## Abstract

Local unipolar electrograms (UEGs) permit assessment of local activation and repolarization times at multiple sites simultaneously. However, UEG-based indexes of local repolarization are still debated, in particular for positive T waves. Previous experimental and computer modeling studies have not been able to terminate the debate. In this study we validate a simple theoretical model of the UEG and use it to explain how repolarization statistics in the UEG relate to those in the action potential. The model reconstructs the UEG by taking the difference between an inverted local action potential and a position-independent remote signal. In normal tissue, this extremely simple model predicts T-wave morphology with surprising accuracy while explaining in a readily understandable way why the instant of repolarization is always related to the steepest upstroke of the UEG, both in positive and negative T waves, and why positive T waves are related to early repolarizing sites, whereas negative T waves are related to late repolarizing sites.

- ventricular repolarization
- activation recovery interval
- computer model

measurement of repolarization time from the unipolar electrogram (UEG) is important for clinical studies of repolarization abnormalities, as well as for experimental studies. It is, therefore, important to understand how the T wave in the electrogram is generated and how it relates to local repolarization time. The ongoing debate on repolarization measurement in positive T waves (8, 9, 23, 46, 47) demonstrates that this understanding is incomplete.

Wyatt et al. (45) proposed to use the instant of steepest upstroke (*T*_{up}) of the T wave in the UEG as an index of local repolarization. Several studies have confirmed the validity of this method theoretically (5, 6, 17, 34) and experimentally (3, 8, 17, 21). Others have proposed that for positive T waves the instant of steepest downstroke (*T*_{down}) should be used (2, 13, 48). The T-wave apex has also been used as an estimate for repolarization time (32).

The current mechanistic insight in the relation between *T*_{up} and repolarization is based on theoretical studies in a one-dimensional cable model (5, 6, 17, 34). This explanation cannot be easily extended to arbitrary repolarization patterns in the three-dimensional (3-D) heart. Numerical evaluations with realistic 3-D heart models have confirmed that theoretically *T*_{up} is the best index of repolarization even in the complex situation of a complete 3-D heart (5, 6, 28, 29), but the overwhelming complexity of the computational techniques does not allow for a clear explanation.

We propose a simple UEG model that is more suitable to give insight in repolarization phenomena. This model approximates the UEG by a linear combination of the local action potential (AP) and a position-independent signal, which we name the remote component. The purpose of this article is to demonstrate that this model is accurate enough to predict the most important repolarization statistics and that it can be used to understand how these statistics relate to the underlying AP. To show the validity and limitations of the simple model, simulated UEGs were compared with UEGs simulated by one of the most complete numerical models of the intact human heart currently available during various stimulation modalities.

## MATERIALS AND METHODS

APs were simulated with a large-scale computer heart model that has been described and validated previously (27, 38). This model has anisotropic myocardium with transmurally rotating fiber orientation. Sinus rhythm was mimicked by stimulating the ventricles at the early activation sites published by Durrer et al. (12, 19). Propagating AP were computed at 15 million points in the ventricular myocardium with a monodomain reaction-diffusion equation, based on ionic currents computed with the Ten Tusscher–Noble–Noble–Panfilov (TNNP) membrane model for the human ventricular myocyte (37), which distinguishes endocardial, midmyocardial, and epicardial cells. Differences between the left ventricle and right ventricle (11, 43) were implemented as detailed in Table 1.

The anatomy of the model is illustrated in Fig. 1. The atria were not activated but were included in the simulation of electrograms to allow the reference potential to be set to zero at the roof of the right atrium (27). Ventricular and atrial cavities were filled with blood.

Two different models were used for the computation of electrograms: a realistic model and a simple model.

