Download Flow through Collapsible Tubes at High Reynolds Numbers

988
Flow through Collapsible Tubes at High
Reynolds Numbers
CAROL K. LYON, JERRY B
SCOTT, DONALD K. ANDERSON, AND C.Y. W A N G
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SUMMARY The pressure-flow relationships of collapsible tubes were studied utilizing the Starling
resistor model. Reynolds numbers much higher than previously reported were used to simulate high
cardiac output states. Alterations which occur in vivo, including longitudinal tension, stretch, tubing
diameter, length, and outflow resistance were also simulated and systematically investigated. The
pressure-flow curves showed an initial rising phase, a plateau phase, as well as a late-rising phase
which has not been reported previously. Self-induced oscillations occurred during the plateau phase
and persisted throughout the late-rising phase. These perturbations were markedly increased by
longitudinal tension and stretch, but were attenuated by increased diameter, length, and outflow
pressure. These instabilities may prove to be an explanation for the "venoiia hum." Circ Res 49: 988996, 1981
VEINS are non-self-supporting structures that collapse when a critical transmural pressure is reached.
Collapse of a vein greatly increases its resistance to
flow, so that when collapse occurs, a non-linear
pressure-flow relationship results. Forty years ago,
Holt (1941) resorted to the use of a Starling resistor
model to investigate and demonstrate the mechanics of the non-linear pressure-flow relationship of
the inferior vena cava. His technique introduced
into vascular research the use of a physical model
which obviated many of the experimental difficulties encountered with in vivo venous research. The
Starling resistor model has subsequently been
adopted by many investigators to demonstrate the
principles underlying the pressure-flow relationships which they have observed for various collapsible vessels: Brecher et al. (1952), the superior vena
cava system; Swann et al. (1952), intra-renal venous
pressures; Doppman et al. (1966), radiographic appearance of vena cava collapse; Rodbard et al.
(1971), the vasculature of contracting muscle; Lopez-Muniz et al. (1968), the pulmonary circulation;
and Fung and Sobin (1972), the pulmonary microcirculation. For all of these investigators, physical
modeling has proven to be an important means of
quantifying pressure-flow relationships and investigating the mechanisms of vascular collapse.
In a previous paper (Lyon, Scott and Wang,
1980), we have shown that the flow properties of
collapsible vessels depend heavily on the Reynolds
number. This is a measure of the importance of the
From the Department of Physiology, Michigan State University, East
Laming, Michigan.
This investigation was supported in part by a grant from the Michigan
Heart Association, and by National Institutes of Health Grants HL10879
and HL243S3.
Presented in part at the 62nd Annual Meeting of Fed. Am. Soc. Eip.
Biol at Atlantic City, 197a
Address foi reprints: Jerry B Scott, Department of Physiology, Michigan State University, East Lansing, Michigan 48824
Received December 17, 1980: accepted for publication June 5, 1981.
inertial forces relative to the importance of viscous
forces in determining flow properties. In the microvasculature, the Reynolds number is much less than
one, and so viscous forces take predominance over
the inertial forces. However, in vessels of larger
diameter, such as arteries and veins, where the
Reynolds number is large, inertial forces greatly
influence the pressure-flow relationships.
The experiments of Conrad (1969), Katz et al.
(1969), Moreno et al. (1969), and our laboratory
(Lyon et al., 1980) show that the properties and
pressure-flow relationships of flow through collapsible vessels at high Reynolds numbers differ from
those of the "waterfall model" of Permutt et al.
(1962). The waterfall model predicts a total cessation of flow at low inflow pressures not observed by
the above researchers. Flow at high Reynolds numbers also shows marked unsteady self-induced oscillations which have a significant effect on the
pressure-flow relationships. The waterfall model as
originally presented by Permutt et al. (1962) was an
approximation of flow through the canine pulmonary micro vasculature. Such flow is described by
low Reynolds numbers. Our data (Lyon et al., 1980)
showed that when the Reynolds number is low
enough to simulate the microcirculation, flow properties of the waterfall model are achieved. This
result is indifferent to the mechanical properties of
the collapsible tube, tube size, or the method of
tube mounting.
