstream from the center of stenosis (begin of the constriction),
and 1D (end of the constriction), 4D, and 6D downstream from the center of the
stenosis, and plotted in Fig. 3. The last two downstream locations correspond to
the position of the maximum pressure fluctuation for 75% and 50% constriction
cases, respectively. For the 50% case, the temporal pressure fluctuation is found
to be the superposition of the overall pulsation of pressure gradient and the
fluctuations caused by the post-stenotic vortex motion. For the 75% case, the
magnitude of the pressure fluctuation induced by the vortex motion is about 10
times larger than the 50% case, especially at 4D downstream from the stenosis,
and the most severe pressure fluctuations are observed at around the maximum
flow rate phase.
12
a
b
5
x 10
-14
|v'|2
4
3
2
1
5
10
15
20
x/D
Figure 4. a) Root mean squared acoustic pressure fluctuation (prms) for 75% constriction case. b)
Intensity of vertical velocity fluctuation on the epidermal surface.
Arterial Bruits
The root-mean-squared (rms) acoustic pressure fluctuation field is shown in Fig.
4a for 75% constriction case. In this plot, the origin of the acoustic waves seems
to be at the post-stenotic region. The vertical velocity fluctuation on the epidermal
surface which represents the arterial bruit is monitored at a number of positions
and the stream wise variation of the intensity is plotted in Fig. 4b. The bruit is
strongest over the post-stenotic region and a shallow peak is observed at 5-6D
downstream from the stenosis which is consistent with Fig. 4a. However, finding
that the spectral characteristics at different locations are nearly indistinguishable,
we analyze the signal at one location: 6D downstream from the stenosis where the
maximum acoustic energy is measured. The monitored time signals are plotted in
Fig. 5a for two cases. Since some transducers sense acceleration which is
proportional to the force or pressure, the epidermal acceleration (dv/dt) is also
shown. For the 50% constriction, small fluctuations are superimposed on the
overall sinusoidal profile which is caused by the pressure gradient pulsation, but
the amplitude of these fluctuations is relatively small. For the 75% case however,
stronger high frequency fluctuations are observed, especially during the peak
phase of the sinusoidal variation. This additional higher frequency fluctuation is
expected to produce a distinct arterial murmur and this is more clearly represented
in the acceleration signal.
13
a
0
0
0.0001
v'
75%
0
0
dv'/dt
-2E-05
1
dv'/dt
2E-05
v'
v'
1E-05
v'
50%
dv'/dt
2E-05
dv'/dt
2
3
4
-1E-05
-2E-05
1
2
3
4
-0.0001
t/T
t/T
b
1
10
Frequency [Hz]
2
10
-6
10
10
-4
1
10
2
10-5
10-7
10-5
10-6
10-7
10-8
10-6
10-7
10
-8
10
10-9
-9
10-10
<v'>
10-6
<dv'/dt>
<v'>
10
10
10
-7
10-8
50 %
10
-10
10
-11
10
-8
<dv'/dt>
Frequency [Hz]
-5
10-9
75 %
<v'>
<dv'/dt>
10-1
100
St
10
-11
10
101
-12
10
10
<v'>
<dv'/dt>
-9
-10
10-1
100
10
-10
10
101
-11
St
c
Figure 5. Vertical velocity fluctuations (v) and acceleration (dv/dt) on the epidermal surface
monitored 6D downstream from the center of the stenosis. a) Time series, b) frequency spectrum;
Frequency in Hertz (Hz) is estimated by assuming the heart beat rate to be 75 BPM (beat-per-min).
Vertical dashed lines indicate break-frequencies c) time-frequency spectrogram for | v|.
The frequency spectra of v and dv/dt are shown in Fig. 5b. The peak at
the origin represents the pulsation frequency (St=0.024) and this peak is followed
by a broad-band spectrum for St>0.1, which represents the bruit. For 50% case,
this broad-band spectrum is extends from St=0.1 to 1 but for the 75% case, the
amplitude of the broad-band spectrum is significantly higher and the frequency
14
range extends up to St~5. The vertical dashed lines indicate the breakfrequency[10] where the slope of spectrum changes significantly, and the
secondary peak is observed around the break-frequency in the acceleration
spectrum. The bruit spectrum for the acceleration is very similar to the in-vivo
measurement on the skin surface reported by Miller et al.[23]. Time-frequency
spectrograms of epidermal velocity fluctuation computed by a short-term-Fourier
transform[3] are also plotted in Fig. 5c for 50% and 75% cases and show the
intensity and frequency content of the arterial bruit with respect to the phase of
pulsation.
upper
lower
4
50%
- dF y /dt
0
-4
4
75%
0
-4
1
2
3
4
t/T
a
<v'>
10
-2
10
-8
10
10
-3
-6
75%
-7
50%
10
10
-1
10
75%
10
10
-6
-7
50%
<v'>
10
10-5
0
<dFy/dt>
10
-5
-9
10-8
10
-9
10-4
10
10
-10
10
-1
10
St
0
10
-10
Analytical
Computational
-5
-11
10
b
10
dF y/dt
v'
10
1
-11
10-1
c
100
101
St
Figure 6. a) Time-derivative of pressure force integrated along the stream wise direction at the
upper and lower surfaces of arterial wall. b) Comparison of bruit spectrum (v) and the spectrum of
time-derivative of the integrated pressure force (dFy/dt). c) The bruit spectrum as evaluated by Eq.
