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Taper: An important feature of y-bifurcations in porcine renal arteries and human cerebral arteries
J. Biomechanics Vol. 25, No. 9. pp. 1047-1052, Printed in Great Britain
1992. 0
TECHNICAL
0021-9290/92 $.5.00+0.00 1992 Pergamon F’mss Ltd
NOTE
TAPER: AN IMPORTANT FEATURE OF Y-BIFURCATIONS IN PORCINE RENAL ARTERIES AND HUMAN CEREBRAL ARTERIES* NEIL F. MACLEAN,~$ RALPH G. KRATKY, THOMAS W. R. MACFARLANE and MARGOT R. ROACH Department of Medical Biophysics, University of Western Ontario, London, Ontario N6A Xl, Canada Abstract-The geometry of arterial bifurcations has been shown to alter fluid flow and the propagation of both pressure and flow waves. Here we provide a more complete description of the renal artery bifurcation geometry and show that the geometry of the bifurcation is more complex than was believed previously. The objective of this study was to quantify changes in cross-sectional luminal area in systemic arterial bifurcations using the method developed by Macfarlane [Ph.D. thesis, University of Western Ontario, Ontario (1985)] to study the geometry of human cerebral bifurcations. Porcine renal arterial bifurcations from seven young (6-14 weeks) and six old (> 52 weeks) animals were pressure-fixed (P = 140 mmHg) for 3 h with 10% formalin. Bifurcations were embedded in a block of frozen latex paint. Serial sections were cut at 20 pm (k 0.01 pm) using a sledge microtome while the block face was scanned with a video camera and the images were stored on a videotape. The luminal area was measured digitally upon playback. Bifurcations from the two age groups changed in cross-sectional area not only at the flow divider but also along the socalled straight regions. Proximal linear increases in cross-sectional area were observed proximal to the apex of the bifurcation in both young and old vessels, while linear luminal area decreases were measured in the daughter branches of both young and old porcine renals. Taper was defined as the change in luminal area per unit length of parent and daughter branches, respectively. All tapers were significantly different from zero and were used to compare luminal area changes in young and old renal arteries to human cerebral bifurcations. All vessels showed the same trend: negative taper, or expansion in luminal area, was present in the parent branch proximal to the apex of the flow divider while positive taper was present in the daughter branches. The magnitude of the taper present in both the parent and daughter branches from the two age groups of porcine renals and human cerebral bifurcations were not significantly different (P > 0.05). The area ratio, defined as the ratio of the summed luminal area of the daughter branches to that of the parent branch, was not significantly different in the young and the old porcine vessels as well as from the optimum value (1.2). Taper provides a better representation of the changes in luminal cross-sectional area present at bifurcations since area ratios do not take continuous luminal area changes into account and are very dependent on the location of the area measurement. These large luminal area changes, which have been overlooked previously, have significant implications for systemic flow patterns as well as for the propagation of flow and pressure waves. INTRODUCTION The human arterial tree is a complex network of branching and bending vessels. Blood flow in the arteries has been studied intensely since arterial diseases such as atherosclerosis and aneurysms seem to develop preferentially at bends and bifurcations. Since blood flow has been shown to be disturbed in these regions (Karino and Goldsmith, 1985), fluid forces such as shear stress have been implicated as one of the possible factors responsible for the localization of atherosclerosis. Previous studies (Adamson and Roach, 1981; Karino and Goldsmith, 1981; Houle and Roach, 1981; Ku et al., 1985) have measured blood velocity through model bifurcations so that flow forces, such as shear stress, could be
Received in Jinal form4 February 1992.
*Supported by a grant from the Heart and Stroke Foundation of Ontario and bv the iTerence W. Gilmore Memorial Scholarship. * t Author to whom correspondence should be addressed: Neil MacLean, Room 115 Medical Biophysics, Health Sciences Centre, University of Western Ontario, London, Ontario N6A 5C1, Canada. Phone: (519) 663-3627; Fax: (519) 661-2123.
correlated to plaque location. Traditionally, particles or dyes were injected into bifurcation models made from long, straight sections of glass tubing fused at a predetermined angle. These models were used to study the effect of branch angle, branch diameter, Reynolds number [(density x velocity x diameter/viscosity)] and other flow parameters on particle trajectory. Macfarlane et al. (1983) showed that human cerebral arterial bifurcations had a large taper on either side of the bifurcation. Luminal cross-sectional area of the parent branch increased linearly by 94.5% (*5.0% S.E.M.) up to the apex of the bifurcation and decreased linearly by 36.9% (&4.0% S.E.M.) distal to the flow divider. These changes could have a significant effect on both blood velocity and shear stress in regions of bifurcations. Recent bifurcation models have incorporated in uiuo bifurcation geometry by either casting fresh arterial samples with silicone and epoxy resins to produce a rigid, transparent model (Friedman et al., 1980; Rayman et al., 1985) or by dehydrating the sample with alcohol followed by immersion in methylsalicylate, which rendered the artery rigid, yet transparent (Karino and Motomiya, 1983). While numerous studies used correct bifurcation geometry, they did not realize that large tapers may have been present in their models. The effect of these tapers on blood flow has not been examined. We have incorporated methods developed by Macfarlane et al. (1983)
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1048
Technical Note
to provide a more complete description of cross-sectional luminal area changes in porcine renal arterial bifurcations than has been available previously. Since flow parameters such as volume rate of flow and Reynolds number are functions of tube diameter, the results of this study may have implications on the way in which future hemodynamic studies are interpreted. METHODS Tissue selection
Porcine renal arterial bifurcations were chosen as representative of muscular systemic bifurcations. Although the renal artery is elastic at the aorta, the vessel becomes muscular before the first bifurcation, the region studied here. The renal bifurcation has planar geometry (i.e. parent trunk and daughter branches lie in the same plane), which simplifies the analysis. Porcine renal arteries were harvested from two groups of animals; the first group of tissue was collected from animals at the abattoir (age > 52 weeks, n = 6) and the second group was obtained from donor mini-pigs used for bowel transplants here at University Hospital (age 6-14 weeks, n = 7). Since there was a large difference in animal age as well as a difference in animal strain, the tissue was separated into two groups based on age. Changes in cross-sectional luminal area were assessed in the renal arterial bifurcation from the block face as serial sections were removed (Macfarlane et af., 1983).Distortions due to tearing and proceising encountered commonly in histological sections were minimized. Formalin fixation and sectioning