The purpose of the present research is to study
flow through collapsible vessels with particular emphasis on the following areas which have not been
investigated previously:
1. Pressure-flow relationships at high flow rates
and Reynolds numbers. The maximum Reynolds
number for previously published experiments is less
than 1400. This may be adequate to model blood
flow in the vena cava of resting man (Reynolds
number = 900) and the dog (Reynolds number =
FLOW THROUGH COLLAPSIBLE TUBES: HIGH REYNOLDS NUMBERS/Ljyon et al.
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1375) (Cooney, 1976). However, cardiac output and,
consequently, the corresponding Reynolds number,
may increase to five or six times the resting level
during exercise (Wexler et al., 1968).
2, The effects on the pressure-flow relationships
of tubing diameter, length, tension, and stretch of
the conducting vessel. Both diameter and length of
collapsible veins vary greatly within the human
body. When the body is at rest, veins are subject to
constant longitudinal tension due to their anatomical tethering to organs within the body (Moreno et
al., 1970), but they must also stretch to accommodate body movement. Therefore, these parameters
were also systematically investigated.
3. Significance and quantification of the selfinduced oscillations of the tubing. Rodbard (1953)
proposed that such oscillations were the source of
arterial sounds. In consideration of the phenomena
of audible venous "hums" and palpable venous
"thrills" (vibrations) during conditions of high cardiac output, these osculations may be clinically
significant in the determination of venous blood
flow patterns.
Methods
In accord with previous investigations, we used
the "Starling resistor" to study the flow through
collapsible vessels. The experimental apparatus was
very similar to that used in a previous study (Lyon
et al., 1980). The physical model (Fig. 1) consisted
of flexible latex Penrose tubing (American Hospital
Supply) enclosed in an 8,000-ml airtight transparent
Plexiglas box. The Penrose tubing was 1.27 cm in
diameter when expanded, 7.5 cm long, and had a
wall thickness of 0.032 cm. Within the box, the
tubing was mounted at both ends upon rigid cylindrical metal tubes with an inner diameter of 1.27
cm and a wall thickness of 0.05 cm. The tubing was
mounted horizontally with no longitudinal stress
applied unless otherwise noted. Pressure taps for
the measurement of inflow pressure (P,) and outflow pressure (Po) were located outside the box on
the metal tubing, 8.5 cm from either end of the
collapsible tube. Pressure within the box external
to the tubing (P.) was modified by the use of a
pump connected by latex tubing to a port in the
box. The cannula used to measure P e was inserted
into the box through an airtight opening in the box.
Pe, Pi, and Po pressure measuring cannulas were
polyethylene tubing (PE 60), 80 cm long, filled with
distilled water. Each was connected to a separate
Statham low volume displacement transducer
(P23Gb, Hato Rey) which was coupled to a Hewlett-Packard (7796 model 1065C) direct-writing oscillograph and Esterline Angus X-Y recorder
(model 575). All pressure transducers were calibrated against a mercury manometer. The HewlettPackard electronic average was used for all pressure-flow measurements. Oscillations of the tubing
were quantified for pressure amplitude and fre-
RECORDER
989
X-Y PLOTTER
PUMP
DAMPER
ho
FLOW
METER
COLLAPSIBLE
TUBE
AIR-TIGHT BOX
PUMP
BASIN
FIGURE 1 A schematic diagram of the experimental
"Starling resistor" model which consisted of a collapsible latex Penrose tube mounted within an airtight Plexiglas box. Pressure ports for the measurements of inflow
pressure (PJ and outflow pressure (PJ were located just
outside the box on the mounting tubing. The pressure in
the box (Pe) was modified by a hand pump connected to
a port in the box. When water was pumped through the
collapsible tube, pressure measurements and flow (Q)
were charted by the recorder, and pressure differences
(P,-Pa and P*-PJ were computed and graphed by the XY plotter. A l.S-ml latex bulb was used to dampen the
pulse of the Po measurements.
quency by use of the Hewlett-Packard electronic
high filter setting and expansion of the time scale.
