Optical Ultrastructure of Large Mammalian Hearts Recovers Discordant Alternans by In Silico Data Assimilation

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 a fine-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.

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 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 Ca²⁺ from the sarcoplasmic reticulum could lead to Ca²⁺ 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. 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 μm² per pixel for a grid size of 7.7 × 7.7 cm² 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.,

$I_{x,y}\left(t\right)\to F_{x,y}\left(f\right)\text{,}(1)$

where I_x,y(t) is the intensity at the spatial position (x, y) and t is 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% $\mathrm{m}\mathrm{a}\mathrm{x}\left(I_{x,y}\right)$. The temporal difference of subsequent action potentials ΔAPD is then quantified as

$\mathrm{\Delta }{\text{APD}}_n={\text{APD}}_{n+1}-{\text{APD}}_n\text{,}(2)$

where n denotes 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

${\partial }_{t}u=\nabla \cdot \left(D_{ij}\nabla u\right)-\left(J_{fi}+J_{so}+J_{si}\right),(\mathrm{3a})$

$d_{t}v=\left[1-\mathrm{\Theta }\left(u-{\theta }_v\right)\right]\left(v_{\infty }-v\right)/{\tau }_v^{-}-\mathrm{\Theta }\left(u-{\theta }_v\right)v/{\tau }_v^{+},(\mathrm{3b})$

$d_{t}w=\left[1-\mathrm{\Theta }\left(u-{\theta }_w\right)\right]\left(w_{\infty }-w\right)/{\tau }_w^{-}-\mathrm{\Theta }\left(u-{\theta }_w\right)w/{\tau }_w^{+},(\mathrm{3c})$

$d_{t}s=\left[\left(1+\tanh \left(k_s\left(u-u_s\right)\right)\right)/2-s\right]/{\tau }_s,(\mathrm{3d})$

where u is 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:

$J_{fi}=-\mathrm{\Theta }\left(u-{\theta }_v\right)\left(u-{\theta }_v\right)\left(u_u-u\right)v/{\tau }_{fi},(\mathrm{4a})$

$J_{so}=\left[1-\mathrm{\Theta }\left(u-{\theta }_w\right)\right]\left(u-u_o\right)/{\tau }_o+\mathrm{\Theta }\left(u-{\theta }_w\right)/{\tau }_{so},(\mathrm{4b})$

$J_{si}=-\mathrm{\Theta }\left(u-{\theta }_w\right)ws/{\tau }_{si}.(\mathrm{4c})$

Time constants and asymptotic values for gating variables depend on the membrane voltage:

${\tau }_v^{-}\left(u\right)=\left[1-\mathrm{\Theta }\left(u-{\theta }_v^{-}\right)\right]{\tau }_{v1}^{-}+\mathrm{\Theta }\left(u-{\theta }_v^{-}\right){\tau }_{v2}^{-},(\mathrm{5a})$

${\tau }_w^{+}\left(u\right)={\tau }_{w1}^{+}+\left({\tau }_{w2}^{+}-{\tau }_{w1}^{+}\right)\left[\tanh \left[k_w^{+}\left(u-u_w^{+}\right)\right]+1\right]/2,(\mathrm{5b})$

${\tau }_w^{-}\left(u\right)={\tau }_{w1}^{-}+\left({\tau }_{w2}^{-}-{\tau }_{w1}^{-}\right)\left[\tanh \left[k_w^{-}\left(u-u_w^{-}\right)\right]+1\right]/2,(\mathrm{5c})$

${\tau }_{so}\left(u\right)={\tau }_{so1}+\left({\tau }_{so2}-{\tau }_{so1}\right)\left[\tanh \left[k_{so}\left(u-u_{so}\right)\right]+1\right]/2,(\mathrm{5d})$

${\tau }_s\left(u\right)=\left[1-\mathrm{\Theta }\left(u-{\theta }_w\right)\right]{\tau }_{s1}+\mathrm{\Theta }\left(u-{\theta }_w\right){\tau }_{s2},(5e)$

${\tau }_o\left(u\right)=\left[1-\mathrm{\Theta }\left(u-{\theta }_o\right)\right]{\tau }_{o1}+\mathrm{\Theta }\left(u-{\theta }_o\right){\tau }_{o2},(\mathrm{5f})$

