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Wall shear rate measurements in an elastic curved artery model
Biorheology, Vol. 14, No. 1, pp. l-17.1997 Copyright 0 1997 Fkvicr Science Ltd Printed in the USA. All rights reserved ooo6s55x/97 $17.00 + .oo
PIISOOO6355X(97)OOOO1-2
WALL SHEAR RATE MEASUREMENTS CURVED ARTERY MODEL M. W. WESTON Physiological Engineering, USA
IN
AN
ELASTIC
AND J. M. TARBELL
Transport Studies Laboratory, The Pennsylvania State University,
Department of Chemical University Park, Pennsylvania,
Reprint requests to: John M. Tarbell, Physiological Transport Studies Laboratory, Department of Chemical Engineering, The Pennsylvania State University, 155 Fenske Laboratory, University Park, PA 168024400 USA; Fax: (814) 865-7846; e-mail: [email protected]
ABSTRACT Since atherosclerotic lesions tend to be localized at bends and branching points, knowledge of wall shear rate patterns in models of these geometries may help elucidate the mechanism of atherogenesis. This study uses the photochromic method of flow visualization to determine both the mean and amplitude of the wall shear rate waveform in straight and curved elastic arterial models to demonstrate the effects of curvature, elasticity, and the phase angle between the flow and pressure waveforms (impedance phase angle). Under sinusoidal flow conditions characteristic of large arteries, the mean shear rate at the inner wall of the curved tube is reduced 40-56% from its steady flow value, depending on the phase angle. Wall shear rate amplitudes in the curved tube are significantly reduced by wall motion (3655% of the Womersley amplitude for a straight rigid tube). The shear rate amplitude at the outer wall decreases 30% as the phase angle is reduced from -20” to -66’, while the shear rate amplitude at the inner wall increases 45%. As a result, the oscillatory nature of flow at the outer waLl decreases with decreasing negative phase angle, but flow at the inner wall becomes much more oscillatory. At large negative phase angles, characteristic of hypertension or vasoactive agents, the shear rate at the inner wall has a small mean and cycles through positive and negative values; the shear rate at the outer wall remains positive throughout the flow cycle. Thus, the impedance phase angle could affect atherogenesis along the inner wall if temporal and directional changes in wall shear rate play a role. 0 1997~~~irrsdnrc1ld
Introduction
Individual risk factors such as diet, fitness, and heredity play a major role in the development of atherosclerosis. Recent studies though, have found that atherosclerotic lesions tend to be localized at branching points and bends in
KEYWORDS:
Wall shear rate; photochromic
phase
angle;
I
curvature;
elastic;
2
WaUshear rate in a curved artery model
Vol. 34, No. 1
the circulation regardless of the underlying risk factors involved (see review by Nerem, 1995). Fox and Seed (1981) reported the distribution of early atheroma in human coronary arteries and noted that fatty plaques were localized predominantly at the inner wall of the curvature, where the vessel contacts the curved surface of the heart. Asakura and Karino (1990) visualized both steady and pulsatile flow in human coronary arteries rendered transparent through postmortem treatment. Sites of atherosclerosis experienced low flow velocities, recirculation zones, or strong secondary flows, all of which may be indicative of low wall shear stress. Low wall shear has been linked to atherosclerosis in the aortic (Friedman et al., 1981) and carotid (Ku et al., 1985) bifurcations and in end-to-side (Rittgers et al., 1978; LoGerfo et aZ., 1983) and end-to-end (Binns et al., 1989) vascular grafts. Other studies indicate that temporal and directional variations in shear rate may contribute to atherogenesis (Walden et al., 1980; Ku et aZ., 1985; Okadome et al., 1989). A uniform curved tube (Fig. 1) is a simple model of a curved artery, such as the aortic arch or the coronary arteries. The first theoretical considerations of fully developed steady flow behavior in a rigid curved tube far from the entrance were made by Dean (1927, 1928). Curvature has a significant effect on flow, inducing a symmetric pair of secondary flow vortices (Dean vortices) that sweep flow toward the outer wall, causing the peak axial velocity to occur closer to the outer wall, not at the centerline as in straight tube flow. The intensity of curved tube flows is characterized by a modified Reynolds number (Re) referred to as the Dean number, Dn (E Re / h112, where 3Lis the aspect ratio, Wall shear measurements under steady flow conditions R/ r). (Dn = 139 - 2868) in a rigid curved pipe have been reported by Choi et al. (1979). Near the tube entrance, shear stress was circumferentially uniform at low Dn but became higher near the inner wall for Dn above 400. However, by the exit of the curved tube (El = 180”), the shear rate at the outer wall was roughly 2.5 times the value predicted for Poiseuille flow (yp), while the shear rate at the inner wall was close to yr. This skewing of the wall shear rate Outer
wall
Fig. 1. Schematic drawing of the curved tube geometry. A tube of radius r is curved through a radius of curvature R, the degree of curvature is described by the aspect ratio h = R / r. In this model, flow enters at 8 = 0” and exits at 180”. The inner wall refers to the wall closest to the center of curvature:
the outer wall is farther
from the center.
Vol. 34, No. 1
Wall shear rate in a curved
artery model
3
distribution, observed at all Dn, results from the progression of the secondary flow. Theoretical work by Lyne (1970) demonstrated a transition from two to cour vortex secondary flow at elevated frequencies in oscillatory flow. Combining numerical simulations with experiments, Lin and Tarbell (1980) found a resonance condition between the sinusoidal flow frequency and the frequency of the secondary flow rotation that produced elevated peak shear stresses at the outer wall and attenuated shear stresses at the inner wall. Physiological pulsatile flow in a rigid curved tube simulating the aortic arch was studied experimentally by Chandran and Year-wood (1981) and numerically by Chang and Tarbell (1985). Both observed high peak shear rates at the inner wall but also significant flow reversal, resulting in time-averaged mean shear rates that were lower than at the outer wall. Chang and Tarbell (1988) reported similar results for simulations of coronary artery conditions. The temporal phase angle between the pressure and flow waveforms (impedance phase angle) may also affect the wall shear rate distribution. If the pressure and flow waveforms are resolved into Fourier harmonics, then the impedance phase angle of each harmonic can be determined. Under physiological flow conditions, the impedance phase angles of the lower harmonics are usually negative, and flow leads pressure. A non-zero impedance phase angle is caused by wave reflections in the circulation from branching points, bends, and the high resistance of peripheral vessels. Typically, the physiologic value of the phase angle of the first harmonic of the input impedance (measured in the ascending aorta) is near -45”. However, hypertensive people have more negative phase angles (Merillon et al., 1982), while vasoactive drugs may change the phase angle in either direction, depending on the drug used (White et al., 1994). A relationship between phase angle and wall shear was first suggested by Klanchar et al. (1990). They measured wall shear stress at the exit (180”) of a curved elastic tube under large artery flow conditions using hot-film anemometry. Peak shear rates were lower at the inner wall of the curve than at the outer wall, but mean shear rates could not be determined by hot-film anemometry due to local flow reversal during part of the cycle. Peak wall shear rate was insensitive to phase angle for values above -6O”, but as the impedance phase angle decreased from -60” to -BOO, the peak shear rate at both walls increased significantly. However, they did not measure the local flow rate, which probably also varied with phase angle. Recently developed theories for straight elastic tubes predict that by increasing the phase angle (toward 0”). the mean shear rate will increase and the shear rate amplitude will decrease (Wang and Tarbell, 1992; 1995). These theories have been supported by Rhee and Tarbell (1994) and Weston et al. (1996), who used the photochromic method to study the wall shear rate waveform in end-to-end anastomosis models. By administering different vasoactive drugs in dogs, White et al. (1994) found that the wall shear rate waveform in viuo is sensitive to changes in the phase angle, but their pressure and flow waveforms varied significantly in each experiment. In the present study, more recent experimental techniques were used to clarify the results of Klanchar et al. (1990). The local flow rate was measured accurately at any site in the elastic artery model using ultrasound. The photochromic method, a relatively new technique for determining wall shear rate, allowed reversed flow to be accurately quantified so that both the mean and the amplitude of the wall shear rate waveform could be determined. The complete wall shear rate waveforms on the inner and outer walls at the exit of an elastic curved tube were determined. The experiments were repeated with the tube straightened to verily the accuracy of the technique by comparison
4
Wall shear rate in a curued artery model
Vol. 34, No. 1
with theory (Wang and Tarbell, 1992; 1995) and to demonstrate the effect of curvature on the wall shear rate waveform. In both of these configurations, the impedance phase angle was varied over a large enough range to evaluate its physiological influence. Methods The photochromic method, originally developed by Popovich and Hummel (1967), has been used to visualize flow in suaight and curved elastic tube models. A photochromic substance changes color through an isomerization reaction when it is excited by light of an appropriate wavelength. Conversely, when the light is removed, the color gradually disappears. In flow visualization, the photochromic material is dissolved in a host fluid. By focusing a laser light source of an appropriate wavelength on a flowing fluid at the location of interest, a color change occurs in the thin path of the beam. As flow proceeds, this thin path of excited particles moves with the flow, indicating their displacement from their initial location. This technique has been used recently for flow visualization in rigid arterial models (Ojha et al., 1988), and for wall shear rate measurements in rigid (Ojha, 1994) and elastic models (Rhee and Tarbell, 1994; Weston et al., 1996). The flow loop used in this study (Fig. 2) is similar to that of Rhee and Tarbell (1994). The test fluid was heated to 45’C in a water bath and circulated around the closed flow loop by a peristaltic roller pump (Masterflex 7019, Cole Parmer). A coil of acrylic tubing was used to damp out flow perturbations from the peristaltic pumping, producing a steady flow. Sinusoidal flow could then be imposed through a piston pump (1423, Harvard Apparatus). The stroke volume was adjusted to produce a diameter variation [ (dMm - dMIN) / dAVG] of lo%, while the beat frequency was set at 1 Hz to simulate the cardiac pulse. While not truly physiologic, sinusoidal flow was used so that the phase angle of only one harmonic was present, allowing its effect to be isolated. Flow from the piston pump passed through 152 cm of flexible latex tubing, 1.9 cm ID, (Primeline Industries), which matched the diameter and compliance of the arterial model to ensure that the flow entering the model was fully developed. Flow exited the model through an additional 30.5 cm of the same tubing to allow for wave propagation. Distal to this tubing, another 472 cm of the same
piston
Pump
Ir
IEi
Static fluid column Roller
1
DumD
Ultrasonic flow meter
-ml
I
Catheter-tip pressure transducer
I
Tem’peratur< bath Distal latex tubing
Proximal latex tubing
tube model
Fig. 2 The flow loop, shown here for the curved tube experiments, allows both steady and sinusoidal flow to be studied. The impedance phase angle is adjusted by altering the downstream resistance.