Porcine renal arterial bifurcations were harvested from the animal with the abdominal aorta and kidney still attached, and excess connective tissue was removed. A rigid plastic tubing connected the proximal end of the aorta to a pressure-fixation apparatus. The bifurcation was made pressure-tight by tying-off the distal end of the aorta, the renal vein and the ureter. In oioo arterial geometry was obtained by pressure-fixing the vessel with 10% buffered formalin at mean porcine arterial pressure (140 mmHg; Pond and Houpt, 1978) for a period of 3 h. The effect of smooth muscle tone in these vessels was reduced by leaving the samples in saline (T= 4” C) for 1 day prior to pressure-fixing the vessels. Kratky et al. (1991) found that recoil in pressurefixed canine carotid arteries reached a minimum of 5% in diameter and 10% in length following 3 h with this fixative if the values were compared to physiological zero pressure. The precise location of the luminal surface of the artery was enhanced in two ways: (1) white latex paint was used as the embedding compound, and (ii) the sample was stained with methylene blue (100 mg/lOO ml) for 1 h and excess dye removed with saline. A Kryomat unit (Leitz) was used to freeze the latex paint on the stage of the sledge microtome by circulating refrigerated methanol (T= -35°C) through both the stage and the knife of the microtome. The parent trunk of the bifurcation was embedded perpendicular to the microtome stage to ensure that cross sections were produced. The samples were mounted vertically so that the daughter branches were closest to the blade. Initially, the lumen of the stained artery was filled with latex paint prior to building-up of the embedding block around the exterior of the vessel. The block was allowed to freeze for 1 h prior to sectioning to achieve thermal equilibrium within the block. Macfarlane et aZ.(1983) showed that volume changes of the latex block due to freezing were negligible ( f 0.03%). The length of the samples sectioned was limited by the vertical distance between the blade and the stage, which was 2.5 cm. Serial sections were cut at 20.00 + 0.01 pm increments from the block face by sliding the embedded artery and stage under a refrigerated knife (T= -35°C). During the
sectioning process, an image of the top surface of the block was recorded with an inverted video camera (Sony, AVC 3260) and a 55 mm Nikkon close-up lens which was mounted at a constant height (Fig. 1). The video signal from the camera was interlaced with a time/date generator and stored on a videocassette. The videotape was replayed through an IBM-compatible computer equipped with a frame-grab board. Due to the large number of sections produced by this process (800-lOOO), we chose to examine every tenth slice. The luminal area was measured digitally using the existing software (ImagePro II), which required the operator to outline the lumen with a mouse (Logitech). Approximately 50 points were used to trace the luminal surface of the artery. Each of the two lumens produced in slices through the daughter region were analyzed separately. Since the digitized image was traced from a monitor composed of pixels which are not square, it was necessary to calibrate the area measurements by video recording a scale bar which was mounted at blade level and then rotated by 90” and recorded a second time. Calibration with the scale bar allowed us to convert area measurements in (pixels)’ to units of (mm)2. Data analysis
A careful mounting of the planar bifurcations prevented oblique sectioning of the parent trunk; however, sections of the daughter branches distal from the flow divider were cut obliquely. The oblique cross-sectional area is related to the transverse area or the ‘true’ cross-sectional area through the cosine of the half branch angle, defined as the angle subtended by the axis ofsymmetry of the parent trunk and one of the daughter branches. The maximum error introduced by not correcting oblique sections in bifurcations which possessed half-angles less than 25” was ~9%. This error was insignificant compared to the large changes present in luminal cross-sectional area. Any method of correcting crosssectional area caused a discontinuity in the area-location curve. In addition, curvature in the branches made determination of the branch angle difficult. As a result, we chose
Digitized
hago
D.,‘.
h7 Fig. 1. Schematic diagram of the apparatus used to section porcine renal arterial bifurcations at 20.00 pm (+ 0.01 pm). An inverted video camera recorded the top face of the tissue block onto a videotape. The luminal areas were measured digitally upon playback.
1049
Technical Note not to correct our data for branch angle and excluded samples with half-angles greater than 25”. Only four samples were rejected from this study. Geometrical parameters Area ratio. The area ratio (AR) is defined as the ratio of the summed luminal area of the daughter branches to that of the parent branch measured one parent-tube diameter from the apex (Zamir and Phipps, 1987). The parent tube diameter was calculated assuming that the luminal area was circular. Taper: slope ofthe area-location curves. The luminal-area measurements obtained from digitized serial sections of porcine renal arterial bifurcations were plotted as a function of distance from the flow divider (in mm). The cross-sectional area was largest at the flow divider. The luminal-area measurements were divided by the value at the flow divider to normalize the data. Sections of the parent trunk were assigned negative distances from the flow divider while daughter sections were assigned positive distances. The values reported for the luminal area of the daughter branches were the sum of the two branches. This normalization process was also performed on Macfarlane’s data (1985). Taper was defined as the slope of the area-location curve for both parent and daughter branches, and was calculated as the change in area (in mm’) from the apex to its minimum value divided by the distance (in mm) over which the change occurred. Tapers in the parent branches were negative and corresponded to an expansion in the cross-sectional area.
Sample
z”
-8
-6
-4 Sample
Changes in the luminal cross-sectional area of porcine renal arterial bifurcations of two age groups (age 6-14 weeks, n = 7; >52 weeks, n = 6) were compared to each other as well as to changes measured in human cerebral arterial bifurcations (Macfarlane, 1985) based on the taper from parent and daughter vessels, separately, using an analysis of variance (ANOVA, P ~0.05). The significance of the taper was determined from the correlation coefficient (?) and P-value using a P-value cutoff of 0.01 (McClave and Dietrich, 1986). The mean area ratios obtained from seven young and six old porcine vessels were compared using a two-tailed r-test (P
RESULTS
Cross-sectional luminal area. Cross-sectional luminal areas proximal to the flow divider increased while those distal to the flow divider decreased [Fig. 2(a) and (b)]. The percentage change in the luminal cross-sectional area in the parent branch (d.$) as well as the summed luminal-area change in the daughter branches (dA,) was calculated using the maximum area at the apex and the minimum area value in the parent and daughter vessels, respectively. The luminal cross-sectional area increased in the parent branch by 47.0% (*7.6% S.E.M.) in the old pig renal and 35.5% (+7.6% S.E.M.) in the ‘young’ vessels (Tables land 2). Macfarlane showed that the luminal area of the human cerebral arteries increased by 94.5% ( f 5.0%) proximal to the flow divider (Table 3). Since the magnitude of the area change depended on the length of the vessel, we compared the ratio of the diameter change to the vessel length (i.e. taper) in both young and old samples as well as in the human cerebral vessels. Taper. All tapers were significantly different from zero for both parent and daughter vessels in all the vessels studied (P < 0.05). All vessels showed the same trend: negative taper proximal to the flow divider, followed by positive taper distal to the flow divider. No significant difference was found in the magnitude of the taper between the parent branches of the young and old porcine renal and human cerebral arteries (ANOVA, P > 0.05). The taper present in the daughter bran-
0
Length
4
2
(mm)
0.5
-10
Statistical analysis
-2
-4
-2
0 Length
2
4
6
8
(mm)
Fig. 2. (a) Linear changes in the cross-sectional luminal area of the bifurcation were observed in the parent trunk as well as in the daughter region. Cross-sectional luminal area of the daughter branches represents the sum of the two branches. The method used to calculate tapers in the parent region is indicated on figure. The data has been normalized with respect to the location and luminal area of the apex. (b) The variation of luminal area present between samples of the old group was small. Daughter branches in samples OP-02 and OP-03 did not follow the trend observed since they were entering a second bifurcation.
