(Circulation. 2000;101:1819.)
© 2000 American Heart Association, Inc.
Clinical Investigation and Reports |
Assessment of Aortic Line of Elasticity Using Polynomial Regression Analysis
From Hippokration Hospital, Department of Cardiology, University of Athens, Greece.
Correspondence to Christodoulos Stefanadis, MD, FESC, 9 Tepeleniou St, 15452 Athens, Greece. E-mail cstefan{at}cc.uoa.gr
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Abstract |
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Background—The aim of this study was to assess the sigmoid line of elasticity in the human aorta.
Methods and Results—The pressure-diameter relation was measured in the descending aorta in 120 subjects. In an additional group of 6 subjects, transient vena caval occlusion produced 5 sets of pressure-diameter data. We found that the best fit curve of the pooled pressure-diameter data was a third-order polynomial. A polynomial equation was used to calculate the sigmoid line of elasticity in the entire population and after the administration of diltiazem (15 patients) or enalaprilat (10 patients). The sigmoid line of elasticity was significantly different with respect to age (P<0.001), history of hypertension (P<0.004), and hypercholesterolemia (P<0.02). The difference between the transition point and the peak systolic pressure was increased in normal subjects compared with patients (P<0.0001). The sigmoid line shifted leftward and upward with diltiazem, but it remained unchanged with enalaprilat. During an average of 3 years of follow-up, 19 of 88 patients developed stroke (n=4), unstable angina (n=8), acute myocardial infarction (n=4), or acute pulmonary edema (n=3).
Conclusions—This approach provides a quantitative evaluation of the aortic line of elasticity, which can differentiate the intrinsic from the extrinsic aortic elastic properties. Furthermore, it is a powerful and independent risk factor for cardiovascular events.
Key Words: aorta • elasticity • nonlinear dynamics
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Introduction |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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The analysis of aortic function by the sigmoid curve of elasticity provides good differentiation of intrinsic aortic properties from extrinsic changes.1 2 Active changes in the elastic properties of the aorta are related to changes in the intrinsic elastic properties of the aorta and are characterized by the shifting of the pressure-diameter loop to a different line of elasticity. Passive changes in aortic elastic properties are characterized by the sliding of the pressure-diameter loop upward or downward along the same sigmoid line of elasticity.2
The purpose of the present study was to use pressure-diameter relationships to assess the aortic line of elasticity in humans. (1) We tested the best-curve fit to the pressure-diameter relation of the aorta over a physiological range of aortic pressures, which were modified by manipulating preload using transient inferior vena caval occlusion in 6 patients. (2) Using the analysis of nonlinear behavior of the thoracic aorta in 120 patients who underwent cardiac catheterization, we investigated the factors that influence the aortic curve of elasticity. (3) We calculated the transition point of the third-order polynomial and the relative position of each aortic loop on the aortic line of elasticity. (4) We studied the effects of diltiazem and enalaprilat on the sigmoid line of elasticity. (5) Finally, we examined the prognostic benefit of the aortic line of elasticity in predicting cardiovascular events in patients with hypertension, congestive heart failure, and coronary artery disease.
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Methods |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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Study Population
A total of 120 patients were selected from those who were routinely referred to undergo cardiac catheterization at our institution. Of these patients, 49 had coronary artery disease, which was diagnosed by a positive result on coronary arteriography, and 28 had moderate or severe hypertension. The remaining 32 were normal subjects who underwent coronary angiography to evaluate symptoms compatible with angina pectoris.
In 25 of the hypertensive patients, the effects of vasodilating drugs
(diltiazem or enalaprilat) on the aortic line of elasticity were
studied. Eleven of these 120 patients had congestive heart failure (7
due to ischemic heart disease and 4 due to idiopathic dilated
cardiomyopathy). We also examined 6 additional
subjects (5 men and 1 women) who had coronary artery
disease (2 patients), congestive heart failure (1 patient), or
hypertension (1 patient) or who were normal subjects (2 patients). In
these 6 subjects, volume load was altered by transient
inferior vena caval occlusion. All patients discontinued
medications, if any, for 
5 half-lives before the study, except for
intravenous furosemide or sublingual
nitroglycerine, which were given when indicated.