Vol. 34, No. 1
Wall shear rate in a curued artery
model
5
tubing and a vertical section of stiff Tygon tubing along with an adjustable screw clamp, were used to vary the downstream impedance and phase angle. Flow rate and pressure measurements were taken within one diameter distal to the wall shear rate measurement site. The flow rate was measured locally by an ultrasonic flow meter (TlOl, Transonic Systems)and the flow probe did not compress the tube during any part of the flow cycle. The pressure was measured by a catheter-tip transducer (TCD 500, Millar Instruments). Both the flow and pressure signals were recorded using an analog-to-digital data acquisition unit (500A, Keithley) and software (Soft500, Keithley) installed in a personal computer (PS2, IBM). A fast Fourier transform (Asyst, MacMillan Software) was performed on the flow and pressure waveforms to resolve them into harmonics of sine waves with a base frequency (0). Since a simple sinusoidal pressure waveform drives the flow, only the first harmonics were significant. From the Fourier analysis, the phase difference between the pressure and flow waveforms (the impedance phase angle) could be calculated. The phase angle was varied through the length of the downstream latex section and the position of the adjustable screw clamp which increased the resistance by compressing (but not constricting) the diameter available to flow. More positive phase angles were achieved by clamping farther from the model along the 472 cm latex section placed before the vertical Tygon tubing column. More negative impedance phase angles were achieved when the latex section was removed and the clamp was placed on the column. Impedance phase angles between -70” and +lO” could be obtained. The operating conditions used in each experiment are listed in Table 1. The straight elastic tube was made in several steps. A 1.9 cm ID polyvinyl chloride (PVC) pipe (L = 49.3 cm) was cut in half lengthwise. The pieces were taped together and mounted inside a vise to hold their alignment. Melted polyethylene glycol wax (PEG Compound 20M, Union Carbide, melting point 60°C) was then poured into the pipe. After cooling, the pipe halves were separated from the wax rod. MetaI caps (cap ID = 1.9 cm, end post OD = 0.32 cm) were placed on the ends of the wax rod to allow it to be held in a rotating spit. A transparent silicone elastomer (Sylgard 184, Dow Corning, index of refraction 1.43) was poured onto the rotating wax rod and cured for 9 hr below the melting point of the wax. After curing, the rod was soaked in boiling water to melt the wax, leaving behind the flexible silicone rubber tubing with a wall thickness of about 0.05 cm. The straight tube was mounted inside an optical correction chamber. This chamber was filled with the test fluid to eliminate any optical distortions caused by the tube curvature, providing an accurate view of the dye lines (see Rhee and Tarbell, 1994 or Weston et al., 1996). The side walls of the chamber had “windows” of thin slide glass to allow the laser beam to passthrough into the tube. Small supports on the bottom of the chamber kept the tube level, and thin-walled metal connectors (OD = 1.9 cm, L = 1.9 cm) joined the model tubing to the latex tubing on the proximal and distal sides. The wave reflection caused by these short connectors was small (Rbee and Tarbell, 1994). For the curved elastic tube experiments, the same silicone rubber tube was bent into a curved configuration and mounted inside of another optical correction chamber. The radius of curvature of the tube was fixed by the chamber design at 9.85 cm, giving an aspect ratio of 10, which is typical of coronary arteries. The operating conditions used (Table 1) are more representative of flow in the aortic arch than in the coronary arteries , but are similar to values used in previous studies (Klanchar et aL, 1990). Shear rate data were taken at both the inner and outer walls, 180” from the curvature entrance.
each
curved
Experimental
tube
conditions.
experiment.
operating The
Each curved
experiment tube
had a different
measurements
were
impedance taken 180”
phase angle. A steady from the tube entrance.
flow
197
Dn
4.07 600
(L/min)
75.60
Re
rate
(mm
Steadv flow
Pressure
Flow
Hg)
1015
587
Curved
sequence
10.38
11.4
2.92
4.00
334
-
1172
623
9.41
9.7
3.42
3.88
26.01
76.50
79.75 26.98
0.97
-66.4
0.99
5.1
Peak Dn
1130
639
9.33
10.2
3.04
3.96
27.92
78.48
0.99
-23.5
193
1100
639
9.57
10.1
2.92
4.05
28.26
78.40
0.99
-45.6
elastic tube
Dn
1011
Peak Re
Mean
614
9.74
11.6
2.54
3.93
29.04
77.40
0.97
-65.0
Re
parameter
(%)
(L/min)
(L/min)
Hg)
Hg)
(mm
(mm
(Hz)
variation
Mean
Unsteadiness
Diameter
amplitude
Flow
rate
flow
amplitude
pressure
Mean
Pressure
Mean
Frequency
Sinusoidal flow
Phase angle (deg)
Straight
Table 1
was
199
606
4.12
recorded
75.86
363
193
1104
588
10.37
10.8
3.52
4.01
25.75
76.99
0.97
-55.7
elastic tube
before
199
608
4.10
75.60
362
194
1105
591
10.30
10.5
3.49
4.01
24.27
77.48
0.98
-20.0
Vol. 34, No. 1
Wall shear rate in a curved
artery model
7
The working fluid employed was a polyalkylene glycol ether (UCON 50-HE55, Union Carbide, specific gravity 0.971, index of refraction 1.43, kinematic viscosity 6.8 cs at 45°C). This Newtonian fluid demonstrates good photochromic behavior for sharp dye line contrast. It has the same index of refraction as the transparent silicone rubber used in the artery model, and the two are chemically compatible. The photochromic dye, 1’,3’,3’-trimethylnitroindoline-2-Spiro-2-benzopyran (TNSB) , prepared as described by Koelsch and Workman (1952), was dissolved in this fluid. This dye can be excited by light of 337 nm wavelength. For good color contrast and beam penetration, a dye concentration of 0.01% by weight was suggested by Rhee and Tarbell (1994). To generate dye lines, a pulsed nitrogen laser (VSL-33’7 ND, Laser Science, 337 nm) was focused precisely at the measurement location, perpendicular to the tube wall at its centerline. Since the power (4 mW) of the laser was not sufficient to generate a dye line across the entire width of the tube, the beam had to enter the tube at the measurement location. To generate traces at the outer wall (Fig. 3, upper panel), a lens with a focal length of 7.6 cm was mounted in front of the laser. The position of the laser and lens was adjusted until sharp dye lines were observed. To focus the beam at the inner wall (Fig. 3, lower panel), the laser was elevated so that the beam passed over the model to a beam director (Edmund Scientific) positioned inside the curve. This beam director hati two rectangular mirrors mounted at 45” angles to deflect the beam first downward and then back toward the inner wall. The lens focused the deflected beam to produce a sharp dye line. The flow field was observed with a black-and-white video camera (TM-745E, Pulnix) equipped with a zoom lens (D.O. Industries, 18-108 mm) and viewed on a monitor (Trinitron PVM-1342Q Sony). The camera was focused at the measurement site to provide a close view of the dye lines. Images were recorded by a super-VHS video recorder (BV-1000, Mitsubishi). Back-lighting was provided by an externally triggered strobe (1538-A Strobotac, General Radio) placed underneath the model. The laser and the flash were triggered from a data acquisition unit (500, Keithley) and software (Soft500, Keithley). The time delay between the laser and the flash was built into the triggering program through extraneous commands and measured by passing the signals to an oscilloscope (2230, Tektronix). Two different triggering programs were used, providing delays of 6.07 + 0.11 and 12.03 + 0.09 msec. Thus each image appeared at a known displacement time after the initial laser pulse. Traces were generated by the laser every 66 msec, roughly once every other frame on the videotape. Each image was digitized by a frame grabber board (DT2953, Data Translation) installed in a personal computer. Using an image processing package (Optimas, Bioscan) , these frames could be enhanced by a stop motion filter to smooth out a “bouncy” image or by edge filters to make grayscale gradients more distinct. The size of the image was calibrated by recording a few frames with a ruler placed just above the flow field. A line of known length could then be drawn, and the imaging package calculated the size of each pixel accordingly. To set the orientation of the image, the software imposed a rectangular coordinate system from an origin set by the user, chosen to coincide with the point where the laser beam path intersected the tube wall. The dye lines captured represented instantaneous displacement profiles as a function of radial distance. The sharp trailing edge of these lines was manually traced from the wall within a boundary layer of approximate thickness ~(v/w) 1/Z, which was about 0.2 cm in these experiments. The imaging package
8
WaU shear rate in a curved
artery model
Video
Vol. 34, No. 1
camera
Zoom
lens I
Lens 0 Nitrogen
laser
Optical
correction
Strobe
(b)
Video
Optical
correction
box
light
camera
box
Fig. 3 Optical arrangement for focusing the laser beam at the centerline on the walls of the curved tube. At both the outer and the inner walls, the beam passes into the optical correction chamber through a thin slide glass “window”. A video camera with a zoom lens records the displacement of the dye lines. Illumination is provided by a strobe light triggered at a known delay time after the laser pulse. (A) At the outer wall, the focusing lens is mounted in front of the laser, and the two are moved together on a translational stage to focus the beam at the outer wall. (B) Since the laser cannot be aimed directly at the inner wall, the beam path is manipulated by a beam director. The beam is shot over the tube and deflected first down and then back toward the inner wall. Fixed in a ring mount, the focusing lens is positioned to accept the beam and focus it at the inner wall.
