# Deconvolution: A Novel Signal Processing Approach for Determining Activation Time From Fractionated Electrograms and Detecting Infarcted Tissue

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## Abstract

*Background* Two important signal processing applications in electrophysiology are activation mapping and characterization of the tissue substrate from which electrograms are recorded. We hypothesize that a novel signal-processing method that uses deconvolution is more accurate than amplitude, derivative, and manual activation time estimates. We further hypothesize that deconvolution quantifies changes in morphology that detect electrograms recorded from regions of myocardial infarction.

*Methods and Results* To determine the accuracy of activation time estimation, 600 unipolar electrograms were calculated with a detailed computer model using various degrees of coupling heterogeneity to model infarction. Local activation time was defined as the time of peak inward sodium current in the modeled myocyte closest to the electrode. Deconvolution, minimum derivative, and maximum amplitude were calculated. Two experienced electrophysiologists blinded to the computer-determined activation times marked their estimates of activation time. F tests compared the variance of activation time estimation for each method. To evaluate the performance of deconvolution to detect infarction, 380 unipolar electrograms were recorded from 10 dogs with infarcts resulting from ligation of the left anterior descending coronary artery. The amplitude, duration, number of inflections, peak frequency, bandwidth, minimum derivative, and deconvolution were calculated. Metrics were compared by Mann-Whitney rank-sum tests, and receiver operating curves were plotted.

*Conclusions* Deconvolution estimated local activation time more accurately than the other metrics (*P*<.0001). Furthermore, the algorithm quantified changes in morphology (*P*<.0001) with superior performance, detecting electrograms recorded from regions of myocardial infarction. Thus, deconvolution, which incorporates a priori knowledge of electrogram morphology, shows promise to improve present clinical metrics.

Interpretation of cardiac electrograms is an important aspect of clinical electrophysiological procedures. A frequent application of electrogram interpretation is activation mapping, the goal of which is to determine the spatiotemporal pattern of electrical propagation through myocardium. Although automated systems to determine activation times at several locations and to create activation maps have been devised,^{1} ^{2} ^{3} ^{4} marking activation time is most likely to be reliable and accurate when electrograms exhibit narrow, biphasic deflections. However, neither the time of maximum amplitude nor the time of minimum derivative of the electrogram is necessarily an accurate reflection of local activation time in the setting of abnormal, fractionated electrograms.^{5} Although manual adjustments in the determination of local activation times may augment automated activation mapping,^{1} ^{2} ^{3} ^{4} as the number of sites increases, this leads to a tedious and laborious process.

A second goal of electrogram interpretation is to examine morphology to characterize tissue at the recording site. In patients with prior healed myocardial infarction, the origin of ventricular tachycardia is frequently located at the border zone, where there is interdigitation of surviving cells and scar tissue.^{6} Although electrograms from such regions (during sinus rhythm) are typically described as “fractionated,” a method to describe the statistical characteristics of ventricular tachycardia substrate in the presence of healed infarction more quantitatively is desirable.

Because the basic biophysical process by which electrograms are generated is well understood, signal processing that incorporates this a priori knowledge should provide improved activation time estimates and metrics for characterization of tissue at the recording site. With modeling techniques, it is possible to calculate the electrogram arising from an arbitrary activation pattern in the substrate myocardium.^{7} ^{8} ^{9} In this study, we reformulate the volume conductor equations as a stereotypical current source and then, in essence, pass the current source through a filter that represents the effect of distance and impedance between the current sources and the recording electrode. Mathematically, the way the current source is transformed by the effects of distance and impedance to the voltage recorded at an electrode is formulated as a convolution function. Once the convolution equation is established to describe recorded electrograms, we can use deconvolution of the electrogram to determine the parameters, such as local activation time, that give rise to a calculated electrogram that most closely fits the observed electrogram. The purpose of this study is to test our hypothesis that incorporating a priori knowledge of electrogram generation as a deconvolution is more accurate than empirical methods such as amplitude, derivative, and manual activation time estimates. We further hypothesize that the difference between an actual recorded electrogram and the electrogram calculated by deconvolution quantifies changes in morphology, which can be used to determine whether a given electrogram has been recorded from regions of myocardial infarction.

