Mathematical birth of Early Afterdepolarizations in a cardiomyocyte model.

Early Afterdepolarizations (EADs) are abnormal behaviors that can lead to cardiac failure and even cardiac death. In this paper we investigate the occurrence and development of these phenomena in a reduced Luo-Rudy cardiac model. Through a comprehensive dynamical analysis, we map out the distinct patterns observed in the parametric plane, differentiating between normal beats without EADs and pathological beats with EADs. By examining the bifurcation structure of the model, we elucidate the dynamical elements associated with these patterns and their transitions. Using a fast-slow analysis, we explore the emergence and evolution of EADs in the model. Notably, our approach combines the two commonly used fast-slow approaches (1-slow-2-fast and 2-slow-1-fast), and we show how both approaches together provide a more complete understanding of this phenomenon.

ECG Patient Simulator Based on Mathematical Models.

In this work, we propose a versatile, low-cost, and tunable electronic device to generate realistic electrocardiogram (ECG) waveforms, capable of simulating ECG of patients within a wide range of possibilities. A visual analysis of the clinical ECG register provides the cardiologist with vital physiological information to determine the patient's heart condition. Because of its clinical significance, there is a strong interest in algorithms and medical ECG measuring devices that acquire, preserve, and process ECG recordings with high fidelity. Bearing this in mind, the proposed electronic device is based on four different mathematical models describing macroscopic heartbeat dynamics with ordinary differential equations. Firstly, we produce full 12-lead ECG profiles by implementing a model comprising a network of heterogeneous oscillators. Then, we implement a discretized reaction-diffusion model in our electronic device to reproduce ECG waveforms from various rhythm disorders. Finally, in order to show the versatility and capabilities of our system, we include two additional models, a ring of three coupled oscillators and a model based on a quasiperiodic motion, which can reproduce a wide range of pathological conditions. With this, the proposed device can reproduce around thirty-two cardiac rhythms with the possibility of exploring different parameter values to simulate new arrhythmias with the same hardware. Our system, which is a hybrid analog-digital circuit, generates realistic ECG signals through digital-to-analog converters whose amplitudes and waveforms are controlled through an interactive and friendly graphic interface. Our ECG patient simulator arises as a promising platform for assessing the performance of electrocardiograph equipment and ECG signal processing software in clinical trials. Additionally the produced 12-lead profiles can be tested in patient monitoring systems.

Spatiotemporal Cardiac Statistical Shape Modeling: A Data-Driven Approach.

Clinical investigations of anatomy's structural changes over time could greatly benefit from population-level quantification of shape, or spatiotemporal statistic shape modeling (SSM). Such a tool enables characterizing patient organ cycles or disease progression in relation to a cohort of interest. Constructing shape models requires establishing a quantitative shape representation (e.g., corresponding landmarks). Particle-based shape modeling (PSM) is a data-driven SSM approach that captures population-level shape variations by optimizing landmark placement. However, it assumes cross-sectional study designs and hence has limited statistical power in representing shape changes over time. Existing methods for modeling spatiotemporal or longitudinal shape changes require predefined shape atlases and pre-built shape models that are typically constructed cross-sectionally. This paper proposes a data-driven approach inspired by the PSM method to learn population-level spatiotemporal shape changes directly from shape data. We introduce a novel SSM optimization scheme that produces landmarks that are in correspondence both across the population (inter-subject) and across time-series (intra-subject). We apply the proposed method to 4D cardiac data from atrial-fibrillation patients and demonstrate its efficacy in representing the dynamic change of the left atrium. Furthermore, we show that our method outperforms an image-based approach for spatiotemporal SSM with respect to a generative time-series model, the Linear Dynamical System (LDS). LDS fit using a spatiotemporal shape model optimized via our approach provides better generalization and specificity, indicating it accurately captures the underlying time-dependency.

Daily Rhythm of Fractal Cardiac Dynamics Links to Weight Loss Resistance: Interaction with CLOCK 3111T/C Genetic Variant.