Vol. 34, No. 1
Wall shear rate in a curved artety model
9
automatically segmented this tracing into 64 points. These displacement profiles were then fit to second-order polynomials (r2 > 0.98) using a statistical software package (Tablecurve, Jandel Scientific). This displacement occurred over the delay time (At) between the laser and the flash, so dividing this polynomial by At gave an average velocity profile. However, this time interval was small enough, that the error in assuming this average velocity profile to be an instantaneous velocity profile, should be less than 0.2% for the frequencies studied. The derivative of the velocity profile evaluated at the wall gave the wall shear rate. Due to the manual nature of the steps involved in this analysis, Rhee and Tarbell (1994) estimated the error in the wall shear rate to be within +5%. The procedure described above provided an estimate of the axial component of the wall shear rate. The secondary flow should have only a minor effect on this axial wall shear rate estimate. Since the secondary flow was symmetric about the central plane of the curved tube, the azimuthal (out-ofplane) component of velocity was zero at the inside and outside wall locations where our measurements were made, even though it may have been significant near the top (or bottom) of the tube. Only the radial and axial velocity components influenced the displacement of the dye line in our setup. Simple boundary layer estimates of velocity magnitudes (e.g., Ito, 1969) suggested that for our flow conditions, the radial velocity near the wall will only be a few percent of the axial velocity, and therefore the dye line displacement would be due predominantly to axial flow. For each steady flow condition, 10 images were recorded and analyzed. The wall shear rates obtained were averaged, and a sample standard deviation was calculated. For each sinusoidal flow condition, 18 images were analyzed to cover one entire flow cycle at 1 Hz. Using the statistical package, wall shear rates (Y) were fit to a sine wave through linear regression. The equation was fit , where t is time elapsed from the beginning as Y(t)= Ym + Yamp sin [w (t+nr)] of the cycle, ‘yrn (the intercept) is the mean wall shear rate, yamp (the slope) is the wall shear rate amplitude, w is the frequency of the first harmonic (1 Hz), r is the time interval between successive images (66 msec), and n = 0, fl, f2, etc. The phase term nz accounts for beginning the analysis a few images before or after the onset of forward flow; several values of n were compared to find the best regression fit. The sine wave fits to the wall shear rate data (r* = 0.84 - 0.90) were somewhat low due to the slight contribution of higher harmonic components present in the flow loop. In addition to determining ym and yamp, the statistical package also calculated the standard deviations of these parameters. The videotape was also analyzed to determine the diameter variation. Instead of focusing the camera near the wall, as for shear rate measurements, the camera was focused on the whole tube so the motion of the walls during the flow cycle could be followed. Wall motion appeared to be uniform, implying that no difference in elasticity existed between the inner and outer walls. The outer surfaces of the tube were marked automatically by the imaging software through differences in grayscale. Diameters were measured throughout one flow cycle and fit to a sine wave (r * = 0.95 - 0.98) as described above. Wall motion was described as the diameter variation DV = (dm - dM1N) / dAvG. Results and Discussion The wall shear rate results are shown in Table 2. The straight elastic tube results under sinusoidal flow conditions serve only to verify our techniques. Several features of the straight elastic tube results agree with expectations. The
angle
YP
Ym
1 YP
/
amp
(s-l)
tube
193.2
177.3 0.86 1.74 1.99
151.8 0.80 1.40 1.82
2.08
1.72
0.79
151.7
152.5
121.6
0.95
93.1
88.4
-23.5
elastic
2.40
2.10
0.84
215.3
180.6
0.96
89.7
86.0
5.1
2
0.36
2.22
2.12
0.33
0.37
149.9
2.21
2.60
0.52
0.40
0.53 7.19
183.9
72.7
0.98
1.99
70.7
140.6
2.02
73.2
elastic
tube
2.60
3.30
0.36
183.2
66.6
0.63
0.29
70.4
20.2
0.46
73.6
33.6
Inner
-55.7
148.1
Outer
155.6
82.0
0.45
1.06 54.9
0.16
70.4
2.34
70.7
11.4
72.0
73.3
165.4
25.8
Inner
162.5
Outer
-66.4
Curved
2.65
0.65
0.55
187.3
103.3
1.06
2.24
70.8
158.2
2.12
74.2
2.49
4.15
0.38
180.2
67.7
0.48
0.22
72.5
16.3
0.47
75.7
35.4
Inner
-20.0
157.4
Outer
Wall shear rate results for each experiment. The steady flow (yS) and mean (y,) wall shear rates are normalized by the appropriate Poiseuille value (yp), while the shear rate amplitudes (“la,,,& are normalized by the Womersley theory (yw) for a straight rigid tube. Normalized results are denoted by r. No phase angle exists for steady flow; the values merely indicate the phase angle trial with which they were recorded.
Yw
‘lamp
l-
Yw
tssl)
amolitude
/
rm
Ys
0.98
1.04
Ym
89.0
83.6
(s-l)
YP
45.6
87.6
WSR
-65.0
86.7
rm
(s-9
WSR
wall
(deg)
Straight
Ym 6-l)
‘lamp
WSR
flow
Ys (s-l)
Mean
Steadv
Measurement
Phase
Model
Table
Vol. 34, No. 1
Wclll shear rate in a curved artery
model
11
normalized mean wall shear rates (I,) generally agree within one experimental standard deviation of the predictions of Wang and Tarbell (1992) for a straight elastic tube (Fig. 4). The normalized mean wall shear rate is expected to be less than unity because of the convective acceleration due to wall motion (Wang and Tarbell, 1992). The impedance phase angle has a relatively small effect on the mean wall shear rate. This is consistent with the work of Wang and Tarbell (1992) and Dutta et al. (1992), which predict only a modest decrease in yri, with decreasing negative phase angles in straight elastic tubes. The wall shear rate amplitudes are only 79-86% of the Womersley amplitudes for a rigid pipe (Fig. 5), also in general agreement within one standard deviation of the predictions of the elastic tube theory (Wang and Tarbell, 1995). Under the conditions of our experiments, the impedance phase angle also has only a modest effect on the wall shear rate amplitude in the straight elastic tube, which is consistent with the theory of Wang and Tarbell (1995). Accordingly, the ratio of the amplitude to the mean of the wall shear rate waveform is consistent with the elastic tube theory (see Table 2). The consistent deviation of the ratio of yam to ym from the rigid tube theory (Table 2) indicates that the phase ang Pe does not influence the oscillatory nature of flow greatly, in a straight elastic tube, under these conditions. Rhee and Tarbell (1994) demonstrated the accuracy of the photochromic method for determining wall shear rates in a straight rigid tube. They obtained steady flow shear rates consistent with Poiseuille flow and sinusoidal flow shear rate amplitudes consistent with the Womersley theory. However, several steps involved in the tracing of each dye line can affect the results obtained, specifically, the placement of the origin at the exact measurement site in each frame and the manual tracing of the trailing edge of the dye line. Also, the actual time delay between the laser pulse and the strobe flash may not be uniform within a particular flow cycle, varying by 2%. Rhee and Tarbell (1994) estimated the uncertainty in the wall shear rate waveform parameters to be about 5%. To better quantify the uncertainty in the present results, standard deviations of the curve fit parameters have been reported with the results obtained. In the straight elastic tube, the uncertainty (the ratio of the standard deviation to the experimental value) in y,,, was ll%, while the uncertainty in “lampwas 9%. These estimates are limited, however, in that only one cycle from one experiment was analyzed. A more thorough analysis of multiple cycles in repeated photochromic experiments using a bifurcation model, has been performed by Lee and Tarbell (1997). For the curved tube, 180’ from the entrance, the normalized wall shear rate under steady flow conditions is compared to the normalized mean wall shear rate for sinusoidal flow in Fig. 6. Skewing of the velocity profile was observed as a result of curvature. Under steady flow conditions, the shear rate is 4.4-6.2 times greater at the outer wall than at the inner wall as a result of skewing of the velocity profile. Under sinusoidal flow conditions, wall motion caused the timeaveraged mean shear rate to be reduced greatly (37-55%) from its steady flow value at the inner wall, but no significant effects were observed at the outer wall. The wall shear rate amplitude is shown as a function of phase angle in Fig. 7. The amplitudes are normalized by the Womersley amplitude (yw) for flow in a straight rigid pipe. Curvature by itself has a slight effect on yamp, with values within 10% of yw (Chang and Tarbell, 1985). Considering wall motion, however, the shear rate amplitudes in the curved elastic tube are similar in magnitude at both the inner and outer walls but are only 36-55% of the Womersley amplitude. Comparing this to the straight elastic tube results,
Vol. 34, No. 1
W&U shear mte in a curved artery model 15-
Mean
---*--,
wang/rarben
Phase angle (deg) Fig. 4. Normahzed mean watl shear rates for sinusoidal flow as a function of phase angle in the straight elastic tube; values of Tm predicted from the straight elastic tube theory (Wang and Tarbell, 1992) are also shown. Error bars represent one standard deviation in curve fit parameters. The time-averaged mean shear rates generally fall within one standard deviation of the predictions of the straight elastic tube theory (Wang and Tarbell, 1992).
0 6-
Phase angle (deg) Fig. 5. Normalized wall shear rate amplitude as a function of phase angle in the straight elastic tube. Predictions for the shear rate amplitudes in an elastic tube (Wang and Tarbell, 1995) generally agree with the experimental results within one standard deviation.
