Some illustrative examples of permutability of fuzzy operators and fuzzy relations

Composition of fuzzy operators often appears and it is natural to ask when the order of composition does not change the result. In previous papers, we characterized permutability in the case of fuzzy consequence operators and fuzzy interior operators. We also showed the connection between the permutability of the fuzzy relations and the permutability of their induced fuzzy operators. In this work we present some examples of permutability and non permutability of fuzzy operators and fuzzy relations in order to illustrate these results.Postprint (published version

Huge reduction of defibrillation thresholds using four electrode defibrillators

In the absence of a better solution, ventricular fibrillation is treated by applying one or several large electrical shocks to the patient. The question of how to lower the energy required for a successful shock is still a current issue in both fundamental research and clinical practice. In the study presented here we will compare defibrillation applied through a four electrode device with the standard procedure using two electrodes. The method is tested through intensive numerical simulations. Here we have used a one dimensional geometry. At the level of the cardiac tissue, the bidomain and the modified Beeler-Reuter models were used. Three different shock waveforms are tested: monophasic and two types of biphasic shocks. The results are compared with those obtained with standard two electrode device. A significant reduction in defibrillation thresholds is achieved for all the three tested waveforms when we use a four electrode device.Postprint (published version

Huge reduction of defibrillation thresholds using four electrode defibrillators

In the absence of a better solution, ventricular fibril- lation is treated by applying one or several large electri- cal shocks to the patient. The question of how to lower the energy required for a successful shock is still a cur- rent issue in both fundamental research and clinical prac- tice. In the study presented here we will compare defib- rillation applied through a four electrode device with the standard procedure using two electrodes. The method is tested through intensive numerical simulations. Here we have used a one dimensional geometry. At the level of the cardiac tissue, the bidomain and the modified Beeler- Reuter models were used. Three different shock waveforms are tested: monophasic and two types of biphasic shocks. The results are compared with those obtained with stan- dard two electrode device. A significant reduction in de- fibrillation thresholds is achieved for all the three tested waveforms when we use a four electrode device.Postprint (published version

Extended bidomain modeling of defibrillation: quantifying virtual electrode strengths in fibrotic myocardium

Defibrillation is a well-established therapy for atrial and ventricular arrhythmia. Here, we shed light on defibrillation in the fibrotic heart. Using the extended bidomain model of electrical conduction in cardiac tissue, we assessed the influence of fibrosis on the strength of virtual electrodes caused by extracellular electrical current. We created one-dimensional models of rabbit ventricular tissue with a central patch of fibrosis. The fibrosis was incorporated by altering volume fractions for extracellular, myocyte and fibroblast domains. In our prior work, we calculated these volume fractions from microscopic images at the infarct border zone of rabbit hearts. An average and a large degree of fibrosis were modeled. We simulated defibrillation by application of an extracellular current for a short duration (5 ms). We explored the effects of myocyte-fibroblast coupling, intra-fibroblast conductivity and patch length on the strength of the virtual electrodes present at the borders of the normal and fibrotic tissue. We discriminated between effects on myocyte and fibroblast membranes at both borders of the patch. Similarly, we studied defibrillation in two-dimensional models of fibrotic tissue. Square and disk-like patches of fibrotic tissue were embedded in control tissue. We quantified the influence of the geometry and fibrosis composition on virtual electrode strength. We compared the results obtained with a square and disk shape of the fibrotic patch with results from the one-dimensional simulations. Both, one- and two-dimensional simulations indicate that extracellular current application causes virtual electrodes at boundaries of fibrotic patches. A higher degree of fibrosis and larger patch size were associated with an increased strength of the virtual electrodes. Also, patch geometry affected the strength of the virtual electrodes. Our simulations suggest that increased fibroblast-myocyte coupling and intra-fibroblast conductivity reduce virtual electrode strength. However, experimental data to constrain these modeling parameters are limited and thus pinpointing the magnitude of the reduction will require further understanding of electrical coupling of fibroblasts in native cardiac tissues. We propose that the findings from our computational studies are important for development of patient-specific protocols for internal defibrillators

Phase-2 reentry in cardiac tissue: role of the slow calcium pulse

Phase-2 re-entry is thought to underlie many causes of idiopathic ventricular arrhythmias as, for instance, those occurring in Brugada syndrome. In this paper, we study under which circumstances a region of depolarized tissue can re-excite adjacent regions that exhibit shorter action potential duration (APD), eventually inducing reentry. For this purpose, we use a simplified ionic model that reproduces well the ventricular action potential. With the help of this model, we analyze the conditions that lead to very short action potentials (APs), as well as possible mechanisms for re-excitation in a cable. We then study the induction of re-entrant waves (spiral waves) in simulations of AP propagation in the heart ventricles. We show that re-excitation takes place via a slow pulse produced by calcium current that propagates into the region of short APs until it encounters excitable tissue. We calculate analytically the speed of the slow pulse, and also give an estimate of the minimal tissue size necessary for allowing reexcitation to take place.Peer ReviewedPostprint (published version

Integral behaviour for localized synchronization in nonidentical extended systems

We report the synchronization of two nonidentical spatially extended fields, ruled by one-dimensional complex Ginzburg-Landau equations. The two fields are prepared in different dynamical regimes, and interact via an imperfect coupling consisting of a given number of local controllers Nc . The strength of the coupling is ruled by the parameter «. We show that, in the limit of three controllers per correlation length, the synchronization behavior is not affected if the product «Nc /N is kept constant, providing a sort of integral behavior for localized synchronization

Using high-resolution voltage maps to predict “redo” in the treatment of atrial fibrillation (AF)

The aim of this work is to use biomarkers extracted from high-resolution voltage maps of atrial fibrillation (AF) patients in order to make predictions about future “redo” procedures. We collected maps of the left atrium of 122 patients, prior of being treated for AF The bipolar voltage maps were extracted with the Rythmia system from Boston Scientific and subsequently analyzed in the MATLAB environment. The present study focuses on three biomarkers extracted from those maps. Two are associated with the bipolar voltage measurements on the map, i.e., the mean voltage and the voltage dispersion on the map. The third indicator is the area of the atrium evaluated from the map. The data are used for feeding a supervised classification algorithm. The output variable is a binary variable that is set to 1 if the patient will need a “redo” procedure in the twelve months following the cardiac intervention and 0 otherwise. We show that the biomarkers have some statistical power in predicting future outcomes. Especially the mean voltage on the map is the best predictor of the future outcome. We determine the cutoff value for the mean voltage based on the best prediction accuracy of Vm=0.542 mV in agreement with previous studies. We discuss some extensions of this study that could allow improvements in predictive power.Peer ReviewedPostprint (published version

Influence of gap junction dynamics on the stability of reentrant waves in cadiac tissue

Constant conductances are often assumed when model- ing cardiac tissue. However experimental evidences have shown that gap junctions (GJ) actually connect adjacent cardiac myocytes. These GJ are complex proteins of the connexin family (Cx40; Cx43; Cx45 are the most common in human). These GJ modify the conductances between cardiac cell through a dynamical process. The aim of this study is to develop a bidomain model of the cardiac tis- sue where the dynamics of the connexins is also included. In particular we will compare the differences associated with the use of a monodomain versus bidomain formula- tion in inducing intra-cellular conductance variations. We have found that the monodomain formulation gives con- ductance variations a factor four to five larger with respect to the bidomain formulationPeer ReviewedPostprint (published version