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Recent studies have demonstrated conflicting mechanisms
underlying persistent atrial fibrillation (AF), with the spa-
tial resolution of data often cited as a potential reason for the
disagreement. The hierarchical model of AF states that dis-
turbances are sustained by drivers, in the form of rotors or
focal sources.1 Evidence for rotors as drivers of human AF is
inferred from termination through ablation of putative stable
rotor sites, mapped with basket catheters,2,3 as well as ablation
of regions with a high probability of transient rotors, identi-
fied using a noninvasive body surface mapping technology.4
Despite these data, the rotor paradigm is neither confirmed
nor universally accepted,5–7 with recent studies raising ques-
tions about the efficacy of rotor-targeted ablation.8,9 The con-
trasting multiple-wavelet hypothesis of AF, proposed by Moe
et al10 in the 1960s, states that AF is sustained by multiple,
self-perpetuating, randomly propagating activation wavelets.
This is supported by Allessie et al11 and de Groot et al12 who
found no evidence for the presence of stable focal sources or
rotors using a small high-resolution spoon-shaped mapping
device. Similarly, the Waldo laboratory found no evidence of
rotational activity using an epicardial electrode array (inter-
electrode spacing, 5.2–7.0 mm); in this case, AF was main-
tained by wavefronts from foci and breakthrough sites.13
These contradictory results have spawned intense debate5,7
with findings attributed to the divergent methods used. One
source of variation arises from differences in scale (global
versus regional) and electrode density and therefore spatial
resolution of the mapping techniques. A second difference is
the approach used to analyze fibrillatory wavefront dynam-
ics, using either phase mapping14 or activation time.12 Correct
Circ Arrhythm Electrophysiol is available at http://circep.ahajournals.org DOI: 10.1161/CIRCEP.116.004899
Original Article
Background—Recent studies have demonstrated conflicting mechanisms underlying atrial fibrillation (AF), with the spatial
resolution of data often cited as a potential reason for the disagreement. The purpose of this study was to investigate
whether the variation in spatial resolution of mapping may lead to misinterpretation of the underlying mechanism in
persistent AF.
Methods and Results—Simulations of rotors and focal sources were performed to estimate the minimum number of
recording points required to correctly identify the underlying AF mechanism. The effects of different data types (action
potentials and unipolar or bipolar electrograms) and rotor stability on resolution requirements were investigated. We also
determined the ability of clinically used endocardial catheters to identify AF mechanisms using clinically recorded and
simulated data. The spatial resolution required for correct identification of rotors and focal sources is a linear function
of spatial wavelength (the distance between wavefronts) of the arrhythmia. Rotor localization errors are larger for
electrogram data than for action potential data. Stationary rotors are more reliably identified compared with meandering
trajectories, for any given spatial resolution. All clinical high-resolution multipolar catheters are of sufficient resolution
to accurately detect and track rotors when placed over the rotor core although the low-resolution basket catheter is prone
to false detections and may incorrectly identify rotors that are not present.
Conclusions—The spatial resolution of AF data can significantly affect the interpretation of the underlying AF mechanism.
Therefore, the interpretation of human AF data must be taken in the context of the spatial resolution of the recordings.
(Circ Arrhythm Electrophysiol. 2017;10:e004899. DOI: 10.1161/CIRCEP.116.004899.)
Key Words: ablation techniques ◼ arrhythmias, cardiac ◼ atrial fibrillation
◼ computational modeling ◼ reentry ◼ rotor
Received December 20, 2016; accepted April 11, 2017.
From the ElectroCardioMaths Programme (C.H.R., C.D.C., N.A.Q., P.B.L., P.K., N.S.P., F.S.N.), and the Department of Bioengineering (J.H.T.), Imperial
College London, United Kingdom; IHU Liryc, Electrophysiology and Heart Modeling Institute, Fondation Bordeaux Université, Pessac-Bordeaux, France
(J.D.B., E.J.V.); and Université de Bordeaux, IMB, UMR 5251, Talence, France (J.D.B., E.J.V.).
*Drs Vigmond and Ng contributed equally to this work.
The Data Supplement is available at http://circep.ahajournals.org/lookup/suppl/doi:10.1161/CIRCEP.116.004899/-/DC1.
Correspondence to Nicholas S. Peters, MD, Imperial College London, 4th Floor Imperial Centre for Translational and Experimental Medicine,
Hammersmith Campus, Du Cane Rd, London W12 0NN, United Kingdom. E-mail n.peters@imperial.ac.uk
© 2017 The Authors. Circulation: Arrhythmia and Electrophysiology is published on behalf of the American Heart Association, Inc., by Wolters Kluwer
Health, Inc. This is an open access article under the terms of the Creative Commons Attribution Non-Commercial-NoDerivs License, which permits
use, distribution, and reproduction in any medium, provided that the original work is properly cited, the use is noncommercial, and no modifications or
adaptations are made.
Spatial Resolution Requirements for Accurate
Identification of Drivers of Atrial Fibrillation
Caroline H. Roney, PhD; Chris D. Cantwell, PhD; Jason D. Bayer, PhD;
Norman A. Qureshi, MRCP, PhD; Phang Boon Lim, MRCP, PhD; Jennifer H. Tweedy, PhD;
Prapa Kanagaratnam, PhD; Nicholas S. Peters, MD; Edward J. Vigmond, PhD*; Fu Siong Ng, MRCP, PhD*
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2 Roney et al Spatial Resolution Requirements During AF
interpretation of AF mechanisms is critical for effective diag-
nosis and delivery of ablation therapy.
In this study, we systematically investigated the hypothesis
that the variation in spatial resolution of mapping systems may
lead to misinterpretation of mechanism in persistent AF. We
determined, through computer simulation, the minimum reso-
lution required to accurately identify rotors and focal sources
and to avoid false detections, using unipolar and bipolar recon-
structed electrograms from 5 clinical catheter configurations.
These were compared against action potential data requirements
for computational modeling data. We considered stationary
versus meandering rotors. Finally, we compared clinical phase
maps and detected singularities for data measured during AF.
Methods
The Methods are briefly described here with full details in the Data
Supplement.
Simulation Data
We initially determined resolution requirements on a regularly
spaced 2-dimensional (2D) homogeneous grid for a stable rotor or
focal source, before testing on more complicated arrhythmias with
spatially varying activation and repolarization properties, realistic ge-
ometries, and catheter electrode arrangements.
Monodomain simulations of rotors and focal sources were per-
formed using the Courtemanche-Ramirez-Nattel human atrial cell
model, with changes representing electrical remodeling in AF.15 To
generate a physiological range of spatial wavelengths in a 10 cm×10
cm sheet, the conduction velocity (CV)16 and local atrial rate17,18 were
varied by modifying tissue diffusivity (0.0005, 0.001, and 0.0015
cm2/ms) and IK1 conductance (gK1; 0.09, 0.135, and 0.18 nS/pF), re-
sulting in CVs of 0.26, 0.36, and 0.43 m/s and action potential (AP)
durations of 121, 142, and 181 ms (considered as 9 combinations;
Tables I and II and Section 1.1.1 in the Data Supplement).
