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Optical Ultrastructure of Large
Mammalian Hearts Recovers
Discordant Alternans by In Silico Data
Assimilation
Alessandro Loppini
1
, Julia Erhardt
2
, Flavio H. Fenton
3
, Simonetta Filippi
1
, Marcel Hörning
2
*
and Alessio Gizzi
1
1
Nonlinear Physics and Mathematical Modeling Laboratory, University Campus Bio-Medico of Rome, Rome, Italy,
2
Biobased
Materials Laboratory, Institute of Biomaterials and Biomolecular Systems, Faculty of Energy, Process and Biotechnology,
University of Stuttgart, Stuttgart, Germany,
3
School of Physics, Georgia Institute of Technology, Atlanta, GA, United States
Understanding and predicting the mechanisms promoting the onset and sustainability of
cardiac arrhythmias represent a primary concern in the scientific and medical communities
still today. Despite the long-lasting effort in clinical and physico-mathematical research, a
critical aspect to be fully characterized and unveiled is represented by spatiotemporal
alternans patterns of cardiac excitation. The identification of discordant alternans and
higher-order alternating rhythms by advanced data analyses as well as their prediction by
reliable mathematical models represents a major avenue of research for a broad and
multidisciplinary scientific community. Current limitations concern two primary aspects: 1)
robust and general-purpose feature extraction techniques and 2) in silico data assimilation
within reliable and predictive mathematical models. Here, we address both aspects. At
first, we extend our previous works on Fourier transformation imaging (FFI), applying the
technique to whole-ventricle fluorescence optical mapping. Overall, we identify complex
spatial patterns of voltage alternans and characterize higher-order rhythms by a frequency-
series analysis. Then, we integrate the optical ultrastructure obtained by FFI analysis within
afine-tuned electrophysiological mathematical model of the cardiac action potential. We
build up a novel data assimilation procedure demonstrating its reliability in reproducing
complex alternans patterns in two-dimensional computational domains. Finally, we prove
that the FFI approach applied to both experimental and simulated signals recovers the
same information, thus closing the loop between the experiment, data analysis, and
numerical simulations.
Keywords: cardiac alternans, cardiac arrhythmias, optical mapping, frequency analysis, mathematical modeling,
data assimilation
1 INTRODUCTION
In nature, a broad variety of pattern formations can be found on very different length scales and
functions, such as collective behavior of fish swarms (Jakobsen and Johnsen, 1988), the animal skin
patterning (Murray, 2003;Miyazawa et al., 2010), the cell dynamics during embryogenesis (Ju et al.,
2017), the formation of neuronal networks in brains (van den Heuvel and Hulshoff Pol, 2010), and
the electromechanical function of the cardiovascular system (Christoph et al., 2018). The latter is
Edited by:
Ulrich Parlitz,
Max Planck Society, Germany
Reviewed by:
Blas Echebarria,
Universitat Politecnica de Catalunya,
Spain
Seth H. Weinberg,
The Ohio State University,
United States
*Correspondence:
Marcel Hörning
marcel.hoerning@bio.uni-stuttgart.de
Specialty section:
This article was submitted to
Networks in the Cardiovascular
System,
a section of the journal
Frontiers in Network Physiology
Received: 30 January 2022
Accepted: 04 March 2022
Published: 13 April 2022
Citation:
Loppini A, Erhardt J, Fenton FH,
Filippi S, Hörning M and Gizzi A (2022)
Optical Ultrastructure of Large
Mammalian Hearts Recovers
Discordant Alternans by In Silico
Data Assimilation.
Front. Netw. Physiol. 2:866101.
doi: 10.3389/fnetp.2022.866101
Frontiers in Network Physiology | www.frontiersin.org April 2022 | Volume 2 | Article 8661011
ORIGINAL RESEARCH
published: 13 April 2022
doi: 10.3389/fnetp.2022.866101
crucial to maintain life as we know but is susceptible to
malfunction due to its complex morphology such as the
vascular system (Luther et al., 2011), cellular orientation
(Papadacci et al., 2017), and mechano-electrophysiological
wave patterning (Hörning et al., 2012). Slight variations in the
organization of those patterns can have fatal consequences, and
thus, cardiovascular diseases are the primary cause of death in
industrial countries.
One of the complex and not fully understood heart behaviors,
possibly inducing cardiac disease, is alternans. It describes a
phase-dependent alternation on either a single-cell or tissue
level and can be described as a beat-to-beat alternation of
short and long heartbeats (membrane potential, intracellular
calcium) or myocyte contractions. Alternans is known to be
involved in a series of cardiovascular conditions as either
cause or symptom. These include, among others, ventricular
fibrillation, arrhythmias, and sudden cardiac death (Adam
et al., 1984;Konta et al., 1990;Pastore et al., 1999), especially
in patients after myocardial infarction (Ikeda et al., 2000). Other
triggers for alternans are ischemia of the myocardium, ectopic
heartbeats, and coronary occlusion (Green, 1935;Dilly and Lab,
1988;Taggart et al., 1996;Ren et al., 2011). In early studies,
alternans was observed in terms of myocardial contractility, aortic
pressure, and stroke volume (Mitchell et al., 1963). In medical
applications, this phenomenon has therefore been widely
employed as a predictive tool for determining risks for
fibrillation, venous thromboembolism, arrhythmia, and sudden
cardiac death (Dilly and Lab, 1988;Kim et al., 2009). Besides, it is
used to assess the necessity and urgency of certain surgical
operations, such as implantation of cardioverter defibrillators
(Merchant et al., 2012).
Several mechanisms have been revealed during three decades
of intensive research that can induce alternans. Early studies
stated the critical role of calcium cycling and electrical restitution
of the action potential in the generation of alternans (Badeer et al.,
1967;Dilly and Lab, 1988;Konta et al., 1990). Repolarization
gradients at the tissue level have further been shown to lead to
complex macroscopic voltage alternans patterns (Pastore et al.,
1999). Later, it was shown that fluctuations in the cyclic release of
Ca
2+
from the sarcoplasmic reticulum could lead to Ca
2+
alternans tightly coupled with voltage repolarization alternans
(Lab and Lee, 1990;Qu et al., 1999;Walker et al., 2003;Diaz et al.,
2004). Similar to that, a fine-scaled initiation of alternans was
linked to the subsequent formation of larger alternating regions
(Jia et al., 2010). Additionally, alternans can be promoted by low
temperature or application of drugs (Xie and Weiss, 2009;Gizzi
et al., 2017;Loppini et al., 2021).