The protocol was approved by the Institutional Ethics Committee of our
institution, and all patients gave informed consent before
participation.
Measurement of Diameter and Pressure
The aortic pressure-diameter relation was calculated as
previously described.1 2 3 4 5 6 7
Protocol of Transient Vena Caval Occlusion
A balloon catheter (Balt, Cristal balloon
valvouloplastie) was inserted via a femoral vein sheath. The
balloon was gradually inflated in the right atrium and
simultaneously and gently drawn back toward the
inferior vena cava until it fit snugly at the right atrium
and inferior vena caval junction. This maneuver led to a
reproducible preload reduction; the fall in peak systolic
pressure averaged 30%.8 After 8 to 10 s, the balloon
was deflated, and venous return was allowed to recover. Heart rate
changed by only 0 to 4 beats/min during the period in which pressure
fell.
Administration of Vasodilating Drugs
In 15 patients, after baseline measurements, diltiazem was
administered intravenously through a peripheral
line in a bolus dose of 0.15 mg/kg over a period of 2
minutes.5 In another group of 10 age- and sex-matched
hypertensive patients, enalaprilat was administered
intravenously through a peripheral line in a
bolus dose of 1.25 mg over a period of 5 minutes. Measurements were
continuously monitored and recorded at baseline, at the end of drug
administration, and repeatedly thereafter for 20 minutes.
Clinical and Laboratory Parameters
Left ventricular ejection fraction and mass index
were calculated using 2D echocardiography (HP Sonos
2500). Hypertension history, smoking (>5 cigarettes per day),
hyperlipidemia (total cholesterol >200
mg/dL), and diabetes mellitus (fasting serum level of glucose >100
mg/dL) were assumed as simple independent variables with 2
categories.
Data Analysis
The aortic pressure-diameter relation was obtained by plotting
the diameter versus pressure of digitized data. We used a polynomial
regression program (SPSS for Windows, version 8) to discover the
best-fit curve. In the transient vena caval occlusion group, there were
5 cardiac cycles of different loading conditions. We performed the
polynomial regression analysis starting with a first-order
polynomial (linear) and then proceeded, one order at a time, to a
third-order polynomial.9 We compared 2 models with the
same number of parameters using the F ratio, as described
in the Appendix. The fit with the lower residual SS was superior
because its curve lay closer to the points. Then, we determined whether
the 2 sets of data of the 2 sequential cardiac cycles differed
significantly. This test was repeated for all sets of data for each
patient. If a higher-order regression model that fit the 5 sets of
data, which were analyzed separately 2x2, did not differ
significantly, the same general model was used to fit the pooled
(combined) sets of data when analyzed
simultaneously (see Appendix). This polynomial describes
the equation of the sigmoid line of elasticity.9 10 11 By
means of this higher-order regression model, the pressure-diameter data
of the 120 subjects of the study population were analyzed, and
the parameters of the polynomial function were
determined.
The transition point is given by the following formula: Ptrans=-b2/3b3. This formula results from the second derivative of the third-order polynomial, which is f"(x)=6b3x+2b2. The point that zeroes the f"(x) represents the pressure (Ptrans) at which the aortic line of elasticity diverges due to changes in aortic elastic behavior. The position of the pressure-diameter relation on the aortic line of elasticity was estimated by the difference between Ptrans and peak systolic pressure (Pdif=Ptrans-peak systolic pressure) in each patient.
Definition of End Points
A total of 88 patients were followed-up for a mean period of 3
years; no patient was lost to follow-up. Information was
obtained by reviewing hospital records, direct contact with the
patient’s primary physician, or contact with the patient by means of a
questionnaire. End points included hospitalization for stroke, unstable
angina, acute myocardial infarction, or acute pulmonary edema.
A cerebrovascular event was documented by a computed tomography scan
and confirmed by a neurologist.
Statistical Analysis
ANOVA and paired t tests were used to compare
continuous variables. Pair-wise comparisons between group means
were done with the Bonferroni t test. The linear regression
lines of the 5 sets of data of the 6 patients who underwent vena caval
occlusion were compared using ANCOVA. Multivariate
analysis of variance for multiple dependent variables by
additional factor variables or covariates was used to test the null
hypothesis of the effects of age, body mass index, sex,
coronary artery disease, left ventricular mass
index, left ventricular ejection fraction, diabetes
mellitus, hypercholesterolemia, smoking habit,
and history of hypertension with the parameters
(b0, b1,
b2, and b3) of the
polynomial equation, which were the dependent variables.
