Reconstruction of instantaneous phase of unipolar ...

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Reconstruction of instantaneous phase of unipolar atrial contact electrogram using a concept of sinusoidal recomposition and Hilbert transform Pawel Kuklik, Stef Zeemering, Bart Maesen, Jos Maessen, Harry J. Crijns, Sander Verheule, Anand N. Ganesan and Ulrich Schotten 

Abstract—The Hilbert transform has been used to characterize wave propagation and detect phase singularities during cardiac fibrillation. Two mapping modalities have been used: optical mapping (used to map atria and ventricles) and contact electrode mapping (used only to map ventricles). Due to specific morphology of atrial electrograms, phase reconstruction of contact electrograms in the atria is challenging and has not been investigated in detail. Here, we explore the properties of Hilbert transform applied to unipolar epicardial electrograms and devise a method for robust phase reconstruction using the Hilbert transform. We applied the Hilbert transform to idealized unipolar signals obtained from analytical approach and to electrograms recorded in humans. We investigated effects of deflection morphology on instantaneous phase. Application of the Hilbert transform to unipolar electrograms demonstrated sensitivity of reconstructed phase to the type of deflection morphology (uni- or bi-phasic), the ratio of R and S waves and presence of the noise. In order to perform a robust phase reconstruction, we propose a signal transformation based on the recomposition of the electrogram from sinusoidal wavelets with amplitudes proportional to the negative slope of the electrogram. Application of the sinusoidal recomposition transformation prior to application of the Hilbert transform alleviates the effect of confounding features on reconstructed phase. Index Terms— Biomedical signal processing, Phase estimation, Fibrillation, Cardiology.

I. INTRODUCTION

C

ARDIAC fibrillation is a complex phenomenon requiring sophisticated approaches to characterize wave propagation dynamics. One of the suggested approaches is reconstruction of the instantaneous phase of local electrical activity introduced by A.T. Winfree [1]. Instantaneous phase denotes at which point of the local cycle a given oscillatory process is. The spatial distribution of the phase across the mapped field provides insight into wave propagation dynamics and enables localization of phase singularities corresponding with the tips of the functional reentry waves (so called “rotors”). For review of the concept of phase in cardiac electrophysiology, see a paper by Umapathy et al. [2]. Phase reconstruction has been introduced as a technique to understand complex spatiotemporal dynamics of cardiac fibrillation in experimental models and humans. Phase reconstruction has been first applied to transmembrane signals using both temporal encoding [3] and state space encoding using time-delay embedding [4] followed by applications to contact electrograms [2, 5]. More recently, phase analysis based on Hilbert transform has been introduced to calculate instantaneous phase and to facilitate mapping of cardiac fibrillation in humans [5-8]. However, to our knowledge, with an exception of a study by Shors et al. [7], all studies focus on analysis of ventricular signals leaving application of Hilbert transform to atrial electrograms unaddressed. Application of the Hilbert transform to ventricular unipolar electrograms during ventricular tachycardia has been successful because of the sinusoidal morphology of the signal (lack of significant iso-electric intervals between consecutive deflections). Atrial electrograms, because of the smaller volume of the atrial musculature, are characterized by long iso-electric intervals between consecutive deflections. We hypothesized that the presence of such intervals could result in a higher sensitivity of phase reconstruction to noise in the signal – even small amplitude variations may result in artifacts in reconstructed phase. In addition, we further hypothesized that morphology features of atrial unipolar electrogram could significantly affect phase reconstruction. In this study, we aim to explore basic properties of Hilbert transform applied to unipolar electrograms and suggest a signal preprocessing method leading to a robust phase

Pawel Kuklik is at Department of Physiology, Maastricht University Medical Center, Maastricht, The Netherlands and University Heart Center, Department of Cardiology and Electrophysiology, University Hospital Hamburg-Eppendorf, Hamburg, Germany (e-mail: pawel.kuklik@ maastrichtuniversity.nl). Bart Maesen (email: [email protected]) and Jos Maessen (email: [email protected]) are at Department of Cardio-thoracic Surgery, Maastricht University Medical Centre. Harry Crijns (email: [email protected]) is at Department of Cardiology, Maastricht University Medical Centre. Stef Zeemering (e-mail: [email protected]), Sander Verheule (email: [email protected]) and Ulrich Schotten (email: [email protected]) are at Department of Physiology, Maastricht University, Maastricht, The Netherlands. Anand N. Ganesan is at Centre for Heart Rhythm Disorders (CHRD), South Australian Health and Medical Research Institute (SAHMRI), University of Adelaide and Royal Adelaide Hospital, Adelaide, Australia (email: [email protected]). Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to [email protected].

