Fourier spectral methods for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is computationally demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reactiondiffusion equations. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is show-cased by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models,together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator

Computational probabilistic quantification of pro-arrhythmic risk from scar and left-to-right heterogeneity in the human ventricles

Both scar and left-to-right ventricular (LV/RV) differences in repolarization properties have been implicated as risk factors for lethal arrhythmias. As a possible mechanism for the initiation of re-entry, a recent study has indicated that LV/RV heterogeneities in action potential duration (APD) adaptation can cause a transient increase in APD dispersion following rate acceleration, promoting unidirectional block of conduction at the LV/RV junction. In the presence of an ischemic region and ectopic stimulation, a pathological dispersion in repolarization has been suggested to increase the risk of electrical re-entry. However, the exact location and timing of the ectopic activation play a crucial role in initiation of re-entry, and certain combinations may lead to re-entry even under normal LV/RV dispersion in repolarization. This suggests that the phenomenon needs to be investigated in a quantitative way. In this study we employ a computationally efficient, phenomenological model in order to investigate the proarrhythmic properties of a range of combinations of position and timing of an ectopic activation. This allows us to probabilistically study how increasing interventricular dispersion of repolarization increases arrhythmic risk. Results indicate that a larger LV/RV dispersion in repolarization allows ectopic beats to initiate re-entry during a significantly larger time window and from a greater number of locations compared to the case of smaller LV/RV dispersion

Na/K pump regulation of cardiac repolarization: Insights from a systems biology approach

The sodium-potassium pump is widely recognized as the principal mechanism for active ion transport across the cellular membrane of cardiac tissue, being responsible for the creation and maintenance of the transarcolemmal sodium and potassium gradients, crucial for cardiac cell electrophysiology. Importantly, sodium-potassium pump activity is impaired in a number of major diseased conditions, including ischemia and heart failure. However, its subtle ways of action on cardiac electrophysiology, both directly through its electrogenic nature and indirectly via the regulation of cell homeostasis, make it hard to predict the electrophysiological consequences of reduced sodium-potassium pump activity in cardiac repolarization. In this review, we discuss how recent studies adopting the Systems Biology approach, through the integration of experimental and modeling methodologies, have identified the sodium-potassium pump as one of the most\ud important ionic mechanisms in regulating key properties of cardiac repolarization and its rate-dependence, from subcellular to whole organ levels. These include the role of the pump in the biphasic modulation of cellular repolarization and refractoriness, the rate control of intracellular sodium and calcium dynamics and therefore of the adaptation of repolarization to changes in heart rate, as well as its importance in regulating pro-arrhythmic substrates through modulation of dispersion of repolarization and restitution. Theoretical findings are consistent across a variety of cell types and species including human, and widely in agreement with experimental findings. The novel insights and hypotheses on the role of the pump in cardiac electrophysiology obtained through this integrative approach could eventually lead to novel therapeutic and diagnostic strategies

Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization

Structural heterogeneity constitutes one of the main substrates influencing impulse propagation in living tissues. In cardiac muscle, improved understanding on its role is key to advancing our interpretation of cell-to-cell coupling, and how tissue structure modulates electrical propagation and arrhythmogenesis in the intact and diseased heart. We propose fractional diffusion models as a novel mathematical description of structurally heterogeneous excitable media, as a mean of representing the modulation of the total electric field by the secondary electrical sources associated with tissue inhomogeneities. Our results, validated against in-vivo human recordings and experimental data of different animal species, indicate that structural heterogeneity underlies many relevant characteristics of cardiac propagation, including the shortening of action potential duration along the activation pathway, and the progressive modulation by premature beats of spatial patterns of dispersion of repolarization. The proposed approach may also have important implications in other research fields involving excitable complex media

Atrial fibrillation dynamics and ionic block effects in six heterogeneous human 3D virtual atria with distinct repolarization dynamics

