Mathematical Modeling to Investigate Antiarrhythmic Drug Side Effects: Rate-Dependence Role in Ionic Currents and Action Potentials Shape in the O’Hara Model

Sudden cardiac death due to ventricular arrhythmia is one of the leading causes of mortality in the world. In the last decades, it has proven that anti-arrhythmic drugs, which prolong the refractory period by means of prolongation of the cardiac action potential duration (APD), play a good role in preventing of relevant human arrhythmias. However, it has long been observed that the “class III antiarrhythmic effect” diminish at faster heart rates and that this phenomenon represent a big weakness, since it is the precise situation when arrhythmias are most prone to occur. It is well known that mathematical modeling is a useful tool for investigating cardiac cell behavior. In the last 60 years, a multitude of cardiac models has been created; from the pioneering work of Hodgkin and Huxley (1952), who first described the ionic currents of the squid giant axon quantitatively, mathematical modeling has made great strides. The O’Hara model, that I employed in this research work, is one of the modern computational models of ventricular myocyte, a new generation began in 1991 with ventricular cell model by Noble et al. Successful of these models is that you can generate novel predictions, suggest experiments and provide a quantitative understanding of underlying mechanism. Obviously, the drawback is that they remain simple models, they don’t represent the real system. The overall goal of this research is to give an additional tool, through mathematical modeling, to understand the behavior of the main ionic currents involved during the action potential (AP), especially underlining the differences between slower and faster heart rates. In particular to evaluate the rate-dependence role on the action potential duration, to implement a new method for interpreting ionic currents behavior after a perturbation effect and to verify the validity of the work proposed by Antonio Zaza using an injected current as a perturbing effect.

Electrolyte disorders in the human ventricle. A simulation study.

Ventricular cells are immersed in a bath of electrolytes and these ions are essential for a healthy heart and a regular rhythm. Maintaining physiological concentration of them is fundamental for reducing arrhythmias and risk of sudden cardiac death, especially in haemodialysis patients and in the heart diseases treatments. Models of electrically activity of the heart based on mathematical formulation are a part of the efforts to improve the understanding and prediction of heart behaviour. Modern models incorporate the extensive and ever increasing amounts of experimental data in incorporating biophysically detailed mechanisms to allow the detailed study of molecular and subcellular mechanisms of heart disease. The goal of this project was to simulate the effects of changes in potassium and calcium concentrations in the extracellular space between experimental data and and a description incorpored into two modern biophysically detailed models (Grandi et al. Model; O’Hara Rudy Model). Moreover the task was to analyze the changes in the ventricular electrical activity, in particular by studying the modifications on the simulated electrocardiographic signal. We used the cellular information obtained by the heart models in order to build a 1D tissue description. The fibre is composed by 165 cells, it is divided in four groups to differentiate the cell types that compound human ventricular tissue. The main results are the following: Grandi et al. (GBP) model is not even able to reproduce the correct action potential profile in hyperkalemia. Data from hospitalized patients indicates that the action potential duration (APD) should be shorter than physiological state but in this model we have the opposite. From the potassium point of view the results obtained by using O’Hara model (ORD) are in agreement with experimental data for the single cell action potential in hypokalemia and hyperkalemia, most of the currents follow the data from literature. In the 1D simulations we were able to reproduce ECGs signal in most the potassium concentrations we selected for this study and we collected data that can help physician in understanding what happens in ventricular cells during electrolyte disorder. However the model fails in the conduction of the stimulus under hyperkalemic conditions. The model emphasized the ECG modifications when the K+ is slightly more than physiological value. In the calcium setting using the ORD model we found an APD shortening in hypocalcaemia and an APD lengthening in hypercalcaemia, i.e. the opposite to experimental observation. This wrong behaviour is kept in one dimensional simulations bringing a longer QT interval in the ECG under higher [Ca2+]o conditions and vice versa. In conclusion it has highlighted that the actual ventricular models present in literature, even if they are useful in the original form, they need an improvement in the sensitivity of these two important electrolytes. We suggest an use of the GBP model with modifications introduced by Carro et al. who understood that the failure of this model is related to the Shannon et al. model (a rabbit model) from which the GBP model was built. The ORD model should be modified in the Ca2+ - dependent IcaL and in the influence of the Iks in the action potential for letting it him produce a correct action potential under different calcium concentrations. In the 1D tissue maybe a heterogeneity setting of intra and extracellular conductances for the different cell types should improve a reproduction of the ECG signal.