## Methods

### Deconvolution

Convolution is one of the most powerful principles in linear systems and signal processing. It defines a unique relationship between the input and the output of a linear time-invariant system. For example, the well-known process of bandpass filtering of electrograms represents an instance of convolution whereby the input (raw signal) is convolved with a filtering function to produce an output, the bandpass-filtered signal. Furthermore, methods have been developed to perform deconvolution, an inversion by which output can be used to calculate system parameters. If we frame the mechanism of electrogram generation as a convolution, we can develop a deconvolution by which an observed electrogram can be used to calculate local activation time and to quantify changes in electrogram morphology.

An electrogram recorded from a point electrode in an unbounded, homogeneous, isotropic, conducting medium can be calculated by the volume conductor equation,^{9} which forms the basis of our model-derived convolution equation. If uniform, constant-velocity activation is assumed, the volume conductor equation is recast as a convolution equation. The details of this derivation are given in the “Appendix.”

The convolution equation calculates the electrogram arising from uniform, constant-velocity activation as a function of three parameters: local activation time, electrode height, and activation velocity. The electrogram that most closely fits an observed electrogram recorded from known height is calculated by deconvolution (“Appendix”). The local activation time is estimated by the value that most closely fits the observed electrogram. Changes in electrogram morphology are quantified by calculating the fraction of the power in the observed electrogram that cannot be explained by the best-fitting uniform, constant-velocity activation.

### Electrograms for Testing the Accuracy of Activation Time Estimation

To test the accuracy of activation time estimates, it was necessary to have a set of electrograms for which the precise local activation time was known. Recently,^{10} we described a computer model of nonuniformly uncoupled myocardium that was used to generate electrograms with varying degrees of fractionation. This model was used to define a test set of electrograms on which to assess the accuracy of the deconvolution method.

Briefly, a 10×2.5-mm rectangular sheet of myocytes was simulated by use of 25-μm square elements with Beeler-Reuter ionic kinetics^{7} and incorporation of anisotropy of cell-to-cell connections. In addition to normal anisotropy (longitudinal to transverse coupling ratio, 10.9:1), we simulated patchy uncoupling by a recursive process of subdivision and multiplication by random numbers so as to produce patchy heterogeneity of coupling whereby neighboring myocytes had similar coupling resistivity, whereas those far apart were relatively uncorrelated. The heterogeneity of this patchy uncoupling was quantified by its correlation dimension.^{11}

For the present study, 30 simulated tissues were created, 5 at each value of the correlation dimension, ranging from uniform to the highest possible heterogeneity: 0.000, 0.006, 0.091, 0.223, and 0.591. Electrograms were calculated at five points above each 10×2.5-mm simulated tissue. At each point, an electrogram was generated for heights of 1, 10, 100, and 1000 μm. Thus, the test set consisted of 600 electrograms with various degrees of fractionation.

### Activation Time Estimation by Maximum Amplitude and Minimum Derivative

To compare other methods of activation time estimation with our deconvolution method, for each electrogram we calculated the times of maximum amplitude and minimum derivative. The first time derivative of the electrogram was calculated as the first-order backward difference. The time at which the greatest negative value of the derivative occurred and the time at which the maximum absolute amplitude of the electrogram occurred were calculated for each simulated electrogram.

### Activation Time Estimation by Electrophysiologists

Two experienced electrophysiologists who were blinded to all information about the computer-determined activation times independently marked their estimates of local activation time. Electrograms were presented in random order to eliminate any possibility of inferring activation time from neighboring sites. A Silicon Graphics Indigo workstation was used to display sets of 20 electrograms. Electrograms were scaled to have a peak height of 1 cm and a horizontal scale of 200 mm/s, making the display format consistent with clinical displays of electrograms. The electrophysiologist positioned a crosshair cursor over the electrogram to be marked. Activation time markers could be repositioned if desired. After all 20 electrograms on the screen were marked with local activation times, the locations of all activation time markers were recorded and the next set of 20 electrograms was presented.

