Key aspects for effective mathematical modelling of fractional-diffusion in cardiac electrophysiology: a quantitative study

Simulation of atrial fibrillation in a non-ohmic propagation model with dynamic gap junctions

Gap junctions exhibit nonlinear electrical properties that have been hypothesized to be relevant to arrhythmogenicity in a structurally remodeled tissue. Large-scale implementation of gap junction dynamics in 3D propagation models remains challenging. We aim to quantify the impact of nonlinear diffusion during episodes of arrhythmias simulated in a left atrial model. Homogenization of conduction properties in the presence of nonlinear gap junctions was performed by generalizing a previously developed mathematical framework. A monodomain model was solved in which conductivities were time-varying and depended on transjunctional potentials. Gap junction conductances were derived from a simplified Vogel-Weingart model with first-order gating and adjustable time constant. A bilayer interconnected cable model of the left atrium with 100  μm resolution was used. The diffusion matrix was recomputed at each time step according to the state of the gap junctions. Sinus rhythm and atrial fibrillation episodes were simulated in remodeled tissue substrates. Slow conduction was induced by reduced coupling and by diffuse or stringy fibrosis. Simulations starting from the same initial conditions were repeated with linear and nonlinear gap junctions. The discrepancy in activation times between the linear and nonlinear diffusion models was quantified. The results largely validated the linear approximation for conduction velocities >20 cm/s. In very slow conduction substrates, the discrepancy accumulated over time during atrial fibrillation, eventually leading to qualitative differences in propagation patterns, while keeping the descriptive statistics, such as cycle lengths, unchanged. The discrepancy growth rate was increased by impaired conduction, fibrosis, conduction heterogeneity, lateral uncoupling, fast gap junction time constant, and steeper action potential duration restitution.

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.

A space-fractional bidomain framework for cardiac electrophysiology: 1D alternans dynamics

Cardiac electrophysiology modeling deals with a complex network of excitable cells forming an intricate syncytium: the heart. The electrical activity of the heart shows recurrent spatial patterns of activation, known as cardiac alternans, featuring multiscale emerging behavior. On these grounds, we propose a novel mathematical formulation for cardiac electrophysiology modeling and simulation incorporating spatially non-local couplings within a physiological reaction-diffusion scenario. In particular, we formulate, a space-fractional electrophysiological framework, extending and generalizing similar works conducted for the monodomain model. We characterize one-dimensional excitation patterns by performing an extended numerical analysis encompassing a broad spectrum of space-fractional derivative powers and various intra- and extracellular conductivity combinations. Our numerical study demonstrates that (i) symmetric properties occur in the conductivity parameters' space following the proposed theoretical framework, (ii) the degree of non-local coupling affects the onset and evolution of discordant alternans dynamics, and (iii) the theoretical framework fully recovers classical formulations and is amenable for parametric tuning relying on experimental conduction velocity and action potential morphology.

Thermal effects on cardiac alternans onset and development: A spatiotemporal correlation analysis

Alternans of cardiac action potential duration represent critical precursors for the development of life-threatening arrhythmias and sudden cardiac death. The system's thermal state affects these electrical disorders requiring additional theoretical and experimental efforts to improve a patient-specific clinical understanding. In such a scenario, we generalize a recent work from Loppini et al. [Phys. Rev. E 100, 020201(R) (2019)PREHBM2470-004510.1103/PhysRevE.100.020201] by performing an extended spatiotemporal correlation study. We consider high-resolution optical mapping recordings of canine ventricular wedges' electrical activity at different temperatures and pacing frequencies. We aim to recommend the extracted characteristic length as a potential predictive index of cardiac alternans onset and evolution within a wide range of system states. In particular, we show that a reduction of temperature results in a drop of the characteristic length, confirming the impact of thermal instabilities on cardiac dynamics. Moreover, we theoretically investigate the use of such an index to identify and predict different alternans regimes. Finally, we propose a constitutive phenomenological law linking conduction velocity, characteristic length, and temperature in view of future numerical investigations.