The pressure catheter-transducer-recorder system
was tested on high filter setting according to the
method of Glantz and Tyberg (1969) and found to
have an undampened natural frequency response of
33 Hz.
Resistances upstream and downstream to the
pressure ports (R, and Ro), were controlled by screw
clamps on latex tubing, 1.2 cm i.d. This latex tubing
opened above an outflow reservoir 8.5 cm distal to
the P o port and, when completely open, produced
minimal outflow resistance (approximately 1 mm
Hg/80 ml per sec). Ro was elevated only when
stated, by tightening the screw clamp on the outflow
tubing so that it formed a near elliptical crosssection, the size of which was recorded as the diameter of its inner minor axis.
Two of the components of the phenomenon of
stretch, the effects of longitudinal tension and the
effects of changes in vessel length, were investigated
independently. The length of the collapsible tubing
was taken as the measured separation of the mount-
CIRCULATION RESEARCH
990
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ing tubes within the airtight box. By definition,
longitudinal tension was the force tending to cause
longitudinal deformation (stretch) of the collapsible
tubing. To study the effects of longitudinal tension,
we applied tension to the tubing during the process
of mounting, and shall refer to this initial tension
as applied tension. The effects of longitudinal tension were isolated by keeping the separation of the
mounting tubes (length) constant but mounting the
collapsible tubing at varying levels of applied tension. On the other hand, the effects of vessel length
were isolated by decreasing the distance between
the mounting tubes in 1-cm decrements and
shortening a section of Penrose tubing accordingly
so that there was no applied tension on the collapsible tube. Finally, the simultaneous effects of applied tension and the resultant changes in length
(stretch) were investigated by mounting a 4.5-cm
length of tubing with no applied tension, then increasing the distance between the mounts by 0.5or 1.0-cm increments. This operation increased both
tension and length.
To determine the effect of changes in diameter,
Penrose tubing 0.635 cm in diameter and 3.25 cm
long, was mounted on rigid metal tubes that were
0.635 cm in diameter. This maintained the same
length: diameter ratio that had been used for the
larger (1.27 cm in diameter) Penrose tubing.
A pressure-dependent centrifugal pump (Little
Giant Pump Company) capable of delivering from
less than 1 ml/sec up to 80 ml/sec of virtually nonpulsatile flow was employed to recirculate room
temperature water through the system. The resistance R, served to modify the flow rate (Q). Q was
measured upstream to the P, port by a flow-through
flow probe connected to a Pulsed Logic Flowmeter
(Biotronex Laboratory, Inc.), also coupled to the
oscillograph and X-Y recorder. The zero flow set-
'0
20
VOL. 49, No. 4, OCTOBER 1981
ting was obtained by stopping the pump and waiting
for stabilization; calibration was obtained from 15second timed outflow collections that demonstrated
a linear response. Zero drift was less than 0.1 ml/
sec during the time necessary to plot a curve, and
the zero flow setting was checked immediately before initiating flow. Calibration checked at the end
of the experiments was found to vary less than 1%.
Because there were high amplitude fluctuations
of P o when the Penrose tube oscillated, a "Windkessel" chamber was used to dampen the pulse in
the P o measurements. It consisted of a 1.5-ml waterfilled latex bulb chamber connected in series by a
T-tube to the pressure transducer tubing at its
junction with the P o cannula. This compliance
chamber was tested at low amplitude fluctuations
(±10 mm Hg), and its use resulted in the same
mean pressure as when the Hewlett-Packard electronic mean was used alone. Changes of volume of
the Penrose tube resulted in minimal changes in P e
(maximum of ±1 mm Hg).