(13). The spectrum is plotted along with the present computational result.
15
Sound Source
The integrated pressure force in the y-direction (Fy,2D in Eq. 13, subscript 2D is
dropped hereafter) is calculated for the upper and lower walls of the flow domain
and its time derivative is plotted in Fig. 6a for the 50% and 75% cases. The
pressure forces integrated on the upper and lower wall are almost identical and
this supports our earlier assumption (used in deriving Eq. 13) that there is little
difference between the upper and lower wall pressures. The computed frequency
spectrum of the integrated pressure force is compared with the spectrum of
velocity fluctuation at the epidermal surface in Fig 6b and found to match very
well with the bruit spectrum for both cases. The bruit spectrum is also evaluated
analytically using Eq. (13) and plotted along with the present computational result
in Fig. 6c. Again, the two results agree very well not only for the shape but also
for the amplitude.
10-5
10-5
10
0
100
10-6
10-6
10-1
10
-3
10
-4
10
-5
10-8
10
10-2
<v'>
10
10
10-10
10-11
10-1
-9
10-11
101
b
10-4
v'
dF1/dt
dF2/dt
dF3/dt
10-10
100
St
a
10-8
10-3
-9
v'
dF1/dt
dF2/dt
dF3/dt
-7
<dF/dt>
10
-2
<dF/dt>
<v'>
10
10-1
-7
10-1
10
100
-5
101
St
Figure 7. Comparison of bruit spectrum (v) and the spectrum of time-derivative of integrated
pressure force for the segments of artery; F1: upstream region (x/D=0~10), F2: post-stenotic region
(x/D=10~20), and F3: further downstream region (x/D=20~30). a) 50% constriction and b) 75%
constriction.
To find the region of the flow most responsible for the generation of source,
the pressure integral in Eq. 13 is decomposed into three parts: i) the upstream
region (x/D=0~10), ii) near post-stenotic region (x/D=10~20), and iii) far poststenotic region (x/D=20~30); these are denoted by F1, F2, and F3, respectively.
The frequency spectra of the time-derivatives for each of the three force
16
components are plotted in Fig. 7 along with the bruit spectrum. For both the cases,
the bruit spectrum for St>0.1 coincides best with the spectrum of the force
component from the near post-stenotic region (x/D=10~20) which has an
amplitude that is about an order-of-magnitude higher than that for the upstream
component. For the 50% case, the force on the near and far post-stenotic region
are comparable for higher frequencies (St>0.3), while the force on the near poststenotic region is dominant throughout the frequency range St>0.1 for 75% case.
Discussion
In this study, an analysis of the computed results indicates that the epidermal
velocity fluctuations are correlated well with the time-derivative of the pressure
force on the lumen integrated over the near post-stenotic region. This supports the
view that the primary source of arterial bruits is the vortex inducted perturbations
in the near post-stenotic region.
In the previous 3D LES study of Mittal et al.[24] which also employed a
similar model, the maximum flow disturbance was found to be located near the
flow re-attachment region where the shear layer rolls up and breaks up into
vorticies. In the present simulations, we also find that the shear layer breaks and
rolls up into vorticies around 4-6D downstream from the stenosis, and the
maximum wall pressure fluctuation is observed at these locations.
The present computations show that the acoustic fluctuation induced by
the blood flow has a stronger intensity and higher frequency content for the higher
level of constriction. This tendency is in line with the experimental observation of
Borisyuk[8] and is mainly due to the fact that the jet velocity through the gap
below the stenosis is higher for the larger constriction (smaller gap). The high
frequency, high intensity components are important in auscultation, since they
will make the bruit more audible to human ears. For both cases, however, the
most energetic, high frequency components of the bruit are generated at (or near)
the phase corresponding to the maximum flow rate. This observation is in line
with the in-vivo study of Murgo[27] which addressed systolic ejection murmurs
from heart. Note that while past computational hemodynamic [13,24] and
experimental[19] studies have also found pressure fluctuations increasing with
constriction severity and have constructed a similar connection between
17
constriction and bruit intensity, the current coupled flow-acoustic model proves
this from first-principles.
