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Journal of Electrocardiolog

When deriving the spatial QRS-T angle from the 12-leadelectrocardiogram, which transform is more Frank: regression or

inverse dower?Daniel L. Cortez, BS,a,b,c Todd T. Schlegel, MDc,⁎

aUniversity of Illinois College of Medicine, Champaign and Rockford, IL, USAbNational Space Biomedical Research Institute, Houston, TX, USA

cNASA Johnson Space Center, Houston, TX, USA

Received 20 January 2010

Abstract Introduction: Our primary objective was to ascertain which commonly used 12-to-Frank-lead

⁎ Corresponding aHouston, TX 77058, U

E-mail address: to

0022-0736/$ – see frodoi:10.1016/j.jelectroc

transformation yields spatial QRS-T angle values closest to those obtained from simultaneouslycollected true Frank-lead recordings.Materials and Methods: Simultaneous 12-lead and Frank XYZ-lead recordings were analyzed for 100postmyocardial infarction patients and 50 controls. Relative agreement, with true Frank-lead results, of12-to-Frank-lead–transformed results for the spatial QRS-T angle using Kors' regression versus inverseDower was assessed via analysis of variance, Lin's concordance, and Bland-Altman plots.Results: Spatial QRS-T angles from the true Frank leads were not significantly different thanthose derived from the Kors' regression-related transformation but were significantly smaller thanthose derived from the inverse Dower-related transformation (P b .001). Independent of method,spatial mean QRS-T angles were also always significantly larger than spatial “maximum”(“peaks”) QRS-T angles.Discussion: Spatial QRS-T angles are best approximated by regression-related transforms. Spatialmean and spatial “peaks” QRS-T angles should not be used interchangeably.Published by Elsevier Inc.

Keywords: Vectorcardiography; Spatial ventricular gradient; 3-Dimensional ECG; Lead reconstruction

Introduction

The spatial QRS-T angle has repeatedly been shown tohave diagnostic1-3 and prognostic4-12 value. Changes in spatialQRS-T angles, for example, may be useful in evaluatingefficacy of hypertension treatment,1 diabetes mellitus,3

incident coronary heart disease2,10 and heart failure,2,9,11 andmost importantly the propensity for cardiac events2,4,12 andmortality.4-7,12 Of the various Frank-lead vectorcardiocardio-graphic (VCG) reconstruction methods used for approximat-ing spatial QRS-T angle values from conventional 12-leadelectrocardiogram (ECG) recordings,13–16 arguably the 2mostcommon are the inverse Dower-related reconstructiontechnique13 and the regression-related method of Korset al.14 Of these 2 methods, the latter has tended to

uthor. NASA Johnson Space Center, Mail code SK3SA.dd.t.schlegel@nasa.gov

nt matter. Published by Elsevier Inc.ard.2010.03.010

demonstrate better performance in reconstructing the actualFrank XYZ-lead signals.14,16 However, it remains unclearfrom the literature which of these methods actually bestreconstructs those secondarily derived parameters, such asthe spatial QRS-T angle, that have enough clinicalimportance to effectively drive a clinical need for 12-to-Frank-lead transformations.

Interestingly, in healthy, nonhospitalized individuals,mean values for the spatial mean (SM) QRS-T angle derivedfrom inverse Dower-related reconstructions have typicallyranged from 66 to 81 degrees,8,17,18 whereas those derivedfrom Kors' regression-related reconstructions have typicallyranged from 44 to 65 degrees,19,20 the latter evidently closerto mean values obtained from the true Frank leads (35-51degrees).21,22 To further complicate matters, the spatialQRS-T angle has often been measured as the angulardifference between the “maximums” (peaks) of the 3-dimensional QRS and T loops (spatial “maximum” QRS-Tangle)1,3,23-25 rather than as the angle between the position

Table 1Results for the control Group, post-MI group, and total sample for the true Frank leads and for the Kors' regression-related and inverse Dower-relatedtransformations

Parameter Control group (n = 50) Post-MI group (n = 100) Total sample (n = 150)

SM QRS-T angle (°)-true Frank leads 54.4 ± 25.8 84.3 ± 39.8 74.3 ± 38.3SM QRS-T angle (°)-Kors' regression 53.4 ± 26.0 91.4 ± 39.4 78.8 ± 39.7SM QRS-T angle (°)-inverse Dower 70.0 ± 28a,b 98.3 ± 40.9a 88.9 ± 39.4a,b