Vol. 34, No. 1
Wall shear rate in a curned
atiety
13
model
3.0
1.5 -
1.0-
---*mm.
I -------------_____________________ 0.0 -70
. . ..I’....... -60
-50
.I....,....,.... -40
Sleadyblner) Mean (inner)
3 -xl
-20
. -10
Phase angle (deg) Fig. 6. Normalized wall shear rates for steady flow and mean wall shear rates for sinusoidal flow as a function of phase angle at both the inner and outer walls of the curved elastic tube. The wall shear rate is about 5 times greater at the outer wall under both steady and sinusoidal flow conditions. At the inner wall, the mean wall shear rate is reduced from its steady flow value, but at the outer wall there is no significant difference between the mean and steady flow values.
-50
40
-30
Phase angle (deg)
Fig. 7.
Normalized wall shear rate amplitude as a function of phase angle in the curved elastic tube. The amplitude at the outer wall decreases with decreasing phase angle, while the amplitude at the inner wall increases over this same range. The shear rate amplitude is much less at both walls of the curved tube than in the straight tube.
14
Wall shear rate in a cuwed
artety
model
Vol. 34, No. 1
curvature affects the wall shear rate amplitude in elastic tubes significantly. In the straight tube, the shear rate amplitudes are 1421% less than the rigid tube theory, while the amplitudes in the curved elastic tube are reduced 45-64% at both walls-a very large effect. The shear rate amplitudes at each wall show an opposite dependence on phase angle. At the outer wall, the shear rate amplitude decreases as the phase angle becomes more negative, but at the inner wall, the shear rate amplitude increases as the phase angle becomes more negative. These trends indicate that changes in the phase angle cause changes in local flow patterns which affect the wall shear rate amplitude. Considering the oscillatory nature of the flow as characterized by the ratio yamp / yrn (Fig. S), both the curvature and the phase angle have significant effects. The straight elastic tube theory predicts wall shear reversal (yw / yp greater than unity, see Table 2) under these flow conditions. However, curvature induces changes in this ratio. The experimental values at the outer wall are much less than the straight tube values and much less than unity (0.33-0.65), indicating that no shear reversal occurs at the outer wall, as was apparent from the videotape. Quite the opposite behavior is observed at the inner wall. Shear reversal is observed as the ratio is much greater than unity (3.30-7.19), and much greater than that predicted by the straight tube theory. In the straight elastic tube, this ratio is 13-23% less than predicted by the rigid tube theory. In the curved tube, flow is much less oscillatory at the outer wall (75-84% less than the rigid tube theory) and much more oscillatory at the -
l3pt (outer)
IP 0
I
.
-7Cl
.
,
I
-60
’
.
I
-SO
.
.
I
’
-40
’
’
1
-30
.
’
-
’
-20
.
.
.-
-10
Phase angle (deg)
Fig. 8.
Ratio of the wall shear rate amplitude to the mean wail shear rate as a function of phase angle in the curved elastic tube. The experimental values are compared to predictions from the straight elastic tube theory (Wang and Tarbell, 1992; 1995). The error bars on the outer wall data are too small to be shown. At the outer wall, flow is only slightly oscillatory, with a shear rate amplitude only 33-65% of the mean. At the inner wall, flow is much more oscillatory, with shear rate amplitudes 3.3S7.19 times the mean, showing a significant degree of reversal. Decreasing the phase angle to more negative values causes flow at the outer wall to become less oscillatory and flow at the inner wall to become more oscillatory.
Vol. 34, No. 1
Wall shear rate in a curved artery model
15
inner wall (1.27-3.25 times the rigid tube theory). Phase angle changes affect this ratio. Decreasing the phase angle from -55” to -66” causes the oscillatory nature of flow at the outer wall to be diminished. In contrast to the outer wall, decreasing the phase angle from -55” to -66” causes the degree of shear reversal at the inner wall to increase greatly. The values of the ratio yamp / ym (Fig. 8) show that the phase angle reduces the oscillatory nature of flow at the outer wall while greatly enhancing it at the inner wall. These results differ in some respects from the findings of Klanchar et al. (1990). They found that the peak shear rate at both walls increased significantly as the phase angle was reduced from -60” to -80”. The present results support such an effect only at the inner wall. The major difference between the two studies lies in the techniques used. Unlike Klanchar et al., the present study measured the local flow rate and the entire wall shear rate waveform directly; thus our results should represent a refinement of the earlier study. These experiments demonstrate that while impedance phase angle may not exert a great influence on either the mean or the amplitude of the wall shear rate waveform in straight vessels,it has a significant influence in a curved tube, particularly at very negative phase angles. Vessel curvature induces low mean shear rates and changes in shear direction (oscillatory shear) on the inner wall, where atherosclerotic plaques tend to be localized. Vessel wall motion tends to exaggerate these features, particularly at more negative phase angles. As hypertension is associated with more negative phase angles (Merillon et al., 1982), it may exacerbate atherogenesis in curved vessels by inducing lower and more oscillatory shear stress on the inner curvature through a phase angle mechanism as we have described. Vasorelaxants increase phase angle (White et aZ., 1994) and may be therapeutic for atherosclerosis, at least in part, through a phase angle effect on wall shear.
References ASAKURA, T., and KARINO, T. (1990). Flow patterns and spatial distribution of atherosclerotic lesions in human coronary arteries. Circ. Res.66, 1045-1066. BINNS, R.L., KU, D.N., STEWART, M.T., ANSLEY, J.P., and COYLE, K.A. (1989). Optimal graft diameter: effect of wall shear stress on vascular healing. J. Vast. Surg. 10, 326-337. CHANDRAN, KB., and YEARWOOD, T.L. (1981). Experimental study of physiologic pulsatile flow in a curved tube. J. Fluid Mech. 111, 59-85. CHANG, LJ., and TARBELL, J.M. (1985). Numerical simulation of fully developed sinusoidal and pulsatile (physiological) flow in curved tubes. J l&id Mech. 161, 175-198. CHANG, LJ., and TARBELL, J.M. (1988). A numerical study of in curved tubes simulating coronary arteries. J. Bionwh. 21, 927-93’7.
flow
CHOI, U.S., TALBOT, L., and CORNET, I. (1979). Experimental study of wall shear rates in the entry region of a curved tube. J Fluid Mech. 93, 465489.
DEAN, W.R. Philos. Mag.
(1927).
4, 208-223.
Note on the motion of fluid in a curved pipe.
16
Wall &ear rate in a curved artery model DEAN, W.R. (1928). The streamline Philos. Mug. 5, 673-695.
motion
Vol. 34, No. 1 of fluid in a curved
pipe.
DUTTA, A., WANG, D.M., and TARBELL, J.M. (1992). Numerical analysis of flow in an elastic artery model. J. Biomech. Eng. 114, 2633. FOX, B., and SEED, W.A. (1981). Location of early atheroma human coronary arteries. J. Biomech. Eng. 103; 208-212. FRIEDMAN, M.H., HUTCHINS, G.M., BARGERON, C.B., OJ., and MARK, F.F. (1981). Correlation between intimal and fluid shear in human arteries. Atherosclerosis 39, 425436. ITO, H.
(1969).
Laminar
flow in curved pipes. ZAMM
in the
DETERS, thickness
21, 653-663.
KLANCHAR, M., TARBELL, J.M., and WANG, D.M. (1990). In vitro study of the influence of radial wall motion on wall shear stress in an elastic tube model of the aorta. Circ. Res. 66, 1624-1635. KOELSCH, thermochromic
C.F., and WORKMAN, W.R. (1952). spirans. J Am. Chem. Sot. 74, 6288-6289.
Some
KU, D.N., GIDDENS, D.P., ZARINS, C.K., and GLAGOV, S. (1985). Pulsatile flow and atherosclerosis in the human carotid bifurcation: positive correlation between plaque location and low and oscillating shear stress. Arteriosclerosis 5, 293-302. LEE, C.S., and TARBELL, J.M. (1997). Wall shear rate distribution in an abdominal aortic bifurcation model: effects of vessel compliance and phase angle between pressure and flow waveforms. J. Biomech. Eng., in press. LIN, J.Y., and TARBELL, numerical study of periodic 623-638.
J.M. (1980). flow in a curved
An experimental and tube. J. fluid Mech. 100,
LOGERFO, F.W., QUIST, W.C., NOWAK, M.D., CRAWSHAW, H.M., and HAUDENSCHILD, C.C. (1983). Downstream anastomotic hyperplasia: a mechanism of failure in Dacron arterial grafts. Ann. Surg. 197, 479-483. LYNE, W.H. (1970). Mech. 100, 623-638.
Unsteady
viscous
flow
in a curved
pipe. J. Fluid
J.F., JAFFRIN, MERILLON, J.P., FONTENIER, GJ., LERALLUT, M.Y., MOTTE, G.A., GENAIN, C.P., and GUORGON, R.R. (1982). Aortic input impedance in normal man and arterial hypertension: its modifications during changes in aortic pressure. Cardiovasc. Res. 16, 646d56. NEREM, R.M. (1995). Atherosclerosis and the role of wall shear stress. In: Flow-dependent regulation of vascular function. J.A. Bevan, G. Kaley, and G.M. Rubanyi, Eds., Oxford, New York, pp. 300-319. OJHA, M. (1994). Wall shear stress temporal anastomotic intimal hyperplasia. Circ. Res. 74, 1227-1231.
gradient
and
Vol. 34, No. 1
Walt shear rate in a curved artery model
17
OJHA, M., HUMMEL, R.L., COBBOLD, R.S.C., and JOHNSTON, Development and evaluation of a high resolution K.W. (1988). photochromic dye method for pulsatile flow studies. j Whys. E. Sci. zn.str. 21, 998-1004. OKADOME, K., YUKIZANE, T., MII, S., FUKUDA, A., and Correlation of long-term results of SUGIMACHI , K (1989). femoropopliteal autogenous vein grafts with shear stress variation: in situ or reversed graft?J. Cardiovasc. Surg. 30, 932-935. POPOVICH, non-disturbing 22,21-25.