#### Realistic model.

Let *V*_{m}(**x**, *t*) be the membrane potential at time *t* and position **x** (boldface symbols are used to indicate vector quantities). The local UEG according to the realistic model, φ_{e}(**x**, *t*), was computed from *V*_{m} throughout the heart by solving the bidomain equation (27, 41) in the form of *Eq. 9* given in the appendix, with the additional condition that φ_{e}(**y**, *t*) = 0 at the reference electrode site, **y**, in the right atrium. By solving this equation, φ_{e}(**x**, *t*) was obtained at 35 million model points in the ventricular myocardium, atria, connective tissue, and intracavitary blood. Details of the solution methods used are given in previous work (27). To allow analysis of temporal derivatives, error tolerance levels were chosen 100 times lower than previously reported (27). To limit the consequent increase in computation time, we chose a lower spatial resolution of 0.25 mm for computation of both AP and electrograms. The resulting errors in propagation velocity were compensated by choosing 15% higher values for the conductivity in the intracellular domain. Nominal conductivity values were taken from literature (31). The adapted conductivities were *g*_{eT} = 0.12, *g*_{eL} = 0.30, *g*_{iT} = 0.035, and *g*_{iL} = 0.33 Siemens/m, where the letter e represents the extracellular domain, the letter i represents the intracellular domain, the letter T represents transverse, and the letter L represents longitudinal. The intracavitary blood had conductivity 0.6, and connective tissue and atrial myocardium had conductivity 0.2 Siemens/m.

#### Simple model.

We define *S*(**x**, *t*), the simple model for the UEG at a point **x** as a function of time *t*, as the difference between a position-dependent local component, *L*(**x**, *t*), and a position-independent remote component, *R*(*t*): (1)

The names of these components reflect that *L*(**x**, *t*) is completely determined by local activity, whereas *R*(*t*) is dominated by remote activity. The local component, *L*(**x**, *t*), is defined as a scaled mirror image of the local membrane potential: (2) where *g*_{e} and *g*_{i} represent the conductivities of the extracellular and intracellular domains, respectively. For the simple model, these conductivities are assumed to be isotropic. The rationale for *Eq. 2* is given in the appendix. The remote component is obtained by comparison with φ_{e}(**x**, *t*) computed by the realistic model, as follows. First we compute a residual signal, *r*(**x**, *t*), for each location: (3)

This signal contains the effects of anisotropy, inhomogeneities, and tissue boundaries, but it is dominated by the differential effect of activity throughout the myocardium on the positive (exploring) and negative (reference) electrodes. To single out the global effect, we take the average, *R*(*t*), of *r*(**x**, *t*) obtained from a large number of different locations **x**. We assume that local effects cancel out in averaging. We will show that *r*(**x**, *t*) from different sites are similar, so that approximating them by the average *R*(*t*) has little effect on *S*(**x**, *t*) even if this assumption is incorrect. Alternatively, *R*(*t*) can be computed directly from membrane potentials throughout the heart. A derivation based on lead field theory (14, 15, 20) is presented in the appendix.

To facilitate the discussion, we introduce a signal *V*_{R} that can be compared with membrane potentials: (4)

In the appendix we demonstrate that *V*_{R}(*t*) can be considered as a (weighted) average of *V*_{m}(**x**, *t*) over the surface that bounds the myocardium. We now reformulate our definition of the simple model as follows: (5)

For the fraction *g*_{i}/(*g*_{i} + *g*_{e}), the value 0.25 was chosen. This value is close to the ratio *g*_{iT}/(*g*_{iT} + *g*_{eT}) in the realistic model. This model is based on ideas that were independently published by Colli Franzone et al. (6) and by us (28) and is similar to a UEG model derived by Geselowitz (15) (see appendix).

It is important to note that the two components of the simple model, *L* and *R*, have no direct physical meaning, i.e., they do not correspond to signals that can be observed independently. Only the difference between the two is meaningful.

#### Statistics.

The AP and UEG were analyzed at 10^{4} randomly selected sites throughout the ventricular myocardium. From each simulated AP, repolarization time (*T*_{R}) and the instant of 90% repolarization (*T*_{90}) were determined and used as reference timings. *T*_{R} was defined as the instant of minimum first derivative (steepest downstroke) of the AP. In addition, the activation time *T*_{A} was defined as the instant of maximum first derivative (steepest upstroke) of the AP. AP duration (APD) was computed as *T*_{R} − *T*_{A}.