The method of experimentation used in this
study was to initiate a low flow rate through the
system. Pressure in the box was held at a predetermined level of P.-Po for each curve. All data points
were graphed immediately via the X-Y recorder,
which also served as an analog computer of P,-Po
and Pe-Po. The pressure-flow relationships were
obtained by plotting Q as a function of Pi-Po and
Pe-Po, as explained in a previous paper (Lyon et al.,
1980).
Results
A typical graph of the pressure-flow relationships
of Penrose tubing (1.27 cm in diameter, 7.5 cm long)
ia presented in Figure 2. For many of these curves,
there are three distinct phases: an initial sharp-
40
60
Q cc/sec
FIGURE 2 A typical graph of the pressure-flow relationships of the collapsible tube, 1.27 cm in diameter, 7.5 cm long.
For these curves, P.-Po was held constant at 50, 30, 20, 10, and 5 mm Hg, respectively. Maximum Reynolds number
achieved was 6,000. There are three distinct phases represented: an initial rising phase at low Q, a plateau phase at
moderate Q, and a late rising phase for Q > 20 ml/sec.
FLOW THROUGH COLLAPSIBLE TUBES: HIGH REYNOLDS NUMBERS/Z/yon et al.
991
1 The Effect of Pr-Po and Flow on Frequency and Pressure Amplitude of
Oscillations
TABLE
50 ml/sec
16 ml/sec
P.'Po
(mm Hg)
Amplitude
(mm Hg)
Frequency
(Hi)
P,-Po
(mmHg)
Amplitude
(mm Hg)
Frequency
(Hi)
P.Pe
(mm Hg)
0
10
30
50
0
1
1
1
0
5.5
10.5
14.5
1.4
110
31.5
50.7
31
16
1
1
5 25
7.0
10.25
13.0
11.0
16.8
33.6
52.9
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affected both the initial and late-rising phases. During the initial rising phase, the shortest (2.5 cm)
length of tubing was held partially open by the end
constraints, resulting in a very low pressure gradient
for flows less than 25 ml/sec. For this length, no
oscillations, only a four-lobed constriction, was
noted at all flow rate3. The effect of the end constraints was also apparent in the initial rising phase
of the curve for the 3.5-cm segment of tubing. The
pressure difference (P,-Po) of the late-rising phase
reached a maximum at a length of 5.5 cm, and for
lengths greater than 5.5 cm, the slope of the late
rising phase was less. Representative data for the
effect of vessel length on frequency and pressure
amplitude of oscillations is presented in Table 2.
Two sets of curves representing the effects of
longitudinal tension are shown in Figure 5. Applied
tension decreased the slope of the initial rising
phase, but increased the oscillations and the slope
of the late-rising phase. Representative data of the
effect of longitudinal tension on frequency and pressure amplitude of oscillations is presented in Table
3.
The effect of stretch was quantified by taking a
4.5 cm length of tubing, and stretching it to 5.5, 6.5,
7.5, 8.5, and 9 cm. Whereas, for the previous experiments, the pressure-flow relationships were extremely consistent, the results obtained for stretch
were less so, and indicated that the effects of length
rising phase, a plateau phase, and a late-rising phase
of lesser slope than the initial rising phase.
At flow rates of less than 6 ml/sec (the initial
rising phase), the tube was collapsed flat except for
two small round side channels. As flow approached
6-8 ml/sec, the flattened area gradually receded
until only the outflow region of the tube appeared
to be "pinched" closed, and the tube began to open
and close intermittently (the plateau phase). During
the plateau phase, P,-Po stabilized at approximately
the level of Pe-Po. At flows greater than 20 ml/sec,
the curves again began to rise. This late-rising phase
occurred at higher flow rates for higher Pe-Po. The
Reynolds number for this data, based on tubing
diameter, maximum flow rate, and average velocity
of flow, increased linearly with flow to a Reynolds
number of 6,000 at a flow rate of 60 ml/sec. The
amplitude of the pressure fluctuations caused by
the self-excited oscillations increased during the
late rising phase (Table 1).