ches of the vessels studied were not significantly different (Table 4). Area ratio. The mean area ratio, calculated one parenttube diameter proximal and distal to the flow divider, for the old pig group (1.2 IO.1) was not significantly different (P z 0.05) from that of the young pig group (0.8 kO.2). DISCUSSION
The goal of this study was to determine if systemic Ybifurcations and cerebral Y-bifurcations had comparable changes in geometry. Since our measurements showed that large changes in luminal area occurred over relatively short distances, we concluded that the measurement of area ratio described by others could be subject to large variations if the location of the measurement was not precise (because of the assumption that the vessels were cylindrical). For this reason we concluded that tapers were probably more important than area ratios. We concluded, based on the magnitude and the variance of the taper in the parent and daughter branches, that the young and the old porcine renal arterial bifurcations were not different. The difference between these groups might have become significant if the sample sizes of each of the three groups were increased. In both young and old parent branches. a large, linear increase in luminal area was observed
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Technical Note Table 1. Data from old pig group (> 1 year) Area change (%) Sample
Parent diameter (mm)
Parent
Daughter
OP-01 OP-02 OP-03 OP-05 OP-12 OP-19
5.0 5.6 5.4 3.6 3.4 7.1
45.5 32.9 68.9 83.6 37.1 13.9
12.7 3.1 0.7 31.6 19.5 27.9
(i::)
47.0 (10.5)
15.9 (5.2)
Average (+S.E.M.)
Taper Area ratio
Parent
Daughter
-1.8 -1.3 -2.5 -2.2 -2.1 -1.1
1.8 1.6 1.4 4.0
- 1.8 (0.2)
2.2 (0.5)
1.1 1.2 1.1
-
(i::,
Note. Parent diameters were calculated from the minimum cross-sectional luminal area in the parent branch. Area change represents the percentage change of luminal area in either parent or daughter branches relative to the area at the apex. Area ratios are defined as the ratio of the summed luminal area of the daughter branches to that of the parent branch measured one parenttube diameter from the apex. Taper was defined as the change in luminal area divided by the distance from the apex over which this change occurred.
Table 2. Data from young pig group (6-14 weeks) Area change (%) Daughter
Parent
Daughter
48.8 66.3 47.8 33.3 10.1 27.9 14.5
14.3 17.6 60.4 62.8 79.9 40.8 26.4
1.5 1.1 0.9 0.6 0.3 0.4 0.9
-1.6 - 1.3 -1.7 -0.7 -0.3 -0.7 -0.7
1.2 2.1 1.1 1.0 1.5 2.3 1.4
35.5 (7.6)
43.2 (9.5)
-1.0 (0.2)
(A::)
Parent
3.9 4.2 2.2 2.1 1.7 3.2 2.7
SP-16 SP-17 SP-21 SP-24 SP-30 SP-40 SP-60 Average (+ S.E.M.)
Taper Area ratio
Parent diameter (mm)
Table 3. Data obtained and calculated from human cerebral arterial bifurcations (from Macfarlane, 1985) Area change (%) Location
Age
Parent diameter (mm)
3-Rt. Mid 4-Rt. Mid. 1-Lt. Ant. 3-Lt. Ant. 2-Rt. Ant. 1-Rt. Ant.
65M 65M 68F 65M 4OM 68F
1.6 0.8 1.8 1.4 1.6 1.7
Average (*S.E.M.)
61.8 (4.4)
(&
Parent
Daughter
87.4 111.9 97.6 102.7 76.9 90.5
31.7 42.8 28.6 50.7 42.1 25.4
94.5 (5.0)
proximal to the flow divider. This would be representative of negative taper, or of an expansion. Distal to the flow divider a linear decrease in cross-sectional area was measured. Gosling et al. (1971) studied the effect of age on the area ratio of aortoiliac bifurcations from aortograms of healthy cockerels, dogs and humans. No variation of the area ratio with age was observed in cockerels and dogs, but the area ratio decreased linearly with age in humans from an optimum value of 1.11 in the first decade to a value of 0.75 by the tifth decade. Gosling’s data support the results of our study since we showed that there was no variation of area ratio with age in porcine renal
36.9 (4.0)
Taper (mm) Area ratio 0.8
Parent
Daughter
-0.7 -0.8 -1.9 -1.0 -2.0 -2.3
1.3 1.1 1.5 1.4 1.4 2.0
-1.5 (0.3)
(A::,
arterial bifurcations. Ludin et al. (1972) examined the alteration in renal artery diameter from abdominal aortograms of both children (n = 127), adults (n =458) and autopsy specimens (n=27) of both sexes which included both normal patients and patients suffering from renal disease. Ludin et a[. defined a tapering parameter to be the relative difference between the diameters measured at 1 and 3 cm from the origin. Roth renal artery diameter (at origin) and tapering parameter were statistically independent of age, sex and blood pressure at the time of admission. Although Ludin’s group showed that taper was present in the parent branch of
1051
Technical Note Table 4. Summary of mean data (+ S.E.M.) from young and old porcine renals and human cerebral bifurcations Young pig Area change (%) Parent Daughter Taper (mm) Parent Daughter Area ratio
35.5 k 1.6 43.2* 9.5 - l.OkO.2 1.5kO.2 0.8 +0.2
Old pig
Human cerebral
45.7 f 10.5 15.9f5.2
94.5 * 5.0 36.9k4.0
- 1.8kO.2 2.2kO.5 1.2*0.1
- 1550.2 1.5kO.2 -
Note. Young and old porcine renals and human cerebral data were compared with an ANOVA with a cutoff at PcO.05. The area ratio was calculated as the ratio of the summed luminal area of the daughter branches to the area of the parent trunk, both measurements made one-tube diameter from the apex of the bifurcation. The area ratios of young and old porcine renal bifurcations were not significantly different (P>O.O5). Taper refers to the slope of area-location curves. No significant difference was found between tapers in young and old porcine renal and human cerebral arteries in the parent and daughter branches, respectively.