The spectra, especially those corresponding to the epidermal acceleration,
are very much inline with the general characteristics of arterial bruit described in
Ref.[23]; the amplitude of spectrum slowly goes up to a discrete peak after which
the intensity falls off rapidly with increasing frequency. The present epidermal
acceleration spectrum shows good qualitative agreement with the in-vivo
measurement of Miller et al.[23]. The “break-frequency”[10] which is a wellknown characteristics of arterial bruits, is also observable in the present results.
The break-frequencies estimated from the Fig. 5b are St=0.68 for the 50% case
and St=2.44 for the 75% case. If the Strouhal number for the break frequency is
computed as St2 = fd/uj , where d is the stenotic diameter (d=D-b, in the present
study) and uj is the volume averaged peak jet velocity through the stenosis[18], it
yields St2 = 0.22 and 0.20 the for 50% and 75% cases, respectively. The two
values are not significantly different and therefore, the break frequencies
identified here scale well with uj and d. This observation also agrees with the
experimental studies of Jones & Fronek[18] and the large-eddy simulation study
of Mittal et al. [24]. It should be noted that the scaling of St2~(d/D)0.26 suggested
in Ref. [18] for a constricted pipe can be recast to St2~(d/D)0.13 for the present 2D
channel case, since the cross sectional area is linearly proportional to the diameter
of channel, and the present results indicate a scaling of St2~(d/D)0.138 which is
commensurate with the above scaling.
The wall pressure fluctuations has long been believed to be responsible for
the generation of arterial bruits[8,13,24]. In the previous study of Mittal et al.[24],
the strongest wall pressure fluctuations were observed in the location where the
shear layer and vortices interacted with the wall. The analytical evaluation of
sound source indicates that while bruits are connected with pressure fluctuations
(as expressed in Eq. (11)), the sound detected by a stethoscope results from the
integrated contributions from all locations in the lumen in the vicinity of the
stenosis. This is confirmed by the present computational results, which show that
the bruit spectrum coincides very well with the spectrum of the time-derivative of
integrated pressure force. Furthermore, the analytically evaluated bruit spectrum
agrees well with the computational result. Therefore, for the present canonical
18
configuration, it seems quite clear that the bruit sound is governed by the timederivative of integrated pressure force on the wall of the blood vessel.
Having established that the bruit is associated with the fluctuating force on
the vessel wall, the focus is turned towards determining the local region of the
flow that is most responsible for the bruit sound generation. The most widely
accepted notion in this context is that bruits are associated with the “disturbed”
flow and the associated pressure fluctuation in the post-stenotic region. We
investigated this issue by decomposing the total integrated pressure force and
found that for the 50% constriction case, both the near and far post stenotic
regions contribute equally to the bruit sound generation, but for 75% case, the
bruit mostly originates from the near-post stenotic region. From these
observations, we can conclude that the time-derivative of the integrated pressure
force on the post-stenotic region is the dominant source of the bruit and this
confirms the conjectures in some previous studies[8,13,24].
The present coupled hemodynamic-acoustic computational study enables
us to establish that arterial bruits from stenosed arteries are directly related with
the time-derivative of the integrated pressure force on the vessel wall, and that the
most dominant contribution to this force comes from the post-stenotic pressure
fluctuations that are caused by strong vortex motions and their interaction with the
wall. Although the present study is limited to two-dimensional analysis, the
source mechanism found in this study is not expected to change significantly for
more realistic three-dimensional turbulent flows. Fredberg[12] has shown that the
contribution of turbulence-associated wall pressure fluctuations is significantly
diminished by the integration along the streamwise direction, especially for the
higher frequencies associated with turbulence. The LES study of Mittal et al. [24]
also found that the wall pressure fluctuations were strongest in the near-post
stenotic region and were produced by the interaction of the shear layer with the
wall. Taking all of this into consideration, it may be concluded that the
contribution of turbulence to the bruit is very small and this supports the
conjecture of Bruns[9] .
We also note that other effects such as those due to the viscoelastic nature
of the arterial wall, shear wave propagation[31], and the presence of arterial
branches[20] downstream of the stenosis are not considered. Despite these
limitations, the current study provides a clear perspective on the generation of
19
bruits from stenosed arteries. Three dimensional effect and other mechanisms will
be considered in a future study. The current approach is also being applied to the
analysis of cardiac sounds including sounds associated with diastolic
dysfunction[33] and systolic murmurs[27].
Acknowledgement
This research is partially supported by the CDI program at NSF through grant IOS1124804. This work used the Extreme Science and Engineering Discovery Environment
(XSEDE), which is supported by NSF grant number TG-CTS100002.
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George Dotts