A.T., and HUMMEL, flow measurements
R.L. (1967). A new method for very close to a wall. Cbm. Eng. Sci.
RHEE, K., and TARBELL, J.M. (1994). A study of the wall shear rate distribution near the end-to-end anastomosis of a rigid graft and a compliant artery. J. Birch. 27, 329-338. RI’ITGERS, S.E., KARAYANNACOS, P.E., GUY, J.F., NEREM, R.M., SHAW, G.M., HOSTETLER, J.R., and VASKO, J.S. (1978). Velocity distribution and intimal proliferation in autologous vein grafts in dogs. Circ. Res. 42, 792-801. WALDEN, R., L’ITALIEN, G.J., MEGERMAN, J., and ABBOTT, W.M. (1980). Matched elastic properties and successful arterial grafting. Arch. SUTg. 115, 1166-1169. WANG, D.M., and TARBELL, J.M. (1992). Nonlinear analysis of flow in an elastic tube (artery): steady streaming effects. J. Z&id Mech. 239, 341-358. WANG, D.M., and TARBELL, J.M. (1995). Nonlinear analysis of oscillatory flow, with a non-zero mean, in an elastic tube (artery). J Biomech. Eng. 117, 127-135.
WESTON, M.W., RHEE, K., and TARBELL, J.M. (1996). Compliance and diameter mismatch affect the wall shear rate distribution near an end-to-end anastomosis. J Biomech. 29, 187-198. WHITE, K.C., KAVANAUGH, J.F., WANG, D.M., and TARBELL, J.M. (1994). Hemodynamics and wall shear rate in the abdominal aorta of dogs: effects of vasoactive drugs. Circ. Res. 75, 637-649. Acknowledgments This study was supported in part by NIH grant HL35549. The authors thank Dr. Russell Heikes of the Georgia Institute of Technology in Atlanta for discussion of statistical issues. Received 17 October 1995;
accepted
in
revised form
1 February
1997.
PIISOOO6355X(97)OOOO1-2
WALL SHEAR RATE MEASUREMENTS CURVED ARTERY MODEL M. W. WESTON Physiological Engineering, USA
IN
AN
ELASTIC
AND J. M. TARBELL
Transport Studies Laboratory, The Pennsylvania State University,
Department of Chemical University Park, Pennsylvania,
Reprint requests to: John M. Tarbell, Physiological Transport Studies Laboratory, Department of Chemical Engineering, The Pennsylvania State University, 155 Fenske Laboratory, University Park, PA 168024400 USA; Fax: (814) 865-7846; e-mail: [email protected]
ABSTRACT Since atherosclerotic lesions tend to be localized at bends and branching points, knowledge of wall shear rate patterns in models of these geometries may help elucidate the mechanism of atherogenesis. This study uses the photochromic method of flow visualization to determine both the mean and amplitude of the wall shear rate waveform in straight and curved elastic arterial models to demonstrate the effects of curvature, elasticity, and the phase angle between the flow and pressure waveforms (impedance phase angle). Under sinusoidal flow conditions characteristic of large arteries, the mean shear rate at the inner wall of the curved tube is reduced 40-56% from its steady flow value, depending on the phase angle. Wall shear rate amplitudes in the curved tube are significantly reduced by wall motion (3655% of the Womersley amplitude for a straight rigid tube). The shear rate amplitude at the outer wall decreases 30% as the phase angle is reduced from -20” to -66’, while the shear rate amplitude at the inner wall increases 45%. As a result, the oscillatory nature of flow at the outer waLl decreases with decreasing negative phase angle, but flow at the inner wall becomes much more oscillatory. At large negative phase angles, characteristic of hypertension or vasoactive agents, the shear rate at the inner wall has a small mean and cycles through positive and negative values; the shear rate at the outer wall remains positive throughout the flow cycle. Thus, the impedance phase angle could affect atherogenesis along the inner wall if temporal and directional changes in wall shear rate play a role. 0 1997~~~irrsdnrc1ld
Introduction
Individual risk factors such as diet, fitness, and heredity play a major role in the development of atherosclerosis. Recent studies though, have found that atherosclerotic lesions tend to be localized at branching points and bends in
KEYWORDS:
Wall shear rate; photochromic
phase
angle;
I
curvature;
elastic;
2
WaUshear rate in a curved artery model
Vol. 34, No. 1
the circulation regardless of the underlying risk factors involved (see review by Nerem, 1995). Fox and Seed (1981) reported the distribution of early atheroma in human coronary arteries and noted that fatty plaques were localized predominantly at the inner wall of the curvature, where the vessel contacts the curved surface of the heart. Asakura and Karino (1990) visualized both steady and pulsatile flow in human coronary arteries rendered transparent through postmortem treatment. Sites of atherosclerosis experienced low flow velocities, recirculation zones, or strong secondary flows, all of which may be indicative of low wall shear stress. Low wall shear has been linked to atherosclerosis in the aortic (Friedman et al., 1981) and carotid (Ku et al., 1985) bifurcations and in end-to-side (Rittgers et al., 1978; LoGerfo et aZ., 1983) and end-to-end (Binns et al., 1989) vascular grafts. Other studies indicate that temporal and directional variations in shear rate may contribute to atherogenesis (Walden et al., 1980; Ku et aZ., 1985; Okadome et al., 1989). A uniform curved tube (Fig. 1) is a simple model of a curved artery, such as the aortic arch or the coronary arteries. The first theoretical considerations of fully developed steady flow behavior in a rigid curved tube far from the entrance were made by Dean (1927, 1928). Curvature has a significant effect on flow, inducing a symmetric pair of secondary flow vortices (Dean vortices) that sweep flow toward the outer wall, causing the peak axial velocity to occur closer to the outer wall, not at the centerline as in straight tube flow. The intensity of curved tube flows is characterized by a modified Reynolds number (Re) referred to as the Dean number, Dn (E Re / h112, where 3Lis the aspect ratio, Wall shear measurements under steady flow conditions R/ r). (Dn = 139 - 2868) in a rigid curved pipe have been reported by Choi et al. (1979). Near the tube entrance, shear stress was circumferentially uniform at low Dn but became higher near the inner wall for Dn above 400. However, by the exit of the curved tube (El = 180”), the shear rate at the outer wall was roughly 2.5 times the value predicted for Poiseuille flow (yp), while the shear rate at the inner wall was close to yr. This skewing of the wall shear rate Outer
wall
Fig. 1. Schematic drawing of the curved tube geometry. A tube of radius r is curved through a radius of curvature R, the degree of curvature is described by the aspect ratio h = R / r. In this model, flow enters at 8 = 0” and exits at 180”. The inner wall refers to the wall closest to the center of curvature:
the outer wall is farther
from the center.
