Maria Siebes
Department of Biomedical Engineering,
The University of Iowa,
Iowa City, lA 52242
Charles S. Campbell
Department of Mecfianical Engineering.
David Z. D'Argenio
Department of Biomedical Engineering.
University of Southern California,
Los Angeles, CA 90089-1451
Fluid Dynamics of a Partially
Collapsible Stenosis in a Flow
Model of the Coronary
Circulation
The influence of passive vasomotion on the pressure drop-flow (AP-Q) characteristics of a partially compliant stenosis was studied in an in vitro model of the coronary
circulation. Twelve stenosis models of different severities (50 to 90 percent area
reduction) and degrees of flexible wall (0 to 2 of the wall circumference) were
inserted into thin-walled latex tubing and pressure and flow data were collected
during simulated cardiac cycles. In general, the pressure drop increased with increasing fraction of flexible wall for a given flow rate and stenosis severity. The magnitude
of this effect was directly dependent upon the underlying stenosis severity. The diastolic AP-Q relationship of severe, compliant models exhibited features of partial
collapse with an increase in pressure drop at a decreasing flow rate. It is concluded
that passive vasomotion of a normal wall segment at an eccentric stenosis in response
to periodic changes in intraluminal pressure causes dimensional changes in the
residual lumen area which can strongly affect the hemodynamic characteristics of
the stenosis during the cardiac cycle. This mechanism may have important implications for the onset of plaque fracture and the prediction of the functional significance
of a coronary stenosis based on quantitative angiogram analysis.
Introduction
The formation of an atherosclerotic plaque within the arterial
wall leads to a stenosis that progressively compromises the
vascular lumen and ultimately may cause malperfusion of the
down-stream tissue. Traditionally, researchers concerned with
blood flow through diseased arteries have considered these stenoses as rigidly fixed obstructions in a rigid vessel. On the basis
of this assumption, a fluid dynamic model has been developed
to describe the pressure drop-flow (AP-Q) relationship of an
arterial stenosis [49]. This model has been adapted to a practical
method for the quantitative analysis of coronary arteriograms
that is used to predict the fluid dynamic severity of a coronary
stenosis [14, 39, 47]. For this analysis, the absolute dimensions
of a coronary artery stenosis are determined from a selected
angiographic frame using computer-assisted image processing
methods. These geometric data then serve as input to the fluid
dynamic model that is used to predict the hemodynamic characteristics of the analyzed stenosis in terms of its AP-Q relationship.
Several pathological studies have shown that the majority of
coronary artery stenoses is eccentric and contains an arc of
normal wall circumference [12, 31, 45, 46]. There is mounting
experimental and clinical evidence that both active and passive
mechanisms may alter the shape of such a compliant coronary
stenosis and thereby influence its hemodynamic characteristics.
A number of in vitro studies in excised human [25] and canine
[16, 32] arteries have demonstrated that a compliant arterial
stenosis may show a paradoxical flow behavior when a decrease
in intraluminal pressure is elicited either directly by a change
in input pressure or indirectly through changes in downstream
' This work was supported in part by NIH grant #P41-RR01861, DOE grant
#DE-FG03-91ER14223, NSF Research Initiation Grant #BCS-9011761, and by
a grant from the Whitaker Foundation.
Contributed by the Bioengineering Division for publication in the JOURNAL
OF BiOMECHANiCAL ENGINEERING. Manuscript received by the Bioengineering
Division July 14, 1992; revised manuscript received Novemhier 5, 1995. Associate
Technical Editor; J. M. Tarbell.
resistance. In contrast to a rigid stenosis in a rigid vessel, these
partially flexible obstructions showed an increase in pressure
drop at a given flow rate for either of the above interventions,
i.e., stenosis resistance increased. Similar observations have
been described from in vivo studies [4, 5, 21, 34, 48].
This dynamic behavior of compliant stenoses has been related
to purely passive changes in stenotic dimensions in response to
changes in positive transmural pressure. Results from steadyflow experiments in hydraulic models employing a partially
flexible stenosis support this hypothesis [10, 16, 19]. In analogy
to the mechanisms involved in the collapse of veins and airways,
the concept of a Starling resistor, for which negative transmural
pressure leads to partial wall collapse, flutter, and flow limitation, has also entered the ongoing discussion of this paradoxical
arterial flow phenomenon [1, 2, 10, 21, 22].
The present study was undertaken to investigate in more
detail the effect of passive wall motion on the fluid dynamic
behavior of a stenosis with a compliant wall segment during
simulated cardiac cycles in an in vitro model of the coronary
circulation. Selected parameters of stenosis geometry were investigated with respect to their influence on the AP-Q relationship in a specially designed flow system which allowed the
reproduction of physiological coronary pressure and flow waveforms.
JMethods
Model Specification. In analogy to a hydraulic model developed by Mates and Gupta [28], the coronary circulation was
considered as a series combination of a pulsatile pressure source
representing the aortic input pressure, and a time-varying distal
resistance simulating the impediment of coronary flow during
myocardial contraction. While the distal coronary bed resistance
is distributed in nature and includes capacitive and inductive
elements due to arterial compliance and the inertia of blood
[18], good agreement has been achieved with a lumped, purely
resistive model for the purpose of describing the flow waveNOVEMBER 1996, Vol. 1 1 8 / 4 8 9
Journal of Biomechanical Engineering
Copyright © 1996 by ASME
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Table 1 Comparison of average parameters for the normal
coronary circulation and tlie in vitro model
Variable
Coronary
artery
In vitro
model
diameter, D [mm]
wall thickness, h [mm]
incr. elast. modulus, Ei„^ [10' Pa]
viscosity, p. [cP]
density, p [g/cm'j
input pressure, P [kPa] (mmHg)
sys/dias ratio
flow rate, Q [ml/min]
sys/dias ratio
distal resist., R [Pa/ml/min]
sys/dias ratio
cycle period, T [sec]
sys/dias ratio
frequency parameter, a
Reynolds number. Re
Euler number, Eu
nondira. resistance, R'
nondim. elasticity, E'
3.0
0.3
12.2*
3.6
1.06
13.1 (98)
1.16
240
0.25
95.4
4.62
0.8
0.67
2.281
500
38.5
85.2
359.1
6.35
0.3
10.82
3.55
1.24
13.1 (98)
1.16
424
0.25
53,1
4.62
4.25
0.67
2.281
500
207.3
473.4
811.5
• This value was estimated for normal intracoronary pressure [15].
Fig. 1 Synthesized input data for the unobstructed tubing. Vertical solid
and dashed lines indicate the beginning of systole and diastole, respectively.
forms in the epicardial arteries [20, 28]. Blood flow in the large
coronary arteries is, as a first approximation, assumed to be
quasi-steady and is determined by the instantaneous ratio of
distal coronary pressure to the lumped myocardial resistance.
