Validity and Reliability of Diastolic Pulse Contour Analysis (Windkessel Model) in Humans
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Abstract
The present study assessed (1) the impact of the measurement site (lower versus upper extremity) on the corresponding compliance variables and (2) the overall reliability of diastolic pulse contour (Windkesselderived) analysis in normal and hypertensive subjects. Arterial tonograms were recorded in the supine position from the radial and posterior tibial arteries in 20 normotensive (116±12/68±8 mm Hg) and 27 essential hypertensive subjects (160±16/94±14 mm Hg). Ensembleaveraged data for each subject were fitted to a firstorder lumpedparameter model (basic Windkessel) to compute wholebody arterial compliance (C_{A}) and to a thirdorder lumpedparameter model (modified Windkessel) to compute proximal compliance (C_{1}) and distal compliance (C_{2}). Despite highfidelity waveforms in each subject, the firstorder Windkessel model did not yield interpretable (positive) values for C_{A} in 50% of normotensives and 41% of hypertensives, whereas the thirdorder model failed to yield interpretable C_{1} or C_{2} results in 15% of normotensives and 41% of hypertensives. No betweensite correlations were found for the firstorder time constant, 2 of the 3 thirdorder model curvefitting constants, or C_{A}, C_{1}, or C_{2} (P>0.50). Mean values for all 3 compliance variables were higher for the leg than the arm (P<0.05 each). We conclude that differences in Windkesselderived compliance values in the arm and leg invalidate wholebody model assumptions and suggest a strong influence of regional circulatory properties. The validity and utility of Windkesselderived variables is further diminished by the absence of betweensite correlations and the common occurrence of uninterpretable values in hypertensive subjects.
Increased stiffness (decreased compliance) of the aorta and large blood vessels is associated with wide pulse pressure, systolic hypertension, and increased cardiovascular risk.^{1–3}⇓⇓ Because wide pulse pressure and systolic hypertension are late manifestations of the arteriosclerotic process, there is significant interest in the development of more sensitive compliance measurements that can detect premature vascular stiffening at an earlier stage of the process. Our laboratory has undertaken a systematic analysis of 3 of these noninvasive compliance methods: systolic pulse contour analysis, plethysmography, and diastolic pulse contour analysis.^{4} Using these 3 methods in over 100 subjects, we have found that the respective compliance and reflectance values correlate poorly across methods and have concluded that either the biologic information that each method provides is intrinsically different or that major methodological artifacts exist.^{4}
The first method that we have subjected to additional methodological validation is diastolic pulse contour analysis.^{5–8}⇓⇓⇓ Diastolic pulse contour analysis is based on the assumption that the circulation can be represented by 1 (firstorder model) or 2 (thirdorder model, including an interposed inertance function) capacitors arranged in parallel with a resistor (Figure 1). In these models, vascular elastic recoil (compliance) is represented by the capacitor(s). By fitting the diastolic decay portion of an individual’s arterial waveform to a firstorder model (basic Windkessel) or thirdorder model (modified Windkessel), compliance variables can, theoretically, be derived.
The specific purpose of the present studies was to address issues of validity and reliability of Windkesselderived compliance variables. We tested a specific assumption of this lumpedparameter model: that compliance estimates obtained from all peripheral measurement sites should be equal.^{6,7}⇓ Different compliance estimates obtained from 2 different sites would imply that regional, as well as systemic, factors influence the results. We also investigated the reliability of the system. This last issue is significant because analysis criteria and clinical reliability have not been published during the development of existing proprietary systems that measure arterial compliance.
Methods
Subjects
The study was approved by an institutional review committee, and the subjects gave written informed consent. The study procedures followed were in accordance with institutional guidelines. A samplesize calculation was performed before the study. Based on an effect size equal to 0.5, an α level equal to 0.05 for a nondirectional test and 1β equal to 0.9, the required sample size is approximately 44 subjects. Twenty normotensive subjects (median age 35, range 26 to 79 years) and 27 hypertensive subjects (median age 56, range 43 to 78 years) volunteered for testing. Systolic blood pressure ranged from 96 to 136 mm Hg (mean±SD 116±12 mm Hg) in the normotensive subjects and from 142 to 194 mm Hg (mean±SD 160±16 mm Hg) in the hypertensive subjects. Diastolic blood pressure ranged from 52 to 84 mm Hg (mean±SD 68±8 mm Hg) in the normotensive subjects and from 70 to 118 mm Hg (mean±SD 94±14 mm Hg) in the hypertensive subjects.