$v_{\infty }=1-\mathrm{\Theta }\left(u-{\theta }_v^{-}\right),(\mathrm{5g})$

$w_{\infty }=\left[1-\mathrm{\Theta }\left(u-{\theta }_o\right)\right]\left(1-u/{\tau }_{{\mathrm{w}}_{\infty }}\right)+\mathrm{\Theta }\left(u-{\theta }_o\right)w_{\infty }^{*}.(\mathrm{5h})$

D_ij is the two-dimensional diffusion tensor, defined as

$D_{ij}\equiv \frac{{\sigma }_{ij}}{S_o{C}_m}=\left(\begin{array}{cc}\hfill D_{11}\hfill & \hfill D_{12}\hfill \\ \hfill D_{21}\hfill & \hfill D_{22}\hfill \end{array}\right),(6)$

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

$D_{11}=D_{\parallel }{\cos }^2\left(\alpha \left(x,y\right)\right)+D_{\perp }{\sin }^2\left(\alpha \left(x,y\right)\right),(\mathrm{7a})$

$D_{22}=D_{\parallel }{\sin }^2\left(\alpha \left(x,y\right)\right)+D_{\perp }{\cos }^2\left(\alpha \left(x,y\right)\right),(\mathrm{7b})$

$D_{12}=D_{21}=\left(D_{\parallel }-D_{\perp }\right)\cos \left(\alpha \left(x,y\right)\right)\sin \left(\alpha \left(x,y\right)\right).(\mathrm{7c})$

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 (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.

TABLE 1

TABLE 1. Model parameters for the ventricular action potential. The membrane voltage u and related parameters are dimensionless. The scaling u_m = (85.7u − 84) mV can be used to recover the membrane potential in mV.

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).

FIGURE 1

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 f and 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.

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 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 $f_{1\text{/}2}^1$ and $f_{1\text{/}2}^2$. The well-defined spatial distribution of the two different AP rhythms is visualized at the phase and amplitude reconstructions in Figure 2B. Here, $f_{1\text{/}2}^2$ and f_3/4 show additional spatial information to the frequencies at f_p and $f_{1\text{/}2}^1$ in 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 $f_{1\text{/}2}^1$ and $f_{1\text{/}2}^2$, 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 $f_{1\text{/}2}^1$ (top, right side, Figure 2B), the phase wave accelerates at the nodal line from 1 to 2 (Figure 2C), while contrarily at $f_{1\text{/}2}^2$, 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 $(f_{1\text{/}2}^2)$ 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 $f_{1\text{/}2}^1$, $f_{1\text{/}2}^2$, and f_3/4 describe the periodic temporal variations.

FIGURE 2

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. f indicates the entrainment frequency, and f_3/4 indicates the presence of a 4:4 AP rhythm with its two corresponding peaks, $f_{1/2}^1$ and $f_{1/2}^2$ 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 $f_{1/2}^1$. $f_{1/2}^2$ 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 $f_{1/2}^1$ and $f_{1/2}^2$ indicate the phase waves that are shifted by π.

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 $(f_{1\text{/}2}^2)$. 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 = 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.

FIGURE 3

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.

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).

FIGURE 4

FIGURE 4. Stabilization of nodal line formation at higher entrainment frequencies. (A) shows the evolution 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.

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 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.

FIGURE 5

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 = 3 Hz 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).

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 (${\tau }_w^{+}$, 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

$H\left(x,y\right)=\delta \frac{r}{\mathrm{m}\mathrm{a}\mathrm{x}\left(|r|\right)}+1,(8)$

where $r=s-\bar{s}$, s = |F_x,y (0.5 Hz)|, and δ is 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 H₁(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 H₁ (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 H₂(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_⊥, ${\tau }_w^{+}$, τ_so, τ_si), by using the following constitutive law:

$p\left(x,y\right)=\bar{p}{H}_i\left(x,y\right).(9)$

Here, p (x, y) denotes a spatial dependent parameter, $\bar{p}$ is 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 P₁ as the set of diffusivity parameters and P₂ as that of APD-regulating time constants, we investigated all possible combinations of H₁ and H₂ maps on P₁ and P₂, 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 H₁ (x, y) and H₂ (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 H₁ (x, y) and H₂ (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 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) H₁ maps applied on diffusivity (H1 model), 3) H₂ maps applied on APD-regulating parameters (H2 model), and 4) H₁ and H₂ 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 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.