A three-compartment non-linear model of myocardial cell conduction block during photosensitization

This study constructed a new non-linear model of myocardial electrical conduction block during photosensitization reaction to identify the vulnerable cell population and generate an index for recurrent risk following catheter ablation for tachyarrhythmia. A three-compartment model of conductive, vulnerable, and blocked cells was proposed. To determine the non-linearity of the rate parameter for the change from vulnerable cells to conductive cells, we compared a previously reported non-linear model and our newly proposed model with non-linear rate parameters in the modeling of myocardial cell electrical conduction block during photosensitization reaction. The rate parameters were optimized via a bi-nested structure using measured synchronicity data during the photosensitization reaction of myocardial cell wires. The newly proposed model had a better fit to the measured data than the conventional model. The sum of the error until the time where the measured value was higher than 0.6, was 0.22 in the conventional model and 0.07 in our new model. The non-linear rate parameter from the vulnerable cell to the conductive cell compartment may be the preferred structure of the electrical conduction block model induced by photosensitization reaction. This simulation model provides an index to evaluate recurrent risk after tachyarrhythmia catheter ablation by photosensitization reaction. A three-compartment non-linear model of myocardial cell conduction block during photosensitization.

On the Role of Ionic Modeling on the Signature of Cardiac Arrhythmias for Healthy and Diseased Hearts

Computational cardiology is rapidly becoming the gold standard for innovative medical treatments and device development. Despite a worldwide effort in mathematical and computational modeling research, the complexity and intrinsic multiscale nature of the heart still limit our predictability power raising the question of the optimal modeling choice for large-scale whole-heart numerical investigations. We propose an extended numerical analysis among two different electrophysiological modeling approaches: a simplified phenomenological one and a detailed biophysical one. To achieve this, we considered three-dimensional healthy and infarcted swine heart geometries. Heterogeneous electrophysiological properties, fine-tuned DT-MRI -based anisotropy features, and non-conductive ischemic regions were included in a custom-built finite element code. We provide a quantitative comparison of the electrical behaviors during steady pacing and sustained ventricular fibrillation for healthy and diseased cases analyzing cardiac arrhythmias dynamics. Action potential duration (APD) restitution distributions, vortex filament counting, and pseudo-electrocardiography (ECG) signals were numerically quantified, introducing a novel statistical description of restitution patterns and ventricular fibrillation sustainability. Computational cost and scalability associated with the two modeling choices suggests that ventricular fibrillation signatures are mainly controlled by anatomy and structural parameters, rather than by regional restitution properties. Finally, we discuss limitations and translational perspectives of the different modeling approaches in view of large-scale whole-heart in silico studies.

Sensitivity analysis of an electrophysiology model for the left ventricle

Modelling the cardiac electrophysiology entails dealing with the uncertainties related to the input parameters such as the heart geometry and the electrical conductivities of the tissues, thus calling for an uncertainty quantification (UQ) of the results. Since the chambers of the heart have different shapes and tissues, in order to make the problem affordable, here we focus on the left ventricle with the aim of identifying which of the uncertain inputs mostly affect its electrophysiology. In a first phase, the uncertainty of the input parameters is evaluated using data available from the literature and the output quantities of interest (QoIs) of the problem are defined. According to the polynomial chaos expansion, a training dataset is then created by sampling the parameter space using a quasi-Monte Carlo method whereas a smaller independent dataset is used for the validation of the resulting metamodel. The latter is exploited to run a global sensitivity analysis with nonlinear variance-based indices and thus reduce the input parameter space accordingly. Thereafter, the uncertainty probability distribution of the QoIs are evaluated using a direct UQ strategy on a larger dataset and the results discussed in the light of the medical knowledge.

Excitable dynamics in neural and cardiac systems

The application of mathematics, physics and engineering to medical research is continuously growing; interactions among these disciplines have become increasingly important and have contributed to an improved understanding of clinical and biological phenomena, with implications for disease prevention, diagnosis and treatment. This special issue presents examples of this synergy, with a particular focus on the investigation of cardiac and neural excitability. This issue includes 24 original research papers and covers a broad range of topics related to the physiological and pathophysiological function of the brain and the heart. Studies span scales from isolated neurons and small networks of neurons to whole-organ dynamics for the brain and from cardiac subcellular domains and cardiomyocytes to one-dimensional tissues for the heart. This preface is part of the Special Issue on "Excitable Dynamics in Neural and Cardiac Systems".