Author + information
- Received July 5, 2016
- Revision received December 3, 2016
- Accepted December 15, 2016
- Published online July 17, 2017.
- aDepartment of Medicine, Division of Cardiology, University of British Columbia, Vancouver, British Columbia, Canada
- bEmmes Canada, Burnaby, British Columbia, Canada
- cFaculty of Health Sciences, Simon Fraser University, Vancouver, British Columbia, Canada
- ↵∗Address for correspondence:
Dr. Simon W. Rabkin, Department of Medicine, Division of Cardiology, University of British Columbia, Level 9, 2775 Laurel Street, Vancouver, British Columbia V5Z 1M9, Canada.
Objectives This study sought to develop a formula from a large population-based study that best fit the QT-heart rate (HR) relationship independent of the standard mathematical relationships.
Background Attempts to adjust or correct for the impact of HR on the QT interval (QTc) have applied various mathematical equations to electrocardiographic (ECG) data rather than allowing the data to determine the form of the relationship.
Methods A spline correction function was developed using the ECG data from NHANES (National Health and Nutrition Examination Surveys) II and III. The magnitude of linear, quadratic, and cubic trends in the relationship between HR and each QTc was quantified using an F-statistic with differences between QTcs compared using a permutation procedure.
Results The effect of HR on QT was obliterated by the spline QT for both men and women. The cross-validated spline QTc was superior (i.e., flatter) to 6 other formulae, including ones proposed previously. In ECGs from the clinic with HRs faster than 70 beats/min, the QTcs from different formulae were significantly (p < 0.0001) different from one another. Individual values suggest the use of the longest and shortest QTc intervals as developed originally. The new QTc and its population percentile ranking can be provided for clinical ECGs.
Conclusions A new QTc formula was developed which eliminates the relationship between QT and HR. At faster HRs, the 2 most commonly used QTcs provide numerical values at the extremes of QTc. Compared to existing formulae, the new formula had the best performance.
Assessment of the QT interval has attracted considerable attention as a marker for potential drug-induced cardiac toxicity or drug-induced sudden cardiac death (1), electrolyte abnormalities (2), and identification of individuals with genetic mutations that carry a high risk of sudden death (3). A critical aspect of the evaluation of the QT interval is its adjustment for heart rate because of the recognized inverse relationship between QT interval and heart rate. The formulae most frequently used to produce a QT interval that corrects (QTc) for the changes in heart rate were developed almost 100 years ago by Bazett (4) (QTcBZT) and Fridericia (5) (QTcFRD). Bazett (4), in 1920, concluded that the relationship between the duration of systole shown on the electrocardiograph (ECG) was a function of the square root of the heart rate, a concept which was proposed earlier in studies by others on the duration of mechanical systole with different heart rates. In the same year, Fridericia (5) published his conclusion, on the basis of 50 apparently normal individuals, that QT interval was a function of the cubed root of the RR interval. Subsequently, many other QT adjustment formulae have been proposed (6). The Framingham baseline ECG data were used to develop a linear regression model (7). A power function QT correction formula was derived from ECGs that were baseline recordings, before drug treatment in Eli Lilly clinical trials by Dmitrienko et al. (8) (QTcDMT). Most recently, Rautaharju et al. (9) developed 2 formulae (QTcRTHa and QTcRTHb) on the basis of data pooled from population surveys as well as ECGs from baseline before drug testing. These studies used linear or power-based equations and found the best parameters from the chosen equation. The mathematical equation, linear, exponential, cubic, and so forth, have been applied to ECG data rather than allowing the data to determine the form of the relationship. The first objective of this study was to develop a formula from a large population based study which best fit the QT-heart rate relationship independent of the standard mathematical relationships, which presuppose a linear, power, or logarithmic function. The second objective was to compare the developed formula to previously derived QTc formulae. The third objective was to define age- and sex-based mean ± SD for the formulae; the fourth objective was to test the new formula using ECGs from a clinical setting.