The effects of simulated data type and rotor stability were test-
ed using an atrial bilayer model.19,20 These simulations included
interstitial fibrosis as microstructural discontinuities,20 with distribu-
tions based on late-gadolinium intensity values from patients with
persistent AF21 used to infer probabilities for fibrosis inclusion in
the model, resulting in heterogeneous anisotropic conduction. For 1
simulation, areas of fibrosis also included reduced conductivity and
changes to the ionic properties. Unipolar electrograms were calculat-
ed 1 mm off the endocardium with bipolar electrograms calculated as
differences between paired unipoles with 4 mm spacing. Full model
details are given in Section 1.1.2 in the Data Supplement.
High-density catheters were simulated, including a circular
(Lasso), spiral (AFocus II), and 2 variations of a 5-spline (PentaRay)
catheter with different interelectrode spacings, all of diameter 2 cm.
Lower resolution basket catheters (median interelectrode spacing,
10.2 mm; lower quartile, 5.9 mm; upper quartile, 16.2 mm) were
simulated in an anatomically accurate human left atrial model for 30
seconds of AF for 2 parameter sets, corresponding to short (45.2 mm)
and long (75.2 mm) wavelength activity (Section 1.1.3 in the Data
Supplement).
Clinical Data
All data were obtained with informed consent under ethical ap-
proval from the UK Health Research Authority Ref 13/LO1169.
Electrograms and electrode locations were recorded during AF from
the left atrium of 11 patients (6–17 catheter recording locations per
patient; 127 total) at the beginning of ablation procedures, using mul-
tipolar AFocus II catheters and the Ensite Velocity electroanatomic
mapping system (St Jude Medical, Inc). Unipolar and bipolar electro-
grams were recorded for 16 to 106 seconds (mean, 34 seconds). To in-
vestigate the effects of resolution on phase singularity (PS) detection,
analysis was performed for random subsets of 4 to 19 electrograms,
and the number of missing and false PS detections were calculated.
Identifying Rotors and Focal Sources
Figure 1A outlines our methodology. AP and bipolar and unipolar
electrogram data were downsampled, phase was calculated for each
modality22 and interpolated, singularities were identified, and statis-
tics were calculated on a regional basis. PSs were located by calculat-
ing the topological charge23 and were tracked over time, with those
lasting >120 ms defined to be rotors.20
Resolution requirements were determined for the 10 cm×10 cm
sheet by uniformly spatially downsampling voltage data to different
resolutions, ranging from 1 to 25 mm. For the atrial bilayer model,
we considered subsets of nodes corresponding to the average distance
between nodes, termed mesh resolution (MR), of 1.62 to 17.1 mm.
To compare results between different resolutions, downsampled
phase (uniformly downsampled resolutions: 1–25 mm) was in-
terpolated using cubic splines to full grid resolution (0.1 mm) for
the 2D sheet (Figure 1B) or to 1.62 mm MR for the bilayer model
(MR=1.62–17.1 mm, 4813–36 points). Phase rather than voltage was
interpolated (Section 1.2 in the Data Supplement) because electro-
grams vary in magnitude (particularly bipoles) making their interpo-
lation challenging.
For focal source identification, we calculated the divergence of
the CV field24 (Figure I in the Data Supplement). For each AP, activa-
tion time was calculated as the location of the maximum temporal
derivative. CV vectors were calculated by differencing the activation
times of four neighboring points.25 The point of maximum divergence
of the normalized CV field identified the origin of focal sources.
Criteria for Determining Required Resolutions
The accuracy of rotor identification was assessed using 2 measures:
(1) visual inspection of isopotential plots over time and (2) error in the
center of the rotor trajectory calculated using phase (time-averaged
center error criterion; success if within an ablation catheter diameter
of 4 mm). For (2), PS locations were calculated as detailed above.
To separate these PSs into rotor PSs and false detections, a rotor PS
was seeded in an initial frame of the simulation and tracked over time
subject to a movement threshold to detect rotor PSs over the simu-
lation duration. Other PSs were then defined to be false detections
WHAT IS KNOWN
• It is unclear whether the different reported causes of
persistence of atrial fibrillation—focal and rotational
drivers, and multiple wavelets—are the result of dif-
ferent underlying mechanisms or result from different
scales and resolutions of recording devices and inter-
pretations of the electrographic data they produce.
WHAT THE STUDY ADDS
• This study determined the minimum resolution re-
quired to accurately identify rotors and focal sources,
and to avoid false detections, as a function of the spa-
tial wavelength (the distance between wavefronts) of
the arrhythmia.
• Stationary rotors are more reliably identified com-
pared to meandering trajectories, for any given spa-
tial resolution.
• All clinical high-resolution multipolar catheters are
of sufficient resolution to accurately detect and track
rotors when placed over the rotor core, though the
low-resolution basket catheter is prone to false de-
tections and may incorrectly identify rotors that are
not present.
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3 Roney et al Spatial Resolution Requirements During AF
(Figure 2B). To assess the influence of false detections on correct
rotor identification, both the number and distribution of falsely identi-
fied PSs were assessed.
A methodology for determining an appropriate threshold for the
number of permissible false detections was developed by consider-
ing the number of PSs as a function of distance from the true rotor
core location, which was taken to be the time-averaged full-resolution
rotor core location (Figure 2C). A resolution is considered to fail the
false PS detection histogram criterion if the resulting histogram con-
tains multiple peaks (Figure 2F), corresponding to additional spatial
clusters of PSs that represent false detections. These spatial clusters
could be misidentified as rotor locations.
Example resolutions for which identification is successful and un-
successful for each of the 3 criteria are shown in Figure 2.
Figure 1. Methods schematic. A, Action potential (AP) data were computed at a mesh resolution (MR) of 0.34 mm edge length (93 927
points). Data were then downsampled: 1.62 to 17.1 mm (4813–36 points). Voltages were interpolated (to MR=1.62 mm), and phase was
calculated. Unipolar electrograms were calculated at AP point distribution. Bipolar electrograms were calculated from paired unipolar
electrograms with 4-mm interelectrode spacing. Phase of unipolar and bipolar electrograms was calculated and interpolated to MR=1.62
mm. Phase singularities were tracked over time (>120 ms trajectories tagged as rotors), and regional assessment was performed. B, A
mapping is introduced for phase interpolation. Direct interpolation of the phase angle θ leads to issues when interpolating, in the instance
that neighboring points are close to π and −π (left). Mapping to the exponential form (eiϑ), interpolating this and then converting back to a
phase angle, removes the issue with phase angle discontinuities (right). The errors become larger as the grid spacing is increased (bot-
tom). The domain size shown here is 10 cm-by-10 cm.