Previous studies have shown that the quasi-steady assumption
is adequate for describing flow in stenosed coronary arteries in
vivo [20].
The coronary circulation was modeled in the maximally vasodilated state, since vasodilation occurs as a physiological response to compensate for the loss in perfusion pressure caused
by an arterial obstruction [17]. This implies that the pressureflow relationship is essentially linear over a wide range of flow
rates [17, 43 ] and that the periodic variations in distal resistance
remain constant regardless of changes in load caused by a stenosis. The effect of a stenosis is thus reflected by a pressure loss
and a decrease in flow. Typical coronary flow waveforms were
synthesized based on data available from the literature, assuming a mean flow rate under vasodilated conditions of 240 ml/
min with a mean systolic to mean diastolic flow ratio of 1:4
[8] and a ratio of the systolic to diastolic interval of 2:3 [11],
Based on a heart rate of 75 beats per minute (period T = 0.8
s), coronary artery diameter, Z), of 3 mm, blood viscosity, ji,
of 3.6 cP, and blood density, p, of 1.06 g/cm' the frequency
parameter ( a = Z)/2(27rp/r/x)"^) was calculated to be 2.281.
A set of dimensionless parameters describing geometric variables, inertial and viscous force ratios, and timing ratios of the
pressure and flow waveforms was used to scale the hydraulic
model. Figure 1 illustrates the scaled synthesized waveforms.
A comparison of coronary and in vitro flow model parameters
is provided in Table 1. Similitude to relevant parameters describing the flow conditions in a normal (unstenosed) epicardial
coronary artery was achieved by maintaining the instantaneous
Reynolds number (Re = Dpvljjb) and the frequency parameter.
Based on a model diameter of 0.635 cm, these requirements
were satisfied by a flow rate 2.1 times the coronary flow rate
and a period 5.3 times as long as the physiological heart beat.
A comparison of the values for the Euler number (Eu = P/pv^),
dimensionless wall elasticity parameter ( £ ' = £,„c/j/pu^D), and
non-dimensional resistance (/?' = RD^Ipv) indicates that these
parameters were not well simulated. However, both in the coronary artery and the in vitro test section, these numbers are much
larger than unity. For example, the Euler number in the coronary
equals 38.5, implying that dynamic pressure is only 2.6 percent
of the static pressure. In the model, these values are 207.3
and 0.5 percent, respectively. Hence, the dynamic pressure is
underestimated in the flow model, but since in the coronary
artery the relative contribution of the dynamic pressure is small,
the results from the model will not deviate much from those in
the normal coronary artery in this respect. Similar reasoning
applies to the dimensionless wall elasticity and resistance. Differences between these model and coronary parameters may
become more relevant at the stenosis where an increased flow
velocity would enhance the importance of dynamic pressure
variations. The magnitudes of the driving pressure and pulse
pressure were chosen based on the ability to achieve the desired
flow rates and taking into account the distensibiUty of the test
section and stenosis models. Considerations regarding the elastic properties of the wall are described in more detail subsequently.
Hydraulic Model. A schematic diagram of the experimental setup is shown in Fig. 2. It was designed as a recirculating
flow system with two reservoir tanks, two in-line regulating
valves (models SS6L-RS6 and SS-5PDF8, Whitey Co., Highlands Heights, OH), and a pump for fluid return. Dimensionally
stable plastic tubing (Nalgene, Nalge Co., Rochester, NY) with
/
COHPUTER-CONTROLLEDX P
»AUE ^
-^
f7=Xxc
Fig. 2
f
CDHPUTED-CONTHOaED
L
»«LVE
Vdllt
iJL
=0=0=
Schematic diagram of the recirculating flow system
490 / Vol. 118, NOVEMBER 1996
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Transactions of the ASME
an inner diameter of 6.35 mm (1.59 mm wall thickness) was
used to connect the components, with the exception of the flexible test section. No external pressure was applied to the test
section. The downstream end of the tubing was open to atmospheric pressure at the same height as the test section in order
to avoid suction effects. An overflow in the upper reservoir
provided a constant pressure head of 24 kPa (180 mm Hg).
Pressures and flow rates upstream and downstream of the test
section were measured with two cannulating electromagnetic
flow probes (6 mm inner diameter) that had been modified to
accommodate a pressure tap in close vicinity to the flow measuring coil. The pressure taps were connected to two pressure
transducers (models P23-Db and P23-BB, Gould-Statham Inc.,
Oxnard, CA), and the flow probes were connected to a twochannel flowmeter (model BL-610, Biotronix, Kensington,
MD). The valves were located 110 cm upstream and 53 cm
downstream of the test section to allow an entrance length sufficient for the development of laminar flow before the test section. Each valve stem was coupled to the shaft of a stepper motor
(model 23D-6103H, Rapidsyn, American Precision Industries,
Oceanside, C A). Movement of these motors was coordinated by
a two-axis bilevel driver and indexer (model DPF-38, Anaheim
Automation, Anaheim, CA) which was controlled via software
commands from a personal computer (model XT, IBM Coip.,
Boca Raton, FL). AH signals were ampUfled and sampled into
12 bits at 50 Hz after analog to digital conversion (model Labmaster TM40-PGL, Scientific Solutions, Inc., Solon, OH). Pressure and flow data were sampled for at least two consecutive
cycles.
A solution of 36 percent glycerin and 15 percent potassium
iodide in water, with a viscosity of 3.55 cP and a density of
1.24 g/cm' was used as the operating fluid. The kinematic
viscosity, i/, was measured with a calibrated capillary viscometer (Cannon-Fenske model 75-F276, Cannon Instrument Co.,
State College, PA) in a water bath at the controlled operating
temperature of the experimental fluid (28°C). The fluid density,
p, was found by weighing a known volume and the dynamic
viscosity, /i, was determined as fx = up.
Motor Control. Time-varying pressure and flow waveforms representative of an epicardial coronary artery were generated in the test section by controlled opening and closing of
the upstream and downstream valve. The digital nature of the
stepper motors required an open-loop control mode to achieve
the desired test conditions (constant periodic input pressure and
downstream resistance to simulate vasodilation) regardless of
the hydraulic load, i.e., the stenosis model. Appropriate motor
positions or valve orifice openings were determined under
steady flow conditions for each of 40 equally spaced times
during the synthesized cardiac cycle. The corresponding control
strings for the velocity, direction, and acceleration parameters
required to reach the next motor position during a given time
interval were then determined for each motor. They were stored
in a file for later continuous "playback" from the computer to
the motor driver in order to generate the desired pulsatile pressure and flow waveforms.