### Electrograms for Characterization of Tissue at the Recording Site

To use the deconvolution method to characterize substrate myocardium, we recorded electrograms from 10 dogs with 2-week-old myocardial infarctions. All animal care procedures followed guidelines approved by the University of California, San Francisco. The dogs were preanesthetized with Innovar-Vet, 0.1 mL/kg SC (fentanyl/droperidol mixture), and then anesthetized with sodium pentobarbital 10 mg/kg IV. Myocardial infarctions were created by ligation of the left anterior descending coronary artery just below the first diagonal branch by the Harris two-stage procedure.^{12} Two weeks after the ligation, an epicardial sock with 56 circumferential unipolar electrodes (Bard) was placed on the surface of the heart. During normal sinus rhythm, unipolar electrograms were simultaneously recorded with 12-bit analog-to-digital conversion at a sampling rate of 1000 samples per second. The dogs were injected with 2 mCi of thallium. The heart was removed en bloc with the sock in place. Electrode positions were carefully registered, and the hearts were sectioned, photographed, and imaged with a gamma camera. Scintigraphic images were examined, and each electrode site was designated as normal or infarcted tissue. Sites were determined to be normal or infarcted by visual analysis of the thallium image. Any regionally reduced thallium uptake was judged to represent infarcted myocardium. Regions of reduced thallium occurred in the vascular territory of the left anterior descending coronary artery, whereas remaining areas of myocardium were well perfused.

A 128-ms segment of each electrogram was selected for deconvolution. Segments were chosen to include the waveform resulting from ventricular activation while avoiding artifacts due to repolarization or atrial activation. The fraction of electrogram power unexplained by the deconvolution was recorded for all electrograms. In addition, the amplitude, duration, number of inflections, peak frequency, bandwidth, and minimum derivative were calculated for each electrogram. The peak-to-peak amplitude was the difference between the maximum and minimum values, and the duration was the time from the first excursion 0.1 mV from baseline to the last excursion 0.1 mV from baseline, excluding T waves.^{13} The number of inflections was the number of fluctuations in the electrogram >0.5 mV.^{14} To determine frequency of peak power and bandwidth, a fast Fourier transform of the 128-ms segment was calculated, yielding a frequency resolution of 7.8 Hz. The frequency of peak power was the Fourier transform coefficient with the largest squared magnitude. Bandwidth was determined to be the minimum frequency such that the power of all lower-frequency components was 90% of the total power.

### Statistical Analysis: Modeled Electrogram Test Set

Simulated electrograms were sorted into two groups, fractionated and normal. Fractionated electrograms were defined to be those with more than two inflections (peaks>0.5 mV) or low amplitude (<1.0 mV peak to peak) and duration >60 ms.^{14} Twenty electrograms were rejected from the data set because they were recorded above modeled myocytes that did not activate. Five electrograms were rejected from the data set because their amplitude was <0.1 mV, making them too small for classification. A total of 575 electrograms were used in the statistical analysis.

The estimation errors, defined as the difference between the time of peak inward sodium current in the myocyte closest to the electrode and the corresponding estimates of activation time, were calculated for each electrogram. F tests were used to determine the statistical power of the differences in the variation of each estimation error distribution. Linear regressions were calculated for the SDs of each estimate as a function of electrode height and tissue coupling heterogeneity as measured by the correlation dimension. The slopes of the regressions were recorded to determine the sensitivity of each estimate to electrode height and tissue coupling heterogeneity.

### Statistical Analysis: Canine Infarct Electrograms

A total of 380 electrograms were recorded from electrodes overlying the ventricles. For nonparametric statistical comparison of metrics between normal and infarcted myocardium, the Mann-Whitney rank-sum test for unpaired observations (StatView version 4.02) was used, with a value of *P*<.05 considered to be significant.

To evaluate the ability of each metric to detect infarcted tissue, a receiver operating curve was plotted by varying the threshold for detection and plotting the probability of a true positive as a function of the probability of a false positive. For purposes of this analysis, a true positive was defined to be an electrogram recorded from infarcted tissue that resulted in a metric exceeding the threshold for detection. A false positive was an electrogram recorded from normal tissue that resulted in a metric exceeding the threshold for detection.