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4 Roney et al Spatial Resolution Requirements During AF
Focal sources were identified using the same measures as for ro-
tors, except the center of the focal source was identified using the
maximum divergence of the velocity. There were no false detections
of positive divergence.
Wavelength Estimation
We express resolution requirements in terms of the number of record-
ing points (N) needed within 1 spatial wavelength (λ), the distance
between consecutive wavefronts.
Figure 2. Methodology for defining success or failure of rotor identification. Left column (A, C, and E): successful identification at 7-mm
spacing; right column (B, D, and F): failed identification at 17-mm spacing. A and B, Phase singularity (PS) locations corresponding to the
rotor core (green) and false detections (red and blue, coloured depending on spin). C and D, Rotor core PSs (green), showing the time-
averaged center of the full-resolution rotor trajectory (black) and the time-averaged center of the given resolution rotor trajectory (purple).
The distance between these gives the time-averaged center error (C: 0.9 mm, success; D: 4.3 mm, failure of the time-averaged center
error criterion). E and F, Histogram of number of PSs plotted as a function of distance from the full-resolution time-averaged rotor center.
At 7 mm (E), there is a single peak corresponding to the true rotor center, whereas at 17 mm (F), there are 2 peaks in the histogram corre-
sponding to a failure of the false PS detection histogram criterion because the false detections may be misidentified as a rotor core.
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5 Roney et al Spatial Resolution Requirements During AF
The wavelength associated with each parameter set was automati-
cally determined from full-resolution data by calculating the distance
between arms of spiral wavefronts of a rotor or consecutive circular
wavefronts of a focal source, using isopotential lines26 (Figure 3A;
Section 1.4.1 in the Data Supplement).
Where measurements are sparse, we define λ as the product of
mean CV (Section 1.4.2 and Figure II in the Data Supplement) and
mean cycle length: λ≈CV×cycle length. For bilayer simulations, λ
was estimated for all nodes at MR=1.62 mm by calculating mean CV
and cycle length over the simulation duration for data within a 2-cm
diameter.
Results
Resolution Required for Correct Identification of
Rotors and Focal Sources Is a Function of Spatial
Wavelength
For each assessment criteria, the minimum measuring points,
N, per wavelength was determined for each sheet simulation
parameter set, as the reciprocal of the gradient of the line of
best fit for each identification criterion. Figure 3B illustrates
that resolution and wavelength clearly influence the accuracy
of rotor core detection. We found that N=2.5 for visual iden-
tification, N=2.7 for the time-averaged center error criterion,
and N=3.1 for the false PS detection histogram criterion, as
shown in Figure 3C.
There must necessarily be a 3×3 grid of measuring points
between consecutive wavefronts for focal source identification
using maximum divergence to be successful. Because the dis-
tance between wavefronts decreases for shorter wavelengths,
correspondingly finer grid spacing is necessary (Figure III in
the Data Supplement). For accurate identification, N=3.3 for
visual inspection and N=1.6 when using the maximum diver-
gence criterion (Figure 3D).
Rotor Localization Errors Are Larger for
Electrogram Data Than for AP Data
Figure 4A shows an area of high PS density in an area of
high fibrosis in an anatomically accurate simulation of 2
rotors. Wavelength varies spatially (range, 21.5–108.1; mean
67.8±15.5 mm) because of the heterogeneous CV (range,
0.12–0.60 m/s; mean 0.37±0.09 m/s), where slow conduction
is seen in areas of high fibrosis. The 3 modeled elements of
fibrosis all decreased CV. As such, resolution requirements
also varied spatially.
For a given resolution, Figure 4B shows that PS distribu-
tions were visually similar across data types, as were the num-
ber of PSs, number of rotors and rotor duration, as shown in
Figure 4C. For computational efficiency, electrograms were
only calculated at MR≥1.6 mm, whereas AP interpolation was
only calculated for MR≥3.5 mm. The mean localization error
was generally higher for both types of electrogram phase than
for AP phase. Results for AP phase were similar when using
either voltage or phase interpolation.
Stationary Rotors Are More Reliably Identified
Compared With Meandering Trajectories
We analyzed simulation data in which 1 rotor anchored to an
area of high fibrosis intensity on the posterior wall (Figure 5A,
compare PS density and late gadolinium enhancement maps),
and a second rotor meandered across the anterior wall cover-
ing a larger area (Figure 5B). The CV is again heterogeneous
(range, 0.21–0.59 and 0.44±0.08 m/s), leading to heteroge-
neous wavelength (39.7–110.1 and 81.2±13.9 mm), with
shorter wavelengths in areas of fibrosis (Figure 5A).
On reducing resolution, PSs are still identified near the
stable rotor, but the meandering rotor trajectory breaks up
with both AP and unipolar data (Figure 5B). This is apparent
in the regional analysis (Figure 5C) in which region 3, cor-
responding to the stable rotor, is a high driver region across
all resolutions (top PS region for AP data for all resolutions),
whereas regions 5 and 6, corresponding to the meandering
rotor, decrease in importance for MR≥11.9 mm for AP data.
The average number of PSs and rotors detected decreased
with coarser MR (Figure 5D) as did rotor duration (Figure 5E).
PS location error increased at coarser MR for all data types.
Multipolar Catheters Are of Sufficient Resolution to
Accurately Detect and Track Rotors If Placed Over
the Rotor Core
We investigated whether electrode arrangements of com-
monly used high-density clinical mapping catheters satisfy
the resolution requirements identified above for reliably iden-
tifying rotors at the shortest wavelength (33.6 mm). Illustra-
tive isophase maps and rotor core PS trajectories are shown in
Figure 6A.
For 20 unipole configurations, the circular (Lasso) catheter
produced the largest time-averaged center location error (3.5
mm) with respect to full-resolution (0.1 mm) simulated data.
Other catheters gave significantly lower errors (Figure 6B).
Corresponding frame-wise errors in PS location are shown in
Figure 6C, where the circular catheter again had the largest
error.
For the 10 bipole configuration, formed from 20 unipolar
signals, the spiral (AFocus II) catheter produced the smallest
location errors (quantified in Figure 6B and 6C). The circu-
lar catheter gave similar errors with either 20 unipoles or 10
bipoles, whereas the accuracy of the other catheters decreased
as the number of data points was reduced.
