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

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

In Vivo Human Left-to-Right Ventricular Differences in Rate Adaptation Transiently Increase Pro-Arrhythmic Risk following Rate Acceleration

Left-to-right ventricular (LV/RV) differences in repolarization have been implicated in lethal arrhythmias in animal models. Our goal is to quantify LV/RV differences in action potential duration (APD) and APD rate adaptation and their contribution to arrhythmogenic substrates in the in vivo human heart using combined in vivo and in silico studies. Electrograms were acquired from 10 LV and 10 RV endocardial sites in 15 patients with normal ventricles. APD and APD adaptation were measured during an increase in heart rate. Analysis of in vivo electrograms revealed longer APD in LV than RV (207.8±21.5 vs 196.7±20.1 ms; P<0.05), and slower APD adaptation in LV than RV (time constant τs = 47.0±14.3 vs 35.6±6.5 s; P<0.05). Following rate acceleration, LV/RV APD dispersion experienced an increase of up to 91% in 12 patients, showing a strong correlation (r2 = 0.90) with both initial dispersion and LV/RV difference in slow adaptation. Pro-arrhythmic implications of measured LV/RV functional differences were studied using in silico simulations. Results show that LV/RV APD and APD adaptation heterogeneities promote unidirectional block following rate acceleration, albeit being insufficient for establishment of reentry in normal hearts. However, in the presence of an ischemic region at the LV/RV junction, LV/RV heterogeneity in APD and APD rate adaptation promotes reentrant activity and its degeneration into fibrillatory activity. Our results suggest that LV/RV heterogeneities in APD adaptation cause a transient increase in APD dispersion in the human ventricles following rate acceleration, which promotes unidirectional block and wave-break at the LV/RV junction, and may potentiate the arrhythmogenic substrate, particularly in patients with ischemic heart disease

Population of human ventricular cell models calibrated with in vivo measurements unravels ionic mechanisms of cardiac alternans

Cardiac alternansis an important risk factor in cardiac physiology, and is related to the initiation of many pathophysiological conditions. However, the mechanisms underlying the generation of alternans remain unclear. In this study, we used a population of computational human ventricle models based onthe O’Hara model [1] to explore the effect of 11 key factors experimentally reported to be related to alternans. In vivo experimental datasets coming from patients undergoing cardiac surgery were used in the calibration of our in silico population of models. The calibrated models in the population were divided into two groups (Normal and Alternans) depending on alternans occurrence. Our results showed that there were significant differences in the following 5 ionic currents between the two groups: fast sodium current, sodium calcium exchanger current, sodium potassium pump current, sarcoplasmic reticulum (SR) calcium release flux and SR calcium reuptake flux. Further analysis indicated that fast sodium current and SR calcium uptake were the two most significant currents that contributed to voltage and calcium alternans generation, respectively

Effects and underlying mechanisms of refractory period pacing on repolarization dynamics in the human heart

Repolarization alternans is related to the initiation of life threatening cardiac arrhythmias. Experimental and computational studies suggest that the abolishment of alternans using dynamic pacing protocols may prevent abnormal heart rhythms. In a recent animal study, refractory period pacing (RPP) on every other beat has shown promising results in alternans reduction. However, the cellular mechanisms underlying this therapy and its efficiency in human patients remain unclear. In this study, in vivo unipolar electrograms acquired during RPP from 240 epicardial sites from one patient were analysed. Current clamp of 18 channels was performed in silico to elucidate the ionic mechanisms underlying action potential modulation by RPP. Its efficacy with positive and negative polarities was tested on a population of 87 calibrated human ventricular models exhibiting alternans. In vivo electrograms showed significant changes in T-wave alternans when applying RPP. In silico, results showed APD shortening for RPP with positive polarity and APD prolongation with RPP negative. Under current clamp protocols, voltage rectification of L-type Ca(2)+ (ICaL) and inward rectifier K+ (IK1) currents were identified as the key determinants for the observed changes. RPP pacing successfully reduced alternans on the in silico models using a negative polarity stimulus in the short beat