Because no “gold standard,” such as floating microelectrode recordings, was available, accuracy of activation time estimation was not assessed for the canine electrograms.

## Results

### Modeled Electrogram Test Set

Electrogram sorting resulted in 388 normal and 187 fractionated electrograms. Deconvolution determined the least-mean-squares fit and resulting activation time estimate for each electrogram (Fig 1⇓). For normal electrograms from homogeneously coupled tissue, the electrogram and the convolution fit very closely with the fraction of unexplained power <0.04 and a local activation time estimate with an error of 0.02±2.23 ms. For fractionated electrograms, the deconvolution method provided a robust and accurate method of determining local activation time, although the performance was degraded somewhat compared with normal electrograms. The error of the local activation time estimate by deconvolution was 0.92±6.18 ms.

Table 1⇓ compares the results of the deconvolution method of determining local activation time with those for other automated methods and manual marking by electrophysiologists. Deconvolution estimates had very low bias, indicated by a mean error of 0.31 ms when applied to all electrograms. Maximum amplitude and minimum derivative estimates had larger biases, as did both electrophysiologists' estimates. Although it is generally agreed that a consistent bias can be corrected, examination of Table 1 shows that the bias can differ by 2.2 ms between normal and fractionated electrograms, suggesting that the bias cannot be fully corrected for all electrograms.

Although uncorrected bias introduces error, the SD of the error indicates inaccuracies that more seriously distort the construction of an activation map, because they indicate the uncertainty of local activation time estimates. For normal electrograms, the deconvolution estimates had a significantly smaller SD than maximum amplitude and electrophysiologists' estimates. The difference between deconvolution and minimum derivative was not significant for normal electrograms; both methods had small SDs. For fractionated electrograms, the deconvolution estimates had a significantly smaller SD than any other method. The SD for all electrograms was also lowest for the deconvolution method. Larger SDs indicated that other estimates were spread more broadly around the time of peak inward sodium current and were therefore less accurate than the deconvolution.

The sensitivity of each estimate to changes in electrode height and coupling heterogeneity of the underlying tissue was calculated (Table 2⇓). As expected, the relative increase of the contribution of distant currents to the electrogram with increases in electrode height reduced the accuracy of local activation time estimates. Local activation time estimation by deconvolution was more robust to changes in height, as indicated by a sensitivity less than half that of any other method. As expected, activation time estimation accuracy also decreased with increasing coupling heterogeneity, which produced fractionated electrograms. Local activation time estimation by deconvolution was less sensitive to tissue coupling heterogeneity than any other method. Thus, deconvolution provides both a robust and an accurate estimate of local activation time.

A detailed plot of a fractionated electrogram illustrates two characteristics of fractionated electrogram that introduced errors (Fig 2⇓). Broad peaks and jagged downslopes disrupted amplitude and derivative estimates. The location of the maximum amplitude varied across broad peaks, and jagged downslopes caused the location of the minimum derivative to vary. These localized distortions may similarly reduce the accuracy of manual marking. Fractionated electrograms also exhibit multiple inflections and split potentials, which create further ambiguity. The inflection with maximum amplitude or the downslope with minimum derivative did not necessarily correspond to local activation. Both electrophysiologists' estimates were often aligned with inflections and baseline crossings that did not correspond to local activation. On the other hand, deconvolution determined the least-mean-squares fit on the basis of the assumption of a uniform, constant-velocity activation, which nevertheless closely fits one inflection and baseline crossing. Because the fit aligns itself with the entire electrogram, it most often matched the correct inflection and was relatively undisturbed by localized distortions, providing an accurate estimate of local activation time.

