| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
(Circulation. 1996;93:1123-1132.)
© 1996 American Heart Association, Inc.
Articles |
Contributions of Frequency Distribution Analysis to the Understanding of Coronary Restenosis
A Reappraisal of the Gaussian Curve
From the Thoraxcenter (P.W.S.) and Cardialysis (R.M.), Erasmus University, Rotterdam, Netherlands, and the University of Washington School of Medicine (K.G.L.), Seattle, Wash.
Correspondence to Kenneth G. Lehmann, MD, Section of Cardiology (111C), Seattle Veterans Affairs Medical Center, 1660 S Columbian Way, Seattle, WA 98108.
 |
Abstract |
|---|
 Top Abstract IntroductionMethods Results Discussion References |
|---|
Background Clinical restenosis after balloon angioplasty can be categorized by use of dichotomous terms based on the presence or absence of recurrent myocardial ischemia. In contrast, recent investigations have concluded that late luminal renarrowing, documented through angiographic imaging, occurs to a variable extent in nearly all stenoses. This process has been characterized by a gaussian or normal frequency distribution, with restenosis simply representing an extreme form of this delayed remodeling. In the current study, frequency distribution analysis was used to examine the process of coronary restenosis in a large cohort of patients at risk.
Methods and Results Quantitative coronary
angiographic analysis was applied to 9279
cineangiograms obtained in 3093 patients before and
immediately after angioplasty and after 6-month follow-up. Late
loss, defined as the change in minimum lumen diameter of the target
stenosis from postdilation to follow-up, did not
statistically conform to a normal distribution (P<.0001 by
both 
2 statistic and Kolmogorov-Smirnov test),
even after the exclusion of the 236 stenoses that displayed
total occlusions at follow-up angiography. Examination of
deviations from a normal curve revealed an excessively high frequency
of stenoses that experienced either little change (0.0±0.3 mm)
or marked change (1.0 to 2.0 mm) in late loss, with a low frequency of
stenoses with intermediate values (0.3 to 1.0 mm). Similarly,
although the distribution of percent diameter stenosis of the
target lesion was statistically normal immediately after dilation, this
gaussian distribution disappeared during the follow-up period.
Other angiographic indexes of restenosis also failed to
approximate a normal curve. In an attempt to improve the goodness
of fit, a probabilistic model of late loss was created on the basis of
deconvolution of the observed data distribution. Two theoretical,
discrete populations of stenoses were identified, one with and
one without overall late luminal narrowing. Unlike the gaussian
distribution, this model provided a good representation of the
observed data (P=NS for lack of fit).
Conclusions The frequency distributions of angiographic indexes of restenosis often superficially resemble a gaussian curve, an appearance that is artifactually enhanced by the measurement imprecision of current quantitative techniques. Nevertheless, standard indexes of coronary restenosis fail to conform statistically to a normal distribution. The pattern of deviations observed supports the possible existence of discrete subpopulations of lesions, each with a different propensity toward the development of restenosis after coronary intervention.
Key Words: angioplasty • coronary disease • restenosis
|
Introduction |
|---|
|
TopAbstractIntroductionMethods Results Discussion References |
|---|
Shortly after the introduction of coronary angioplasty 18 years ago,1 it became evident that a certain proportion of patients would experience recurrent ischemic symptoms in the early months after their balloon procedure. Moreover, repeat angiography documented these recurrent symptoms to be accompanied by luminal renarrowing at the site of the previous dilation.2 The presence or absence of recurrent symptoms provided investigators with a convenient dichotomous clinical outcome and led to a search for a dichotomous angiographic correlate. A number of potential angiographic candidates were proposed,2 all of which were capable of providing a restenosis "rate" but none of which proved to be an ideal measure for all stenoses.
Although the manifestations of clinical restenosis may be inherently dichotomous, recent work has concluded that this binary division is not true for angiographic restenosis. Several studies3 4 reported that angiographic measures of luminal renarrowing are not only continuous but also appear to fit well with a normal or gaussian distribution. With this model, late luminal narrowing after angioplasty can best be described as a singular process that occurs to a variable extent in all stenoses. Restenosis simply represents an extreme form of this delayed remodeling process.5
In reporting the results of angioplasty trials and other medical investigations, continuous data are regularly presented with the mean used as a single measure of location, generally in conjunction with a measure of variability such as a standard deviation or standard error. This routine practice is quite useful in the reporting of trial results in which the random variables under study possess a normal distribution. In fact, the mean and the SD, in conjunction with sample size, are fully sufficient to provide a complete characterization of a specific normal probability distribution curve. However, published uses of a sample mean as a surrogate for a population are rarely accompanied by either graphic or statistical attempts to verify the goodness of fit relative to a normal distribution. Moreover, certain populations, particularly those with skewed distributions, cannot be adequately represented by the sample mean alone. Frequency tabulations represent an alternative form of data analysis that is free of these limitations, with the data generally displayed in graphic format by use of a frequency polygon or histogram.