the bifurcation. To our knowledge, the effect of a linear taper on pressure and flow waves has not been examined. Distal to the flow divider, the propagation of the waveforms becomes even more complicated by the presence of a linear decrease in area. Although recent flow models have maintained in uivo geometry, the effect of luminal cross-sectional area changes on flow profiles has been overlooked. We have reported the change in luminal area in both parent and daughter branches of porcine renal arterial bifurcations of two age groups. Although the magnitude of tapers in the parent and daughter branches of porcine renals and human cerebral bifurcations were not significantly different, this study has shown that large changes in the luminal area were present in all planar Ybifurcations examined. We believe that the taper present in muscular Y-bifurcations could have serious implications for the way future flow studies are designed and interpreted. Acknowledgements-We are grateful to both surgeons and technicians, Department of Surgery, University Hospital for providing the vessels from the young pigs; to Dr I. C. MacDonald for obtaining the old pig vessels from the abbatoir. Dr A. Buchan allowed us the use of his digitizing equipment.
REFERENCES the renal artery, our study showed that area changes were linear and were also present in the daughter branches. If we compared the ratios of the mean taper in the daughter branches to the mean taper in the parent trunk for both young and old samples, the young bifurcations would have a ratio of -2.22 (k 0.68 S.E.M., n = 7) and the old pigs would have a value of - 1.54 (kO.76 S.E.M., n=4). The taper ratios were compared using a t-test and were not significantly different (P > 0.05). The ratio of the tapers provides a measure of the asymmetry present between the area changes in the parent and the daughter branches of one sample. The luminal cross-sectional area changes of the porcine renal and human cerebral bifurcations (Macfarlane, 1985) were not significantly different. Since both bifurcations are classified as muscular, Y-bifurcations, we believe that large tapers could be present in other arteries, such as coronary and carotid arterial bifurcations. These tapers could have a significant effect on blood flow through them. Many studies have examined the effects of mean fluid shear stress as well as time-varying shear stress on the localization of atherosclerosis (Ku et al., 1985). Friedman et al. (1981) found an inverse relationship between the intimal thickness observed in mildly atherosclerotic aortoiliac bifurcations and fluid shear stress measured in models of this bifurcation. Friedman found the minimum shear stress to be present on the lateral walls opposite to the tip of the flow divider. This is the region which was found to have the largest cross-sectional area and where we would expect both flow velocity and shear stress to be minimized. The flow entering the parent branch of the renal artery would enter a zone which increased in area linearly, thereby causing it to decelerate, be divided into two branches at the apex, and then accelerate through the daughter branches which decrease linearly in luminal area. Our hypothesis is that the expansion zone proximal to the flow divider can alter the location of both separation and reattachment points of recirculation zones, but the minimum taper required to produce such an effect on flow remains to be studied. Since systemic blood flow is pulsatile, the expansion in the luminal cross-sectional area which we have reported could have an effect on both pressure and flow waves. Waves are reflected at points of impedance mismatching; however, since the luminal area increases linearly proximal to the bifurcation, we suspect that the impedance would change in a similar fashion, thereby reducing the mismatch at the apex of
Adamson, S. L. and Roach, M. R. (1981) Measurement of wall shear stress in a glass model renal bifurcation by a technique that monitors the rate of erosion of an opaque coating layer. Biorheology 18, 9-21. Friedman, M. H., Bargeron, C. B., Hutchins, G. M. and Mark, F. F. (1980) Hemodynamic measurements in human arterial casts and their correlation with histology and luminal area. J. biomech. Engng 102, 247-251. -Friedman, M. H.. Hutchins. Cl. M.. Bareeron. C. B.. Deters. 0. J. and Mark, F. F. (1981) CorrelaGon between intimal thickness and fluid shear in human arteries. Atherosclerosis 39,425-436. Gosling, R. Cl., Newman, D. L., Bowden, N. L. R. and Twinn, K. W. (1971) The area ratio of normal aortic junctions: aortic configuration and pulse-wave reflection. Br. J. Radiol. 44, 850-853.
Houle, S. and Roach, M. R. (1981) Flow studies in a rigid model of an aorto-renal junction: a case for high shear as a cause of the localization of sudanophilic lesions in rabbits. Atherosclerosis 40, 231-244. Karino, T. and Goldsmith, H. L. (1981) Role of blood cellwall interactions in thrombogenesis and atherogenesis: a microrheological study. Biorheology 21, 587-601. Karino, T. and Goldsmith, H. L. (1985) Particle flow behaviour in models of branching vessels: II. Effects of branching angle and diameter on flow patterns. Biorheology 22, 87-104. Karino, T. and Motomiya, M. (1983) Flow visualization in isolated transparent natural blood vessels. Biorheology 20, 119-127. Kratky, R. G., Lo, D. K. and Roach, M. R. (1991) Quantitative measurement of fixation rate and dimension changes in the aldehyde/pressure-fixed canine carotid artery. Blood Vessels 28, 386-395. 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.
Ludin, H., Juabin, E. J., Kaufmann, H. J. and vonPein, W. (1972) Significance of alterations in main renal artery calibre and configuration. Acta Radial. Diagnosis 12, 803-832.
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Technical Note
Macfarlane, T. W. R. (1985) A computer-based quantitative image analysis of the geometry of human cerebral arterial bifurcations. Ph.D. thesis, University of Western Ontario, London, Ontario, pp. 62-94. Macfarlane, T. W. R., Petrowski, S., Rigutto, L. and Roach, M. R. (1983) Computer based video analysis of cerebral arterial geometry using the natural fluorescence of the arterial wall and contrast enhancement techniques. Blood Vessels 20, 161-171. McClave, J. T. and Dietrich, F. H. (1986) A First Course in
Statistics, p. 296. Dellen, San Francisco, CA. Pond, W. G. and Houpt, K. A. (1978) The Biology ofthe Pig, p. 31. Comstock. Ithaca, NY. Rayman, R., Kratky, R. G. and Roach, M. R. (1985) Steady flow visualization in a rigid canine aortic cast. J. Biomechanics 18, 863-875. Zamir, M. and Phipps, S. (1987) Morphometric analysis of the distributing vessels of the kidney. Can. J. Physiol. Pharmacol. 65,2433-2440.
