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Journal of Electrocardiology 47 (2014) 144–150www.jecgonline.com

The case of the QRS-T angles versus QRST integral mapsAdriaan van Oosterom, PhD⁎

Medical Faculty, Radboud University Nijmegen Medical and Biophysics, Nijmegen, The Netherlands

Abstract This contribution discusses the QRS-T angle as well as the QRST integral map. Both of these topics

⁎ Medical FaculBiophysics, Geert GNetherlands.

E-mail address: av

0022-0736/$ – see frohttp://dx.doi.org/10.10

have been tested in their application in extracting the major features of depolarization andrepolarization: their spatio-temporal behaviour, and how much of their global or local nature mightbe deduced from signals that can be observed clinically. Recently, it is in particular the QRS-T anglethat has received considerable attention, a method that stems directly from vectorcardiography, asubdomain of electrocardiography. The QRST integral map is a display of a map on the body surfaceof the integrals over time of the ECG signals observed at sets of electrodes. The common biophysicalbackground of both techniques is highlighted. In particular it is explained why, in healthymyocardium, both provide a similar view on the global timing of the depolarization andrepolarization of all cardiac myocytes, more specifically, on the dispersion of their action potentialdurations. In the presence of ischemia, the view obtained is of the integral over time of thetransmembrane potentials, which comprises a 'mixture' of their timing and magnitude. The analysisof results of a simulation study emphasizes the large discrepancies that may be observed between theQRS-T angle in the frontal plane and its 3D variant. It is shown that the required vectorrepresentation of the signals may be derived from the 12-lead ECG by using the transfer matrixproposed in 1990 by Kors and colleagues.© 2014 Elsevier Inc. All rights reserved.

Keywords: QRS-T angle; The QRST-integral map; Frontal plane vectors; 3D vectors

Introduction

The discussion about the way the electrical sources of theheart are expressed in the potential differences on the thorax(ECGs) demands the formulation of a model of the activeelectrical cardiac sources as well as of the effect of the(electrically passive) exterior body tissues. The techniques ofboth the QRS-T angle and the QRST integral map are basedon one and the same rudimentary source description.

First, the elements of this source description that areessential for the main part of the discussion are brieflyintroduced. Next, both techniques are summarized. Finally,some illustrations are presented of the interpretation of thelink between the observed ECG data, i.e. the QRS-T angleand the QRST-integral, and global depolarization andrepolarization markers. It is shown that the analysis basedon QRST related integrals may be viewed as a generalizationof the source dipole vector in 3D physical space in the formof the L-dimensional signal space of L simultaneouslyrecorded ECG signals. The material presented here is aimed

ty, Radboud University Nijmegen Medical androotrplein-Noord 21 6500HB, Nijmegen, The

o-linden@home.nl

nt matter © 2014 Elsevier Inc. All rights reserved.16/j.jelectrocard.2013.10.006

at emphasizing and expanding some of the theory describedand developed in Section 5.9.4 of Volume One of the secondedition of Comprehensive Electrocardiology [1]. The samematerial is also available as a paperback publication [2]. Thevector and matrix notations as used here, as well as adescription of their handling can be found in Appendix A ofeither of these books.

Theory

The rudiments

The rudimentary model of all currents generated bybioelectric sources can be expressed as

Jm r→; t

� �¼ S−1

v ∇⋅σ r→

� �∇Vm r

→; t

� �: ð1Þ

It describes how the current density Jm r→; t

� �(with unit

A/m2) flowing into the passive external medium at anymembrane patch is linked to the local transmembranepotentials (TMPs) Vm r

→; t

� �at location r

→ at time t. TheSy denotes the area of the cell patch divided by the cellvolume, σ r

→� �

the electric conductivity of the intracellularspace. Moreover, ∇ denotes the gradient operator of the

145A. van Oosterom / Journal of Electrocardiology 47 (2014) 144–150

scalar field Vm r→; t

� �and ∇ the divergence of the resulting

vector field (Section 2.3.4 of [2]). In its application tothe computation of the potential fieldφ r

→; t

� �in an external,

passive medium, the current density Jm r→; t

� �acts as

the source.The electric volume conduction properties of the external

medium are time independent and linear. As a consequencethe external potential field φ r

→; t

� �can be found by

integrating (adding up) all contributions of Jm r→; t

� �to the

external field. The computation involved is known as solvingthe Forward Problem (Chapter 8 of [2]). The resultingexpression that is essential for the subsequent discussion canbe cast in the format of matrix algebra (Appendix A ofChapter 2 of [2]). Expressing the Vm r