In addition, interactions between factors and the effects of individual factors were investigated. All statistical assumptions were met, and no multicollinearity problems existed our analysis. The probability of survival in various patient subgroups was estimated by the Kaplan-Meier method. Multivariate comparisons of the influence of the aforementioned factors on survival were performed using the Cox proportional hazard method. The cumulative hazard function, as well as the partial residuals and the differences in ß values for the covariates, were used to check the distribution and the proportionality of the hazard over time (SPSS-Save-Diagnostics). All assumptions were valid. Relative risk and 95% confidence limits were calculated from the proportional hazard model as parameter estimates. All tests were considered significant at P=0.05.
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Results |
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TopAbstractIntroductionMethodsResultsDiscussionAppendix 1References |
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Sigmoid Line of Elasticity
The analyses of the 5 cardiac beats in the vena caval occlusion group revealed that the linear regression lines were different in all patients (ANCOVA, with P between 0.01 and 0.001). The third-order regression model provided the best curve fit; its residual sum of squares (SS) was significantly less than that of the second-order polynomial for every beat in all patients (P between 0.01 and 0.001). Thus, the sigmoid line of elasticity is best described by the following equation: D=b0+(b1P)+(b2P2)+(b3P3), where P and D are the instantaneous aortic pressure and diameter, respectively, and b0, b1, b2, and b3 are the parameters of the polynomial. In each patient, the cubic regression lines of the 5 sets of data did not differ in the 2x2 comparison using the F test (P=NS for all comparisons). In Figure 1
, the 5 sets of
pressure-diameter data that were produced by transient vena caval
occlusion are shown; the third-order regression line represents
the sigmoid line of elasticity.
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Effects of Clinical and Laboratory Parameters on the
Aortic Line of Elasticity
In Table 1, the demographic data,
clinical characteristics, and hemodynamic
parameters of the subjects are shown. In the pooled
population of 120 subjects, multivariate tests revealed
that the sigmoid line of elasticity of the aorta was significantly
different with respect to age, history of hypertension, and
hypercholesterolemia, both in
multivariate and univariate tests (Table 2). Significant interactions were found
between age and hypertension (Table 2).
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Transition Point
We found that the pressure at the transition point
(Ptrans) increased with age, history of
hypertension, coronary artery disease, and congestive heart
failure (P<0.001). Nevertheless, the difference in pressure
(Pdif) decreased with these variables
(P<0.01). Moreover, Pdif was
significantly different between groups of subjects (ANOVA,
P<0.0001; Figure 2).
Pair-wise comparisons revealed that a significant difference existed
between each pair (P<0.05), except for the pair of
hypertensives versus patients with coronary artery disease
(P=NS).
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Effects of Vasodilating Drugs on the Aortic Line of
Elasticity
No difference existed at baseline between the 2 subgroups of
hypertensive patients who took either diltiazem or enalaprilat with
respect to demographic and hemodynamic data. Diltiazem
resulted in a significant upward and leftward shift of the aortic line
of elasticity in all patients (Figure 3).
Furthermore, the mean values of the polynomial parameters
showed significant changes between groups (Table 3). Although the transition point did not
change significantly, Pdif significantly
increased (Table 3 and Figure 3). In contrast, the aortic
line of elasticity did not alter in any patient after enalaprilat
administration (Figure 4). The aortic
loop slid downward along the same sigmoid line. Furthermore, no
polynomial parameter showed significant changes during the
time that Pdif was significantly increased
(Table 4).
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Predictors of End Points
During follow-up, 19 of 88 patients reached an end point. Of these
19 patients, 4 had hypertension and developed stroke; 12 had
coronary artery disease and developed either unstable angina
(n=8) or acute myocardial infarction (n=4); and 3 had congestive heart
failure and developed acute pulmonary edema. The event-free
survival rate at 59 months of follow-up was 73% (Figure 5). Specifically, the stroke-free
survival rate in hypertensives was 79%; the acute coronary
event–free survival rate in patients with coronary artery
disease was 71%; and the acute pulmonary edema–free survival
rate in patients with congestive heart failure was 70% (Figure 5). No patient died of a noncardiac cause.