While alternans can be observed at single cells for the action
potential duration (APD) and the calcium transient amplitude
(CTA), in tissue, those oscillations can synchronize and lead to
spatial concordant alternans (CA) or discordant alternans (DA)
(Uzelac et al., 2017). CA is observed when the entire tissue
alternates in phase, while DA is classified with at least two
out-of-phase oscillating regions (Xie and Weiss, 2009;Gizzi
et al., 2013;Gizzi et al., 2017) spatially separated by nodal
lines, i.e., non-alternating domains (Hörning et al., 2017). The
conduction velocity (CV) plays a crucial role in developing
alternans. Usually, concordant or discordant APD and CTA
depend on CV restitution (Karagueuzian et al., 2013). A
slowing of the CV, caused by the incomplete recovery of the
fast sodium current, concurs to promote large gradients of
repolarization, thus sustaining DA patterns. Furthermore,
alternans can be studied in terms of electromechanically out-
of-phase regions. In this case, larger CTAs are triggered by shorter
APDs and vice versa (Sato et al., 2006).
Based on the experimental finding, numerous computational
models have been developed that can show the onset and
transition of alternans in both single cells and tissue (Karma,
1993,1994;Qu et al., 1999;Watanabe et al., 2001;Cherry and
Fenton, 2004;Tao et al., 2008;Shiferaw et al., 2003;Huertas et al.,
2010). However, the striking limitation of the current modeling
approaches consists in the capability of reproducing complex DA
patterns in anatomically realistic computational domains. In
practice, the appearance of CA and DA in numerical
simulations requires, up to now, an ad hoc tuning of
physiological parameters, usually deviating from the optimal
set obtained from experimental CV and restitution curves
(Cairns et al., 2017). Innovative multiscale and multiphysics
formulations of cell–cell couplings aim at filling this gap. Non-
linear, stress-assisted, and fractional diffusion (Lin and Keener,
2010;Hurtado et al., 2016;Cherubini et al., 2017;Cusimano et al.,
2020;Cusimano et al., 2021), ephaptic and gap
junction–mediated couplings (Lenarda et al., 2018;Weinberg,
2017), cellular automata, and coarse-grained homogenized gap
junction approaches (Treml et al., 2021;Irakoze and Jacquemet,
2021) represent the state-of-the-art in this direction.
Furthermore, within the specific context of cardiac
electrophysiology, recent studies are proposing novel methods
of data estimation, data assimilation, and uncertainty
quantification (Barone et al., 2020a;Barone et al., 2020b;
Pathmanathan et al., 2020;Marcotte et al., 2021) to reproduce
complex cardiac dynamics with a reduced computational cost.
On such a ground, we propose an innovative data assimilation
technique using the optical ultrastructure obtained from a
frequency analysis of voltage fluorescence activations on intact
canine ventricles demonstrating its potential role in recovering
complex alternans patterns in silico. The results presented in this
study fundamentally advance the understanding of alternans.
Furthermore, the proposed observation strategy may enable
possible applications to personalized medicine, such as
quantifying alternans and higher-other rhythms without heavy
computational resources or massive experimental campaigns. As
the ultrastructure of the heart is unique for every subject, it may
be used as the base for studying possible diseased states and
treatments. Thus, this study lays the promising foundation for
such approaches in the near future.
This manuscript is organized as follows. Section 2 introduces
the Fourier-based method and the experimental data assimilation
technique in electrophysiological mathematical models. Section 3
demonstrates the ability of our frequency technique to recover
alternans in cardiac tissue at high-frequency pacing rates, and we
identify the optical ultrastructure to assimilate in silico. Besides,
the optimal combination of data assimilation heterogeneities
matches complex experimental alternans patterns at best.
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Loppini et al. Ultrastructure Recovers Alternans In-Silico
Section 4 closes the work with a discussion of limitations and
perspectives of the current approach.
2 MATERIALS AND METHODS
2.1 Experimental Setup
Right ventricle wedges from canine were prepared according to
the experimental protocols approved by the Institutional Animal
Care and Use Committee of the Center for Animal Resources and
Education at Cornell University. Fluorescence optical mapping
recordings of the membrane potential were recorded at a spatial
resolution of 600 × 600 μm
2
per pixel for a grid size of 7.7 ×
7.7 cm
2
and a temporal resolution of 2 ms at physiological
conditions. For details of the experimental setup information,
we refer to the previous studies (Fenton et al., 2009;Luther et al.,
2011;Gizzi et al., 2013;Gizzi et al., 2017).
2.2 Data Analysis
2.2.1 Fourier Transformation Imaging
Fourier transformation imaging was applied to the fluorescence
optical mapping recordings, as introduced before (Hörning et al.,
2017). The optical recordings were pixel-wise decomposed and
transformed to the mathematically complex Fourier space, F
x,y
(f),
as a function of the frequency f, i.e.,
Ix,y t
()→Fx,y f
,(1)
where I
x,y
(t) is the intensity at the spatial position (x,y) and tis the
time. From that, the amplitude |F
x,y
(f)| and the phase arg(F
x,y
(f))
are calculated and spatially recomposed to two respective Fourier
frequency-series.
2.2.2 APD Alternans Maps
Alternans maps were pixel-wise calculated on pre-analyzed
signals. Pre-analysis involves detrending, nearest-neighbor
averaging in time with a rectangular window (7 frames width),
and space filtering with Gaussian kernel (4 pixels radius). The
APD is the extracted by threshold crossing at 20% max(Ix,y ). The
temporal difference of subsequent action potentials ΔAPD is then
quantified as
ΔAPDnAPDn+1−APDn,(2)
where ndenotes the beat number. ΔAPD maps were recomposed,
and a functional color scheme was applied that indicates non-
alternating tissue and nodal lines when ΔAPD = 0 ± 2 ms, which
is defined by the temporal resolution of the recordings. A larger or
smaller ΔAPD shows phase-dependent alternans, as introduced
before (Gizzi et al., 2013).