With Pe-Po held constant, increasing R^ had the
effect of increasing the value of all pressure measurements. This outflow resistance reduced the frequency and the pressure amplitude of the oscillations, and decreased the slope of the late rising
phase. A graph of a representative experiment at
p e .p o = 10 mm Hg is shown in Figure 3.
An experiment using different lengths of tubing
resulted in the gTaph of Figure 4. Changes in length
40
X
E
E
I 20
4O
20
Q
60
CC/sec
3 A graph of a representative experiment using high, moderate, and no applied outflow resistance (RJ at
constant P.-Po = 16 mm Hg. Ro decreased the frequency and pressure amplitude of the self-induced oscillations, and
decreased the slope and the resistance of the late rising phase.
FIGURE
CIRCULATION RESEARCH
992
VOL. 49, No. 4, OCTOBER 1981
20
0
20
40
Q
60
cc/sec
4 Representative results of an experiment using different lengths of tubing at constant Pr-P,, = 16 mm Hg.
The shorter lengths of tubing were held partially open by the openings of the rigid mounting tubes, resulting in low
pressure gradients for flows < 25 ml/sec. The slope and the resistance of the late rising phase were maximal at a
length of 5xh cm. The slope of the late rising phase was decreased for lengths greater than 5V> cm.
FIGURE
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and tension were not additive. Therefore, a statistical analysis was undertaken (Table 4). At high
flow rates, the resistance increased as the tube was
stretched from 4.5 to 7.5 cm. When stretched to 7.5
cm, the tube existed in either of two distinct states:
a high resistance state of vigorous oscillations, or a
lower resistance state of decreased oscillations.
When stretched to 8.5 cm and 9 cm, the resistance
of the tube increased incrementally from its lower
resistance state at 7.5 cm. Analysis of variance was
performed on the raw data of the stable stretched
lengths. Tukey's multiple comparisons between all
stable means proved significant (P < 0.05). Table 5
summarizes the data obtained from a representative
experiment. It shows that stretch increased the
pressure amplitude of the oscillations.
The results for water perfusion of the smaller
Penrose tubing, 0.635 cm in diameter, 3.25 cm long,
demonstrated an interesting effect of tubing diameter (Fig. 6). Compared to the tubing that was 1.27
cm in diameter, the initial rising phase of the curves
was steeper, and the plateau phase began at lower
flow rates. Oscillations of the tubing were noted at
the beginning of the plateau phase, and they continued as flow was increased. The steep curve of the
late-rising phase also began at lower flow rates. The
maximum Reynolds number for water perfusion of
this smaller Penrose tubing for flow of 20 ml/sec
was 4,000.
TABLE 2
Discussion
The use of the Starling resistor model to describe
pressure-flow relationships avoids the biological inconsistencies that are encountered during in vivo
research of collapsible blood vessels. Because flow
rates and experimental pressures, especially tissue
pressure, are extremely difficult to maintain and
reproduce from one animal experiment to the next,
only a statistical quantification of these puzzling
relationships can be established in vivo. The Starling resistor model has proven to be a reliable tool
that can be used to quantify these relationships so
that the underlying mechanisms of blood flow
through collapsible vessels can be better understood.
For this research, we have used a segment of
Penrose tubing as the collapsible tube. The reasons
are as follows: (1) Penrose tubing has been used to
study the properties of collapsible vessels during
the past 40 years. Thus, its use facilitates the comparison of our data with previous results. (2) The
elastic properties of the Penrose tube do not change
with time, whereas in vitro vein segments perfused
with artificial medium deteriorate progressively. (3)
The experimental results using the Penrose tubing
are highly reproducible.
In recent careful experiments, Baird and Abbott
(1977) compared the elastic properties of various
The Effect of Vessel length on Frequency and Pressure Amplitude of
Oscillations (P<-Pa = 16 mm Hg)
75 ml/sec
27ml/»ec
Length
(cm)
Frequency
(Hz)
Amplitude
(mm Hg)
P.P.
