Available via license: CC BY 4.0
Content may be subject to copyright.
ESTIMATING CARDIAC TISSUE CONDUCTIVITY FROM
ELECTROGRAMS WITH FULLY CONVOLUTIONAL NETWORKS
A PREPRINT
Konstantinos Ntagiantas1*, Eduardo Pignatelli1, Nicholas S. Peters2, Chris D. Cantwell3, Rasheda
A. Chowdhury2$, Anil A. Bharath1$
1 Department of Bioengineering, Imperial College London, London SW7 2AZ, United Kingdom
2 National Heart and Lung Institute, Imperial College London, London W12 0NN, United Kingdom
3 Department of Aeronautics, Imperial College London, London SW7 2AZ, United Kingdom
$ Authors contributed equally.
* Corresponding Author
E-mail: konstantinos.ntagiantas19@imperial.ac.uk
Address: Royal School of Mines, Department of Bioengineering, Imperial College London, London SW7 2AZ, United
Kingdom
December 7, 2022
ABS TRAC T
Atrial Fibrillation (AF) is characterized by disorganised electrical activity in the atria and is known to
be sustained by the presence of regions of fibrosis (scars) or functional cellular remodeling, both of
which may lead to areas of slow conduction. Estimating the effective conductivity of the myocardium
and identifying regions of abnormal propagation is therefore crucial for the effective treatment of AF.
We hypothesise that the spatial distribution of tissue conductivity can be directly inferred from an
array of concurrently acquired contact electrograms (EGMs). We generate a dataset of simulated
cardiac AP propagation using randomised scar distributions and a phenomenological cardiac model
and calculate contact electrograms at various positions on the field. A deep neural network, based on
a modified U-net architecture, is trained to estimate the location of the scar and quantify conductivity
of the tissue with a Jaccard index of
91
%. We adapt a wavelet-based surrogate testing analysis to
confirm that the inferred conductivity distribution is an accurate representation of the ground truth
input to the model. We find that the root mean square error (RMSE) between the ground truth and
our predictions is significantly smaller (
pval = 0.007
) than the RMSE between the ground truth and
surrogate samples.
Keywords
Electrogram
·
Convolutional network
·
Action potential
·
Tissue conductivity
·
Fibre orientation
·
Atrial
fibrillation
1 Introduction
The normal propagation of electrical signals through the myocardium leads to coordinated contraction of the heart
muscle. The cardiac action potential (AP) reflects the movement of ions between the interior of the myocytes and
the extracellular space. When the transmembrane potential of these cells increases above the threshold of activation,
an AP takes place [
1
]. The propagation of APs through the heart is the result of the collective expression of ion
channels and gap junctions (connexin proteins) whose roles are, respectively, to signal and initiate the cellular AP and
to propagate this AP to neighboring cells. The elongated shape of myocytes that form the cardiac fibres, and the polar
positioning of gap junctions in the direction of the fibre orientation leads to anisotropic conduction. Pathological loss
arXiv:2212.03012v1 [cs.LG] 6 Dec 2022
APREPRINT - DECEMBER 7, 2022
of gap junctions or fibre disarray, for example in AF, can lead to changes in the anisotropic ratio. Physiological and
pathological heterogeneities and changes in the expression of these proteins in the myocardial cells result in a reduction
in effective conductivity and the substrate being non-homogeneous throughout the myocardium [2].
In addition to channel abnormalities, areas devoid of myocytes (scars) exhibit lower conductivity and abnormal
propagation of the cardiac AP at the macro-scale, potentially generating re-entrant electrical waves which can initiate
atrial fibrillation (AF) [
3
]. Destroying the partially conductive tissue (ablation) may provide an effective way to
eliminate slow conduction pathways and reduce the likelihood of reentrant circuits forming [4].
EGMs can be measured clinically during cardiac ablation procedures, to investigate arrhythmias and steer treatment [
5
].
In contrast to an electrocardiogram (ECG) which is recorded by electrodes places on the skin, the electrogram (EGM) is
recorded by electrodes in direct contact with the myocardium. It measures the superposition of electric fields generated
by the aforementioned movement of ions in the local field of view of the electrode; it is consequently affected by the
heterogeneity and anisotropy of the myocardial substrate and the expression of gap junctions. Furthermore, it provides
information about localised changes in conduction. The morphology of EGMs has previously been qualitatively
binarised (complex fractionated atrial electrograms vs. simple electrograms) or quantified, using techniques such
as dominant frequency analysis [
6
], by clinicians to identify possible ablation targets with ambiguous results [
7
,
8
].
This lack of methodological efficacy has prevented widespread implementation of such techniques. Consequently, the
current success rate of AF treatment through catheter ablation remains considerably low. A better understanding of the
underlying substrate, and robust methods for identification of abnormalities, may lead to more successful treatments to
combat AF.
In this study, we propose to use modern deep learning techniques to infer the structural properties of the tissue, namely
the fibre orientation and the tissue conductivity, underlying a set of electrodes used to acquire unipolar EGMs. Deep
learning is being increasingly explored, in place of more traditional techniques, for inferring the solution to inverse
problems that quickly become intractable because of ill-posedness or complexity [
9
,
10
,
11
,
12
]. With classical machine
learning algorithms like Principal Component Analysis, high-dimensional signals are represented by low-dimensional
feature vectors. By inferring general laws from experimental data, deep learning allows for the enrichment and/or
re-evaluation of many of the existing heuristics that are considered informative to solve the inverse problem. Specifically,
in the problem we are addressing, there is no consensus about which features of unipolar EGMs are informative of
substrate properties and how these signal characteristics can be used to estimate properties of the myocardium. Deep
neural networks work by compressing the raw, high-dimensional observations into a lower-dimensional space, allowing
to treat previously intractable problems. Finally, there is rising evidence that the operator represented by the deep
network projects the raw signal not only to a smaller space, but also one that makes the problem linear [
13
]. Therefore,
we propose to fit to a user-generated dataset, a deep neural network, whose input is the entire raw EGM signal. This
allows the transition from relying to manually engineered features [
14
], to identifying important EGM parts and
morphology during the training process [
15
]. Furthermore, the network is trained to learn the spatial relationship
between neighbouring electrodes, through the use of convolutions, to infer the local anisotropy of the substrate. In
cardiology, deep learning techniques have been successfully used with electrocardiogram (ECG) data for automatic
analysis and diagnosis [
16
]. Hybrid datasets of simulated and clinical intracardiac EGMs have been explored in their
ability to classify patient tissue as fibrotic or non-fibrotic [
17
]. However to the best of our knowledge, there has been no
published attempt to quantitatively predict the anisotropic and heterogeneous diffusivity tensor at every point of an
unseen field/tissue, in simulated or biological data. In the present work, we predict unseen, simulated diffusivity tensor
fields from the respective simulated EGMs.