2.3 Mathematical Model
We based our numerical simulations on the four-variable
minimal model for ventricular action potentials (Bueno-
Orovio et al., 2008), solved on two-dimensional anisotropic
heterogeneous spatial domains according to the fine-tuning
performed by Fenton et al. (2013). The model includes a
phenomenological description of main transmembrane ion
currents and is properly generalized with a heterogeneous
diffusion contribution to account for spatial effects. The
model’s equations are
ztu∇·Dij∇u
−Jfi +Jso +Jsi
,(3a)
dtv1−Θu−θv
()[]
v∞−v
()τ−
v−Θu−θv
()
vτ+
v,(3b)
dtw1−Θu−θw
()[]
w∞−w
()τ−
w−Θu−θw
()
wτ+
w,(3c)
dts1+tanh ksu−us
()()()2−s
τs,(3d)
where uis the dimensionless cell membrane potential, v,w, and s
are gating variables regulating ion current activation, and Θ(x)
denotes the Heaviside step function. J
fi
,J
so
,J
si
represent fast-
inward, slow-outward, and slow-inward transmembrane
currents:
Jfi −Θu−θv
()
u−θv
()
uu−u
()
vτfi,(4a)
Jso 1−Θu−θw
()[]
u−uo
()τo+Θu−θw
()τso,(4b)
Jsi −Θu−θw
()
wsτsi. (4c)
Time constants and asymptotic values for gating variables depend
on the membrane voltage:
τ−
vu
()
1−Θu−θ−
v
τ−
v1+Θu−θ−
v
τ−
v2,(5a)
τ+
wu
()
τ+
w1+τ+
w2−τ+
w1
tanh k+
wu−u+
w
+1
2,(5b)
τ−
wu
()
τ−
w1+τ−
w2−τ−
w1
tanh k−
wu−u−
w
+1
2,(5c)
τso u
()
τso1+τso2−τso1
()
tanh kso u−uso
()[]
+1
[]2,(5d)
τsu
()
1−Θu−θw
()[]
τs1+Θu−θw
()
τs2,(5e)
τou
()
1−Θu−θo
()[]
τo1+Θu−θo
()
τo2,(5f)
v∞1−Θu−θ−
v
,(5g)
w∞1−Θu−θo
()[]
1−uτw∞
+Θu−θo
()
w∞
* . (5h)
D
ij
is the two-dimensional diffusion tensor, defined as
Dij ≡
σij
SoCm
D11 D12
D21 D22
,(6)
where σ
ij
is the conductivity tensor, S
o
represents the cell surface-
to-volume ratio, and C
m
is the membrane capacitance. The tensor
elements are defined in the two-dimensional Cartesian domain as
D11 Dcos2αx, y
+D⊥sin2αx, y
,(7a)
D22 Dsin2αx, y
+D⊥cos2αx, y
,(7b)
D12 D21 D−D⊥
cos αx, y
sin αx, y
. (7c)
Here, the function α(x,y) represents the local fiber orientation
and D
and D
⊥
denote diffusivities along the directions parallel
and perpendicular to the fibers. We used the same anisotropy
settings as in previous computational studies on cardiac
activation maps (Loppini et al., 2019). Model parameters are
reported in Table 1. The parameter set is consistent with the one
originally fine-tuned by Fenton et al. (2013), related to
endocardial tissue at 37°Celsius. Specifically, parameter values
are set to reproduce AP features, conduction velocity, and
restitution curves as observed in canine cardiac tissues, the
same considered in this study. Furthermore, we assume the
anisotropy ratio as 1:3 in line with previous modeling analyses
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Loppini et al. Ultrastructure Recovers Alternans In-Silico
(Loppini et al., 2019). For a more detailed description of model
parameters and for a comparison between the four-variable
model results and experiments, we refer the reader to the two
abovementioned studies.
As detailed in Section 3.4, the generalization to heterogeneous
modeling by data assimilation is obtained by imposing a spatial
variation of selected parameters, opportunely sorted around their
reference values (i.e., Table 1) based on experimentally informed
profiles. On this basis, the heterogeneous model is obtained by
perturbing parameters of the homogeneous model so that global
features in the evoked electrical activity are still correctly
reproduced. We numerically integrated the model with an
explicit Euler scheme implemented in Fortran, discretizing the
spatial operators to account for heterogeneous diffusion and
phase-field boundary conditions. We solved the model in both
1D and 2D domains, assuming zero-flux boundary conditions.
Space and time discretization is Δx= 0.025 cm, Δt= 0.01 ms.
Stability and conduction velocity convergence was verified upon
mesh refinement and testing higher-order discretization schemes
in time (second- and fourth-order Runge–Kutta), achieving non-
significant variations in the computed results.
3 RESULTS
Alternans in cardiac tissues is observed at high-frequency
entrained (Gizzi et al., 2013;Loppini et al., 2021) and self-
sustained freely rotating and heterogeneity-bound spiral waves
(Hörning et al., 2017). In the past, those dynamics were difficult
to visualize without the use of heavy spatial–temporal filters and
thus hindering the fine-scale visualization and study of nodal
lines that are observed in discordant alternans. Here, we apply the
spatial-filter–free FFI analysis method that was recently
introduced by Hörning et al. (2017). It is worth mentioning
that unique alternative methods assessing cardiac alternans are
still required due to the critical differences in electrophysiological
signals. The action potential amplitude (Chen et al., 2017) and
calcium transient duration and amplitude (Clusin, 2008;
Visweswaran et al., 2013) represent, in fact, different
approaches that require a meticulous comparison.
3.1 Alternans in Intact Canine Ventricles
Simultaneous recordings of the epicardium and endocardium in
RV canine preparations were observed (Gizzi et al., 2013).