Vol. 34, No. 1
Wall shear rate in a curved
artery model
3
distribution, observed at all Dn, results from the progression of the secondary flow. Theoretical work by Lyne (1970) demonstrated a transition from two to cour vortex secondary flow at elevated frequencies in oscillatory flow. Combining numerical simulations with experiments, Lin and Tarbell (1980) found a resonance condition between the sinusoidal flow frequency and the frequency of the secondary flow rotation that produced elevated peak shear stresses at the outer wall and attenuated shear stresses at the inner wall. Physiological pulsatile flow in a rigid curved tube simulating the aortic arch was studied experimentally by Chandran and Year-wood (1981) and numerically by Chang and Tarbell (1985). Both observed high peak shear rates at the inner wall but also significant flow reversal, resulting in time-averaged mean shear rates that were lower than at the outer wall. Chang and Tarbell (1988) reported similar results for simulations of coronary artery conditions. The temporal phase angle between the pressure and flow waveforms (impedance phase angle) may also affect the wall shear rate distribution. If the pressure and flow waveforms are resolved into Fourier harmonics, then the impedance phase angle of each harmonic can be determined. Under physiological flow conditions, the impedance phase angles of the lower harmonics are usually negative, and flow leads pressure. A non-zero impedance phase angle is caused by wave reflections in the circulation from branching points, bends, and the high resistance of peripheral vessels. Typically, the physiologic value of the phase angle of the first harmonic of the input impedance (measured in the ascending aorta) is near -45”. However, hypertensive people have more negative phase angles (Merillon et al., 1982), while vasoactive drugs may change the phase angle in either direction, depending on the drug used (White et al., 1994). A relationship between phase angle and wall shear was first suggested by Klanchar et al. (1990). They measured wall shear stress at the exit (180”) of a curved elastic tube under large artery flow conditions using hot-film anemometry. Peak shear rates were lower at the inner wall of the curve than at the outer wall, but mean shear rates could not be determined by hot-film anemometry due to local flow reversal during part of the cycle. Peak wall shear rate was insensitive to phase angle for values above -6O”, but as the impedance phase angle decreased from -60” to -BOO, the peak shear rate at both walls increased significantly. However, they did not measure the local flow rate, which probably also varied with phase angle. Recently developed theories for straight elastic tubes predict that by increasing the phase angle (toward 0”). the mean shear rate will increase and the shear rate amplitude will decrease (Wang and Tarbell, 1992; 1995). These theories have been supported by Rhee and Tarbell (1994) and Weston et al. (1996), who used the photochromic method to study the wall shear rate waveform in end-to-end anastomosis models. By administering different vasoactive drugs in dogs, White et al. (1994) found that the wall shear rate waveform in viuo is sensitive to changes in the phase angle, but their pressure and flow waveforms varied significantly in each experiment. In the present study, more recent experimental techniques were used to clarify the results of Klanchar et al. (1990). The local flow rate was measured accurately at any site in the elastic artery model using ultrasound. The photochromic method, a relatively new technique for determining wall shear rate, allowed reversed flow to be accurately quantified so that both the mean and the amplitude of the wall shear rate waveform could be determined. The complete wall shear rate waveforms on the inner and outer walls at the exit of an elastic curved tube were determined. The experiments were repeated with the tube straightened to verily the accuracy of the technique by comparison
4
Wall shear rate in a curued artery model
Vol. 34, No. 1
with theory (Wang and Tarbell, 1992; 1995) and to demonstrate the effect of curvature on the wall shear rate waveform. In both of these configurations, the impedance phase angle was varied over a large enough range to evaluate its physiological influence. Methods The photochromic method, originally developed by Popovich and Hummel (1967), has been used to visualize flow in suaight and curved elastic tube models. A photochromic substance changes color through an isomerization reaction when it is excited by light of an appropriate wavelength. Conversely, when the light is removed, the color gradually disappears. In flow visualization, the photochromic material is dissolved in a host fluid. By focusing a laser light source of an appropriate wavelength on a flowing fluid at the location of interest, a color change occurs in the thin path of the beam. As flow proceeds, this thin path of excited particles moves with the flow, indicating their displacement from their initial location. This technique has been used recently for flow visualization in rigid arterial models (Ojha et al., 1988), and for wall shear rate measurements in rigid (Ojha, 1994) and elastic models (Rhee and Tarbell, 1994; Weston et al., 1996). The flow loop used in this study (Fig. 2) is similar to that of Rhee and Tarbell (1994). The test fluid was heated to 45’C in a water bath and circulated around the closed flow loop by a peristaltic roller pump (Masterflex 7019, Cole Parmer). A coil of acrylic tubing was used to damp out flow perturbations from the peristaltic pumping, producing a steady flow. Sinusoidal flow could then be imposed through a piston pump (1423, Harvard Apparatus). The stroke volume was adjusted to produce a diameter variation [ (dMm - dMIN) / dAVG] of lo%, while the beat frequency was set at 1 Hz to simulate the cardiac pulse. While not truly physiologic, sinusoidal flow was used so that the phase angle of only one harmonic was present, allowing its effect to be isolated. Flow from the piston pump passed through 152 cm of flexible latex tubing, 1.9 cm ID, (Primeline Industries), which matched the diameter and compliance of the arterial model to ensure that the flow entering the model was fully developed. Flow exited the model through an additional 30.5 cm of the same tubing to allow for wave propagation. Distal to this tubing, another 472 cm of the same
piston
Pump
Ir
IEi
Static fluid column Roller
1
DumD
Ultrasonic flow meter
-ml
I
Catheter-tip pressure transducer
I
Tem’peratur< bath Distal latex tubing
Proximal latex tubing
tube model
Fig. 2 The flow loop, shown here for the curved tube experiments, allows both steady and sinusoidal flow to be studied. The impedance phase angle is adjusted by altering the downstream resistance.
Vol. 34, No. 1
Wall shear rate in a curued artery
model
5
tubing and a vertical section of stiff Tygon tubing along with an adjustable screw clamp, were used to vary the downstream impedance and phase angle. Flow rate and pressure measurements were taken within one diameter distal to the wall shear rate measurement site. The flow rate was measured locally by an ultrasonic flow meter (TlOl, Transonic Systems)and the flow probe did not compress the tube during any part of the flow cycle. The pressure was measured by a catheter-tip transducer (TCD 500, Millar Instruments). Both the flow and pressure signals were recorded using an analog-to-digital data acquisition unit (500A, Keithley) and software (Soft500, Keithley) installed in a personal computer (PS2, IBM). A fast Fourier transform (Asyst, MacMillan Software) was performed on the flow and pressure waveforms to resolve them into harmonics of sine waves with a base frequency (0). Since a simple sinusoidal pressure waveform drives the flow, only the first harmonics were significant. From the Fourier analysis, the phase difference between the pressure and flow waveforms (the impedance phase angle) could be calculated. The phase angle was varied through the length of the downstream latex section and the position of the adjustable screw clamp which increased the resistance by compressing (but not constricting) the diameter available to flow. More positive phase angles were achieved by clamping farther from the model along the 472 cm latex section placed before the vertical Tygon tubing column. More negative impedance phase angles were achieved when the latex section was removed and the clamp was placed on the column. Impedance phase angles between -70” and +lO” could be obtained. The operating conditions used in each experiment are listed in Table 1. The straight elastic tube was made in several steps. A 1.9 cm ID polyvinyl chloride (PVC) pipe (L = 49.3 cm) was cut in half lengthwise. The pieces were taped together and mounted inside a vise to hold their alignment. Melted polyethylene glycol wax (PEG Compound 20M, Union Carbide, melting point 60°C) was then poured into the pipe. After cooling, the pipe halves were separated from the wax rod. MetaI caps (cap ID = 1.9 cm, end post OD = 0.32 cm) were placed on the ends of the wax rod to allow it to be held in a rotating spit. A transparent silicone elastomer (Sylgard 184, Dow Corning, index of refraction 1.43) was poured onto the rotating wax rod and cured for 9 hr below the melting point of the wax. After curing, the rod was soaked in boiling water to melt the wax, leaving behind the flexible silicone rubber tubing with a wall thickness of about 0.05 cm. The straight tube was mounted inside an optical correction chamber. This chamber was filled with the test fluid to eliminate any optical distortions caused by the tube curvature, providing an accurate view of the dye lines (see Rhee and Tarbell, 1994 or Weston et al., 1996). The side walls of the chamber had “windows” of thin slide glass to allow the laser beam to passthrough into the tube. Small supports on the bottom of the chamber kept the tube level, and thin-walled metal connectors (OD = 1.9 cm, L = 1.9 cm) joined the model tubing to the latex tubing on the proximal and distal sides. The wave reflection caused by these short connectors was small (Rbee and Tarbell, 1994). For the curved elastic tube experiments, the same silicone rubber tube was bent into a curved configuration and mounted inside of another optical correction chamber. The radius of curvature of the tube was fixed by the chamber design at 9.85 cm, giving an aspect ratio of 10, which is typical of coronary arteries. The operating conditions used (Table 1) are more representative of flow in the aortic arch than in the coronary arteries , but are similar to values used in previous studies (Klanchar et aL, 1990). Shear rate data were taken at both the inner and outer walls, 180” from the curvature entrance.
each
curved
Experimental
tube
conditions.
experiment.
operating The
Each curved
experiment tube
had a different
measurements
were
impedance taken 180”
phase angle. A steady from the tube entrance.
flow
197
Dn
4.07 600
(L/min)
75.60
Re
rate
(mm
Steadv flow
Pressure
Flow
Hg)
1015
587
Curved
sequence
10.38
11.4
2.92
4.00
334
-
1172
623
9.41
9.7
3.42
3.88
26.01
76.50
79.75 26.98
0.97
-66.4
0.99
5.1
Peak Dn
1130
639
9.33
10.2
3.04
3.96
27.92
78.48
0.99
-23.5
193
1100
639
9.57
10.1
2.92
4.05
28.26
78.40
0.99
-45.6
elastic tube
Dn
1011
Peak Re
Mean
614
9.74
11.6
2.54
3.93
29.04
77.40
0.97
-65.0
Re
parameter
(%)
(L/min)
(L/min)
Hg)
Hg)
(mm
(mm
(Hz)
variation
Mean
Unsteadiness
Diameter
amplitude
Flow
rate
flow
amplitude
pressure
Mean
Pressure
Mean
Frequency
Sinusoidal flow
Phase angle (deg)
Straight
Table 1
was
199
606
4.12
recorded
75.86
363
193
1104
588
10.37
10.8
3.52
4.01
25.75
76.99
0.97
-55.7
elastic tube
before
199
608
4.10
75.60
362
194
1105
591
10.30
10.5
3.49
4.01
24.27
77.48
0.98
-20.0
Vol. 34, No. 1
Wall shear rate in a curved
artery model
7
The working fluid employed was a polyalkylene glycol ether (UCON 50-HE55, Union Carbide, specific gravity 0.971, index of refraction 1.43, kinematic viscosity 6.8 cs at 45°C). This Newtonian fluid demonstrates good photochromic behavior for sharp dye line contrast. It has the same index of refraction as the transparent silicone rubber used in the artery model, and the two are chemically compatible. The photochromic dye, 1’,3’,3’-trimethylnitroindoline-2-Spiro-2-benzopyran (TNSB) , prepared as described by Koelsch and Workman (1952), was dissolved in this fluid. This dye can be excited by light of 337 nm wavelength. For good color contrast and beam penetration, a dye concentration of 0.01% by weight was suggested by Rhee and Tarbell (1994). To generate dye lines, a pulsed nitrogen laser (VSL-33’7 ND, Laser Science, 337 nm) was focused precisely at the measurement location, perpendicular to the tube wall at its centerline. Since the power (4 mW) of the laser was not sufficient to generate a dye line across the entire width of the tube, the beam had to enter the tube at the measurement location. To generate traces at the outer wall (Fig. 3, upper panel), a lens with a focal length of 7.6 cm was mounted in front of the laser. The position of the laser and lens was adjusted until sharp dye lines were observed. To focus the beam at the inner wall (Fig. 3, lower panel), the laser was elevated so that the beam passed over the model to a beam director (Edmund Scientific) positioned inside the curve. This beam director hati two rectangular mirrors mounted at 45” angles to deflect the beam first downward and then back toward the inner wall. The lens focused the deflected beam to produce a sharp dye line. The flow field was observed with a black-and-white video camera (TM-745E, Pulnix) equipped with a zoom lens (D.O. Industries, 18-108 mm) and viewed on a monitor (Trinitron PVM-1342Q Sony). The camera was focused at the measurement site to provide a close view of the dye lines. Images were recorded by a super-VHS video recorder (BV-1000, Mitsubishi). Back-lighting was provided by an externally triggered strobe (1538-A Strobotac, General Radio) placed underneath the model. The laser and the flash were triggered from a data acquisition unit (500, Keithley) and software (Soft500, Keithley). The time delay between the laser and the flash was built into the triggering program through extraneous commands and measured by passing the signals to an oscilloscope (2230, Tektronix). Two different triggering programs were used, providing delays of 6.07 + 0.11 and 12.03 + 0.09 msec. Thus each image appeared at a known displacement time after the initial laser pulse. Traces were generated by the laser every 66 msec, roughly once every other frame on the videotape. Each image was digitized by a frame grabber board (DT2953, Data Translation) installed in a personal computer. Using an image processing package (Optimas, Bioscan) , these frames could be enhanced by a stop motion filter to smooth out a “bouncy” image or by edge filters to make grayscale gradients more distinct. The size of the image was calibrated by recording a few frames with a ruler placed just above the flow field. A line of known length could then be drawn, and the imaging package calculated the size of each pixel accordingly. To set the orientation of the image, the software imposed a rectangular coordinate system from an origin set by the user, chosen to coincide with the point where the laser beam path intersected the tube wall. The dye lines captured represented instantaneous displacement profiles as a function of radial distance. The sharp trailing edge of these lines was manually traced from the wall within a boundary layer of approximate thickness ~(v/w) 1/Z, which was about 0.2 cm in these experiments. The imaging package
8
WaU shear rate in a curved
artery model
Video
Vol. 34, No. 1
camera
Zoom
lens I
Lens 0 Nitrogen
laser
Optical
correction
Strobe
(b)
Video
Optical
correction
box
light
camera
box
Fig. 3 Optical arrangement for focusing the laser beam at the centerline on the walls of the curved tube. At both the outer and the inner walls, the beam passes into the optical correction chamber through a thin slide glass “window”. A video camera with a zoom lens records the displacement of the dye lines. Illumination is provided by a strobe light triggered at a known delay time after the laser pulse. (A) At the outer wall, the focusing lens is mounted in front of the laser, and the two are moved together on a translational stage to focus the beam at the outer wall. (B) Since the laser cannot be aimed directly at the inner wall, the beam path is manipulated by a beam director. The beam is shot over the tube and deflected first down and then back toward the inner wall. Fixed in a ring mount, the focusing lens is positioned to accept the beam and focus it at the inner wall.