The U.S. NHANES (National Health and Nutrition Examination Survey) II and III studies, conducted by the U.S. Centers for Disease Control and Prevention (CDC), were chosen to evaluate the QT interval in different age groups. These studies were selected because each NHANES survey was conducted using a representative sample of the civilian U.S. population, and importantly, weighting factors were available, meaning that the results are more representative than voluntary survey samples or volunteers in a drug trial. NHANES II and III 2012 and 2015 data were downloaded from the CDC (10,11) and from U.S. Department of Housing and Urban Development 2008 data. Data were imported and processed using Excel software (2013, Microsoft, Redmond, Washington). The possibility that the same individuals were selected for both studies was considered not significant because of the randomness of the participant selection process. Subject sampling weights for NHANES I and II provided by the CDC were used in all analyses. Data from NHANES II and III were pooled to create one large data set. Then, exclusion criteria were applied to exclude ECG factors, known to affect the duration of the QT interval. Briefly, the exclusion criteria removed subjects with: 1) no valid QT interval duration or heart rate data; 2) probable or possible myocardial infarction or major ECG abnormalities; 3) rhythm not being in sinus; 4) probable left ventricular hypertrophy; and 5) left or right bundle branch block. The full definitions of all ECG abnormalities are available in the NHANES documentation (10,11). The heart rate and QT interval measurements were made by a computerized ECG analysis algorithm which eliminated intraobserver variability (10,11).
Development of a new QTc formula
The new QT correction function, the spline QT correction, was developed using R statistical software (2014, R Core Team, Vienna, Austria). The spline correction function was modeled using a cubic regression spline with 4 knots and a fixed adjustment for sex, with each observation weighted by the respective NHANES sampling weight with spline parameters selected as those that minimized the squared vertical distance to the fitted line (i.e., least squares estimate). The spline QTc was then computed by taking the difference between QT measurements and the predicted value of the regression spline at the observed heart rate and sex and then adding the predicted spline value at a heart rate of 60 beats/min for men. The predicted spline value at a heart rate of 60 beats/min for men was used as the “baseline,” where the spline-predicted value was considered to be deviation from this baseline.
An age-adjusted spline QTc was computed similarly by adding a fixed adjustment for age to the spline correction function regression model. To compute the age-adjusted QTc, the difference between QT measurement and predicted value of the regression spline at the observed heart rate, sex, and age was computed, and then the predicted spline value at a heart rate of 60 beats/min for men at 50.3 years of age was added. The age of 50.3 years was chosen as the baseline because it was the weighted estimate of mean age in the NHANES data.
Although we were able to compute an age-adjusted spline QTc formula, existing QTc formulae do not account for age, and for fair comparison purposes, we compared the performance of the spline QTc, which accounts for sex but not age, to the existing QTc formulae. Some results from the age-adjusted spline QTc are included to illustrate the benefit of additionally adjusting for age.
An applet to compute the spline QTc from user input was made using the shiny package, a software package which builds web applets for data analysis and computation using user-inputted data, in R software (12). Percentiles of the input data were computed by comparing their QTc to the QTcs of the NHANES subjects of the same sex as the reference; those who had higher QTc percentiles are indicative of greater QT prolongation than usual.
To evaluate the spline-based QT conversion function, we used a 10-fold cross-validation procedure to obtain the spline QTc for individual subjects not included in the training set for the spline QTc. In this approach, spline QTcs were computed by randomly splitting the data into 10 folds of roughly equal size, called “test sets.” For each test set, all observations in the data that were not selected to be in the test set were included in a corresponding training set (i.e., 10 training sets and 10 test sets). The spline correction function was fit to each training set, as described earlier, and cross-validated spline-QTcs were computed for the test set by using the training set fit. Because some data were held out of the training set in each fold, the cross-validated spline QTcs were computed using data that were not used to fit the correction function and hence approximate performance in an independent data set.
Comparison of different QTc formulae
Six different heart rate correction formulae were evaluated either because of their long-term usage or their recent introduction from large population studies. The formulae include the equations proposed by Bazett (4) (QTcBZT), Fridericia (5) (QTcFRD), Hodges et al. (13) (QTcHDG), Framingham (Sagie et al. ) (QTcFRM), Dmitrienko et al. (8) (QTcDMT), and Rautaharju et al. (9) (QTcRTHa), which were identified by a proposed standardized nomenclature (6). Two formulae were described in Rautaharju et al. (9).