2 Methods
The outline of the steps we follow are described below and are illustrated diagrammatically in Figure 1. First, we
generate a scar maps set of synthetic myocardial tissue, consisting of fields with a variety of diffusivity tensors (scar
maps). A description of how we generate the scar maps is detailed in Section 2.4. Next, on each of these generated
fields, we simulate AP propagation by solving a cardiac EP model, described below in Section 2.1, and compute
synthetic EGMs by placing virtual electrodes on top of the field, in a grid arrangement consistent with a clinical HD
grid catheter. The final dataset is a set of input-output pairs, where one input is a set of EGM time series, and the output
its corresponding scar-map. The mathematical formulation of this inverse problem is:
F(D(x); Vm(x, t)) = 0,x∈Ω,(1)
B(Vm(x)) = 0,x∈∂Ω,(2)
where,
Ω
is the finite computational domain with boundary
∂Ω
,
F(·)
is the monodomain model,
Vm
is the transmem-
brane potential,
D(x)
is the true diffusivity tensor field, and
B(·)
are the imposed boundary conditions. The goal is to
2
APREPRINT - DECEMBER 7, 2022
ti
ti
a) Cardiac cell
model simulation
b) Synthetic EGM
generation
d) Estimation of
scar map from EGMs
HD grid
c) Network training
with EGMs as input
AP propagation
Predicted
scar map
Synthetic data generation Network training and inference
Scar map
Electrode grid e) Comparison to
ground truth and
statistical evaluation
of prediction
EGM signals
example at time
Figure 1: Schematic of the pipeline.
a-b
: First we generate the data by solving an AP propagation model in a specific
domain characterized by a diffusivity tensor. From the AP simulation, we calculate the EGMs on an electrode grid,
which has a density similar to the HD grid as shown in the magnification. The calculation is described in detail in
Section 2.1. Virtual electrodes record EGMs simultaneously.
c-e
: We use the generated EGMs in the training of a
neural network that estimates the scar map of the domain from which the EGMs have been recorded, and evaluate the
predictions using statistical methods.
then train a network,
h:E(Vm(x, t)) 7→ ˜
D(x),(3)
where
E(·)
is the EGM calculation shown in Equation 6, and
h
is a deep neural network with parameter set
θ
that
maps the EGMs to an estimate of the diffusivity tensor
˜
D(x)
. We train the network using the EGM time series,
E(Vm(x;t)) ∈Rc×c×T
, to find
h∗(E, θ∗)
with the optimal set of parameters
θ∗
that predicts the diffusion tensor fields
˜
D(x)∈Rn×n×3
, where the third dimension corresponds to the three components of the symmetric diffusion tensor.
Finally,
n×n
is the dimension of the discrete field and
c×c
is the number of electrodes, each of which provides an
EGM sequence, and Tis the duration of the input. EGMs are sampled from multiple electrodes concurrently.
2.1 Cardiac EP model
We aim to validate our hypothesis in a simulated setting. For this purpose, we use the simplified three-variable Fenton-
Karma-Cherry model [
18
], to produce simulations in a synthetic
12 ×12 cm
square region,
Ω
. The monodomain
partial differential equation that governs the action potential propagation, denoted by the function
F
in Equation 4, is
given by:
∂Vm(x, t)
∂t =∇ · (D(x)∇Vm) + Iion +Istim
Cm
(4)
where the diffusion term
∇ · (D(x)∇Vm)
propagates the AP through the myocardium, and the reaction term
Iion
is
responsible for generating the APs by modelling the opening and closing of the ion gates. The diffusivity tensor is
related to the conductivity as
D(x) = 1
βCmG(x)
, where
β
is the cellular surface-to-volume ratio and
Cm= 1 µF/cm2
[
19
,
20
] represents the membrane capacitance.
Istim
corresponds to the injected stimuli current to initiate and maintain
AP propagation. In the Fenton-Karma-Cherry model,
Iion
is modelled with three currents and three state variables,
one of which is the transmembrane voltage
Vm
. Although mathematically simple, this model can accurately reproduce
ventricular AP propagation and so provides a favourable starting point from a methodological perspective.
It is given by the equations:
3
APREPRINT - DECEMBER 7, 2022
0 50 100
x [mm]
0
20
40
60
80
100
120
y [mm]
Conductivity [
10
−
2
cm
2
/ms]
0 50 100
0
20
40
60
80
100
120
100 ms
0 50 100
0
20
40
60
80
100
120
200 ms
0 50 100
0
20
40
60
80
100
120
300 ms
0 50 100
0
20
40
60
80
100
120
400 ms
0 50 100
0
20
40
60
80
100
120
500 ms
0.0
0.2
0.4
0.6
0.8
1.0
Figure 2: Diffusivity field and corresponding AP propagation in different instances of a simulation. The modelled
scar is shown in white, and is
90%
less conducting than healthy tissue (
D= 10−3cm2/ms
for healthy tissue and
D= 10−4cm2/ms
for modelled scar respectively. The effect of the low conductivity region can be seen in all of the
frames shown, slowing down the wavefront. For this particular example, we stimulate the tissue with a point stimulus at
the top left corner of the domain, resulting in a circular wavefront. All values are normalized from zero to one.
Iion =−(Jfi(u, v) + Jso(u) + Jsi (u, w)),
∂v
∂t =H(uc−u)(1 −v)/τ −
v(u)− H(u−uc)v/τ +
v,
∂w
∂t =H(uc−u)(1 −w)/τ −
w− H(u−uc)w/τ +
w,
Jfi(u, v) = −H(u−uc)(1 −u)(u−uc)(v/τd),
Jso(u) = H(uc−u)(u/τ0) + H(u−uc)(1/τr),
Jsi(u, w) = −(1 + tanh(k(u−usi
c)))w/(2τsi),
where
Jfi
is a fast inward current responsible for the depolarisation of the membrane,
Jso
is a slow outward current
responsible for the repolarisation of the membrane, and
Jsi
is a slow inward current that opposes
Jso
in the recovery
phase. We use
H
to denote the Heaviside step function.
u
is the dimensionless form of
Vm
, while
v
and
w
are auxiliary
variables that represent the biophysical state of each unit of myocardium alongside
u
. To revert from non-dimensional
forms, the relation is given by
Vm=u(Vfi −V0) + V0
, where
V0
is the membrane resting potential and
Vfi
the reversal
potential of Ifi. Similarly, Ji=Ii/(Cm(Vi−V0)) for i=f i, so, si.
We apply Neumann boundary conditions Bat the boundaries of the square area:
dVm
dn
∂Ω= 0 (5)
where nis the direction normal to the boundary.