Low-Resolution Basket Catheters Are Prone to
False Detections
In contrast to the high-density catheters examined above,
basket catheters provide global coverage at a lower electrode
density.2 Geodesic distances between each basket electrode
and its 4 neighboring electrodes are shown in Figure 7A. The
majority of interelectrode distances satisfy our requirements
for accurately locating rotor cores (time-averaged center error
criterion): 99.1% for the longer wavelength (75.2 mm) resolu-
tion requirement of 27.9 mm (75.2/2.7=27.9) and 79.3% for
the shorter wavelength (45.2 mm) resolution requirement of
16.7 mm. Fewer interelectrode distances satisfied the require-
ments to avoid false detections (false PS detection histo-
gram criterion): 96.4% for the longer wavelength resolution
requirement of 24.4 mm and 64.0% for the shorter wavelength
resolution requirement of 14.5 mm.
Interpolated phase maps were qualitatively similar to
the high-resolution phase maps, as shown in Figure 7C
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6 Roney et al Spatial Resolution Requirements During AF
and 7E. The rotor core was accurately located for the short
wavelength simulation (1.3 mm time-averaged center
error). For the long wavelength simulation, 2 rotor cores
were present in the mapping area for much of the simu-
lation. The first was located with sufficient accuracy (3.6
mm time-averaged center error; 2.6% of frames missing
Figure 3. Resolution requirements for spiral wave detection and focal source detection depend on spatial wavelength. A, Technique to
calculate wavelength of a spiral or focal wavefront. Isopotential lines at −60 mV with positive (green) and negative (blue) gradient. Inter-
sections of the ray (white line) with the isopotential lines of positive gradient are shown (purple dots). B, Distributions of PSs over time for
rotor simulations at different resolution and wavelengths. Phase singularities corresponding to a rotor core location are shown in green.
Number of false detections (chirality shown in blue and red) increased as wavelength decreased and as grid spacing increased. C, Mini-
mum N necessary to identify a rotor for each criterion. D, Minimum N necessary to identify a focal source for each criterion.
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7 Roney et al Spatial Resolution Requirements During AF
rotor core detections), whereas the second rotor had a time-
averaged center error greater than the 4-mm threshold (5.4
mm) because many PSs were along the edge of the full-
resolution area of coverage and as such were not picked up
by the basket arrangement (40.2% of frames missing rotor
core detections).
For the short wavelength simulation, many false detections
were observed. For example, Figure 7G shows an additional
cluster of PSs close to the main rotor for the short wavelength
simulation. This aligned with a larger interelectrode spacing
between electrodes vertically. Subsequently, this led to a sec-
ondary peak in the PS distribution histogram (Figure 7H).
When a basket catheter with double the number of splines
(ie, 16 splines of 8 electrodes) was simulated, the cluster of
false detections was no longer present, as shown in Figure 7J
and 7K.
In addition, the average rotor path is accurate; however, the
PS trajectory showed a larger rotor meander area for the bas-
ket resolution data than for the high-resolution data (Section
2.3 and Figure IV in the Data Supplement).
Figure 4. Phase singularity (PS) distributions and characteristics for different data modalities. A, Normalized late gadolinium enhance-
ment (LGE)-magnetic resonance imaging data for a patient with persistent atrial fibrillation was used to infer probabilities for fibrosis
inclusion in the model; high PS density is seen to coincide with high fibrosis density; PS locations over time show rotor trajectories; wave-
length varies spatially. B, Comparison of detected PS locations for mesh resolutions (MRs) of 3.52 mm (top) and 13.6 mm (bottom), for
different AP interpolations and electrogram modalities. C, Number of PSs (solid lines) and rotors (dashed lines), (D) rotor durations, and (E)
distance errors as a function of MR.
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8 Roney et al Spatial Resolution Requirements During AF
Figure 5. Stationary rotors are more reliably identified compared with meandering trajectories. A, Average late gadolinium enhancement
(LGE)-magnetic resonance imaging map, phase singularity (PS) density, and local wavelengths, as well as numbered regions used for
regional analysis. B, PS distributions shown on the posterior (top) and anterior (bottom) walls for different resolutions and modalities. C,
Regional analysis showing mean number of phase singularities and rotors in each region (error bars indicate SD for the number of phase
singularities). D, Number of PSs (solid lines) and rotors (dashed lines), (E) rotor durations, and (F) distance errors as a function of mesh
resolution (MR).
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Figure 6. Multipolar catheters are of sufficient resolution to accurately detect and track rotors. A, Top: Example isophase maps interpo-
lated from the recording points shown in black (I–IV), with the phase from the full-resolution simulation data shown in (V). Bottom: Rotor
core phase singularity (PS) trajectories for each catheter type calculated using the interpolated phase. Examples are shown for spiral
(AFocus II), circular (Lasso), and 2 five-spline electrode arrangements (PentaRay I and PentaRay II). B, Errors in the time-averaged esti-
mated center location compared with the time-averaged location computed from the raw simulation data. Catheters are configured as
either 20 unipoles or 10 bipoles. C, Box plots to show frame-wise difference in estimated PS location compared with the location com-
puted from raw simulation data. The boxes indicate the interquartile range (IQR) and median (red line) of the data; the whiskers extend to
a maximum of 1.5×IQR; and the crosses represent outliers.
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10 Roney et al Spatial Resolution Requirements During AF
For Clinical AF Data, Reducing the Number of
Electrodes in Mapping Catheters Increased the
Number of Missing and False PS Detections
We determined the ability of multipolar catheters to detect
PSs as electrodes were removed. Clinical catheter recordings
with different degrees of rotational activity were analyzed,
ranging from planar activity to curved rotor cores: overall
mean number of PSs for unipolar catheters 0.47±0.20, range
0 to 0.91 and for bipolar 0.36±0.16, range 0 to 0.73. Figure 8
shows box plots for the percentage of missing PSs (percent-
age of full-resolution PSs not present in downsampled data)
and the percentage of false detections (percentage of downs-
ampled data PSs not present in full-resolution data), which
both increase as the number of recording points is reduced.
Figure 7. Low-resolution basket catheters are prone to false detections. A, Interelectrode distances in an 8-spline basket catheter. Reso-
lution requirements for avoiding false detections for the 2 wavelengths (45.2 and 75.2 mm) are marked. B, Example isopotential map for
longer wavelength simulation with basket electrodes marked. C, High-resolution phase map generated from phase at mesh vertices.
Phase singularity (PS) marked as a black dot. D, Phase of electrodes as arranged on a regular 8×8 grid. E, Interpolated phase from bas-
ket arrangement of electrodes. F, Rotor PS locations computed from high-resolution data (blue) and the 8-spline basket electrodes (red)
for the short wavelength simulation. Only PSs that correspond to the rotor are shown. G, All detected PSs of 8-spline basket catheter. H,
PS detection histogram for 8-spline catheter. I, Rotor PS locations, (J) all PSs, and (K) PS detection histogram for a simulated 16-spline
catheter—double the clinical resolution.