Physiological alternans-free wave conduction and alternans
states could be observed depending on the pacing site position
and pacing frequency. At lower pacing frequencies, no alternans
is observed, as the APD is sufficiently short to prevent the
interaction of subsequent waves. Figure 1A shows such an
example observed at the entrainment frequency f
p
= 3.2 Hz on
the epicardium. The phase and amplitude of the pixel-wise FFI-
analyzed recordings show a continuously evolving phase and
amplitude in the entire tissue. No spatially correlated phase or
amplitude is observed at f
1/2
=f
p
/2 = 1.6 Hz that would indicate
alternans. Also, closer inspection of the local signaling does not
indicate alternans (Figure 1A, right panels). The normalized APs
at P1 (pink square, top) and P2 (cyan square, bottom) show no
alteration in peak height or APD, as confirmed by the respective
amplitude in the Fourier space. Only a single peak at the
entrainment frequency f
p
is observed. The lower amplitude
peak at 2f
p
shows the second mode and does not carry
relevant information. Contrarily, alternans is observed when
the epicardium is entrained with a higher frequency.
Figure 1B shows the same analysis and local signaling
recordings at f
p
= 8.0 Hz. In this case, the pacing frequency is
sufficiently high so that subsequent waves influence each other.
The phase and amplitude information at f
1/2
= 4 Hz shows a
typical pattern of discordant 2:2 alternans. That means that two
subsequent waves lead to two different APDs in time. In space,
the APD of each wave can transiently switch between the two
APDs that are spatially confined by nodal lines. The latter can be
identified by the spatial phase jump of about π, and the amplitude
valley. The normalized APs at P1 (cyan) and P2 (pink) show APD
alternation between shorter and longer APDs. Every second AP is
shown in either black or red to facilitate visualization. For those
time series, a second peak at f
1/2
= 4 Hz is visible in the amplitude
spectrum, since a second underlying frequency is present that
correlates with two times the wave period (2T=f
1/2
).
3.2 Visualization of Higher-Order
Discordant Alternans
Although 2:2 alternans is the most commonly observed AP
rhythm, other higher-order AP rhythms exist (see, e.g.,
Figure 6 in Gizzi et al. (2013)). The epicardium that is shown
in Figure 1 was additionally paced from the bottom (base) of the
heart at f
p
= 8.5 Hz, which led to a spatially mixed mode of 2:2 and
4:4 alternans (Figure 2A). While the 2:2 AP rhythm shows a
TABLE 1 | Model parameters for the ventricular action potential. The membrane voltage uand related parameters are dimensionless. The scaling u
m
= (85.7u−84) mV can
be used to recover the membrane potential in mV.
u
o
=0 k+
w=8 τ+
v= 1.4506 ms τ
fi
= 0.10 ms D
= 0.010 cm
2
/ms
u
u
= 1.56 k−
w=20 τ−
v1=55ms τ
si
= 2.9013 ms D
⊥
= 0.003 cm
2
/ms
u
s
= 0.9087 k
s
= 2.0994 τ−
v2=40ms τ
s1
= 2.7342 ms —
u
so
= 0.65 k
so
=2 τ+
w1= 175 ms τ
s2
=2ms —
u−
w= 0.00615 θ
v
=0.3 τ−
w1=40ms τ
so1
=40ms —
u+
w= 0.0005 θ−
v=0.2 τ+
w2= 230 ms τ
so2
=1.2ms —
wp
∞= 0.78 θ
w
= 0.13 τ−
w2= 115 ms τ
o1
= 470 ms —
θ
o
= 0.006 τw∞= 0.0273 ms τ
o2
=6ms —
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Loppini et al. Ultrastructure Recovers Alternans In-Silico
single amplitude peak at f
1/2
= 4.25 Hz, two additional amplitude
peaks are observed in the Fourier space for the 4:4 AP rhythm:
one very close to f
1/2
and one at f
3/4
. As the two peaks at around f
1/
2
are very close to each other but implicate different information,
they are from here on defined as f1
1/2and f2
1/2. The well-defined
spatial distribution of the two different AP rhythms is visualized
at the phase and amplitude reconstructions in Figure 2B. Here,
f2
1/2and f
3/4
show additional spatial information to the
frequencies at f
p
and f1
1/2in the Fourier space. The 2:2 (left
side, RV anterior) and 4:4 (right side, RV anterior) alternans
regimes are spatially separated at f
3/4
. The right side shows
synchronized phase and correspondingly elevated amplitude
signaling that is present on the left side. At f1
1/2and f2
1/2,
more complex phase dynamics are observed in addition to the
nodal lines that separate different entrained alternating regions in
the tissue. Figure 2C shows the corresponding phase wave
directions. Numbers 1 and 2 indicate the temporal order of
occurring phase waves. In this framework, nodal lines and
nodal areas carry more information compared to the
classically analyzed temporal differences in APD (Pastore
et al., 1999). At the phase of f1
1/2(top, right side, Figure 2B),
the phase wave accelerates at the nodal line from 1 to 2
(Figure 2C), while contrarily at f2
1/2, the phase waves
propagate in the opposite direction and jump by 2π(top, right
side, Figure 2B). The latter indicate the typical phase waves, as
observed in the Fourier space (f2
1/2)at 2:2 AP rhythms (see also
Figure 1). For the sake of completeness, Figure 2 shows
additionally f
1/4
that corresponds to the duration of four AP
rhythms. However, f
1/4
is a rather non-significant frequency and
appears only because of the intrinsic noise of the system, as the
peak at f
p
describes already the main oscillation of the system, and
the three other amplitude peaks at f1
1/2,f2
1/2, and f
3/4
describe the
periodic temporal variations.