FIGURE 6

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, H₁ (x, y), and time constants regulating the APD, H₂ (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.

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 ≃ π arises 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.

FIGURE 7

FIGURE 7. Data assimilation and cardiac alternans maps for right ventricle anterior stimulation. (A) Heterogeneity maps for both diffusivity, H₁ (x, y), and time constants regulating the APD, H₂ (x, y). (B) Modeled alternans maps at pacing frequencies f_p = 4.0 Hz and 5.6 Hz and corresponding 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.

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.

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 specific choice considering the invariance of the emergent FFI spatial 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 investigation on this hypothesis as a first 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).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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 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 are able to induce CA–DA transitions, still not recovering complex shapes of nodal lines. When the heterogeneity fields H₁ and H₂ 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.

References

Adam, D. R., Smith, J. M., Akselrod, S., Nyberg, S., Powell, A. O., and Cohen, R. J. (1984). Fluctuations in T-Wave Morphology and Susceptibility to Ventricular Fibrillation. J. Electrocardiol. 17, 209–218. doi:10.1016/s0022-0736(84)80057-6

PubMed Abstract | CrossRef Full Text | Google Scholar

Ashikaga, H., and James, R. G. (2018). Inter-Scale Information Flow as a Surrogate for Downward Causation that Maintains Spiral Waves. Chaos 28, 075306. doi:10.1063/1.5017534

PubMed Abstract | CrossRef Full Text | Google Scholar

Badeer, H., Ryo, U., Gassner, W., Kass, E., Cavaluzzi, J., Gilbert, J., et al. (1967). Factors Affecting Pulsus Alternans in the Rapidly Driven Heart and Papillary Muscle. Am. J. Physiology-Legacy Content 213, 1095–1101. doi:10.1152/ajplegacy.1967.213.5.1095

PubMed Abstract | CrossRef Full Text | Google Scholar

Barone, A., Carlino, M. G., Gizzi, A., Perotto, S., and Veneziani, A. (2020a). Efficient Estimation of Cardiac Conductivities: A Proper Generalized Decomposition Approach. J. Comput. Phys. 423, 109810. doi:10.1016/j.jcp.2020.109810

CrossRef Full Text | Google Scholar

Barone, A., Gizzi, A., Fenton, F., Filippi, S., and Veneziani, A. (2020b). Experimental Validation of a Variational Data Assimilation Procedure for Estimating Space-Dependent Cardiac Conductivities. Comput. Methods Appl. Mech. Eng. 358, 112615. doi:10.1016/j.cma.2019.112615

CrossRef Full Text | Google Scholar

Bueno-Orovio, A., Cherry, E. M., and Fenton, F. H. (2008). Minimal Model for Human Ventricular Action Potentials in Tissue. J. Theor. Biol. 253, 544–560. doi:10.1016/j.jtbi.2008.03.029

PubMed Abstract | CrossRef Full Text | Google Scholar

Cairns, D. I., Fenton, F. H., and Cherry, E. M. (2017). Efficient Parameterization of Cardiac Action Potential Models Using a Genetic Algorithm. Chaos 27, 093922. doi:10.1063/1.5000354

PubMed Abstract | CrossRef Full Text | Google Scholar

Chen, D. D., Gray, R. A., Uzelac, I., Herndon, C., and Fenton, F. H. (2017). Mechanism for Amplitude Alternans in Electrocardiograms and the Initiation of Spatiotemporal Chaos. Phys. Rev. Lett. 118, 168101. doi:10.1103/PhysRevLett.118.168101

PubMed Abstract | CrossRef Full Text | Google Scholar

Cherry, E. M., and Fenton, F. H. (2004). Suppression of Alternans and Conduction Blocks Despite Steep Apd Restitution: Electrotonic, Memory, and Conduction Velocity Restitution Effects. Am. J. physiology-Heart circulatory Physiol. 286, H2332–H2341. doi:10.1152/ajpheart.00747.2003