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reconstruction of atrial electrograms. II. METHODS A. Definition of Hilbert transform and instantaneous phase Hilbert transform is a linear operator transforming a function u(t) into a function H(u)(t):

1



u( t ) (1) d  t  where P is the Cauchy principal value of the integral. The Hilbert transform has found a widespread use, especially in signal processing applications where it is used to obtain an analytic representation of the signal, allowing calculation of instantaneous frequency and phase [9]. Phase is defined as an angle between the original signal and the Hilbert transform of the signal. Exact formulations of the phase vary between studies. In our study, we define instantaneous phase as follows: H (u )(t ) 

P



  (u (t )  u*)    H (u )(t )  u * 

 (t )  arctan

(2)

where u* sets the origin of the phase plane with respect to which phase is computed.

Fig. 1 Definition of the instantaneous phase using the motion of an object along the circle as an example (a). Position of the object along Y axis (b). Corresponding instantaneous phase (c). Completion of one full rotation corresponds with a change of the phase from a minimum to maximum value. Instantaneous phase increases monotonically within consecutive cycles of oscillation, reverting to a base value after completion of each cycle. This property results in a “saw tooth” appearance of the instantaneous phase plot. An illustration of the definition of the phase is shown in Fig. 1, using as an example a motion of an object with constant velocity along circular trajectory (Fig. 1.a). The position of such object along y-axis is a sinusoid (Fig. 1.b) in which one completion of the rotation corresponds with one period of the sinusoid. Instantaneous phase of this sinusoid is shown in Fig. 1.c. Each period of the sinusoid corresponds with one segment in the phase “saw tooth” plot. B. Construction of idealized unipolar signals In order to assess effects of specific features of the unipolar deflection morphology, we constructed idealized signals using analytic considerations. This approach enabled us to control

the morphology of the electrograms (such as the amplitude of R and S waves and the level of noise) and assess their effect on reconstructed phase. Details of the mathematical approach are presented in Appendix A. C. AF electrograms acquisition Unipolar electrograms were collected in forty patients (12 pts with Paroxysmal AF (PAF) and 9 with Persistent AF (PersAF)) during cardiac surgery. Rectangular plaques (16x16 electrodes, 1.5 mm inter-electrode spacing) were placed on the epicardial surface of the atria. The plaque was positioned on the left atrial posterior wall and the right atrial free wall. Atria were mapped sequentially with a 256-channel computerized mapping system (bandwidth, 0.5 to 500 Hz; sampling rate 1 kHz, resolution 12 bits). Far-field QRS complexes were subtracted from the unipolar fibrillation electrograms by a single beat cancellation method [10, 11]. After exclusion of poor contact recordings, we included 23 recordings in PAF group (11 in left and 12 in right atria) and 17 recordings in PersAF group (8 in left and 9 in right atria). D. Calculation of Hilbert transform and instantaneous phase All signal processing was conducted in MATLAB (version 7.12, Mathworks Inc., Natick, MA, USA). The Hilbert transform was calculated using the ‘hilbert’ unction. Phase was calculated using Eq. 2 using four-quadrant inverse tangent function 'atan2'. E. Electrogram transformation: Sinusoidal Recomposition We propose a transformation of atrial unipolar electrograms that should be applied prior to application of the Hilbert transform. We designed this transformation based on following assumptions: (i) Due to the mathematical properties of the Hilbert transform, phase reconstruction of the instantaneous phase performs best in case of a sinusoidal morphology of the signal. (ii) Local activity in unipolar signals related with a beginning of new cycle is proportional to the signal slope. Based on these considerations, we proposed the following transformation: (i) The transformed signal is a sum of sinusoidal waves of one period length (called ‘sinusoidal wavelets’ below). (ii) For each time point of the original signal, one sinusoidal wavelet is created. (iii) The amplitude of the sinusoidal wavelet is proportional to the slope of the signal at a given time point. (iv) A wavelet is generated only if a derivative of the signal is negative (since a negative slope in unipolar electrogram corresponds with the passing of a wave). (v) The period of the sinusoidal wavelet is equal to the mean cycle length of the electrogram derived

Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to [email protected].