3.3 Spatial Synchronization of Alternans
Patterns
The frequency response observed at a single recorded pixel is
useful to get an overview of the local alternans offset (in analogy
to the well-known restitution curves). Figure 3A shows frequency
maps with the normalized amplitudes depending on the
entrainment f
p
for top (base, left panels) and left (RV
posterior, right panels) paced canine ventricles. The top and
bottom panels show data recorded at the epicardium (EPI) and
endocardium (ENDO). The main peaks (bright yellow peaks)
indicate f
p
. Above f
p
appears a second peak from about 5 Hz that
indicates alternans (f2
1/2). Additionally, only for very high
frequencies, here f
p
= 9.2 Hz, a peak at f
3/4
is visible that
indicates 4:4 alternans. Figure 3B shows a generalized scheme
of the frequency maps for comparisons. The offset of 2:2 alternans
differs between the EPI and the ENDO, while the peak f
3/4
is
observed for all four frequency maps at the same entrainment
frequency f
p
= 9.2 Hz. At frequencies higher than 10.0 Hz,
fibrillation is observed, as additionally shown in the right (f
p
=
FIGURE 1 | Fourier analysis in a high-frequency entrained canine heart. (A,B) show the epicardium of a canine heart that is paced from the top (RV anterior)
entrained with f
p
= 3.2 Hz (no alternans) and f
p
= 8.0 Hz (2:2 DA), respectively. Shown is the Fourier space (phase and amplitude) of frequencies fand f
1/2
. The white
arrows indicate the direction of the propagation wave. Two AP rhythms measured at two independent locations (P1 and P2, 6 × 6 pixel FOV) and their respective Fourier
spectra are shown exemplarily. (B) shows a typical example where nodal lines are visible at f
1/2
. Every second AP time course is shown in red to facilitate
visualization of the 2:2 AP rhythm. The second peak 2f
p
in the Fourier space of the upper AP rhythms is a typical higher-order frequency mode. The positions P1 and P2
are highlighted by pink and cyan squares in (A) and (B). Three waves are marked by the wave numbers, as n−1, n, and n+1.
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Loppini et al. Ultrastructure Recovers Alternans In-Silico
10.7 Hz, RV posterior) paced EPI and ENDO frequency maps.
The comparison between f
p
= 9.2 Hz and 10.7 Hz is shown in
Figure 3C. Fibrillation shows an elevated baseline and a broad
spectrum for frequencies above f
p
.
While a critical frequency induces fibrillation, the complex
spatiotemporal alternans patterns stabilize with the increasing
entrainment frequency (Gizzi et al., 2013)(Figure 4A). Figure 4B
shows selected snapshots of the EPI and ENDO from two
different pacing sites. Initially, no alternans is observed at the
EPI at a lower f
p
≃4 Hz, but the initiation of alternans at the
ENDO can be seen (endocardium base paced, Figure 4).
Interestingly, this occurs in larger speckled patches rather than
in defined areas, which indicates the alternans-offset difference
among individual cells. This speckled-like early fine-scale
initiation of alternans was suggested previously by Jia et al.
(2010). With increasing f
p
, those patterns synchronize spatially
and lead to distinct phase areas that are spatially separated by
nodal lines, as best visible at the EPI. Although the ENDO shows
comparable stabilization of alternans in the phase, the amplitude
shows more spatial variations. This is most likely caused by the
influence of the Purkinje fibers that are confined to the
subendocardial layer and believed to be responsible for the
initiation of ventricular fibrillation (Fox et al., 2002;Muñoz
et al., 2018).
3.4 Data Assimilation From Optical
Ultrastructure
As the differences of the electrophysiological properties of
individual cells also lead to differences in the alternans-offset
and restitution characteristics, it is useful to take pixel-based
differences into account when modeling alternans dynamics in
silico. The advantage of the Fourier analysis of heart tissues is that
the optical ultrastructure can be revealed in the low-frequency
regime, as shown in Figure 5. Especially, the amplitude
information at f= 0.5 Hz is a stable indicator for
morphologically restricted differences that are independent of
the pacing location (Figure 5A) and pacing frequency
FIGURE 2 | Simultaneous DA of different AP rhythms in a high-frequency entrained epicardium of a canine heart. (A) shows two AP rhythms (P1 and P2, 6 × 6 pixel
FOV) of 2:2 and 4:4 alternans and their Fourier spectra, respectively. The stimulation site is on the bottom (base) of the heart with f
p
= 8.50 Hz. findicates the entrainment
frequency, and f
3/4
indicates the presence of a 4:4 AP rhythm with its tw o corresponding peaks, f1
1/2 and f2
1/2 around f
p
/2. The peak at f
1/4
corresponds to a wavelet of four
APs that is fully expressed by f
p
. Every second or fourth AP is shown in red to facilitate visualization of the 2:2 or 4:4 AP rhythms. (B) shows the corresponding
spatial phase and amplitude information of the five frequency peaks. The Fourier data shown at f
3/4
indicate a 4:4 AP rhythm on the right side (RV anterior) of the heart
only. The respective counter phase is illustrated at f1
1/2.f2
1/2 shows the typical 2:2 phase of a 2:2 AP rhythm, similarly to that illustrated in Figure 1B. The positions P1 and
P2 are highlighted by pink and cyan squares. (C) shows the direction of the respective phase information indicated by red arrows that are shown in (B). Nodal lines are
outlined by black dotted lines. Numbers 1 and 2 shown in f1
1/2 and f2
1/2 indicate the phase waves that are shifted by π.
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Loppini et al. Ultrastructure Recovers Alternans In-Silico
FIGURE 3 | Pacing-site–dependent frequency maps. (A) shows frequency maps obtained from the top (base, left panels) and left (RV posterior, right panels) paced
canine ventricles, respectively. The top and bottom panels show data recorded at the epicardium (EPI) and endocardium (ENDO). 2:2 AP rhythms (f
1/2
) are observed from
about 4.5 Hz. 4:4 AP rhythms (f
3/4
) are observed only in base paced canine recordings at a pacing of about 9.2 Hz. (B) shows a guide of the eye for (A) with the main
frequencies (solid lines), secondary peaks (dashed lines), and higher-ordered peaks (dotted lines). (C) shows a comparison of the normalized amplitudes for 4:4
alternans (9.2 Hz, red line) and fibrillation (10.7 Hz, black line) that is observed at the ENDO paced from the RV posterior.