In the isotropic setting where the tensor field is a scalar multiple
d
of the identity, we use
d= 10−3cm2/ms
for the
healthy tissue [
21
], while for the scar regions of low conductivity
d= 10−4cm2/ms
. An example of a diffusion
field and the corresponding propagation of the AP through the domain can be seen in Figure 2. Our numerical
solver is implemented using the software JAX for GPU-accelerated computations [
22
]. A second-order accurate finite
4
APREPRINT - DECEMBER 7, 2022
0 25 50 75 100 125 150 175 200
Time [ms]
−5
0
5
Voltage [mV]
x=15 mm ,y=15 mm
x=85 mm, y=28 mm
x=85 mm, y=85 mm
Figure 3: Contact electrogram signals acquired from different positions in the field shown in Figure 2. The second
channel from position
x= 85 mm, y = 28 mm
has a significantly lower output, since the electrode is positioned
within the scar region.
differences scheme is used for computing spatial gradients, and a forward Euler scheme for temporal derivatives, with
dx =dy = 0.01 cm
and
dt = 0.01 ms
respectively. The resulting simulations have a spatial resolution of
1200 ×1200
grid points, representing a tissue of 12 cm ×12 cm.
2.2 Electrogram calculation
In the clinical setting, EGM signals are recorded from the endocardium using mapping catheters. For the purposes of
this study, the EGM signals are calculated from the solutions to the monodomain equation, at specific positions on the
field. The extracellular potential
φe
is the sum of currents in the field, weighted by the inverse square distance from the
electrode [1], given by the surface integral:
E(Vm(x, t)) = φe(x, t) = ZΩ
∇Vm(x0, t)·(x−x0)
4πσe||x−x0||3dΩ(6)
where
∇Vm(x0, t)
is the spatial gradient of the transmembrane voltage in
Ω
, at time
t
,
x
is the location of the probe, and
σe
is the conductivity of the domain, with a typical value
σe= 20 mS/cm
[
23
,
20
]. The virtual probes are assumed to
have infinitesimal spatial extent (i.e. the virtual electrode radius is zero). For the calculation of the simulated EGM, a
second-order accurate finite differences scheme is used for the gradient of the voltage field, and the rectangle rule for
numerically calculating the integral over
Ω
. The virtual electrode is placed at a vertical distance of
z= 1 mm
above
the field to reduce computational artefacts and avoid a singularity. An example of EGM signals acquired at different
positions for the simulation shown in Figure 2, can be seen in Figure 3.
We calculate the EGM signal at a uniform rectangular grid of points, at a sampling rate of
1 ms
. This electrode grid
is coarser than the finite differences grid used to solve the monodomain model, and the choice of an inter-electrode
spacing of
4 mm
is inspired by modern grid catheters used in the clinical setting for contact EGM acquisition [
24
,
25
].
Each simulation results in 841 EGMs signals in total
2.3 Fully Convolutional Networks
We use a feed-forward fully convolutional neural network. A convolutional layer convolves multiple, parameterised
filters (kernels) with the input field, a process that extracts local spatial information from the input data. Since
convolution operations are invariant to the size of the inputs, the method is applicable to cardiac tissue of arbitrary sizes.
Convolutional networks have been very successful in tasks such as image and instance segmentation for scene parsing
or medical imaging analysis [
26
], due to their ability to represent and encode information from spatial fields. We use an
Encoder-Decoder architecture, similar to a U-Net [
26
], without the skip connections. The encoder projects the input
into subsequent lower-dimensional feature spaces that extract useful spatial and temporal information. These are then
parsed through the decoder for upscaling through transposed convolutions and upsampling. The architecture is shown
in Figure 4.
We use the
H1×W1
grid of electrogram signals as the input to the model, together with the normalised coordinate of
each electrode. The three-dimensional input has height
H1
and width
W1
corresponding to the size of the electrode
grid. Each of the
N
time points from these signals is a separate channel. Since we are dealing with spatiotemporal
data, the absolute
x
and
y
coordinates of both the EGM probes and the predicted diffusion tensor values are important
information to capture in the model. We use a CoordConv [
27
] layer instead of the traditional convolutional layer in
the first and last layers of the model to support this. The CoordConv variation is implemented by adding two extra
channels to the input and output of the network, that contain the
x
and
y
coordinates of the electrode positions (in
5
APREPRINT - DECEMBER 7, 2022
the input) and the diffusion tensor field units (in the output) respectively. The first, second and third layers of the
encoder extract information from the signal to higher-order feature spaces of depth 60, 120 and 240 respectively, while
the depth of the decoder layers is 120, 60, 30, 15 and finally the output has a depth of 5 channels; the first three
channels represent the fields
Dxx
,
Dyy
and
Dxy
of the diffusion tensor, and the last two channels contain the
x
and
y
coordinates of the electrode probe corresponding to each grid unit (CoordConv variation). The decoder also upsamples
the information to a resolution of
96
by
96
. Every convolutional layer is followed by a batch normalization layer and
a ReLU activation layer. The EGM signals are normalised, and Gaussian white noise with a standard deviation of
0.05,
ˆ
φe=φe+N(0,0.05)
, is added to improve the robustness of the model, where
φe
is the z-score normalized
EGM signal and
N(0,0.05)
represents the Gaussian noise. Noise is similarly also added to the CoordConv layers. We
train the network for 100 epochs, and use the variation of Adam [
28
] with weight decay [
29
], with a learning rate of
0.001
and weight decay of
0.01
. We use the root mean square error (RMSE) as the loss function. We formalise the
optimisation problem as follows:
h∗(E, θ) = argmin
θ
E
(E,D(x))∼D qkh(E, θ)−D(x)k2
2(7)
where
h∗(E, θ) = h(E, θ∗)
denotes the network after 100 iterations of Adam, and
D
is the dataset as described above.
To evaluate the accuracy of scar estimation, the Jaccard index is used,
J(s, ˆs) = |s∩ˆs|
|s∪ˆs|
, where
s, ˆs
is the true scar
map and its prediction respectively, expressing the percentage of scar tissue that is identified correctly by the network.
During training, we use 10-fold cross validation to assess the performance of the model [30].
2.4 Scar and Fibre Maps
22 × 𝐻!× 𝑊
!
60 × 𝐻"× 𝑊
"
120 × 𝐻#× 𝑊
#
240 × 𝐻$× 𝑊
$
120 × 𝐻%× 𝑊
%
60 × 𝐻&× 𝑊
&
30 × 𝐻'× 𝑊
'
15 × 𝐻(× 𝑊
(
5 × 𝐻)× 𝑊
)
𝐶𝑜𝑛𝑣 3𝑥3 +𝐵𝑎𝑡𝑐ℎ𝑁𝑜𝑟𝑚 +𝑅𝑒𝐿𝑈
𝑇𝐶𝑜𝑛𝑣 3𝑥3 +𝐵𝑎𝑡𝑐ℎ𝑁𝑜𝑟𝑚 +𝑅𝑒𝐿𝑈
Figure 4: Architecture of the Encoder-Decoder network that
was used. In this configuration, each electrogram signal
contains 20 time points. The extra 2 channels of the input
are the normalized xand ycoordinates of the probes.