1992. 0
TECHNICAL
0021-9290/92 $.5.00+0.00 1992 Pergamon F’mss Ltd
NOTE
TAPER: AN IMPORTANT FEATURE OF Y-BIFURCATIONS IN PORCINE RENAL ARTERIES AND HUMAN CEREBRAL ARTERIES* NEIL F. MACLEAN,~$ RALPH G. KRATKY, THOMAS W. R. MACFARLANE and MARGOT R. ROACH Department of Medical Biophysics, University of Western Ontario, London, Ontario N6A Xl, Canada Abstract-The geometry of arterial bifurcations has been shown to alter fluid flow and the propagation of both pressure and flow waves. Here we provide a more complete description of the renal artery bifurcation geometry and show that the geometry of the bifurcation is more complex than was believed previously. The objective of this study was to quantify changes in cross-sectional luminal area in systemic arterial bifurcations using the method developed by Macfarlane [Ph.D. thesis, University of Western Ontario, Ontario (1985)] to study the geometry of human cerebral bifurcations. Porcine renal arterial bifurcations from seven young (6-14 weeks) and six old (> 52 weeks) animals were pressure-fixed (P = 140 mmHg) for 3 h with 10% formalin. Bifurcations were embedded in a block of frozen latex paint. Serial sections were cut at 20 pm (k 0.01 pm) using a sledge microtome while the block face was scanned with a video camera and the images were stored on a videotape. The luminal area was measured digitally upon playback. Bifurcations from the two age groups changed in cross-sectional area not only at the flow divider but also along the socalled straight regions. Proximal linear increases in cross-sectional area were observed proximal to the apex of the bifurcation in both young and old vessels, while linear luminal area decreases were measured in the daughter branches of both young and old porcine renals. Taper was defined as the change in luminal area per unit length of parent and daughter branches, respectively. All tapers were significantly different from zero and were used to compare luminal area changes in young and old renal arteries to human cerebral bifurcations. All vessels showed the same trend: negative taper, or expansion in luminal area, was present in the parent branch proximal to the apex of the flow divider while positive taper was present in the daughter branches. The magnitude of the taper present in both the parent and daughter branches from the two age groups of porcine renals and human cerebral bifurcations were not significantly different (P > 0.05). The area ratio, defined as the ratio of the summed luminal area of the daughter branches to that of the parent branch, was not significantly different in the young and the old porcine vessels as well as from the optimum value (1.2). Taper provides a better representation of the changes in luminal cross-sectional area present at bifurcations since area ratios do not take continuous luminal area changes into account and are very dependent on the location of the area measurement. These large luminal area changes, which have been overlooked previously, have significant implications for systemic flow patterns as well as for the propagation of flow and pressure waves. INTRODUCTION The human arterial tree is a complex network of branching and bending vessels. Blood flow in the arteries has been studied intensely since arterial diseases such as atherosclerosis and aneurysms seem to develop preferentially at bends and bifurcations. Since blood flow has been shown to be disturbed in these regions (Karino and Goldsmith, 1985), fluid forces such as shear stress have been implicated as one of the possible factors responsible for the localization of atherosclerosis. Previous studies (Adamson and Roach, 1981; Karino and Goldsmith, 1981; Houle and Roach, 1981; Ku et al., 1985) have measured blood velocity through model bifurcations so that flow forces, such as shear stress, could be
Received in Jinal form4 February 1992.
*Supported by a grant from the Heart and Stroke Foundation of Ontario and bv the iTerence W. Gilmore Memorial Scholarship. * t Author to whom correspondence should be addressed: Neil MacLean, Room 115 Medical Biophysics, Health Sciences Centre, University of Western Ontario, London, Ontario N6A 5C1, Canada. Phone: (519) 663-3627; Fax: (519) 661-2123.
correlated to plaque location. Traditionally, particles or dyes were injected into bifurcation models made from long, straight sections of glass tubing fused at a predetermined angle. These models were used to study the effect of branch angle, branch diameter, Reynolds number [(density x velocity x diameter/viscosity)] and other flow parameters on particle trajectory. Macfarlane et al. (1983) showed that human cerebral arterial bifurcations had a large taper on either side of the bifurcation. Luminal cross-sectional area of the parent branch increased linearly by 94.5% (*5.0% S.E.M.) up to the apex of the bifurcation and decreased linearly by 36.9% (&4.0% S.E.M.) distal to the flow divider. These changes could have a significant effect on both blood velocity and shear stress in regions of bifurcations. Recent bifurcation models have incorporated in uiuo bifurcation geometry by either casting fresh arterial samples with silicone and epoxy resins to produce a rigid, transparent model (Friedman et al., 1980; Rayman et al., 1985) or by dehydrating the sample with alcohol followed by immersion in methylsalicylate, which rendered the artery rigid, yet transparent (Karino and Motomiya, 1983). While numerous studies used correct bifurcation geometry, they did not realize that large tapers may have been present in their models. The effect of these tapers on blood flow has not been examined. We have incorporated methods developed by Macfarlane et al. (1983)
1047
1048
Technical Note
to provide a more complete description of cross-sectional luminal area changes in porcine renal arterial bifurcations than has been available previously. Since flow parameters such as volume rate of flow and Reynolds number are functions of tube diameter, the results of this study may have implications on the way in which future hemodynamic studies are interpreted. METHODS Tissue selection
Porcine renal arterial bifurcations were chosen as representative of muscular systemic bifurcations. Although the renal artery is elastic at the aorta, the vessel becomes muscular before the first bifurcation, the region studied here. The renal bifurcation has planar geometry (i.e. parent trunk and daughter branches lie in the same plane), which simplifies the analysis. Porcine renal arteries were harvested from two groups of animals; the first group of tissue was collected from animals at the abattoir (age > 52 weeks, n = 6) and the second group was obtained from donor mini-pigs used for bowel transplants here at University Hospital (age 6-14 weeks, n = 7). Since there was a large difference in animal age as well as a difference in animal strain, the tissue was separated into two groups based on age. Changes in cross-sectional luminal area were assessed in the renal arterial bifurcation from the block face as serial sections were removed (Macfarlane et af., 1983).Distortions due to tearing and proceising encountered commonly in histological sections were minimized. Formalin fixation and sectioning
