Mechanism of ventricular vulnerability to single premature stimuli in open-chest dogs

Are physiological oscillations physiological?

Despite widespread and striking examples of physiological oscillations, their functional role is often unclear. Even glycolysis, the paradigm example of oscillatory biochemistry, has seen questions about its oscillatory function. Here, we take a systems approach to argue that oscillations play critical physiological roles, such as enabling systems to avoid desensitization, to avoid chronically high and therefore toxic levels of chemicals, and to become more resistant to noise. Oscillation also enables complex physiological systems to reconcile incompatible conditions such as oxidation and reduction, by cycling between them, and to synchronize the oscillations of many small units into one large effect. In pancreatic β-cells, glycolytic oscillations synchronize with calcium and mitochondrial oscillations to drive pulsatile insulin release, critical for liver regulation of glucose. In addition, oscillation can keep biological time, essential for embryonic development in promoting cell diversity and pattern formation. The functional importance of oscillatory processes requires a re-thinking of the traditional doctrine of homeostasis, holding that physiological quantities are maintained at constant equilibrium values, a view that has largely failed in the clinic. A more dynamic approach will initiate a paradigm shift in our view of health and disease. A deeper look into the mechanisms that create, sustain and abolish oscillatory processes requires the language of nonlinear dynamics, well beyond the linearization techniques of equilibrium control theory. Nonlinear dynamics enables us to identify oscillatory ('pacemaking') mechanisms at the cellular, tissue and system levels.

The physics of heart rhythm disorders

The global burden caused by cardiovascular disease is substantial, with heart disease representing the most common cause of death around the world. There remains a need to develop better mechanistic models of cardiac function in order to combat this health concern. Heart rhythm disorders, or arrhythmias, are one particular type of disease which has been amenable to quantitative investigation. Here we review the application of quantitative methodologies to explore dynamical questions pertaining to arrhythmias. We begin by describing single-cell models of cardiac myocytes, from which two and three dimensional models can be constructed. Special focus is placed on results relating to pattern formation across these spatially-distributed systems, especially the formation of spiral waves of activation. Next, we discuss mechanisms which can lead to the initiation of arrhythmias, focusing on the dynamical state of spatially discordant alternans, and outline proposed mechanisms perpetuating arrhythmias such as fibrillation. We then review experimental and clinical results related to the spatio-temporal mapping of heart rhythm disorders. Finally, we describe treatment options for heart rhythm disorders and demonstrate how statistical physics tools can provide insights into the dynamics of heart rhythm disorders.

R-on-T and the initiation of reentry revisited: Integrating old and new concepts

Initiation of reentry requires 2 factors: (1) a triggering event, most commonly focal excitations such as premature ventricular complexes (PVCs); and (2) a vulnerable substrate with regional dispersion of refractoriness and/or excitability, such as occurs during the T wave of the electrocardiogram when some areas of the ventricle have repolarized and recovered excitability but others have not. When the R wave of a PVC coincides in time with the T wave of the previous beat, this timing can lead to unidirectional block and initiation of reentry, known as the R-on-T phenomenon. Classically, the PVC triggering reentry has been viewed as arising focally from 1 region and propagating into another region whose recovery is delayed, resulting in unidirectional conduction block and reentry initiation. However, more recent evidence indicates that PVCs also can arise from the T wave itself. In the latter case, the PVC initiating reentry is not a separate event from the T wave but rather is causally generated from the repolarization gradient that manifests as the T wave. We call the former an "R-to-T" mechanism and the latter an "R-from-T" mechanism, which are initiation mechanisms distinct from each other. Both are important components of the R-on-T phenomenon and need to be taken into account when designing antiarrhythmic strategies. Strategies targeting suppression of triggers alone or vulnerable substrate alone may be appropriate in some instances but not in others. Preventing R-from-T arrhythmias requires suppressing the underlying dynamic tissue instabilities responsible for producing both triggers and substrate vulnerability simultaneously. The same principles are likely to apply to supraventricular arrhythmias.

OUP accepted manuscript

Aims

Mapping data of human ventricular fibrillation (VF) are limited. We performed detailed mapping of the activities underlying the onset of VF and targeted ablation in patients with structural cardiac abnormalities.