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11 Roney et al Spatial Resolution Requirements During AF
Discussion
Main Findings
In this study, we demonstrated that sufficient spatial resolu-
tion is essential for the accurate detection of rotors and focal
sources and propose that insufficient resolution may be respon-
sible for the conflicting findings of recent human studies.2,12,27
An estimate of the resolution requirements as a function of the
spatial wavelength was found for spiral wavefronts (rotors)
and circular wavefronts (focal sources) using different criteria.
For regularly spaced grids, the minimum resolution required
is a ratio of spatial wavelength to number of measuring points
per wavelength (λ/N). For rotors, N=2.5 (visual inspection),
N=2.7 (rotor core time-averaged center error), and N=3.1
(to avoid false detections). For focal sources, N=3.3 (visual
inspection) and N=1.6 (maximum divergence calculation of
focal source origin location). The results suggest that although
the basket catheter has adequate resolution to track rotors, it
has inadequate resolution to avoid false detections.
We found that although stationary rotors may be identi-
fied at coarse resolutions, meandering rotors are lost. For atrial
bilayer simulations, regional analyses at all resolutions consid-
ered identified the same region as having the highest PS density,
whereas rotor localization error was unacceptable for MR≥11.9
mm. This suggests that standard mapping modalities offer suffi-
cient resolution for ablation guided by regional driver density27
although localization of meandering rotors may not be possible.
In addition, resolution requirements are similar for unipolar and
bipolar electrogram data. Correct PS identification for clinical
spiral (AFocus II) mapping catheter recordings is sensitive to
the number of electrodes used in the analysis.
Spatial Wavelengths in Human AF
We simulated 9 different wavelengths for rotors and focal
sources to determine the relationship between resolution
requirements and wavelength. Based on previous reports, the
expected range of spatial wavelengths in human AF is 44 to
127 mm, because of the varying degree of electric remodeling
in patients with AF. This range was estimated as CV divided
by dominant frequency, where CVs are in the range 0.38±0.1
to 0.61±0.06 m/s,16 and dominant frequency ranges from 4.8
to 8.6 Hz.17,18 The wavelengths of the spiral waves simulated
in this study cover a subset of this range from 33 to 78 mm.
Wavelength may vary spatially (Figures 4A and 5A) because
of conduction or repolarization heterogeneities, leading to
spatially varying resolution requirements. This is particularly
important as rotors may anchor to areas of slow conduction.
Away from a rotor core, 7 points were required for an accu-
rate and reliable estimate of spatial wavelength if located within
1 wavelength (Section 2.1 in the Data Supplement). High-density
mapping catheters fulfill this criterion because wavelengths in
human AF are estimated to be longer than their diameters.
Required Resolution for Regular Grids
The Nyquist criterion states that interelectrode spacing must
be less than half the smallest spatial wavelength of interest,28
corresponding to N=2. This study aimed to extend the work
of Rappel and Narayan,29 where a theoretical approach deter-
mined that the resolution required to identify stable rotors and
focal sources is of the form λ/N; their study identified wave
patterns visually and the required value of N was not quanti-
tatively determined. In our study, we find that the resolution
requirements are linear in λ, suggesting that the resolution
required does follow λ/N.
Four of the identification criteria suggest a slightly higher
value of N than the theoretical Nyquist criterion is needed in
practice, whereas the maximum divergence location suggests
a smaller value. This criterion was applied for focal sources
where the grid was centered over the focal source, which is the
optimal arrangement; off-center arrangements and placements
away from the source will require a higher N.
Required Resolution for Clinically Used Catheters
The most stringent spatial resolution requirement found for
identification of rotors in human AF is 44/3.1=14.2 mm. The
interelectrode spacings of all high-density mapping catheters
considered (AFocus II 4 mm, Lasso 6 mm, PentaRay 4 mm,
Figure 8. For clinical atrial fibrillation (AF) data, reducing the number of electrodes in high-density mapping catheters increased the
number of missing and false phase singularity (PS) detections. A, Box plots to show the percentage of full-resolution PSs not present in
downsampled data measured across 127 catheter recordings, for unipolar and bipolar electrode recordings. B, Box plots to show the per-
centage of PSs in downsampled data not present in full-resolution data. In all cases, the boxes indicate the interquartile range (IQR) and
median of the data (red line); the whiskers extend to a maximum of 1.5×IQR; and the crosses represent outliers.
by guest on May 17, 2017http://circep.ahajournals.org/Downloaded from
12 Roney et al Spatial Resolution Requirements During AF
or 6 mm) are smaller than this distance, suggesting the ability
of these catheters to accurately locate PSs if placed over the
rotor core. For 20 recording points, the circular Lasso cath-
eter gave the largest error in estimating rotor center location
(Figure 6B). Similarly, Weber30 found that a simulated circu-
lar catheter performed worse than spiral and 5-spline cath-
eters because it could not identify focal sources, but rather,
the radial basis function interpolation showed a planar wave.
For clinical data, correct PS identification was sensitive to the
number of points used for interpolation from a high-density
spiral AFocus II mapping catheter (Figure 8).
A major disadvantage of mapping catheters is their local-
ized coverage; as such, rotor tracking is only possible if
the catheter is fortuitously placed over a rotor that does not
meander outside the margins of the catheter poles. If the cath-
eter does not lie over the rotor core, techniques presented by
Roney et al31 could be used to direct the catheter toward the
rotor, but these techniques are dependent on some degree of
organization of wavefronts remote from the driver. In addi-
tion, it may be necessary to consider the activity of surround-
ing electrograms to differentiate rotors from interactions
around lines of block.32
Unlike the catheters mentioned above, basket catheters
provide global coverage, which is a possible reason why stud-
ies using them2,33 were able to detect rotors in human AF,
whereas studies using catheters with only regional coverage7
were not. Our results confirm that basket catheters can accu-
rately detect rotors (Figure 7) and faithfully track PS trajecto-
ries (Figure IV in the Data Supplement).
Berenfeld and Oral33 comment that some areas of interpo-
lation for basket mapping have interspline difference of >20
mm; for the basket catheter used in this study, 12.6% of inter-
electrode distances are >20 mm. Laughner et al34 found that
equatorial bunching of basket catheter splines often occurred,
leading to a wide range of interspline distances within the bas-
ket, and this varied between patients. In addition, coverage of
the pulmonary veins, left septum, and left lateral wall was lim-
ited, with only 55% of the atrial surface covered, as observed
by Benharash et al,8 explaining the large number of missing
rotor detections in our study.
Low-Resolution Basket Catheters Are Prone to
False Detection of PSs
The basket catheter, however, was found to be inadequate
to avoid spurious rotors. Only 63.1% of the interelectrode
distances are less than the resolution requirement of 14.2
mm, corresponding to 3.1 points per spatial wavelength.
This is likely the cause of the false PS detections, where
the simulated basket data failed the false PS detection his-
togram criterion.