Stenosed Artery Model. The test section with a partially
compliant stenosis was created by placing an eccentric, rigid
obstruction in a thin-walled latex tubing of 6.35 mm nominal
diameter (Penrose drain tubing, Davol Inc., Cranston, RI).
Changes in the outer diameter of this tubing, fixed at a constant
length under 20 percent longitudinal tension, were measured
over a static pressure range from 0-200 mm Hg. The resulting
pressure-diameter relationship was linear (r = 0.997) with a
slope of 0.6 mm/100 mm Hg change in intraluminal pressure.
The incrementar elastic modulus, £,„c, of the latex tubing is
constant and matches that of coronary arteries at the diastolic
range of normal physiological pressures [29]. The wall modulus of elasticity was 10.82 X 10' Pa, which is close to the value
of 12.2 X 10' Pa estimated for the incremental elastic modulus
Journal of Biomechanical Engineering
i flevibk
no f'txibl
Fig. 3 Plexiglas plaque models before insertion into the tubing. The
letters a, b, and c refer to the 50, 80, and 90 percent area stenosis,
respectively. These models were glued into the latex tubing (not shown
here) to form a partially compliant stenosis with 0 (rigid), j , j , and ~
flexible wall circumference.
of human coronary arteries in vivo [15]. The complex material
properties of the arterial wall which are highly nonlinear and
include viscoelastic elements are difficult to model. For a thinwalled tube the stiffness of the vessel wall is more directly
described in terms of volume distensibility C = 0.15DIEi„Ji,
which is inversely proportional to the wave speed [6, 44],
Under normal pressure conditions for which the elastic moduli
are similar, this parameter is larger for the latex tubing by
a factor roughly equivalent to the ratio of model-to-coronary
diameter. However, due to the nonlinearity of the arterial stressstrain relationship the coronary wall becomes increasingly more
compliant at lower pressures [9], thereby matching or possibly
exceeding the distensibility of the latex tubing. For the purposes
of this study the linear elastic properties and distensibility of
this tubing were therefore considered appropriate to demonstrate
the effect of a partially flexible wall segment on the pressure
drop-flow relationship of an eccentric stenosis where intraluminal pressure is greatly reduced.
Twelve plaque models were fabricated from plexiglas. All
models were blunt-ended and 12.7 mm long, with a length-todiameter ratio LID = 2. Three arcs of flexible wall remaining
after insertion of the plaque model (j, 5, and 5 of the circumference) were chosen for each of three percent area reductions
(50, 80, and 90 percent, referenced to a nominal diameter of
6.35 mm). Rigid, concentric models of corresponding area reduction served as controls for the partially compliant models.
The different desired degrees of flexible wall for a given area
reduction resulted in various cross-sectional shapes of these
plaque models, which are shown in Fig. 3. Table 2 lists the
geometric dimensions of these plaque models. The models are
referenced in the text by their severity (nominal percent area
stenosis) and the fraction of flexible wall circumference, e.g.,
'50-3' refers to an eccentric 50 percent stenosis with one third
flexible wall circumference, while '90-0' refers to a rigid concentric 90 percent stenosis.
The plaque models were fastened to the inside of the latex
tubing by applying a thin layer of epoxy glue to the "rigid"
part of the plaque circumference and inserting it into the tubing,
which had previously been turned inside out for about half of
its length. Since the diameter of the unpressurized latex tubing
was less than 6.35 mm, the plaque models were inserted by
slightly stretching the tubing. Based on the pressure-diameter
relationship the actual resulting stenosis severities were estimated to correspond to a 52, 84, and 95 percent area reduction.
The tubing was then slipped back over the stenosis and the glue
was allowed to dry for 24 hours. The final length of the tubing
NOVEMBER 1996, Vol. 1 1 8 / 4 9 1
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Table 2 Nominal geometry of stenosis models. Percent
area stenosis and circumference of the residual area are
referenced to the nominal diameter D = 6.35 mm. The
length-to-diameter ratio was LID = 2
Model
Fraction
of
flexible
wall
Plaque
area
[mm^]
% Area
stenosis
Circumference
of residual
area [mm]
0-1
50-0
50-4
50-3
50-2
80-0
80-4
80-3
80-2
90-0
90-4
90-3
90-2
1/1
0
1/4
1/3
1/2
0
1/4
1/3
1/2
0
1/4
1/3
1/2
0
15.78
15.82
15.87
15.84
25.43
25.33
25.48
25.35
28.74
28.79
28.77
28.77
0
49.87
49.96
50.10
50.00
80.28
79.99
80.45
80.03
90.76
90.92
90.84
90.84
19.95
14.12
14.23
14.47
16.32
8.86
10.17
12.15
18.00
6.06
9.48
12.52
19.04
Nq StgnosI?
~
_
i /^\
if \
°' / \
°^. 1
] 3
\
\
1 1
i
'\ I I
\
was 16 cm, with the stenosis located 7 cm from the beginning.
The flexible test section was then mounted between the ends
of the rigid tubing under 20 percent longitudinal stretch to prevent it from sagging.
Protocol. The existence of fully developed flow was verified in preliminary experiments by the presence of a linear
pressure drop-flow relationship for the unobstructed test section
over the flow range used in this study. After careful calibration
of all instruments, experiments were carried out for each of the
twelve stenosed test sections and for the unobstructed flexible
tubing. The previously determined motion control parameters
for each stenosis model were continuously supplied by the computer to the motor controller. Data acquisition was performed
in "background" mode without disrupting control of the motor
movements. After conversion of the recorded data into physical
pressure and flow units the instantaneous pressure loss and
downstream resistance were calculated. The sampled and derived variables were plotted as a function of time. In addition,
the pressure drop was plotted as a function of the upstream
flow rate. Systolic, diastolic, and overall time-averages were
calculated for each variable and compared between models.
Results
Typical coronary pressure and flow waveforms could be well
reproduced with this hydraulic system, as is illustrated in Fig.
4 for the unobstructed flexible tubing. The average input pulse
pressure was 5.5 kPa (41 mmHg) and the flow waveform
showed an inverse relationship to the pressure typical for the
coronary circulation, exhibiting a minimum at maximum systolic pressure and being highest during the diastolic pressure
decay.
Examples of pressure and flow recordings are shown in Figs.
5 - 7 for the 50-2, 80-2, and 90-2 compliant stenosis models,
respectively. The pulsatile flow waveform displayed extensive
changes with increasing stenosis severity and compliance. Flow
became generaUy less pulsatile, which was mainly due to a
disproportionate fall in diastolic volume flow. For severe, compliant stenoses (models 90-3 and 90-2) mean systolic flow was
larger than diastolic flow, in contrast to the dominant diastolic
flow for all other models. Capacitive effects, as indicated by
the ratio of upstream {Q\) and downstream {Q2) flow rate, were
small and were mainly restricted to the systolic time period.