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from dominant frequency of given electrogram [12]. These steps can be summarized in a following equation: t

w( t ) 

T 2



T t 2

dt ' sin( t  t ' )

dv dt '

1  sign(

dv ) dt '

(3)

2

where w(t) is a transformed signal, v(t) is an original electrogram and T is a mean cycle length of the original electrogram (derived from dominant frequency of the electrogram) and sign() is the signum function. A schematic explaining the construction of the transformed signal is shown in Fig. 2. Examples of phase reconstruction are shown in Fig. 3. Since we use sinusoidal wavelets to construct transformed signals, we term this transformation a “sinusoidal recomposition”. The phase of the recomposed signal was calculated using Equation 2.

Fig. 2 Schematic of the “sinusoidal recomposition” transformation. Original signal (a). Sinusoidal wavelets are created for each time point of the original signal (signal was down-sampled for clarity) (b). Recomposed signal is a sum of all sinusoidal wavelets (c). Corresponding instantaneous phase (d).

obtained applying Hilbert transform to original electrogram, electrogram after sinusoidal recomposition, phase of the recomposed electrogram. Dots in original electrogram denote positions of intrinsic deflections calculated using template matching. Dots in instantaneous phase plots denote positions of phase inversions (time point at which phase changes from π to - π). Since the recomposed signal is a sum of sinusoidal wavelets with a mean value equal to zero and wavelets of the greatest amplitude are clustered around the negative slope of the local deflection, the resultant recomposed signal also has a sinusoidal morphology oscillating around zero value. Based on this consideration, we set u* (origin of the phase space with respect to which phase is computed, see Eq. 2) to zero. We used a specific definition of phase presented in Eq.2 in order to obtain timing of the phase inversion (time point at which phase changes value from maximum to minimum denoting a beginning of a new cycle) coinciding with the timing of the local deflection in the electrogram. Formulation in Eq.2 results, in case of a sinusoidal signal, in phase inversion occurring at position of the maximum negative slope of sinusoid. Since during sinusoidal recomposition individual sinusoidal wavelets are triggered according to the timing of the negative slope in electrogram, this will result in timings of the phase inversions centered at timings of the local deflections in electrogram. F. Study design In order to assess the properties of the Hilbert transform applied to contact electrograms, we employed two approaches: (i) analysis of idealized morphologies obtained from mathematical modeling of unipolar electrograms and (ii) analysis of contact unipolar electrograms obtained during epicardial mapping of human AF. Taking advantage of the customizability of idealized electrograms, we investigated how instantaneous phase is affected by: (i) monophasic and biphasic morphology of the deflection, (ii) R vs. S wave ratio of the unipolar electrogram and (iii) noise added to the signal. We used contact electrograms recorded during human AF to: (i) assess agreement between estimation of mean and standard deviation of cycle length and (ii) investigate a possibility of using timings of phase inversions to assess wave propagation complexity. We compared number of reconstructed waves and their mean size based on timings of phase inversions vs. timings based on intrinsic deflections (using previously validated wave reconstruction algorithm [11]). III. RESULTS

Fig. 3 Instantaneous phase of atrial unipolar electrograms recorded in paroxysmal (left) and persistent (right) atrial fibrillation. From top to bottom: original signal, phase

A. Instantaneous phase of monophasic and biphasic deflection morphology Ideally, a deflection in a unipolar electrogram corresponding with local conduction event should be biphasic, i.e. contain two peaks (R and S waves; see Fig. 2.a) [13]. However, in experimental recordings, a monophasic morphology of deflections is often observed (see Fig. 3.b). Instantaneous phase in case of the monophasic and biphasic deflection morphology is shown in Fig. 4. In the case of biphasic deflection, phase morphology obtained by application of


TBME-00668-2014 Hilbert transform directly to electrogram is distorted, showing a sudden increase just prior to end of the cycle (see red curve in bottom plot in Fig. 4.b). This effect is also present in case of monophasic morphology (Fig. 4.a) but much less pronounced. Phase inversion of the recomposed signal (green line) is located in the middle of down slope and has triangular morphology in both cases.