The tendency of basket catheters with inadequate resolu-
tion to detect nonexistent PSs may explain the discrepancy
between recent clinical studies, where studies using basket
catheters report stable rotors,2 whereas regional, higher-
resolution mapping do not report stable rotors.12,35 This may
explain, in part, the large incidence of rotors reported by
Narayan et al,2 a low termination rate,8 and poor long-term
success9 for ablating rotors detected by basket catheters. The
modeled 16-spline basket catheter did not suffer from false PS
detections although good endocardial contact of such a cath-
eter may be difficult to achieve in practice.
Our study comparing resolution requirements for station-
ary and meandering rotors found that rotor trajectories may be
lost at resolutions for which stable rotors are still identifiable
(Figure 5), which may explain differences in findings on rotor
stability with basket catheters identifying stable rotors and
noninvasive electrocardiographic imaging identifying tran-
sient meandering rotors.27,36
Effect of Datatype
Resolution requirements for AP, unipolar electrogram, and
bipolar electrogram data (Figure 4) were similar. Localization
errors were larger for electrogram data than for AP data and
always larger than the 4-mm threshold, corresponding to an
ablation catheter diameter, used for rotor location error, per-
haps also because of rotor meander and irregular point spacing
on the surface mesh (compared with the regular 2D grid).
Limitations
The limitations of our study include (1) we assume the pres-
ence of rotors, (2) our tissue is simplified and we do not model
endocardial–epicardial dissociation. Furthermore, in the sim-
ulations for the clinically used catheters, all electrograms were
noise free, representing perfect data. In reality, electrograms
will contain noise, motion artifacts, and may have unsatisfac-
tory tissue contact.33
Conclusions
We determined the minimum spatial resolution requirements,
as a function of AF wavelength, to correctly identify the under-
lying AF mechanism. All clinically used catheters assessed in
our study possess adequate spatial resolution to identify and
track rotor core location for the range of wavelengths occur-
ring in human AF if covering the location of the rotor PS.
However, the low resolution of basket catheters renders them
prone to false detections. Resolution requirements depend on
rotor meander and AF spatial wavelength, but are similar for
AP, unipolar electrogram, and bipolar electrogram data. Over-
all, the spatial resolution of AF data can significantly affect the
interpretation of the underlying AF mechanism.
Acknowledgments
We thank Dr Hubert Cochet for the late gadolinium enhancement
(LGE)-magnetic resonance imaging data used in this study.
Sources of Funding
This work was supported by funding awarded from the British
Heart Foundation (FS/11/22/28745 and RG/16/3/32175); the
ElectroCardioMaths Programme of the Imperial BHF Centre of
Research Excellence; the National Institute for Health Research.
Dr Ng is funded by National Institute for Health Research Clinical
Lectureship (1716). Dr Roney is funded by a Lefoulon-Delalande
Foundation fellowship administered by the Institute of France. In
addition, this study was supported through the Investment of the
Future grant, ANR-10-IAHU-04, and the grant Equipex MUSIC
ANR-11-EQPX-0030. Computer time for this study was provided
by the computing facilities Mésocentre de Calcul Intensif Aquitain
of the Université de Bordeaux and of the Université de Pau et des
Pays de l’Adour.
by guest on May 17, 2017http://circep.ahajournals.org/Downloaded from
13 Roney et al Spatial Resolution Requirements During AF
Disclosures
None
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Ng SiongJennifer H. Tweedy, Prapa Kanagaratnam, Nicholas S. Peters, Edward J. Vigmond and Fu
Caroline H. Roney, Chris D. Cantwell, Jason D. Bayer, Norman A. Qureshi, Phang Boon Lim,
Fibrillation
Spatial Resolution Requirements for Accurate Identification of Drivers of Atrial
Print ISSN: 1941-3149. Online ISSN: 1941-3084
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! !
1!
Supplemental Material
Expanded Methods and Results
Methods:
We initially determined resolution requirements for a regularly spaced two-
dimensional grid of data with homogeneous properties with a stable rotor or focal
source. We then tested these requirements on more complicated arrhythmias in a
bilayer model with spatially varying activation and repolarization properties, realistic
geometries and catheter electrode arrangements.
1.1 Simulations
1.1.1 Simulations: Generating a range of wavelengths
The wavelength of a sinusoidal wave is the wave speed v divided by the wave
frequency f. We assume the spatial wavelength in cardiac tissue varies linearly with
speed or frequency: λ =v/f.
The monodomain tissue model was used for excitation propagation in a 10cm x 10cm
two-dimensional sheet of tissue and the Courtemanche-Ramirez-Nattel human atrial
cell model 1 was used to represent transmembrane ionic currents. Electrical
remodelling in AF was incorporated by reducing the maximal ionic conductances of
Ito, IKur and ICaL by 50%, 50% and 70% respectively, following 2 .
Conduction velocity (CV) of the wavefront (wave speed) was modified using the
diffusion coefficient D. Three values of D were chosen (D=0.0005, 0.0010, 0.0015
cm2/ms) based on preliminary simulations that showed these values result in CVs that
span the physiological range in humans 3. Conductivities for homogeneous two-
dimensional sheet simulations were isotropic.
To modify the maximum frequency of propagation for simulated spiral waves, IK1
conductance was multiplied by 1.0, 1.5 or 2.0. Increasing this parameter has been
! !
2!
shown to reduce the action potential duration (APD), and so increase the maximum
frequency of propagation, as well as to increase spiral wave stability in modelling
studies 4.
Spiral wave simulations were run for parameter set combinations of the three
diffusion coefficients with the three values of the IK1 conductance, resulting in nine
different wavelengths (see S-Table 1). This was repeated using the same parameter
sets for a focal source stimulus applied at a frequency close to the corresponding
spiral wave frequency (see S-Table 2). Initial conditions for each value of IK1
conductance and pacing frequency were obtained by prepacing a single cell for 100
beats in order to space clamp the 2D model with single cell conditions. In addition,
the sodium and potassium concentrations were treated as fixed constants to eliminate
drift, following 5 .
The monodomain equations were solved using a finite difference operator splitting
scheme in space 6 and an alternating-direction implicit scheme in time 7, with a space
step of 0.1mm and a time step of 0.01ms.
S-Table 1: Wavelengths for spiral wavefronts of rotor simulations (top lines: full
resolution calculated wavelengths (median and interquartile range); bottom lines:
estimated wavelength (CV x cycle length (CL))).
! !
3!
S-Table 2: Wavelengths for circular wavefronts of focal source simulations (top
lines: full resolution calculated wavelengths (median and interquartile range); bottom
lines: estimated wavelength (CV x cycle length (CL))).