From the UEG at the same location we determined *T*_{up} and *T*_{down}, the instant of the T-wave apex (*T*_{apex}) (32), and the instant of minimum second derivative (*T*_{d2}), which has recently been reported to be related to *T*_{90} (5). First, T-wave signs were determined. There is a continuous spectrum of positive, biphasic, and negative T waves in the UEG. To classify T waves as positive or negative, we evaluated the area under the electrogram, from 50 ms after the end of depolarization (the latest *T*_{A}) to the end of the simulation. A T wave was considered positive when the positive area exceeded the negative area. *T*_{up} and *T*_{down} were evaluated as the instants of maximum and minimum first derivative of the UEG, respectively, in the interval from 50 ms after the end of depolarization to the end of the simulation. For positive T waves, *T*_{down} was evaluated in the interval from *T*_{up} to the end of the simulation. The repolarization markers are illustrated in Fig. 2. Analysis of the UEG was the same for the simple and realistic models. All statistics were determined automatically by a computer program. No manual corrections were made.

Computation and analysis of φ_{e} and *S* was done at a 1-ms sampling rate. Therefore, timing results are expressed with 1-ms accuracy. The difference between the realistic and simple UEGs at a site **x** was quantified using the relative difference (RD) (40) defined as: (6) where the time index *t* = 1,…, 500 ranges over all samples (depolarization and repolarization). The RD is a dimensionless measure of the difference between two signals (*S* and φ_{e} in our case); it is zero if the signals are identical.

## RESULTS

#### Predictions of the realistic model.

To show the rationale for developing the simple model, we briefly review results obtained with the realistic model. These results summarize and expand previously reported findings (28, 29) and confirm results from studies with smaller-scale models (5, 6, 17, 34).

Figure 3 shows a representative sample of UEGs from the ventricular myocardium. Local repolarization times (*T*_{R}) are indicated with solid dots in the electrograms. These are invariably located on the upslope of the T wave.

In sinus rhythm, positive T waves were found in 44% of the analyzed positions (Table 2). Their average *T*_{R} was 40 ms earlier than that of negative T waves. In Fig. 4*A*, *T* _{R} distribution is shown separately for positive and negative T waves. Clearly, positive T waves are generally associated with earlier repolarization than negative T waves. There is overlap between the distributions, reflecting the fact that repolarization time is associated with a continuous spectrum of T-wave configurations from entirely positive, through biphasic, to entirely negative. This is illustrated in Fig. 4*B*, using (signed) T-wave area as a single parameter to describe the configuration.

The performance of *T*_{up} in sinus rhythm is illustrated in Fig. 4*C*. On 1,000 points shown in the figure, eight outliers are seen; the other points lie within a few milliseconds from the identity line.

The performance of *T*_{up}, *T*_{down}, *T*_{d2}, and *T*_{apex} on a sample of 10^{4} randomly chosen sites throughout the ventricular myocardium, in sinus rhythm, is summarized in Table 2. Table 2 shows that *T*_{up} is a very accurate substitute for *T*_{R}. An alternative for *T*_{up} in positive T waves, *T*_{d2} predicts *T*_{90} reasonably well. In positive T waves, the performance of *T*_{apex} is similar to that of *T*_{d2} but with an opposite bias, whereas *T*_{down} performs very poorly. Pacing in the left ventricular apex caused repolarization to move predominantly from apex to base. This led to 85% positive T waves and increased the difference in average *T*_{R} between positive and negative T waves to 70 ms. The accuracy of *T*_{up}, *T*_{down}, and *T*_{apex} as predictors for repolarization time was not affected (changes <1 ms). In a model with uniform intrinsic APD (all cells set to type endocardial; see Table 1), 49% of the T waves was positive and the difference in average *T*_{R} between positive and negative T waves was reduced to 31 ms (from 40 ms in sinus rhythm), but the accuracy of the repolarization indexes was again not affected.

#### Remote component for the simple model.

Residuals *r*(**x**, *t*) computed according to *Eq. 3* for 100 randomly chosen myocardial sites are plotted in Fig. 5*A*, together with their average. Although individual signals differ, the average gives a reasonable estimate of their individual shapes, especially during repolarization. To obtain a smooth signal, we used the average, *R*(*t*), from 10^{4} randomly chosen sites to compute UEGs with the simple model.