The effectiveness of weight loss treatment displays dramatic inter-individual variabilities, even with well-controlled energy intake/expenditure. This study aimed to determine the association between daily rhythms of cardiac autonomic control and weight loss efficiency and to explore the potential relevance to weight loss resistance in humans carrying the genetic variant C at CLOCK 3111T/C. A total of 39 overweight/obese Caucasian women (20 CLOCK 3111C carriers and 19 non-carriers) completed a behaviour-dietary obesity treatment of ~20 weeks, during which body weight was assessed weekly. Ambulatory electrocardiographic data were continuously collected for up to 3.5 days and used to quantify the daily rhythm of fractal cardiac dynamics (FCD), a non-linear measure of autonomic function. FCD showed a 24 h rhythm (p < 0.001). Independent of energy intake and physical activity level, faster weight loss was observed in individuals with the phase (peak) of the rhythm between ~2-8 p.m. and with a larger amplitude. Interestingly, the phase effect was significant only in C carriers (p = 0.008), while the amplitude effect was only significant in TT carriers (p < 0.0001). The daily rhythm of FCD and CLOCK 3111T/C genotype is linked to weight loss response interactively, suggesting complex interactions between the genetics of the circadian clock, the daily rhythm of autonomic control, and energy balance control.

Heart Rate Dynamics During Acute Recovery From Maximal Aerobic Exercise in Young Adults.

INTRODUCTION: Resting heart rate (HRrest), heart rate variability (HRV), and HR recovery (HRR) from exercise provide valuable information about cardiac autonomic control. RR-intervals during acute recovery from exercise (RRrec) are commonly excluded from HRV analyses due to issues of non-stationarity. However, the variability and complexity within these trends may provide valuable information about changes in HR dynamics. PURPOSE: Assess the complexity of RRrec and determine what physiologic and demographic information are associated with differences in these indices in young adults. METHODS: RR-intervals were collected throughout maximal treadmill exercise and recovery in young adults (n = 92). The first 5 min of RRrec were (1) analyzed with previously reported methods that use 3-interval lengths for comparison and (2) detrended using both differencing(diff) and polynomial regression(res). The standard deviation of the normal interval (SDNN), root mean square of successive differences (rMSSD), root mean square (RMS) of the residual of regression, and sample entropy (SampEn) were calculated. Repeated measures analysis of covariance (ANCOVA) tested for differences in these indices for each of the methodological approaches, controlling for race, body fat, peak oxygen uptake (VO2p eak), and resting HR (HRrest). Statistical significance was set at p < 0.05. RESULTS: VO2p eak and HRrest were significantly correlated with traditional measures of HRR and the variability surrounding RRrec. SampEndiff and SampEnres were correlated with VO2p eak but not HRrest or HRR. The residual-method provided a significantly (p = 0.04) lower mean standard error (MSE) (0.064 ± 0.042) compared to the differencing-method (0.100 ± 0.033). CONCLUSIONS: Complexity analysis of RRrec provides unique information about cardiac autonomic regulation immediately following the cessation of exercise when compared to traditional measures of HRR and both HRrest and VO2peak influence these results.

Assessment of Nocturnal Autonomic Cardiac Imbalance in Positional Obstructive Sleep Apnea. A Multiscale Nonlinear Approach.

Positional obstructive sleep apnea (POSA) is a major phenotype of sleep apnea. Supine-predominant positional patients are frequently characterized by milder symptoms and less comorbidity due to a lower age, body mass index, and overall apnea-hypopnea index. However, the bradycardia-tachycardia pattern during apneic events is known to be more severe in the supine position, which could affect the cardiac regulation of positional patients. This study aims at characterizing nocturnal heart rate modulation in the presence of POSA in order to assess potential differences between positional and non-positional patients. Patients showing clinical symptoms of suffering from a sleep-related breathing disorder performed unsupervised portable polysomnography (PSG) and simultaneous nocturnal pulse oximetry (NPO) at home. Positional patients were identified according to the Amsterdam POSA classification (APOC) criteria. Pulse rate variability (PRV) recordings from the NPO readings were used to assess overnight cardiac modulation. Conventional cardiac indexes in the time and frequency domains were computed. Additionally, multiscale entropy (MSE) was used to investigate the nonlinear dynamics of the PRV recordings in POSA and non-POSA patients. A total of 129 patients (median age 56.0, interquartile range (IQR) 44.8-63.0 years, median body mass index (BMI) 27.7, IQR 26.0-31.3 kg/m2) were classified as POSA (37 APOC I, 77 APOC II, and 15 APOC III), while 104 subjects (median age 57.5, IQR 49.0-67.0 years, median BMI 29.8, IQR 26.6-34.7 kg/m2) comprised the non-POSA group. Overnight PRV recordings from positional patients showed significantly higher disorderliness than non-positional subjects in the smallest biological scales of the MSE profile (τ = 1: 0.25, IQR 0.20-0.31 vs. 0.22, IQR 0.18-0.27, p < 0.01) (τ = 2: 0.41, IQR 0.34-0.48 vs. 0.37, IQR 0.29-0.42, p < 0.01). According to our findings, nocturnal heart rate regulation is severely affected in POSA patients, suggesting increased cardiac imbalance due to predominant positional apneas.