→; t

� �values of all

myocytes by matrix Vm, with its rows representing the timecourses of the individual values, and its columns theirinstantaneous magnitudes, the integration involved, com-pactly denoted by a matrix multiplication, is

Φ ¼ AVm ð2Þ

This equation expresses the potentials in the extra-cellulardomain,Φ, as the (matrix) multiplication of the TMPsVm bya transfer matrix A of size L × N. Matrix A expresses allvolume conduction effects, be they inhomogeneous orhomogenous, isotropic or anisotropic. In words: the potentialℓ(t) at any observation point ℓ (electrode position ℓ = 1…L) is found to be a linear combination of the TMPs at locationn at that same instant of time t (t = 1….T). The weightingfactors are the element of row ℓ of matrix A.

As is well known [3,4], if all membranes are polarizeduniformly throughout space the generated external potentialfield is zero. Correspondingly, in the numerical approxima-tion, Eq. (2), the sums of all elements of any row ℓ ofthis matrix are zero. Hence, when “feeding” transfer matrixA with any constant column vector c, a vector having n = 1…N identical elements cn, the result expressed in matrixterms is

Ac ¼ 0 ð3Þwith 0 denoting a column vector of size L having zeroelements only.

Vectors and integral maps

Vectors

The vector representing a current dipole D→

tð Þ placedinside a bounded volume conductor with homogeneouselectric conductivity is a classic, so-called equivalent sourcedescription of the potential field on the thorax. It can beviewed as a global representation of the source strength of allcardiac myocytes, resulting from the integration over theentire myocardial volume of the elementary contributions asspecified by Eq. (1) (Section 10.3 of [2]).

Its strength over time of the dipole can be specified by itsoverall strength (magnitude) and the orientation. Alterna-tively, it can be represented by its projections [D1(t)D2(t) D3(t)] on a set of three orthogonal directional unit

vectors (Section 11.4 of [2]). Initially, just two unit vectorswere considered, lined up in the frontal plane of the thorax [5].

Later, the necessity for including the third dimension wasstressed, e.g. by Burger and van Milaan [6]. Although thepopularity of the dipole as used in vectorcardiography haswaned, in clinical electrocardiography its concept, in particularin the application to the frontal plane, is still ubiquitous.

The variation over time of the strength of a current dipolevector can be represented by vector loops, such as illustratedin Fig. 1.

The vector loops shown in Fig. 1 were derived by usingthe Gabor-Nelson equations [7]. This method compliesentirely with the basic definition of the dipole as stated inthe beginning of this section, and hence can be taken as thegold standard. It requires the specification of the closedsurface of the body surface as well as of the potential fieldover this surface.

The loops in Fig. 1 were based on a numerical (300nodes) representation of the geometry of the thorax of thesubject studied, derived from magnetic resonance images(MRIs), and the potentials at these nodes based on an inverseprocedure [8] applied to 64 measured ECGs at 65 electrodelocations. The data required for applying the Gabor-Nelsonare rather demanding and have led to estimations of thevector derived from different limited lead systems, whileusually not taking into account individual thorax geometry.

The QRS-T angle may be derived from the three vectorcomponents at the moment of maximum magnitude of theQRS loop, say a = [a1 a2 a3] and those of the STT loop, sayb = [b1 b2 b3].

An alternative that has been proposed is the use ofthe mean values over time of [D1(t) D2(t) D3(t)] i. e.,[―D1

―D2

―D3] over the respective two intervals. In both cases,

the QRS-T angle (unit: radians), denoted here as γ can becomputed as

γ ¼ acosa1 � b1 þ a2 � b2 þ a3 � b3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 þ a22 þ a23

p�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21 þ b22 þ b23

q ¼ acosab

a bð4Þ

where the argument of the inverse cosine (acos) is also knownas the normalized dot product of the vectors a and b, withmagnitudes denoted as a and b as is shown on the right of Eq.(4). The normalized dot product has the property of lyingwithin the range [−1 1] provided that neither of two lengths ofthe vectors is zero (Section A.1.2.2 of Chapter 2 of [2]). Theinverse cosine function (acos) maps the normalized dotproduct in a continuous, non-increasing fashion onto theinterval [0 π] ([0° 180°]). For any two vectors that have zeromean value elements, the expression defining the normalizeddot product yields the same value as does the linearcorrelation coefficient (CC). When applied to the estimatedvectors in the frontal plane γ can be computed by discarding(in Eq. (4)) the terms involving a3 and/or b3.