In the current study, the frequency distributions of several angiographic indexes of restenosis were examined in a large cohort of patients undergoing balloon angioplasty. Each patient received follow-up angiography to delineate the extent of late luminal narrowing, with each film subjected to quantitative coronary angiographic analysis. Deviations from a normal distribution were examined in detail in the hope that they might provide insights into the processes underlying the phenomenon of coronary restenosis.
|
Methods |
|---|
|
TopAbstractIntroductionMethods Results Discussion References |
|---|
Study Population
Patients included in the current study received balloon angioplasty of a de novo stenosis in a native coronary artery for clinical indications other than treatment of an evolving acute myocardial infarction. Potential candidates were selected from participants in four recent multicenter, pharmacological trials of restenosis: CARPORT,6 MERCATOR,7 PARK,8 and MARCATOR.9 In each trial, the study medication failed to show either clinical or angiographic benefit, which permitted pooling of the placebo and active treatment groups in the present study. All patients agreed before enrollment to undergo repeat coronary angiography at 6 months or sooner if clinically indicated. Each patient provided written informed consent in a form that was approved by the institutional review committee. Of the 3568 potential candidates, 475 were excluded from the current study on the basis of the following sequentially applied criteria: (1) inability to dilate the target lesion to a diameter stenosis <50% as judged by visual assessment (n=138); (2) major complication (death, q-wave myocardial infarction, or emergent coronary bypass surgery) accompanying the dilation procedure (n=72); (3) death during the follow-up period (n=18); (4) inability or refusal to receive repeat angiography during the 6-month follow-up period (n=221); or (5) poor quality angiographic image that precluded accurate quantitative analysis of any of the 35-mm cinefilms recorded before or after angioplasty or at 6-month follow-up (n=26). The rate of follow-up angiography in this selected group of eligible patients was 93.4%. Patients were not excluded if the angioplasty procedure was deemed successful by visual assessment but failed to achieve a specific minimum improvement in stenosis diameter as determined by quantitative analysis. The remaining 3093 patients received coronary angioplasty of 3799 stenoses; this group serves as the basis of this report.
Angiographic and Angioplasty Procedures
Selective coronary
angiography was performed in all
patients before (preangioplasty) and immediately after
(postangioplasty) the angioplasty procedure, as well as at the end of
the 6-month follow-up period (follow-up). Overall, 641 patients
developed symptoms suggestive of restenosis before their
6-month anniversary and underwent early angiography. If this early
study was performed within 3 to 4 months of the initial angioplasty and
failed to show definite restenosis, the angiogram was
repeated at 6 months as previously planned. Angiography was undertaken
after the administration of intracoronary
nitroglycerin or isosorbide dinitrate, with the images
recorded on 35-mm cinefilm. The angle and skew of each baseline
view were charted, and these exact views were duplicated during all
subsequent angiographic procedures. Other measures detailed
elsewhere10 11 were also applied prospectively to
ensure
standardized image acquisition methods and the best possible image
quality.
Balloon coronary angioplasty was performed in all patients by use of conventional techniques. The balloon size, balloon material, catheter manufacturer, inflation pressure, duration of inflation, and number of inflations were left to the discretion of the investigator.
Quantitative Angiographic Analysis
All 9279 cineangiograms
were forwarded to a central
core laboratory for blinded analysis. Stenosis
dimensions were determined by use of the Coronary Artery
Analysis System, which has been described previously and
validated repeatedly.12 13 Images used by a
computer-based system were taken from a selected 6.9x6.9-mm area
of a single cineframe (representing 11% of the total frame
area) and digitized into a 512x512-pixel matrix that encompassed the
area of interest. Scan lines of video density were computed every 0.1
mm along the centerline of the stenosis and adjacent
coronary segments. The weighted sum of the first and second
derivative functions of the video density profile were used to
construct vessel lumen contours. With rare exception,
computer-derived contours were accepted for measurement without
user intervention. Magnification correction was accomplished with the
shaft of the empty angiographic catheter used as a scaling device. To
avoid the variability of catheter dimensions between manufacturers,
models, and lots,14 the diameter of each angiographic
catheter used for imaging was measured directly with a
micrometer. We used the radial distance from the center of
each catheter and arterial segment to correct pincushion
distortion. Using a porcine phantom model and catheter calibration,
this system has a published accuracy of 0.09 mm and precision of 0.23
mm.15
Angiographic Parameters
All angiographic parameters were
derived from
multiple views, with a minimum of two views required for
analysis of the right coronary artery and its branches
and three views for the left coronary artery and its branches.
The smallest distance between contours at the site of the
stenosis, averaged over multiple views, was defined as the
minimum lumen diameter (MLD). The reference diameter was computed by
interpolation of the luminal dimensions of the arterial
segments adjacent to the stenosis. By applying curvature
analysis to the descending and ascending limbs of the luminal
diameter function curve, stenosis end points were derived for
use in computing lesion length. Plaque area was computed as the
integral of the difference between the lumen contours that spanned the
stenosis and a computer-derived reconstruction of the same
site assuming the absence of disease. In addition to measuring
stenosis severity in absolute dimensions (MLD), severity was
expressed in relative terms as the percent diameter stenosis,
with this latter parameter computed as the difference
between the reference diameter and the MLD, normalized to the reference
diameter. Four definitions used in the study summarized changes in
angiographic measurements over time. Acute gain reflected the magnitude
of the initial success of the procedure and referred to the change in
MLD, expressed in millimeters, between the postangioplasty and
preangioplasty angiograms. Similarly, late loss, defined as the change
in MLD from postangioplasty to follow-up, served as a measure of
the restenosis process. The difference between these two
parameters was net gain, an index of the long-term (ie,
6-month) success of the procedure. Finally, loss index, computed as the
ratio of late loss to acute gain, corrected observed luminal
renarrowing by the magnitude of the initial success.