Waveform Acquisition
Highfidelity pressure waveforms were obtained noninvasively from the radial and posterior tibial artery by applanation tonometry using a Millar tonometer (Millar Instruments, Inc). The SphygmoCor Blood Pressure Analysis System (PWV Medical, Ltd) was used to amplify the signal and ensemble average the waveform data. Applanation tonometry can be used to record pressure waveforms from any peripheral artery that can be supported by a bony structure. Gently flattening but not collapsing an artery with a tonometer balances circumferential forces in the arterial wall, and the resulting contact force between the artery and the tonometer is equal to intraarterial pressure.^{9} To minimize the effects of movement artifact on wave morphology, pressure waveforms were first assessed visually by a technician and then submitted to analysis by the SphygmoCor software, which reports a quality control parameter for diastolic waveform variability. If diastolic waveform variability exceeded 10%, new data were obtained immediately. Pressure waveforms that qualified for data analysis were ensembleaveraged over 11 seconds.
Subjects were supine for all measurements, which were obtained from the right side of the body. Brachial blood pressure was measured on the right arm by auscultation before each tonometry measurement. The order of tonometry measurements was alternated among subjects.
FirstOrder Model (Basic Windkessel)
The basic Windkessel model, a firstorder lumped parameter model, assumes that the time constant (τ) of the monoexponential pressure decay is determined by the product of systemic vascular resistance and aortic compliance.^{8} In the electrical analogue (Figure 1A), voltage (v) is analogous to pressure in the aorta, the capacitor (C_{A}) to wholebody arterial compliance, electrical current (i) to blood flow, and the resistor (R) to systemic vascular resistance. The standard firstorder model equation for diastolic pressure (voltage) as a function of time is
where (A_{1}+A_{3}) represents endsystolic pressure, A_{3} is mean circulatory pressure and t is time. The fitted parameter of interest is the time constant (τ). When the resistance (R) is known, the wholebody arterial compliance is calculated as
ThirdOrder Model (Modified Windkessel Model)
The thirdorder lumpedparameter model^{5–7}⇓⇓ assumes that compliance of the arterial system can be partitioned into central and distal compartments, where central compliance is distinct from distal compliance. In the electrical analogue (Figure 1B), voltage (v) is analogous to mean pressure in the aorta, the first capacitor (C_{1}) to central, proximal, or large artery compliance, electrical current (i) to blood flow, inductance (L) to inertance of a column of blood, the second capacitor (C_{2}) to distal or small artery compliance, and the resistor (R) to systemic vascular resistance.^{6,7}⇓ The thirdorder model equation for diastolic pressure (voltage) as a function of time is
The first term of the equation (A) is derived from the exponential decay of the entire diastolic waveform. The second term (B) is a decaying sinusoid function; the decaying sinusoid accounts for the dicrotic notch and the subsequent damped oscillations. The constants A_{1}, A_{3}, and A_{6} may vary between measurement sites.^{5–7}⇓⇓ In the standard model, the constants A_{2}, A_{4}, and A_{5} should not vary between measurement sites because they are intended to represent the physical structure of the electrical analogue. Constants A_{2} and A_{4} are damping constants, whereas A_{5} is the frequency of oscillation. C_{1} and C_{2} were calculated using the following equations:
In both the first and thirdorder model equations, R is an estimate of systemic vascular resistance and demonstrates the high degree of theoretical dependence of C_{2} on R. For this study, it was not necessary to calculate R because intraindividual comparisons were used; accordingly, an arbitrary constant R value of 1500 dyne · s · cm^{−5} was used to calculate C_{A}, C_{1}, and C_{2} at each site.