Each simulation of the generated dataset is character-
ized by a different diffusivity tensor field, representing
a distribution of scar and fibre orientation. Isotropic het-
erogeneous, anisotropic homogeneous and anisotropic
heterogeneous virtual substrates are considered. In the
first two cases, our architecture is evaluated on the estima-
tion of scar location and fibre orientation independently;
Figure 5 and Figure 6 show examples of generated maps
for these two cases. In the third case, both the scar loca-
tion and fibre orientation are estimated in combination.
For the isotropic scar maps, the low-conductivity regions
are representative of compact fibrosis patterns [
31
]. Such
anatomical barriers can be critical for the onset and perpet-
uation of AF, as they can be the cause of single reentrant
circuits [
32
]. Other types of cardiac fibrosis patterns, in-
cluding the more common interstitial and patchy types,
result from collagen bundles separating the excitable my-
ocytes. The degree of fibrosis depends on the ratio of
collagen bundles to myocytes. For such cases, the local
average of the conductivity would need to be represented
in the diffusivity map, but this configuration is beyond
the scope of the current work.
Non-scar
Scar
Figure 5: Examples of heterogeneous isotropic conductivity fields with compact regions of fibrosis. White regions
represent scar and have a lower conductivity value compared to the black regions, which represent healthy tissue.
Homogeneous anisotropic maps model the fibre orienta-
tion of cardiomyocytes and are used to test the ability of the neural network to estimate it from sparse EGM recordings.
6
APREPRINT - DECEMBER 7, 2022
Orientation
90o
0o
-90o
1
2
3
4
5
Figure 6: Examples of homogeneous anisotropic fibre orientation fields. The lines depict local fibre orientation. On the
first example, we show the five control points and resulting spline which is used to generate the scar map.
Anisotropy is introduced by the differentiation of longitudinal versus transversal conduction in cardiac myocytes;
dl
and
dt
respectively. To produce the anisotropic maps, first the longitudinal conduction
dl(x)
is defined across the
field, together with the ratio
λ(x) = dl(x)
dt(x)
and the fibre angle
α(x)
in degrees. The angle
α(x)
is generated through a
rule-based approach [
33
]. First, a path between two points on the field is constructed randomly, defined by a spline
passing through five control points
(xi, yi)i= 1, ..., 5
, where
xi
are equidistantly spaced, with
x1, x5
being the two
initial points, and
yi∼ N (0,0.09)
. The fibre orientation of the whole field is then calculated by extending the path
perpendicularly to the line passing through the two initial points. The selection of five control points for the spline
produces fields that are complex enough to robustly asses the performance of the network. Examples of generated
anisotropic fields can be seen in the bottom row of Figure 5. In the homogeneous tissue
λ(x)=4
throughout the field,
and dl(x) = 10−3cm2/ms. In the heterogeneous cases, dl(x)is equivalent to a heterogeneous isotropic scar map.
The diffusion tensor of each simulation is represented by three matrices
Dxx
,
Dyy
and
Dxy
, with dimensions equal
to the dimensions of the field, where each
i, j
-entry contains the corresponding
dxx
,
dyy
and
dxy =dyx
values of the
diffusion tensor at that location. For isotropic fields,
Dxx =Dyy
and
Dxy = 0
. These three matrices are the output of
the deep neural network.
2.5 Surrogate Testing Analysis
Although the loss metric RMSE can provide some evidence that the model generalizes well, it is not in itself sufficient
to evaluate the predictive performance of the network. In order to statistically validate that the predictions made by
the network are significant, they are compared to other feasible solutions [
34
]. We use surrogates to do this [
35
]. A
simple approach when using surrogate testing methods is to produce surrogates by randomly permuting the elements of
the vectors or matrices. However, this method does not preserve the spatial autocorrelation of the scar maps, which
in both isotropic and anisotropic cases is high and must be taken into consideration; a random permutation results in
non-feasible noisy solutions, leading to an overestimation of the significance of our predictions [
36
]. To create surrogate
samples that preserve autocorrelation, we use the dual-tree complex wavelet transform (DT-CWT) which has been
successfully used in similar tests [37].
We produce proxy fields from the predictions as feasible outputs of the network, as mentioned in section 2.5. The
proxies represent surrogate data that permit a form of permutation test; each proxy field represents a potential draw of
possible values of
Dxx
values over the locations of space corresponding to the output map. Like a permutation test,
each of these draws represents the data under the null hypothesis, which is that the model produces estimates of
Dxx
that are unrelated to its true spatial distribution. However, unlike a permutation test, we do not merely scramble the
values, but instead select only permutations that have a similar spatial structure to the real target spatial distribution
(ground truth map).
3 Results
3.1 Dataset Generation
The generated dataset consists of 330 simulations, each of which is defined by a different diffusion tensor field: 107
heterogeneous isotropic; 186 homogeneous anisotropic; and 36 heterogeneous anisotropic fields. Examples of the
fields used in the simulations are shown in Figure 5 and Figure 6. Heterogeneous anisotropic fields are obtained as
superpositions of heterogeneous isotropic and homogeneous anisotropic fields (scar and variable fibre orientation).
7
APREPRINT - DECEMBER 7, 2022
Each simulation has a duration of
1 s
, paced from a
10×10 mm
rectangular region the top left corner of the domain with
a pulse-wave signal of period
150 ms
. The pacing region can be seen in the bottom-left frame of Fig 2. Electrograms
are recorded at a frequency of
1 kHz
and so the interval between time points is
∆t= 1 ms
. Therefore, the available
data for every simulation consists of
841
unipolar electrogram recordings arranged spatially in a
29 ×29
electrode
grid, as shown in Fig 1, with each recording including
1000
time points. This data is structured in a three-dimensional
1000 ×29 ×29 array.
Input samples for the network are then obtained by extracting subsets of these data. Three parameters define how
these subsets are selected:
N
, the number of time points in each sample;
Nt
, the number of electrogram time points
between the time points of the sample; and
Nτ
, the number of time points between the starting points of subsequent
samples from the same simulation. A choice of
Nt= 1
will retain each consecutive time-point of the electrograms,
while
Nt= 2
will retain every alternate sample from the signal, for example. An example, with
N= 17
,
Nt= 3
,
and
Nτ= 22
can be seen in Fig 7. Suppose the electrogram signal at electrode
(i, j)
of simulation
k
, is denoted by
Figure 7: Parameters N,Nt, and Nτ, defining how we sample from the EGM time series.
φk
e(xij , t), then the m-th sample of simulation k,Sk
mis defined as
Sk
m=φk
e(xij ,(mNτ+nNt)∆t)|n∈[1..N], i, j ∈[1..29]
where
[a..b]
denotes the range of integers between
a
and
b
inclusive. Three EGMs from one such sample can be seen in
Fig 8, where Nt= 1,N= 600.