Fig. 4 Instantaneous phase in case of monophasic (a) and biphasic (b) deflection morphology. B. Effect of R/S amplitude asymmetry in unipolar electrogram In order to assess the effect of R/S amplitude asymmetry in unipolar electrograms on phase reconstruction, we multiplied left and/or right portion of the deflection by a constant value, which resulted in asymmetric R and S waves (see blue signals in Fig. 5.a and Fig. 5.b for the definition of R and S waves). In general, different ratios of R and S waves resulted in distinct shapes of reconstructed phase (see red curves in Fig. 5.a).

4 Closer inspection of the relation of the phase to the morphology of the electrogram (blue) shows that the general shape of the phase follows the shape of the electrogram: it is fully convex if electrogram is convex (see Fig. 5.a, R=0), fully concave in case of a concave electrogram for S=0 (see Fig. 5.a, S=0) and it exhibits both a convex and concave shape in between the R=0 and S=0 extremes (e.g. see Fig. 5.a, S=2/3R: first and second half of the phase (red) between consecutive deflections). Phase reconstruction based on recomposed electrogram was not affected by R/S ratio (see see Fig. 5.a, green curves). C. Effect of noise added to the unipolar electrogram In order to investigate the effect of noise on phase reconstruction, we added low- or high-frequency noise to the signal. High-frequency noise was defined as a Gaussian noise of a specified standard deviation σ. Low-frequency noise was also obtained from Gaussian noise, however after applying a low pass filter at a specified cutoff frequency. Since we aim at qualitative exploration, we selected just one frequency cutoff, 40 Hz, resulting in an electrogram fractionation resembling morphology observed in experimental setting. The amplitude of the noise was controlled by the standard deviation of the noise σ and expressed as a proportion of the R-wave amplitude. Results are shown in Fig. 6 and Fig. 7. In case of high-frequency noise, with an increasing amplitude, additional phase inversions occur with a significant increase in number at σ>0.01|R| (see Fig. 6).

Fig. 6 Effect of Gaussian noise added to the signal on the reconstruction of instantaneous phase. Electrogram in blue. Instantaneous phase of the original signal in red. Instantaneous phase of the recomposed signal in green. For σ> 0.01|R|, additional phase inversion occurrences appear.

Fig. 5 Effect of varying ratio of R and S waves amplitude on morphology of reconstructed phase (a). Electrogram in blue. Instantaneous phase of the original signal in red. Instantaneous phase of the recomposed signal in green. Definition of R and S amplitude (b).

Low-frequency noise also caused an occurrence of additional phase inversions (for σ>0.2|R|), however with a broader morphology and fewer in number (see Fig. 7). Phase reconstruction based on recomposed electrograms did not exhibit additional phase inversions with increasing noise. Instead, for σ>0.8|R| a timing of the phase inversion began to be affected (see green line in Fig. 7 with especially pronounced shift of first deflection for σ=1.4|R|). D. Assessment of local cycle length using phase reconstruction Since the timing of a phase inversion corresponds with a position of intrinsic deflection in the electrogram (see Fig. 3),


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we explored the possibility of using timings of the phase inversions to reconstruct local cycle length and compare to previously validated assessment of cycle length based on timing of intrinsic deflections (see Fig. 3 for both definitions of cycle lengths). We used algorithms previously developed and validated in our group to determine timings of the intrinsic deflections [11]. We compared the mean and SD of the cycle length per recorded episode (23 paroxysmal and 17 persistent AF recordings) using both methods. Results are shown in Fig. 8. Both methods show very good agreement with R2=0.99 in PAF group and R2=0.76 in PersAF group for mean cycle lengths, and R2=0.90 in PAF group and R2=0.92 in PersAF group for SD of cycle lengths.

Fig. 8 Comparison of mean local cycle length in PAF group (a) and in PersAF group (c), standard deviation of local cycle length in PAF group (b) and in PersAF group (d) assessed using phase reconstruction vs. intrinsic deflection annotation.