Data and Statistical Analysis
In both models, the beginning of diastole was defined as the lowest point of the dicrotic notch (ie, the point after which the pressure began to rise again before decaying exponentially). The end of diastole was defined as the point where the diastolic pressure was no longer monotonically decreasing. The curvefitting constants for each model were estimated by an iterative procedure using Marquardt’s algorithm (nonlinear regression). Curve fits were accepted if the coefficients were significantly different from zero (P<0.05). The variables C_{A}, C_{1}, and C_{2} were calculated using Eq. 2, 4, and 5, respectively. Only data sets where both radial and posterior tibial artery waveforms could be fitted were used.
The χ^{2} statistic was used to test equality of proportions for the first and thirdorder models. Pearson correlation coefficients were calculated to assess the linear relationship between the radial and posterior tibial sites for time constant (τ) (firstorder model), curvefitting constants A_{2}, A_{4}, and A_{5} (thirdorder model), and the model parameter variables C_{A}, C_{1}, and C_{2}. BlandAltman plots^{10} were created to quantify the agreement of C_{A}, C_{1}, and C_{2} values between sites. Means for each variable were compared between measurement sites using paired t tests (α=0.05). For hypothesis testing, effect size (partial ETA squared) was calculated using the following equation:^{11}
where df_{h} denotes degrees of freedom for hypothesis and df_{e} denotes degrees of freedom for error. The values used to characterize small, medium, and large effect sizes are 0.01, 0.06, and 0.14, respectively.^{11} For this study, η^{2}_{p} represents the proportion of the total variability attributable to choosing the posterior tibial artery as a measurement site to estimate model parameters C_{A}, C_{1}, and C_{2}.
Results
Waveforms obtained from the radial artery were distinctly different from corresponding waveforms obtained from the posterior tibial artery (Figure 2), as expected. Coefficients of variation of ensembleaveraged blood pressures within a given individual were less than 5% (radial artery 3.7±1.7%, posterior tibial artery 4.2±2.2%). The mean values, standard deviations, and correlation coefficients for the firstorder time constant (τ), thirdorder curve fitting constants (A_{2}, A_{4}, and A_{5}), and each model’s compliance estimates (C_{A}, C_{1}, and C_{2}) are presented in Table 1. C_{A}, C_{1}, and C_{2} estimates are presented graphically in Figures 3A, 4A, and 5⇓⇓A, respectively.
Data Correspondence to Model
As shown in Table 2, diastolic waveforms from 11 of 27 hypertensive subjects and from 10 of 20 normotensive subjects could not be described adequately by the firstorder model. Waveforms from 11 of 27 hypertensive subjects (not the same individuals) and 3 of 20 normotensive subjects could not be described adequately by the thirdorder model. The thirdorder model fit the diastolic waveform data better than the firstorder model (χ^{2}[1]=13.55, P<0.05). Within the normotensive and hypertensive groups, the ability of the first and thirdorder models to fit the data were independent of blood pressure and age. The lack of fit in either model was not related to the nominal quality control parameters from the SphygmoCor system, which did not eliminate any tracings from further analysis.
Time Constant (FirstOrder Model)
The correlation coefficient between sites for τ was not statistically significant (P=0.97), and a significant difference in mean values was observed between the radial and posterior tibial artery (P=0.027, power=0.62, =0.18).
A_{2}, A_{4}, and A_{5} (ThirdOrder Model)
The correlation coefficients between sites for A_{2} and A_{4} were not statistically significant (A_{2} P=0.08, A_{4} P=0.72), whereas significant differences were observed between the mean values obtained from the radial and posterior tibial artery for A_{2} (P=0.002, power=0.89, = 0.26) and A_{4} (P=0.0001, power=0.99, =0.47). A significant correlation was observed between radial A_{5} and posterior tibial A_{5} (P=0.001). There was no significant difference between sites for A_{5} (P=0.21, power=0.24,=0.05).