Vol. 34, No. 1
Wall shear rate in a curved artety model
9
automatically segmented this tracing into 64 points. These displacement profiles were then fit to second-order polynomials (r2 > 0.98) using a statistical software package (Tablecurve, Jandel Scientific). This displacement occurred over the delay time (At) between the laser and the flash, so dividing this polynomial by At gave an average velocity profile. However, this time interval was small enough, that the error in assuming this average velocity profile to be an instantaneous velocity profile, should be less than 0.2% for the frequencies studied. The derivative of the velocity profile evaluated at the wall gave the wall shear rate. Due to the manual nature of the steps involved in this analysis, Rhee and Tarbell (1994) estimated the error in the wall shear rate to be within +5%. The procedure described above provided an estimate of the axial component of the wall shear rate. The secondary flow should have only a minor effect on this axial wall shear rate estimate. Since the secondary flow was symmetric about the central plane of the curved tube, the azimuthal (out-ofplane) component of velocity was zero at the inside and outside wall locations where our measurements were made, even though it may have been significant near the top (or bottom) of the tube. Only the radial and axial velocity components influenced the displacement of the dye line in our setup. Simple boundary layer estimates of velocity magnitudes (e.g., Ito, 1969) suggested that for our flow conditions, the radial velocity near the wall will only be a few percent of the axial velocity, and therefore the dye line displacement would be due predominantly to axial flow. For each steady flow condition, 10 images were recorded and analyzed. The wall shear rates obtained were averaged, and a sample standard deviation was calculated. For each sinusoidal flow condition, 18 images were analyzed to cover one entire flow cycle at 1 Hz. Using the statistical package, wall shear rates (Y) were fit to a sine wave through linear regression. The equation was fit , where t is time elapsed from the beginning as Y(t)= Ym + Yamp sin [w (t+nr)] of the cycle, ‘yrn (the intercept) is the mean wall shear rate, yamp (the slope) is the wall shear rate amplitude, w is the frequency of the first harmonic (1 Hz), r is the time interval between successive images (66 msec), and n = 0, fl, f2, etc. The phase term nz accounts for beginning the analysis a few images before or after the onset of forward flow; several values of n were compared to find the best regression fit. The sine wave fits to the wall shear rate data (r* = 0.84 - 0.90) were somewhat low due to the slight contribution of higher harmonic components present in the flow loop. In addition to determining ym and yamp, the statistical package also calculated the standard deviations of these parameters. The videotape was also analyzed to determine the diameter variation. Instead of focusing the camera near the wall, as for shear rate measurements, the camera was focused on the whole tube so the motion of the walls during the flow cycle could be followed. Wall motion appeared to be uniform, implying that no difference in elasticity existed between the inner and outer walls. The outer surfaces of the tube were marked automatically by the imaging software through differences in grayscale. Diameters were measured throughout one flow cycle and fit to a sine wave (r * = 0.95 - 0.98) as described above. Wall motion was described as the diameter variation DV = (dm - dM1N) / dAvG. Results and Discussion The wall shear rate results are shown in Table 2. The straight elastic tube results under sinusoidal flow conditions serve only to verify our techniques. Several features of the straight elastic tube results agree with expectations. The
angle
YP
Ym
1 YP
/
amp
(s-l)
tube
193.2
177.3 0.86 1.74 1.99
151.8 0.80 1.40 1.82
2.08
1.72
0.79
151.7
152.5
121.6
0.95
93.1
88.4
-23.5
elastic
2.40
2.10
0.84
215.3
180.6
0.96
89.7
86.0
5.1
2
0.36
2.22
2.12
0.33
0.37
149.9
2.21
2.60
0.52
0.40
0.53 7.19
183.9
72.7
0.98
1.99
70.7
140.6
2.02
73.2
elastic
tube
2.60
3.30
0.36
183.2
66.6
0.63
0.29
70.4
20.2
0.46
73.6
33.6
Inner
-55.7
148.1
Outer
155.6
82.0
0.45
1.06 54.9
0.16
70.4
2.34
70.7
11.4
72.0
73.3
165.4
25.8
Inner
162.5
Outer
-66.4
Curved
2.65
0.65
0.55
187.3
103.3
1.06
2.24
70.8
158.2
2.12
74.2
2.49
4.15
0.38
180.2
67.7
0.48
0.22
72.5
16.3
0.47
75.7
35.4
Inner
-20.0
157.4
Outer
Wall shear rate results for each experiment. The steady flow (yS) and mean (y,) wall shear rates are normalized by the appropriate Poiseuille value (yp), while the shear rate amplitudes (“la,,,& are normalized by the Womersley theory (yw) for a straight rigid tube. Normalized results are denoted by r. No phase angle exists for steady flow; the values merely indicate the phase angle trial with which they were recorded.
Yw
‘lamp
l-
Yw
tssl)
amolitude
/
rm
Ys
0.98
1.04
Ym
89.0
83.6
(s-l)
YP
45.6
87.6
WSR
-65.0
86.7
rm
(s-9
WSR
wall
(deg)
Straight
Ym 6-l)
‘lamp
WSR
flow
Ys (s-l)
Mean
Steadv
Measurement
Phase
Model
Table
Vol. 34, No. 1
Wclll shear rate in a curved artery
model
11
normalized mean wall shear rates (I,) generally agree within one experimental standard deviation of the predictions of Wang and Tarbell (1992) for a straight elastic tube (Fig. 4). The normalized mean wall shear rate is expected to be less than unity because of the convective acceleration due to wall motion (Wang and Tarbell, 1992). The impedance phase angle has a relatively small effect on the mean wall shear rate. This is consistent with the work of Wang and Tarbell (1992) and Dutta et al. (1992), which predict only a modest decrease in yri, with decreasing negative phase angles in straight elastic tubes. The wall shear rate amplitudes are only 79-86% of the Womersley amplitudes for a rigid pipe (Fig. 5), also in general agreement within one standard deviation of the predictions of the elastic tube theory (Wang and Tarbell, 1995). Under the conditions of our experiments, the impedance phase angle also has only a modest effect on the wall shear rate amplitude in the straight elastic tube, which is consistent with the theory of Wang and Tarbell (1995). Accordingly, the ratio of the amplitude to the mean of the wall shear rate waveform is consistent with the elastic tube theory (see Table 2). The consistent deviation of the ratio of yam to ym from the rigid tube theory (Table 2) indicates that the phase ang Pe does not influence the oscillatory nature of flow greatly, in a straight elastic tube, under these conditions. Rhee and Tarbell (1994) demonstrated the accuracy of the photochromic method for determining wall shear rates in a straight rigid tube. They obtained steady flow shear rates consistent with Poiseuille flow and sinusoidal flow shear rate amplitudes consistent with the Womersley theory. However, several steps involved in the tracing of each dye line can affect the results obtained, specifically, the placement of the origin at the exact measurement site in each frame and the manual tracing of the trailing edge of the dye line. Also, the actual time delay between the laser pulse and the strobe flash may not be uniform within a particular flow cycle, varying by 2%. Rhee and Tarbell (1994) estimated the uncertainty in the wall shear rate waveform parameters to be about 5%. To better quantify the uncertainty in the present results, standard deviations of the curve fit parameters have been reported with the results obtained. In the straight elastic tube, the uncertainty (the ratio of the standard deviation to the experimental value) in y,,, was ll%, while the uncertainty in “lampwas 9%. These estimates are limited, however, in that only one cycle from one experiment was analyzed. A more thorough analysis of multiple cycles in repeated photochromic experiments using a bifurcation model, has been performed by Lee and Tarbell (1997). For the curved tube, 180’ from the entrance, the normalized wall shear rate under steady flow conditions is compared to the normalized mean wall shear rate for sinusoidal flow in Fig. 6. Skewing of the velocity profile was observed as a result of curvature. Under steady flow conditions, the shear rate is 4.4-6.2 times greater at the outer wall than at the inner wall as a result of skewing of the velocity profile. Under sinusoidal flow conditions, wall motion caused the timeaveraged mean shear rate to be reduced greatly (37-55%) from its steady flow value at the inner wall, but no significant effects were observed at the outer wall. The wall shear rate amplitude is shown as a function of phase angle in Fig. 7. The amplitudes are normalized by the Womersley amplitude (yw) for flow in a straight rigid pipe. Curvature by itself has a slight effect on yamp, with values within 10% of yw (Chang and Tarbell, 1985). Considering wall motion, however, the shear rate amplitudes in the curved elastic tube are similar in magnitude at both the inner and outer walls but are only 36-55% of the Womersley amplitude. Comparing this to the straight elastic tube results,
Vol. 34, No. 1
W&U shear mte in a curved artery model 15-
Mean
---*--,
wang/rarben
Phase angle (deg) Fig. 4. Normahzed mean watl shear rates for sinusoidal flow as a function of phase angle in the straight elastic tube; values of Tm predicted from the straight elastic tube theory (Wang and Tarbell, 1992) are also shown. Error bars represent one standard deviation in curve fit parameters. The time-averaged mean shear rates generally fall within one standard deviation of the predictions of the straight elastic tube theory (Wang and Tarbell, 1992).