FIGURE 4 | Stabilization of nodal line formation at higher entrainment frequencies. (A) shows the evol ution of alternans from lower to higher pacing frequencies. The
Fourier space, phase and amplitude at f
1/2
and ΔAPD, is shown from f
p
= 2 Hz to 7.2 Hz. (B) shows the concordant alternans evolution of the frequency maps shown in
Figures 3A,B. The top (base, left panels) and left (RV posterior, right panels) paced canine hearts are shown on the left and right sides, and the respective epicardium
(EPI) and endocardium (ENDO) are shown at the bottom and top. The regimes of no alternans, concordant alternans (CA), and discordant alternans (DA) are
indicated on the top and bottom of the figures. Red arrows indicate the position of the electrode.
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Loppini et al. Ultrastructure Recovers Alternans In-Silico
(Figure 5B). Here, we utilize the low-frequency regime, as it is
also an indirect measure of the signal height, i.e., the observed
baseline of the AP rhythms. Therefore, we assume that the
strength of the emitted signal depends on the local tissue
properties and thus relates to the heart ultrastructure.
In order to validate this hypothesis, we propose a novel data
assimilation approach using the ultrastructure observed at f=
0.5 Hz assuming the influence in the diffusive term, Eq. 2.3 (D
,
D
⊥
), and the time constants that shape the AP, Eq. 5 (τ+
w,τ
so
, and
τ
si
). The tissue heterogeneities are applied on a pixel-based scale,
so that the local variations of conduction velocity and action
potential are accounted through specific parameters having a
strong impact on the resulting APD. In detail, a mask encoding
the actual tissue of the experimental samples was extracted by
analyzing the signal-to-noise ratio, and the spatial map of the
Fourier amplitude spectrum at f= 0.5 Hz was computed. The
resulting ultrastructure profile was smoothed by using a Gaussian
kernel, restricted to a radius of 6 pixels with a variance equal to 5.
The heterogeneity field was evaluated by normalizing the
ultrastructure mask as
Hx,y
δr
max |r|
()
+1,(8)
where rs−
s,s=|F
x,y
(0.5 Hz)|, and δis the parameter denoting
the desired maximal variation. We set this value in the range [0,1]
by assuming heterogeneity variations of the local parameters up
to 100%. Eventually, the heterogeneity map is combined with the
tissue map to include also information on tissue boundaries
(i.e., phase-field). We refer to this heterogeneity field as an
H
1
(x,y) map. Furthermore, we considered a second
heterogeneity field that enhances the data assimilation
procedure from low-amplitude areas in the Fourier spatial
map. We evaluated the reciprocal of the H
1
(x,y) map and
normalized the result according to Eq. 8, still constraining the
parameter δ∈[0, 1]. This second spatial scalar field is denoted as
the H
2
(x,y) map with average 1 and variation δ. It is worth noting
that the proposed procedure represents a generalization of the
phase-field method (Fenton et al., 2005) that permits both the
inclusion of tissue boundary and specific modulations of model
parameters. The resulting heterogeneity maps are used as a
multiplication factor on selected model parameters to achieve
a constitutive heterogeneity shaped on a tissue ultrastructure. In
our analyses, we focused on heterogeneities related to diffusivity
and APD parameters (D
,D
⊥
,τ+
w,τ
so
,τ
si
), by using the following
constitutive law:
px,y
pH
ix, y
.(9)
Here, p(x,y) denotes a spatial dependent parameter,
pis the
original model parameter, and H
i
(x,y) is the heterogeneity field
(with i= 1, 2). A visual representation of the adopted data
assimilation technique is thoroughly provided in the next
sections. Naming P
1
as the set of diffusivity parameters and P
2
as that of APD-regulating time constants, we investigated all
possible combinations of H
1
and H
2
maps on P
1
and P
2
, i.e., a
specific heterogeneity profile was applied to diffusivities and
APD-related time constants. It is worth noting that, with this
setting, we are assuming a correlation between diffusivity and
APD, with such parameters non-linearly coupled through
complex electronic effects in cardiac tissue.
3.5 In Silico Data Assimilation and Alternans
Model Prediction
We performed an extensive in silico study on both one-
dimensional cables and two-dimensional tissues to test the
heterogeneity effects on alternans onset and severity. In
particular, we computed H
1
(x,y) and H
2
(x,y) maps from a
selected experimental tissue to shape model parameters’
heterogeneity in 1D and 2D domains, investigating all possible
combinations of the heterogeneity fields on diffusivity and APD-
regulating time constants, at δ= 0.25, 0.5, 1. For 1D simulations,
we extracted one-dimensional cuts of H
1
(x,y) and H
2
(x,y) maps
along the experimental propagating wavefronts (not shown). This
preliminary set of numerical simulations was used as a first
benchmark of the data assimilation procedure. We observed
that the heterogeneous model is able to 1) recover the
expected average CV and AP features and 2) emphasize
alternans onset and severity, also inducing conduction block
phenomena not observed in the homogeneous case. We then
tested data assimilation within 2D computational domains
observing notable differences with respect to the homogeneous
case. In the following, we show two representative examples
comparing the overall results for the same ventricle stimulated
FIGURE 5 | Optical ultrastructure extracted from the low-frequency
regime of the epicardium (EPI). (A) shows the Fourier space—phase and
amplitude—at f= 0.5 Hz for f
p
=3Hz frequency entrained tissues that are
paced on the endocardium from four different directions, as indicated by
the white arrows. (B) shows the amplitudes of the Fourier space at f= 0.5 Hz
for different entrainment frequencies f
p
that are stimulated at the base. The
white arrows in (A,B) indicate the respective direction of wave propagation.
Below the amplitude images are indicated the regimes of no alternans and
discordant 2:2 alternans (DA).
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Loppini et al. Ultrastructure Recovers Alternans In-Silico
with a pacing-down protocol both at the ventricle base and in the
right anterior ventricular region. The pacing-down protocol
consists in stimulating the tissue starting from a low frequency
and progressively reducing the pacing frequency. In particular, at
each frequency, we delivered a stimulation train of 10 beats to
ensure the tissue reached a stationary regime. This protocol
reproduces the experimental one, and in our analyses, we
computed alternans patterns on the last two beats to avoid
transient effects.