#### Validation of the simple model.

Examples of UEGs computed according to the simple model (*Eq. 5*) are shown in Fig. 5, *B*–*E*. These examples were chosen to illustrate a representative set of RDs between the simple and realistic models. Figure 5*F* shows that the vast majority of the 1,000 sampled sites had acceptable RD values. As expected with this highly simplified model, there were several bad matches as well. These typically occurred in thin trabeculae, where realistic UEGs have small amplitudes and are more influenced by remote activity. The simple model cannot account for this. The simple and realistic models agreed on the polarity of the T wave in 90% of the analyzed positions (*N* = 10^{4}).

The performance of the simple model was similar for different activation and repolarization sequences. Whereas for sinus rhythm the median RD was 0.29 (*N* = 1,000), it was 0.23 for apical pacing and 0.31 for left ventricular epicardial pacing. In a model with uniform intrinsic APD, the median RD was 0.32 for sinus rhythm.

#### Insights from the simple model.

Figure 6 (top 2 rows) shows how the UEG is constructed according to the simple model. The UEG is computed as the inverted and downscaled difference between *V*_{m} and *V*_{R} (*Eq. 5*); it is positive when *V*_{m} is more negative than *V*_{R} (blue area) and negative when *V*_{m} is more positive than *V*_{R} (red area). Thus positive T waves must occur at early repolarizing sites. The third row of signals in Fig. 6 shows how the simple model compares with the realistic model.

The fourth row in Fig. 6 shows how the simple model can help us to understand the proposed repolarization indexes *T*_{up} and *T*_{down}. It shows the temporal derivatives of the *V*_{m} and *V*_{R} signals. We will use the notation *V̇* to indicate the temporal derivative d*V*/d*t*. A dashed trace shows *V̇*_{R}, and a solid trace shows *V̇*_{m}. The minimum of *V̇*_{m} defines *T*_{R}, which is indicated by a red vertical line. Coincident with this line is a dashed black vertical line, which indicates *T*_{up}, the instant of maximum *Ṡ*. The (near) equality of *T*_{R} and *T*_{up} can be understood by observing that the maximum of *Ṡ*, which is proportional to *V̇*_{R} − *V̇*_{m}, always occurs near minimum *V̇*_{m}, due to the soft slopes of *V*_{R}. Thus, in normal tissue, the remote component does not perturb *T*_{up} much.

The fifth row in Fig. 6 shows *Ṡ* itself. Its extrema, which define *T*_{up} and *T*_{down}, occur near the minima of *V̇*_{m} and *V̇*_{R}, respectively, but generally do not coincide exactly with these.

*T*_{down} occurs where the difference *V̇*_{R} − *V̇*_{m} is most negative. At very early repolarizing sites (Fig. 6*A*), *T*_{down} occurs well after the end of the local AP and corresponds with the minimum in *V̇*_{R}. At less early sites, it still occurs after the end of the AP, because *V̇*_{R} can only dominate the difference when *V*_{m} is nearly back at resting potential. At sites with intermediate repolarization times, *V̇*_{R} and *V̇*_{m} together determine *T*_{down}. Both experimental (48) and numerical (28) studies have previously shown a (weak) correlation between *T*_{down} and repolarization time. However, a theoretical explanation has been lacking. The simple model readily explains this relation, also showing why it has a variable, nonunitary slope. According to the simple model, we should expect that for very early repolarizing sites, where the downslopes of *V*_{m} and *V*_{R} occur far away from each other, the local component cannot influence *T*_{down}. Thus at these sites *T*_{down} should be independent of *T*_{R}. At very late repolarizing sites (Fig. 6*C*), *T*_{down} is again very close to the minimum in *V̇*_{R}.