Consecutive resting ECGs from a clinical reading session that fulfilled the criteria of sinus rhythm and without bundle branch block, left ventricular hypertrophy, ST-segment elevation myocardial infarction, or significant ST-T wave changes were considered. The study was approved by our Institutional Research Ethics Board. There were 44 ECGs (50% men) that were obtained anonymously. Because most QTc formulae provide similar QTc values around a heart rate of 60 beats/min, ECGs with heart rate of 70 beats/min or greater were selected. No clinical information was available, similar to the usual clinical ECG interpretation setting. ECGs were acquired and digitally analyzed. Electrocardiographic waveforms were sampled at approximately 500 samples per second, using the Marquette 12SL analysis program (GE Healthcare, Milwaukee, Wisconsin). The QT interval was measured “from the earliest detection of depolarization in any lead (QRS onset) to the latest detection of repolarization in any lead (T offset) (Marquette 12SL ECG Analysis Program; GE Healthcare). The QT interval and heart rate measured by the analysis program was used in the heart rate adjustment formulae.
On the basis of the requirement to adjust the QT interval for heart rate, a superior QT correction function is a QTc that has no relationship to heart rate; that is, mean QTc will be the same (graphically flat) for all heart rates. To compare the performance of the spline QTc to other QTc formulae, we quantified the extent to which trends remained between heart rate and QTc by using an F-statistic from separate, weighted regressions on each QTc formulae and sex involving linear, quadratic, and cubic functions of heart rate as predictors (14). The magnitude of the statistic is then indicative of the strength of linear, quadratic, and cubic trends in the relationship between heart rate and QTc. If the correction formula succeeds in completely removing these trends, F will be 0. Large values of F indicate that at least one of these relationships (linear, quadratic, or cubic) exists and that the QTc is not flat for all heart rates. To quantify differences in F-statistic, a permutation testing procedure was used to test whether the F-statistic from each QTc formula was significantly higher than that of the cross-validated spline QTc (15) in the N = 13,627 ECGs. Because some of the ECGs in the data have very extreme heart rates, a similar trimmed analysis was conducted on the n = 13,610 ECGs obtained at heart rates between 40 and 120 beats/min.
The relationship between QT interval and heart rate demonstrated a consistent reduction in QT interval with each incremental increase in heart rate. This inverse relationship between the duration of the QT interval and heart rate was evident in men (Figure 1A) and women (Figure 1B). The extremes of heart rate were associated with a difference of at least 200 ms in QT interval. The QT interval was longer at each heart rate in women than in men. The spline QT correction formulae (see the equations in the Online Appendix) obliterated the effect of heart rate on QT in both men and women (Figure 2), where the regression lines up to a cubic trend between QTc and HR are almost flat.
Comparison of spline QTc with other QT correction formulae
The cross-validated spline correction function was compared with 6 different heart rate correction formulae QTcBZT, QTcFRD, QTcHDG, QTcFRM, QTcDMT, and QTcRTHa. A superior QT correction function is one in which the QTc has no relationship to heart rate; that is, the mean QTc will be the same or flat for all heart rates. Boxplots by heart rate for each QTc were ordered by the p value comparing flatness of the slope for each QTc formula compared to the cross-validated spline formula for men (Figure 3) and women (Figure 4). The fitted regression lines used to calculate the F-statistic for each conversion formula that demonstrated the QTc-heart rate relationship are displayed as the black and dashed lines. The regression lines displayed over the spline QTc and cross-validated spline QTcs are very flat, demonstrating the lack of relationship between HR and the spline QTc.
The permutation test indicates that in women the cross-validated spline QTc was superior (i.e., is flatter) to the sex-specific QTcRTH (p = 0.045), QTCBZT, QTcDMT, QTcFRD, QTcFRM, and QTcHDG formulae (p < 0.0001). There was no evidence that the cross-validated spline QTc was superior to the general Rautaharju formula (p = 0.49). The trimmed analysis found similar results, that is, there was no evidence that the cross-validated spline QTc was superior to the sex-specific QTcRTH (ptrim = 0.18).
In men, the cross-validated spline QTc was superior to the QTcFRD (p = 0.042), QTcBZT, QTcDMT, QTcHDG, and QTcRTH formulae (p < 0.0001). There was no evidence that the cross-validated spline QTc was superior to the sex-specific QTcRTH (p = 0.49) or the QTcFRM (p = 0.13) formulae in men. The cross-validated spline QTc succeeded for both men and woman at removing the linear, quadratic, and cubic trends relative to most of the other formulae. The trimmed analysis found more convincing results of superiority of the cross-validated spline QTc than for the QTcFRD (ptrim = 0.0041) and QTcFRM (ptrim = 0.047) formulae.