### Characterization of Tissue at the Recording Site

Electrograms recorded in vivo from normal tissue and from infarcted tissue were successfully deconvolved. The assumed normal electrogram calculated by the deconvolution closely fit most electrograms from normal tissue (Fig 3A⇓). The electrogram calculated during the deconvolution was biphasic, resulting from uniform, constant-velocity activation. Some of the electrograms recorded from normal tissue were asymmetrical or monophasic, and as a result the deconvolution fit was not as close, resulting in a large fraction of unexplained power and potentially a false positive (Fig 3B⇓). When applied to fractionated electrograms from infarcted tissue, the deconvolution fit produced large fractions of unexplained power (Fig 3C⇓).

The fraction of electrogram power unexplained by the deconvolution increased significantly from a mean of 41% in normal tissue to a mean of 68% in infarcted tissue. For comparison, an identical statistical analysis was calculated for other electrogram metrics (Table 3⇓). Electrogram amplitude and duration did not differ significantly between normal and infarcted tissue. The number of inflections, peak frequency, bandwidth, and minimum derivative differed significantly between normal and infarcted tissue, indicating that these metrics are sensitive to infarction. The decrease in number of inflections for infarcted tissue was unexpected but was most likely caused by the threshold used to measure inflections.

To evaluate the ability of each metric to detect infarcted tissue, receiver operating curves were plotted (Fig 4⇓). An optimal detection algorithm would have a probability of 1 for true positives and a 0 probability of false positives. The curve for the deconvolution algorithm suggests that electrograms from infarcted tissue can be detected with an 80% probability of true positives and only a 20% probability of false positives. On the other hand, all of the receiver operating curves for the other metrics are below the deconvolution curve, with higher probabilities of false positives for the same probability of true positives.

## Discussion

### Summary of Results

The results of this study confirm our hypothesis that deconvolution is more accurate than amplitude, derivative, and manual estimates of local activation time, at least for a test set of realistic computer-generated normal and fractionated electrograms. The results of this study also confirm our hypothesis that the difference between an actual recorded electrogram and the electrogram calculated by deconvolution quantifies changes in morphology, which can be used to determine whether a given electrogram has been recorded from regions of myocardial infarction.

### Comparison With Previous Results

Previous studies have shown that the minimum of the time derivative of a unipolar electrogram correlates with the peak inward sodium current, indicating local activation time.^{15} Some have advocated that the baseline crossing of the rapid deflection with maximum amplitude should be used as a landmark for marking local activation.^{16} Although minimum derivative and maximum amplitude activation time estimates based on these landmarks were relatively accurate when applied to normal electrograms, fluctuations present in fractionated electrograms reduce their accuracy. Smith and Ideker^{4} advocated that a more thorough understanding of the mechanism by which electrograms are generated could result in a more rational approach to improve the accuracy of local activation time estimation. In a study comparing activation time estimates applied to bipolar electrograms and using the minimum of the derivative of a unipolar electrogram as a “gold standard,” a morphological method was found to be superior to landmark-based estimates.^{1} Our method, applied to unipolar as opposed to bipolar electrograms, incorporated a priori knowledge of electrogram morphology into a signal-processing algorithm, making the activation time estimates accurate and robust in the presence of multiple inflections from fractionated electrograms.

Although the present study found that amplitude and duration did not vary significantly between normal and infarcted tissue, in a previous study using bipolar epicardial recordings, Richards et al^{13} demonstrated that electrograms from infarcted regions had longer duration and lower amplitude than did those from normal regions (*P*<.05). Our results may differ because of the nature of unipolar compared with bipolar recordings. However, the recording electrodes used by Richards et al were specifically placed over infarcted tissue in a “region with broad fractionated electrograms.” Areas of the myocardium that were classified as infarcted in the present study may give rise to electrograms that are less fractionated than those recorded by Richards et al, suggesting that amplitude and duration are less sensitive to damage due to myocardial infarction.

Maximum amplitude and maximum slope (minimum derivative) have been used previously to discriminate between normal and infarcted tissue.^{17} The results of the present study are in agreement, showing reductions in mean amplitude and minimum derivative. However, comparison of the receiver operating curves indicates that deconvolution is a more efficient detection algorithm, producing fewer false positives than previous metrics.