Physiological and Behavioral Factors in Musicians' Performance Tempo.

Musicians display individual differences in their spontaneous performance rates (tempo) for simple melodies, but the factors responsible are unknown. Previous research suggests that musical tempo modulates listeners' cardiovascular activity. We report an investigation of musicians' melody performances measured over a 12-h day and subsequent changes in the musicians' physiological activity. Skilled pianists completed four testing sessions in a single day as cardiac activity was recorded during an initial 5 min of baseline rest and during performances of familiar and unfamiliar melodies. Results indicated slower tempi for familiar and unfamiliar melodies at early testing times. Performance rates at 09 h were predicted by differences in participants' alertness and musical training; these differences were not explained by sleep patterns, chronotype, or cardiac activity. Individual differences in pianists' performance tempo were consistent across testing sessions: participants with a faster tempo at 09 h maintained a faster tempo at later testing sessions. Cardiac measures at early testing times indicated increased heart rates and more predictable cardiac dynamics during music performance than baseline rest, and during performances of unfamiliar melodies than familiar melodies. These findings provide the first evidence of cardiac dynamics that are unique to music performance contexts.

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

Microscopic structural features of cardiac tissue play a fundamental role in determining complex spatio-temporal excitation dynamics at the macroscopic level. Recent efforts have been devoted to the development of mathematical models accounting for non-local spatio-temporal coupling able to capture these complex dynamics without the need of resolving tissue heterogeneities down to the micro-scale. In this work, we analyse in detail several important aspects affecting the overall predictive power of these modelling tools and provide some guidelines for an effective use of space-fractional models of cardiac electrophysiology in practical applications. Through an extensive computational study in simplified computational domains, we highlight the robustness of models belonging to different categories, i.e., physiological and phenomenological descriptions, against the introduction of non-locality, and lay down the foundations for future research and model validation against experimental data. A modern genetic algorithm framework is used to investigate proper parameterisations of the considered models, and the crucial role played by the boundary assumptions in the considered settings is discussed. Several numerical results are provided to support our claims.

Sistemas dinámicos y teoría de la probabilidad aplicados al diagnóstico de la dinámica cardiaca en dieciséis horas / Dynamic systems and probability theory applied to the diagnosis of cardiac dynamics in 16 hours

Resumen Introducción: se han establecido diagnósticos cuantitativos de los sistemas cardiacos, partiendo de teorías como los sistemas dinámicos, la geometría fractal y la teoría de probabilidad. Objetivo: evaluar la dinámica cardiaca con base en una metodología fundamentada en la teoría de probabilidad y los sistemas dinámicos, en dieciséis horas. Metodología: a partir de ochenta registros electrocardiográficos de dinámicas cardiacas, diez normales y setenta con enfermedad, se tomaron los valores máximos y mínimos de la frecuencia cardiaca y el número de latidos/hora durante cada hora, con los cuales se construyó el atractor. Posteriormente, se calculó la dimensión fractal por el método de box counting, los espacios de ocupación y la probabilidad de los espacios de ocupación del atractor. Se determinó el diagnóstico matemático y se hizo una validación estadística respecto al diagnóstico convencional, tomado como estándar de oro. Resultados: se evidenció que la probabilidad de ocupación espacial de los atractores de dinámicas patológicas estuvo entre 0,029 y 0,144 y para dinámicas en estado de normalidad entre 0,164 y 0,329. Se hallaron valores de sensibilidad, especificidad, valor predictivo positivo y negativo de 100% y coeficiente kappa de 1. Conclusiones: se pudo confirmar la capacidad diagnóstica y predictiva de la metodología para diferenciar estados normales de patológicos a nivel clínico.

Abstract Introduction: Quantitative diagnostics of cardiac systems have been established using theories such as, dynamic systems, fractal geometry, and probability theory. Objective: To evaluate cardiac dynamics using a methodology based on probability theory and dynamic systems in sixteen hours. Methods: Using a total of 80 cardiac dynamic electrocardiograph traces (10 normal and 70 with disease), a record was made of the maximum and minimum heart rate values, as well as the number of heart beats/hour during each hour. These values were used to construct the attractor. The fractal dimension was then calculated using the "box counting" method, the spatial occupation, and the probability of spatial occupation by the attractor. The mathematic diagnosis was determined, and a statistical validation was made as regards the conventional diagnosis, which was taken as the reference standard. Results: It was shown that the probability of spatial occupation of the pathological attractor dynamics was between 0.29 and 0.144, and for dynamics in the normal state it was between 0.164 and 0.329. The sensitivity, specificity, positive and negative predictive values were 100%, and the kappa coefficient was 1. Conclusions: The diagnostic and predictive capacity of the methodology to differentiate normal from disease states at clinical level was demonstrated.

