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Experimental Studies |
1 From the Department of Radiology, University of Michigan Medical Center, TC 2910K, 1500 E Medical Center Dr, Ann Arbor, MI 48109-0326. From the 1996 RSNA scientific assembly. Received July 17, 1998; revision requested August 13; revision received November 30; accepted March 26, 1999. Address reprint requests to R.O.B. (e-mail: ronbude@umich.edu).
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Abstract |
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 TOP Abstract IntroductionMATERIALS AND METHODSRESULTSDISCUSSIONAppendix 1References |
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MATERIALS AND METHODS: An essentially noncompliant in vitro model that used a pulsatile pump, blood-mimicking fluid, and a branching tubing network that could be configured to produce a downstream cross-sectional area one, two, four, or eight times that of the feeding vessel was used to investigate the relationship, if any, between arterial bed cross-sectional area and the RI and ESA.
RESULTS: The mean ESA in the branching network was inversely proportional to cross-sectional area, decreasing by approximately a factor of two for every doubling of the cross-sectional area. The mean RI in the branching network decreased with increasing cross-sectional area, but not as greatly as the ESA did; the mean RI in the bed with eight times the upstream cross-sectional area had an RI that was approximately three-fourths the upstream RI. These relationships are real, as the slopes of the plots (ESA vs cross-sectional area, P = .001; RI vs cross-sectional area, P < .02) are significantly different from zero.
CONCLUSION: RI and ESA decrease as a result of increasing downstream cross-sectional diameter of the arterial bed.
Index terms: Blood, flow dynamics, 9*.1312, 9*.133, 9*.91, 9*.922 • Phantoms, 9*.12984, 9*.131, 9*.133, 9*.91, 9*.92 • Ultrasound (US), Doppler studies, 9*.12984
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Introduction |
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TOPAbstractIntroductionMATERIALS AND METHODSRESULTSDISCUSSIONAppendix 1References |
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The total cross-sectional area of an arterial bed progressively increases as the arteries branch beyond the main trunks (5). We speculated that the cross-sectional area of a vascular bed affects the arterial waveform and therefore the RI and ESA. We designed an in vitro model and performed experiments with it to evaluate the effect of the cross-sectional area of a vascular bed on the arterial waveform.
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MATERIALS AND METHODS |
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TOPAbstractIntroductionMATERIALS AND METHODSRESULTSDISCUSSIONAppendix 1References |
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Pressure was measured in the rubber tubing with a pressure transducer (Transpac IV; Abbott Laboratories, Hospital Products, North Chicago, Ill) connected to a pressure monitor (model 78354A; Hewlett-Packard, Bad Homburg, Germany). Flow returned to a reservoir that resupplied the pump, with the reservoir fluid level held constant at approximately 30 cm above the model. The reservoir was situated on top of a magnetic stirrer (Magnestir; A.S. Aloe, St Louis, Mo), which kept the scatterers suspended. A variable clamp downstream of the downstream US site was used to ensure that pressure in the model was within the physiologic range.
The pump settings were constant through the whole experiment, as follows: 30% phase ratio (amount of the "cardiac" cycle occupied by systole), 36 beats per minute, and 20 mL per stroke. To ensure that flow was laminar at all measurement sites throughout the experiment, Reynolds numbers for each experimental trial were calculated. The peak systolic velocity was used for this determination as it was the highest velocity that occurred at each measurement site, which ensured that the largest Reynolds number was calculated. These data are displayed in Table 1. Since flow makes the transition from laminar to turbulent at a Reynolds number of about 2,300 (6), flow was considered to be laminar if the Reynolds number was less than this value (the largest Reynolds number was 2,100).