The *T*_{d2} index can be understood by considering second-order derivatives, which correspond to curvature in the original *V*_{m} and *V*_{R} signals. *T*_{d2} is defined as the instant of minimum d^{2}φ_{e}/d*t*^{2}, i.e. most downward curvature in φ_{e}. According to the simple model, this corresponds to the most upward curvature in *V*_{m}, if distortion from *V*_{R} is negligible. This upward curvature occurs at the end of the AP, near *T*_{90} in healthy myocardium. Because *V*_{R} is a very smooth signal, its curvature is indeed negligible. This explains the small difference between *T*_{d2} and *T*_{90} (Table 2). In positive T waves, the sharp turn in φ_{e} leads in most cases to the apex of the wave. Thus, in positive T waves, *T*_{apex} is close to *T*_{d2}.

We reported above that during apical pacing in the realistic model, 85% of the myocardium showed positive T waves, whereas this was 44% of sinus rhythm. The simple model explains this finding too. When the heart was paced at the apex, the basal region repolarized last. Because the reference electrode was near the base, this region was more strongly represented in *V*_{R}, and its late repolarization prolonged *V*_{R}. Because the majority of sites now repolarized earlier than *V*_{R}, a majority of T waves became positive. During sinus rhythm, positive T waves were a minority. An explanation for this is given in the appendix: under the assumption that the myocardium is isotropic, only sites at the surface of the myocardium contribute to *V*_{R}. In the anisotropic heart this is not strictly true, but the surface still dominates. The fact that we used sites from the whole myocardium to determine *V*_{R} does not change this. Thus the midmyocardial region has less influence on *V*_{R}. In sinus rhythm, this region was the last to repolarize. Due to the small influence of this late repolarizing tissue, repolarization of *V*_{R} during sinus rhythm occurred somewhat earlier than the average of the myocardium, making a larger volume of myocardium appear late repolarizing and having negative T waves.

Although *T*_{up} and *T*_{R} are closely related, they do not coincide exactly. This can be understood in terms of the simple model as follows. In Fig. 7, the signals *V̇*_{m} and *V̇*_{R} are plotted in a short interval around *T*_{R}. These signals correspond to the first column in Fig. 6. In this case, *T*_{R} (the instant of minimum *V̇*_{m}) was 277 ms and *T*_{up} (the instant of maximum *Ṡ*) was 276 ms. The reason for this difference is that *Ṡ* is determined by both *V̇*_{m} and *V̇*_{R}: from *Eq. 5* it follows that *Ṡ*(**x**, t) is proportional to *V̇*_{R}(*t*) − *V̇*_{m}(**x**, *t*). In the case of Fig. 7, the second derivative of *V*_{m} was smaller than the second derivative of *V*_{R} during a full sample interval before *T*_{R}, as shown by the tangent lines. Therefore, the difference *V̇*_{R} − *V̇*_{m} was slightly larger at *t* = 276 ms than at *t* = *T*_{R}.

## DISCUSSION

We have shown that in healthy tissue the local contribution to the UEG is essentially a downscaled inverted AP. The electrogram is positive when local *V*_{m} is more negative than *V*_{R}. This explains why T waves are positive for early repolarizing cells. Cells with intermediate repolarization times have a *V*_{m} that initially stays above *V*_{R} but descends below it in the final stage of repolarization. This explains their biphasic T waves. As observed experimentally (8, 17), such biphasic waves always have a negative part followed by a positive part. In late repolarizing regions, the T wave is negative because the local membrane is still depolarized, whereas it is repolarized elsewhere.

Due to the relatively soft slopes of *V*_{R}, first- and second-order derivatives of the UEG are dominated by the derivatives of *V*_{m}. Thus, where *V*_{m} makes its steepest descent, the UEG makes its steepest ascent. This explains the near equality of *T*_{up} and *T*_{R}. Where *V*_{m} turns most sharply upward, the UEG turns most sharply downward. This explains the small difference between *T*_{d2}, *T*_{90}, and, in positive T waves, *T*_{apex}. Only when *V*_{m} is very flat the slope of *V*_{R} can dominate the slope of the UEG. This leads to a coincidence of *T*_{down} and the instant of minimum d*V*_{R}/d*t* for very late and very early repolarizing sites. At sites with intermediate repolarization times, this relation is modulated by the slope of the local *V*_{m}, leading to a (weak) correlation between *T*_{R} and *T*_{down}.