Integral maps

The idea of studying the QRST integrals dates back to theseminal work done by Wilson et al. [9]. Some decades laterAbildskov and co-workers adopted this concept in an

Fig. 1. Vector loops; three orthogonal directional unit vectors are pointing to the front of the thorax, the left and the head. Heavy solid lines point to the maximummagnitudes (length) of the vectors. The dotted, heavy line indicates the direction of the long axis of the ventricles (apex to base) as observed in MR-images of thesubject. Left panel: QRS vector loop. Right panel: STT vector loop. Both loops are inscribed counter clock-wise when viewed in the plane of the page. Both linesindicating the maximum magnitudes are replicated by thin lines in the accompanying panels. The angle (in 3D space) between these two lines is defined as theQRS-T angle.

146 A. van Oosterom / Journal of Electrocardiology 47 (2014) 144–150

application to the interpretation of Body Surface PotentialMaps (BSPMs). Following the development in the simulta-neous recording of multiple ECGs, of computer technologyand the development of numerical analysis, the inspection ofthe QRST became readily available.

In the sequel, the integral of the signals over the QRSTinterval is denoted as iQRST with the L separate integralsrepresented by the elements of a column vector of size L, thenumber of ECG signals observed on the thorax. In addition,the integration over just the QRS interval as well as over thesubsequent STT interval is carried out. The results aredenoted as iQRS and iSTT. Based on the linear properties ofintegration it follows that we have

iQRST ¼ iQRS þ iSTT: ð5Þ

Provided that these integrals relate to a sufficiently largenumber of signals, the data of all three integrals may be

+

iQRS iSTT

Fig. 2. Maps of the integrals of the ECG lead signals over different time intervalsmiddle panel: the STT interval, and right panel: the QRST interval. The dots represe65 electrode locations at which the signals were measured [10]. The integral valuinterval. Map values at locations not sampled were filled in by a dedicated interpo

mapped on the body surface. An example of this is presentedin Fig. 2.

The pattern of the iQRS map was strongly correlated withthe BSP map at the timing of the peak magnitude of theQRS vector loop (CC = 0.996). Similarly, the correlationbetween the iSTT map and the BSP map at the timing of thepeak magnitude of the STT-loop was also high (CC =0.997), with the CC values based on the 64 recordedsignals only.

Linking rudiments and integrals over time of lead signals

The linking of the rudiments as described in Section 1and both the various angles and integral maps described inSection 2 can be based on Eq. (2). To this end both sidesof the equation are integrated over time, which involvesthe integration over time of the rows of matrices Φ andVm only.

=

iQRST

, depicted over the anterior part of the thorax. Left panel: the QRS interval,nt the 9 electrode locations of the standard 12-leads. These are a subset of thees were scaled by the max_abs value of all recorded signals over the QRSTlation algorithm [11]. Isofunction lines were drawn at steps of 0.2 (a.u.).

147A. van Oosterom / Journal of Electrocardiology 47 (2014) 144–150

As explained in Section 2 the rows of Vm are the timecourses of the TMPs. The shape of a typical transmembranepotential is shown in the left panel of Fig. 3. The intervalbetween the upstroke of the depolarization process, denotedas δ, and of the inflection point of the downslope, which istaken to be marker for the timing repolarization, denoted as ρ,is the activation recovery interval (ARI), denoted as α. Thepanel on the right is a butterfly plot of an ensemble of N =1500 of such TMPs. These data are used in the referencesimulation used in Section 5. Note that for each TMPn (n =1… N), we have αn = ρn − δn. The set of αn values is denotedas a column vectorα of size N and the column vectors δ andρ denote the timing of all depolarization and repolarization,respectively. In vector notation this leads to

α ¼ ρ−δ ð6ÞThe values of the integrals over time of any of these

curves (areas between the curves and base line) shown in theright panel are closely proportional to the corresponding αnvalues (linear regression: area(n) = 0.85 αn − 11; CC =0.9999). As a consequence, the integration over time of bothsides of Eq. (2) leads to

iQRST ¼ A c1 þ c2eð Þ ð7ÞOn the right, matrix A is multiplied by column vector c1

α + c2e, in which e denotes a column vector of size N havingunit elements only. The scaling coefficients c1 and c2 accountfor the size of the upstroke of the TMPs and the value of theresting potential, respectively, in combination with the linearregression constants of area(n) as a function of αn.