### Limitations

The results of this study tested activation time estimation by use of simulated tissues that were anatomically detailed, incorporating anisotropy and coupling heterogeneity. Because the test set used to evaluate the accuracy of activation time estimation was itself generated by a computer model, the results depend on the accuracy of the underlying model. Although more recent ion kinetic models exist,^{8} they differ from the Beeler-Reuter model used in this study principally in their calculations of currents controlling the recovery period. Because the present study considers only activation, the results should be independent of these differences. Although our approach assumes uniform extracellular space resistivity, a bidomain model has been used by some researchers.^{18} However, we have previously shown that our model of “fractally uncoupled” myocardium realistically generates electrograms as might be recorded from regions of infarction.

Although the deconvolution method is more accurate than other methods of activation time estimation in this computer study and has superior performance for tissue characterization in vivo, limitations may arise because of the underlying assumptions in the convolution equation. The present equations assume a point electrode at a known height in an unbounded conducting medium. For the in vivo recordings, the space between the electrode and the epicardial surface was occupied by conducting fluid, but the electrode was mounted with an insulated backing. The recording medium was also confined by the three-dimensional structure of the myocardium. Confinement of the conducting medium may affect the impedance between individual myocytes and the electrode. In addition, the electrode was not a point but a larger surface pressed against the epicardium. In vivo recordings may also be affected by potentials from regions not coupled to the recording site. Although these factors could affect the accuracy of deconvolution applied in vivo, the small fraction of false positives in our results indicates good agreement between calculated and observed electrograms. Geselowitz and coworkers^{19} calculated electrograms arising from point electrodes recording uniform, constant-velocity activation and also reported good agreement with measured potentials from canine ventricular muscle.^{20} If electrograms are more substantially affected by other electrode configurations, future deconvolutions could use volume-conductor equations for an electrode of finite size, which have been calculated previously.^{21} Modification of the volume-conductor equations and reformulation may improve the results of the algorithm, but the fundamental utility of the convolution equation to express a priori knowledge of electrogram generation has been shown.

In addition to reformulation of the volume conductor, the results of this study suggest that the deconvolution algorithm may be expanded for improved tissue characterization. The deconvolution algorithm detects tissue substrate using unexplained power in observed electrograms. If the assumed normal electrogram for the deconvolution cannot sufficiently explain the power contained in electrograms arising from normal tissue, the resulting unexplained power increases the probability of false positives. The electrogram calculations for the convolution equation assume that activation is uniform and has constant velocity, which results in a symmetrical, biphasic electrogram. Although this assumption fit many of the observed electrograms from normal tissue, a substantial number of electrograms were asymmetrical and, in extreme cases, monophasic. In a previous study, similar electrogram morphologies were observed when impulse conduction exhibited curvature or variations in velocity.^{15} Expansion of the deconvolution algorithm to include curvature and acceleration may therefore improve tissue characterization.

The present study examines the ability of deconvolution to detect regions of infarction identified by scintigraphic images. Thallium has been shown to be an excellent marker of viable myocardium. Reduced thallium uptake, particularly after injection at rest without ongoing myocardial ischemia, correlates with infarcted or scarred myocardium determined histologically.^{22} The results of the present study should be comparable for detecting histologically determined regions of infarction.

### Future Implications and Conclusions

The present study demonstrates successful application of deconvolution, a powerful signal-processing technique. Although underlying assumptions were required, our results indicate that these assumptions are compatible with in vivo recordings. Time-frequency techniques,^{23} such as wavelet analysis, may be used to further examine the structure of fractionated electrograms and to improve clinical electrogram interpretation. A second-order deconvolution could estimate local activation time from bipolar electrograms, canceling distant events, which may result in better recognition of local activation.^{3} In other disciplines, more complex deconvolutions have been used to successfully resolve multiple waveforms, thereby solving inverse problems. For example, in geophysical studies analyzing the echo of seismic waves, deconvolution resolves multiple echo waveforms and produces detailed images of underground rock formations.^{24} The accurate description of the velocity and distance to the electrode for each of several activation waves would be of great utility as an automated method to describe asynchronous activation of strands of myocardium, in a manner similar to that proposed by Spach et al.^{25}