3.5.1 Base Ventricle Stimulation
The phase-field ultrastructure and heterogeneity maps are shown
in Figures 6A,B. In this case, we tested the model with 1) spatially
homogeneous parameters, 2) H
1
maps applied on diffusivity (H1
model), 3) H
2
maps applied on APD-regulating parameters (H2
model), and 4) H
1
and H
2
maps applied simultaneously (H3
model). Figure 6C shows simulated alternans maps for a selected
frequency f
p
= 6.2 ± 0.4 Hz. On the left, the homogeneous model
could not reproduce complex and discordant alternans maps
during the pacing-down protocol. Both H1 and H2 models
(center and right columns) succeeded in reproducing
transition into the discordant alternans regime, though
showing regular spatial boundaries. Interestingly, the H2
model produced multiple transitions between concordant and
discordant alternans during pacing-down (Supplementary
Figure S1). However, such a high number of transitions are
FIGURE 6 | Data assimilation procedure and cardiac alternans maps for stimulation at the base of the ventricle. (A) Spatial map of the Fourier spectrum at f=0.5 Hz
and tissue boundary. (B) Computed heterogeneity maps from the tissue ultrastructure (see the text) for both diffusivity, H
1
(x,y), and time constants regulating the APD,
H
2
(x,y). Black dashed lines represent one-dimensional cuts of the heterogeneity maps (top panels). (C) Modeled alternans maps: homogeneous case, heterogeneity in
diffusivity (H1 model), and heterogeneity in APD (H2 model). (D) Comparison between the modeled alternans map, with combined heterogeneities in APD-
regulating time constants and diffusivity (H3 model), and experimental alternans. Top row: ΔAPD maps. Bottom row: FFI phase maps at f=f
p
/2 (f
1/2
). Alternans maps are
obtained at a pacing frequency f
p
= 6.2 ± 0.4 Hz.
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Loppini et al. Ultrastructure Recovers Alternans In-Silico
not observed in experimental activation maps, suggesting that the
H2 model is not the optimal choice. The optimal match was
finally obtained with the H3 model, capable of recovering a
consistent number of CA-DA transitions and complex
alternans patterns, i.e., irregular nodal line shape (Figure 6D).
The accuracy of the model was also checked by comparing the FFI
phase maps computed at f=f
p
/2 (f
1/2
). In particular, the π-out-of-
phase regions of the simulated tissue recovered the shapes
obtained with the standard ΔAPD analysis. As detailed in
previous paragraphs, such a phase shift is typical of 2:2
alternans. This result shows the applicability of the FFI phase
maps on in silico data as well to reveal DA spatial patterns and
also confirms the accuracy of the data assimilation model in
reproducing experimental activation maps.
3.5.2 Anterior Right Ventricle Stimulation
We further investigated the H3 model behavior in response to
anterior right ventricle stimulation. The adopted heterogeneity
maps for this case are shown in Figure 7A. During pacing-down,
the model reproduced both CA and DA alternans patterns as well
as multiple transitions between the two regimes. Figure 7B shows
simulation results at two pacing frequencies, f
p
= 4.0 Hz and f
p
=
5.6 Hz, corresponding to two representative cases of CA and DA
maps characterized by complex alternans patterns. Also in this
case, FFI phase maps at f
1/2
extracted from simulated data are in
agreement with the ΔAPD maps and further verify the accuracy
of the method in grasping both CA and DA patterns. In
particular, CA FFI phase maps show a less severe phase shift
compared to the DA case (less than π). Indeed, in Figure 7,a
change in phase around the blue–yellow transition denotes a
minimal phase shift, given the 2π-periodicity. In contrast, a phase
shift of ≃πarises in the case of DA patterns. As shown in
Figure 7C, simulated maps are in close agreement with the
experimental ones in terms of both ΔAPD and FFI phase, and
similar spatial alternans shapes are recovered both for CA and
for DA.
4 CONCLUSION
We have shown that single-pixel Fourier imaging of high-frequency
entrained intact canine RV preparations is a valuable tool to visualize
action potential alternans. Besides 2:2 DA, as observed in stable spiral
waves in vitro (Hörning et al., 2017), we have also shown that higher-
order DA, e.g., 4:4, can be observed and analyzed in an ex vivo heart
preparation (Gizzi et al., 2013,2017). This indicates the possibility of
fast and reliable full heart analysis in vivo to detect electrical
instabilities in cardiac tissues and thus enables the application to
the medical field. The unnecessity of spatial filtering of the recorded
signals further opens the possibility of detecting ultra-fine structured
early alternans that is only restricted by the optical recording device.
Contrarily, Fourier imaging needs a specific time window of periodic
oscillations to fully take advantage of the Fourier analysis. Subsequent
action potentials cannot be compared and visualized as for the
established analysis of action potential duration difference,
i.e., ΔAPD (see Figure 4B). So, depending on the purpose,
Fourier imaging is a powerful alternative to detect and visualize
alternans.
A second useful application is the use of the optical
ultrastructure that can be extracted in the low-frequency
regime in the Fourier space (see Figure 3). As the
ultrastructure is related to the morphological properties of the
tissue, we assimilated this frequency and pacing site–independent
structure to recover alternans in silico. Using a phenomenological
model tuned on CV and restitution curves (Fenton et al., 2013),
we were able to reproduce strikingly similar CA and DA patterns
as we have observed experimentally. In this context, experimental
tissue heterogeneities included in the model could induce
CA–DA transitions and complex shapes of alternating tissue
areas and nodal boundary lines, not recovered in the fine-
tuned homogeneous model. Furthermore, our analysis proved
the FFI method to be a practical approach to uncover alternans on
in silico data, showing phase maps in close agreement with ΔAPD
dispersion.
FIGURE 7 | Data assimilation and cardiac alternans maps for right ventricle anterior stimulation. (A) Heterogeneity maps for both diffusivity, H
1
(x,y), and time
constants regulating the APD, H
2
(x,y). (B) Modeled alternans maps at pacing frequencies f
p
= 4.0 Hz and 5.6 Hz and corresp onding FFI phase maps at f=f
p
/2 (f
1/2
). (C)
Experimental alternans corresponding to modeled maps shown in panel (B) and corresponding experimental FFI phase maps at f
1/2
.