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-inch] outer diameter, 0.64-cm [
-inch] inner diameter) and was configured as shown in the schematic diagram (Fig 1). Y-type plastic connectors (Nalgene, Nalge Nunc International, Rochester, NY) connected end-to-end through three orders of branching resulted both in expansion from one vessel to a network of eight branches and in reconstitution of flow to a single exit vessel. US was performed upstream and downstream (at A and B, respectively, in Fig 1) to the network and within the network in stable regions of flow, as defined later (in "Inlet Lengths of Tubing"). Blood-mimicking fluid.—The model was filled with 1,500 mL of a solution with the mean viscosity of blood (2.5 centipoise [7]) at 20°C that was composed of a mixture of glycerol and water in the following proportions: 35 mL of glycerol for every 100 mL of water (8). One gram of microparticles (Sephadex G-50; Sigma Chemical, St Louis, Mo) was added as ultrasound scatterers and produced adequate Doppler signals from the branched network that was set up with one, two, or four branches. With eight branches, however, it was necessary to add an additional gram of scatterers for adequate Doppler US because of the relatively slow flow in each branch.
Inlet lengths of tubing.—A flow perturbation (bifurcation, stenosis, tight turn, etc) transiently alters the flow profile for a variable distance, up to a maximum length known as the inlet length (9, pp 38–43). Since the nonpulsatile, laminar flow inlet length is longer than the inlet lengths for other types of flow, it was used to ensure our Doppler measurements were performed in areas of stable flow, with all measurements performed at least one inlet length from any flow perturbation.
The nonpulsatile, laminar flow inlet length is defined as follows: L = 2kVr2/
, where L is inlet length in centimeters, k is an experimentally derived constant with a value of 0.08, V is mean velocity in centimeters per second, r is tube radius in centimeters, and is kinematic viscosity in stokes (centimeters squared per second) (9, pp 38–43). The inlet length for the input and output regions for all experimental trials and also for the branch in the one-branch mode was 26.2 cm. The inlet lengths for the branches in the two-, four-, and eight-branch modes were, respectively, 13.1, 6.6, and 3.3 cm.
Doppler US
A freestanding aluminum framework above the model supported the US transducer and allowed it to be moved among all insonation sites with ease. Doppler waveforms were obtained at 4.0 MHz with a 5.0-MHz curvilinear transducer (Spectra; Diasonics Vingmed Ultrasound, Santa Clara, Calif) at the following settings: At the input region upstream and the output region downstream to the branching vascular network, the wall filter was 75 or 105 Hz; pulse repetition frequency, 2.9 or 4.0 kHz; and Doppler angle, 51° or 53°. At the branches of the network, the wall filter was 35 or 105 Hz; pulse repetition frequency, 1.4 or 4.0 kHz; and Doppler angle, 45°–53°.
The Doppler sample volume included the entire vessel lumen. Doppler gains were optimized by scanning at gains just below those at which background noise first became apparent. Water-filled plastic bags coupled with US gel to the transducer and tubing facilitated US. A sound-absorbent material (SOAB [MKI] nonresonant acoustic absorber; B.F. Goodrich, Richfield, Ohio) interposed between the tubing and the tabletop reduced ultrasound reverberations.
Waveform RIs were calculated by using measurements taken by hand with calipers (Absolute Digimatic; Mitutoyo, Tokyo, Japan) according to the following formula: RI = (S - D)/S, where S is the height of the systolic peak and D is the height of the end-diastolic trough. All reported RIs are the means of four waveform RIs, with the waveform sweep speed set at four waveforms per waveform strip for RI measurement.
ESAs were calculated by using measurements taken with calipers as the slope of a tangent of the initial systolic upsweep (1), with the waveform sweep speed set at two waveforms per waveform strip. All reported ESAs are the means of four waveform ESAs. For both RIs and ESAs, the Doppler velocity scale that gave the largest possible waveforms without aliasing was used to decrease measurement error.
Validaton That the Model Had Extremely Low and Essentially Zero Compliance
To determine if our model had appreciable compliance, the input RI proximal to and the output RI distal to the branching segments for each mode of the experiment were determined. The rationale for this is described in "Discussion."
Experimental Trials
Experimental trials were performed by using one, two, four, or eight branches of the network, by appropriately placing or removing the clamps at sites 1, 2, and/or 3 in the model (Fig 1). Placement of clamps at sites 1, 2, and 3 produced a model without branching; placement of clamps at sites 1 and 2 produced two branches; placement of clamps at site 1 produced four branches; and absence of clamps produced eight branches.