Porcine renal arterial bifurcations were harvested from the animal with the abdominal aorta and kidney still attached, and excess connective tissue was removed. A rigid plastic tubing connected the proximal end of the aorta to a pressure-fixation apparatus. The bifurcation was made pressure-tight by tying-off the distal end of the aorta, the renal vein and the ureter. In oioo arterial geometry was obtained by pressure-fixing the vessel with 10% buffered formalin at mean porcine arterial pressure (140 mmHg; Pond and Houpt, 1978) for a period of 3 h. The effect of smooth muscle tone in these vessels was reduced by leaving the samples in saline (T= 4” C) for 1 day prior to pressure-fixing the vessels. Kratky et al. (1991) found that recoil in pressurefixed canine carotid arteries reached a minimum of 5% in diameter and 10% in length following 3 h with this fixative if the values were compared to physiological zero pressure. The precise location of the luminal surface of the artery was enhanced in two ways: (1) white latex paint was used as the embedding compound, and (ii) the sample was stained with methylene blue (100 mg/lOO ml) for 1 h and excess dye removed with saline. A Kryomat unit (Leitz) was used to freeze the latex paint on the stage of the sledge microtome by circulating refrigerated methanol (T= -35°C) through both the stage and the knife of the microtome. The parent trunk of the bifurcation was embedded perpendicular to the microtome stage to ensure that cross sections were produced. The samples were mounted vertically so that the daughter branches were closest to the blade. Initially, the lumen of the stained artery was filled with latex paint prior to building-up of the embedding block around the exterior of the vessel. The block was allowed to freeze for 1 h prior to sectioning to achieve thermal equilibrium within the block. Macfarlane et aZ.(1983) showed that volume changes of the latex block due to freezing were negligible ( f 0.03%). The length of the samples sectioned was limited by the vertical distance between the blade and the stage, which was 2.5 cm. Serial sections were cut at 20.00 + 0.01 pm increments from the block face by sliding the embedded artery and stage under a refrigerated knife (T= -35°C). During the
sectioning process, an image of the top surface of the block was recorded with an inverted video camera (Sony, AVC 3260) and a 55 mm Nikkon close-up lens which was mounted at a constant height (Fig. 1). The video signal from the camera was interlaced with a time/date generator and stored on a videocassette. The videotape was replayed through an IBM-compatible computer equipped with a frame-grab board. Due to the large number of sections produced by this process (800-lOOO), we chose to examine every tenth slice. The luminal area was measured digitally using the existing software (ImagePro II), which required the operator to outline the lumen with a mouse (Logitech). Approximately 50 points were used to trace the luminal surface of the artery. Each of the two lumens produced in slices through the daughter region were analyzed separately. Since the digitized image was traced from a monitor composed of pixels which are not square, it was necessary to calibrate the area measurements by video recording a scale bar which was mounted at blade level and then rotated by 90” and recorded a second time. Calibration with the scale bar allowed us to convert area measurements in (pixels)’ to units of (mm)2. Data analysis
A careful mounting of the planar bifurcations prevented oblique sectioning of the parent trunk; however, sections of the daughter branches distal from the flow divider were cut obliquely. The oblique cross-sectional area is related to the transverse area or the ‘true’ cross-sectional area through the cosine of the half branch angle, defined as the angle subtended by the axis ofsymmetry of the parent trunk and one of the daughter branches. The maximum error introduced by not correcting oblique sections in bifurcations which possessed half-angles less than 25” was ~9%. This error was insignificant compared to the large changes present in luminal cross-sectional area. Any method of correcting crosssectional area caused a discontinuity in the area-location curve. In addition, curvature in the branches made determination of the branch angle difficult. As a result, we chose
Digitized
hago
D.,‘.
h7 Fig. 1. Schematic diagram of the apparatus used to section porcine renal arterial bifurcations at 20.00 pm (+ 0.01 pm). An inverted video camera recorded the top face of the tissue block onto a videotape. The luminal areas were measured digitally upon playback.
1049
Technical Note not to correct our data for branch angle and excluded samples with half-angles greater than 25”. Only four samples were rejected from this study. Geometrical parameters Area ratio. The area ratio (AR) is defined as the ratio of the summed luminal area of the daughter branches to that of the parent branch measured one parent-tube diameter from the apex (Zamir and Phipps, 1987). The parent tube diameter was calculated assuming that the luminal area was circular. Taper: slope ofthe area-location curves. The luminal-area measurements obtained from digitized serial sections of porcine renal arterial bifurcations were plotted as a function of distance from the flow divider (in mm). The cross-sectional area was largest at the flow divider. The luminal-area measurements were divided by the value at the flow divider to normalize the data. Sections of the parent trunk were assigned negative distances from the flow divider while daughter sections were assigned positive distances. The values reported for the luminal area of the daughter branches were the sum of the two branches. This normalization process was also performed on Macfarlane’s data (1985). Taper was defined as the slope of the area-location curve for both parent and daughter branches, and was calculated as the change in area (in mm’) from the apex to its minimum value divided by the distance (in mm) over which the change occurred. Tapers in the parent branches were negative and corresponded to an expansion in the cross-sectional area.
Sample
z”
-8
-6
-4 Sample
Changes in the luminal cross-sectional area of porcine renal arterial bifurcations of two age groups (age 6-14 weeks, n = 7; >52 weeks, n = 6) were compared to each other as well as to changes measured in human cerebral arterial bifurcations (Macfarlane, 1985) based on the taper from parent and daughter vessels, separately, using an analysis of variance (ANOVA, P ~0.05). The significance of the taper was determined from the correlation coefficient (?) and P-value using a P-value cutoff of 0.01 (McClave and Dietrich, 1986). The mean area ratios obtained from seven young and six old porcine vessels were compared using a two-tailed r-test (P
RESULTS
Cross-sectional luminal area. Cross-sectional luminal areas proximal to the flow divider increased while those distal to the flow divider decreased [Fig. 2(a) and (b)]. The percentage change in the luminal cross-sectional area in the parent branch (d.$) as well as the summed luminal-area change in the daughter branches (dA,) was calculated using the maximum area at the apex and the minimum area value in the parent and daughter vessels, respectively. The luminal cross-sectional area increased in the parent branch by 47.0% (*7.6% S.E.M.) in the old pig renal and 35.5% (+7.6% S.E.M.) in the ‘young’ vessels (Tables land 2). Macfarlane showed that the luminal area of the human cerebral arteries increased by 94.5% ( f 5.0%) proximal to the flow divider (Table 3). Since the magnitude of the area change depended on the length of the vessel, we compared the ratio of the diameter change to the vessel length (i.e. taper) in both young and old samples as well as in the human cerebral vessels. Taper. All tapers were significantly different from zero for both parent and daughter vessels in all the vessels studied (P < 0.05). All vessels showed the same trend: negative taper proximal to the flow divider, followed by positive taper distal to the flow divider. No significant difference was found in the magnitude of the taper between the parent branches of the young and old porcine renal and human cerebral arteries (ANOVA, P > 0.05). The taper present in the daughter bran-
0
Length
4
2
(mm)
0.5
-10
Statistical analysis
-2
-4
-2
0 Length
2
4
6
8
(mm)
Fig. 2. (a) Linear changes in the cross-sectional luminal area of the bifurcation were observed in the parent trunk as well as in the daughter region. Cross-sectional luminal area of the daughter branches represents the sum of the two branches. The method used to calculate tapers in the parent region is indicated on figure. The data has been normalized with respect to the location and luminal area of the apex. (b) The variation of luminal area present between samples of the old group was small. Daughter branches in samples OP-02 and OP-03 did not follow the trend observed since they were entering a second bifurcation.