We can therefore obtain
ν=jL−(N−1)Nt
Nτ+ 1k
EGM input samples, where
L= 1000
is the total number of time
points in the simulated electrograms. The model produces one estimation of the diffusion field for every sample, so the
final prediction is calculated as the average estimate across all samples from the same simulation.
3.2 Network Performance
The performance of the network in four different cases is evaluated, depending on the type of diffusion field included in
the training and testing datasets: we have four different modes where we train and test on heterogeneous isotropic (HeI);
homogeneous anisotropic (HoA); heterogeneous anisotropic (HeA); and the combined case where all simulations are
used (C). In each of these cases, we do a grid-search optimization for the sampling parameters
N
,
Nt
,
Nτ
mentioned in
the previous section 3.1. Based on the average testing set error in the final predicted diffusion tensor fields across all
folds, the best performing combination in all the cases is
N= 10, Nt= 5, Nτ= 25
. Given the parameter set and the
dimensions of the field, the wavefront created from a point stimulus originating on the border of the field takes
200 ms
to travel to the opposite side. This implies that a sample spanning
50 ms
contains information about approximately a
quarter of the field, so the parameter choice makes sense from a physical perspective. The training and loss curves
for this best-performing parameter set are presented in Figure 10A, showing a small generalization error across all
cases. As previously mentioned in section 2.3, the CoordConv layers are implemented to explicitly include spatial
information about the position of the electrodes. To validate our selection, the error in the predictions of the network
when CoordConv is not implemented is increased by 55%, and the respective loss curves are shown in Fig 10B.
Reconstructed isotropic and anisotropic fields are shown in Figures 11 and 12, respectively. In the heterogeneous
isotropic case a Jaccard index of
j= 0.91
is obtained. To further validate the model, we compare the predicted fields
against the average scar map that is produced from our scar map generators, to ensure that the estimator does not simply
converge to predict the mean of the scar maps set. The predictions are significantly different from the average with
pval = 6 ×10−8
; as this can be an overestimation of the performance of the network, we present in the next section a
more conservative evaluation.
8
APREPRINT - DECEMBER 7, 2022
0 60 120
x [mm]
0
60
120
y [mm]
200ms 400ms 600ms 800ms
200 300 400 500 600 700 800
Time [ms]
−10000
−5000
0
5000
Voltage [mV]
x=15 mm ,y=15 mm
x=85 mm, y=28 mm
x=85 mm, y=85 mm
0.00
0.25
0.50
0.75
1.00
A
B
Figure 8: Sample from the training dataset.
A
: Sequence of AP propagation sampled from one simulation. The point
stimulus is introduced in the top left corner of the field as seen in the first frame.
B
: Examples of electrogram signals
acquired from the simulation shown above. The electrograms are sampled every 1
ms
and for a total of 600
ms
for
every sample.
3.3 Surrogate testing analysis
For each testing simulation, 100 surrogate fields are produced from the network output. Every surrogate and prediction
is compared to the corresponding ground truth conductivity map. The RMSE is used as the discriminating statistic. We
then calculate the percentile of the prediction in the surrogate distribution. The median percentile
P
for the testing set
against the surrogate distributions is
0.0
, and the distribution of
dprediction
compared to
dsurrogates
has a significantly
smaller mean with a pval = 0.007.
4 Discussion
The results presented in this paper serves a dual purpose. From a machine learning perspective, it provides an initial
proof of concept that deep neural networks can be used to solve inverse problems for unsteady, previously unseen,
cardiac action potential systems modelled by partial differential equations, such as those contained in the specific
cardiac model used in this work. From a biological perspective, it confirms the hypothesis that electrogram signals can
be used to recover structural information about the underlying cardiac substrate in an in silico setting despite the level
of spatial information content being lower than that of all individual contributing APs within the whole tissue. This
includes identifying compact non-conducting fibrotic regions, and fibre orientation, using a spatial EGM resolution that
is consistent with clinical high-density mapping catheters, opening potential avenues for more practical applications.
Although similar networks are being increasingly used with impressive performance as solvers, to predict future states
of dynamical systems from past states [
38
], their ability to solve inverse problems [
39
] has not been explored as much,
especially for time-dependent PDEs. Similar inverse problems have been solved using a denser electrogram mapping
grid and traditional optimization techniques [
40
], however, the investigated cases have been significantly simpler and
required a large number of electrograms. Other methods make use of activation times or APs rather than electrogram
recordings [
41
], [
42
] which is the clinical raw data recorded and allows hypothesis-free interrogation by the network.
Furthermore, this work allows for the estimation of conductivity fields in unknown domains by relying only on the
electrogram signals, in contrast to the other existing methods which interpolate within the same domain, and may
require information like system-specific properties [43].
The choice of working with electrogram signals instead of the AP is a practical one and for the purposes of future
translation; in a biological or clinical setting, contact electrograms can be acquired from the surfaces of the myocardium
by mapping catheters. Conversely, directly acquiring the AP propagating through the cardiac muscle requires the use of
9
APREPRINT - DECEMBER 7, 2022
HeI HoA HeA C
Training Mode
N
= 10
,
Nt
= 5
,
Nτ
= 25
N
= 5
,
Nt
= 10
,
Nτ
= 25
N
= 20
,
Nt
= 5
,
Nτ
= 50
N
= 10
,
Nt
= 10
,
Nτ
= 50
N
= 5
,
Nt
= 20
,
Nτ
= 50
N
= 5
,
Nt
= 10
,
Nτ
= 50
N
= 50
,
Nt
= 5
,
Nτ
= 100
N
= 10
,
Nt
= 20
,
Nτ
= 100
N
= 10
,
Nt
= 10
,
Nτ
= 100
N
= 20
,
Nt
= 10
,
Nτ
= 100
N
= 20
,
Nt
= 5
,
Nτ
= 100
N
= 20
,
Nt
= 20
,
Nτ
= 100
N
= 5
,
Nt
= 20
,
Nτ
= 100
N
= 50
,
Nt
= 10
,
Nτ
= 50
N
= 10
,
Nt
= 20
,
Nτ
= 200
N
= 20
,
Nt
= 10
,
Nτ
= 200
Setting
0.18 0.16 0.2 0.17
0.18 0.17 0.2 0.17
0.17 0.19 0.21 0.17
0.17 0.19 0.2 0.17
0.17 0.19 0.2 0.17
0.17 0.18 0.21 0.17
0.17 0.21 0.23 0.18
0.17 0.22 0.24 0.18
0.17 0.21 0.24 0.18
0.17 0.22 0.23 0.18
0.17 0.21 0.24 0.18
0.17 0.23 0.24 0.18
0.17 0.21 0.25 0.18
0.17 0.22 0.2 0.18
0.18 0.25 0.3 0.21
0.18 0.25 0.3 0.21
RMSE
0.18
0.20
0.22
0.24
0.26
0.28
0.30
Figure 9: Average RMSE across all folds in the testing set. Modes refer to the training and testing scar map dataset
used: C = Combined cases of all fields, HeA = Heterogeneous Anisotropic fields, HoA = Homogeneous Anisotropic
fields, HeI = Heterogeneous Isotropic fields.