Data Analysis
Frequency tabulations were computed for the
quantitative
angiographic parameters, with data distributions examined
by use of frequency histograms. On the basis of the means and SDs
derived from these groups, a normal frequency function was computed and
compared with the observed data distribution. The normal probability
density function f(x) was defined as follows:
 |
where
x represents the variate, µ the mean,
and 
the standard deviation of the population under study, and
e represents the base of the natural logarithm. The
goodness of fit between the normal theoretical distribution and the
observed data was computed in two ways. The first, the
2 statistic, compared observed and expected
frequencies per group and therefore was dependent on the number of
groupings selected. The second, the Kolmogorov-Smirnov (KS)
one-sample test, measured the maximum amount by which the
cumulative distribution function differed from that of the fitted
distribution and therefore was independent of the method of
grouping.16
Two other parameters were derived to help describe the observed frequency distribution in comparison with a theoretical normal curve. The skewness coefficient was used to evaluate the symmetry of the data. A value of zero indicates that the data are symmetrically distributed, with positive values revealing a shift toward the upper tail and negative values a shift toward the lower tail. The coefficient used in the present study was defined as follows:
 |
where n represents the number of stenoses. The kurtosis coefficient describes the steepness of the observed data distribution compared with a gaussian or normal distribution. For normally distributed data, this coefficient is zero. A value >0 indicates increased steepness of the curve. The kurtosis coefficient was defined by the following function with the same variables described above:
 |
Plus-minus values represent mean±SD unless otherwise stated. A probability value of .05 was accepted as the limit of statistical significance.
|
Results |
|---|
|
TopAbstractIntroductionMethods Results Discussion References |
|---|
Clinical and Procedural Characteristics of the Study Group
The baseline characteristics of the study group are presented in Tables 1
and 2. Of
3093 patients, 81.5% were male, and the mean age at the time of the
initial procedure was 52.7 years. Although prior coronary
artery surgery was uncommon (4.2%), prior myocardial infarction was
relatively frequent (42.3%). Nearly two thirds of the group had
significant disease limited to a single coronary vessel, and
half of the overall population experienced substantial angina pectoris
(Canadian Cardiovascular Society class III or IV).
|
|
Based on quantitative analysis, the mean diameter stenosis before attempted dilation was 61.4±14.6%. Total occlusions were present at the start of the procedure in 7.5% of stenoses. Overall success was achieved in 94.3% of target stenoses based on the quantitative (as opposed to visual) assessment of the postprocedural angiogram. The average time from the initial dilation to follow-up angiography was 161±45 days, with the mean skewed by a small number of early, clinically indicated procedures.
Frequency Distribution of Late Loss
The data distribution for
all 3799 stenoses included in
the study is graphically depicted in Fig 1, with late
loss (MLD postangioplasty minus MLD at follow-up) as the
independent variable and a representative index of
the restenosis process. For this histogram, the data were
divided into classes equally spaced over abscissa values that ranged
from -1.5 to +2.0 mm. Note the superficial resemblance to a
normal distribution. However, when the normal probability density
function was computed on the basis of the precise mean and SD of the
study population and was superimposed on this histogram (Fig
1),
substantial and statistically important deviations from the normal
curve were apparent (P<.0001 by both
2 statistic and KS test for lack of fit).
|
One possible
explanation for this poor fit is the inclusion of the 236
stenoses with total occlusions that were observed on the
follow-up angiogram. This group may represent a unique
subpopulation of the study, with both the rate and mechanism of late
luminal narrowing differing from that which operates within
nonoccluding stenoses. Because of the unknown effects of this
potential confounding variable, the data were reexamined after
excluding these stenoses. The frequency distribution for the
3563 remaining stenoses is presented in Fig 2. Note that
although the distribution of data more
closely fits a theoretical gaussian curve, deviations are still
present that statistically reveal the data to not be distributed
normally (P<.0001 by 2 statistic and
P<.002 by KS test for lack of fit). Furthermore, although
the magnitude differs, the pattern of these deviations is similar to
that observed in Fig 1
. Specifically, the frequency of stenoses
that show little or no change in MLD over the follow-up period
(late loss 
0 mm) exceeds that predicted by a normal
distribution. A second, smaller peak of excess frequency is observed at
a late loss of 1 mm.
|
This pattern is more clearly demonstrated when
hanging histobars are
used as an alternate graphic format (Fig 3). As in a
frequency histogram, the height of each histobar corresponds to the
frequency of observed data in that specific class. However, the
histobars are suspended from the superimposed normal frequency curve so
that the position of their lower borders reflects the magnitude of
deviation from a normal curve. As graphically depicted in Fig
3, the
frequency of late loss is greater than expected over the range of –0.3
to +0.3 mm and again over the range of about 1 to 2 mm. This effect
is apparent with (Fig 3A) or without (Fig 3B)
the inclusion of
stenoses that were no longer patent at follow-up
angiography. Subsequent analyses present only the group
with patent vessels at follow-up to avoid the potential confounding
effect of total occlusions. However, the results appeared similar when
the analyses were also applied to the total population of 3799
stenoses.
|
Fig 4 is a scattergram depicting the frequency
of
deviations from a normal distribution for the 3563 stenoses
patent at follow-up angiography. The curve shown in the figure
represents this same data smoothed with an 11-term,
nonweighted, moving average. Two frequency peaks clearly are evident,
one centered near a late-loss value of 0 mm and the other centered
near 1.2 mm. In addition, a trough separating these two peaks is also
seen, representing a lower-than-expected frequency
of intermediate values of late loss.
|
An attempt was made to determine
whether this distribution pattern was
isolated to specific categories of stenoses. Frequency
distributions of late loss were generated and analyzed after
subgrouping by the following parameters, with mean values
for dichotomization of continuous variables: (1) artery containing
the target stenosis (left anterior descending branch, left
circumflex branch, or right coronary artery); (2) location of
target stenosis (proximal or distal); (3) severity of the
target stenosis (MLD <1 mm or 
1 mm); (4) intrinsic size of
the target vessel (reference diameter <2.6 mm or 2.6 mm); (5)
initial improvement after dilation (acute gain <0.73 mm or 0.73 mm);
(6) sex (male or female); (7) angina classification (stable or
unstable); (8) treatment group (active drug or placebo); and (9) timing
of follow-up angiography (6 months or earlier). None of these
subgroupings revealed a pattern of distribution that was statistically
normal or appeared substantially different from that observed with all
stenoses combined.
Frequency Distribution of Other Quantitative Angiographic
Parameters
Quantitative measures of fit that relate to a normal
frequency
distribution are provided in Table 3 for many of the
angiographic parameters used in the present study.
