Effects of spatial segmentation in the continuous model of excitation propagation in cardiac muscle

Late phase of repolarization is autoregenerative and scales linearly with action potential duration in mammals ventricular myocytes: a model study

Scaling of action potential (AP) duration (APD) in mammals of different size is a rather complex phenomenon, dominated by a regulatory type mechanism of ion channels expression. By means of simulations performed on six published mathematical models of cardiac ventricular APs of different mammals, it is shown that AP repolarization is autoregenerative in its later phase (ARRP) and that the duration of such phase scales linearly with APD. For each AP, a 3-D instantaneous time-voltage-current surface is constructed, which has been recently described in a more simplified model. This representation allows us to measure ARRP and to study the contribution to it for different ion currents. It has been found that the existence of an ARRP is not intrinsic to cardiac models formulation; one out of the six models does not show this phase. A linear correlation between ARRP duration and APD in the remaining models is also found. It is shown that ARRP neither simply depend on AP shape nor on APD. Though I(K1) current seems to be the main responsible for determining and modulating this phase, the mechanism by which ARRP scales linearly with APD remains unclear and raises further questions on the scaling strategies of cardiac repolarization in mammals.

3D current-voltage-time surfaces unveil critical repolarization differences underlying similar cardiac action potentials: A model study

The number of mathematical models of cardiac cellular excitability is rapidly growing, and compact graphical representations of their properties can make new acquisitions available for a broader range of scientists in cardiac field. Particularly, the intrinsic over-determination of the model equations systems when fitted only to action potential (AP) waveform and the fact that they are frequently tuned on data covering only a relatively narrow range of dynamic conditions, often lead modellers to compare very similar AP profiles, which underlie though quite different excitable properties. In this study I discuss a novel compact 3D representation of the cardiac cellular AP, where the third dimension represents the instantaneous current-voltage profile of the membrane, measured as repolarization proceeds. Measurements of this type have been used previously for in vivo experiments, and are adopted here iteratively at a very high time, voltage, current-resolution on (i) the same human ventricular model, endowed with two different parameters sets which generate the same AP waveform, and on (ii) three different models of the same human ventricular cell type. In these 3D representations, the AP waveforms lie at the intersection between instantaneous time-voltage-current surfaces and the zero-current plane. Different surfaces can share the same intersection and therefore the same AP; in these cases, the morphology of the current surface provides a compact view of important differences within corresponding repolarization dynamics. Refractory period, supernormal excitability window, and extent of repolarization reserve can be visualized at once. Two pivotal dynamical properties can be precisely assessed, i.e. all-or-nothing repolarization window and membrane resistance during recovery. I discuss differences in these properties among the membranes under study, and show relevant implications for cardiac cellular repolarization.

How different two almost identical action potentials can be: a model study on cardiac repolarization

Spatial heterogeneity in the properties of ion channels generates spatial dispersion of ventricular repolarization, which is modulated by gap junctional coupling. However, it is possible to simulate conditions in which local differences in excitation properties are electrophysiologically silent and only play a role in pathological states. We use a numerical procedure on the Luo-Rudy phase 1 model of the ventricular action potential (AP1) in order to find a modified set of model parameters which generates an action potential profile (AP2) almost identical to AP1. We show that, although the two waveforms elicited from resting conditions as a single AP are very similar and belong to membranes sharing similar passive electrical properties, the modified membrane generating AP2 is a weaker current source than the one generating AP1, has different sensitivity to up/down-regulation of ion channels and to extracellular potassium, and a different electrical restitution profile. We study electrotonic interaction of AP1- and AP2- type membranes in cell pairs and in cable conduction, and find differences in source-sink properties which are masked in physiological conditions and become manifest during intercellular uncoupling or partial block of ion channels, leading to unidirectional block and spatial repolarization gradients. We provide contour plot representations that summarize differences and similarities. The present report characterizes an inverse problem in cardiac cells, and strengthen the recently emergent notion that a comprehensive characterization and validation of cell models and their components are necessary in order to correctly understand simulation results at higher levels of complexity.

Design of a predictive targeting error simulator for MRI-guided prostate biopsy

Multi-parametric MRI is a new imaging modality superior in quality to Ultrasound (US) which is currently used in standard prostate biopsy procedures. Surface-based registration of the pre-operative and intra-operative prostate volumes is a simple alternative to side-step the challenges involved with deformable registration. However, segmentation errors inevitably introduced during prostate contouring spoil the registration and biopsy targeting accuracies. For the crucial purpose of validating this procedure, we introduce a fully interactive and customizable simulator which determines the resulting targeting errors of simulated registrations between prostate volumes given user-provided parameters for organ deformation, segmentation, and targeting. We present the workflow executed by the simulator in detail and discuss the parameters involved. We also present a segmentation error introduction algorithm, based on polar curves and natural cubic spline interpolation, which introduces statistically realistic contouring errors. One simulation, including all I/O and preparation for rendering, takes approximately 1 minute and 40 seconds to complete on a system with 3 GB of RAM and four Intel Core 2 Quad CPUs each with a speed of 2.40 GHz. Preliminary results of our simulation suggest the maximum tolerable segmentation error given the presence of a 5.0 mm wide small tumor is between 4-5 mm. We intend to validate these results via clinical trials as part of our ongoing work.