Fig. 7 Effect of low-frequency noise added to the unipolar signal on instantaneous phase reconstruction. Electrogram in blue. Instantaneous phase of the original signal in red. Instantaneous phase of the recomposed signal in green. E. Reconstruction of wave propagation using instantaneous phase We used timings of phase inversions as a starting point of an individual activation wave reconstruction. Wave reconstruction was performed using an algorithm previously developed and validated in our group [11]. We compared the number and mean size of reconstructed waves using timings of intrinsic deflections vs. timings corresponding with phase inversions. The mean number of the waves and mean wave size were estimated per AF episode (23 paroxysmal AF and 17 persistent AF recordings). An example of individual waves reconstructed using both approaches is shown in Fig. 9.a-b. The total number of waves was correlated with R2=0.95 in PAF group and R2=0.94 in PersAF group (Fig. 9.c and Fig. 9.e respectively). Mean wave size correlated with R2=0.68 in PAF group and R2=0.96 in PersAF group (Fig. 9.d and Fig. 9.f respectively). Fig. 9 Example identification of individual waves (example case from PersAF group) based on clustering electrodes with similar activation times (see ref. [11]) using timings of local Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to [email protected].

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deflections in electrogram (a) and using timings of phase inversions (b). Each color denotes an individual wave at given time point (whole square corresponds with 16x16 electrodes plaque). Number of individual waves and their mean size was calculated for each AF recording using both methods. Number of reconstructed waves for PAF group (c) and PersAF group (e). Mean wave size for PAF group (d) and PersAF group (f). IV. DISCUSSION In this manuscript we aimed to achieve two goals: (i) explore the properties of instantaneous phase reconstruction using the Hilbert transform applied to atrial unipolar contact electrograms and (ii) devise a method for robust phase reconstruction of such electrograms. With respect to first goal, we applied the Hilbert transform to idealized electrograms derived from analytic considerations and demonstrated that the reconstructed phase is influenced by electrogram morphology (mono-/biphasic shape, R/S wave ratio) and is very sensitive to high frequency noise added to the signal (which is very relevant in case of atrial electrograms characterized by long isoelectric line between consecutive deflections during which noise can be interpreted as a new deflection). Those properties make application of the Hilbert transform directly to electrograms recorded in experimental settings inappropriate due to great variability of above-mentioned factors and presence of noise in broad range of frequencies. In order to overcome this difficulty, we devised a signal preprocessing method (which we coined “sinusoidal recomposition”) that should be applied prior to application of the Hilbert transform. Phase reconstruction of electrograms after sinusoidal recomposition showed large robustness and little dependence on all confounding factors we identified (mono-/biphasic shape, R/S ratio variability, noise added to signal) resulting in the expected triangular shape of the phase morphology. We applied our method to electrograms recorded during AF in humans and demonstrated its ability to robustly estimate the mean and SD of cycle length, the number of fibrillation waves and their mean size. There are two important factors related with phase reconstruction: (1) the timing of the phase inversion and (2) the morphology of the reconstructed phase. Correct timing of the phase inversions (with respect to a feature in the signal related with change of the cycle) is important since it defines starting and ending point of each cycle. The exact position of the phase inversion depends on the definition of phase (see Eq.2). We chose the definition presented in Eq.2 since it results in timing of the phase inversion simultaneous with timing of the greatest negative-slope in electrogram. With respect to the morphology of the reconstructed phase, the ideal phase reconstruction should result in a “saw-tooth” shape. Such morphology denotes linear change of the phase between consecutive cycles. A linear phase change is especially important in assessment of coherence and synchronization between different signals. Such measures rely on the difference in phase between two electrograms, therefore each deviation from linear morphology (e.g. related with R/S asymmetry as shown in Fig. 5) will confound the obtained measures. We found that phase computed on recomposed signals robustly exhibits linear shape in presence of such