The average systolic ratio of 22/fii was 0.89 with a minimum
of 0.69 occurring for the 90-2 model, indicating a larger flow
into than out of the stenosis during systole. During diastole the
average flow ratio Q^IQx was 1.02, with a maximum of 1.28
1 1 1 1 '1
1 1 1 1 1 1
1 1 i
1 1 1 1 1
Time (sec)
Fig. 4 Measured upstream (1) and downstream (2) pressure (P) and
flow rate (Q) for the unobstructed flexible tubing. Vertical solid and
dashed tines indicate the beginning of systole and diastole, respectively.
again occurring for the 90-2 model. For the severe, compliant
stenosis models, the mean flow rate was reduced to less than
20 percent of the flow rate through the unobstructed tube.
For a given stenosis compliance, a progressive fall in the
mean flow rate was obtained with increasing stenosis severity,
which was predominantly due to a disproportionate decrease in
flow during diastole (Fig. 8). In addition, for a given stenosis
severity, mean flow generally fell with increasing degree of
compliance. As compared to the corresponding rigid model,
mean flow for the most flexible model decreased by 12, 34, and
SO-2 Stenosis
o
Time (sec;
Fig. 5 IVIeasured upstream (1) and downstream (2) pressure (P) and
flow rate (Q) for the 50-2 model. Vertical solid and dashed lines indicate
the beginning of systole and diastole, respectively.
492 / V o l . 118, NOVEMBER 1996
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Transactions of the ASIVIE
80-2 Stenosis
Average Flow
800 -1
-0-
diastole
-•-
overall
systole
400 -
1
-
i
200-
i
Q1
Q2
~
;
•
O
\
if
'
CO Co (D 00
C3 O C3 O
Ol O) O) 0>
Stenosis Model
w/
0
oo oY o*? o^
CS O Q O
to lo lo in
\
\ ^
Fig. 8 Average flow rates for all stenosis models
, , 1 , , , 1 ' ' ' 1 ' ' ' i ' ' '
2
4
6
8
10
Time (sec)
Fig. 6 IMeasured upstream (1) and downstream (2) pressure (P) and
flow rate (Q) for the 80-2 model. Vertical solid and dashed lines indicate
the beginning of systole and diastole, respectively.
80 percent for the 50, 80, and 90 percent stenoses, respectively.
This suggests that the effect of a flexible wall segment at the
stenosis increased with the underlying stenosis severity.
Pressure loss across the stenotic section was also mainly a
diastolic phenomenon (Fig. 9). The mean pressure drop increased with stenosis severity from about 0.7 kPa (5 mmHg)
to 3.8 kPa (28 mmHg) for the rigid models. Pressure losses
were clearly augmented by the existence of a partially flexible
wall. As compared to the corresponding rigid model, an additional mean pressure gradient of about 0.1 kPa (1 mmHg) re-
9n.2 stenosis
suited for the 50-2 model, which increased to about 0.8 kPa (6
mmHg) and 4.8 kPa (36 mmHg) for the 80-2 and 90-2 models,
respectively.
These changes in the mean and the pulsatile nature of the
pressure and flow were reflected in the instantaneous AP-Q
relationship of the stenosis models. The AP-Q relationships for
the rigid and most compliant models of the 50, 80, and 90
percent area stenoses are shown in Figs. 10-12. The AP-Q
relationships for models of intermediate compliance {\ and |
flexible wall circumference) lie between these curves and have
been omitted for clarity. The pressure drop, A P , is plotted
versus the upstream flow rate, Q\, since this is the flow that
can be measured in vivo, as opposed to the downstream flow
rate, Q2, which is generally inaccessible. Overall, a higher stenosis severity caused an increase in the slope of the AP-Q
relationship, reflecting higher pressure losses for a given flow
rate. Increasing the degree of flexible wall circumference for a
given stenosis severity had only a small effect on the AP-Q
characteristics of the 50 percent models (Fig. 10), while it
substantially changed the fluid dynamics of the 80 and 90 percent models (Figs. 11 and 12). The maximum attainable dia-
Averaqe Pressure Drop
-0—
diastole
-%—
overall
—£x— systole
1
1
Q1
CO
Q.
I
U^
1
a
<
o
X
/''"*'••
'
f
1
1
1 'l
1
1 1
'
1 1 1 1
1 -I I -r
ec-^'-ir^
Time (sec)
10 to 10 10
Fig. 7 Measured upstream (1) and downstream (2) pressure (P) and
flow rate (Q) for the 90-2 model. Vertical solid and dashed lines indicate
the beginning of systole and diastole, respectively.
Journal of Biomechanical Engineering
o Y m py
ci o ci ci
CO
CO
03
CO
O Y (*3 CM
6 o t i C5
o} O) o en
stenosis Model
Fig. 9 Average pressure losses for all stenosis models
NOVEMBER 1996, Vol. 118/493
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90% Stenosis
50% Stenosis
1-90
0
start syst
•
start diast
9
start syst
A
start (jiast
-60
8 -
9
i
a
*
6-
^//
<
-30
50-0
<«^_
—I
0
f
,
200
1
1
400
1
600
r—
SOO
Q, (ml/min)
0
200
400
600
800
Q, (ml/min)
Fig. 10 Pressure drop-flow relationship of the 50-0 rigid (dotted line)
and 50-2 compliant (solid line) models. The symbols marl< the start of
systole ( • ) and diastole (A). The open arrows Indicate the time course
during the flow cycle. The mid-diastolic period is marked by a thick line.
The larger degree of flexible wall circumference reduced the maximum
flow rate and Increased the diastolic pressure gradient.
stolic flow rate was markedly reduced for these compliant models, which also exhibited a greatly enhanced pressure drop,
thereby shifting the AP-Q curves up and to the left.
For all rigid models as well as for the compliant 50 percent
models, the existence of unsteady inertial effects during periods
of rapid flow changes was indicated by an "opening" of the
AP-Q relationship into a clockwise loop. Additional pressure
losses during late systolic and early diastolic flow acceleration
caused the ascending limb of the AP-Q loop to be higher than
the descending limb representing late diastolic flow deceleration. In accordance with a fall in flow pulsatility, either due to a
higher stenosis severity for the rigid models or due to increasing
compliance the contribution of these unsteady inertial pressure
losses became smaller and the AP-Q loops became more and
more closed.