Multivariate Cox proportional hazards analysis
revealed that the only independent predictors of risk of the multiple
cardiovascular end points were the polynomial
parameters of the aortic line of elasticity
(b0, b1,
b2, and b3), as well as
Pdif (Table 5).
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Discussion |
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There are 5 main findings in the present study. (1) The sigmoid line of elasticity of the human aorta was described in vivo; this line represents a third-order polynomial and, for the normal range of aortic pressures, it could be calculated from a single beat. This approach can differentiate the intrinsic from the extrinsic properties of the aorta. (2) Age, history of hypertension, and hypercholesterolemia affect the aortic line of elasticity; the impact of these clinical parameters is seen in the intrinsic elastic properties of the aorta. (3) The aorta operates at a point lower than the transition point in normal subjects, but in patients, the aorta functions near or above the transition point. (4) Vasodilating drugs have a differential effect on the aortic line of elasticity. Diltiazem shifts the aortic loop onto a different sigmoid line, whereas enalaprilat slides the aortic loop downward along the same sigmoid line of elasticity. (5) The parameters of the third-order polynomial and the position of the aortic loop on the aortic line of elasticity are independent predictors of cardiovascular events.
Consideration of the Methods
In this study, we showed that a nonlinear polynomial models the
pressure-diameter relation of the aorta better than a linear one.
Indeed, the third-order polynomial had a lower residual SS compared
with the linear and second-order polynomial. Furthermore, the linear
regression lines of the aortic loops during altered preload were
significantly different while these loops lay on the same curve.
Finally, Lanne et al12 13 also found that the
aortic pressure-diameter curve is nonlinear, with one transition point.
A curve with one transition point is described with a third-order
polynomial.
Transition Point
The transition point should reflect the critical pressure
(Ptrans) required to expand all aortic elastin
fibers; at higher aortic pressures, collagen fibers and smooth
muscle cells are recruited. Lanne et al12 13 found that
above Ptrans, the vessel is stiffer. They also
found that the individual pressure-diameter curves show hysteresis.
Those findings agree with the results of our study and are ascribed to
the viscoelastic properties of the aorta. Both elastin and collagen
contribute to wall mechanics; elastin is preferentially load-bearing at
small distensions, and collagen is load-bearing at large distensions.
Thus, the steep increase in diameter below the transition point mainly
reflects the stretching of elastin, whereas both elastin and collagen
contribute to the wall tension above the transition
point.12 13 Thus, Ptrans, a
functional index derived in vivo, provides useful anatomical and
pathophysiological information on the aorta.
Indeed, the tensile modulus of an elastin fiber is 
750 mm Hg,
which is relatively low for a connective tissue
fiber.14 15 In contrast, collagen fibers are much stiffer.
The tensile modulus of individual collagen fibers may be as high as
3.7x106 mm Hg, which is 5000 times that of
elastin.16 Experimental studies have indicated that
vascular smooth muscle cells from several species all generate maximal
stresses of 750 to 1500 mm Hg. The vascular smooth muscle cell
contractile state is influenced by a variety of agents that may alter
vascular stiffness.
We found that the aortic loop of normal subjects lies beneath the transition point in an area in which the blood pressure load is totally imposed on the elastin fibers of the aorta and a lot of pressure lag exists from peak systolic pressure to the transition point. In patients, the aortic loop operates near the transition point and, in some, the peak systolic blood pressure is higher than Ptrans. Under these circumstances, the elastin fibers are maximally extended and, over time, they degenerate.
Effects of Clinical Variables on the Aortic Line of
Elasticity
The aortic line of elasticity was significantly related to age,
history of hypertension, and
hypercholesterolemia. These variables
change the intrinsic elastic properties of the aorta independently of
loading conditions.5 17 18 Furthermore, the interaction
between age and hypertension suggested that the presence of
hypertension in elderly patients has additional deleterious effects on
the intrinsic properties of the aorta.