Next, in anticipation of the effect of the properties of Aspecified in Eq. (3), the differences are introduced betweenall of the individual ARI values αn and their mean value (ascalar), viz., α̃n = αn− α. These values represent the fulldocumentation of the commonly used term dispersion. Thecolumn vector variant of the dispersion is α̃ ¼ α−α e. Byreplacing α in Eq. (6) accordingly, we find

iQRST ¼ A c1 α̃þ c1α eþ c2 eð Þ ¼ c1A α̃: ð8ÞThe two terms involving e have disappeared due to the

properties of A specified in Eq. (3). By absorbing scaling

time [ms]0 50 100 150 200 250 300 350 400 450 50

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Vm

[a.u

.]

α=372

α=242

Fig. 3. Left panel: Stylized wave forms of two normalized ventricular TMPs. Therepresent the activation recovery intervals, here denoted by α. Right panel: ensem

factor c1 in A the final expression, forming the basis of thesubsequent discussion, is

iQRST ¼ A α̃; ð9Þexpressing that the integrals over time of the signalsobserved on the thorax are a linear combination (a mixexpressed by A) of the dispersion of the ARI values of allmyocytes, note: not the ARI values as such. Both the meanvalue of the ARI values and the constant c2, which reflectsthe (mean) values of the resting potential values are notexpressed in iQRST.

By virtue of the linear properties of integration and of themean value of any set of numbers, a combination of theresults expressed in Eqs. (5), (6) and (9) indicated that

iQRS þ iSTT ¼ iQRST ¼ Aα̃¼ −Aδ̃ þAρ̃; ð10Þthus linking iQRS to −A δ̃ and iSTT to A ρ̃, with thesame interpretations and restrictions as discussed followingEq. (9).

Results; QRS-T angles and QRS-T integrals

In this section some illustrative results are describedobtained by the application of the concepts discussed appliedin a forward simulation based on values of depolarizationtimes δn and repolarization times ρn of the TMPs assigned toN = 1500 nodes of a dense mesh. The mesh is a numericalrepresentation of the closed surface bounding the entireventricular myocardium obtained by means of MR imaging.The transfer matrix A was based on the equivalent doublelayer (EDL) type source description, the strength of whichwas proportional to the local TMPs and an inhomogeneous,volume conductor model bounded by the torso surface. Thegeometry of which was based on matching MR images. Thelatter included the effect of the higher electric conductivity ofthe blood inside the cavities and the lower electricconductivity of lung tissue.

The reference sets of timings, δ and ρ, were derivedfrom an inverse procedure described in [8]. A softwarepackage of ECGSIM, available from www.ecgsim.org, freeof charge, enables the reader to study the topics discussed in

0 50 100 150 200 250 300 350 400 450 5000

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

intervals between the inflection points of the up slopes and the down slopesble of 1500 such TMPs, each having a different timing.

148 A. van Oosterom / Journal of Electrocardiology 47 (2014) 144–150

this paper. Originally published in [12], the most recentversion of ECGSIM is based on the same biophysical modelas the one used here. Collected results for different valuesof f are listed in Table 1. The QRS-T angles in columns 2and 3 were derived using the gold standard: the Gabor-Nelson equations.

Application to QRS-T angles

Figs. 1, 2 and 3 were based on the analysis of the signalsobserved on 65 electrodes placed at 'strategic' positions onthe torso surface [10,13]. These relate to the application ofthe set of reference timings, δ and ρ. A collection of ameaningful set of perturbations of these data wasconstructed, which enables the study of the behaviour ofthe QRS-T angle under varying conditions. To this end, theset of depolarization times was set at their reference values,and the set specifying the ARI values, α (= ρ − δ), wasscaled by factors f: αf = f × (α − α e) + αe thus merelyscaling the dispersion of the ARI values while retainingtheir spatial distribution.

Correspondingly, the assigned repolarization timing is:ρf = δ + f × (α − α e) + αe. Due to the basic property oftransfer A formulated by Eq. (3), the constant column vectoris not expressed in the dispersion of the QRST integrals ofthe BSP data.