In addition to demonstrating the feasibility of research with a novel and powerful signal-processing technique, our results provide a method to improve current clinical techniques of electrogram interpretation. It is not uncommon for clinical activation maps to be plotted with a resolution of 10 ms. Our results suggest that the current accuracy of the underlying activation time estimates is on the order of 10 ms. With its increased accuracy, deconvolution could be used to generate higher-resolution, more accurate activation maps. Deconvolution also shows promise to facilitate clinical studies by minimizing the physician time required to manually mark activation time estimates and the number of false positives encountered during the localization of infarcted tissue. However, further studies are needed to validate the use of deconvolution during clinical electrophysiological studies. Substantial in vivo validation will be required before these signal-processing techniques so useful in other disciplines can be generally applied to cardiac electrophysiology.

## Appendix

### Volume-Conductor Equation Expressed as a Convolution

An electrogram results from the superposition of the transmembrane current of each myocyte, I_{M,myo}(t). For an electrogram, E(t), recorded from a point electrode in an unbounded, homogeneous, isotropic, conducting medium, this superposition is expressed by the following volume-conductor equation^{9} :E(t)|<|=|>||<|\sum_|<|myo|>||>| \frac|<||<|\rho|>||>||<|4|<|\pi|>||<|\vert|>|r|<|\bar|>|_|<|myo|>||<|\vert|>||>| I_|<|M,myo|>|(t)where ρ is the conductivity of the extracellular space and is the distance from each myocyte to the electrode.

Formulating the weighted sum in Equation 1 as a convolution requires the assumption of a linear, time-invariant system. Specifically, the transmembrane current, I_{M,myo}(t), of any myocyte is determined uniquely by the time shift, d_{myo}, and a stereotypical transmembrane current, I_{M}(t). Thus, I_{M,myo}(t)=I_{M}(t−d_{myo}).

The Beeler-Reuter model of a single myocyte was used to calculate the discrete time stereotypical transmembrane current, I_{M}(nΔt).^{7}

In discrete time, Equation 1 can now be expressed as a convolution.E(n|<|\Delta|>|t)|<|=|>|I_|<|M|>|(n|<|\Delta|>|t)|<|\ast|>|H(n|<|\Delta|>|t)|<|=|>||<|\sum_|<|q|>||>| H(q|<|\Delta|>|t)I_|<|M|>||<|[|>|(n|<|-|>|q)|<|\Delta|>|t|<|]|>|where * indicates convolution.

H(nΔt) is expressed as a function of local activation time. Furthermore, H(nΔt) is constructed with an assumed “normal” activation pattern: uniform, constant velocity (Fig 5⇓). Myocytes are arranged in a two-dimensional sheet defined by *x,y* axes intersecting under a point electrode at height h. The distance from the electrode to a myocyte at location (*x,y*) is .

We define T_{a} as the local activation time when the myocyte directly under the electrode produces an action potential. All myocytes activating with a time shift of n samples relative to the myocyte directly under the electrode are located in a row such that x=VΔt(n−T_{a}), where V is the conduction velocity and Δt is the discrete sample time. These assumptions and reformulations having been made, the volume conductor equation (Equation 1) can be recast as the following convolution:E(n|<|\Delta|>|t)|<|=|>|I_|<|M|>|(n|<|\Delta|>|t)|<|\ast|>|H(n|<|\Delta|>|t)|<|=|>|I_|<|M|>|(n|<|\Delta|>|t)|<|\ast|>|\left(|<|\sum_|<|y|<|=|>||<|-|>||<|\infty|>||>|^|<||<|\infty|>||>||>|\frac|<||<|\rho|>||>||<|4|<|\pi|>|\sqrt|<|h^|<|2|>||<|+|>||<|[|>|V|<|\Delta|>|t(n|<|-|>|T_|<|a|>|)|<|]|>|^|<|2|>||<|+|>|y^|<|2|>||>||>|\right)