In contrast, the AP-Q loops for the compliant severe models
opened again, although the flow waveform was extremely
damped (c.f. Fig. 7). The time course of these loops was
counter-clockwise, i.e., there was less pressure loss during late
0
Start syst
A
start diast
90% Compliant Stenoses
, / » , 90-2
/M ^ ,
1
6 -
0
systolic flow acceleration than during late diastolic flow deceleration (Figs. 11 and 12). Figure 13 shows the AP-Q loops for
the compliant 90 percent stenosis models at an expanded flow
scale together with several theoretical curves for rigid stenoses
of increasing severity. The slope of the diastolic AP-Q relationship became increasingly steeper and was even reversed for the
90-3 and 90-2 models (solid arrows), i.e., flow decreased with
a concomitant increase in pressure drop. For these models, a
small flow elevation in mid-systole was accompanied by a large
fall in pressure drop. During mid-diastole, the pressure loss
rapidly increased by up to 10.7 kPa (80 mmHg) for the 90-2
model while the flow, after an initial small increase, fell by
about 30 ml/min. This reversal in the direction of the APQ loops is consistent with partial systolic wall expansion and
diastolic collapse of the flexible wall segment at the stenosis,
thereby causing a higher stenosis severity during diastole than
during systole. During the course of a single cardiac cycle, the
AP-Q relationships of these compliant models cross several
theoretical AP-Q curves for rigid models of different severities.
Inflation and deflation of the stenotic section and the tubing
90-3
8P% Stsnpgls
^.
Fig. 12 Pressure drop-flow relationships of the 90-0 rigid (dotted line)
and 90-2 compliant (solid line) stenosis models. Increasing the degree
of flexible wall circumference substantially worsened the fluid dynamic
characteristics. (See Fig. 10 for explanation of symbols.)
200
400
600
300
Q, (ml/min)
Fig. 11 Pressure drop-flow relationships of the 80-0 rigid (solid line)
and 80-2 compliant (dotted line) models. Increasing the degree of flexible
wall circumference caused a steeper slope and reversed the direction
of the AP-Q relationship. (See Fig. 10 for explanation of symbols.)
Fig. 13 Pressure drop-flow relationships of the compliant 90 percent
area stenosis models 90-4,90-3, and 90-2. The thin lines show theoretical
relationships for fixed stenoses of increasing severity. The solid arrows
mark a mid-diastolic reversal in slope, featuring a decreasing flow with
increasing pressure drop. (See Fig. 10 for explanation of symbols.)
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downstream of the stenosis was visible for the more severe
compliant models during the flow cycle. No self-excited oscillations or total collapse were observed during the course of these
experiments.
Discussion
The influence of two geometric parameters, stenosis severity
and arc of flexible wall at a stenosis, was investigated to address
the issue of the hemodynamic significance of passive vasomotion of a normal wall segment opposite an eccentric plaque in
an epicardial coronary vessel. Although a relatively simple,
purely resistive flow system was employed to model the coronary circulation, the results indicate that the main features of
coronary pressure and flow waveforms could be adequately
reproduced. Similitude to significant parameters describing flow
conditions in the normal coronary artery was maintained by
matching the instantaneous Reynolds number and the frequency
parameter. In the stenotic section, the Euler number and dimensionless wall elasticity may become relevant (approach unity).
However, the contribution of dynamic pressure (pv^) is underestimated in the flow model and thus, the measured changes in
pressure drop are also likely to be underestimated compared to
a stenosed coronary artery. This suggests that elastic effects due
to pressure reduction by convective acceleration at the stenosis
may be more accentuated in vivo.
Several assumptions were made in modeling blood flow in a
stenosed epicardial coronary vessel. A fluid with constant viscosity was used instead of blood which is a suspension of
formed elements and exhibits non-Newtonian behavior. However, at shear rates greater than 200 s"' the viscosity of blood
(hematocrit 44 percent) is constant and independent of the capillary radius (r > 0.1 mm) [7]. Considering the relatively high
shear rates in a stenosed artery and the fact that even for the
most severe stenosis models the minimum hydraulic diameter
was larger than 0.2 mm, treating blood as a homogeneous,
Newtonian fluid appears a reasonable assumption for the purposes of this study.
A plexiglas plug in latex tubing leaving a partially flexible
wall segment was chosen to model a coronary artery stenosis.
While this model is consistent with the predominantly eccentric
nature of most coronary stenoses, it does not reproduce the
nonlinear viscoelastic properties of the arterial wall. As outlined
before, the distensibility of the model tubing was higher than
that of a coronary artery at normal physiological pressures.
However, the tube wall was most likely less flexible than the
coronary wall at the greatly reduced pressures within the stenosis, in which case our results would tend to underestimate the
corresponding effect in vivo. The chosen model therefore appears adequate for the purpose of this study which focussed
on changes in the AP-Q characteristics caused by a partially
compliant stenosis. Our results are consistent with observations
of other investigators that have demonstrated an increased resistance of a flexible eccentric coronary stenosis as a result of low
perfusion pressures [25, 35] which was contributed to partial
collapse of the wall at the stenotic segment.
The blunt-ended configuration of the models used in this
study is not considered to be a significant factor, since the
detailed entrance and exit configuration of a stenosis mainly
affects viscous losses, which are small compared to exit losses
for higher flow rates or severe stenoses for which the total
pressure drop is primarily a function of the minimum area [28,
36]. Our model also assumed a completely rigid underlying
obstruction whereas atherosclerotic plaques may deform under
pressure [3, 23]. Future work may be carried out using, for
example, a "soft," pliable plaque model.
The rigid tubes used in the in vitro setup provided some
structural support to the ends of the compliant tubing which
may have had an effect on the results. However, the length of
the downstream compliant section was sufficient (10 cm) to
Journal of Biomechanical Engineering
allow dimensional changes including complete collapse. This
was only observed during preliminary steady flow tests at distal
resistances lower than those used in the pulsatile experiments
reported here. It is therefore unlikely that the results seen in
this study would have qualitatively been much different with a
longer downstream flexible section. Tethering of the epicardial
coronary arteries to the surface of the heart was not modeled,
with the exception of longitudinal stretch. Connective tissue
pressure is minimal for epicardial coronary arteries whose outer
surface is surrounded by fluid with external pressures close to
zero [27].
Although myocardial function is independent of perfusion
pressure in the presence of adequate blood flow [41], pressure
losses of the magnitude demonstrated across the most severe
stenoses in this in vitro study may not be sustainable by the in
vivo coronary circulation due to the much reduced blood flow.
However, pressure losses reported in our study are comparable
to those observed by Gould et al. [13] in the stenosed left
circumflex coronary artery of chronically instrumented awake
dogs. To test whether the mechanisms demonstrated in this
study occur to a similar magnitude in humans would require
pressure-flow measurements at elevated flow rates (maximal
vasodilation) in patients with a similar stenosis.