Frontiers in Network Physiology | www.frontiersin.org April 2022 | Volume 2 | Article 86610110
Loppini et al. Ultrastructure Recovers Alternans In-Silico
Pros and cons of the present study shall be mentioned. As for the
data assimilation, alternative methods can be used for parameter
inference. Genetic algorithms or variational approaches aim at fitting
recorded spatiotemporal cardiac activity targeting diffusive properties
encoded in the conductivity tensor (Cairns et al., 2017;Barone et al.,
2020b;Irakoze and Jacquemet, 2021). If these methods mostly work
in the time domain, the data assimilation technique here proposed
focuses on the frequency domain instead. It allows, in fact, to account
for changes in cardiac tissue properties not considered in other
parameter fitting techniques. We believe that our method,
combined with other procedures, can enrich data assimilation
toward customized models with high predictive power. The
present numerical model, in fact, was limited to two-dimensional
computational domains (though based on ventricular geometries).
An additional level of predictability is expected to appear once the
whole ventricular thickness is considered. In such a scenario, the
mathematical characterization of intramural rotational anisotropy,
combined with a surface-based FFI data assimilation, may open the
path toward a multiscale assessment and control of alternans, as well
as to scale-transitioning information theories (Garzón et al., 2009;
Ashikaga and James, 2018).
We remark that various approaches could be used to derive
heterogeneity fields from the FFI spectrum. Indeed, slight variations
in the selected frequency or alternative transformation laws can lead
to different H(x,y) maps. In the present study, we performed a
specificchoiceconsideringtheinvarianceoftheemergentFFIspatial
structure and the non-random organization of the amplitude
dispersion. Besides, the adopted scaling can be interpreted as a
“perturbed”version of the homogeneous model allowing
investigating heterogeneity effects without additional parameter
optimization. Although different interpretations of FFI peaks and
valleys can be pursued to derive optimized heterogeneity maps and
maximize data assimilation, we remark that the present method is
generally applicable to multiple cardiac surfaces
(endocardium–epicardium, atria–ventricles) and integrated with
both biophysical and phenomenological models. Furthermore, one
can sort parameters other than diffusivity and APD-regulating time
constants based on heterogeneity fields and pursue different
assumptions on their correlation. In this context, we assumed that
diffusivity and APD-regulating time constants followed correlated
spatial heterogeneity profiles. Accordingly, we developed our
investigationonthishypothesisasafirst explorative study on the
effect of a frequency-based data assimilation procedure on cardiac
modeling. We hope our study could be further tested and validated in
future numerical analyses.
Tissues undergoing fluorescence optical mapping are
inherently wet, and they must be kept without drying out to
retain physiologically realistic activity. The wet tissue reflects
directional light into the camera, causing bright patches in the
image known as specular reflection. Regions with specular
reflection do not contain information on the tissue texture.
Furthermore, these bright spots could produce unrealistic
distortion due to the change in angle between the surface and
the light source during small residual deformations. On the
contrary, diffusion only contains the wavelengths that were
not absorbed by the tissue and therefore carry texture
information. In such a perspective, including specialized
lighting setups would concur to reduce specular reflection. In
particular, a cross-polarized lighting setup may provide the best
quality images with the least specular reflection and most detailed
textures (Lentle and Hulls, 2018). The appearance of optical
ultrastructure further connects the present study with a major
and multidisciplinary research effort in high-resolution imaging
of large biological tissues (Kuruppu et al., 2021).
To conclude, the FFI method outlined in this contribution
represents a new and effective method to investigate alternans
onset and development in whole-ventricle optical experiments.
Accordingly, it can be potentially applied to both calcium and
voltage data and does not require excessive pre-analysis, such as
the APD-based approaches. Moreover, we have shown that
spectral analysis of experimental data at low frequencies can
be used to uncover invariant and coherent spatial structures
associated with the underlying cardiac tissue
properties—ultrastructure—which can serve as input for data
assimilation in numerical simulations.
DATA AVAILABILITY STATEMENT
The raw data supporting the conclusions of this article will be
made available by the authors, without undue reservation.
ETHICS STATEMENT
The animal study was reviewed and approved by the Institutional
Animal Care and Use Committee of the Center for Animal
Resources and Education at Cornell University.
AUTHOR CONTRIBUTIONS
AL, MH, and AG conceived the study. FF and AG conducted the
experiments. JE and MH conceived and conducted the data
analysis. AL conceived and conducted the numerical study. SF
provided facilities and infrastructure. AL, JE, MH, and AG
drafted the original manuscript. All authors contributed to the
article and approved the submitted version.
FUNDING
This study was partially funded by Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation,
No. 442207423).
ACKNOWLEDGMENTS
We thank Dr. Christian Cherubini for his fruitful discussions and
helpful comments on the manuscript and Dr. Alessandro Barone
for his helpful comments on data assimilation procedures. AL
and AG acknowledge the support of the Italian National Group
for Mathematical Physics (GNFM-INdAM). MH thanks the
Frontiers in Network Physiology | www.frontiersin.org April 2022 | Volume 2 | Article 86610111
Loppini et al. Ultrastructure Recovers Alternans In-Silico
GNFM-INdAM for the visiting support to University Campus
Bio-Medico of Rome.
SUPPLEMENTARY MATERIAL
The Supplementary Material for this article can be found online at:
https://www.frontiersin.org/articles/10.3389/fnetp.2022.866101/
full#supplementary-material
Supplementary Figure S1 | Modeled alternans maps during the pacing-down
protocol with stimulation at the base of the ventricle: A) homogeneous model,
B) H1 model, C) H2 model, and D) H3 model (see the text). The homogeneous
model is not able to reproduce the transition into the DA regime and complex
alternans boundaries. As for the H1 and H2 models, the heterogeneity fields
areabletoinduceCA–DA transitions, still not recovering complex shapes of
nodal lines. When the heterogeneity fields H
1
and H
2
are appropriately
combined in the H3 model to shape, respectively, diffusivity and APD-
regulating parameters, both the recovered number of CA–DA transitions
and alternating tissue shapes are consistent with the experimental
observations.
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