Accelerating simulations of cardiac electrical dynamics through a multi-GPU platform and an optimized data structure.

Simulations of cardiac electrophysiological models in tissue, particularly in 3D require the solutions of billions of differential equations even for just a couple of milliseconds, thus highly demanding in computational resources. In fact, even studies in small domains with very complex models may take several hours to reproduce seconds of electrical cardiac behavior. Today's Graphics Processor Units (GPUs) are becoming a way to accelerate such simulations, and give the added possibilities to run them locally without the need for supercomputers. Nevertheless, when using GPUs, bottlenecks related to global memory access caused by the spatial discretization of the large tissue domains being simulated, become a big challenge. For simulations in a single GPU, we propose a strategy to accelerate the computation of the diffusion term through a data-structure and memory access pattern designed to maximize coalescent memory transactions and minimize branch divergence, achieving results approximately 1.4 times faster than a standard GPU method. We also combine this data structure with a designed communication strategy to take advantage in the case of simulations in multi-GPU platforms. We demonstrate that, in the multi-GPU approach performs, simulations in 3D tissue can be just 4× slower than real time.

Evaluación de la dinámica cardiaca con la aplicación de metodologías basadas en entropía proporcional y la ley Zipf-Mandelbrot.

INTRODUCCIÓN: Las metodologías fisicomatemáticas son de utilidad para el diagnóstico de la dinámica cardiaca. OBJETIVO: Comparar la aplicación de dos métodos matemáticos de evaluación de la dinámica cardiaca. Una basada en proporciones de la entropía y otra en la ley de Zipf-Mandelbrot. MÉTODO: Se tomaron 10 registros Holter, cinco de pacientes con enfermedad aguda y cinco normales. Se construyó un atractor numérico, se evaluó la probabilidad, entropía y proporciones de entropía. Para aplicar la segunda metodología se agruparon los valores de frecuencia cardiaca en rangos de 15 latidos/minuto y se aplicó la ley de Zipf-Mandelbrot para obtener la dimensión fractal estadística. Finalmente se comparó la evaluación matemática obtenida por ambas metodologías. RESULTADOS: La metodología basada en las proporciones de la entropía diferenció normalidad, enfermedad y estados intermedios. La segunda metodología diferenció normalidad de enfermedad aguda mediante el grado de complejidad. CONCLUSIÓN: Ambas metodologías establecen evaluaciones de ayuda diagnóstica de la dinámica cardiaca de forma objetiva y reproducible. La entropía proporcional permite cuantificar normalidad, enfermedad y evolución entre estados con carácter predictivo y mayor precisión. INTRODUCTION: Physical-mathematical methodologies have been useful for the diagnosis of cardiac dynamics. OBJECTIVE: To compare the application of two mathematical methodologies for cardiac dynamics evaluation, one of them based on entropy proportions and the other based on of Zipf-Mandelbrot law. METHOD: 10 Holter, 5 acute disease dynamics and 5 normal records were taken. A numerical attractor was constructed; probability, entropy and entropy proportions were evaluated. To apply the second methodology, heart rate values were grouped in 15-beat/min ranges, and Zipf-Mandelbrot's law was applied in order for the statistical fractal dimension to be obtained. Finally, the mathematical evaluation obtained by both methodologies was compared. RESULTS: The methodology based on entropy proportions differentiated normality, disease and intermediate states. The second methodology differentiated normality from acute disease through the degree of complexity. CONCLUSION: Both methodologies establish diagnostically helpful evaluations of cardiac dynamics in an objective and reproducible way. Proportional entropy allows normality, disease and evolution between states to be quantified in a predictive manner and with higher accuracy.