Mean, median, and mode represent measures of location and
should be nearly identical for data that closely approximate a normal
distribution. The interquartile range provides an estimate of the
extent of data dispersion. The skewness coefficient reflects the
symmetry of the data and the kurtosis coefficient the steepness of the
data distribution compared with a normal curve.
|
As there is no single,
universally accepted angiographic measure of the
restenosis process, it is important to examine other
angiographic parameters besides late loss. A relevant
example is provided in Fig 5, displayed in the format of
hanging histobars. The upper curve presents the distribution of
percent diameter stenosis of the target lesion immediately
after dilation, whereas the lower curve presents this same measure
in the same group of stenoses examined at the time of
follow-up angiography. Only stenoses patent at
follow-up were included. Overall, mean diameter stenosis
after the procedure was 34.6±9.5%. At follow-up, the mean
diameter stenosis increased to 43.1%, with a broader range of
values reflected in a larger SD of 14.0%. An important aspect of this
temporal change, however, cannot be appreciated through the use of
simple summary statistics alone. The frequency distribution of the
percent diameter stenosis after the procedure fits well, both
visually and statistically, with a theoretical normal curve
(P=NS for both 2 and KS tests for lack
of fit). At follow-up, not only does the curve broaden, but the
random and uniform deviations from a normal distribution appear to be
replaced by two broad peaks of stenoses that exceed the
expected frequency. These peaks are approximately centered around 30%
diameter stenosis, a value commonly observed
postprocedure, and 65% diameter stenosis, a value that was
frequently seen predilation but was rarely encountered after the
procedure.
|
Frequency histograms with fitted distributions are shown in
Fig 6 for four other quantitative angiographic
parameters that might be related to restenosis.
Again, only stenoses patent at follow-up were included. One
potential criticism with the use of late loss as an index of
restenosis is that it reflects only absolute changes in
MLD, irrespective of the size of the vessel undergoing dilation.
Dividing late loss by the reference diameter of the initial
stenosis permits the formulation of relative late loss. As
shown in Fig 6A, this parameter displays a distribution
pattern somewhat similar to absolute late loss (Fig 2). It did
not
conform to a gaussian distribution as assessed by either a
2 statistic (P<.01) or KS
analysis (P<.005). Late loss can also be corrected
for the amount of initial gain achieved during the procedure by taking
the ratio of these two values.17 This
parameter, known as loss index, is graphically displayed in
Fig 6B. Although a ratio such as this might not be expected to
fit a
normal curve, the distribution is additionally noteworthy for an
apparent skew toward a loss index of zero. Quantitative
analysis permits the measurement of lumen dimensions not only
at its smallest point, but along the entire length of the
stenosis. Integrating this information will provide an estimate
of the planar area occupied by the atherosclerotic lesion. The
distribution of late gain in plaque area between the postangioplasty
angiogram and follow-up angiogram is depicted in Fig 6C. Again,
the
data do not fit with a normally distributed model. Finally, net gain is
displayed (Fig 6D), representing the long-term
(6-month) outcome of the angioplasty procedure. The same recurring but
nonnormal pattern of frequency distribution is evident.
|
Alternate Models of Distribution
It appears that a gaussian
or normal probability density function
cannot account statistically for the observed data distribution for
late loss or other related angiographic parameters.
Consequently, the goodness of fit of 17 additional standard
distribution functions was evaluated.18 None of these
provided a statistically acceptable model for the actual data, either
due to a poor fit or to violation of critical data limitations (such as
a requirement for only positive or integer values).
In an attempt to
improve the goodness of fit, a probabilistic model of
late loss was constructed next. Parameter estimates
incorporated in the model were derived from deconvolution of the
original histogram (Fig 2), a process accomplished by use of a
maximum
likelihood estimation based on the Newton-Raphson algorithm (BMDP LE,
version 7.0). The model presumed the existence of two discrete groups
of stenoses, each displaying a normal distribution, with 67%
of the overall population assigned to the first group. The combined
distribution of these two groups is graphically depicted in Fig
7A. The larger group showed on average little or no
change (µ=0.08 mm) in true MLD from postprocedure to follow-up
angiography; this group was labeled the "no
restenosis" group. Virtually all of the stenoses
in the smaller group, designated the "restenosis"
group, showed loss of MLD averaging 0.51 mm during the same period.
Only minimal (5.3%) overlap between groups was present, with a
statistically significant difference in mean values observed
(P<.0001). In this model, the two groups of
stenoses were then subjected to simulated quantitative
coronary angiographic analysis. Some of the errors
associated with this technique have been well characterized and
carefully measured, such as the precision of repeated sequential
measures of MLD obtained from a single angiographic
frame.15 Other errors, although just as real, have been
less completely quantified, such as those involved with cineframe
selection,19 misregistration of repeated views, and film
exposure variability. When the inherent imprecision of the quantitative
analysis scheme used for the actual measurements was entered
into the model, the distributions of the two populations were broadened
substantially, as shown in Fig 7B. These two populations were
then
combined, generating a new distribution curve for this total group,
labeled "sum" on Fig 7B. Note its superficial
resemblance to a
normal distribution curve, a resemblance mathematically at odds with
its origin. Its differences from a gaussian curve become more apparent
when the normal probability density function is superimposed, as in Fig
7C. This normal curve represents the exact function depicted in
Fig 2, and a close look at these two graphs (one theoretical,
one
observed) shows them to be quite similar. Finally, the theoretical
curve generated by the model and the distribution created by the
observed data are displayed simultaneously in Fig 7D. The
two curves are nearly superimposed over most of the data range,
which suggests that the model provides a good fit of the actual
angiographic data.