In Col. 2 of Table 1 the QRS-T angles are listed asestimated by applying the Gabor-Nelson equations appliedto the simulated potential field at the 300 nodes of thethorax geometry. Their values (unit: degrees) were derived

Table 1Collected results for different values f. Col.1: the factor f; Col.2: QRS-Tangles (unit: degrees) derived from the respective integrals over time of theVCG signals; Col.3, as in Col 2, now derived at the timings of two maxima(QRS, STT) of the spatial magnitude vector; Col.4: RMS value of the signalsat apex T (unit: mV); Col.5 STD of the (1500) ρn values (unit: ms), Col.6:linear correlation coefficient between the L = 65 elements of the iQRS andiSTT vectors; Col.7: slope of the linear regression line of the elements of iQRSand iSTT; Col. 8: Application of the inverse cosine function to the correlationcoefficients listed in Col.6 (unit: degrees).

1 2 3 4 5 6 7 8

1.0 85.2 82.2 0.29 15.5 0.07 0.03 86.20.9 90.3 88.1 0.27 15.3 −0.02 −0.01 91.40.8 96.6 94.7 0.24 15.2 −0.14 −0.05 97.80.7 104.3 102.4 0.22 15.3 −0.27 −0.09 105.60.6 113.7 111.7 0.20 15.6 −0.42 −0.13 115.10.5 124.8 122.6 0.18 16.0 −0.59 −0.16 126.20.4 137.2 134.6 0.17 16.5 −0.75 −0.20 138.40.3 149.8 146.8 0.18 17.1 −0.87 −0.24 150.70.2 161.5 158.5 0.19 17.9 −0.95 −0.28 162.10.1 171.7 168.8 0.20 18.8 −0.99 −0.32 172.00 179.8 177.1 0.23 19.7 −1.00 −0.35 179.8−0.1 172.9 172.7 0.25 20.8 −0.99 −0.39 173.2−0.2 167.4 167.6 0.28 21.8 −0.98 −0.43 167.8−0.3 162.9 163.0 0.31 23.0 −0.96 −0.47 163.4−0.4 159.2 158.9 0.34 24.2 −0.94 −0.50 159.7−0.5 156.1 155.9 0.37 25.4 −0.92 −0.54 156.7−0.6 153.6 153.5 0.40 26.6 −0.90 −0.58 154.2−0.7 151.4 151.6 0.43 27.9 −0.88 −0.61 152.0−0.8 149.6 149.9 0.46 29.2 −0.87 −0.65 150.1−0.9 148.0 148.6 0.50 30.6 −0.85 −0.69 148.5−1.0 146.7 147.4 0.53 31.9 −0.84 −0.73 147.1

from the respective integrals over time of the VCG signals.In Col. 3 corresponding results are listed now derived fromthe three vector values at the timing of the two maxima(QRS, STT) of the spatial magnitude vector. Note that forf = 0 (row 11 of Table 1), ρf = δ + αe and therepolarization timing follows depolarization at uniformintervals, yielding apex T values having reversed polarityto those of the QRS complexes. Correspondingly, columnentries [2 3] of this row are close to 180° and entry 6 (thecorrelation coefficient) is −1.

The results were compared by those derived with each ofthree of the major limited lead systems that are availablefrom the literature. Applied to the measured, rather than thesimulated ECGs, the results are: Gabor-Nelson: 78.7°, Frank[14] (7 electrodes) 93.8°, Kors et al. [15] (the 9 electrodes-common reference signals of the Standard 12-lead system)77.8° and Edenbrandt and Pahlm [16] (9 electrodes of theStandard 12-lead system) 94.7°.

The interest in the QRS-T angle has tempted severalgroups to study the properties of the QRS-T angle in thefrontal plane only, thus ignoring front-to-back component ofthe vector. The frontal QRS-T angle found when discardingthe front-to-back vector component, that can be observed inFig. 1, was 16.4°.

Application to QRST integrals

As discussed in Section 3, the integrals over time ofany number L of BSP signals, observed by any leadsystem, may be represented by numerical vectors havingL elements. The basic three VCG signals may be viewedas a special case: L = 3. Derived from any weightingmatrix M of size (3,K), with K the number of electrodessampling the field, is the pseudo inverse of the involvedlead field matrix of size (K,3). Based on the generalformulation Eq. (2), the genesis of the three VCG signalscan be formulated as Vxyz = MAVm, with the matrixproduct B = M A replacing A, while inheriting theproperty Be = 0.

When comparing any pair of L BSP signals observed attwo different time instances, or the integrals over two non-overlapping time intervals, such as the QRS- and STTintervals, the linear correlation coefficient CC is anobvious tool. Its definition renders their value to beindependent of the magnitudes of the two numericalvectors (their norms) as well as of their mean values. As aconsequence, in the application to the interpretation toECG signals, this marker is independent of inter-individualdifferences in overall signal magnitude, as well as of thepotential reference involved.