In this way, since the tubes had the same diameter, cross-sectional areas of the vascular network of one, two, four, and eight times that of the input were obtained. (Cross-sectional areas of three, five, six, and seven times that of the input were not studied because this would have resulted in different orders of branching at some parts of the model compared with others, which would have altered the distribution of flow so that it was no longer symmetric.)
Doppler waveforms were obtained at the input, at each branch of the network (RIs and ESAs were averaged to obtain means for all branches for each trial), and at the output. For all trials, the input RI was set at 0.6 (a "generic" in vivo organ RI), as measured on the display screen by using machine calipers; subsequent caliper measurement taken by hand from the film hard-copy images showed the input RIs varied slightly (see Results). Pressures were within the physiologic range, with mean pressures ranging from 9.9 x 103 to 1.2 x 104 Pa (74–87 mm Hg) for all trials.
Data Analysis
To analyze the mean branch RI data and to compare it from trial to trial, it was necessary to compensate for the slight variability of the input RIs. To accomplish this, we wished to normalize the mean branch RIs to the input RI for each trial. However, because the input and output RIs for two of the four trials were not the same, we decided to first average the input and output RIs and then to normalize the mean branch RIs of any trial to the mean of the input and output RIs. An analogous procedure was performed for the ESAs.
Because the velocity of flow decreases as the cross-sectional area of a tube increases, it seemed that the acceleration should also decrease with increasing cross-sectional area. A theoretic derivation of the relationship between cross-sectional area and acceleration was performed (Appendix) to see if the experimental results agreed with the theoretic results.
RI data were plotted with Cartesian coordinates. ESA data were plotted in log-log coordinates to obtain a linear plot. Linear least-square fits were performed for both RI and ESA data. To determine if there were functional relationships between RI and cross-sectional area and between ESA and cross-sectional area, (ie, to show that the slopes of the plots of RI or ESA vs cross-sectional area were not zero), regression analyses (analysis of variance) of the slopes of the plots were performed. The 95% CIs of the slopes were calculated to determine if they included zero. A P value less than .05 was considered to indicate a statistically significant difference.
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RESULTS |
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TOPAbstractIntroductionMATERIALS AND METHODSRESULTSDISCUSSIONAppendix 1References |
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The mean ESA of a vascular bed branch was inversely proportional to cross-sectional area, decreasing by approximately a factor of two for every doubling of the cross-sectional area (Table 3, Fig 3). The mean RI of a vascular bed branch also decreased with increasing cross-sectional area of the vascular bed (Table 2, Fig 2), but the decrease was not as great as the decrease in ESA.
The following data applied to the RI versus cross-sectional area relationship. The R2 value of the linear-curve fit of the data was 0.972. The slope of the linear-curve fit of the data was -0.0315 (P = .014; 95% CI of the slope: -0.0153, -0.0477).
The following data applied to the ESA versus cross-sectional area relationship. The R2 value of the linear curve fit of the data was 0.998. The slope of the linear curve fit of the data was -1.16 (P = .001; 95% CI of the slope: -1.00, -1.32).
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DISCUSSION |
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TOPAbstractIntroductionMATERIALS AND METHODSRESULTSDISCUSSIONAppendix 1References |
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In the dog, the total cross-sectional area increases approximately 3–5 times from arteries approximately 4 mm in diameter to arterioles approximately 100 µm in diameter and increases approximately 11 times from arteries approximately 4 mm in diameter to arterioles approximately 50 µm in diameter. For vessels that branch dichotomously (most do), the mean total cross-sectional area of the branches is 1.26 times that of the feeding artery (5). It has been inferred that similar increases occur in humans.
Our model was designed to achieve an increase in cross-sectional area similar to that just described, with incremental increases up to eight times the cross-sectional area of the feeding artery evaluated. Since it is currently possible to obtain power Doppler images and spectral Doppler waveforms from very small arteries (the renal interlobular arteries are seen with power Doppler US, especially in transplants), we believe this is a reasonable range of cross-sectional area to study, especially since technologic advances may allow the future study of smaller and smaller arteries.
Because vascular compliance is one of the factors that alters waveform pulsatility and morphology (1,2) and because we desired to study only the effect of cross-sectional area on the waveforms, it was necessary for our model to possess a compliance so low that it would not substantially affect the waveforms. Vascular compliance, however, is a dynamic phenomenon that we could not measure directly. Therefore, we used the following indirect method to determine if our model had appreciable compliance.