|
|
Discussion |
|---|
|
TopAbstractIntroductionMethods Results Discussion References |
|---|
Findings of the Current Study
Despite a superficial resemblance, late loss and other indexes of coronary restenosis do not conform statistically to a normal distribution. This conclusion is derived from analyses that used the impressive statistical power afforded by a very large study population. The patterns of deviations from a normal curve were also of interest. Two frequency peaks of late loss were evident (Fig 4
). The first and larger peak of excess frequency was centered
around
zero, which represented a higher-than-expected
number of stenoses that showed little change in luminal
dimensions during the follow-up period. The second, smaller peak
comprised late loss values that ranged from 1.0 to 1.5 mm. Intermediate
values of late loss (0.3 to 1.0 mm) occupied the area of the histogram
between these two peaks and showed a frequency of stenoses
lower than predicted by a normally distributed model. This pattern of
deviations raises the possibility of the existence of two subgroups of
stenoses, one that shows little tendency toward late luminal
narrowing and the second a propensity toward substantial
renarrowing. It is possible that these two subgroups consist of stenoses that completely occluded and stenoses that remained patent during the follow-up period. There are theoretical reasons why the pattern and rate of late luminal narrowing might differ between these groups. Investigations into dynamic coronary flow models have estimated a severe reduction in coronary flow through a stenosis with an MLD of <0.5 mm caused by the imposition of a nonphysiologically achievable pressure drop across the narrowing.20 21 This in turn leads to relative stasis near the stenosis, which, when combined with local heightened shear rates, predisposes the coronary lesion to accelerated thrombosis and complete occlusion.22 This theory is supported by the findings in the current data set that reveal the diameter stenosis of the tightest patent lesion at follow-up to be 86.6% (corresponding MLD=0.4 mm), with only 32 patent lesions (0.84% of total population) possessing a diameter stenosis >75%. Another contributory factor is the relative inability of both eye and computer to identify and therefore measure the true luminal boundaries of tight stenoses, an effect of the low video density associated with the narrow stream of contrast that outlines the lesion. Because of these concerns, all subsequent analyses performed in this study utilized only the 3563 stenoses that remained patent at follow-up. The same pattern of deviations remained, however, even after this exclusion was applied.
Attempts at fitting the observed data to other standard
distribution
functions were equally unrewarding. However, it was possible to
construct a probabilistic model of late loss that fit well with the
observed data (Fig 7). Curve deconvolution provides a
mathematical
technique useful in separating overlapping data distributions. This
process identified the potential presence of two populations of
stenoses from a distribution that may have been artifactually
smoothed by measurement "noise." In this model, 3563 theoretical
stenoses were dichotomized into two groups, the larger showing
no important change in MLD during follow-up and the smaller being
those that experienced a definite decrease in MLD. Each of these two
groups was assumed to be normally distributed. These theoretical
stenoses were then subjected to simulated quantitative
coronary analysis with its known limitations. In
contrast to true late loss, the distribution of measured late loss
showed substantial overlap between groups, resulting in a combined
distribution curve that appeared continuous, unimodal, and
approximately bell shaped. Statistically, this combined distribution
did not approximate a normal curve. It did, however, fit well with the
actual observed data, with the curves representing the
theoretical model and observed data nearly superimposed over most of
the range of abscissa values. The only obvious separation in the curves
occurred at extreme values of negative late loss. Stenoses
located in this area of the frequency histogram showed substantial
improvement rather than the more typical deterioration of luminal
dimensions during follow-up. This situation could occur, for
example, when a major dissection that has partially occluded the artery
after angioplasty remodels with time to provide a larger smooth lumen.
Perhaps these stenoses should rightly be assigned to a third
subgroup, providing a further refinement of the model and an even
better fit of the actual data.
Importance of the Normal Distribution
The normal curve, also
known as the gaussian or probability curve,
occupies a key position in modern statistics and probability theory.
Its underlying concept is not new, with its equation first published by
De Moivre in 1733.23 However, there are several reasons
for its enduring prominence in statistical practice.24
First, a number of random variables found in science and medicine
appear to be well described by a normal distribution. Variables
that do not initially approximate a normal distribution sometimes can
be made to fit through the use of a data transformation, such as a
logarithm. Second, the normal probability function possesses many
features that make mathematical manipulations relatively easy. As a
result, its properties have been accurately and extensively tabulated,
further promoting its use. Third, a large number of statistical tools
used in medical research are most directly applicable to data that
approximate a normal distribution. The wide range of common
parametric tests includes the Student's t test,
ANOVA, and multiple linear regression. The use of these tests with data
derived from small samples and that display markedly nonnormal
distributions is mathematically and therefore scientifically
unsound.
Relation to Prior Work
In 1991, King et al25
presented a preliminary
report of a bimodal distribution of percent diameter stenosis
observed during follow-up angiography. A cautious interpretation
was advised by the authors, however, because of the unblinded
interpretation of the films combined with the potential subjective
influence of handheld caliper measurements. In contrast, more recent
studies derived from blinded, computer-derived measurements
reported that measured parameters of luminal dimensions
obtained at follow-up angiography appear continuous rather than
dichotomous in distribution. In response, most investigators have begun
computing, describing, and reporting coronary
restenosis in terms of continuous
variables.26 27 28 This observation,
however, does not
necessarily imply that the mechanism involved in the
restenosis process is continuous, with all lesions
experiencing luminal renarrowing that differs only in degree. An
alternative explanation lies in the effects of measurement variability
and other errors that represent an obligate confounder of
clinical investigations in restenosis.29
Errors such as these, both known and unknown, tend to be random and
near-normally distributed. Values for all of the angiographic
variables under study represent a composite of true
arterial dimensions distorted by a number of independent
component errors, most of which influence the data by altering its
distribution to more closely mimic a normal curve. Hence, minor
errors present at any or all stages of a clinical study of
restenosis can falsely impute a normal or near-normal
distribution to the results that in reality does not exist. Fig
7
provides an example based on application of a test with a known
measurement precision. As shown in the model, quantitative angiographic
measurement error alone is sufficient to mask the true bimodal
distribution underlying the late loss parameters used in
the model.