(mm Hg)
Frequency
iHx)
Amplitude
(mm Hg)
2.5
4.5
5.5
6.5
7.5
10.5
13.5
0
0
0
0
1
3
1.5
2
2
16
18
18.5
19
18
17.5
0
11
12
12
11.5
8
9
0
7
26
24
25
28
16
0
0
8
8
8
8
PrPo
(mm Hg)
16
24
43
41
39
31.5
27
FLOW THROUGH COLLAPSIBLE TUBES: HIGH REYNOLDS NUMBERS/L>orc et al.
993
40r
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O
20
40
Q
60
cc/sec
FIGURE 5 The effect of longitudinal tension at constant Pr-P0 = 16 mm Hg. Penrose tubing with a manufactured
length of' 4h cm was placed on mounting tubes positioned 7M> cm apart. The curve generated by this tubing is labeled
4Vi + 3 cm, and should be compared to the curve labeled 7M>, the tubing of which had no applied tension. The curve
labeled l&h + 3 also demonstrates the effect of longitudinal tension when compared to that labeled 13lh. Applied
tension decreased the slope of the initial rising phase, but increased the elope of the late rising phase.
creases so that it is less likely to collapse. The slope
was increased by decreasing the transmural pressure or the length of the vessel. It is probable that
for small veins, which are less compliant because
their wall thickness-to-diameter ratio is greater, the
initial rising phase would be present.
During the plateau phase (P, > P e > Po), low
amplitude oscillations of the tubing occurred. In
this phase, resistance decreased remarkably as Q
increased. The use of increased R, served to decrease these oscillations and prolong the plateau
phase. Interestingly, the plateau phase always occurred at approximately Pi-Po = Pe-Po, even with
the use of various types of Penrose tubing. The
major effect of our experimental maneuvers upon
the plateau phase was to modify the flow rates
during which this phase occurred.
At higher flow rates, the experimental graphs
show a late rising phase not previously reported by
other researchers. It is characterized by a distinct
collapsible vessels. It was found that Penrose tubing
possesses a linear stress-strain relationship, whereas
that of veins is non-linear. However, the Young's
modulus (slope of the stress-strain curve) of the
Penrose tube closely approximates that of the polytetrafluoroethylene (PTFE) synthetic graft used
clinically, and also those of the femoral vein and
carotid artery at high strains. Thus the results of
our experiments apply qualitatively to in vivo situations.
The pressure-flow relationships of the Penrose
tubing Starling resistor model show that at low flow
rates, where P e > Pi > Po, the initial steep slope and
high resistance of the curves increased as the parameter Pe-Po increased. These data confirm the
findings of Conrad (1969), Katz et al. (1969), and
Moreno et al. (1969). The slope of the initial rising
phase was decreased by increasing longitudinal tension or stretch of the vessel. This occurred because,
when the vessel is pulled taut, its compliance de-
TABLE 3 The Effect of Applied Tension on Frequency and Pressure Amplitude of
Oscillations
P,-P. - 16 mm Hg
Manufactured
Experimental
length
length
(cm)
(cm)
13.5
10.5
10.5
7.5
7.5
4.5
13.5
13.5
10.5
10.5
7.5
7.5
Flow r»U - 75 ml/sec
Frequency
(Hi)
Amplitude
(mm Hgl
8.8
8.5
29
63
38
9.3
10.1
10.8
11.2
43-68
20-30
52-54
Pi-P.
(mm Hg)
27.0
300
31.0
415
33.0
64.5
CIRCULATION RESEARCH
994
VOL. 49, No. 4, OCTOBER
1981
TABLE 4 The Effect of Stretch on Resistance (n - 25)
P.-Pp - 16 mm Hg
Flow rate
m
74 ml/sec
Manufactured
length
(cm)
Experimental
length
(era)
Mean
Pi-P.
(mmHg)
P,-Po
(mm Hg)
Resistance
(mm Hg/ml per set)
4.5
4.5
45
4.5
45
4.5
25.9*
38.5*
56.1*
42.0*
43.9*
2.4
2.5
3.2
1 6
0.9
0.35
0.52
0.76
0.57
0.59
5.5
6.5
8.5
9.0
SD
• P < 0.05, Tukey's multiple comparison-
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stream influence cannot propagate upstream beyond the "pinched" segment of the collapsible tube.