CrossRef Full Text | Google Scholar

Cherubini, C., Filippi, S., Gizzi, A., and Ruiz-Baier, R. (2017). A Note on Stress-Driven Anisotropic Diffusion and its Role in Active Deformable Media. J. Theor. Biol. 430, 221–228. doi:10.1016/j.jtbi.2017.07.013

PubMed Abstract | CrossRef Full Text | Google Scholar

Christoph, J., Chebbok, M., Richter, C., Schröder-Schetelig, J., Bittihn, P., Stein, S., et al. (2018). Electromechanical Vortex Filaments during Cardiac Fibrillation. Nature 555, 667–672. doi:10.1038/nature26001

PubMed Abstract | CrossRef Full Text | Google Scholar

Clusin, W. T. (2008). Mechanisms of Calcium Transient and Action Potential Alternans in Cardiac Cells and Tissues. Am. J. Physiology-Heart Circulatory Physiol. 294, H1–H10. doi:10.1152/ajpheart.00802.2007

CrossRef Full Text | Google Scholar

Cusimano, N., Gerardo-Giorda, L., and Gizzi, A. (2021). A Space-Fractional Bidomain Framework for Cardiac Electrophysiology: 1d Alternans Dynamics. Chaos 31, 073123. doi:10.1063/5.0050897

PubMed Abstract | CrossRef Full Text | Google Scholar

Cusimano, N., Gizzi, A., Fenton, F. H., Filippi, S., and Gerardo-Giorda, L. (2020). Key Aspects for Effective Mathematical Modelling of Fractional-Diffusion in Cardiac Electrophysiology: A Quantitative Study. Commun. Nonlinear Sci. Numer. Simulation 84, 105152. doi:10.1016/j.cnsns.2019.105152

PubMed Abstract | CrossRef Full Text | Google Scholar

Diaz, M. E., O’Neill, S. C., and Eisner, D. A. (2004). Sarcoplasmic Reticulum Calcium Content Fluctuation Is the Key to Cardiac Alternans. Circ. Res. 94, 650–656. doi:10.1161/01.RES.0000119923.64774.72

PubMed Abstract | CrossRef Full Text | Google Scholar

Dilly, S. G., and Lab, M. J. (1988). Electrophysiological Alternans and Restitution during Acute Regional Ischaemia in Myocardium of Anaesthetized Pig. J. Physiol. 402, 315–333. doi:10.1113/jphysiol.1988.sp017206

PubMed Abstract | CrossRef Full Text | Google Scholar

Fenton, F. H., Cherry, M. E., Karma, A., and Rappel, W.-J. (2005). Modeling Wave Propagation in Realistic Heart Geometries Using the Phase-Field Method. Chaos 15, 013502. doi:10.1063/1.1840311

PubMed Abstract | CrossRef Full Text | Google Scholar

Fenton, F. H., Gizzi, A., Cherubini, C., Pomella, N., and Filippi, S. (2013). Role of Temperature on Nonlinear Cardiac Dynamics. Phys. Rev. E 87, 042717. doi:10.1103/PhysRevE.87.042717

PubMed Abstract | CrossRef Full Text | Google Scholar

Fenton, F. H., Luther, S., Cherry, E. M., Otani, N. F., Krinsky, V., Pumir, A., et al. (2009). Termination of Atrial Fibrillation Using Pulsed Low-Energy Far-Field Stimulation. Circulation 120, 467–476. doi:10.1161/CIRCULATIONAHA.108.825091

PubMed Abstract | CrossRef Full Text | Google Scholar

Fox, J. J., Riccio, M. L., Hua, F., Bodenschatz, E., and Gilmour, R. F. (2002). Spatiotemporal Transition to Conduction Block in Canine Ventricle. Circ. Res. 90, 289–296. doi:10.1161/hh0302.104723

PubMed Abstract | CrossRef Full Text | Google Scholar

Garzón, A., Grigoriev, R. O., and Fenton, F. H. (2009). Model-Based Control of Cardiac Alternans on a Ring. Phys. Rev. E 80, 021932. doi:10.1103/PhysRevE.80.021932