Atrial fibrillation (AF) usually manifests as reentrant circuits propagating through the whole atria creating chaotic activation patterns. Little is yet known about how differences in electrophysiological and ionic properties between patients modulate reentrant patterns in AF. The goal of this study is to quantify how variability in action potential duration (APD) at different stages of repolarization determines AF dynamics and their modulation by ionic block using a set of virtual whole-atria human models. Six human whole-atria models are constructed based on the same anatomical structure and fiber orientation, but with different electrophysiological phenotypes. Membrane kinetics for each whole-atria model are selected with distinct APD characteristics at 20, 50, and 90% repolarization, from an experimentally calibrated population of human atrial action potential models, including AF remodeling and acetylcholine parasympathetic effects. Our simulations show that in all whole-atria models, reentrant circuits tend to organize around the pulmonary veins and the right atrial appendage, thus leading to higher dominant frequency (DF) and more organized activation in the left atrium than in the right atrium. Differences in APD in all phases of repolarization (not only APD90) yielded quantitative differences in fibrillation patterns with long APD associated with slower and more regular dynamics. Long APD50 and APD20 were associated with increased interatrial conduction block and interatrial differences in DF and organization index, creating reentry instability and self-termination in some cases. Specific inhibitions of IK1, INaK, or INa reduce DF and organization of the arrhythmia by enlarging wave meandering, reducing the number of secondary wavelets, and promoting interatrial block in all six virtual patients, especially for the phenotypes with short APD at 20, 50, and/or 90% repolarization. This suggests that therapies aiming at prolonging the early phase of repolarization might constitute effective antiarrhythmic strategies for the pharmacological management of AF. In summary, simulations report significant differences in atrial fibrillatory dynamics resulting from differences in APD at all phases of repolarization

Continuous adjoint approach for the spalart-allmaras model in aerodynamic optimization

In this paper, the continuous adjoint method to compute shape sensitivities in aerodynamic design with turbulence modeling is described and developed. The focus is on compressible flows described by the Reynolds-averaged Navier-Stokes equations and the classical Spalart-Allmaras model for turbulence. Turbulence modeling usually requires, in particular, computation of the distance to the surface. Here, this distance is incorporated to the system as a new variable, solving the Eikonal equation. The accuracy of the sensitivity derivatives obtained with the complete turbulent approach is assessed by comparison with finite difference computations and the classical continuous adjoint with frozen viscosity, showing substantial improvements in the convergence properties of the method and in the quality of the obtained gradients. The validity of the overall methodology is illustrated with several design examples, including the optimization of three-dimensional geometries in combination with advanced freeform techniques for mesh deformation

Experimentally-calibrated population of models predicts and explains inter-subject variability in cardiac cellular\ud electrophysiology

Cellular and ionic causes of variability in the electrophysiological activity of hearts from individuals of the same species are unknown. However, improved understanding of this variability is key to enable prediction of the response of specific hearts to disease and therapies. Limitations of current mathematical modeling and experimental techniques hamper our ability to provide insight into variability. Here we describe a methodology to unravel the ionic determinants of inter-subject variability exhibited in experimental recordings, based on the construction and calibration of populations of models. We illustrate the methodology through its application to rabbit Purkinje preparations, due to their importance in arrhythmias and safety pharmacology assessment. We consider a set of equations describing the biophysical processes underlying rabbit Purkinje electrophysiology and we construct a population of over 10,000 models by randomly assigning specific parameter values corresponding to ionic current conductances and kinetics. We calibrate the model population by closely comparing simulation output and experimental recordings at three pacing frequencies. We show that 213 of the 10,000 candidate models are fully consistent with the experimental dataset. Ionic properties in the 213 models cover a wide range of values, including differences up to ±100% in several conductances. Partial correlation analysis shows that particular combinations of ionic properties determine the precise shape, amplitude and rate dependence of specific action potentials. Finally, we demonstrate that the population of models calibrated using data obtained under physiological conditions quantitatively predicts the action potential duration prolongation caused by exposure to four concentrations of the potassium channel blocker dofetilide