techniques which, in some cases, are impossible in a clinical setting, such as ex vivo optical mapping. As mentioned
in section 2.1, the density of the electrogram grid is selected, so that each smaller grid of 4 by 4 electrode probes is a
representation of a modern state-of-the-art catheter mapping system (HD grid, Abbott Medical). While generating the
simulations, a grid discretization of
dx = 0.01 cm
is used, while in the EGM grid, the spacing is
dxEGM = 0.4 cm
,
and the output has a discretization of
dx = 0.125 cm
. Therefore, by considering only the EGM signals, we are working
with a coarse grain approximation of the field, with the input grid containing
292/962= 0.91
or
9.1%
of the spatial
information of the output tensor. Given this reduction in information and resolution, the accuracy in the identification of
the compact scar regions demonstrates the effectiveness of our approach.
In this in silico investigation, the generated data is produced using a specific set of parameters for the Fenton-Karma AP
model. We do not consider the performance of the model in predicting conductivity fields from simulations generated
from a range of AP parameter values, as this is beyond the scope of this paper. Subsequent experimentation could be
conducted to investigate how much impact the range of the model parameters has on the predictive ability of the model.
Furthermore, it can be seen quantitatively and qualitatively in section 3.2 that isotropic fields with compact patches are
better estimated than the fibre orientations of the anisotropic fields. One explanation for this is the observation that the
compact fibrotic areas have a greater impact in wave propagation than gradually changing fibre orientations. As the
next step, implementation on biologically informed fibre orientation fields will verify that the model can perform as
well as it does in the case of compact substrate.
There is not a complete understanding of how beneficial contact EGMs can be in the identification of target regions for
ablation, or how much of an improvement they provide over standard procedures such as pulmonary vein isolation
(PVI). For a while, many clinicians considered complex fractionated atrial electrograms (CFAEs) to indicate candidate
target sites for AF [
5
], although that is now met with skepticism [
8
,
7
]. There have been studies that suggest that
EGM-guided ablation provide no added benefit over PVI [
44
]. In this work, the hypothesis is not that the morphology
of EGMs directly indicates candidate regions for ablation, but rather that an array of concurrent EGMs can be used to
determine properties of the underlying substrate and therefore indirectly steer ablation targeting of pathophysiological
myocardium[15] that disrupts the correct propagation of the cardiac electrical signal.
10
APREPRINT - DECEMBER 7, 2022
0 20 40 60 80 100
Epochs
0.1
0.2
0.3
0.4
0.5
Loss
(RMSE)
A
Train
Test
C
HeA
HoA
HeI
0 20 40 60 80 100
Epochs
0.2
0.3
0.4
0.5
Loss (RMSE)
B
Train
Test
w/o CoordConv
with CoordConv
Figure 10: Training and validation loss curves for the model. (Left) The different colors correspond to the performance
of the model when we consider specific subsets of the scar map dataset for the training and testing: C = Combined cases
of all fields, HeA = Heterogeneous Anisotropic fields, HoA = Homogeneous Anisotropic fields, HeI = Heterogeneous
Isotropic fields.
0 100
0
50
100
Ground Truth
0 100
0
50
100
Prediction
0 100
0
50
100
Absolute error
0.5
1.0
10
−
3
cm
2
/ms
0.25
0.50
0.75
10
−
3
cm
2
/ms
0.5
1.0
10
−
3
cm
2
/ms
Figure 11: Predicted isotropic fields with fibrotic regions. The ground truth is shown for comparison. The absolute
error is larger near the boundaries of the scar.
In our work, only unipolar EGMs are considered. The reasoning behind this choice is that the raw signal is used as the
input into the deep learning models. Most state-of-the-art techniques that seek to estimate the direction of the wavefront,
which are used in the clinic usually consider the bipolar electrogram or the relatively new omnipolar mapping [
24
]. The
bipolar EGM is the result of the difference of two unipolar EGMs where the electrodes are placed in close proximity to
one another, with the usual inter-electrode distance currently being close to
2 mm
. The popularity of the bipolar EGM is
based on its filtering properties; when properly positioned and oriented, a bipolar EGM will cancel out far-field signals
or other sources of noise [
45
] due to simultaneous detection at both electrodes. While widely used for catheter mapping
and ablation, the morphology and properties of the bipolar EGM, like its unipolar components, is still not yet entirely
understood. It has been suggested that bipolar and unipolar EGMs can be used together for optimal identification of
ablation targets [
45
]. Should the approach proposed here work in a clinical setting, signals from electrogram arrays
will be used for the estimation of scar location. In our model the network considers the juxtaposition of neighbouring
electrodes, thereby, while not directly considering bipolar EGMs, may benefit from similar advantages of proximity.
5 Conclusion
This work verifies the hypothesis that, in an in silico model, electrogram recordings can be used in conjunction with
deep neural networks to estimate the conduction properties of the underlying myocardium. Although the electrogram
recordings, conductivity and fibre orientation of biological samples are significantly more complex and noisy, the
principle of the deep neural network as an inverse solver remains the same. Applying the predictive model to biological
and/or clinical settings requires further investigation and data and is beyond the scope of this paper.
11
APREPRINT - DECEMBER 7, 2022
0 100
0
50
100
Ground Truth
0 100
0
50
100
Prediction
0 100
0
50
100
Absolute Error
0 100
0
50
100
0 100
0
50
100
0 100
0
50
100
−50
0
50
Degrees
[
◦
]
0
100
Degrees
[
◦
]
−50
0
50
Degrees
[
◦
]
−50
0
50
Degrees
[
◦
]
0
100
Degrees
[
◦
]
−50
0
50
Degrees
[
◦
]
Figure 12: Examples of two predicted anisotropic fields (top and bottom). Lines have been superimposed to make the
visualization of the prediction easier.
0 50 100
0
25
50
75
100
0 50 100
0
25
50
75
100
0 50 100
0
25
50
75
100
0 50 100
0
25
50
75
100
0 50 100
0
25
50
75
100
0 50 100
0
25
50
75
100
0.2
0.4
0.6
0.8
1.0
10
−
2
cm
2
/ms
D
(
x
)
Figure 13: Examples of autocorrelation-preserving surrogates.
Top
: Three examples of scar maps, with different
degrees of spatial autocorrelation. Bottom: The respective surrogates produced via the DT-CWT method.