C_{A} (FirstOrder Model)
The correlation coefficient between sites for C_{A} was not statistically significant (r=−0.006, P=0.97, r^{2}=0.00004; Figure 3A), but a significant difference was observed between the radial and posterior tibial artery (P=0.027, power=0.62, =0.18). The BlandAltman plot for C_{A} (Figure 3B) quantifies the limits of agreement between measurement sites: 4.9×10^{−4} to 3.1×10^{−4} dyne · s · cm^{5}. The 95% confidence intervals for the upper and lower limits of agreement are 7.4×10^{−4} to −2.5×10^{−4} dyne · s · cm^{5} and −0.66×10^{−4} to −2.5×10^{−4} dyne · s · cm^{5}, respectively.
C_{1} and C_{2} (ThirdOrder Model)
The correlation coefficients between sites were not statistically significant for either C_{1} (r=−0.11, P=0.55, r^{2}=0.01; Figure 4A) or C_{2} (r=0.04, P=0.81, r^{2}=0.002; Figure 5A). There were significant differences in C_{1} (P=0.04, power=0.54, = 0.12) and C_{2} (P=0.0001, power=1.0, =0.61) between the radial and posterior tibial artery. The BlandAltman plots for C_{1} (Figure 4B) and C_{2} (Figure 5B) quantify the limits of agreement between measurement sites: 2.44×10^{−4} to −4.29×10^{−4} dyne · s · cm^{5} and 2.19×10^{−4} to −9.41×10^{−4} dyne · s · cm^{5}, respectively. For C_{1}, the 95% confidence intervals for the upper and lower limits of agreement are 1.32 to 3.56 dyne · s · cm^{5} and −5.41 to −3.17 dyne · s · cm^{5}, respectively. For C_{2}, the 95% confidence intervals for the upper and lower limits of agreement are 0.39 to 3.99 dyne · s · cm^{5} and −7.61 to −11.21 dyne · s · cm^{5}, respectively.
Discussion
Precise quantitation of arterial compliance variables is the cornerstone of future identification of individuals in whom premature vascular stiffening signals increased systolic blood pressure and increased cardiovascular risk. Present data, however, strongly suggest that the problems encountered in using Windkesselderived diastolic pulse contour analysis in humans are similar to those reported in dogs, where the thirdorder model has been found to yield “unreliable estimates of arterial compliance.”^{12} Our results show that the lumpedparameter Windkessel models yield different results from the upper and lower limb. These differences probably represent the influences of regional circulatory properties and suggest that a simple “systemic measurement” of wholebody, proximal or distal compliance cannot be reliably obtained from peripheral tonography and diastolic pulse contour analysis.
From a theoretical perspective, diastolic pulse contour analysis indirectly estimates arterial compliance by fitting the diastolic portion of an arterial waveform to a lumpedparameter model. This lumpedparameter model of wholebody compliance is valid only if (1) the pressure wave speed is high enough so that all segments of the large and small arteries are pressurized simultaneously, and (2) no reflection sites exist. Under these 2 conditions, peripheral pressure waveforms at different arterial sites differ only in scale, and thus, compliance variables calculated from any site would be equivalent. However, pressure changes do not occur instantaneously throughout the arterial tree and the arterial tree is not a reflectionless system.^{13} Instead, after each systole, a pressure wave travels downstream with variable intrinsic velocities that depend on local arterial wall properties. Reflected waves emanate from points of significant impedance mismatch, and the summation of antegrade and retrograde waves determines the morphology of the composite arterial waveform at any given point along the arterial tree. Because of differences in the length of individual arteries, the number of regional reflection sites, and the stiffness of the individual arterial walls, the morphology, timing, and magnitude of reflected waves are intrinsically different in the wrist and ankle.^{14} Thus, Windkesselderived compliance values in the upper and lower extremities would not be expected to be similar because they represent local, as well as systemic, vascular properties. The validity of these statements is supported by the current data.