Methods and results

We evaluated 54 patients (50 ± 16 years) with VF in the setting of ischaemic (n = 15), hypertrophic (n = 8) or dilated cardiomyopathy (n = 12), or Brugada syndrome (n = 19). Ventricular fibrillation was mapped using body-surface mapping to identify driver (reentrant and focal) areas and invasive Purkinje mapping. Purkinje drivers were defined as Purkinje activities faster than the local ventricular rate. Structural substrate was delineated by electrogram criteria and by imaging. Catheter ablation was performed in 41 patients with recurrent VF. Sixty-one episodes of spontaneous (n = 10) or induced (n = 51) VF were mapped. Ventricular fibrillation was organized for the initial 5.0 ± 3.4 s, exhibiting large wavefronts with similar cycle lengths (CLs) across both ventricles (197 ± 23 vs. 196 ± 22 ms, P = 0.9). Most drivers (81%) originated from areas associated with the structural substrate. The Purkinje system was implicated as a trigger or driver in 43% of patients with cardiomyopathy. The transition to disorganized VF was associated with the acceleration of initial reentrant activities (CL shortening from 187 ± 17 to 175 ± 20 ms, P < 0.001), then spatial dissemination of drivers. Purkinje and substrate ablation resulted in the reduction of VF recurrences from a pre-procedural median of seven episodes [interquartile range (IQR) 4–16] to 0 episode (IQR 0–2) (P < 0.001) at 56 ± 30 months.

Conclusions

The onset of human VF is sustained by activities originating from Purkinje and structural substrate, before spreading throughout the ventricles to establish disorganized VF. Targeted ablation results in effective reduction of VF burden.

Key question

The initial phase of human ventricular fibrillation (VF) is critical as it involves the primary activities leading to sustained VF and arrhythmic sudden death. The origin of such activities is unknown.

Key finding

Body-surface mapping shows that most drivers (≈80%) during the initial VF phase originate from electrophysiologically defined structural substrates. Repetitive Purkinje activities can be elicited by programmed stimulation and are implicated as drivers in 37% of cardiomyopathy patients.

Take-home message

The onset of human VF is mostly associated with activities from the Purkinje network and structural substrate, before spreading throughout the ventricles to establish sustained VF. Targeted ablation reduces or eliminates VF recurrence.

Dominant vector changes during early wavebreak/spiral wave (Wiggers stage 1) in ventricular fibrillation: insights from the analysis of 100 electrophysiology studies

PURPOSE

To describe electrocardiographic vector patterns during early VF transition (Wiggers stage 1).

METHODS

In 100 electrophysiology studies with VF induction, the first 3 beats of VF were analyzed in lead I for left/right axis (LA/RA), V1 for left/right bundle (LB/RB), and aVF for superior/inferior axis (SA/IA). Correlation with demographic/clinical factors was performed using regression analyses and mixed effect modeling.

RESULTS

VF initiated more likely with LA than RA (P < 0.001) and LB than RB (P = 0.04) suggesting original wavebreak in the right ventricle. The 3-dimensional morphology changed in 69% of VF during the first 3 beats, with predominant increase in RB, suggesting a transition of QRS-originating vector to septum/left ventricle. Conservation of morphology (31%) was favored by initial RB (P = 0.002) and LA morphology (P = 0.01). Initiation of VF with LA vs RA was more likely in African-Americans (P = 0.016) and increasing age (P = 0.032). Ischemic cardiomyopathy favored VF initiation with RB 6.7-fold (P = 0.025), possibly linking LV myocardial scar to initial VF wavebreak location. Male gender and ischemic cardiomyopathy prolonged time-to-loss of predominant vector by 119% (P = 0.002) and 71% (P = 0.017), respectively, suggesting more preserved anatomic/functional reentry.

CONCLUSION

The predominant QRS vectors during early Wiggers stage 1 VF are not random and suggest an initial wavebreak more commonly in the right ventricle, followed by a transitional shift to the septum/left ventricle. Ethnicity, male gender, age, and co-morbidities result in directional preservation of initiating VF vectors possibly due to myocardial mass/fibrosis. Findings may allow new treatment/ablation approaches.

Effects of high-frequency biphasic shocks on ventricular vulnerability and defibrillation outcomes through synchronized virtual electrode responses

Electrical defibrillation is a well-established treatment for cardiac dysrhythmias. Studies have suggested that shock-induced spatial sawtooth patterns and virtual electrodes are responsible for defibrillation efficacy. We hypothesize that high-frequency shocks enhance defibrillation efficacy by generating temporal sawtooth patterns and using rapid virtual electrodes synchronized with shock frequency. High-speed optical mapping was performed on isolated rat hearts at 2000 frames/s. Two defibrillation electrodes were placed on opposite sides of the ventricles. An S1-S2 pacing protocol was used to induce ventricular tachyarrhythmia (VTA). High-frequency shocks of equal energy but varying frequencies of 125–1000 Hz were used to evaluate VTA vulnerability and defibrillation success rate. The 1000-Hz shock had the highest VTA induction rate in the shorter S1-S2 intervals (50 and 100 ms) and the highest VTA defibrillation rate (70%) among all frequencies. Temporal sawtooth patterns and synchronous shock-induced virtual electrode responses could be observed with frequencies of up to 1000 Hz. The improved defibrillation outcome with high-frequency shocks suggests a lower energy requirement than that of low-frequency shocks for successful ventricular defibrillation.