Electrotonic influences on action potential duration dispersion in small hearts: a simulation study

Intrinsic spatial variations in repolarization currents in the heart can produce spatial gradients in action potential duration (APD) that serve as possible sites for conduction block and the initiation of reentrant activity. In well-coupled myocardium, however, electrotonic influences at the stimulus site and wavefront collision sites act to modulate any intrinsic heterogeneity in APD. These effects alter APD gradients over an extent larger than that suggested by the length constant associated with propagation and, thus, are hypothesized to play a greater role in smaller hearts used as experimental models of human disease. This study uses computer simulation to investigate how heart size, tissue properties, and the spatial assignment of cell types affect functional APD dispersion. Simulations were carried out using the murine ventricular myocyte model of Pandit et al. or the Luo-Rudy mammalian model in three-dimensional models of mouse and rabbit ventricular geometries. Results show that the spatial extent of the APD dispersion is related to the dynamic changes in transmembrane resistance during recovery. Also, because of the small dimensions of the mouse heart, electrotonic effects on APD primarily determine the functional dispersion of refractoriness, even in the presence of large intrinsic cellular heterogeneity and reduced coupling. APD dispersion, however, is found to increase significantly when the heart size increases to the size of a rabbit heart, unmasking intrinsic cell types.

Simulation of QRST Integral Maps With a Membrane-Based Computer Heart Model Employing Parallel Processing

The simulation of the propagation of electrical activity in a membrane-based realistic-geometry computer model of the ventricles of the human heart, using the governing monodomain reaction-diffusion equation, is described. Each model point is represented by the phase 1 Luo-Rudy membrane model, modified to represent human action potentials. A separate longer duration action potential was used for the M cells found in the ventricular midwall. Cardiac fiber rotation across the ventricular wall was implemented via an analytic equation, resulting in a spatially varying anisotropic conductivity tensor and, consequently, anisotropic propagation. Since the model comprises approximately 12.5 million points, parallel processing on a multiprocessor computer was used to cut down on simulation time. The simulation of normal activation as well as that of ectopic beats is described. The hypothesis that in situ electrotonic coupling in the myocardium can diminish the gradients of action-potential duration across the ventricular wall was also verified in the model simulations. Finally, the sensitivity of QRST integral maps to local alterations in action-potential duration was investigated.

Nonlinear effects in subthreshold virtual electrode polarization

Introduction of the virtual electrode polarization (VEP) theory suggested solutions to several century-old puzzles of heart electrophysiology including explanation of the mechanisms of stimulation and defibrillation. Bidomain theory predicts that VEPs should exist at any stimulus strength. Although the presence of VEPs for strong suprathreshold pulses has been well documented, their existence at subthreshold strengths during diastole remains controversial. We studied cardiac membrane polarization produced by subthreshold stimuli in 1) rabbit ventricular muscle using high-resolution fluorescent imaging with the voltage-sensitive dye pyridinium 4-[2-[6-(dibutylamino)-2-naphthalenyl]-ethenyl]-1-(3-sulfopropyl)hydroxide (di-4-ANEPPS) and 2) an active bidomain model with Luo-Rudy ion channel kinetics. Both in vitro and in numero models show that the common dog-bone-shaped VEP is present at any stimulus strength during both systole and diastole. Diastolic subthreshold VEPs exhibited nonlinear properties that were expressed in time-dependent asymmetric reversal of membrane polarization with respect to stimulus polarity. The bidomain model reveals that this asymmetry is due to nonlinear properties of the inward rectifier potassium current. Our results suggest that active ion channel kinetics modulate the transmembrane polarization pattern that is predicted by the linear bidomain model of cardiac syncytium.