Vascular compliance, in conjunction with vascular resistance, progressively damps waveform pulsatility as flow progresses from the highly pulsatile central arteries to the essentially nonpulsatile capillaries (1,10–12), which have an RI of essentially zero. Therefore, our model can be assumed to have a very low compliance if the pulsatility is not substantially damped, as would be manifested with an output RI equal to or very nearly the same as the input RI. The input and output RIs in each mode of our experiment varied from 0.00 to 0.04 (Table 2).
The upstream measurements were not independent of each other. The downstream measurements were also not independent of each other. Therefore, it was impossible to prove statistically that the upstream and downstream RIs were not significantly different. However, the minute RI changes just described indicate the compliance of our model was very low, and for these experiments, it was considered to be essentially zero.
Our results show the ESA decreased with increasing cross-sectional area of an arterial bed, independently of vascular compliance and resistance (Table 3, Fig 3), by a factor of approximately two for every doubling of the cross-sectional area. (Since the P value for the slope indicates a significant difference and the 95% CIs for the slope do not intersect zero, this relationship is real.) This agrees with theoretic results (Appendix).
Thus, the ESA measured in the feeding artery of an organ is likely to be substantially different than the ESA in an arteriole owing to cross-sectional area alone. This is independent of other factors such as vascular compliance, which also considerably affects the ESA (1,2) since the difference in cross-sectional area between these two sites is as much as a factor of 10, as described earlier. These results corresponded to the results of Halpern et al (1) from a study in the human kidney in which they noted the ESA decreases from the central arteries to more peripheral ones. Thus, our results provide a basis for understanding why ESA criteria developed from more central arteries to detect disease may not apply to ESAs obtained from arteries substantially downstream from these central arteries.
Of additional interest is that the RI is dependent on cross-sectional area (Table 2, Fig 2), independent of the effects of vascular resistance and compliance, which also alter the RI (3,4,13–17). (Since the P value for the slope indicates a significant difference and the 95% CIs for the slope do not intersect zero, this relationship also is real.) This helps explain why the RI varies with the site of measurement in the normal renal arterial tree (decreasing centrally to peripherally), as shown by Knapp et al (18). Thus, as with ESA, on the basis of cross-sectional area alone, RI criteria developed in one region of the arterial tree may not necessarily apply to substantially different regions of the arterial tree.
The cause for the RI dependence on cross-sectional area is unknown. We presumed that an essentially incompressible fluid flowing through an essentially noncompliant system would proportionately transfer the velocities downstream at each time increment, with these velocities scaled by the ratio of the upstream cross-sectional area to the cross-sectional area at the site of measurement downstream, as happens with acceleration (Eq [A4], Appendix). If so, the ratio would be present in both the numerator and denominator of the RI expression and would cancel out, leaving the RI unchanged.
We speculate the RI dependence on cross-sectional area may be due to the effects of velocity and pulsatility on laminar flow. Steady laminar flow produces a parabolic flow profile, with flow absent at the wall and maximal at the center of the vessel. When laminar flow is pulsatile, a true parabolic flow profile is no longer present, and the flow profile alters with time as a function of pulsatility (9, pp 35–38). As the cross-sectional area increases and as the bulk flow rate slows, the laminar flow profile may alter, changing the relationship between peak systole and end diastole. Further experimentation is required to evaluate this hypothesis.
The following are potential limitations of our study. First, the ESA of the reconstituted flow waveforms (at B in Fig 1) was higher than the ESA of the input waveforms (at A in Fig 1), although the effect was not great (Table 3). If cross-sectional area is the only factor affecting the waveform, the reconstituted downstream ESA should equal the upstream ESA. To counteract this effect, we normalized the mean branch ESAs to the mean of both the upstream and downstream ESAs. This was done on the assumption that the effect of this factor on the branch ESAs, which were obtained at a site essentially halfway between the input and output waveform measurement sites, is likely nullified by normalizing the mean branch ESAs to the mean of the input and output ESAs.