Despite these limitations, the present findings provide useful
insight into the hemodynamic changes that may occur chnically
due to passive vasomotion of a normal wall segment at an
eccentric plaque in a coronary artery during vasodilation. The
changes in the pressure and flow waveforms observed with
increasing severity of a fixed stenosis are consistent with those
described by Gould et al. [13 ]. Phasic changes in flow decreased
with increasing stenosis severity and the flow pattern was reversed for the most severe models with systolic flow being
higher than diastolic flow. In contrast, phasic changes in the
pressure gradient were enhanced, with consequential changes
in the AP-Q relationship. As in their in vivo studies, unsteady
inertial effects of rapid flow acceleration and deceleration were
reflected as an open loop in the AP-Q presentation, extending
in a clockwise direction from mid-systole in the lower left to
mid-diastole in the upper right of the curve.
A higher wall flexibility for a given stenosis severity generally produced an increase in stenosis resistance. The maximum
attainable flow rate decreased while the pressure loss increased.
The corresponding AP-Q relationships became steeper and
shifted upwards and to the left. Some discrepancy from this
trend existed for the 80-3 and 90-3 models, which was most
likely due to the difficulty in maintaining the exact desired
dimensions when gluing the plaque models into the latex tubing.
However, the external boundary conditions (time-varying input
pressure and downstream resistance) were similar for all models, and the observed changes in the fluid dynamic characteristics thus reflected the differences in stenosis configuration.
The pressure drop across a stenosis for a given flow rate is
dominated by the inverse fourth power of the minimum diameter, which leads to a higher sensitivity of the fluid dynamic
characteristics of more severe stenoses to small changes in lumenal dimensions [36, 47, 49]. Our results showed that the
upward displacement of the AP-Q relationship with increasing
wall compliance became more pronounced with increasing stenosis severity, indicating that a longer arc of compliant wafl
circumference produced a larger adverse effect on the fluid
dynamics of more severe stenoses. An alternate reason for the
increase in pressure drop for a given flow rate could be the
shape of the residual area. The desired constant plaque area at
various degrees of flexible wall for each stenosis severity resulted in an increasing deviation from a circular cross-section
for the residual area. In addition, the increase in flexible waU
circumference created a progressively more eccentric stenosis.
While both of these geometric variables (eccentricity and noncircular cross-section) have been shown to increase the pressure
loss across a rigid stenosis, their relative effect decreased with
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increasing stenosis severity [40, 50] and was negligible for
severe stenoses [28, 36].
Moreover, an additional pressure drop due to any fixed geometric parameter should have had a proportional effect at all
flow rates and thus should have merely shifted the whole APQ relationship up and to the left. Instead, the resulting curves
for the severe compliant models indicate a disproportionate increase in the diastolic pressure drop. An example is given for
the 90-4 stenosis model (Fig. 13). Shortly after the beginning
of diastole, near maximum flow, the pressure drop rises
abruptly, with little change in flow. This causes the mid-diastolic
limb of the AP-Q relationship to be shifted upwards and to the
left, resulting in a counter-clockwise orientation of the loop, in
contrast to the clockwise direction of the less severe and/or
less compliant models. This behavior was seen for all compliant
severe models which demonstrated a pressure loss that, for the
same flow rate, was higher during mid-to end-diastole (after
maximum flow had been reached) than during early diastole
(flow acceleration). As illustrated in Fig. 13, the resulting APQ loops moved across a family of curves, each representing a
different fixed stenosis severity. The results of the present study
therefore suggest that the geometric characteristics of a partially
compliant stenosis are continuously changing during the cardiac
cycle. This effect cannot be caused by a fixed stenosis geometry,
regardless of the cross-sectional shape. We therefore consider
a narrowing of the stenotic lumen area during diastole due to
partial collapse of the flexible wall segment to be responsible
for the worsened hemodynamic effect.
Several experimental studies have shown that a partially compliant stenosis may act as a flow-limiting resistor, i.e., during
interventions that reduce the downstream resistance flow choking or even reduction is observed despite an increase in the
driving pressure [2, 10, 19, 21, 25, 33, 42]. This paradoxical
flow behavior has been described by Shapiro [38] for flow past
a pressurizing external cuff around a thin-walled tube. In this
situation the flow through the throat of the stenosis may become
supercritical, i.e., the local flow velocity becomes larger than
the local wave speed, causing flow limitation. These events are
associated with partial collapse of the flexible wall segment in
response to the loss of intraluminal pressure within the throat
of a stenosis due to conservation of energy (Bernoulli principle), until an equilibrium between the wall forces and the distending pressure is reached. Stenotic dimensions then remain
stable until the distending pressure changes. Due to the pulsatile
nature of the simulated coronary flow waveforms in the present
study, flow limitation was restricted to the diastolic period of
the simulated cardiac cycles, during which the downstream pressure and resistance were lowest. Features indicative of flow
limitation were present in the AP-Q curves of the 80-3, 80-2,
90-4, 90-3, and 90-2 models. They are most evident for the 903 and 90-2 models, for which flow actually decreased during
diastole while the pressure drop increased markedly at the same
time (solid arrows in Fig. 13). The maximum flow rate was
reached despite a continued decrease in downstream pressure,
which is consistent with flow limitation due to partial collapse.
In contrast, flow during mid-systole increased with a concomitant decrease in the pressure drop, which is indicative of a
passive expansion of the stenosis lumen area. This behavior is
similar to that observed by Judd and Mates [19] in steady flow
experiments involving an eccentric, partially flexible stenosis.
Binns et al. [2] reported systolic wall collapse downstream of
a concentric rigid stenosis for pulsatile flow through a model
of a stenosed systemic (carotid) artery. Under physiological
conditions, arterial collapse may occur due to transition to supercritical flow at the throat of a concentric stenosis with associated negative transmural pressure [22]. The likelihood of collapse in stenotic arteries is further enhanced in the presence of
an eccentric plaque [1],
Based on the results of this study, it is concluded that partially
compliant coronary artery stenoses may undergo periodic geo-
metric changes during the cardiac cycle in passive response to
pulsatile changes in intraluminal distending pressure. For severe
stenoses with a flexible wall segment, the diastolic fall in distal
resistance results in substantial changes in the pressure dropflow relationship, which does no longer follow a single quadratic (curvilinear) relation and exhibits features that are consistent with partial collapse of the compliant wall segment during
diastole. This behavior may well be amplified for in vivo stenoses due to the enhanced compliance of the arterial wall at low
intraluminal pressures.