One parameter that is often cited as an index of
restenosis is late loss.30 This variable,
presented in units of length, is computed as the difference in
the stenosis MLD between the angiogram taken immediately after
the procedure and the angiogram obtained after 6 months of
follow-up. Recently, Rensing et al3 and Kuntz et
al4 independently published frequency histograms of late
loss that they concluded fit a distribution that is normal or near
normal. Although this would appear at first to conflict with the
results obtained in the current study, a close examination of the
histograms provided in each of the previous investigations shows their
pattern of distribution to be surprisingly similar to Figs 1
and 2.
There are two principal reasons why opposite conclusions might have
been drawn from similar-appearing histograms. First, the sample
size of stenoses contributing to the current study is
substantially greater than those used previously. In fact, the size of
the cohort of stenoses used by Kuntz et al4 is
<5% of that used in the current study. Because of their lower
statistical power, analyses of these smaller samples could
easily fail to resolve subtle but important deviations from a gaussian
curve. Second, Rensing et al3 based their conclusion on
the goodness of fit between the data and a theoretical normal function
using visual inspection of the data alone without applying formal
statistical analyses. It is clear from the current study that a
superficial resemblance to a normal curve often may be accompanied by a
statistically unacceptable fit.
Potential Limitations
There are several possible limitations
with the current study.
First, although the population selected appears to be
representative of most angioplasty candidates, the
findings obtained in other populations may differ. The use of a
different dilation strategy or different device might yield disparate
results. In addition, entry into the study was limited to de novo
stenoses that had not been subjected to prior angioplasty.
Although this exclusion helped maximize the uniformity of the
population under study, it precluded further insights into the behavior
of previously dilated lesions undergoing treatment for
restenosis.
Second, unknown factors could have systematically
influenced the data
and distorted an otherwise normally distributed parameter.
Arguing against this are the data presented in Fig 5. Although
the percent diameter stenosis at follow-up is not normally
distributed, this same parameter examined immediately after
the completion of dilation is visually and statistically normal. As
each of these measurements should be subjected to the same pattern and
magnitude of errors associated with quantitative coronary
analysis, the change in distribution observed is more likely
the result of the underlying biological process than measurement
artifact.
Third, although the model presented in Fig 7
does provide a
good statistical fit with the actual empirical data, this should by no
means imply superiority over other potential models. Indeed, it is
likely that other paradigms could be envisioned that fit the data well.
The intent of the proposed model was to illustrate that the observed
data in fact could have originated from two independent populations of
stenoses, one with and one without significant late luminal
renarrowing.
Clinical Implications
It is not possible from the current
study to reach a definitive
conclusion about the process or processes underlying coronary
restenosis. Nevertheless, the concept of
restenosis as an extreme form of a ubiquitous
postprocedural luminal narrowing may be overly simplistic or actually
incorrect. In fact, in the current study, the best fit with the
observed data occurred with a model comprising two discrete populations
of stenoses, one that experienced restenosis and
the other that did not.
Although this issue is fundamental to the scientific investigations and clinical practice of interventional cardiology, a definitive answer may not be obtainable at this time given the limitations of modern angiography and its quantitative assessment. Perhaps alternative anatomic measurement techniques, such as intravascular ultrasound, or the nonanatomic assessment of stenosis severity by use of physiological principles will provide the improved measurement precision required. In the meantime, basic investigations may be especially useful in providing important clues into the underlying mechanism of coronary restenosis.
|
Acknowledgments |
|---|
We appreciate the contributions of Drs Theo Steijnen, Eric Boersma, and David Foley in their suggestions and review of the manuscript.
Received August 1, 1995; revision received October 19, 1995; accepted October 23, 1995.
|
References |
|---|
|
TopAbstractIntroductionMethods Results Discussion References |
|---|
-
Gruentzig A. Transluminal dilatation of
coronary-artery stenosis.
Lancet. 1978;1:263. Letter. [Medline]
[Order article via Infotrieve]
-
Holmes DR, Vliestra RE, Smith HC, Vetrovec GW, Kent
KM, Cowley MJ, Faxon DP, Gruentzig AR, Kelsey SF, Detre KM, van Raden
MJ, Mock MB. Restenosis after
percutaneous transluminal coronary angioplasty
(PTCA): a report from the PTCA registry of the National Heart, Lung,
and Blood Institute. Am J Cardiol. 1984;53:77C-81C. [Medline]
[Order article via Infotrieve]
-
Rensing BJ, Hermans WRM, Deckers JW, de Feyter PJ,
Tijssen JGP, Serruys PW. Lumen narrowing after
percutaneous transluminal coronary balloon
angioplasty follows a near Gaussian distribution: a quantitative
angiographic study in 1,445 successfully dilated lesions.
J Am Coll Cardiol. 1992;19:939-945. [Abstract]
-
Kuntz RE, Safian RD, Levine MJ, Reis GJ, Diver DJ,
Baim DS. Novel approach to the analysis of
restenosis after the use of three new coronary
devices. J Am Coll Cardiol. 1992;19:1493-1499. [Abstract]
-
Kuntz RE, Baim DS. Defining coronary
restenosis: newer clinical and angiographic
paradigms. Circulation. 1993;88:1310-1323.
[Free Full Text]
-
Serruys PW, Rutsch W, Heyndrickx GR, Danchin N, Mast
EG, Wijns W, Rensing BJ, Vos J, Stibbe J, for the Coronary
Artery Restenosis Prevention on Repeated
Thromboxane-Antagonism Study Group. Prevention of
restenosis after percutaneous transluminal
coronary angioplasty with thromboxane
A2-receptor blockade: a randomized, double-blind,
placebo-controlled trial. Circulation. 1991;84:1568-1580.