6 Acknowledgements
This work was supported by the Wellcome Trust under Grant 222845/Z/21/Z. For the purpose of open access, the author
has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
12
APREPRINT - DECEMBER 7, 2022
7 Author Contributions
K.N., C.C, R.C, and A.B conceptualized the study. K.N. generated the simulated data, built and trained the model
and analyzed the results. K.N., and E.P. wrote the software to generate the simulated data. N.P. and R.C. contributed
biological expertise. K.N., E.P., C.C., R.C. and A.B. wrote the manuscript.
8 Declaration of interests
The authors declare no competing interests.
9 Data availability
Code to reproduce all the simulations, models, and analysis is provided upon request, and will be made available upon
acceptance of this manuscript.
References
[1] R. Plonsey and Roger Barr. Bioelectricity: A Quantitative Approach. Springer, 2000.
[2]
Rasheda A. Chowdhury, Konstantinos N. Tzortzis, Emmanuel Dupont, Shaun Selvadurai, Filippo Perbellini,
Chris D. Cantwell, Fu Siong Ng, Andre R. Simon, Cesare M. Terracciano, and Nicholas S. Peters. Concurrent
micro- to macro-cardiac electrophysiology in myocyte cultures and human heart slices. Scientific Reports,
8(1):6947, 2018.
[3]
Flavio Fenton and Alain Karma. Vortex dynamics in three-dimensional continuous myocardium with fiber rotation:
Filament instability and fibrillation. Chaos: An Interdisciplinary Journal of Nonlinear Science, 8(1):20–47, 1998.
[4]
Ramanathan Parameswaran, Ahmed M. Al-Kaisey, and Jonathan M. Kalman. Catheter ablation for atrial
fibrillation: current indications and evolving technologies. Nature Reviews Cardiology, 18(3):210–225, Mar 2021.
[5]
Atul Verma, Roberto Mantovan, Laurent Macle, Guiseppe De Martino, Jian Chen, Carlos A. Morillo, Paul Novak,
Vittorio Calzolari, Peter G. Guerra, Girish Nair, Esteban G. Torrecilla, and Yaariv Khaykin. Substrate and Trigger
Ablation for Reduction of Atrial Fibrillation (STAR AF): a randomized, multicentre, international trial. European
Heart Journal, 31(11):1344–1356, 03 2010.
[6]
Wenhai Li, Cuiwei Yang, Yanlei Wang, Dexi Wang, Ying Chen, and Zhong Wu. Several insights into the
preprocessing of electrograms in atrial fibrillation for dominant frequency analysis. BioMedical Engineering
OnLine, 15(1):38, Apr 2016.
[7]
Omer Berenfeld and José Jalife. Complex fractionated atrial electrograms: is this the beast to tame in atrial
fibrillation? Circulation. Arrhythmia and electrophysiology, 4(4):426–428, 08 2011.
[8]
Jane Caldwell and Damian Redfearn. Ablation of complex fractionated atrial electrograms in catheter ablation for
AF; where have we been and where are we going? Current cardiology reviews, 8, 2012.
[9]
Lu Lu, Raphael Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, and Steven G. Johnson. Physics-
informed neural networks with hard constraints for inverse design. 2021.
[10]
Housen Li, Johannes Schwab, Stephan Antholzer, and Markus Haltmeier. Nett: solving inverse problems with
deep neural networks. Inverse Problems, 36(6):065005, jun 2020.
[11]
Kyong Hwan Jin, Michael T. McCann, Emmanuel Froustey, and Michael Unser. Deep convolutional neural
network for inverse problems in imaging. IEEE Transactions on Image Processing, 26(9):4509–4522, 2017.
[12]
Makoto Takamoto, Timothy Praditia, Raphael Leiteritz, Dan MacKinlay, Francesco Alesiani, Dirk Pflüger, and
Mathias Niepert. PDEBench: An extensive benchmark for scientific machine learning. In Thirty-sixth Conference
on Neural Information Processing Systems Datasets and Benchmarks Track, 2022.
[13]
Bethany Lusch, J. Nathan Kutz, and Steven L. Brunton. Deep learning for universal linear embeddings of nonlinear
dynamics. Nature Communications, 9(1):4950, Nov 2018.
[14]
Joseph Brook, Min young Kim, Simos Koutsoftidis, David Pitcher, Danya Agha-Jaffar, Annam Sufi, Catherine
Jenkins, Konstantinos Tzortzis, Suofeiya Ma, Richard J. Jabbour, Charles Houston, Balvinder S. Handa, Xinyang
Li, Ji Jian Chow, Anand Jothidasan, Poppy Bristow, Justin Perkins, Sian Harding, Anil A. Bharath, Fu Siong
Ng, Nicholas S. Peters, Chris D. Cantwell, and Rasheda A. Chowdhury. Development of a pro-arrhythmic ex
13
APREPRINT - DECEMBER 7, 2022
vivo intact human and porcine model: cardiac electrophysiological changes associated with cellular uncoupling.
Pflugers Archiv European Journal of Physiology, 472(10):1435–1446, 2020.
[15]
Chris D. Cantwell, Yumnah Mohamied, Konstantinos N. Tzortzis, Stef Garasto, Charles Houston, Rasheda A.
Chowdhury, Fu Siong Ng, Anil A. Bharath, and Nicholas S. Peters. Rethinking multiscale cardiac electrophys-
iology with machine learning and predictive modelling. Computers in Biology and Medicine, 104:339 – 351,
2019.
[16]
Zahra Ebrahimi, Mohammad Loni, Masoud Daneshtalab, and Arash Gharehbaghi. A review on deep learning
methods for ECG arrhythmia classification. Expert Systems with Applications: X, 7:100033, 2020.
[17]
Jorge Sánchez, Giorgio Luongo, Mark Nothstein, Laura A. Unger, Javier Saiz, Beatriz Trenor, Armin Luik, Olaf
Dössel, and Axel Loewe. Using Machine Learning to Characterize Atrial Fibrotic Substrate From Intracardiac
Signals With a Hybrid in silico and in vivo Dataset. Frontiers in Physiology, 12(July):1–15, 2021.
[18]
Flavio H. Fenton, Elizabeth M. Cherry, Harold M. Hastings, and Steven J. Evans. Multiple mechanisms of spiral
wave breakup in a model of cardiac electrical activity. Chaos, 12(3):852–892, 2002.
[19]
Silvio Weidmann. Electrical constants of trabecular muscle from mammalian heart. The Journal of Physiology,
210(4):1041–1054, 2020/09/01 1970.
[20]
Karoline Horgmo Jæger, Andrew G Edwards, Andrew McCulloch, and Aslak Tveito. Properties of cardiac
conduction in a cell-based computational model. PLoS computational biology, 15(5):e1007042–e1007042, 05
2019.
[21]
R.H. Clayton, O. Bernus, E.M. Cherry, H. Dierckx, F.H. Fenton, L. Mirabella, A.V. Panfilov, F.B. Sachse,
G. Seemann, and H. Zhang. Models of cardiac tissue electrophysiology: Progress, challenges and open questions.
Progress in Biophysics and Molecular Biology, 104(1):22 – 48, 2011. Cardiac Physiome project: Mathematical
and Modelling Foundations.