A Phase Space Box-counting based Method for Arrhythmia Prediction from Electrocardiogram Time Series

Arrhythmia is one kind of cardiovascular diseases that give rise to the number of deaths and potentially yields immedicable danger. Arrhythmia is a life threatening condition originating from disorganized propagation of electrical signals in heart resulting in desynchronization among different chambers of the heart. Fundamentally, the synchronization process means that the phase relationship of electrical activities between the chambers remains coherent, maintaining a constant phase difference over time. If desynchronization occurs due to arrhythmia, the coherent phase relationship breaks down resulting in chaotic rhythm affecting the regular pumping mechanism of heart. This phenomenon was explored by using the phase space reconstruction technique which is a standard analysis technique of time series data generated from nonlinear dynamical system. In this project a novel index is presented for predicting the onset of ventricular arrhythmias. Analysis of continuously captured long-term ECG data recordings was conducted up to the onset of arrhythmia by the phase space reconstruction method, obtaining 2-dimensional images, analysed by the box counting method. The method was tested using the ECG data set of three different kinds including normal (NR), Ventricular Tachycardia (VT), Ventricular Fibrillation (VF), extracted from the Physionet ECG database. Statistical measures like mean (μ), standard deviation (σ) and coefficient of variation (σ/μ) for the box-counting in phase space diagrams are derived for a sliding window of 10 beats of ECG signal. From the results of these statistical analyses, a threshold was derived as an upper bound of Coefficient of Variation (CV) for box-counting of ECG phase portraits which is capable of reliably predicting the impeding arrhythmia long before its actual occurrence. As future work of research, it was planned to validate this prediction tool over a wider population of patients affected by different kind of arrhythmia, like atrial fibrillation, bundle and brunch block, and set different thresholds for them, in order to confirm its clinical applicability.

The effects of hypocalcemia on spatial alternans and ventricular fibrillation studied with the optical mapping technique

The heart is a wonderful but complex organ: it uses electrochemical mechanisms in order to produce mechanical energy to pump the blood throughout the body and allow the life of humans and animals. This organ can be subject to several diseases and sudden cardiac death (SCD) is the most catastrophic manifestation of these diseases, responsible for the death of a large number of people throughout the world. It is estimated that 325000 Americans annually die for SCD. SCD most commonly occurs as a result of reentrant tachyarrhythmias (ventricular tachycardia (VT) and ventricular fibrillation (VF)) and the identification of those patients at higher risk for the development of SCD has been a difficult clinical challenge. Nowadays, a particular electrocardiogram (ECG) abnormality, “T-wave alternans” (TWA), is considered a precursor of lethal cardiac arrhythmias and sudden death, a sensitive indicator of risk for SCD. TWA is defined as a beat-to-beat alternation in the shape, amplitude, or timing of the T-wave on the ECG, indicative of the underlying repolarization of cardiac cells [5]. In other words TWA is the macroscopic effect of subcellular and celluar mechanisms involving ionic kinetics and the consequent depolarization and repolarization of the myocytes. Experimental activities have shown that TWA on the ECG is a manifestation of an underlying alternation of long and short action potential durations (APDs), the so called APD-alternans, of cardiac myocytes in the myocardium. Understanding the mechanism of APDs-alternans is the first step for preventing them to occur. In order to investigate these mechanisms it’s very important to understand that the biological systems are complex systems and their macroscopic properties arise from the nonlinear interactions among the parts. The whole is greater than the sum of the parts, and it cannot be understood only by studying the single parts. In this sense the heart is a complex nonlinear system and its way of working follows nonlinear dynamics; alternans also, they are a manifestation of a phenomenon typical in nonlinear dynamical systems, called “period-dubling bifurcation”. Over the past decade, it has been demonstrated that electrical alternans in cardiac tissue is an important marker for the development of ventricular fibrillation and a significant predictor for mortality. It has been observed that acute exposure to low concentration of calcium does not decrease the magnitude of alternans and sustained ventricular Fibrillation (VF) is still easily induced under these condition. However with prolonged exposure to low concentration of calcium, alternans disappears, but VF is still inducible. This work is based on this observation and tries to make it clearer. The aim of this thesis is investigate the effect of hypocalcemia spatial alternans and VF doing experiments with canine hearts and perfusing them with a solution with physiological ionic concentration and with a solution with low calcium concentration (hypocalcemia); in order to investigate the so called memory effect, the experimental activity was modified during the way. The experiments were performed with the optical mapping technique, using voltage-sensitive dye, and a custom made Java code was used in post-processing. Finding the Nolasco and Dahlen’s criterion [8] inadequate for the prediction of alternans, and takin into account the experimental results, another criterion, which consider the memory effect, has been implemented. The implementation of this criterion could be the first step in the creation of a method, AP-based, discriminating who is at risk if developing VF. This work is divided into four chapters: the first is a brief presentation of the physiology of the heart; the second is a review of the major theories and discovers in the study of cardiac dynamics; the third chapter presents an overview on the experimental activity and the optical mapping technique; the forth chapter contains the presentation of the results and the conclusions.