This theory has been challenged recently by the
experiments of Conrad et al. (1978). Still another
method of analysis assumed a slowly varying crosssection of the tube, such that the viscous Poiseuille's
law can be applied to the "pinched" section of
tubing. This assumption was used by Rubinow and
Keller (1972) for the circular tube, Wild et al. (1977)
for the elliptic tube, and Brower and Noordergraaf
(1978) for the rectangular tube. However, only the
time-independent first and second phase can be
obtained by this method. Since these various theories axe based on conflicting principles and because
none predicts the late-rising phase, it appears that
the theory of flow in collapsible tubes at high Reynolds numbers remains controversial and incomplete.
Finally, we would like to comment on the physiological implications of our study. In all of our
experiments except one, low outflow resistance and
low outflow pressure were used. This is similar to
the hemodynamics of venous flow into the low
pressure system of the thorax and atria. The one
experiment in which outflow resistance was applied
to increase P o represents less physiological states,
such as heart failure, pericardial tamponade, or the
Valsalva maneuver. The collapsible tubing was connected to rigid mounting tubes which may be considered to represent the anatomical tethering that
tends to hold blood vessels open. Examples of anatomical tethering are the junctions of the venae
cavae and the right atrium. The non-collapsible
venous sinuses of the brain and the blood vessels
within the bones also serve as anatomical tethering
increase in oscillations as well as an increase in
slope. The oscillations have been noted by previous
investigators, but their systematic quantification
has remained elusive until now. In general, our
experiments show that the amplitude of the oscillations increases when there is an increase in flow,
tension, and/or stretch. The amplitude of the oscillations is decreased by increasing P e -P o or increasing Ro and PQ. The effect of tension, length, and
stretch on the frequency of the oscillations is variable.
The late-rising phase is more prominent with
experimental manipulations that increase the amplitude of the oscillations: low Pe-Po, low P o resulting from low Ro, increasing tension, and increasing
stretch. The slope of the late-rising phase represents
a relative constancy of resistance. Although we
cannot rule out the development of turbulence as a
mechanism for the late-rising phase, dye injections
failed to demonstrate this.
There have been several theoretical attempts to
describe the flow in collapsible tubes at high Reynolds numbers. Approximation of the system by
lumped parameters (resistors, capacitors, inductors,
and nonlinear elements) was done by Conrad
(1969), Kresch and Noordergraaf (1969), Katz et aL
(1969), and Mahrenholtz (1974). Some success was
obtained in the explanation of the early rising phase
and the oscillations of the plateau phase. A completely different approach was taken by Lambert
(1969), Griffiths (1975), Oates (1975), and Shapiro
(1977), who assumed essentially one-dimensional
inviscid flow. By using a compressible flow analogy,
they theorized a critical speed (possibly related to
the self-induced oscillations) above which down-
TABLE 5 The Effect of Stretch on Frequency and Pressure Amplitude of Oscillations
P,-P. - 16 mm Hg
Flow rau =• 7S ml/sec
Manufactured
length
(cm)
Experimental
length
(cm)
Frequency
(cm)
Amplitude
(mm Hg)
P.-P.
(mm Hg)
4.5
4.5
4.5
4.5
4.S
4.5
4.5
4.5
5.5
6.5
7.5
7.5
8.5
9.0
11
10.25
12.25
11.25
9.7
11
4.1
6-S
24-30
68
47
9-26
12-47
7-50
26.8
33.5
63
51.8
33.3
40
44.3
FLOW THROUGH COLLAPSIBLE TUBES: HIGH REYNOLDS NUMBERS/Lyonetal.
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7
E
E
a?