### Deconvolution by Iterative Minimization

With the convolution equation (Equation 3), the parameters h, V, and T_{a} fully describe an extracellular electrogram resulting from uniform, constant-velocity activation. The deconvolution was designed to minimize the mean-squares difference between the convolution equation and the observed electrogram, E_{0}(nΔt):|<|\sum_|<|n|>||>|\left[E_|<|0|>|(n|<|\Delta|>|t)|<|-|>|I_|<|M|>|(n|<|\Delta|>|t)|<|\ast|>|\left(|<|\sum_|<|y|<|=|>||<|-|>||<|\infty|>||>|^|<||<|\infty|>||>||>|\frac|<||<|\rho|>||>||<|4|<|\pi|>|\sqrt|<|h^|<|2|>||<|+|>||<|[|>|V|<|\Delta|>|t(n|<|-|>|T_|<|a|>|)|<|]|>|^|<|2|>||<|+|>|y^|<|2|>||>||>|\right)\right]^|<|2|>|

Physiological considerations limit the possible values of V to a narrow range, so we use a constrained minimization algorithm. For each value of V, beginning with V=0.5 m/s (the velocity observed in normal myocardium^{26} ), a matched filter^{27} is used to calculate the cross-correlation of E_{0}(n) and E(n), and the peak time is used as the value of T. For the best pair of V and T, we perform a second minimization using parabolic interpolation^{28} to determine the values of V and T that minimize the mean-squares difference between the convolution equation and the observed electrogram.

The deconvolution algorithm provides a metric for electrogram morphology. The fraction of the power of the observed electrogram that is unexplained by uniform, constant-velocity activation is f_{U}.f_|<|U|>||<|=|>|\frac|<||<|\sum_|<|n|>||>|\left[E_|<|0|>|(n|<|\Delta|>|t)|<|-|>|I_|<|M|>|(n|<|\Delta|>|t)|<|\ast|>|\left(|<|\sum_|<|y|<|=|>||<|-|>||<|\infty|>||>|^|<||<|\infty|>||>||>|\frac|<||<|\rho|>||>||<|4|<|\pi|>|\sqrt|<|h^|<|2|>||<|+|>||<|[|>|V|<|\Delta|>|t(n|<|-|>|T_|<|a|>|)|<|]|>|^|<|2|>||<|+|>|y^|<|2|>||>||>|\right)\right]^|<|2|>||>||<||<|\sum_|<|n|>||>||<|[|>|E_|<|0|>|(n|<|\Delta|>|t)|<|]|>|^|<|2|>||>|

## Acknowledgments

This work was supported by National Institutes of Health grants HL-25847 and HL-45664, by American Heart Association Grant-in-Aid 9000914, and by the National Science Foundation/San Diego Supercomputer Center. Dr Ellis was initially supported by a National Science Foundation graduate fellowship. This work was done during the tenure of a research fellowship from the American Heart Association, California Affiliate. The authors are grateful to Carol A. Stillson and Michael Chin for their assistance during the animal experiments and to Bridget O'Meara for her assistance in preparing the manuscript. The authors wish to acknowledge the electrophysiology fellows of the University of California, San Francisco, who reviewed the electrograms.

- Received April 22, 1996.
- Revision received June 11, 1996.
- Accepted June 17, 1996.

- Copyright © 1996 by American Heart Association

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- Deconvolution: A Novel Signal Processing Approach for Determining Activation Time From Fractionated Electrograms and Detecting Infarcted TissueWillard S. Ellis, Susan J. Eisenberg, David M. Auslander, Michael W. Dae, Avideh Zakhor and Michael D. LeshCirculation. 1996;94:2633-2640, originally published November 15, 1996https://doi.org/10.1161/01.CIR.94.10.2633
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- Deconvolution: A Novel Signal Processing Approach for Determining Activation Time From Fractionated Electrograms and Detecting Infarcted TissueWillard S. Ellis, Susan J. Eisenberg, David M. Auslander, Michael W. Dae, Avideh Zakhor and Michael D. LeshCirculation. 1996;94:2633-2640, originally published November 15, 1996https://doi.org/10.1161/01.CIR.94.10.2633