1.1.2 Simulations: Bilayer model
To test the effects of rotor stability and data type on resolution requirements,
simulations were run using a previously published atrial bilayer model 8. The finite
element model includes 2D endocardial and epicardial layers that are discretely
connected for the left atrium, as well as fast conducting pathways (Bachmann’s
bundle, crista terminalis, pectinate muscles). The Courtemanche AF cellular model
was used, tuned to match monophasic action potential duration of persistent AF
patients 9,10. Further rescalings were used to incorporate regional repolarisation
heterogeneity 9,11,12. Fiber orientation was included in the model following the rule
based approach of Labarthe et al. 13, and regional conductivity values were tuned to
match the activation data of Lemery et al 14. Simulations were run using the Cardiac
Arrhythmia Research Package (CARP) simulator 15.
These simulations included interstitial fibrosis as microstructural discontinuities in the
mesh 9,16, with distributions based on late-gadolinium intensity values for persistent
AF patients. In particular, edges were probabilistically selected as fibrotic based on
normalized LGE intensity and longitudinal fiber direction. Mesh element edges
parallel to the longitudinal fiber direction were taken to be four times more likely to
be fibrotic than edges that are transverse (using a scaling factor: α(4 cos2(θ) + sin2(θ)),
for which θ is the angle between a given edge and the longitudinal fiber direction for
the mesh element face). A uniformly distributed random number in the range (0,1)
was generated for each mesh element edge and compared to the product of the
normalised LGE intensity value and fiber direction scaling factor. Edges for which the
random number is less than this product were assigned to be fibrotic. These fibrotic
edges were arranged into connected networks so that no flux boundary conditions
could be applied, following Costa et al. 16. Mesh element faces for which all edges
were selected as fibrotic were removed from the mesh. For one of the simulations (Fig
4, main manuscript), we also modeled changes in tissue properties and cellular ionic
properties based on our previous publication 17.
! !
4!
The LGE distribution used for the first simulation to test the effects of datatype were
for an individual patient with persistent AF; for the second simulation to compare
stable and meandering rotors, the distribution used is from Cochet et al. and
represents the likelihood of LGE intensity averaged across 26 patients with persistent
AF 18.
Unipolar electrograms were calculated at node locations projected 1mm endocardially
along the surface normal vectors; bipolar electrograms were calculated as the
difference of paired unipoles, with 4mm spacing, based on a PentaRay catheter
(BioSense Webster, South Diamond Bar, CA). Regional analysis was performed as in
our previous study 9, with the motivation that ablation strategies may target regions of
high PS density 19. For this analysis, the left atrium was divided into eight regions (see
Fig 1 of the main manuscript).
1.1.3 Simulations: Basket
The resolution requirements of a basket catheter were assessed by simulating a
realistic human left atrial geometry with unipolar electrograms calculated at basket
catheter measurement locations. The geometry was segmented (using ITK-SNAP 20)
from cardiac magnetic resonance imaging. The surface was opened at the four
pulmonary veins and at the mitral valve (using Blender 21), and re-meshed to create
triangular elements of characteristic size suitable for use with spectral/hp element
discretisations (using gmsh 22).
The locations of the electrodes of a basket catheter were exported from an electro-
anatomic mapping system used during a clinical case. This basket catheter was 48mm
in diameter and consisted of eight splines, with eight electrodes on each spline. The
electrode locations were shifted to be centred in the simulated atrial chamber and
rotated such that the largest gap between splines was located at the mitral valve. The
electrode locations were projected 0.2mm inside the blood cavity 23, along the surface
normal to the closest vertex.
! !
5!
To establish the true location of rotors for comparison, unipolar electrograms were
calculated at every vertex of the mesh within the area covered by the basket catheter
and again projected 0.2mm inside the blood cavity (491 measurement points).
Simulations were run using the cardiac electrophysiology solver 24 in the Nektar++
spectral/hp element framework 25, with the Courtemanche-Ramirez-Nattel AF model
2. An extra-stimulus pacing protocol was employed to generate spiral wave re-entry in
the simulation. Unipolar electrograms were calculated at the electrode location points,
following 26. Phase of the unipolar electrograms were calculated as described
previously 17. After flattening to a two-dimensional representation 27,28, phase was
interpolated to a regular grid of 0.5mm spacing for identifying the true rotor location,
and 2mm spacing for the basket recording points. Interpolating the basket phase data
onto a finer grid than 2mm spacing caused wavefronts to artificially break-up.
Basket rotor locations were taken to be the closest PS location of the correct chirality
that was within a 10mm distance threshold of the high-resolution rotor location, on a
frame-by-frame basis. The moving-average rotor core path was estimated using a
window of length 1000ms that shifted in 100ms increments. For each window of data,
the width of the path (the diameter of the window) was calculated as the greatest
distance between the rotor core at any two times within that window.
1.2 Phase interpolation
To avoid the issue of interpolation across the phase angle branch cut, it was necessary
to convert the phase angle (θ) to exponential form (eiθ) before interpolation. The
mapped data was then interpolated, and finally converted back to phase angles
between -π and π. This is shown in Fig 1B of the main manuscript, comparing
interpolating θ (left) with mapping to eiθ, interpolating and then mapping back to θ
(right). Errors associated with the phase discontinuities in the direct interpolation
become significantly more pronounced as the resolution is reduced.
In order to assess the effects of spatial resolution on the observed wavefront
dynamics, the phase calculated using the downsampled data (resolutions ranging from
1mm to 25mm) were interpolated to the full grid resolution (0.1mm).
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The data were interpolated using the Matlab interp2 function, with spline
interpolation, since this resulted in the fewest number of false phase singularity
detections.
For low-resolution surface mesh data, an inverse distance squared weighting
interpolation was used (using neighbours within a 7mm radius sphere) 17. For the
atrial bilayer model, downsampled data of mesh resolution 1.62mm to 17.1mm (4813-
36 points) were interpolated to 1.62mm.
1.3 Using divergence to identify focal sources
For focal source identification, we calculated the divergence of the conduction
velocity field since peaks in divergence indicate locations of sources of electrical
activity 29. An outline of the steps involved is shown in S-Fig 1.
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S-Fig 1: Methodology for calculating divergence, used to identify focal sources from
activation time data. (A) Activation times for a simulated focal source; (B) times for
one focal beat downsampled to 2mm spacing; (C) normalised CV vectors; (D)
divergence of the CV field. The point of maximum divergence was used to identify
the origin of focal sources; seen clearly in (D). The CV vector field was normalised so
that the divergence depended only on direction, and not speed.
1.4 Wavelength Calculations
1.4.1 Estimating Wavelength
An automated algorithm to calculate the wavelength as the distance between arms of a
spiral wavefront or subsequent circular wavefronts of a focal source (Fig 3A, main
manuscript) works as follows. For each frame, an isopotential line was calculated (-
60mV) using the method from 30: points were selected for an isopotential line if and
only if, firstly, the potential of the node was less than the isopotential value and,
secondly, the values of between one and three of its four neighbouring nodes were
greater than the isopotential value. The isopotential line was split into regions of
positive and negative time derivatives. For spiral waves, the centre was defined as the
pixel where such gradients meet (!"