SP QRS-T angle (°)-true Frank leadsc 41.7 ± 27.1 68.9 ± 40.9 59.8 ± 38.9SP QRS-T angle (°)-Kors' regressionc 40.7 ± 23.5 76.4 ± 42.1 64.5 ± 40.6SP QRS-T angle (°)-inverse Dowerc 58.4 ± 27.8a,b 87.9 ± 45.7a 78.1 ± 42.9a,b

SVG magnitude (mV ⁎ ms)-true Frank leads 98.3 ± 44.7 46.7 ± 22.0 63.9 ± 39.7SVG magnitude (mV ⁎ ms)-Kors' regression 101.9 ± 44.1 51.6 ± 25.2 68.4 ± 40.3SVG magnitude (mV ⁎ ms)-inverse Dower 97.3 ± 44.3 54.0 ± 28.0 68.4 ± 39.8SVG elevation (°)-true Frank leads 32.1 ± 11.0 28.3 ± 33.6 29.2 ± 28.2SVG elevation (°)-Kors' regression 32.6 ± 9.9 24.7 ± 29.1 27.3 ± 24.7SVG elevation (°)-inverse Dower 28.5 ± 11.4 20.1 ± 26.9 22.9 ± 23.2a

SVG azimuth (°)-true Frank leads 20.0 ± 21.7 −8.1 ± 62.3 1.3 ± 53.9SVG azimuth (°)-Kors' regression 19.9 ± 17.6 −0.5 ± 64.7 6.3 ± 54.5SVG azimuth (°)-inverse Dower 11.5 ± 20.0 −5.4 ± 70.9 0.3 ± 59.4

a Derived VCG result significantly differs from true Frank-lead result by ANOVA.b Derived VCG results significantly differ from one another by ANOVA.c SP QRS-T angle results significantly differ from SM QRS-T angle results.

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vectors defined by the areas of these same loops (spatial“mean” QRS-T angle4-11,17-22,26-29—see Appendix 1 forour own understanding of the difference). Even moreconfusingly, healthy subjects have also typically had highermean values for the spatial “mean” QRS-T angle than for thespatial “maximum” QRS-T angle (means for the latterranging from 11 to 21 degrees for inverse Dowertransformations1,3,23 and from 42 to 51 degrees for the trueFrank leads).24,25 In the present study, we used Lin'sconcordance correlation coefficient to ascertain the relativeconcordance, to true Frank-lead results, of 12-to-Frank-lead–transformed results derived from Kors' regression-related versus inverse Dower-related reconstructions. Basedon the comparatively smaller mean quadratic deviationsfrom the true XYZ leads that have been noted for the Kors'regression method in the past,16 we hypothesized that thismethod would also better approximate important secondarilyderived VCG parameters such as the spatial QRS-T angle.We also sought to better define any differences betweenresults for the SM versus spatial “maximum” (“peaks”) QRS-T angles using both the true Frank leads and the 2 differentreconstruction methods.

Methods

Data collection

The data were obtained from a publicly available source,the Physikalisch-Technische Bundesanstalt (PTB) Diagnos-tic ECG Database30 available at http://www.physionet.org/physiobank/database/ptbdb/. The PTB ECG data werecollected in the 1990s by Dr Michael Oeff et al at theDepartment of Cardiology of University Clinic BenjaminFranklin in Berlin, Germany, using a noncommercialprototype recorder that allowed the simultaneous acquisitionof both 12-lead ECG and true Frank-lead VCG data stored at1000 samples per second per channel. We focused our own

analyses on data from: (1) PTB patients 001 through 101, allof whom were being evaluated for a recent myocardialinfarction (MI) and (2) the first 50 “healthy controls” in thePTB database, beginning with PTB subject 0104, asspecified by the PTB demographic files that accompanythe recorded raw data. In the recent MI group, the mean (SD)age was 58.8 (11.3) years, and 72% were males. In thehealthy control group, the mean (SD) age was 43.5 (14.7)years, and 76% were males.

