An analytical study of the physiology and pathology of the propagation of cardiac action potentials

The conundrum of the complex relationship between acute kidney injury and cardiac arrhythmias

Acute kidney injury (AKI) is defined by a rapid increase in serum creatinine levels, reduced urine output, or both. Death may occur in 16%-49% of patients admitted to an intensive care unit with severe AKI. Complex arrhythmias are a potentially serious complication in AKI patients with pre-existing or AKI-induced heart damage and myocardial dysfunction, fluid overload, and especially electrolyte and acid-base disorders representing the pathogenetic mechanisms of arrhythmogenesis. Cardiac arrhythmias, in turn, increase the risk of poor renal outcomes, including AKI. Arrhythmic risk in AKI patients receiving kidney replacement treatment may be reduced by modifying dialysis/replacement fluid composition. The most common arrhythmia observed in AKI patients is atrial fibrillation. Severe hyperkalemia, sometimes combined with hypocalcemia, causes severe bradyarrhythmias in this clinical setting. Although the likelihood of life-threatening ventricular arrhythmias is reportedly low, the combination of cardiac ischemia and specific electrolyte or acid-base abnormalities may increase this risk, particularly in AKI patients who require kidney replacement treatment. The purpose of this review is to summarize the available epidemiological, pathophysiological, and prognostic evidence aiming to clarify the complex relationships between AKI and cardiac arrhythmias.

A condition for setting off ectopic waves in computational models of excitable cells

The purpose of this paper is to study the stability of steady state solutions of the Monodomain model equipped with Luo-Rudy I kinetics. It is well established that re-entrant arrhythmias can be created in computational models of excitable cells. Such arrhythmias can be initiated by applying an external stimulus that interacts with a partially refractory region, and spawn breaking waves that can eventually generate extremely complex wave patterns commonly referred to as fibrillation. An ectopic wave is one possible stimulus that may initiate fibrillation. Physiologically, it is well known that ectopic waves exist, but the mechanism for initiating ectopic waves in a large collection of cells is poorly understood. In the present paper we consider computational models of collections of excitable cells in one and two spatial dimensions. The cells are modeled by Luo-Rudy I kinetics, and we assume that the spatial dynamics is governed by the Monodomain model. The mathematical analysis is carried out for a reduced model that is known to provide good approximations of the initial phase of solutions of the Luo-Rudy I model. A further simplification is also introduced to motivate and explain the results for the more complicated models. In the analysis the cells are divided into two regions; one region (N) consists of normal cells as model by the standard Luo-Rudy I model, and another region (A) where the cells are automatic in the sense that they would act as pacemaker cells if they where isolated from their surroundings. We let delta denote the spatial diffusion and a denote a characteristic length of the automatic region. It has previously been shown that reducing diffusion or increasing the automatic region enhances ectopic activity. Here we derive a condition for the transition from stable resting state to ectopic wave spread. Under suitable assumptions on the model we provide mathematical and computational arguments indicating that there is a constant eta such that a steady state solution of this system is stable whenever delta approximately > etaa(2), and unstable whenever delta approximately < etaa(2).

Asymptotic analysis and analytical solutions of a model of cardiac excitation

We describe an asymptotic approach to gated ionic models of single-cell cardiac excitability. It has a form essentially different from the Tikhonov fast-slow form assumed in standard asymptotic reductions of excitable systems. This is of interest since the standard approaches have been previously found inadequate to describe phenomena such as the dissipation of cardiac wave fronts and the shape of action potential at repolarization. The proposed asymptotic description overcomes these deficiencies by allowing, among other non-Tikhonov features, that a dynamical variable may change its character from fast to slow within a single solution. The general asymptotic approach is best demonstrated on an example which should be both simple and generic. The classical model of Purkinje fibers (Noble in J. Physiol. 160:317-352, 1962) has the simplest functional form of all cardiac models but according to the current understanding it assigns a physiologically incorrect role to the Na current. This leads us to suggest an "Archetypal Model" with the simplicity of the Noble model but with a structure more typical to contemporary cardiac models. We demonstrate that the Archetypal Model admits a complete asymptotic solution in quadratures. To validate our asymptotic approach, we proceed to consider an exactly solvable "caricature" of the Archetypal Model and demonstrate that the asymptotic of its exact solution coincides with the solutions obtained by substituting the "caricature" right-hand sides into the asymptotic solution of the generic Archetypal Model. This is necessary, because, unlike in standard asymptotic descriptions, no general results exist which can guarantee the proximity of the non-Tikhonov asymptotic solutions to the solutions of the corresponding detailed ionic model.

Inherited Conduction System Abnormalities-One Group of Diseases, Many Genes

The cardiac conduction system can be anatomically, developmentally, and molecularly distinguished from the working myocardium. Abnormalities in cardiac conduction can occur due to a variety of factors, including developmental and congenital defects, acquired injury or ischemia of portions of the conduction system, or less commonly due to inherited diseases that alter cardiac conduction system function. So called "idiopathic" conduction system degeneration may have familial clustering, and therefore is consistent with a hereditary basis. This "Molecular Perspectives" will highlight several diverse mechanisms of isolated conduction system disease as well as conduction system degeneration associated with other cardiac and non-cardiac disorders. The first part of this review focuses on channelopathies associated with conduction system disease. Human genetic studies have identified mutations in the sodium channel SCN5A gene causing tachyarrhythmia disorders, as well as progressive cardiac conduction system diseases, or overlapping syndromes. Next, the importance of embryonic developmental genes such as homeobox and T-box transcription factors are highlighted in conduction system development and function. Conduction system diseases associated with multisystem disorders, such as muscular and myotonic dystrophies, will be described. Last, a new glycogen storage cardiomyopathy associated with ventricular preexcitation and progressive conduction system degeneration will be reviewed. There are a myriad of mutations identified in genes encoding cardiac transcription factors, ion channels, gap junctions, energy metabolism regulators, lamins and other structural proteins. Understanding of the molecular and ionic mechanisms underlying cardiac conduction is essential for the appreciation of the pathogenesis of conduction abnormalities in structurally normal and altered hearts.

Non-Tikhonov asymptotic properties of cardiac excitability

Models of electric excitability of cardiac cells can be studied by singular perturbation techniques. To do this one should take into account parameters appearing in equations in nonstandard ways. The physical reason for this is near-perfect switch behavior of ionic current gates. This leads to a definition of excitability different from the currently accepted one. The asymptotic structure revealed by our analysis can be used to devise simplified caricature models, to obtain approximate analytical solutions, and to facilitate numerical simulations.

Dissipation of the excitation wave fronts

An excitation wave in cardiac tissue will fail to propagate if the transmembrane voltage at its front rises too slowly and does not excite the tissue ahead of it. Then the sharp voltage profile of the front will dissipate, and the subsequent spread of voltage will be purely diffusive. This mechanism is impossible in FitzHugh-Nagumo type systems. Here a simplified mathematical model for this mechanism is suggested. The model has exact traveling front solutions, and gives conditions for the front dissipation. In particular, a front will dissipate if it is not allowed to propagate faster than a certain nonzero speed. This critical speed depends only on the properties of the fast sodium current.