#### The activation-recovery interval and alternatives.

Wyatt et al. (45) proposed the activation-recovery interval (ARI) as a substitute for the APD, which cannot easily be measured in vivo. The ARI was measured from the time of steepest downstroke of the UEG to the time of steepest upstroke of its T wave. Comparison with transmembrane APD and with effective refractory period demonstrated the validity of this method (3, 8, 17, 21). A simulation study with a cable model also confirmed the method and demonstrated some of the circumstances in which it is less accurate (34).

Other authors proposed that, for positive T waves, *T*_{down} should be used instead of *T*_{up} (2, 13, 48). The original rationale for this alternative method was that in some studies its results correlated better with the duration of the monophasic action potential (2, 48). The intuition that the end of the local T wave (i.e., *T*_{down} for positive T waves) should correspond to local *T*_{R} may also play a role.

Our simple model shows that in normal tissue, *T*_{up} is temporally linked to *T*_{R}. This (simple) model is not suitable for all conditions. For example, it cannot faithfully simulate UEGs in case of fibrosis or intracavitary measurements. It is possible that the relation fails there, e.g., because the active tissue is so sparse or so far from the exploring electrode that the slope of *V*_{R} can dominate the slope of the UEG or because the tissue is too anisotropic. However, it is difficult to imagine that under these conditions the relation between *T*_{R} and *T*_{up} would be replaced by a relation between *T*_{R} and *T*_{down}. Such circumstances can only be studied with a more realistic model (29).

Another possible alternative for *T*_{up} is the T-wave area, which correlates strongly with *T*_{R} (28, 42). This relation may be useful as a last resort when *T*_{up} cannot be measured. However, caution is necessary because other factors may influence T-wave area. It may be a useful index if such problems can be ruled out or in order to assess local changes in APD. When there is no opportunity for a quantitative analysis, the polarity of the T wave can give a coarse estimate of local repolarization time with respect to the average of the ventricular myocardium.

In model studies, *T*_{d2} too has shown potential as an estimate for *T*_{R} or *T*_{90}, but its sensitivity to noise may limit its applicability in practice (5). For positive T waves, *T*_{apex} is equally accurate.

#### Sources of inaccuracy in the ARI.

Generally, *T*_{up} and *T*_{R} will not be exactly the same, because the UEG contains a remote component. At *T*_{R} itself, where *V̇*_{m} is minimal, the second derivative of *V*_{m}, *V̈*_{m}, is zero, so any signal component with a nonzero second derivative at *T*_{R} can modulate *T*_{up}. This displacement is limited to the interval in which the magnitude of the second derivative of *V*_{m} is smaller than that of the other components of the UEG. According to the simple model, the only other component is *V*_{R}. Its second derivative, *V̈*_{R}, is generally small but not zero at *T*_{R}. If *V̈*_{R} < 0 at *T*_{R}, *T*_{up} will be earlier than *T*_{R}, and if *V̈*_{R} > 0 at *T*_{R}, *T*_{up} will be later than *T*_{R}. In this study, the interval in which the absolute value of *V̈*_{m} was small enough to allow a difference between *T*_{up} and *T*_{R} was generally less than 2 ms long. However, in failing (30) or ischemic myocytes, the AP can have a long, linear downslope, i.e., a long interval in which the absolute value of *V̈*_{m} is small. This can lead to considerable differences between *T*_{up} and *T*_{R} (17, 34). The same problem occurs with atrial myocytes that lack a plateau in their AP (42).

#### Other ways to explain the UEG.

Previous explanations of the validity of the ARI were based on propagation in a uniform cable, which is equivalent to plane-wave propagation in a uniform isotropic 3-D medium (17, 34). Under the assumption that the repolarization wavefront travels with a fixed velocity and the final phase of the AP is temporally symmetric about *T*_{R}, a strict equivalence of *T*_{up} and the instant of minimum third temporal derivative of *V*_{m} can be demonstrated (34). A disadvantage of this approach is that the generalization to arbitrary repolarization sequences is not obvious. Another UEG model that has been successful in providing insight is the oblique dipole layer (4). Whereas this model is very useful in explaining anisotropic effects on the QRS complex, we think that it is not convenient for an explanation of the ARI and T-wave polarity.