Data analyses

The raw binary data files from the PTB database wereprocessed using software developed by the authors atNASA's Johnson Space Center.20,31 Initial analyses revealedthat all of the selected files except recent MI-patient file 079(which demonstrated a paced rhythm and was not analyz-able) had, at a minimum, 40 QRS-T complexes that wereacceptable for signal averaging in all channels when using aminimum cross-correlation cutoff of 97% against the signal-averaged QRS templates formed for each channel in eachfile, as previously described.20,31 Thus, 40-complex signalaverages were ultimately constructed for each patient's file,the principal purpose of signal averaging being to helpeliminate any transient or nonreproducible effects that wouldmore likely influence single complexes than signal averages,such as the precise location of a given complex within therespiratory cycle.

VCG parameters from the true and derived Frank leads

The principal VCG parameters studied were the SM andspatial “peaks” (SP) QRS-T angles, the magnitude of thespatial ventricular gradient (SVG), and the elevation andazimuth angles of the SVG. Time integrals (areas) for the X,Y, and Z signals were first determined separately for theQRS and T complexes by measuring the areas of thecomplexes above and below the baseline and subtractingnegative areas from positive areas. The time integrals of the

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X, Y, and Z signals for the given complex (QRS or T) thenyielded the spatial QRS and T position vector components,respectively. The magnitude of the sum of the spatial QRSand T vectors in turn yielded the SVG, whereas the anglebetween the same 2 vectors yielded the SM QRS-T angle.27

The SP QRS-T angle was calculated in a fashion similar tothe SM QRS-T angle except that instead of using areas asposition vector components, the vector magnitude peakvalues in the QRS and T regions (“loops”) constituted thevector components for the calculation of the angle. (Forfurther details, including the mathematical derivations of allthe above parameters, see Appendix 1.) For the directionalaspects of the SVG vector (ie, azimuth and elevation), thecoordinate system of Draper et al27 was used. Results for allparameters were calculated both directly, from the trueFrank XYZ lead recordings themselves (which served as the“gold standard”), and indirectly, from the inverse Dower13

and Kors' regression-related14 Frank-lead reconstructionsthat used the simultaneously collected standard 12-leadECG recordings.

Fig. 1. Representative 3-dimensional vectorcardiographic loops with accompanyinL, left, A, anterior. The arrows shown (heavily dashed for the QRS loop, lightly dapeak of the given loop, not the vector magnitudes. The SP QRS-T angle from themethod (B) but notably smaller than that obtained from the inverse Dower method (Dower method may have been driven in part by a QRS loop that was more posteriregression method.

Statistics

Statistical evaluations were performed for the entiregroup and separately for the 100 post-MI patients and the50 healthy control subjects using Stata version 10 (Stata,College Station, TX). Analysis of variance (ANOVA) wasused to study differences in results from the true Frankleads versus those from the Kors' regression-related andinverse Dower-related reconstructions. All nonnormalizeddata were log-normalized before applying the ANOVA,with subsequent use of Bartlett's test to confirm equalvariances and Bonferroni pairwise post hoc testing whenthe ANOVA revealed significant (P b .05) differences.Wilcoxon rank-sum and matched pairs signed-ranked testswere also used with appropriate adjustments for multiplecomparisons to assess differences between the post-MI andhealthy control groups, and between the SM and SP QRS-Tangles, respectively.

We also used Lin's concordance correlation coefficient32

to evaluate the relative agreement with the true Frank-lead–

g diagrammatic SP QRS-T angles for 1 healthy subject. R indicates rightshed for the T loop) express only the vector direction from the origin to thetrue Frank leads (A) was similar to that obtained from the Kors' regressionC). In this particular case, the larger SP QRS angle obtained from the inverseorly directed than that obtained from either the true Frank leads or the Kors

;

'

Fig. 2. Line of concordance for the SM QRS-T angle as obtained from the true Frank leads versus the Kors' regression-related (A) and inverse Dower-related (Breconstructions, respectively, for the entire data set. The greater concord of the true Frank-lead results to the Kors' regression-related results than to the inverseDower-related results is readily apparent.

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related results of the Kors' regression-related versus theinverse Dower-related transformation results. Lin's con-cordance correlation coefficient ρ combines measures ofboth precision and accuracy to determine how far theobserved data deviate from the line of perfect concordance(ie, the line at 45 degrees on a square scatterplot). ρ has avalue between 0 and 1 and increases as a function of thenearness of the data's reduced major axis to the line ofperfect concordance (the accuracy of the data) and of thetightness of the data about its reduced major axis (theprecision of the data). A ρ value of 1 would indicatecomplete agreement with the results from the “goldstandard.” As a complement to the relationship-scaleapproach of Lin, we also used Bland and Altman graphsto assess limits of agreement.33

Fig. 3. Bland-Altman plots for the SM QRS-T angle as obtained from the true Frank leads versus the Kors' regression-related (A) and inverse Dower-related (Breconstructions, respectively, for the entire data set. Along with smaller absolute differences from the true Frank-lead results (Y axis values, middle lines), theKors' regression-related results also demonstrated closer 95% Bland-Altman limits of agreement (dashed lines at the top and bottom) to the true Frank-leadresults than did the inverse Dower-related results.