ches of the vessels studied were not significantly different (Table 4). Area ratio. The mean area ratio, calculated one parenttube diameter proximal and distal to the flow divider, for the old pig group (1.2 IO.1) was not significantly different (P z 0.05) from that of the young pig group (0.8 kO.2). DISCUSSION
The goal of this study was to determine if systemic Ybifurcations and cerebral Y-bifurcations had comparable changes in geometry. Since our measurements showed that large changes in luminal area occurred over relatively short distances, we concluded that the measurement of area ratio described by others could be subject to large variations if the location of the measurement was not precise (because of the assumption that the vessels were cylindrical). For this reason we concluded that tapers were probably more important than area ratios. We concluded, based on the magnitude and the variance of the taper in the parent and daughter branches, that the young and the old porcine renal arterial bifurcations were not different. The difference between these groups might have become significant if the sample sizes of each of the three groups were increased. In both young and old parent branches. a large, linear increase in luminal area was observed
1050
Technical Note Table 1. Data from old pig group (> 1 year) Area change (%) Sample
Parent diameter (mm)
Parent
Daughter
OP-01 OP-02 OP-03 OP-05 OP-12 OP-19
5.0 5.6 5.4 3.6 3.4 7.1
45.5 32.9 68.9 83.6 37.1 13.9
12.7 3.1 0.7 31.6 19.5 27.9
(i::)
47.0 (10.5)
15.9 (5.2)
Average (+S.E.M.)
Taper Area ratio
Parent
Daughter
-1.8 -1.3 -2.5 -2.2 -2.1 -1.1
1.8 1.6 1.4 4.0
- 1.8 (0.2)
2.2 (0.5)
1.1 1.2 1.1
-
(i::,
Note. Parent diameters were calculated from the minimum cross-sectional luminal area in the parent branch. Area change represents the percentage change of luminal area in either parent or daughter branches relative to the area at the apex. Area ratios are defined as the ratio of the summed luminal area of the daughter branches to that of the parent branch measured one parenttube diameter from the apex. Taper was defined as the change in luminal area divided by the distance from the apex over which this change occurred.
Table 2. Data from young pig group (6-14 weeks) Area change (%) Daughter
Parent
Daughter
48.8 66.3 47.8 33.3 10.1 27.9 14.5
14.3 17.6 60.4 62.8 79.9 40.8 26.4
1.5 1.1 0.9 0.6 0.3 0.4 0.9
-1.6 - 1.3 -1.7 -0.7 -0.3 -0.7 -0.7
1.2 2.1 1.1 1.0 1.5 2.3 1.4
35.5 (7.6)
43.2 (9.5)
-1.0 (0.2)
(A::)
Parent
3.9 4.2 2.2 2.1 1.7 3.2 2.7
SP-16 SP-17 SP-21 SP-24 SP-30 SP-40 SP-60 Average (+ S.E.M.)
Taper Area ratio
Parent diameter (mm)
Table 3. Data obtained and calculated from human cerebral arterial bifurcations (from Macfarlane, 1985) Area change (%) Location
Age
Parent diameter (mm)
3-Rt. Mid 4-Rt. Mid. 1-Lt. Ant. 3-Lt. Ant. 2-Rt. Ant. 1-Rt. Ant.
65M 65M 68F 65M 4OM 68F
1.6 0.8 1.8 1.4 1.6 1.7
Average (*S.E.M.)
61.8 (4.4)
(&
Parent
Daughter
87.4 111.9 97.6 102.7 76.9 90.5
31.7 42.8 28.6 50.7 42.1 25.4
94.5 (5.0)
proximal to the flow divider. This would be representative of negative taper, or of an expansion. Distal to the flow divider a linear decrease in cross-sectional area was measured. Gosling et al. (1971) studied the effect of age on the area ratio of aortoiliac bifurcations from aortograms of healthy cockerels, dogs and humans. No variation of the area ratio with age was observed in cockerels and dogs, but the area ratio decreased linearly with age in humans from an optimum value of 1.11 in the first decade to a value of 0.75 by the tifth decade. Gosling’s data support the results of our study since we showed that there was no variation of area ratio with age in porcine renal
36.9 (4.0)
Taper (mm) Area ratio 0.8
Parent
Daughter
-0.7 -0.8 -1.9 -1.0 -2.0 -2.3
1.3 1.1 1.5 1.4 1.4 2.0
-1.5 (0.3)
(A::,
arterial bifurcations. Ludin et al. (1972) examined the alteration in renal artery diameter from abdominal aortograms of both children (n = 127), adults (n =458) and autopsy specimens (n=27) of both sexes which included both normal patients and patients suffering from renal disease. Ludin et a[. defined a tapering parameter to be the relative difference between the diameters measured at 1 and 3 cm from the origin. Roth renal artery diameter (at origin) and tapering parameter were statistically independent of age, sex and blood pressure at the time of admission. Although Ludin’s group showed that taper was present in the parent branch of
1051
Technical Note Table 4. Summary of mean data (+ S.E.M.) from young and old porcine renals and human cerebral bifurcations Young pig Area change (%) Parent Daughter Taper (mm) Parent Daughter Area ratio
35.5 k 1.6 43.2* 9.5 - l.OkO.2 1.5kO.2 0.8 +0.2
Old pig
Human cerebral
45.7 f 10.5 15.9f5.2
94.5 * 5.0 36.9k4.0
- 1.8kO.2 2.2kO.5 1.2*0.1
- 1550.2 1.5kO.2 -
Note. Young and old porcine renals and human cerebral data were compared with an ANOVA with a cutoff at PcO.05. The area ratio was calculated as the ratio of the summed luminal area of the daughter branches to the area of the parent trunk, both measurements made one-tube diameter from the apex of the bifurcation. The area ratios of young and old porcine renal bifurcations were not significantly different (P>O.O5). Taper refers to the slope of area-location curves. No significant difference was found between tapers in young and old porcine renal and human cerebral arteries in the parent and daughter branches, respectively.