The cable model can also be used to explain T-wave polarity. For depolarization wavefronts, this has been treated, e.g., by Spach et al. (33). With the signs inversed, this treatment is perfectly applicable to repolarization wavefronts. Just like the amplitude ratio of the R and S waves in the UEG contains information about depolarization time (10), the positive-to-negative amplitude ratio of the T wave gives information about repolarization time.

#### The remote component.

The signal *V*_{R} was obtained in an ad hoc way. In the appendix, we argue that it can be interpreted as an average membrane potential. This average is not calculated over the entire myocardium, but only over its surface. In addition, different cells contribute to it with different weights, and these weights depend on the position of the reference electrode. This is to be expected, since the location of the reference is known to influence the UEG shape substantially. However, the amplitude of *V*_{R} is always similar to the AP amplitude, and its shape always represents a weighted average of remote AP.

#### Modeling techniques used.

In our realistic model, dispersion of AP shape was based on the inhomogeneous expression of ion channels and subunits. Presently, these inhomogeneities are insufficiently characterized to explain the dispersion of AP shape in the human heart and to obtain normal T waves in the ECG (7, 36, 37). However, the similarity of results obtained with the realistic model for different repolarization sequences demonstrates that a faithful representation of the true dispersion of AP shapes is not important for our study.

#### Conclusion

In normal myocardium, the UEG can be understood (approximately) as the difference between an inverted AP and a remote component. With this simple model, the mechanism and limitations of all proposed repolarization markers can be readily explained. The relation between *T*_{up} and *T*_{R}, which has previously been established both theoretically and experimentally, can now be understood intuitively. In positive T waves, there is a weak correlation between *T*_{down} and *T*_{R}. In contrast with the (near) identity relation between *T*_{up} and *T*_{R}, the relation between *T*_{down} and *T*_{R} has a variable, nonunitary slope. A theoretical basis for their correlation has been lacking. The simple model explains the correlation, and the nonunitary slope of the relation, making clear that this relation should not be exploited for the determination of repolarization times. We confirm the conclusion that *T*_{up} is the best estimate of *T*_{R}, regardless of T-wave sign. *T*_{d2} and *T*_{apex} may sometimes be useful. *T*_{down} should never be used.

## APPENDIX

#### Rationale for the simple model.

The bidomain model is the theoretical basis for almost all current modeling work in cardiac electrophysiology (14, 22, 24, 41). It approximates the myocardium with two continuous domains. The intracellular domain represents the interior of myocytes and gap junctions, whereas the extracellular domain represents the interstitium. Intracavitary blood and connective tissue can be thought of as an extension of the extracellular domain (27). The electric conductivity of each domain is represented by conductivity tensor fields **G**_{i} and **G**_{e}, respectively. The bidomain model states that all currents that leaves one domain at a given location must enter the other domain at the same location: (7) where φ_{i} and φ_{e} are the intracellular and extracellular potential fields. With the use of the transmembrane potential (8) the bidomain equation can be rewritten as (9)

This equation allows the computation of φ_{e}(**x**, *t*) from a known distribution of *V*_{m}. It only determines φ_{e} up to an arbitrary offset potential, which is fixed by defining φ_{e}(**y**, *t*) to be zero at the location, **y**, of the reference electrode. Since *Eq. 9* is an implicit equation that must be satisfied for all **x**, it can only be solved in general by computing φ_{e} throughout the heart at once by solving a large system of linear equations (27). The realistic model employed in this study uses this method. This solution process cannot be intuitively understood. To arrive at an equation that can be easily understood, we assume **G**_{i} and **G**_{e} to be homogeneous and isotropic, so that we can represent them by scalars *g*_{i} and *g*_{e} and write (10)

In the absence of boundaries, the general solution of this equation is (11) where φ_{off}(*t*) is an undefined offset potential (25). Due to the assumption of isotropy, this equation for 3-D tissue is the same as the expression for φ_{e} in the one-dimensional cable (16, 18, 25). The first term in *Eq. 11* is our expression for the local component, *L*(**x**, *t*).