Diagnostic methodology of cardiac dynamics based on the Zipf-Mandelbrot law: evaluation with 50 patients / Metodología diagnóstica de la dinámica cardiaca basada en ley de Zipf-Mandelbrot: evaluación con 50 pacientes

Abstract: Objective: Zipf-Mandelbrot law has been used to assess the complexity of cardiac systems. The objective of this work is to corroborate the clinical applicability of a diagnostic methodology developed from Zipf-Mandelbrot law, in the differentiation of normality and acute cardiac disease. Material and methods: there were taken 50 continuous electrocardiographic Holter monitoring records, 20 normal and 30 with acute alterations of the cardiac dynamics. The frequencies of occurrence of heart rates in ranges of 15 lat/min were organized hierarchically to demonstrate the hyperbolic behavior of dynamics and to apply the Zipf-Mandelbrot law. A linearization was performed and the statistical fractal dimension of each dynamic was obtained, giving rise to the mathematical diagnosis. Sensitivity, specificity and Kappa coefficient were calculated. Results: The values of the statistical fractal dimension of the acute cardiac dynamics were between 0.7123 and 0.9327, whereas for the normal dynamics were found between 0.4253 and 0.6698, evidencing quantitative differences between states of normality and disease. Sensitivity and specificity values of 100% were found and the kappa coefficient was 1. Conclusions: The clinical and diagnostic utility of the mathematical methodology based on Zipf-Mandelbrot law was verified, observing a decrease of dynamics complexity in cases of acute heart disease.(AU)

Resumen: Objetivo: La ley de Zipf-Mandelbrot ha sido utilizada con el fin de evaluar la complejidad de los sistemas cardiacos. El objetivo de este trabajo es corroborar la aplicabilidad clínica de una metodología diagnóstica desarrollada a partir de la ley de Zipf-Mandelbrot, en la diferenciación de normalidad y enfermedad cardiaca aguda. Material y métodos: Se tomaron 50 Holter cardiacos (monitoreo electrocardiográfico continuo ambulatorio), 20 normales y 30 con alteraciones agudas de la dinámica cardiaca. Se organizaron jerárquicamente las frecuencias de aparición de frecuencias cardiacas en rangos de a 15 lat/min, para evidenciar el comportamiento hiperbólico de las dinámicas y aplicar la ley de Zipf-Mandelbrot. Se realizó una linealización y se obtuvo la dimensión fractal estadística de cada dinámica, dando lugar al diagnóstico matemático. Fueron calculadas la sensibilidad, especificidad y el coeficiente Kappa. Resultados: Los valores de la dimensión fractal estadística de las dinámicas cardiacas agudas se encontraron entre 0.7123 y 0.9327, mientras que para las dinámicas normales se hallaron entre 0.4253 y 0.6698, evidenciando diferencias cuantitativas entre estados de normalidad y enfermedad. Se encontraron valores de sensibilidad y especificidad del 100% y el coeficiente kappa fue de 1. Conclusiones: Fue comprobada la utilidad clínica y diagnóstica de la metodología matemática basada en la ley de Zipf-Mandelbrot, observando un decremento de la complejidad de la dinámica en casos de enfermedad cardiaca aguda.(AU)

Extracting cardiac dynamics within ECG signal for human identification and cardiovascular diseases classification.

Cardiac characteristics underlying the time/frequency domain features are limited and not comprehensive enough to reflect the temporal/dynamical nature of ECG patterns. This paper proposes a dynamical ECG recognition framework for human identification and cardiovascular diseases classification via a dynamical neural learning mechanism. The proposed method consists of two phases: a training phase and a test phase. In the training phase, cardiac dynamics within ECG signals is extracted (approximated) accurately by using radial basis function (RBF) neural networks through deterministic learning mechanism. The obtained cardiac system dynamics is represented and stored in constant RBF networks. An ECG signature is then derived from the extracted cardiac dynamics along the periodic ECG state trajectories. A bank of estimators is constructed using the extracted cardiac dynamics to represent the trained gait patterns. In the test phase, recognition errors are generated and taken as the similarity measure by comparing the cardiac dynamics of the trained ECG patterns and the dynamics of the test ECG pattern. Rapid recognition of a test ECG pattern begins with measuring the state of test pattern, and automatically proceeds with the evolution of the recognition error system. According to the smallest error principle, the test ECG pattern can be rapidly recognized. This kind of cardiac dynamics information represents the beat-to-beat temporal change of ECG modifications and the temporal/dynamical nature of ECG patterns. Therefore, the amount of discriminability provided by the cardiac dynamics is larger than the original signals. This paper further discusses the extension of the proposed method for cardiovascular diseases classification. The constructed recognition system can distinguish and assign dynamical ECG patterns to predefined classes according to the similarity of cardiac dynamics. Experiments are carried out on the FuWai and PTB ECG databases to demonstrate the effectiveness of the proposed method.