Col. 6 of Table 1 lists the CC values of the iQRS and iSTTvectors. By 'feeding' these CC data to the inverse cosinefunction, a set of corresponding angles is generated (Col.8 of Table 1).

Discussion

A major, highly significant result from the analysiscarried out in Section 4, is the formulation of Eq. (10), which

149A. van Oosterom / Journal of Electrocardiology 47 (2014) 144–150

identifies how the integrals over the intervals QRST, QRSand STT relate to the full dispersion of the ARI values, thetiming of depolarization, and the timing of repolarization,respectively. The adjective ''full'' is used to emphasize thatthe entire set of spatial differences between these variablesand their mean is at stake.

As stated in Section 3, the idea of studying the QRSTintegrals dates back to the seminal work done by Wilson etal. [9]. Initially derived from the extremity leads, this led tothe concept of the Ventricular Gradient [17]. The latterwould be zero if the dispersion of the ARI values (uniformaction potential durations), and consequently, so would bethe iQRST maps, which is not what is observed generally. Theanalysis of the iQRST maps observed in dogs observed duringdifferent stimulation protocols suggested that local changesin the ARI values resulting from induced differences in theactivation sequences are reflected in a minor manner only inthe iQRST maps [18].

In Section 4 the elements of the three vectors α̃,−δ̃ and ρ̃represent the deviations from the mean of the integrals overtime of the transmembrane potentials of the myocytes. Theassumption that the TMP upstrokes were uniform led to theidentification of the three vectors as representing thedispersion of time intervals (ARI) and of the timing ofdepolarization and repolarization. However, if local regionsof reduced TMP magnitude exist, the formulation shown inEq. (10) still holds true but the values of the elements ofα̃, − δ̃ and ρ̃ no longer can be interpreted as representingtiming only, but rather the full TMP wave form,including wave local differences in upstroke magnitudeand duration or timing. This is in agreement with theanalysis of the significance of QRST integrals asformulated by Geselowitz [19]. His interpretation of theQRST, QRS and STT integrals has a direct bearing onthe electrophysiological interpretation of the correspond-ing QRS-T, QRS and STT angles, which corresponding-ly, is not straightforward [20].

The results shown in Section 5 indicate a close linkbetween the QRS-T angles in 3D (real world space) andthose in the abstract L dimensional space of linear algebraapplied to the signal analysis of the multi-lead ECG. Aclose correspondence between the data in Cols. 2 and 3 isobserved, derived from two different selections of thepaired vector elements, mean values over time vs. peakmagnitudes, respectively.

One may speculate that using a higher number ofsignals may result in an even higher diagnostic power(specificity) of the use of the QRS-T angle or thecorresponding CC. Since the application of the Gabor-Nelson method may not be in everybody's reach, asuitable approximation may be considered. Of the threelimited lead approximations tested, the clear winner is thetransfer matrix M designed by Kors and colleagues [15]. Avariant of this matrix can be set up in which the potentialdifferences observed at nine of the electrode locations ofthe standard 12-lead [14] system are used with respect toany arbitrary reference.

As listed in the results section, a huge difference wasobserved between the QRS-T angle in the frontal plane

and the 3D variant. For the measures data the result was16.4° when estimated from the frontal plane, for the 3Dvariant the Frank leads showed 93.8°, Kors et al. indicated77.8° and the Edenbrandt and Pahlm estimate yielded94.7°. This indicates that the QRS-T angle estimated fromthe frontal plane is incomplete, and most likely suboptimalin diagnostic applications.

In the QRS-T angle approach, the relationship betweentwo orientations in 3D space, one related to the QRSinterval, the other to the STT interval. Each of theseorientations demands two variables (azimuth and eleva-tion) for their specification. In contrast, the QRS-T angleis expressed by a single variable: the angle betweenvectors in 3D space. This single variable is insensitive tointer-individual differences in heart orientation. Thisrobustness may explain some of the apparent effectivenessof the QRS-T angle [21–23] estimated using the vectorsrepresenting integrals of the XYZ vector signals, or thoseof the individual signals involved in the QRST integralmaps. Being just a single variable, the angles may play arole an important, robust variable in any multivariate studyof the diagnostic use of the cardiac signals observed onthe thorax.

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