)

Results

Table 1 shows the mean ± SD results for all ECGparameters in both groups. Not surprisingly, independentlyof the method, SM and SP QRS-T angles were smallerand SVG magnitudes larger in the control group than in thepost-MI group (P b .001). On the other hand, there wereno significant differences in SVG elevations or azimuthsbetween the 2 groups, regardless of method. Moreimportantly, Table 1 also shows that all SM and SP QRS-T angles from the true Frank leads were significantly smallerthan those derived from the inverse Dower method but notsignificantly different from those derived from the Kors'regression method. The SM and SP QRS-T angles from theKors' regression method were also often significantly

)

Table 2Lin's concordance coefficient (ρ) and Bland-Altman 95% limits of agreement versus the true Frank leads for, respectively, the Kors' regression and inverseDower-related transformations

Kors' regression versus true Frank leads Inverse Dower versus true Frank leads

ρ (95% CI) Bland-Altman 95%limits of agreement

ρ (95% CI) Bland-Altman 95%limits of agreement

Control group (n = 50)SM QRS-T angle 0.88 (0.81-0.94) −26.2 to 24.2 0.71 (0.59-0.83) −15.3 to 46.4SP QRS-T angle 0.77 (0.66-0.88) −34.7 to 32.7 0.62 (0.48-0.77) −21.8 to 55.4SVG magnitude 0.99 (0.98-0.99) −10 to 17 0.98 (0.97-0.99) −19 to 17SVG elevation 0.76 (0.64-0.88) −12.7 to 15.5 0.73 (0.60-0.86) −18.0 to 12.8SVG azimuth 0.70 (0.57-0.84) −30.0 to 29.8 0.63 (0.48-0.79) −40.9 to 23.9Post-MI group (n = 100)SM QRS-T angle 0.89 (0.85-0.93) −26.8 to 41.1 0.77 (0.69-0.84) −34.2 to 62.3SP QRS-T angle 0.83 (0.77-0.89) −38.3 to 53.3 0.59 (0.47-0.71) −52.5 to 90.4SVG magnitude 0.91 (0.88-0.94) −12 to 22 0.78 (0.71-0.85) −23 to 37SVG elevation 0.93 (0.91-0.95) −25.7 to 18.6 0.86 (0.81-0.90) −36.6 to 20.3SVG azimuth 0.74 (0.65-0.83) −81.3 to 96.4 0.60 (0.48-0.73) −113.7 to 119.1Total sample (n = 150)SM QRS-T angle 0.91 (0.88-0.93) −27.7 to 36.6 0.78 (0.73-0.84) −28.6 to 57.7SP QRS-T angle 0.84 (0.80-0.89) −38.1 to 47.4 0.64 (0.55-0.72) −44.1 to 80.6SVG magnitude 0.97 (0.97-0.98) −11 to 20 0.93 (0.91-0.95) −23 to 32SVG elevation 0.92 (0.90-0.94) −22.2 to 18.4 0.85 (0.81-0.89) −31.7 to 19.0SVG azimuth 0.75 (0.68-0.82) −69.8 to 79.7 0.62 (0.52-0.72) −98.3 to 96.2

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smaller than those derived from the inverse Dower method.On the other hand, SVG magnitudes, elevations, andazimuths derived from the 3 different methods were notsignificantly different, except that for the entire data set (N =150); the SVG elevation derived from the inverse Dowerreconstruction was significantly smaller than that obtainedfrom the true Frank leads. Finally, the SM QRS-T angleswere also significantly larger than the SP QRS-T angles forboth groups and for the data set as a whole. Fig. 1 showsexample 3-dimensional VCG loops and accompanyingdiagrammatic SP QRS-T angles from a healthy individual.Consistent with data from both the healthy and post-MIgroups, this individual's SP QRS-T angle obtained from thetrue Frank leads (Fig. 1A) was very close to that obtainedfrom the Kors' regression method (Fig. 1B) but was notablysmaller than that obtained from the inverse Dower method(Fig. 1C).