Effects of prolonged and repeated immersions on heart rate variability and complexity in military divers.

BACKGROUND: The influence of prolonged and repeated water immersions on heart rate variability (HRV) and complexity was examined in 10 U.S. Navy divers who completed six-hour resting dives on five consecutive days. Pre-dive and during-dive measures were recorded daily. METHODS: Dependent variables of interest were average heart rate (HR), time-domain measures of HRV [root mean square of successive differences of the normal RR (NN) interval (RMSSD), standard deviation of the NN interval (SDNN)], frequency-domain measures of HRV [low-frequency power spectral density (psd) (LFpsd), low-frequency normalized (LFnu), high-frequency psd (HFpsd), high-frequency normalized (HFnu), low-frequency/ high-frequency ratio (LF/HF)], and non-linear dynamics of HRV [approximate entropy (ApEn)]. A repeated-measures ANOVA was performed to examine pre-dive measure differences among baseline measures. Hierarchical linear modeling (HLM) was performed to test the effects of prolonged and repeated water immersion on the dependent variables. RESULTS: Pre-dive HR (P=0.005) and RMSSD (P⟨0.001) varied significantly with dive day while changes in SDNN approached significance (P=0.055). HLM indicated that HR decreased during daily dives (P=0.001), but increased across dive days (P=0.011); RMSSD increased during daily dives (P=0.018) but decreased across dive days (P⟨0.001); SDNN increased during daily dives (P⟨0.001); LF measures increased across dive days (LFpsd P⟨0.001; LFnu P⟨0.001), while HF measures decreased across dive days (HFpsd P⟨0.001; HFnu P⟨0.001); LF/HF increased across dive days (P⟨0.001); ApEn decreased during daily dives (P⟨0.02) and across dive days (P⟨0.001). CONCLUSIONS: These data suggest that the cumulative effect of repeated dives across five days results in decreased vagal tone and a less responsive cardiovascular system.

Ley matemática exponencial aplicada a la evaluación de la dinámica cardíaca en 18 horas / Mathematical exponential law applied to the evaluation of the cardiac dynamics in 18 hours

Antecedentes: la teoría de los sistemas dinámicos estudia la evolución de los sistemas. Mediante esta teoría y la geometría fractal se desarrolló una ley matemática de ayuda diagnóstica a los sistemas dinámicos cardiacos, que permite diferenciar entre normalidad y enfermedad, y la evolución entre los dos estados. Objetivo: confirmar la capacidad diagnóstica de la ley matemática exponencial desarrollada inicialmente para dinámicas cardiacas en 21 horas, para dinámicas evaluadas en 18 horas. Método: se tomaron 400 registros electrocardiográficos, 80 de dinámicas normales y 320 de dinámicas anormales. Se generó una sucesión pseudoaleatoria con el número de latidos/hora y las frecuencias máximas y mínimas cada hora; luego, se construyó el atractor de cada dinámica, para así calcular los espacios de ocupación y la dimensión fractal. Finalmente, se estableció el diagnóstico físico-matemático en 18 y 21 horas y se comparó con el diagnóstico clínico tomado como Gold Standard, obteniendo valores de sensibilidad, especificidad y coeficiente Kappa. Resultados: se encontraron valores de ocupación espacial en la rejilla Kp para normalidad entre 236 y 368 y para estados patológicos entre 22 y 189, lo que permitió diferenciar entre normalidad, enfermedad, y estados de evolución hacia la enfermedad en 18 horas. Se obtuvieron valores de sensibilidad y especificidad del 100 por ciento y coeficiente Kappa igual a 1. Conclusión: la ley matemática permitió dictaminar diagnósticos reduciendo el tiempo de evaluación a 18 horas confirmando así su aplicabilidad clínica(AU)

Dynamical systems theory aims to study the evolution of systems. With this theory and fractal geometry, it was developed a mathematical law of diagnostic utility in cardiac dynamical systems that may differentiate normality from disease and evolution between these two states. Objective: To confirm the diagnostic capacity of the exponential mathematical law initially developed for cardiac dynamics in 21 hours, for dynamics evaluated in 18 hours. Method: There were taken 400 electrocardiographic records, 80 from normal dynamics and 320 from abnormal dynamics. A pseudorandom sequence was generated with the number of beats per hour and the maximum and minimum frequencies each hour; then, the attractors were built for each dynamic, in order to calculate the space occupation and the fractal dimension. Finally the physical and mathematical diagnosis in 18 and 21 hours was established, and compared to clinical diagnosis taken as Gold Standard, obtaining values of sensitivity, specificity and Kappa coefficient. Results: There were found values for spatial occupation in the Kp grid between 236 and 368 for normal cases, and between 22 and 189 for pathological states, which allowed distinguish normality from disease and states of progression to disease in 18 hours. There were obtained values for sensitivity and specificity of 100 percent and a Kappa coefficient equal to 1. Conclusion: The mathematical law allowed to stablish diagnostics by reducing the evaluation time to 18 hours confirming its clinical applicability(AU)

A note on stress-driven anisotropic diffusion and its role in active deformable media.