We speculate the factor responsible for this effect is wave reflection. Wave reflection occurs at changes in vascular impedance and increases with increasing impedance (19). In our model, the clamp downstream of the branching network had a very large impedance analogous to peripheral vascular resistance in vivo. In vivo, peripheral resistance causes a pressure wave that reflects back to the heart. In healthy elderly individuals whose vessels are less compliant than those of younger individuals, this reflected wave returns to the heart during systole, which increases systolic pressure. This is one of the reasons for the increased systolic pressures of healthy older individuals (20,21).
In our essentially noncompliant model, such a reflected wave traveled much more quickly than it would have in vivo because of the lower compliance of our model (the lower the compliance, the faster the transmission of reflected waves) (21). It may, therefore, augment flow velocity earlier in systole downstream (B in Fig 1) than it does upstream (A in Fig 1) ("downstream" and "upstream" refer to the actual direction of fluid flow, not the direction of the reflected waves) and thus may augment the downstream ESA more than the upstream ESA. Further experimentation is required to test this hypothesis. This factor did not affect the RIs, as they were essentially the same upstream as downstream (Table 2).
Second, we did not study turbulent flow, which can occur in vivo and which alters the flow profile differently than laminar flow does.
Third, it may never be possible to determine vascular bed cross-sectional area noninvasively. If so, it will not be possible to correct for this factor when analyzing arterial waveforms, and the results of our study may be limited only to helping explain why disease detection based on analysis of intrarenal arterial waveforms is limited by cross-sectional area without the ability to correct for cross-sectional area.
In summary, the increase in cross-sectional area that normally occurs as flow progresses through the arterial tree lowers both the ESA and the RI independently of other factors such as vascular compliance and resistance. Therefore, our results help explain why ESA and RI criteria developed for disease detection in one site do not necessarily apply to disease detection at another substantially different site in the arterial tree. Since vascular compliance and resistance also alter the arterial waveform, the waveform changes that occur as a consequence of disease are a function of a very complex set of phenomena. Unless these factors can be taken into account, it may not be possible to use Doppler US with arterial waveform analysis to detect disease any better than has currently been shown, and it may be fortuitous that it works as well as it does.
Practical application: A greater understanding of the role of cross-sectional area in altering the Doppler arterial waveform may enable future studies in which this effect is taken into account to better depict pathologic conditions. This may be especially important when intrarenal ESA or RI measurements are used to detect renal arterial stenosis.
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Appendix 1 |
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TOPAbstractIntroductionMATERIALS AND METHODSRESULTSDISCUSSIONAppendix 1References |
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Differentiating Equation (A1) with respect to time t gives dVin/dt = n(dVb/dt). Since V/A = v, where V is volume flow, A is cross-sectional area, and v is velocity, Av can be substituted for V, giving
Since dv/dt equals acceleration a, Equation (A2) reduces to Ainain = nAbab, where ain and ab are the input and branch accelerations, respectively. Rearranging terms gives
Thus, at any instant in time, the acceleration in any branch of the network of equal-size branches is the acceleration of the input branch multiplied by the ratio of the cross-sectional area of the input vessel divided by the total cross-sectional area of the branch vessels.
This theoretic explanation must be regarded as a first approximation only. This is because the distribution of velocities across the diameter of a vessel lumen in pulsatile laminar flow is not constant but changes with time and is influenced by factors including the cross-sectional area of the vessel (9, pp 35–38). Thus, in vivo, it is possible this dependence might further influence the acceleration as calculated from the maximum velocity envelope differently than the ratio of cross-sectional areas does.
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Footnotes |
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Abbreviations: ESA = early systolic acceleration RI = resistive index
Author contributions: Guarantor of integrity of entire study, R.O.B.; study concepts and design, R.O.B.; definition of intellectual content, R.O.B., J.M.R.; literature research, R.O.B.; experimental studies, R.O.B.; data acquisition, R.O.B.; data analysis, R.O.B., J.M.R.; statistical analysis, R.O.B.; manuscript preparation, R.O.B.; manuscript editing and review, R.O.B., J.M.R.
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References |
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TOPAbstractIntroductionMATERIALS AND METHODSRESULTSDISCUSSIONAppendix 1References |
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= values predicted by using theory (Appendix).