Fig. 2 shows Lin's concordance line plots for the SMQRS-T angle as obtained from the true Frank leads versusthe Kors' regression (Fig. 2A) and inverse Dower-related(Fig. 2B) reconstructions, respectively, for the entire data set.The dashed line represents the line of true concordance,whereas the solid line represents a best-fit line to the trueFrank-lead data using the respective reconstruction tech-nique. Fig. 3 shows the corresponding Bland-Altman plotsfor the same comparisons. In Fig. 3, the centrally locateddashed lines represent the best-fit line obtained by the trueFrank leads, whereas the solid lines represent the linesobtained by using the given reconstruction technique. Figs. 2and 3 both visually demonstrate the greater concord of theKors' regression-related results than the inverse Dower-related results to the true Frank-lead results.

Table 2 shows Lin's concordance coefficient (ρ) and itsaccompanying confidence interval (CI) as well as the Bland-Altman 95% limits of agreement for each parameter in eachgroup when the particular reconstruction technique was

tested against the true Frank leads. Of note are theconsistently larger (closer to 1) ρ values as well as tighterρ CIs and Bland-Altman 95% limits of agreement forevery single parameter measured in all 3 groups for the Kors'regression-related method than for the inverse Dower-relatedmethod when using the true Frank-lead results as thecriterion standard.

Discussion

The most important finding from this study is that spatialQRS-T angle results derived from the Kors' regression-related transformation more closely approximate those fromthe true Frank leads than do corresponding results from theinverse Dower-related transformation. Specifically, com-pared with the use of the Kors' regression transformation, theuse of the inverse Dower transformation yields lower Lin'sconcordance correlation coefficients and wider associatedCIs against the true Frank-lead criterion standard. Theinverse Dower transformation, but not the Kors' regressiontransformation, also significantly overestimates the absolutevalue for the SM and SP QRS-T angles both in healthysubjects and post-MI patients. Therefore, results frominverse Dower-related transformations for spatial QRS-Tangle should not be considered tantamount to results fromthe true Frank leads, as has already been at least indirectlydiscovered.17,18 In addition, absolute values for spatialQRS-T angles and SVG that are “abnormal” when usingKors' regression transformations may be falsely construed as“normal” if referencing absolute values from inverse Dowertransformations. Thus, assignments of “normal versusabnormal” should only be made on a transform-specificbasis. Table 3 provides, for rapid reference purposes, asummary of prior studies according to the apparent type ofspatial QRS-T angle measured and to the specific transfor-mation method used.

Table 3Grouping of prior studies according to the apparent type of spatial QRS-Tangle measured and, when transformation was used, its type

Type of anglemeasured

Kors' regressiontransformation

Inverse Dowertransformation

True Frankleads

SM QRS-Tangle

(5-7, 19, 20) (8-11, 17, 18, 29) (21, 22, 28, 34)

SP QRS-Tangle

(1, 3, 23) (24, 25)

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The other finding of interest from this study is that whenusing the true Frank leads or either reconstruction, meanvalues for the SM QRS-T angle are significantly larger thanthose for the SP QRS-T angle, and thus, these 2 measuresshould not be used interchangeably. We prefer to refer tothe angular difference between the “maximum” QRS and Tvectors1,3,23-25 as the spatial “peaks” rather than spatial“maximum” QRS-T angle because the use of the word“maximum” results in confusing nomenclature, inasmuch asin most individuals, the angle between the peaks of thespatial QRS and T loops is actually smaller than the anglebetween the QRS and T position vectors defined by the areasof the same loops. The mathematical derivations for bothtypes of angle are provided in Appendix 1, along with thatfor the SVG.

Limitations

Although the public availability of the PTB database ishighly appreciated and made this study possible, thedatabase's demographic annotations are often not detailed.For example, it is unclear whether risk factors such asdiabetes or hypertension might actually have been present inmany individuals whose data files are presently labeled as“healthy” controls, particularly inasmuch as conventionalECG abnormalities such as Cornell voltage criteria for leftventricular hypertrophy occasionally occur in these controlfiles. The mean (SD) value for the SM QRS-T angle from thetrue Frank leads in the “healthy” controls in this study was54.4 (25.8) degrees, which lies above the limits of thosepresented for nonhospitalized controls in previous studies(35-51 degrees).21,22,28,34 Thus, the relatively increasedabsolute QRS-T angle values in these PTB controlscompared with controls in other studies might be attributableto the former's somewhat higher cardiovascular risk factorprofiles or possibly to a higher group mean age or malegender prevalence.