The assumption of siteindependence has never been fully tested for the basic or modified Windkessel model in man, and existing data are conflicting in animals. In dogs, both similar^{6,7}⇓ and different^{12} compliance values have been reported when aortic and femoral measurement sites were compared. We chose peripheral sites from the upper and lower limbs where the different morphology of the pressure pulses^{14} is due to different regional patterns of wave reflection. In the pulse wave morphology of Figure 2A, the second peak is probably a reflected wave, which violates the assumption of the firstorder model (ie, that the diastolic pressure decay is monoexponential [Eq. 1]) and alters the Windkessel calculation. In the case of the thirdorder model, the presence of a reflected wave exerts an even stronger influence on the decaying sinusoid calculation (B in Eq. 3) than on the overall sinusoid decay (A in Eq. 3).
The reliability of obtaining Windkesselderived compliance values is also of significant concern. Our results are probably not due to methodological artifacts. Both the radial and posterior tibial measurement sites are well situated for tonometry,^{9} with each artery properly supported between a bony structure and the tonometer. Even though we were able to obtain highfidelity arterial waveforms with no apparent artifact, some data could not be fit to the Windkessel equations in many subjects without yielding uninterpretable results, such as negative or “zero” coefficients and compliance values (Table 2). Retrospective analysis of tracings in individuals in whom negative compliance values were obtained demonstrated the presence of latediastolic wave peaks of uncertain origin. These large peaks strongly influence the calculation, converting the sinusoid from a decay pattern (positive compliance value) to an amplification pattern (a negative compliance value).
The model fit was excellent for the data reported. For the firstorder model, the coefficient of determination (r^{2}) averaged 0.99±0.04 and 0.99±0.003 for the radial and posterior tibial artery waveforms, respectively. For the thirdorder model, r^{2} averaged 0.94±0.03 and 0.96±0.02 for the radial and posterior tibial artery waveforms, respectively. The diastolic waveform variability averaged less than 5% at each measurement site, and the curvefitting constants derived from the first and thirdorder lumpedparameter models had low curvefitting errors (Table 1).
The poor reliability is also not due to the slight methodological differences between our system and the commercially available HDI/Pulsewave CR2000 (Hypertension Diagnostics, Inc) system. We have previously reported excellent concordance between the two modified Windkesselbased techniques in individuals in whom interpretable compliance values could be obtained.^{4}
The significant intersite differences observed in this study cannot be attributed to excessive variability in either the radial or posterior tibial diastolic waveforms and probably represent true differences in the respective arterial walls. The firstorder model time constant (τ) and thirdorder model curve fitting constants (A_{2} and A_{4}) were significantly different between the radial and posterior tibial artery. Because the constants were not correlated, the difference between sites was not systematic. The time constant (τ), the curvefitting constants (A_{2} and A_{4}), and their derived compliance values (C_{A}, C_{1}, and C_{2}) were not only uncorrelated between sites, but significantly different as well. The thirdorder model curvefitting constant A_{5}, which describes the frequency of oscillation, was correlated between sites, suggesting that the lack of correlation in other compliance variables could be due to the time delay between the incident and reflected wave. The A_{5} correlation further suggests that the same reflection sites are found in both the upper and lower extremity.
Our results do not rule out the possibility that the compliance variables (C_{A}, C_{1}, and C_{2}) can serve as biomarkers of arterial dysfunction or disease. Abnormal changes in C_{2} have been reported to be altered by aging, hypertension, and congestive heart failure.^{15} However, the physiological meaning of Windkesselderived compliance values remains unclear.
Acknowledgments
This work was supported by Pilot Clinical Pharmacology Training Grant no. FDT000889.
Footnotes

This work was presented in part at the 15th Scientific Meeting of the American Society of Hypertension, New York, NY, May 16–20, 2000.
 Received January 10, 2002.
 Accepted March 15, 2002.
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 Validity and Reliability of Diastolic Pulse Contour Analysis (Windkessel Model) in HumansTimothy S. Manning, Barbara E. Shykoff and Joseph L. IzzoHypertension. 2002;39:963968, originally published May 1, 2002https://doi.org/10.1161/01.HYP.0000016920.96457.7C
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