CrossRef Full Text | Google Scholar

Gizzi, A., Cherry, E. M., Gilmour, R. F., Luther, S., Filippi, S., and Fenton, F. H. (2013). Effects of Pacing Site and Stimulation History on Alternans Dynamics and the Development of Complex Spatiotemporal Patterns in Cardiac Tissue. Front. Physiol. 4, 71. doi:10.3389/fphys.2013.00071

PubMed Abstract | CrossRef Full Text | Google Scholar

Gizzi, A., Loppini, A., Cherry, E. M., Cherubini, C., Fenton, F. H., and Filippi, S. (2017). Multi-Band Decomposition Analysis: Application to Cardiac Alternans as a Function of Temperature. Physiol. Meas. 38, 833–847. doi:10.1088/1361-6579/aa64af

PubMed Abstract | CrossRef Full Text | Google Scholar

Green, H. D. (1935). The Nature of Ventricular Alternation Resulting from Reduced Coronary Blood Flow. Am. J. Physiology-Legacy Content 114, 407–413. doi:10.1152/ajplegacy.1935.114.2.407

CrossRef Full Text | Google Scholar

Hörning, M., Blanchard, F., Isomura, A., and Yoshikawa, K. (2017). Dynamics of Spatiotemporal Line Defects and Chaos Control in Complex Excitable Systems. Sci. Rep. 7, 7757. doi:10.1038/s41598-017-08011-z

PubMed Abstract | CrossRef Full Text | Google Scholar

Hörning, M., Kidoaki, S., Kawano, T., and Yoshikawa, K. (2012). Rigidity Matching between Cells and the Extracellular Matrix Leads to the Stabilization of Cardiac Conduction. Biophysical J. 102, 379–387. doi:10.1016/j.bpj.2011.12.018

CrossRef Full Text | Google Scholar

Huertas, M. A., Smith, G. D., and Györke, S. (2010). Ca2+ Alternans in a Cardiac Myocyte Model that Uses Moment Equations to Represent Heterogeneous Junctional SR Ca2+. Biophysical J. 99, 377–387. doi:10.1016/j.bpj.2010.04.032

CrossRef Full Text | Google Scholar

Hurtado, D. E., Castro, S., and Gizzi, A. (2016). Computational Modeling of Non-linear Diffusion in Cardiac Electrophysiology: A Novel Porous-Medium Approach. Comput. Methods Appl. Mech. Eng. 300, 70–83. doi:10.1016/j.cma.2015.11.014

CrossRef Full Text | Google Scholar

Ikeda, T., Sakata, T., Takami, M., Kondo, N., Tezuka, N., Nakae, T., et al. (2000). Combined Assessment of T-Wave Alternans and Late Potentials Used to Predict Arrhythmic Events after Myocardial Infarction. A Prospective Study. J. Am. Coll. Cardiol. 35, 722–730. doi:10.1016/s0735-1097(99)00590-2

PubMed Abstract | CrossRef Full Text | Google Scholar

Irakoze, É., and Jacquemet, V. (2021). Multiparameter Optimization of Nonuniform Passive Diffusion Properties for Creating Coarse-Grained Equivalent Models of Cardiac Propagation. Comput. Biol. Med. 138, 104863. doi:10.1016/j.compbiomed.2021.104863

PubMed Abstract | CrossRef Full Text | Google Scholar

Jakobsen, P. J., and Johnsen, G. H. (1988). Size-Specific Protection against Predation by Fish in Swarming Waterfleas, Bosmina Longispina. Anim. Behav. 36, 986–990. doi:10.1016/S0003-3472(88)80057-5

CrossRef Full Text | Google Scholar

Jia, Z., Bien, H., and Entcheva, E. (2010). Detecting Space-Time Alternating Biological Signals Close to the Bifurcation point. IEEE Trans. Biomed. Eng. 57, 316–324. doi:10.1109/TBME.2009.2028652

PubMed Abstract | CrossRef Full Text | Google Scholar

Ju, Y. S., Martincorena, I., Gerstung, M., Petljak, M., Alexandrov, L. B., Rahbari, R., et al. (2017). Somatic Mutations Reveal Asymmetric Cellular Dynamics in the Early Human Embryo. Nature 543, 714–718. doi:10.1038/nature21703

PubMed Abstract | CrossRef Full Text | Google Scholar

Karagueuzian, H. S., Stepanyan, H., and Mandel, W. J. (2013). Bifurcation Theory and Cardiac Arrhythmias. Am. J. Cardiovasc. Dis. 3, 1–16.