Conclusions

Spatial QRS-T angle and SVG results derived from Kors'regression-related transformations more closely approximateresults from the true Frank leads than do correspondingresults from inverse Dower-related transformations. Inaddition, inverse Dower-related transformations, but notKors' regression-related transformations, significantly over-estimate absolute values for spatial QRS-T angles, both inhealthy subjects and post-MI patients, such that absolutevalues that are actually abnormal when referencing true

Frank-related or Kors' regression-related population meansmay be falsely construed as “normal” when referencinginverse Dower-related population means. Thus, assignmentsof “normal versus abnormal” should only be made on atransform-specific basis. In addition, population mean valuesfor the SM QRS-T angle are significantly larger than thosefor the spatial “peaks” QRS-T angle such that these 2parameters should not be used interchangeably.

Acknowledgments

The authors especially thank Oeff et al and physionet.orgfor providing access to the PTB Diagnostic ECG Database.They also thank Drs W.B. Kulecz, J.L. DePalma, and E.C.Greco for programming and technical assistance and Dr R.Ploutz-Snyder for statistical advice. Additional support wasprovided by the National Space Biomedical ResearchInstitute and NASA/Wyle Laboratories medical studentinternship programs (DLC) and NASA Technology Invest-ment Funds (TTS).

Appendix 1: Derivation of VCG parameters

A usual first step in combining the signals from the X, Y,and Z leads from the true or derived Frank leads is to form avector magnitude V, which is

V =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX 2 + Y 2 + Z2:

pð1Þ

Denote the area under the curve for the QRS complex inthe X, Y, and Z leads, respectively, as QRSx, QRSy, QRSzand denote the area under the curve for the T wave in the X,Y, and Z leads, respectively, as Tx, Ty and Tz. By using anyone of several possible integration method formulas (eg, theTrapezoidal rule, Simpson's rule, or Simpson's 3/8 rule—thelatter was used for the present investigation),35 the SM QRSarea (time-magnitude) can be calculated as

jQRSareaj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðQRSxÞ2 + ðQRSyÞ2 + ðQRSzÞ2;

qð2Þ

the SM T-wave area (time-magnitude) as

jTarea j¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTxð Þ2 + Tyð Þ2 + Tzð Þ2;

qð3Þ

and the SVG area (time-magnitude) as

SVG ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiQRSx + Txð Þ2þ QRSy + Tyð Þ2þ QRSz + Tzð Þ2;

q

ð4Þ

with |QRSarea|, |Tarea|, and SVG all having units of mV ⁎ ms.

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The SM QRS-T angle is in turn calculated by the dotproduct in units of degrees as

SM QRS−T angle

¼ cosQRSxTx þ QRSyTy þ QRSzTz

jQRSarea j jTarea j� �−1

:

ð5Þ

On the other hand, in the simplest possible terms, the SPQRS-T angle is calculated in units of degrees as

SP QRS−T angle ¼ cosA � BjA j jB j

� �−1

ð6Þ

where A and B are the vectors of maximum magnitude of theQRS and T waves.

To define the vectors of maximum magnitude, first definethe R-wave position vector RP by the values of X, Y, and Zat the location of the 3-dimensional R-wave (QRS loop)peak, where the peak is defined by the maximum value of Vin Eq. (1) over the QRS interval:

RP ¼ RPx + RPy + RPz ð7Þ

Similarly, define the T-wave position vector TP by thevalues of X, Y, and Z at the location of the 3-dimensional T-wave (T-loop) peak, where the peak is defined by themaximum value of V in Eq. (1) over the T interval:

TP ¼ TPx + TPy + TPz ð8Þ

The SP QRS-T angle is then the angle in degrees betweenposition vectors RP and TP as again calculated by the dot-product:

SP QRS−T angle = cosRPxTPx + RPyTPy + RPzTPz

jRP j jTP j� �−1

;

ð9Þ

where

jRP j¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRPxð Þ2 + RPyð Þ2 + RPzð Þ2

qð10Þ

and

jTP j¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiTPxð Þ2 + TPyð Þ2 + TPzð Þ2:

qð11Þ

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