We next sought to examine the effect of age on the new QTc formula because of the significant association of QTc with age (16). There was evidence of linear associations between the cross-validated spline QTc and age in men and women (Figure 5). Weighted linear models were fit with spline QTc as the outcome, and age was considered as a continuous variable and the predictor. The spline QTc was noted to increase by 0.21 (SEM = 0.02) for a 1-year increase in age for men and by 0.16 (SEM = 0.02) for a 1-year increase in age for women, a significant association for both men and women (p < 0.0001).
The methodology of using regression splines to remove linear relationships with QT (such as the relationship between sex and QT) and nonlinear relationships (such as the relationship between heart rate and QT) can be applied to remove effects of other variables on the QT, such as age. By including a linear trend of age in the regression spline model of QT on heart rate and sex, an age-adjusted spline QTc can be computed. If a trend for age is included, the resulting age-adjusted spline QTc is no longer associated with age (p = 0.15 in a test for men; p = 0.25 in a test for women). The age-adjusted spline QTc performs similarly well in removing the relationship between heart rate and QTc compared to other formulae, on the basis of the permutation test (p < 0.0001 compared to QTcBZT, QTcDMT, QTcHDG, and QTcRTHa in men and QTcBZT, QTcDMT, QTcFRD, QTcFRM, and QTcHDG in women; p = 0.02 compared to QTcRTHb in women; p = 0.002 compared to QTcFRD in men; p = 0.25 compared to QTcRTHa in women; p = 0.02 compared to QTcRTHb in women; and p = 0.44 compared to QTcRTHb in men). The age-adjusted spline QTc, however, has an unfair advantage compared to the comparator formulae because it additionally removes the effect of age. For fair comparison, results and visualizations for the spline QTc, which only adjusts for heart rate and sex, are presented.
Application of newer QT correction formulae in the clinic
The values for the heart rate-corrected QT interval and their percentile ranking can vary. As an illustration, the corrected QT interval for a 72-year-old woman with a heart rate of 114 beats/min, the QTc varied widely. QTcBZT was the longest at 469 ms, which was in the 92nd percentile, whereas QTcFRD was 421 ms in the 23rd percentile. In contrast, the spline QTc was 423 ms and was in the 63rd percentile (Table 1). We next examined ECGs from men with a heart rate over 70 beats/min. The group had a mean heart rate of 94.5 ± 13.0 beats/min and a QT of 358.1 ± 23.5 ms. There was evidence that the formulae were not equal (p < 0.0001; Kruskal-Wallis test = 52.9) in the calculated QTc (Figure 6). The individual values suggest that the longest QTc was shown by QTcBZT, whereas the shortest QTC was QTcFRD. Electrocardiographs from women with a heart rate over 70 beats/min were examined. The group had a mean ± SD heart rate of 89.5 ± 9.9 beats/min and a QT interval of 381.5 ± 30.4 ms. The formulae were not equal (p < 0.0001; Kruskal-Wallis test = 34.6) in the calculated QTc (Figure 6). The individual values suggest that the longest QTc interval was shown by QTcBZT, thus the most commonly used QTc formulae provide numbers at the extreme of QTc measurement.
In order to illustrate the effect of heart rate on various QTc formulae compared to the newly proposed one, an individual with several ECGs over a wide range of different heart rates had the QTc calculated with each of the formulae (Figure 7). Not only were there wide differences in QTc among the different formulae at the heart rate of 80 beats/min but the discrepancies among the QTc formulae increased at faster heart rates. For the spline QTc, the slope of the relationship between QTc and heart rate was the smallest of all QTc formulae, and there were no significant (p = 0.84) differences from zero. In contrast, there were significant positive slopes in this relationship for QTcBZT and QTcDMT as well as significant negative slopes for QTcFRM and QTcRTH.