Hypercholesterolemia has effects on the
intrinsic aortic properties; these can be detected as alterations in
the aortic line of elasticity; local atheromatous
change does not necessarily occurr. A similar finding was also reported
in young subjects with familial
hypercholesterolemia.19
Response of Aortic Line to Vasodilating Drugs
Enalaprilat, despite blood pressure and diameter changes, did not
change the intrinsic elastic properties of the aorta (passive
mechanism). In contrast, 2 mechanisms are involved in diltiazem
administration. One is a passive mechanism caused by a shift far away
from the transition point. The other, an active mechanism, contributes
as the aortic loop is shifted to a different sigmoid line of
elasticity. This movement suggests active changes of the elastic
properties of the aorta (change of the intrinsic elastic
properties).
Importance of Aortic Line of Elasticity as a Predictor of
Cardiovascular Events
The development of cardiovascular events can
be predicted among patients with hypertension, coronary artery
disease, and congestive heart failure on the basis of the aortic line
of elasticity and the position of aortic loop on this line. This
finding is consistent with the results of previous studies,
which also concluded that a high pulse pressure or increased aortic
stiffness are independent predictors of cardiovascular,
and especially coronary, mortality.20 21 22 Our
findings are also consistent with studies in which
arterial alterations, as determined from the carotid
elastic modulus, are strong independent predictors of
cardiovascular mortality in patients with end-stage
renal disease who are undergoing hemodialysis.23
This study adds to those of previous investigators, first by
demonstrating the relative prognostic power of the aortic curve of
elasticity and second by demonstrating that the aortic line of
elasticity can independently predict stroke, acute coronary
events, and acute pulmonary edema in patients with
hypertension, coronary artery disease, and heart failure.
Specifically, we found that when b1,
b3, and Pdif increase, the
risk of cardiovascular events will decrease because
these predictors have a relative risk <1 (Table 5
). Thus, these
parameters reflect beneficial aortic elastic
properties.
Clinical Implications
The development of this new approach to analyze the
pressure-diameter relation enabled us to distinguish between the
intrinsic and extrinsic elastic properties of the aortic wall. As a
consequence of this study, it is hoped that the understanding of the
biologic functions of the aorta can be extended and that the behavior
of the aorta can be explained and predicted with greater reliability.
These results give insight into and explain the logic of therapies
currently used for the treatment of hypertension, angina pectoris, and
cardiac failure. They also show that therapeutic and prognostic
benefits may be gained through a study of the intrinsic properties of
the aorta.
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Acknowledgments |
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This study was supported by a grant from the Hellenic Heart Foundation.
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Appendix 1 |
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The goal of polynomial regression is to determine values for the parameters (A, B, C, and D) of the polynomial Y=A+Bx+Cx2+Dx3 that make the curve best fit the data points. The average deviation of the curve from the points is the square root of SS/df, where df indicates degrees of freedom. This is also called the root mean square.11 The statistical significance is obtained by F=SS1/SS2, where both numerator and denominator have NV degrees of freedom (N indicates the number of data points and V, the number of parameters fit by the program).10
To fit 2 sets of data to the same general model and to use the results to determine whether the 2 sets of data differ significantly, the following approach can be used.24 First, the 2 sets of data are analyzed separately. The overall values for the SS for the 2 sets of data analyzed separately are the sums of the individual values from each fit (SSsep=SS1+SS2). Similarly, the number of degrees of freedom (df) is the sum of the values from each fit (dfsep=df1+df2). Next, the 2 sets of data are pooled and analyzed simultaneously. This pooled fit yields values for SSpool and dfpool. The question is whether the separate fit is significantly better than the pooled fit. The significance of the improvement is determined from the F ratio, which is calculated as F=[(SSpool-SSsep)/(dfpool-dfsep)]:(SSsep/dfsep). To interpret the meaning of this F value, a statistical table is used to convert it into a P value. In using such a table, the numerator has (dfpool-dfsep) degrees of freedom, and the denominator has dfsep degrees of freedom. A large F value (with a corresponding low P value) indicates that the separate fit is much better than the pooled fit, ie, that the 2 sets of data are not well fit by 1 curve.
Received July 30, 1999; revision received October 25, 1999; accepted November 15, 1999.
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