Clinical Implications. The findings of this study may have
important implications for the in vivo situation. Periodic motion
of the normal wall circumference opposite an eccentric plaque
may lead to Assuring or tearing of the intima at the junction
between atherosclerotic plaque and normal wall with subsequent
embolization or local thrombus formation [1, 2, 22]. Areas of
local stress concentration have been shown to exist at this junction, which is particularly vulnerable to cyclic loading [24, 26,
30].
These results also point to potential problems regarding the
evaluation of functional stenosis severity from coronary angiograms. Gould et al. [14] have shown that well established fluid
dynamic equations can be used to adequately predict the diastolic AP-Q relationship of a coronary stenosis in vivo for a
given vasoactive state. The necessary stenotic dimensions in
this widely used application are supplied by the quantitative
analysis of a mid-diastolic frame selected from the angiographic
x-ray film. Their conclusions were based on data obtained in
unsedated dogs with a fixed coronary stenosis that exhibited a
single curvilinear diastolic AP-Q relationship. In contrast, the
results of our study show that the fluid dynamics of a partially
compliant stenosis may change continuously during the diastolic
part of the cardiac cycle due to a passive change in the stenotic
lumen area. A diastolic worsening of the minimum stenosis
diameter has been demonstrated in diseased human coronary
arteries by quantitative analysis of sequential angiographic
frames over two cardiac cycles [ 37 ]. In light of the high prevalence of partially compliant, eccentric stenoses in coronary arteries, it should therefore be considered that the validity of fluid
dynamic predictions based on angiographic dimensions may be
limited to the stenosis size present in the particular angiographic
frame that was selected for quantitative analysis. Furthermore,
angiogram-based extrapolations regarding the hemodynamic
characteristics of that stenosis over a wide range of flow rates
should be done with caution due to the passive mobility of the
flexible wall segment. Partial collapse at elevated flow rates
could lead to substantial differences between the predicted and
the actual AP-Q relationship.
References
1 Aoki, T. and Ku, D. N., "Collapse of Diseased Arteries With Eccentric
Cross Section," J. Biomechanics, 26;133-142, 1993.
2 Binns, R. L. and Ku, D. N., "Effect of Stenosis on Wall Motion. A
Possible Mechanism of Stroke and Transient Ischemic Attack." Arteriosclerosis,
9:842-847, 1989.
3 Born, G. V. R. and Richardson, P. D., ' 'Mechanical Properties of Human
Atherosclerotic Lesions,'' in S. Glagov, W. P. Newman, and S. A. Schaffa, editors,
Pathobiology of the Human Atherosclerotic Plaque, New York, 1990. Springer
Veriag, pp. 413-423.
4 Bove, A. A., Santamore, W. P., and Carey, R. A., "Reduced Myocardial
Blood Flow Resulting From Dynamic Changes in Coronary Artery Stenosis," Int
J Cardiol, 4:301-313, 1983.
5 Brown, B. G., Bolson, E. L., and Dodge, H. T., "Dynamic Mechanisms
in Human Coronary Stenosis," Circulation, 70:917-922, 1984.
6 Caro, C. G., Pedley, T. J., Schroter, R. C , and Seed, W. A., The Mechanics
of the Circulation, chapter 12, Oxford University Press, New York, 1978, pp.
278-279,
7 Cerny, L. C , Cook, F. B., and Walker, C. C , "Rheology of Blood," Am
J Physiol, 202:1188-1194, 1962.
8 Chilian, W. M. and Marcus, M. L.,' 'Effects of Coronary and Extravascular
Pressure on Intramyocardial and Epicardial Blood Velocity," Am J Physiol,
248:H170-H178, 1985.
496 / Vol. 118, NOVEMBER 1996
Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 05/27/2015 Terms of Use: http://asme.org/terms
Transactions of the ASME
9 Cox, R. H., "Mechanical Aspects of Large Coronary Arteries," Santamore, W. P., and Bove, A. A., editors. Coronary Artery Disease, Baltimore,
Munich, 1982, Urban & Schwarzenberg, pp. 19-39.
10 Dubill, P. M. and Young, D. P., "Steady Flow Through Models of Compliant Stenoses," Proc. 39th ACEMB, 28:298, 1986.
11 Folkow, B., and Neil, E., Circulation, Oxford University Press, New York,
1971.
12 Freudenberg, H. and Lichtlen, P. R., "The Normal Wall Segment in Coronary Stenoses—A Post-Mortem Study," Z Kardiologie, 70:863-869, 1981.
13 Gould, K. L., "Pressure-Flow Characteristics of Coronary Stenoses in
Unsedated Dogs at Rest and During Coronary Vasodilation," Circ Res, 43:242253, 1978.
14 Gould, K. L., Kelley, K. O., and Bolson, E. L., "Experimental Validation
of Quantitative Coronary Arteriography for Determining Pressure-Flow Characteristics of Coronary Stenosis," Circulation, 66:930-937, 1982.
15 Gow, B. S. and Hadheld, C. D., "The Elasticity of Canine and Human
Coronary Arteries With Reference to Postmortem Changes," Circ Res, 45:588594, 1979.
16 Higgins, D. R., Santamore, W. P., Bove, A. A., and Nemir, P., Jr.,' 'Mechanism for Dynamic Changes in Stenotic Severity," Am J Physiol, 249:H293H299, 1985.
17 Hoffman, J. I. E., "Maximal Coronary Flow and the Concept of Coronary
Vascular Reserve," Circulation, 70:153-159, 1984.
18 Hoffman, J. I. E. and Spaan, J. A. E., "Pressure-Flow Relations in Coronary Circulation," Phys Rev. 70:331-390, 1990.
19 Judd, R. M., and Mates, R. E., "Pressure-Flow Relationships in Partially
Occluded Flexible Tubes," Proc. Sympos. on Biofluid Mechanics and Biorheology. New York, 1989, ASME, pp. 581-590.
20 Kirkeeide, R. L.,' 'Mechanics of Blood Flow Through Normal and Stenotic
Coronary Arteries," PhD thesis, Iowa State University, 1978. Microfilm 7903988.
21 Kirkeeide, R. L., Gould, K. L., and Kelley, K. O., "Fluid Dynamic Behaviour of Severe Coronary Stenoses: Classic Rigid Stenosis or Collapsible Starling
Resistor?" R. E. Mates, R. M. Nerem, and P. D. Stein, eds.. Mechanics of the
Coronary Circulation, New York, 1983, The American Society of Mechanical
Engineers, pp. 91-94.
22 Ku, D. N., Zeigler, M. N., and Downing, J. M., "One-Dimensional Steady
Inviscid Flow Through a Stenotic Collapsible Tube,'' ASME JOURNAL OF BIOMECHANICAL ENGINEERING, 112:444-450, 1990.
