# Contributions of Frequency Distribution Analysis to the Understanding of Coronary Restenosis

## A Reappraisal of the Gaussian Curve

## Jump to

## Abstract

*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 χ^{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.

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 studies^{3} ^{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

### 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 elsewhere^{10} ^{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 6.9×6.9-mm area of a single cineframe (representing 11% of the total frame area) and digitized into a 512×512-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 ς 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 χ^{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

### 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 χ^{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 χ^{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 χ^{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 χ^{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

### 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 al^{25} 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 al^{3} and Kuntz et al^{4} 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 al^{4} 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 al^{3} 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.

- Copyright © 1996 by American Heart Association

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- Contributions of Frequency Distribution Analysis to the Understanding of Coronary RestenosisKenneth G. Lehmann, Rein Melkert and Patrick W. SerruysCirculation. 1996;93:1123-1132, published online before print March 15, 1996http://dx.doi.org/10.1161/01.CIR.93.6.1123
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- Contributions of Frequency Distribution Analysis to the Understanding of Coronary RestenosisKenneth G. Lehmann, Rein Melkert and Patrick W. SerruysCirculation. 1996;93:1123-1132, published online before print March 15, 1996http://dx.doi.org/10.1161/01.CIR.93.6.1123

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