If we define the extracellular potential at a reference location, **y**, to be zero: φ_{e}(**y**, *t*) ≡0, we find the following expression for the offset potential: (12)

Thus bipolar electrograms can be computed with *Eq. 11*. For the UEG, a reference location in a passive medium (inactive tissue, blood, or perfusion fluid) must be used. This situation cannot be modeled realistically with an infinite-medium approach as used above. Therefore, we seek to replace φ_{off}(*t*) in *Eq. 11* by a more general remote component *R*(*t*) that can account for the inhomogeneities and boundaries of the body.

There are essentially two methods to compute UEGs in a bounded and inhomogeneous body. The first is to use a differential equation like *Eq. 9*. This is the approach taken by the realistic model used in this study. As stated above, it has the disadvantage of providing little insight in the mechanisms underlying the UEG. The second method is based on integral rather than differential expressions. A suitable integral-based method was proposed by Geselowitz (15) in a brief conference article; details of his derivations are given in earlier work (14). The method is based on what is now called an equivalent double layer (39). Assume that a current *I*_{app} is injected through an exploring electrode at a point **x** inside the myocardium and leaves through a reference electrode located at a point **y** outside the myocardium. The resulting current field inside the body, including the myocardium, is denoted as **J**_{xy}. The field **J**_{xy}/*I*_{app} is traditionally called the lead field (20) or lead vector field (26) for the given pair of electrodes. With the assumption of isotropic extracellular and intracellular media, the UEG, i.e., the potential difference between the two electrodes, *V*_{xy} can then be expressed as (13) where *S* is the surface of the active myocardium, over which the product of *V*_{m} and **J**_{xy} is integrated (15). Comparison with *Eqs. 5* and *11* confirms the expression for the local component, *L*(**x**, *t*), and identifies the term involving the surface integral as what we named the remote component *R*(**x**, *t*) of the UEG. It is not obvious from this expression that *R* depends little on the position of the exploring electrode. Therefore, we calculated lead fields **J**_{xy} [using the same software as for the realistic model (27)] and evaluated the expression (14) for five different positions {**x**_{n}, *n* = 1,…, 5} of the exploring electrode and a reference electrode at location **y** in the left atrium. The results are shown in Fig. 8 and demonstrate that *R*_{y} defined as the average of several *R*_{xn,y} would be a good approximation for each individual *R*_{xn,y}. For practical reasons, *R* used elsewhere in this article was obtained by subtracting the local component, *L*(**x**, *t*) from φ_{e}(**x**, *t*) computed with a realistic model and averaging the difference over many locations **x**. Figure 8*B* shows that *R*_{y} and *R* are nearly identical. Since the purpose of our simple model is only to give insight, and not to replace the realistic model for numerical simulations, the actual method to arrive at *R* is not important.

From charge conservation, we must have (15)

Thus *Eq. 13* demonstrates that we can think of *V*_{R} for any pair of electrodes at **x** and **y**, which can be expressed as (16) as an average membrane potential, albeit with a weighting that depends on the reference location. Under the assumption of isotropy, only *V*_{m} at the surface of the myocardium contributes. Therefore, *V*_{R} does not reflect late repolarization of the midmyocardium (44). In the anisotropic heart, the deeper myocardium does contribute to *V*_{R}, but the surface still dominates.

In previous work we have computed *V*_{R} as a nonweighted average of *V*_{m} over the whole myocardium (28, 29). The more general method employed here is more accurate, especially for abnormal repolarization sequences.

The simple model is not intended as an alternative for complete bidomain solutions or competition for less crude approximations such as the oblique dipole model (4). It is only a rule of thumb that allows us to explain how the UEG relates to APs. As such, it is similar to models that explain the electrocardiogram as a difference between two APs. Such models are used by many teachers of electrocardiography (1, 35).

## GRANTS

Computational resources for this work were provided by the Réseau québécois de calcul de haute performance. This study was partially supported by the Netherlands Heart Foundation Grant No. 2005B092. M. Potse was partially supported by the Fonds de la recherche en santé du Québec, through the Research Center of Sacré-Coeur Hospital, Montreal, Quebec, Canada.

- Copyright © 2009 the American Physiological Society