This study presents a new formula for correction of the inverse relationship between QT interval and heart rate based on functionally agnostic modeling of population ECG data with flexible regression splines. Previous formulae have used standard mathematical functional forms including exponentials, linear, and logarithmic functions and applied them to the QT-heart rate relationship (6). “Forcing the data” to fit a known relationship may have led to the problem that existing QTc formulae do not accurately adjust for the impact of heart rate on the QT interval. All QTc formulae are good at heart rates close to 60 beats/min, but very few formulae are also good at the higher and lower heart rates. The newly proposed formula did not presuppose a specific functional form (quadratic, logarithmic, or other) but rather used the data to construct a new formula which appears to eliminate the impact of heart rate on QT interval.
The new formula for the adjustment of the QT interval for heart rate was developed on the basis of population data and a flexible regression spline approach, permitting modeling of almost any shape of the QT-heart rate relationship. The history of attempts to minimize or “correct” the effect of heart rate on the QT interval began in 1920 with 2 simple mathematical approaches, one dividing QT interval by the square root (4) and the other dividing the QT interval by the cube root of the heart rate (5). There was little if any electrophysiological basis for these approaches. Subsequent efforts to adjust the QT interval for heart rate also relied heavily on mathematical relationships (6). In addition, a nomogram has been developed using a heart rate of 60 beats/min as the reference value (17). More recent formulae applied linear regression modeling to obtain the parameters of a “best” fit linear relationship between QT and heart rate (7,9). A linear, logarithmic or an exponential best fit of the data will only be valid when the nature of the relationship is truly known, and the relationship is indeed either a linear, logarithmic, or exponential one. This is not the case with the QT-heart rate relationship. Furthermore, the relationship may change at different heart rates, for example, it may be linear in one heart rate range and exponential in another range. A better way to fit such an unknown or potentially variable relationship is to use more flexible functions such as the spline methodology (18).
To address the concern that equations developed from epidemiologic studies are retrospectively accumulated and should have prospective evaluation (19), we cross-validated the spline QTc formula. We used a novel approach in the field of QTc development, specifically, we developed the formula from a data set derived after randomly splitting the population ECGs into 10 folds of approximately equal size. Because some data were held out of the folds, the spline QTc formula was cross-validated by applying it to data that was not used to fit the correction function.
The new formula was compared to a number of different formulae, including both older and more recently proposed formulae. The new formula was found to be an excellent approach to address the issue of the impact of heart rate on the QT interval. We have previously used the approach that the slope of the relationship between heart rate and QT interval should be zero when there is no relationship between QT and heart rate (20). Unfortunately, it is possible for a zero slope of the QT-heart rate relationship to arise if there is a systematic pattern in which positive and negative values exist around the zero sloped QTc-heart rate regression line (e.g., a sinusoidal pattern or a “wobble” around a slope of zero). To quantify this potential pathology, our current approach used the F-statistic from a test of linear, quadratic, and cubic relationships and quantified both linear trends and/or sinusoidal patterns and enabled comparison of “flatness” among QTc formulae. Statistical testing indicated that there are fewer linear, quadratic, or cubic relationships remaining between heart rate and QTc for the cross-validated spline QTc compared to most other formulae.
The comparative data showed that the formula which is most dependent on heart rate is that proposed by Bazett (4). This is apparent from the data considering the QTc at the extremes of heart rate. The obvious differences between the QTcBZT and the relationship without a slope (the spline) show deviations of the overestimation of QTc at lower and higher QT intervals. This problem with QTcBZT has been noted previously (17). Many of the early formulae have been compared (21) but not the recent ones. QTcFRD also failed at high heart rates. QTcFRM was reliable at normal but failed at low and high heart rates. The formula proposed by Hodges (13) was highly variable. These deviations from a linear relationship with zero slope for QTcBZT and QTcHDG are similar to the data for the effect on QT interval with the increases in heart rate produced with exercise (20).
The closest formula to the spline QTc was that proposed by Rautaharju et al. (9). We believe that this is also an excellent formula. It used the entire NHANES database as well as several other data bases to obtain parameters for their formulae. However, some of the QTcRTH performance may be slightly optimistic because it is likely that many of the individuals were included in the QTcRTH conversion function training set, whereas none of the cross-validated spline QTc estimates were used to construct the conversion function. By comparing the cross-validated spline QTcs to the QTcs for competing formulae, we avoided using the best fit conversion function for the whole data as our reference. Doing so would result in overly optimistic results in favor of the spline QTc. The cross-validation method is useful for assessing the performance of the spline approach. However, the entirety of the NHANES data was used to construct the final spline QTc conversion function; the spline conversion formulae calculated in each fold of the cross-validation were very similar (due to the large sample size) and also very similar to the final spline-QTc.