PubMed Abstract | Google Scholar

Karma, A. (1994). Electrical Alternans and Spiral Wave Breakup in Cardiac Tissue. Chaos 4, 461–472. doi:10.1063/1.166024

PubMed Abstract | CrossRef Full Text | Google Scholar

Karma, A. (1993). Spiral Breakup in Model Equations of Action Potential Propagation in Cardiac Tissue. Phys. Rev. Lett. 71, 1103–1106. doi:10.1103/PhysRevLett.71.1103

PubMed Abstract | CrossRef Full Text | Google Scholar

Kim, J. W., Pak, H.-N., Park, J. H., Nam, G.-B., Kim, S. K., Lee, H. S., et al. (2009). Defibillator Electrogram T Wave Alternans as a Predictor of Spontaneous Ventricular Tachyarrhythmias in Defibrillator Recipients. Circ. J. 73, 55–62. doi:10.1253/circj.cj-08-0311

PubMed Abstract | CrossRef Full Text | Google Scholar

Konta, T., Ikeda, K., Yamaki, M., Nakamura, K., Honma, K., Kubota, I., et al. (1990). Significance of Discordant ST Alternans in Ventricular Fibrillation. Circulation 82, 2185–2189. doi:10.1161/01.CIR.82.6.2185

PubMed Abstract | CrossRef Full Text | Google Scholar

Kuruppu, S., Cheng, L. K., Nielsen, P. F., Gamage, T. B., Avci, R., Angeli, T. R., et al. (2021). High-Resolution Spatiotemporal Quantification of Intestinal Motility with Free-form Deformation. IEEE Trans. Biomed. Eng., 1. doi:10.1109/TBME.2021.3135855

CrossRef Full Text | Google Scholar

Lab, M. J., and Lee, J. A. (1990). Changes in Intracellular Calcium during Mechanical Alternans in Isolated Ferret Ventricular Muscle. Circ. Res. 66, 585–595. doi:10.1161/01.RES.66.3.585

PubMed Abstract | CrossRef Full Text | Google Scholar

Lenarda, P., Gizzi, A., and Paggi, M. (2018). A Modeling Framework for Electro-Mechanical Interaction between Excitable Deformable Cells. Eur. J. Mech. - A/Solids 72, 374–392. doi:10.1016/j.euromechsol.2018.06.001

CrossRef Full Text | Google Scholar

Lentle, R. G., and Hulls, C. M. (2018). Quantifying Patterns of Smooth Muscle Motility in the Gut and Other Organs with New Techniques of Video Spatiotemporal Mapping. Front. Physiol. 9, 00338. doi:10.3389/fphys.2018.00338

PubMed Abstract | CrossRef Full Text | Google Scholar

Lin, J., and Keener, J. P. (2010). Modeling Electrical Activity of Myocardial Cells Incorporating the Effects of Ephaptic Coupling. Proc. Natl. Acad. Sci. U.S.A. 107, 20935–20940. doi:10.1073/pnas.1010154107

PubMed Abstract | CrossRef Full Text | Google Scholar

Loppini, A., Barone, A., Gizzi, A., Cherubini, C., Fenton, F. H., and Filippi, S. (2021). Thermal Effects on Cardiac Alternans Onset and Development: A Spatiotemporal Correlation Analysis. Phys. Rev. E 103, L040201. doi:10.1103/PhysRevE.103.L040201

PubMed Abstract | CrossRef Full Text | Google Scholar

Loppini, A., Gizzi, A., Cherubini, C., Cherry, E. M., Fenton, F. H., and Filippi, S. (2019). Spatiotemporal Correlation Uncovers Characteristic Lengths in Cardiac Tissue. Phys. Rev. E 100, 020201. doi:10.1103/PhysRevE.100.020201

PubMed Abstract | CrossRef Full Text | Google Scholar

Luther, S., Fenton, F. H., Kornreich, B. G., Squires, A., Bittihn, P., Hornung, D., et al. (2011). Low-Energy Control of Electrical Turbulence in the Heart. Nature 475, 235–239. doi:10.1038/nature10216