[Abstract/Free Full Text]
-
The Multicenter European Research Trial with
Cilazapril after Angioplasty to Prevent Transluminal Coronary
Obstruction and Restenosis (MERCATOR) Study Group. Does the
new angiotensin converting enzyme inhibitor
Cilazapril prevent restenosis after
percutaneous transluminal coronary angioplasty?
Results of the MERCATOR Study—a multicenter, randomized,
double-blind, placebo-controlled trial.
Circulation. 1992;86:100-110.
[Abstract/Free Full Text]
-
Serruys PW, Klein W, Tijssen JPG, Rutsch W, Heyndrickx
GR, Emanuelsson H, Ball SG, Decoster O, Schroeder E, Liberman MD,
Eichhorn E, Willerson JT, Anderson HV, Khaja F, Alexander RW, Baim D,
Melkert R, van Oene JC, Van Gool R. Evaluation of Ketanserin in
the prevention of restenosis after
percutaneous transluminal coronary angioplasty:
a multicenter randomized, double-blind, placebo-controlled
trial. Circulation. 1993;88:1588-1601.
[Abstract/Free Full Text]
-
Faxon DP, on behalf of the Multicenter American
Research Trial With Cilazapril After Angioplasty to Prevent
Transluminal Coronary Obstruction and Restenosis
(MARCATOR) Study Group. Effect of high dose
angiotensin-converting enzyme inhibition on
restenosis: final results of the MARCATOR study, a
multicenter, double-blind, placebo-controlled trial of
cilazapril. J Am Coll Cardiol. 1995;25:362-369. [Abstract]
-
Reiber JHC. Why and how should QCA systems be
validated? In: Serruys PW, Foley DP, de Feyter PJ, eds.
Quantitative Coronary Angiography in Clinical
Practice. Dordrecht, Netherlands: Kluwer Academic Publishers;
1994:1-6.
-
Serruys PW, Luijten HE, Beatt KJ, Geuskens R, de Feyter
PJ, van den Brand M, Reiber JHC, ten Katen HJ, van Es GA, Hugenholtz
PG. Incidence of restenosis after successful
coronary angioplasty: a time related phenomenon—a
quantitative angiographic study in 342 consecutive patients at 1, 2, 3,
and 4 months. Circulation. 1988;77:361-371.
[Abstract/Free Full Text]
-
Reiber JHC, Serruys PW, Kooijman CJ, Wijns W, Slager
CJ, Gerbrands JJ, Schuurbiers JCH, den Boer A, Hugenholtz PG.
Assessment of short, medium, and long term variations in
arterial dimensions from computer assisted quantification
of coronary cineangiograms.
Circulation. 1985;71:280-288.
[Abstract/Free Full Text]
-
Reiber JHC, Serruys PW. Quantitative
coronary angiography. In: Marcus ML, Schelbert HR, Skorton DJ,
Wolf GL, eds. Cardiac Imaging: A Companion to
Braunwald's Heart Disease. Philadelphia, Pa: WB Saunders;
1990:211-280.
-
Umans VAWM, Hermans WRM, Herrman JPR, Pameyer J,
Serruys PW. Experiences of a quantitative coronary
angiographic core laboratory in restenosis prevention
trials. In: Serruys PW, Foley DP, de Feyter PJ, eds.
Quantitative Coronary Angiography in Clinical
Practice. Dordrecht, Netherlands: Kluwer Academic Publishers;
1994:121-135.
-
Keane D, Haase J, Slager CJ, van Swijndregt E, Lehmann
KG, Ozaki Y, di Mario C, Kirkeeide R, Serruys PW. Comparative
validation of quantitative coronary angiography systems:
results and implications from a multicenter study using a standardized
approach. Circulation. 1995;91:2174-2183.
[Abstract/Free Full Text]
-
Hollander M, Wolfe DA.
Nonparametric Statistical Methods. New York, NY:
Wiley, John, & Sons, Inc; 1973:219.
-
Kuntz RE, Hinohara T, Robertson GC, Safian RD, Simpson
JB, Baim DS. The influence of vessel selection on the observed
restenosis rate following endoluminal stenting or
directional atherectomy. Am J Cardiol. 1992;70:1101-1108. [Medline]
[Order article via Infotrieve]
-
Hastings NAJ, Peacock JB. Statistical
Distributions: A Handbook for Students and
Practitioners. London, England: Butterworth;
1975:29-129.
-
Gibson CM, Sandor T, Stone HS, Pasternak RC, Rosner B,
Sacks FM. Quantitative angiographic and statistical methods to
assess serial changes in coronary luminal diameter and
implications for atherosclerosis regression
trials. Am J Cardiol. 1992;69:1286-1290.[Medline]
[Order article via Infotrieve]
-
Kirkeeide RL, Gould KL, Parsel L. Assessment of
coronary stenoses by myocardial perfusion imaging
during pharmacologic coronary vasodilation, VII: validation of
coronary flow reserve as a single integrated functional measure
of stenosis severity reflecting all its geometric
dimensions. J Am Coll Cardiol. 1986;7:103-113. [Abstract]
-
Gould KL. Pressure flow characteristics of
coronary stenoses in unsedated dogs at rest and during
coronary vasodilation. Circ Res. 1978;43:242-253.
[Abstract/Free Full Text]
-
Folts JD, Crowell EB, Rowe GG. Platelet
aggregation in partially obstructed vessels and its elimination with
aspirin. Circulation. 1976;54:365-370.
[Abstract/Free Full Text]
-
Elderton WP, Johnson NL. Systems of
Frequency Curves. London, England: Cambridge University Press;
1969:70.