!" =0 and V(x,y,t)=-60) 31. Intersections of rays
from this centre with the isopotential line with positive gradient were located, and in
the instance where there was more than one intersection, the distance along the ray
between points was stored. This was repeated every 10 degrees and for frames every
10ms. The wavelength was taken to be the median of these distances.
For focal sources, the calculation was similar but the centre was the point of
maximum divergence.
1.4.2 Automated Conduction Velocity Analysis:
To correctly calculate conduction velocity for a given recording area, the time
window for the activation times analysed must be chosen appropriately such that the
activation times of each of the measuring points are from the same propagating
wavefront.
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For a spiral wave, this means the times should be from the same arm of the spiral; that
is between the same two turns of the wavefront. This is demonstrated in S-Fig 2 in
which the top row shows an inappropriate choice of times for which some points are
on one arm of the spiral and others on another, creating a discontinuity in the
activation time field; the bottom row shows an appropriate choice of times. The
conduction velocity is calculated correctly in the latter case. It can be challenging to
select the correct time window for analysis during complicated rhythms, such as
fibrillation, and so an automated method for selection is proposed here.
First of all, the median interval between subsequent activations of all measuring
points is calculated, to give an estimate of the average cycle length of the activity,
which is then used as the window length, L, for analysis. Activation times for each
recording point are then selected within a window starting at some initial start time, T,
giving [T, T+L] (in the instance that multiple activations occur within the window for
a recording location, the minimum time is selected). The conduction velocity is then
calculated for these times along with the residual of the fit. This is repeated for
intervals of length L, for which the start point is shifted from T, in 10ms increments,
until the end of the recording. The conduction velocity algorithm is applied to each
time window and those with a residual below a threshold value are selected as suitable
time windows, and the conduction velocity estimate is stored. In the example shown
in S-Fig 2, a shift equal to half of the median cycle length from the initial start time
gives the lowest residual and most appropriate choice of activation times.
For analysis of repeating wavefronts, including focal sources and spiral waves, this
technique was used in order to automatically find the conduction velocity, without the
need to pre-specify a time window. For the bilayer simulations, wavelength was
estimated on a downsampled mesh (average edge length 3.52mm). For each node of
this mesh, twenty nodes were selected within a 1cm radius of the node (to
approximate recordings on a high-density catheter), activation times were defined as
the timings of phase 0 for action potential phase, and the mean and standard deviation
conduction velocity estimate was calculated as described above. The wavelength was
then estimated for each node as the mean conduction velocity multiplied by the mean
cycle length at that point. Finally this score was smoothed using an inverse distance
weighting.
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S-Fig 2: Calculation of the conduction velocity of a spiral wavefront. (A, D):
Activation times were analysed for the points shown at the purple dots in the
isochronal maps. (B, E): Locations of the points used for the fit, coloured by their
activation times. (C, F): Plots to show the actual times and fitted times that result from
using these points in the conduction velocity algorithm assuming a circular wavefront.
(A-C) A poor fit is obtained when activation times are not on the same arm of a spiral,
as the first and last measuring points to be activated are both in the centre of the
arrangement of points. (D-F) Shifting by the median cycle length divided by two
gives a time window for which the times are all on the same arm of the spiral, and a
satisfactory fit is obtained. In this case, the first points to be activated are in the
bottom left of the arrangement, and the last points to be activated are in the top right.
The domain size shown here is 10cm x 10cm.
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Results:
2. 1 Estimating wavelength
A range of wavelengths was simulated for spiral wavefronts and circular wavefronts
using combinations of the diffusion co-efficient D and potassium conductance gK1 as
described in Supplemental Material section 1.1.1. Spiral wavefront wavelengths were
within the range 33.6-78.3mm (S-Table 1), while circular wavefront wavelengths
covered the range 32.0-62.5mm (S-Table 2). For focal activation (circular
wavefronts), three out of nine of the simulated parameter combinations produced
wavelengths which were too large (>70mm) to be measured within the computational
domain, which was chosen based on a typical left-atrial surface area (100cm2).
Wavelengths estimated using limited-resolution data, as the product of CV and cycle
length (CL), matched the calculated wavelength to within an average percentage error
of 3.3% for spiral waves and 1.4% for focal activations, as given in S-Tables 1 and 2,
respectively. This was based on a random distribution of twenty points within an area
of diameter 2cm, approximating a high-density multipolar catheter.
The number of measuring points required for successful estimation of rotor or focal
source wavelengths was found to be between six and seven. For spiral waves at the
shortest wavelengths (33.6mm), seven points were required (20.0% of CV estimates
unsuccessful for 6 points; 2.5% unsuccessful for 7 points). For the longest wavelength
spiral wave simulation (78.3mm), six points were required (10.0% of CV estimates
unsuccessful for 5 points; 0% unsuccessful for 6 points).
For focal sources (circular wavefronts), resolution requirements were found to be
similar to those for rotors (spiral wavefronts). Seven points were required for a
wavelength of 53.2mm (12.5% of CV estimates unsuccessful for 6 points; 7.5%
unsuccessful for 7 points).
2.2 Using divergence to identify focal sources
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In order for focal source identification using the maximum divergence to be
successful, there must be a 3x3 grid of measuring points between consecutive
wavefronts. This is illustrated in S-Fig 3.
S-Fig 3: Resolution requirements for focal source identification depend on spatial
wavelength. Having three by three points within a single focal source circular
wavefront gives the minimum information required for successful identification of the
maximum divergence location. A denser grid is required for shorter wavelengths. A-C
are for a shorter wavelength of 32mm, for which a resolution spacing of 20mm or
smaller is required to give three grid points; while D-F are for a longer wavelength of
63mm, for which a minimum resolution spacing of 41mm is required. A, D show
isochronal maps; B, E show the times used for the stated resolutions; C, F show the
interpolated velocity fields from which the point of maximum divergence is located.
2.3 Basket path
S-Fig 4 shows that the path followed by the meandering rotor core estimated using the
basket arrangement is visually similar to the path measured using the dense
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arrangement. The average frame-wise error in the rotor core path is 2.2 ± 1.2mm for
the first rotor and 3.8 ± 2.1mm for the second rotor. The average path is therefore
accurate; however, the PS trajectory shows a larger rotor meander area for the basket
resolution data than for the high-resolution data (diameter increased for basket
compared to high resolution by 8.7 ± 3.5mm and 3.4 ± 7.0mm respectively for each
rotor).
S-Fig 4: Moving average PS path for the basket catheter arrangement. PS locations
for thirty seconds of the longer wavelength simulation (75.2mm) are shown. PSs are
separated by chirality (top and bottom row) to show the locations of the two rotors,
for the high-resolution data and 8-spline basket catheter arrangement. Location of the
moving average core location is marked in black.
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