23 Lee, R. T., Grodzinsky, A. L., Frank, E. H., Kamm, R. D., and Schoen,
F. J.,' 'Structure-Dependent Dynamic Mechanical Behavior of Fibrous Caps From
Human Atherosclerotic Plaques," Circulation, 83:1764-1770, 1991.
24 Lee, R. T., Loree, H. M., Cheng, G. C , Lieberman, E. H., Jaramillo, N.,
and Schoen, F. J., "Computational Structural Analysis Based on Intravascular
Ultrasound Imaging Before In Vitro Angioplasty: Prediction of Plaque Fracture
Locations," J Am Coll Cardiol, IV.lll-l&l,
1993.
25 Logan, S. E., "On the Fluid Mechanics of Human Coronary Artery Stenosis," IEEE Trans Biomed Eng, BME-22:327-334, 1975.
26 Loree, H. M., Kamm, R. D., Stringfellow, R. G., and Lee, R. T., "Effects
of Fibrous Cap Thickness on Peak Circumferential Stress in Model Atherosclerotic
Vessels," Circ Res, 71:850-858, 1992.
27 Lorell, B. H. and Braunwald, E., "Pericardial Disease," Heart Disease,
Philadelphia, 1992. W. B. Saunders, pp. 1465-1516.
28 Mates, R. E., Gupta, R. L., Bell, A. C , and Klocke, F. J.,' 'Fluid Dynamics
of Coronary Artery Stenosis," Circ Res, 42:152-162, 1978.
29 Papageorgiou, G. L. and Jones, N. B., "Physical Modeling of the Arterial
Wall. Part 1: Testing of Tubes of Various Materials," J Biomed Eng, 9:153156, 1987.
Journal of Biomechanical Engineering
30 Richardson, P. D., Davies, M. J., and Born, G. V. R., ' 'Influence of Plaque
Configuration and Stress Distribution on Fissuring of Coronary Atherosclerotic
Plaques," Lancet, 2:941-944, 1989.
31 Saner, H. E., Gobel, F. L., Salomonowitz, E., Erlten, D. A., and Edwards,
J. E.,' 'The Disease-Free Wall in Coronary Atherosclerosis: Its Relation to Degree
of Obstruction," J Am Coll Cardiol, 6:1096-1099, 1985.
32 Santamore, W. P., Bove, A. A., and Carey, R. A., "Hemodynamics of a
Stenosis in a Compliant Artery," Cardiology, 69:1-10, 1982.
33 Schwartz, J. S., "Effect of Distal Coronary Pressure on Rigid and Compliant Coronary Stenoses," Am J Physiol, 245:H1054-H1060, 1983.
34 Schwartz, J. S., Carlyle, P. F., and Cohn, J. N., "Effect of Dilatation of
the Distal Coronary Bed on Flow and Resistance in Severely Stenotic Arteries in
Dog," Am J Cardiol, 43:219-224, 1979.
35 Schwartz, J. S., Carlyle, P. F., and Cohn, J. N., "Effect of Coronary
Arterial Pressure on Coronary Stenosis Resistance," Circulation, 61:70-76,1980.
36 Seeley, B. D. and Young, D. F., "Effect of Geometry on Pressure Losses
Across Models of Arterial Stenoses," J Biomechanics, 9:439-448, 1976.
37 Selzer, R. H., Siebes, M., Hagerty, C , Azen, S. P., Lee, P. L., Blankenhom,
D. H., and Shircore, A., "Effects of Cardiac Phase on Diameter Measurements
from Coronary Cineangiograms," Computers in Cardiology 1988, Los Alamitos,
CA, 1989. IEEE Computer Society, pp. 363-366.
38 Shapiro, A. H., "Steady Flow in Collapsible Tubes," ASME JOURNAL OF
BIOMECHANICAL ENGINEERING, 9:126-147, 1977.
39 Siebes, M., D'Argenio, D. Z., and Selzer, R. H., "Computer Assessment
of Hemodynamic Severity of Coronary Artery Stenosis From Angiograms,'' Comp
Meth Progr Biomed, 21:143-152, 1985.
40 Siebes, M., Tjajahdi, I., Gottwik, M., and Schlepper, M., "Influence of
Ellipitcal and Eccentric Area Reduction on the Pressure Drop Across Stenoses,"
Z Kardiologie, 73:59, 1984.
41 Smalling, R. W., Kelley, K., Kirkeeide, R. L., and Fisher, D. J., "Regional
Myocardial Function is Not Affected by Severe Coronary Depressurization Provided Coronary Blood Flow is Maintained," J Am Coll Cardiol, 5:948-955,
1985.
42 Stergiopulos, N., Moore, Jr., J. E., Strassle, A., Ku, D. N., and Meister,
J.-J.,' 'Steady Flow Tests and Demonstration of Collapse on Models of Comphant
Axisymmetric Stenoses," ASME 1993 Advances in Bioengineering, BED26:455-458, 1993.
43 Van Huis, G. A., Sipkema, P., and Westerhof, N., "Instantaneous and
Steady-State Pressure Flow Relations of the Coronary System in the Canine
Beating Heart," Cardiovasc Res, 19:121-131, 1985.
44 Vatner, S. F., Pasipoularides, A., and Mirsky, I., "Measurement of Arterial
Pressure-Dimension Relationships in Conscious Animals," Ann Biomed Eng,
12:521-534, 1984.
45 Vlodaver, Z. and Edwards, J. W., "Pathology of Coronary Atherosclerosis," Prog Cardiovasc Dis, 14:256-274, 1971.
46 Waller, B. F., "The Eccentric Coronary Atherosclerotic Plaque: Morphologic Observations and Clinical Relevance," Clin Cardiol, 12:14-20, 1989.
47 Wong, W., Kirkeeide, R. L., and Gould, K. L., ' 'Computer Applications
in Angiography," Collins S. M. and Skorton D. J., editors. Cardiac Imaging And
Image Processing, New York, 1986. McGraw-Hill, pp. 206-238.
48 Yasue, H., Takizawa, A., Nagao, M., Nishida, S., Horie, M., Kubota, J.,
and Fujii, H., "Pathogenesis of Angina Pectoris in Patients With One-Vessel
Disease: Possible Role of Dynamic Coronary Obstruction," Am Heart J, 112:263272, 1986.
49 Young, D. P., "Fluid Mechanics of Arterial Stenoses," ASME JOURNAL
OF BIOMECHANICAL ENGINEERING, 101:157-175, 1979.
50 Young, D. F. and Tsai, F. Y., "Flow Characteristics in Models of Arterial
Stenoses—^I. Steady Flow," J Biomechanics, 6:395-410, 1973.
NOVEMBER 1996, Vol. 118/497
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