0 6-
Phase angle (deg) Fig. 5. Normalized wall shear rate amplitude as a function of phase angle in the straight elastic tube. Predictions for the shear rate amplitudes in an elastic tube (Wang and Tarbell, 1995) generally agree with the experimental results within one standard deviation.
Vol. 34, No. 1
Wall shear rate in a curned
atiety
13
model
3.0
1.5 -
1.0-
---*mm.
I -------------_____________________ 0.0 -70
. . ..I’....... -60
-50
.I....,....,.... -40
Sleadyblner) Mean (inner)
3 -xl
-20
. -10
Phase angle (deg) Fig. 6. Normalized wall shear rates for steady flow and mean wall shear rates for sinusoidal flow as a function of phase angle at both the inner and outer walls of the curved elastic tube. The wall shear rate is about 5 times greater at the outer wall under both steady and sinusoidal flow conditions. At the inner wall, the mean wall shear rate is reduced from its steady flow value, but at the outer wall there is no significant difference between the mean and steady flow values.
-50
40
-30
Phase angle (deg)
Fig. 7.
Normalized wall shear rate amplitude as a function of phase angle in the curved elastic tube. The amplitude at the outer wall decreases with decreasing phase angle, while the amplitude at the inner wall increases over this same range. The shear rate amplitude is much less at both walls of the curved tube than in the straight tube.
14
Wall shear rate in a cuwed
artety
model
Vol. 34, No. 1
curvature affects the wall shear rate amplitude in elastic tubes significantly. In the straight tube, the shear rate amplitudes are 1421% less than the rigid tube theory, while the amplitudes in the curved elastic tube are reduced 45-64% at both walls-a very large effect. The shear rate amplitudes at each wall show an opposite dependence on phase angle. At the outer wall, the shear rate amplitude decreases as the phase angle becomes more negative, but at the inner wall, the shear rate amplitude increases as the phase angle becomes more negative. These trends indicate that changes in the phase angle cause changes in local flow patterns which affect the wall shear rate amplitude. Considering the oscillatory nature of the flow as characterized by the ratio yamp / yrn (Fig. S), both the curvature and the phase angle have significant effects. The straight elastic tube theory predicts wall shear reversal (yw / yp greater than unity, see Table 2) under these flow conditions. However, curvature induces changes in this ratio. The experimental values at the outer wall are much less than the straight tube values and much less than unity (0.33-0.65), indicating that no shear reversal occurs at the outer wall, as was apparent from the videotape. Quite the opposite behavior is observed at the inner wall. Shear reversal is observed as the ratio is much greater than unity (3.30-7.19), and much greater than that predicted by the straight tube theory. In the straight elastic tube, this ratio is 13-23% less than predicted by the rigid tube theory. In the curved tube, flow is much less oscillatory at the outer wall (75-84% less than the rigid tube theory) and much more oscillatory at the -
l3pt (outer)
IP 0
I
.
-7Cl
.
,
I
-60
’
.
I
-SO
.
.
I
’
-40
’
’
1
-30
.
’
-
’
-20
.
.
.-
-10
Phase angle (deg)
Fig. 8.
Ratio of the wall shear rate amplitude to the mean wail shear rate as a function of phase angle in the curved elastic tube. The experimental values are compared to predictions from the straight elastic tube theory (Wang and Tarbell, 1992; 1995). The error bars on the outer wall data are too small to be shown. At the outer wall, flow is only slightly oscillatory, with a shear rate amplitude only 33-65% of the mean. At the inner wall, flow is much more oscillatory, with shear rate amplitudes 3.3S7.19 times the mean, showing a significant degree of reversal. Decreasing the phase angle to more negative values causes flow at the outer wall to become less oscillatory and flow at the inner wall to become more oscillatory.
Vol. 34, No. 1
Wall shear rate in a curved artery model
15
inner wall (1.27-3.25 times the rigid tube theory). Phase angle changes affect this ratio. Decreasing the phase angle from -55” to -66” causes the oscillatory nature of flow at the outer wall to be diminished. In contrast to the outer wall, decreasing the phase angle from -55” to -66” causes the degree of shear reversal at the inner wall to increase greatly. The values of the ratio yamp / ym (Fig. 8) show that the phase angle reduces the oscillatory nature of flow at the outer wall while greatly enhancing it at the inner wall. These results differ in some respects from the findings of Klanchar et al. (1990). They found that the peak shear rate at both walls increased significantly as the phase angle was reduced from -60” to -80”. The present results support such an effect only at the inner wall. The major difference between the two studies lies in the techniques used. Unlike Klanchar et al., the present study measured the local flow rate and the entire wall shear rate waveform directly; thus our results should represent a refinement of the earlier study. These experiments demonstrate that while impedance phase angle may not exert a great influence on either the mean or the amplitude of the wall shear rate waveform in straight vessels,it has a significant influence in a curved tube, particularly at very negative phase angles. Vessel curvature induces low mean shear rates and changes in shear direction (oscillatory shear) on the inner wall, where atherosclerotic plaques tend to be localized. Vessel wall motion tends to exaggerate these features, particularly at more negative phase angles. As hypertension is associated with more negative phase angles (Merillon et al., 1982), it may exacerbate atherogenesis in curved vessels by inducing lower and more oscillatory shear stress on the inner curvature through a phase angle mechanism as we have described. Vasorelaxants increase phase angle (White et aZ., 1994) and may be therapeutic for atherosclerosis, at least in part, through a phase angle effect on wall shear.
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CHOI, U.S., TALBOT, L., and CORNET, I. (1979). Experimental study of wall shear rates in the entry region of a curved tube. J Fluid Mech. 93, 465489.
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Walt shear rate in a curved artery model
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OJHA, M., HUMMEL, R.L., COBBOLD, R.S.C., and JOHNSTON, Development and evaluation of a high resolution K.W. (1988). photochromic dye method for pulsatile flow studies. j Whys. E. Sci. zn.str. 21, 998-1004. OKADOME, K., YUKIZANE, T., MII, S., FUKUDA, A., and Correlation of long-term results of SUGIMACHI , K (1989). femoropopliteal autogenous vein grafts with shear stress variation: in situ or reversed graft?J. Cardiovasc. Surg. 30, 932-935. POPOVICH, non-disturbing 22,21-25.
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RHEE, K., and TARBELL, J.M. (1994). A study of the wall shear rate distribution near the end-to-end anastomosis of a rigid graft and a compliant artery. J. Birch. 27, 329-338. RI’ITGERS, S.E., KARAYANNACOS, P.E., GUY, J.F., NEREM, R.M., SHAW, G.M., HOSTETLER, J.R., and VASKO, J.S. (1978). Velocity distribution and intimal proliferation in autologous vein grafts in dogs. Circ. Res. 42, 792-801. WALDEN, R., L’ITALIEN, G.J., MEGERMAN, J., and ABBOTT, W.M. (1980). Matched elastic properties and successful arterial grafting. Arch. SUTg. 115, 1166-1169. WANG, D.M., and TARBELL, J.M. (1992). Nonlinear analysis of flow in an elastic tube (artery): steady streaming effects. J. Z&id Mech. 239, 341-358. WANG, D.M., and TARBELL, J.M. (1995). Nonlinear analysis of oscillatory flow, with a non-zero mean, in an elastic tube (artery). J Biomech. Eng. 117, 127-135.
WESTON, M.W., RHEE, K., and TARBELL, J.M. (1996). Compliance and diameter mismatch affect the wall shear rate distribution near an end-to-end anastomosis. J Biomech. 29, 187-198. WHITE, K.C., KAVANAUGH, J.F., WANG, D.M., and TARBELL, J.M. (1994). Hemodynamics and wall shear rate in the abdominal aorta of dogs: effects of vasoactive drugs. Circ. Res. 75, 637-649. Acknowledgments This study was supported in part by NIH grant HL35549. The authors thank Dr. Russell Heikes of the Georgia Institute of Technology in Atlanta for discussion of statistical issues. Received 17 October 1995;
accepted
in
revised form
1 February
1997.