We introduce a new model to describe diffusion processes within active deformable media. Our general theoretical framework is based on physical and mathematical considerations, and it suggests to employ diffusion tensors directly influenced by the coupling with mechanical stress. The proposed generalised reaction-diffusion-mechanics model reveals that initially isotropic and homogeneous diffusion tensors turn into inhomogeneous and anisotropic quantities due to the intrinsic structure of the nonlinear coupling. We study the physical properties leading to these effects, and investigate mathematical conditions for its occurrence. Together, the mathematical model and the numerical results obtained using a mixed-primal finite element method, clearly support relevant consequences of stress-driven diffusion into anisotropy patterns, drifting, and conduction velocity of the resulting excitation waves. Our findings also indicate the applicability of this novel approach in the description of mechano-electric feedback in actively deforming bio-materials such as the cardiac tissue.

Modelling far field pacing for terminating spiral waves pinned to ischaemic heterogeneities in cardiac tissue.

In cardiac tissue, electrical spiral waves pinned to a heterogeneity can be unpinned (and eventually terminated) using electric far field pulses and recruiting the heterogeneity as a virtual electrode. While for isotropic media the process of unpinning is much better understood, the case of an anisotropic substrate with different conductivities in different directions still needs intensive investigation. To study the impact of anisotropy on the unpinning process, we present numerical simulations based on the bidomain formulation of the phase I of the Luo and Rudy action potential model modified due to the occurrence of acute myocardial ischaemia. Simulating a rotating spiral wave pinned to an ischaemic heterogeneity, we compare the success of sequences of far field pulses in the isotropic and the anisotropic case for spirals still in transient or in steady rotation states. Our results clearly indicate that the range of pacing parameters resulting in successful termination of pinned spiral waves is larger in anisotropic tissue than in an isotropic medium.This article is part of the themed issue 'Mathematical methods in medicine: neuroscience, cardiology and pathology'.

[Evaluation in 18 hours of Cardiac Dynamics with the Mathematical Law of Dynamic Systems] / Evaluación en 18 horas de la dinámica cardiaca con la ley matemática de los sistemas dinámicos.

Introduction: an exponential law has been found for chaotic dynamic cardiac systems, making it possible to quantify the differences between normal and pathological cardiac dynamics. Methodology: 120 electrocardiographic records were analyzed, 40 corresponded to subjects within the limits of normality and 80 with different pathologies. For each holter the attractors generated with the data during 18 hours and throughout the dynamics were analyzed. The fractal dimension of the attractor and its spatial occupation were calculated. To these measures was applied the diagnosis mathematical evaluation previously developed, comparing the evaluation for 18 hours and for the whole registry; sensitivity, specificity and Kappa coefficient were finally calculated. Results: For the normal dynamics, the occupancy spaces in the Kp grid were between 200 and 381 for the evaluation of the whole holter, and between 201 and 384 in the evaluation during 18 hours, showing the closeness in the measurements, which allows that the decrease in the time of the evaluation is consistent, this same proximity was observed for the diseased and acute dynamics. Conclusion: It was evidenced the clinical applicability in 18 hours of the exponential law in the chaotic cardiac dynamics associated with arrhythmias showing to be useful for the prediction of the evolution towards acute states of the dynamics