the bifurcation. To our knowledge, the effect of a linear taper on pressure and flow waves has not been examined. Distal to the flow divider, the propagation of the waveforms becomes even more complicated by the presence of a linear decrease in area. Although recent flow models have maintained in uivo geometry, the effect of luminal cross-sectional area changes on flow profiles has been overlooked. We have reported the change in luminal area in both parent and daughter branches of porcine renal arterial bifurcations of two age groups. Although the magnitude of tapers in the parent and daughter branches of porcine renals and human cerebral bifurcations were not significantly different, this study has shown that large changes in the luminal area were present in all planar Ybifurcations examined. We believe that the taper present in muscular Y-bifurcations could have serious implications for the way future flow studies are designed and interpreted. Acknowledgements-We are grateful to both surgeons and technicians, Department of Surgery, University Hospital for providing the vessels from the young pigs; to Dr I. C. MacDonald for obtaining the old pig vessels from the abbatoir. Dr A. Buchan allowed us the use of his digitizing equipment.
REFERENCES the renal artery, our study showed that area changes were linear and were also present in the daughter branches. If we compared the ratios of the mean taper in the daughter branches to the mean taper in the parent trunk for both young and old samples, the young bifurcations would have a ratio of -2.22 (k 0.68 S.E.M., n = 7) and the old pigs would have a value of - 1.54 (kO.76 S.E.M., n=4). The taper ratios were compared using a t-test and were not significantly different (P > 0.05). The ratio of the tapers provides a measure of the asymmetry present between the area changes in the parent and the daughter branches of one sample. The luminal cross-sectional area changes of the porcine renal and human cerebral bifurcations (Macfarlane, 1985) were not significantly different. Since both bifurcations are classified as muscular, Y-bifurcations, we believe that large tapers could be present in other arteries, such as coronary and carotid arterial bifurcations. These tapers could have a significant effect on blood flow through them. Many studies have examined the effects of mean fluid shear stress as well as time-varying shear stress on the localization of atherosclerosis (Ku et al., 1985). Friedman et al. (1981) found an inverse relationship between the intimal thickness observed in mildly atherosclerotic aortoiliac bifurcations and fluid shear stress measured in models of this bifurcation. Friedman found the minimum shear stress to be present on the lateral walls opposite to the tip of the flow divider. This is the region which was found to have the largest cross-sectional area and where we would expect both flow velocity and shear stress to be minimized. The flow entering the parent branch of the renal artery would enter a zone which increased in area linearly, thereby causing it to decelerate, be divided into two branches at the apex, and then accelerate through the daughter branches which decrease linearly in luminal area. Our hypothesis is that the expansion zone proximal to the flow divider can alter the location of both separation and reattachment points of recirculation zones, but the minimum taper required to produce such an effect on flow remains to be studied. Since systemic blood flow is pulsatile, the expansion in the luminal cross-sectional area which we have reported could have an effect on both pressure and flow waves. Waves are reflected at points of impedance mismatching; however, since the luminal area increases linearly proximal to the bifurcation, we suspect that the impedance would change in a similar fashion, thereby reducing the mismatch at the apex of
Adamson, S. L. and Roach, M. R. (1981) Measurement of wall shear stress in a glass model renal bifurcation by a technique that monitors the rate of erosion of an opaque coating layer. Biorheology 18, 9-21. Friedman, M. H., Bargeron, C. B., Hutchins, G. M. and Mark, F. F. (1980) Hemodynamic measurements in human arterial casts and their correlation with histology and luminal area. J. biomech. Engng 102, 247-251. -Friedman, M. H.. Hutchins. Cl. M.. Bareeron. C. B.. Deters. 0. J. and Mark, F. F. (1981) CorrelaGon between intimal thickness and fluid shear in human arteries. Atherosclerosis 39,425-436. Gosling, R. Cl., Newman, D. L., Bowden, N. L. R. and Twinn, K. W. (1971) The area ratio of normal aortic junctions: aortic configuration and pulse-wave reflection. Br. J. Radiol. 44, 850-853.
Houle, S. and Roach, M. R. (1981) Flow studies in a rigid model of an aorto-renal junction: a case for high shear as a cause of the localization of sudanophilic lesions in rabbits. Atherosclerosis 40, 231-244. Karino, T. and Goldsmith, H. L. (1981) Role of blood cellwall interactions in thrombogenesis and atherogenesis: a microrheological study. Biorheology 21, 587-601. Karino, T. and Goldsmith, H. L. (1985) Particle flow behaviour in models of branching vessels: II. Effects of branching angle and diameter on flow patterns. Biorheology 22, 87-104. Karino, T. and Motomiya, M. (1983) Flow visualization in isolated transparent natural blood vessels. Biorheology 20, 119-127. Kratky, R. G., Lo, D. K. and Roach, M. R. (1991) Quantitative measurement of fixation rate and dimension changes in the aldehyde/pressure-fixed canine carotid artery. Blood Vessels 28, 386-395. 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.
Ludin, H., Juabin, E. J., Kaufmann, H. J. and vonPein, W. (1972) Significance of alterations in main renal artery calibre and configuration. Acta Radial. Diagnosis 12, 803-832.
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Technical Note
Macfarlane, T. W. R. (1985) A computer-based quantitative image analysis of the geometry of human cerebral arterial bifurcations. Ph.D. thesis, University of Western Ontario, London, Ontario, pp. 62-94. Macfarlane, T. W. R., Petrowski, S., Rigutto, L. and Roach, M. R. (1983) Computer based video analysis of cerebral arterial geometry using the natural fluorescence of the arterial wall and contrast enhancement techniques. Blood Vessels 20, 161-171. McClave, J. T. and Dietrich, F. H. (1986) A First Course in
Statistics, p. 296. Dellen, San Francisco, CA. Pond, W. G. and Houpt, K. A. (1978) The Biology ofthe Pig, p. 31. Comstock. Ithaca, NY. Rayman, R., Kratky, R. G. and Roach, M. R. (1985) Steady flow visualization in a rigid canine aortic cast. J. Biomechanics 18, 863-875. Zamir, M. and Phipps, S. (1987) Morphometric analysis of the distributing vessels of the kidney. Can. J. Physiol. Pharmacol. 65,2433-2440.