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20-
i
5
995
P20
P
_______O10
i
10
Q
15
20
25
cc/sec
FIGURE 6 The effect of tubing diameter. The diameter of this tubing was 0.636 cm, tubing length was 3.25 cm, and the
same length: diameter ratio was maintained as that of the larger collapsible tube shown in Figure 1. The pressureflow relationships were tested at Pr-Po of 40, 30, 20, and 10 mm Hg, respectively. The transitions between the three
phases occurred at much lower Q and the slopes were much steeper than those of the larger diameter tubing. At Q =
20 ml /sec, the Reynolds number was 4,000. The initial rising phase, the plateau phase and the late rising phase were
very distinct.
tending to hold open the veins that lead away from
them. An example of a venous system that is held
open at either end is the jugular-subclavian-vena
cava system.
The flow rates of the third phase are representative of flow through the great veins when cardiac
output is high. The self-induced oscillations may be
representative of the "venous hum" that is often
auscultated during high cardiac output states (Hardison, 1975). Probably the most commonly auscultated, although often unrecognized, venous hum is
produced by the cervical veins (Cutforth et al.,
1970). If we examine the dynamics of the production
of this sound, it can be seen that the parameters of
venous pressure and flow concur with those of the
late-rising phase of our study: This "murmur" is
best heard with the patient in the upright position
(low Po), and usually disappears when the patient
is recumbent, when light finger pressure is placed
over the vein at its entrance into the thorax, and
during a Valsalva maneuver (increased Po). Hardison (1975) reported that it ie loudest during diastole
and inspiration (decreased Po), and when the head
is extended and turned away from the examining
side (stretching of the vein being auscultated).
By analysis of the expected values of the various
pressures and flow in each of these sound-producing
situations, one finds that Pi > P e > Po, and P o is
very low or possibly negative. Therefore, it is not
unreasonable to expect that, in all of these instances, the walls of the veins may be oscillating or
vibrating as in our model. Cutforth et al. (1970)
devised a physical model to duplicate the production of the venous hum. Dye injections demonstrated turbulent flow when the hum was reproduced in the model, but his use of relatively noncompliant polyethylene tubing obviated the possibility of non-turbulent oscillations such as we observed with the collapsible Penrose tubing. However, the possibility that oscillatory movement of
the venous walls may be a factor in the production
of the cervical venous hum is given additional credence by the experimental observations of Brecher
et al. (1952). In their study of the effects of respiration on venous collapse of the canine superior
vena cava system, they observed "vibratory oscillations" of canine jugular veins, as well as of their
physical model.
In certain instances venous vibrations are also
palpable. A continuous venous "thrill" (palpable
vibratory phenomenon) accompanied by a "bruit"
(auscultatory phenomenon) is helpful in establishing the diagnosis of an arteriovenous fistula (Young
and deWolfe, 1978). A venous hum is indicative of
systemic pathology when heard over the paraumbilical veins (Ramakriflhnaii, 1978).
Interestingly, Cutforth et aL (1970) found that a
hum could be auscultated over the femoral veins
when individuals were tilted in a 45° head-down
position. Further study of this phenomenon in human subjects is now possible with the use of newer,
more sophisticated non-invasive clinical techniques.
CIRCULATION RESEARCH
996
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It is possible that a better understanding of the
femoral venous hum may lead to a simple bedside
method for monitoring femoral venous flow patterns, which have been shown to be disrupted in
the presence of thrombophlebitis (Young and
de Wolfe, 1978).
In conclusion, our studies have shown that the
pressure-flow relationships of collapsible tubes for
Reynolds numbers comparable to those of high
cardiac output states are markedly affected by instabilities of flow. These instabilities include oscillatory phenomena and higher pressure gradients than
are predicted by the waterfall model. Longitudinal
tension and stretch markedly increased these instabilities. Increased diameter, length, and outflow
pressure had the opposite effect. The pressure gradients were always greater than would be required
for flow through the same vessel in the non-collapsed state. These instabilities markedly increase
the resistance to flow through collapsible tubes and
may prove to be an explanation for the venous hum.
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Circ Res. 1981;49:988-996
doi: 10.1161/01.RES.49.4.988
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