Ischemic modulation of vulnerable period and the effects of pharmacological treatment of ischemia-induced arrhythmias: a simulation study

First identified in the 1930s (Ferris et al., 1936 and Wiggers and Wegria, 1939), the concept of vulnerability applies perfectly to biological oscillators. We can safely say that vulnerability is an inherent property of any excitable media. The duration of vulnerable period (VP) (the time interval during which single stimuli can initiate self-sustaining propagation) is sensitive to medium properties and stimulus parameters (stimulus field, timing behind the conditioning wave, and stimulus amplitude). Apart from medium properties and stimulus characteristics, heart vulnerability is affected by any intervention targeting the excitatory and recovery process. Therefore, we can expect that any pathological condition disturbing heart excitation or tissue recovery will most probably alter the duration of VP. In this paper, we shall explore the implications of ischemia and one of the arrhythmia counteracting methods widely used in clinical practice-antiarrhythmic drugs--in changing the boundaries of VP. The Cardiac Arrhythmia Suppression Trial (CAST) studies, as well as classification based on functional characteristics, revealed the arrhythmogenic potential of both Class I and Class III agents, but failed to identify the proarrhythmic mechanisms. This study presents results from a mathematical model (Cimponeriu et al., 2001) of the ventricle based on Luo-Rudy cellular formulation Luo and Rudy, 1991) modified for studying the ischemic modulation of VP and the effects of pharmacological treatment of ischemia-induced arrhythmia. Simulations revealed the link between the cellular antiarrhythmic properties and the proarrhythmic effect at the multicellular level in the case of Na+ channel blockade. Na+ channel blockade delayed recovery of cellular excitability, but also introduced a nonuniform dispersion of refractoriness along the cardiac fiber that can serve as a substrate for initiating a new arrhythmia. Our initial analysis proved that fast unbinding rates are essential in reducing the proarrhythmic potential of Class I drugs. However, further investigations led us to believe that binding properties are equally important. An antiarrhythmic drug with high affinity for drug-channel complex formation elicits a higher level of blockade per time unit. Under this light, we hypothesize that even the modern, fast unbinding drugs are not necessarily safe.

Triggered alternans in an ionic model of ischemic cardiac ventricular muscle

It has been known for several decades that electrical alternans occurs during myocardial ischemia in both clinical and experimental work. There are a few reports showing that this alternans can be triggered into existence by a premature ventricular contraction. Detriggering of alternans by a premature ventricular contraction, as well as pause-induced triggering and detriggering, have also been reported. We conduct a search for triggered alternans in an ionic model of ischemic ventricular muscle in which alternans has been described recently: a one-dimensional cable of length 3 cm, containing a central ischemic zone 1 cm long, with 1 cm segments of normal (i.e., nonischemic) tissue at each end. We use a modified form of the Luo-Rudy [Circ. Res. 68, 1501-1526 (1991)] ionic model to represent the ventricular tissue, modeling the effect of ischemia by raising the external potassium ion concentration ([K(+)](o)) in the central ischemic zone. As [K(+)](o) is increased at a fixed pacing cycle length of 400 ms, there is first a transition from 1:1 rhythm to alternans or 2:2 rhythm, and then a transition from 2:2 rhythm to 2:1 block. There is a range of [K(+)](o) over which there is coexistence of 1:1 and 2:2 rhythms, so that dropping a stimulus from the periodic drive train during 1:1 rhythm can result in the conversion of 1:1 to 2:2 rhythm. Within the bistable range, the reverse transition from 2:2 to 1:1 rhythm can be produced by injection of a well-timed extrastimulus. Using a stimulation protocol involving delivery of pre- and post-mature stimuli, we derive a one-dimensional map that captures the salient features of the results of the cable simulations, i.e., the {1:1-->2:2-->2:1} transitions with {1:1<-->2:2} bistability. This map uses a new index of the global activity in the cable, the normalized voltage integral. Finally, we put forth a simple piecewise linear map that replicates the {1:1<-->2:2} bistability observed in the cable simulations and in the normalized voltage integral map. (c) 2002 American Institute of Physics.

A computer model of human atria with reasonable computation load and realistic anatomical properties

Atrial fibrillation is the most frequent arrhythmia, provoking discomfort, heart failure and arterial embolisms. The aim of this work is to develop a simplified anatomical computer model of human atria for the study of atrial arrhythmias and the understanding of electrical propagation mechanisms. With the model we propose, up to 40 s of real-time propagation have been simulated on a single-processor computer. The size and the electrophysiological properties of the simulated atria are within realistic values and information about anatomy has been taken into account in a three-dimensional structure. Besides normal sinus beat, pathological phenomena such as flutter and fibrillation have been induced using a programmed stimulation protocol. One important observation in our model is that atrial arrhythmias are a combination of functional and anatomical reentries and that the geometry plays an important role. This virtual atrium can reproduce electrophysiological observations made in humans but with the advantage of showing in great detail how arrhythmias are initiated and sustained. Such details are difficult or impossible to study in humans. This model will serve us as a tool to evaluate the impact of new therapeutic strategies and to improve them.

Computational models of normal and abnormal action potential propagation in cardiac tissue: linking experimental and clinical cardiology

Computational models have the potential to make a huge impact on our understanding of normal and abnormal cardiac function. The aim of this article is to review tools that have been developed to simulate the electrophysiology of cardiac cells and tissue, and to show how computational models have been used to gain insight into normal and abnormal action potential propagation. Some of the practical problems experienced in the development and application of these models are described, and examples are given.