PubMed Abstract | CrossRef Full Text | Google Scholar

Marcotte, C. D., Fenton, F. H., Hoffman, M. J., and Cherry, E. M. (2021). Robust Data Assimilation with Noise: Applications to Cardiac Dynamics. Chaos 31, 013118. doi:10.1063/5.0033539

PubMed Abstract | CrossRef Full Text | Google Scholar

Merchant, F. M., Ikeda, T., Pedretti, R. F. E., Salerno-Uriarte, J. A., Chow, T., Chan, P. S., et al. (2012). Clinical Utility of Microvolt T-Wave Alternans Testing in Identifying Patients at High or Low Risk of Sudden Cardiac Death. Heart Rhythm 9, 1256–1264. doi:10.1016/j.hrthm.2012.03.014

PubMed Abstract | CrossRef Full Text | Google Scholar

Mitchell, J. H., Sarnoff, S. J., and Sonnenblick, E. H. (1963). The Dynamics of Pulsus Alternans: Alternating End-Diastolic Fiber Length as a Causative Factor. J. Clin. Invest. 42, 55–63. doi:10.1172/JCI104696

PubMed Abstract | CrossRef Full Text | Google Scholar

Miyazawa, S., Okamoto, M., and Kondo, S. (2010). Blending of Animal Colour Patterns by Hybridization. Nat. Commun. 1, 66. doi:10.1038/ncomms1071

PubMed Abstract | CrossRef Full Text | Google Scholar

Muñoz, L. M., Gelzer, A. R. M., Fenton, F. H., Qian, W., Lin, W., Gilmour, R. F., et al. (2018). Discordant Alternans as a Mechanism for Initiation of Ventricular Fibrillation In Vitro. J. Am. Heart Assoc. 7, e007898. doi:10.1161/JAHA.117.007898

PubMed Abstract | CrossRef Full Text | Google Scholar

Murray, J. D. (2003). “Mathematical Biology II: Spatial Models and Biomedical Applications,” in Interdisciplinary Applied Mathematics (New York: Springer), 18. doi:10.1007/b98869

CrossRef Full Text | Google Scholar

Papadacci, C., Finel, V., Provost, J., Villemain, O., Bruneval, P., Gennisson, J.-L., et al. (2017). Imaging the Dynamics of Cardiac Fiber Orientation In Vivo Using 3D Ultrasound Backscatter Tensor Imaging. Sci. Rep. 7, 830. doi:10.1038/s41598-017-00946-7

PubMed Abstract | CrossRef Full Text | Google Scholar

Pastore, J. M., Girouard, S. D., Laurita, K. R., Akar, F. G., and Rosenbaum, D. S. (1999). Mechanism Linking T-Wave Alternans to the Genesis of Cardiac Fibrillation. Circulation 99, 1385–1394. doi:10.1161/01.CIR.99.10.1385

PubMed Abstract | CrossRef Full Text | Google Scholar

Pathmanathan, P., Galappaththige, S. K., Cordeiro, J. M., Kaboudian, A., Fenton, F. H., and Gray, R. A. (2020). Data-Driven Uncertainty Quantification for Cardiac Electrophysiological Models: Impact of Physiological Variability on Action Potential and Spiral Wave Dynamics. Front. Physiol. 11, 585400. doi:10.3389/fphys.2020.585400

PubMed Abstract | CrossRef Full Text | Google Scholar

Qu, Z., Weiss, J. N., and Garfinkel, A. (1999). Cardiac Electrical Restitution Properties and Stability of Reentrant Spiral Waves: A Simulation Study. Am. J. Physiology-Heart Circulatory Physiol. 276, H269–H283. doi:10.1152/ajpheart.1999.276.1.H269

PubMed Abstract | CrossRef Full Text | Google Scholar

Ren, L., Fang, X., Wang, Y., and Qi, G. (2011). T-Wave Alternans and Heart Rate Variability: A Comparison in Patients with Myocardial Infarction with or without Diabetes Mellitus. Ann. Noninvasive Electrocardiol. 16, 232–238. doi:10.1111/j.1542-474X.2011.00437.x