Introducción: una ley exponencial se ha hallado para los sistemas dinámicos caóticos cardiacos, logrando cuantificar las diferencias entre dinámicas cardiacas normales y patológicas. Metodología: Se analizaron 120 registros electrocardiográficos, 40 correspondían a sujetos dentro de los límites de normalidad y 80 con diferentes patologías. Para cada holter se analizaron los atractores generados con los datos durante 18 horas y durante toda la dinámica. Se calculó la dimensión fractal del atractor y su ocupación espacial. A estas medidas se aplicó la evaluación matemática diagnostica desarrollada previamente, comparando la evaluación para 18 horas y para todo el registro; finalmente se calculó la sensibilidad, especificidad y coeficiente Kappa. Resultados: Para las dinámicas normales los espacios de ocupación en la rejilla Kp estuvieron entre 200 y 381 en la evaluación de la totalidad del holter, y entre 201 y 384 en la evaluación durante 18 horas, mostrando la cercanía en las medidas, lo que permite que la disminución en el tiempo de la evaluación sea consistente, esta misma cercanía se observó para las dinámicas enfermas y agudas. Conclusión: Se evidenció la aplicabilidad clínica en 18 horas de la ley exponencial en la dinámica cardiaca caótica asociada a arritmias mostrando ser de utilidad para la predicción de la evolución hacia estados agudos de la dinámica.

Ley caótica exponencial de los sistemas dinámicos cardiacos para evaluación del Holter: 16 horas / Exponential chaotic law of cardiac dynamical systems for Holter evaluation: 16 hours

Introducción. En un estudio previo se realizó una reducción a 16 horas en la evaluación de la ley exponencial de la dinámica cardiaca caótica, mostrando su efectividad en la caracterización de enfermedad y normalidad. Objetivo. Confirmar la aplicabilidad clínica de la ley matemática exponencial para evaluar la dinámica cardiaca caótica a partir de los registros Holter en 16 horas, observando su utilidad diagnóstica al disminuir su tiempo de evaluación. Diseño. Estudio observacional de corte trasversal donde se avaluó los parámetros electrocardiográficos mediante metodologías físico matemáticas inductivas con una confirmación estadística. Metodología. Se tomaron 100 registros Holter con diferentes tipos de patología, y 40 Holter que fueron diagnosticados como normales. Para cada Holter se construyó un atractor caótico, y midiendo sus espacios de ocupación y dimensión fractal se aplicó la evaluación matemática para diferenciar normalidad de enfermedad. Finalmente se realizaron medidas de concordancia diagnostica respecto al estándar de oro. Resultados. La ocupación espacial de todos los atractores estuvieron dentro de los valores esperados; los registros normales presentaron en la rejilla Kp valores entre 205 y 423. Para los registros con enfermedad aguda, estos valores oscilaron entre 21 y 65; y para los registros de enfermedad crónica estos valores estuvieron entre 104 y 195. Los valores de sensibilidad y especificidad fueron de 100% y el coeficiente Kappa fue de 1. Conclusión. El presente estudio muestra la aplicabilidad clínica de esta metodología para la evaluación en 16 horas de registros electrocardiográficos continuos o Holter.

Introduction: In a previous study, a 16-hour reduction in the evaluation of the exponential law of chaotic cardiac dynamics was done, showing its effectiveness in the characterization of disease and normality. Objective: To confirm the clinical applicability of the exponential mathematical law to evaluate chaotic cardiac dynamics from the Holter registers in 16 hours, observing its diagnostic utility when reducing its evaluation time. Design: Cross-sectional observational study where the electrocardiographic parameters were evaluated using inductive mathematical methodologies with statistical confirmation. Methodology: We obtained 100 Holter records from patients with different types of pathology, and 40 Holter that were diagnosed as normal. For each Holter, a chaotic attractor was constructed, and measuring their spaces of occupation and fractal dimension, the mathematical evaluation to differentiate normality of disease was applied. Finally, we calculated measures of diagnostic concordance in accordance with the gold standard. Results: The spatial occupation of all the attractors was within the expected values; the normal records had values in the Kp grid between 205 and 423. For the records with acute disease, these values ranged from 21 y 65; and for chronic disease registries these values ranged from 104 y 195. The values of sensitivity and specificity were 100% and the Kappa coefficient was 1. Conclusion: The present study shows the clinical applicability of this methodology for the evaluation in 16 hours of continuous electrocardiographic or Holter registers.

Optogenetic Light Crafting Tools for the Control of Cardiac Arrhythmias.

The control of spatiotemporal dynamics in biological systems is a fundamental problem in nonlinear sciences and has important applications in engineering and medicine. Optogenetic tools combined with advanced optical technologies provide unique opportunities to develop and validate novel approaches to control spatiotemporal complexity in neuronal and cardiac systems. Understanding of the mechanisms and instabilities underlying the onset, perpetuation, and control of cardiac arrhythmias will enable the development and translation of novel therapeutic approaches. Here we describe in detail the preparation and optical mapping of transgenic channelrhodopsin-2 (ChR2) mouse hearts, cardiac cell cultures, and the optical setup for photostimulation using digital light processing.