The spline formula was not only good in samples of the general populations but was also excellent when assessed under conditions in which many QTc formulae perform poorly. A sample of patient ECGs was used to demonstrate the comparative value of the spline QTc. A large sample was not required, as the objective was a comparative demonstration rather than a methodologic evaluation or model development. The spline formulae provided a better heart rate-corrected QTc in a sample of patients with heart rates over 70 beats/min as shown by QTc values which were not as prolonged as QTcBZT or as short as QTcFRD. These data suggest that the widespread use of those 2 formulae (QTcBZT and QTcFRD) should be reassessed.
A limitation of the study is the use of interindividual data to define the QT-heart rate relationship rather than different heart rates in the same individual. Our data in a single individual, however, are illustrative of the stability of the spline formula across different heart rates, especially compared to several other formulae. The power of the cross-sectional data from NHANES is the large number of individuals that were used to develop the spline formula. It is challenging and likely not possible to carry out a large study of the QT interval at different heart rates in the same individual at most institutions for several reasons. First, there are few individuals with multiple ECGs in hospital, and those with multiple ECGs are usually from patients in the coronary care or intensive units whose serial ECGs show ST-segment and T-wave changes which would meet exclusion criteria for accurate QT measurement. Second, most other individuals with multiple ECGs often have the same heart rates in their different ECGs, thus precluding testing of the formulae across meaningful different heart rates. Third, when an individual has a wide range of heart rates, in sinus rhythm, there are often only a few ECGs from which to construct the QT-heart rate relationship. Recognizing these challenges, further research with the new spline formula should be conducted within the same individuals over a wide range of heart rates. We note, however, that the QT correction formulae in widespread use today, which were compared in our study, were constructed from patient samples or populations and examined different heart rates in different individual. In order to address this issue, our study developed the formula in one sample of the population (cohort) and then tested it in different part of the cohort, the validation cohort. To our knowledge, no other study deriving a widely used QTc formula has used this rigorous procedure.
A new approach, independent of the previous mathematical relationships, has been used to develop a QT heart rate correction formula. The formula was developed using one data set and tested on a separate data set from the same large population. The formula is the closest theoretical formula to the goal of a formula with no relationship between QT and heart rate, that is, no impact of heart rate on QTc. The new formula is more “data agnostic” and more robust in its construction than previous QT correction formulae. Furthermore, we used a rigorous statistical approach to compare the spline formula to existing formulae and found that the new formula was usually the best.
COMPETENCY IN MEDICAL KNOWLEDGE: Current QT-heart rate correction (QTc) formulae do not satisfactorily accomplish this objective especially at faster heart rates. A new QTc formula based on functionally agnostic modeling of the NHANES population ECG data using flexible regression splines overcomes this problem. It further provides the data in percentiles of the population considering a person’s age and sex to permit improved assessment of an individual’s probability of QT prolongation.
TRANSLATIONAL OUTLOOK: Research testing the formula in other populations and within individuals at different heart rates should be conducted. This QTc formula should be considered the new standard for adjustment of QT for heart rate.
The authors have reported that they have no relationships relevant to the contents of this paper to disclose.
All authors attest they are in compliance with human studies committees and animal welfare regulations of the authors’ institutions and Food and Drug Administration guidelines, including patient consent where appropriate. For more information, visit the JACC: Clinical Electrophysiology author instructions page.
- Abbreviations and Acronyms
- U.S. National Health and Nutrition Examination Survey
- QT corrections for heart rate
- QTc proposed by Bazett
- proposed by Fridericia
- proposed by Dmitrienko et al.
- QTc based on Framingham data
- proposed by Hodges et al.
- proposed by Rautaharju et al.
- Received July 5, 2016.
- Revision received December 3, 2016.
- Accepted December 15, 2016.
- 2017 American College of Cardiology Foundation
- Bazett H.
- Rabkin S.W.,
- Cheng X.B.
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- ↵Centers for Disease Control and Prevention. National Health and Nutrition Examination Survey III, electrocardiogram data. 2015. Available at: ftp://ftp.cdc.gov/pub/Health_Statistics/NCHS/nhanes/nhanes3/2A/NH3ECG-acc.pdf. Accessed January 2016.
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