PubMed Abstract | CrossRef Full Text | Google Scholar

Sato, D., Shiferaw, Y., Garfinkel, A., Weiss, J. N., Qu, Z., and Karma, A. (2006). Spatially Discordant Alternans in Cardiac Tissue. Circ. Res. 99, 520–527. doi:10.1161/01.RES.0000240542.03986.e7

PubMed Abstract | CrossRef Full Text | Google Scholar

Shiferaw, Y., Watanabe, M. A., Garfinkel, A., Weiss, J. N., and Karma, A. (2003). Model of Intracellular Calcium Cycling in Ventricular Myocytes. Biophysical J. 85, 3666–3686. doi:10.1016/S0006-3495(03)74784-5

CrossRef Full Text | Google Scholar

Taggart, P., Sutton, P. M. I., Boyett, M. R., Lab, M., and Swanton, H. (1996). Human Ventricular Action Potential Duration during Short and Long Cycles. Circulation 94, 2526–2534. doi:10.1161/01.CIR.94.10.2526

PubMed Abstract | CrossRef Full Text | Google Scholar

Tao, T., O'Neill, S. C., Diaz, M. E., Li, Y. T., Eisner, D. A., and Zhang, H. (2008). Alternans of Cardiac Calcium Cycling in a Cluster of Ryanodine Receptors: A Simulation Study. Am. J. Physiology-Heart Circulatory Physiol. 295, H598–H609. doi:10.1152/ajpheart.01086.2007

CrossRef Full Text | Google Scholar

Treml, L. M., Bartocci, E., and Gizzi, A. (2021). Modeling and Analysis of Cardiac Hybrid Cellular Automata via Gpu-Accelerated Monte Carlo Simulation. Mathematics 9, 164. doi:10.3390/math9020164

CrossRef Full Text | Google Scholar

Uzelac, I., Ji, Y. C., Hornung, D., Schröder-Scheteling, J., Luther, S., Gray, R. A., et al. (2017). Simultaneous Quantification of Spatially Discordant Alternans in Voltage and Intracellular Calcium in Langendorff-Perfused Rabbit Hearts and Inconsistencies with Models of Cardiac Action Potentials and Ca Transients. Front. Physiol. 8, 819. doi:10.3389/fphys.2017.00819

PubMed Abstract | CrossRef Full Text | Google Scholar

van den Heuvel, M. P., and Hulshoff Pol, H. E. (2010). Exploring the Brain Network: A Review on Resting-State fMRI Functional Connectivity. Eur. Neuropsychopharmacol. 20, 519–534. doi:10.1016/j.euroneuro.2010.03.008

PubMed Abstract | CrossRef Full Text | Google Scholar

Visweswaran, R., McIntyre, S. D., Ramkrishnan, K., Zhao, X., and Tolkacheva, E. G. (2013). Mechanisms of Calcium Transient and Action Potential Alternans in Cardiac Cells and Tissues. J. Cardiovasc. Electrophysiol. 24, 1287–1295. doi:10.1111/jce.12200

PubMed Abstract | CrossRef Full Text | Google Scholar

Walker, M. L., Wan, X., Kirsch, G. E., and Rosenbaum, D. S. (2003). Hysteresis Effect Implicates Calcium Cycling as a Mechanism of Repolarization Alternans. Circulation 108, 2704–2709. doi:10.1161/01.CIR.0000093276.10885.5B

PubMed Abstract | CrossRef Full Text | Google Scholar

Watanabe, M. A., Fenton, F. H., Evans, S. J., Hastings, H. M., and Karma, A. (2001). Mechanisms for Discordant Alternans. J. Cardiovasc. Electrophysiol. 12, 196–206. doi:10.1046/j.1540-8167.2001.00196.x

PubMed Abstract | CrossRef Full Text | Google Scholar

Weinberg, S. H. (2017). Ephaptic Coupling Rescues Conduction Failure in Weakly Coupled Cardiac Tissue with Voltage-Gated gap Junctions. Chaos 27, 093908. doi:10.1063/1.4999602

PubMed Abstract | CrossRef Full Text | Google Scholar

Xie, L.-H., and Weiss, J. N. (2009). Arrhythmogenic Consequences of Intracellular Calcium Waves. Am. J. Physiology-Heart Circulatory Physiol. 297, H997–H1002. doi:10.1152/ajpheart.00390.2009

CrossRef Full Text | Google Scholar