confounding factors as R/S asymmetry, morphology type and added noise. The question whether such morphology of the phase properly reflects atrial repolarization and allows detection of phase singularities will be addressed in future work. We did not perform any form of signal filtering prior to the application of the Hilbert transform. This may be viewed as a failure to explore the potential benefits proper filtering may bring to the quality of phase reconstruction. However, based on literature review and our own experience, we argue that signal filtering does not have the potential to remedy and compensate for the effects of features specific to atrial electrograms in their extensive range of amplitudes and corresponding frequencies encountered in practice (especially in case of electrograms recorded during atrial fibrillation). Additionally, there is no consensus regarding proper filtering of atrial unipolar electrograms in the context of phase reconstruction and it would be arbitrary to devise a method to compare against. A notable development was suggested by Bray and Wikswo in [6] where authors applied pseudoEmpirical Mode Decomposition (pEMD) to optical signals during cardiac fibrillation in order to properly reconstruct phase and identify phase singularities. However, the pEMD approach assumes an oscillation related with the underlying electrical activity is reflected in the signal amplitude. This condition is true in case of transmembrane voltage during fibrillation, but not in case of atrial unipolar electrograms, where local activity related with a passing wave is reflected in sudden change of electrogram slope, which may be of smaller amplitude than far-field components. Phase singularities in cardiac fibrillation (corresponding with a tip of rotating wave) have been a subject of significant research effort due to their potential role in initiation and maintenance of fibrillation. Their properties and methods of localization became an interest of clinical and basic electrophysiology [14]. Recently, Narayan et al. applied the Hilbert transform to unipolar signals during atrial fibrillation (AF) in order to identify the core of reentrant waves and subsequently terminate arrhythmia by ablation at those sites, [8, 15] reporting very encouraging results. Unfortunately, details of the signal processing used in these studies were not disclosed, therefore hampering their replication and validation by other groups. We hope our contribution addresses this need by providing a simple and robust method of phase mapping using contact mapping of atrial fibrillation electrograms. The main assumption of the proposed method of phase reconstruction is local stability of the underlying cycle length. In the current approach, we used Fast Fourier Transform (see [12] for details) applied to the whole electrogram to estimate dominant frequency, which was then used to derive the length of the sinusoidal wavelet used in sinusoidal recomposition. This assumption may not hold in case of AF with greatly variable frequencies. Several approaches may help to overcome this problem through adapting the length of sinusoidal wavelet to local frequency such as computing FFT in sliding window or application of wavelet transform [16]. The presented method of phase reconstruction could be applied to other types of signals reflecting oscillatory processes beyond unipolar electrograms recorded during atrial fibrillation. There are two conditions such signal should fulfill:


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(i) relatively stable frequency and (ii) existence of specific features corresponding with transition between cycles (such as presence of a large negative slope in unipolar electrogram related with passing wave). A sinusoidal recomposition algorithm could be then adjusted to generate sinusoidal wavelets with amplitude proportional to quantified magnitude of given “change of cycle” feature in the signal. For example, in case of bipolar electrograms it could be an electrogram amplitude (since the spike in a bipolar electrograms is related with passing wave). APPENDIX A: CALCULATION OF THE IDEALIZED UNIPOLAR

EUTRAF, No 261057), and CTMM COHFAR. REFERENCES [1]

[2]

[3]

[4]

ELECTROGRAM

A wavefront of the planar activation wave in Cartesian coordinate system propagating along X axis can be described in parametric form as:

x( p; t )  wt y ( p; t )  p; p   ,  

[5]

(A.1)

[6]

where w is a conduction velocity, t denotes time and p is a parameter. In general, the unipolar electrogram at given point in space is given by:

[7]

 u( re , t )  

 r v( r, t )  3 dV 4 e V r 1

(A.2)

where σe is the extracellular conductance and v(r,t) is the transmembrane voltage distribution in volume V [17]. For simplicity we will assume a homogeneous medium and thus omit 1/4 σe from the equation. In our geometrical approach we assume that only wavefronts of the activation wave contributes to the unipolar electrogram (i.e only phase 0 of the action potential). Therefore, we can replace gradient of the voltage by a constant. Equation (2) then takes the form: p 

 [ x( p )  xe , y ( p )  y e )] Ads

p  

[ x( p )  xe , y ( p )  y e )]

u( xe , y e , t )   

3

(A.3)

where xe and ye are coordinates of the point at which unipolar  electrogram is calculated ( (0,0) in our case) and ds is an infinitesimal vector perpendicular to the wavefront. Parameter A denotes the contribution of a given portion of the wavefront toward unipolar electrogram. We assume this contribution to be constant along the wavefront (therefore A does not depend on parameter p). Since in case of a planar wave propagating in direction of X axis:  ds  [1,0]dp (A.4) Then, the equation for unipolar electrogram expressed using only parameter p is given by:

u ( t )   A

p 

p  

wt [wt , p]

3

dp

(A.5)

Above equation has been integrated in Matlab environment to obtain the morphology of an idealized, single unipolar deflection with a width controlled by the conduction velocity (parameter w).

[8]

[9] [10]

[11]

[12]

[13]

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[16]

[17]

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ACKNOWLEDGMENT This work was funded by the European Network for Translational Research in AF (FP7 collaborative project Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to [email protected].