-
Cramér H. The Elements of
Probability Theory and Some of Its Applications. New York, NY:
Robert E. Krieger Publishing Co; 1973:90-96.
-
King SB III, Weintraub WS, Tao X, Hearn J, Douglas JS
Jr. Bimodal distribution of diameter stenosis 4 to 12 months
after angioplasty: implications for definitions and interpretation of
restenosis. J Am Coll Cardiol. 1991;17:345A. Abstract.
-
Topol EJ, Leya F, Pinkerton CA, Whitlow PL, Hofling B,
Simonton CA, Masden RR, Serruys PW, Leon MB, Williams DO, King SB III,
Mark DB, Isner JM, Holmes DR Jr, Ellis SG, Lee KL, Keller GP, Berdan
LG, Hinohara T, Califf RM, for the CAVEAT study group. A comparison of
directional atherectomy with coronary angioplasty in patients
with coronary artery disease. N Engl J
Med. 1993;329:221-227.
[Abstract/Free Full Text]
-
Umans VA, Hermans WR, Foley DP, de Feyter PJ, Serruys
PW. Restenosis after directional coronary
atherectomy and balloon angioplasty: comparative analysis based
on matched lesions. J Am Coll Cardiol. 1993;21:1382-1390. [Abstract]
-
Popma JJ, De Cesare N, Pinkerton CA, Kerejakes DJ,
Whitlow P, King SB, Topol EJ, Holmes DR, Leon MB, Ellis SG.
Quantitative analysis of factors influencing late lumen loss
and restenosis after directional coronary
atherectomy. Am J Cardiol. 1993;71:552-557. [Medline]
[Order article via Infotrieve]
-
Serruys PW, Foley DP, Kirkeeide RL, King SB III.
Restenosis revisited: insights provided by quantitative
coronary angiography. Am Heart J. 1993;126:1243-1267. [Medline]
[Order article via Infotrieve]
-
Kuntz RE, Safian RD, Levine MJ, Reis GJ, Diver DJ, Baim
DS. Novel approach to the analysis of
restenosis following three new coronary
interventions. J Am Coll Cardiol. 1992;19:1493-1499.
This article has been cited by other articles:
 |
|
 |
|
  J. C.J. Sun, J. W. Eikelboom, P. Agostoni, G. M. Sangiorgi, G. G. Biondi-Zoccai, B. Scheller, M. Bohm, and E. Camenzind Treatment of Restenosis with a Paclitaxel-Coated Balloon Catheter N. Engl. J. Med., March 8, 2007; 356(10): 1071 - 1073. [Full Text] [PDF]
|
|
|
|
 |
|
 P. A. Lemos, N. Mercado, R. T. van Domburg, R. E. Kuntz, W. W. O'Neill, and P. W. Serruys Comparison of Late Luminal Loss Response Pattern After Sirolimus-Eluting Stent Implantation or Conventional Stenting Circulation, November 16, 2004; 110(20): 3199 - 3205. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 D. A. Morrow, E. M. Antman, S. A. Murphy, S. F. Assmann, R. P. Giugliano, C. P. Cannon, C. Michael Gibson, C. H. McCabe, H. V. Barron, F. Van de Werf, et al. The Risk Score Profile: a novel approach to characterising the risk of populations enrolled in clinical studies Eur. Heart J., July 1, 2004; 25(13): 1139 - 1145. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 K. Kwon, D. Choi, S.-H. Choi, Bon Kwon Koo, Y. Jang, W.-H. Shim, and S.-Y. Cho Coronary Stenting After Rotational Atherectomy in Diffuse Lesions of the Small Coronary Artery: Comparison with Balloon Angioplasty Before Stenting Angiology, July 1, 2003; 54(4): 423 - 431. [Abstract] [PDF]
|
|
|
|
|
|
W. R. P. Agema, J. W. Jukema, S. N. Pimstone, and J. J. P. Kastelein Genetic aspects of restenosis after percutaneous coronary interventions;towards more tailored therapy Eur. Heart J., November 2, 2001; 22(22): 2058 - 2074. [PDF]
|
|
|
|
 |
|
 S E Francis, N J Camp, A J Burton, R M Dewberry, J Gunn, A Stephens-Lloyd, D C Cumberland, A Gershlick, and D C Crossman Interleukin 1 receptor antagonist gene polymorphism and restenosis after coronary angioplasty Heart, September 1, 2001; 86(3): 336 - 340. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. Kastrati, A. Schomig, J. Dirschinger, J. Mehilli, F. Dotzer, N. von Welser, and F.-J. Neumann A Randomized Trial Comparing Stenting With Balloon Angioplasty in Small Vessels in Patients With Symptomatic Coronary Artery Disease Circulation, November 21, 2000; 102(21): 2593 - 2598. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 M. Gyongyosi, P. Yang, A. Khorsand, D. Glogar, on behalf of the Austrian Wiktor Stent Study Group, and European Paragon Stent Investigators Longitudinal straightening effect of stents is an additional predictor for major adverse cardiac events J. Am. Coll. Cardiol., May 1, 2000; 35(6): 1580 - 1589. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. Buffon, G. Liuzzo, L. M. Biasucci, P. Pasqualetti, V. Ramazzotti, A. G. Rebuzzi, F. Crea, and A. Maseri Preprocedural serum levels of C-reactive protein predict early complications and late restenosis after coronary angioplasty J. Am. Coll. Cardiol., November 1, 1999; 34(5): 1512 - 1521. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
A. Kastrati, A. Schomig, S. Elezi, H. Schuhlen, M. Wilhelm, and J. Dirschinger Interlesion Dependence of the Risk for Restenosis in Patients With Coronary Stent Placement in Multiple Lesions Circulation, June 23, 1998; 97(24): 2396 - 2401. [Abstract] [Full Text] [PDF]
|
|