[22]
James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, and Skye
Wanderman-Milne. JAX: composable transformations of Python+NumPy programs, 2018.
[23]
Jeroen G. Stinstra, Bruce Hopenfeld, and Rob S. MacLeod. On the passive cardiac conductivity. Annals of
Biomedical Engineering, 33(12):1743–1751, 2005.
[24]
N Karim, N Srinivasan, J Garcia, R Providencia, R Ang, M Dhinoja, R Hunter, P Waddingham, P Lambiase,
F Khan, S Ahsan, W Y Lim, J Behar, M Lowe, R Schilling, and A Chow. Early experience using the adivsor HD
grid to map atrial fibrilation. EP Europace, 20(suppl4):iv33–iv34, 10 2018.
[25]
Kathryn Hong, Damien Redfearn, Sanoj Chacko, Jason Baley, Adrian Baranchuk, and B. Glover. High resolution
mapping of the atria using the high definition grid catheter. HeartRhythm Case Reports, 5, 01 2019.
[26]
Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image
segmentation. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and
Lecture Notes in Bioinformatics), 9351:234–241, 2015.
[27]
Rosanne Liu, Joel Lehman, Piero Molino, Felipe Petroski Such, Eric Frank, Alex Sergeev, and Jason Yosinski. An
intriguing failing of convolutional neural networks and the coordconv solution, 2018.
[28]
Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980,
2014.
[29] Ilya Loshchilov and Frank Hutter. Fixing weight decay regularization in adam. CoRR, abs/1711.05101, 2017.
[30]
Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning. Springer Series in
Statistics. Springer New York Inc., New York, NY, USA, 2001.
[31]
Thao P. Nguyen, Zhilin Qu, and James N. Weiss. Cardiac fibrosis and arrhythmogenesis: The road to repair is
paved with perils. Journal of Molecular and Cellular Cardiology, 70:83–91, 2014.
[32]
M. A. Allessie, F. I.M. Bonke, and F. J.G. Schopman. Circus movement in rabbit atrial muscle as a mechanism
of tachycardia. III. The ’leading circle’ concept: a new model of circus movement in cardiac tissue without the
involvement of an anatomical obstacle. Circulation Research, 41(1):9–18, 1977.
[33]
Martin W. Krueger, Viktor Schmidt, Catalina Tobón, Frank M. Weber, Cristian Lorenz, David U. J. Keller,
Hans Barschdorf, Michael Burdumy, Peter Neher, Gernot Plank, Kawal Rhode, Gunnar Seemann, Damien
Sanchez-Quintana, Javier Saiz, Reza Razavi, and Olaf Dössel. Modeling atrial fiber orientation in patient-specific
geometries: A semi-automatic rule-based approach. In Dimitris N. Metaxas and Leon Axel, editors, Functional
Imaging and Modeling of the Heart, pages 223–232, Berlin, Heidelberg, 2011. Springer Berlin Heidelberg.
14
APREPRINT - DECEMBER 7, 2022
[34]
Véronique Peiffer, Anil A. Bharath, Spencer J. Sherwin, and Peter D. Weinberg. A novel method for quantifying
spatial correlations between patterns of atherosclerosis and hemodynamic factors. Journal of Biomechanical
Engineering, 135(2):1–11, 2013.
[35]
Victor Venema, Steffen Meyer, Sebastián Gimeno Garcìa, Anke Kniffka, Clemens Simmer, Susanne Crewell,
Ulrich Löhnert, Thomas Trautmann, and Andreas Macke. Surrogate cloud fields generated with the iterative
amplitude adapted fourier transform algorithm. Tellus A: Dynamic Meteorology and Oceanography, 58(1):104–
120, 2006.
[36]
Ethan M. Rowland, Yumnah Mohamied, K. Yean Chooi, Emma L. Bailey, and Peter D. Weinberg. Comparison of
Statistical Methods for Assessing Spatial Correlations Between Maps of Different Arterial Properties. Journal of
Biomechanical Engineering, 137(10), 08 2015. 101003.
[37]
Vincent Deblauwe, Pol Kennel, and Pierre Couteron. Testing Pairwise Association between Spatially Autocorre-
lated Variables: A New Approach Using Surrogate Lattice Data. PLoS ONE, 7(11), 2012.
[38]
Stathi Fotiadis, Eduardo Pignatelli, Mario Lino Valencia, Chris Cantwell, Amos Storkey, and Anil A. Bharath.
Comparing recurrent and convolutional neural networks for predicting wave propagation, 2020.
[39]
Haiyi Wu, Hongwei Zhang, Guoqing Hu, and Rui Qiao. Deep learning-based reconstruction of the structure of
heterogeneous composites from their temperature fields. AIP Advances, 10(4):045037, 2020.
[40]
Bahareh Abdi, Richard C. Hendriks, Alle Jan van der Veen, and Natasja M.S. de Groot. A compact matrix model
for atrial electrograms for tissue conductivity estimation. Computers in Biology and Medicine, 107(February):284–
291, 2019.
[41]
Thomas Grandits, Simone Pezzuto, Francisco Sahli Costabal, Paris Perdikaris, Thomas Pock, Gernot Plank,
and Rolf Krause. Learning atrial fiber orientations and conductivity tensors from intracardiac maps using
physics-informed neural networks. pages 1–8, 2021.
[42]
Clara Herrero Martin, Alon Oved, Rasheda A. Chowdhury, Elisabeth Ullmann, Nicholas S. Peters, Anil A. Bharath,
and Marta Varela. Ep-pinns: Cardiac electrophysiology characterisation using physics-informed neural networks.
Frontiers in Cardiovascular Medicine, 8, 2022.
[43]
Carlos Ruiz Herrera, Thomas Grandits, Gernot Plank, Paris Perdikaris, Francisco Sahli Costabal, and Simone
Pezzuto. Physics-informed neural networks to learn cardiac fiber orientation from multiple electroanatomical
maps, 2022.
[44]
Atul Verma, Chen-yang Jiang, Timothy R. Betts, Jian Chen, Isabel Deisenhofer, Roberto Mantovan, Laurent
Macle, Carlos A. Morillo, Wilhelm Haverkamp, Rukshen Weerasooriya, Jean-Paul Albenque, Stefano Nardi, Endrj
Menardi, Paul Novak, and Prashanthan Sanders. Approaches to catheter ablation for persistent atrial fibrillation.
New England Journal of Medicine, 372(19):1812–1822, 2020/08/29 2015.
[45]
Jacques M T de Bakker. Electrogram recording and analyzing techniques to optimize selection of target sites for
ablation of cardiac arrhythmias. Pacing and clinical electrophysiology : PACE, 42(12):1503–1516, 12 2019.
15