Effect of anisotropy on ventricular vulnerability to unidirectional block and reentry by single premature stimulation during normal sinus rhythm in rat heart

Single high-intensity premature stimuli when applied to the ventricles during ventricular drive of an ectopic site, as in Winfree's "pinwheel experiment," usually induce reentry arrhythmias in the normal heart, while single low-intensity stimuli barely do. Yet ventricular arrhythmia vulnerability during normal sinus rhythm remains largely unexplored. With a view to define the role of anisotropy on ventricular vulnerability to unidirectional conduction block and reentry, we revisited the pinwheel experiment with reduced constraints in the in situ rat heart. New features included single premature stimulation during normal sinus rhythm, stimulation and unipolar potential mapping from the same high-resolution epicardial electrode array, and progressive increase in stimulation strength and prematurity from diastolic threshold until arrhythmia induction. Measurements were performed with 1-ms cathodal stimuli at multiple test sites (n = 26) in seven rats. Stimulus-induced virtual electrode polarization during sinus beat recovery phase influenced premature ventricular responses. Specifically, gradual increase in stimulus strength and prematurity progressively induced make, break, and graded-response stimulation mechanisms. Hence unidirectional conduction block occurred as follows: 1) along fiber direction, on right and left ventricular free walls (n = 23), initiating figure-eight reentry (n = 17) and tachycardia (n = 12), and 2) across fiber direction, on lower interventricular septum (n = 3), initiating spiral wave reentry (n = 2) and tachycardia (n = 1). Critical time window (55.1 ± 4.7 ms, 68.2 ± 6.0 ms) and stimulus strength lower limit (4.9 ± 0.6 mA) defined vulnerability to reentry. A novel finding of this study was that ventricular tachycardia evolves and is maintained by episodes of scroll-like wave and focal activation couplets. We also found that single low-intensity premature stimuli can induce repetitive ventricular response (n = 13) characterized by focal activations.NEW & NOTEWORTHY We performed ventricular cathodal point stimulation during sinus rhythm by progressively increasing stimulus strength and prematurity. Virtual electrode polarization and recovery gradient progressively induced make, break, and graded-response stimulation mechanisms. Unidirectional conduction block occurred along or across fiber direction, initiating figure-eight or spiral wave reentry, respectively, and tachycardia sustained by scroll wave and focal activations.

Dynamics and Molecular Mechanisms of Ventricular Fibrillation in Structurally Normal Hearts

Ventricular fibrillation (VF) is the most severe cardiac rhythm disturbance and one of the most important immediate causes of sudden cardiac death. In the structurally normal heart, a small number of stable reentrant sources, perhaps 1 or 2, underlie the mechanism of VF, and the stabilization of the sources, their frequency, and the complexity of the turbulent waves they generate depend on the expression, spatial distribution, and intermolecular interactions of the 2 most important ion channels that control cardiac excitability: the inward rectifier potassium channel, Kir2.1, and the alpha subunit of the main cardiac sodium channel, NaV1.5.

Simulation Methods and Validation Criteria for Modeling Cardiac Ventricular Electrophysiology

We describe a sequence of methods to produce a partial differential equation model of the electrical activation of the ventricles. In our framework, we incorporate the anatomy and cardiac microstructure obtained from magnetic resonance imaging and diffusion tensor imaging of a New Zealand White rabbit, the Purkinje structure and the Purkinje-muscle junctions, and an electrophysiologically accurate model of the ventricular myocytes and tissue, which includes transmural and apex-to-base gradients of action potential characteristics. We solve the electrophysiology governing equations using the finite element method and compute both a 6-lead precordial electrocardiogram (ECG) and the activation wavefronts over time. We are particularly concerned with the validation of the various methods used in our model and, in this regard, propose a series of validation criteria that we consider essential. These include producing a physiologically accurate ECG, a correct ventricular activation sequence, and the inducibility of ventricular fibrillation. Among other components, we conclude that a Purkinje geometry with a high density of Purkinje muscle junctions covering the right and left ventricular endocardial surfaces as well as transmural and apex-to-base gradients in action potential characteristics are necessary to produce ECGs and time activation plots that agree with physiological observations.

Nonlinear and Stochastic Dynamics in the Heart

In a normal human life span, the heart beats about 2 to 3 billion times. Under diseased conditions, a heart may lose its normal rhythm and degenerate suddenly into much faster and irregular rhythms, called arrhythmias, which may lead to sudden death. The transition from a normal rhythm to an arrhythmia is a transition from regular electrical wave conduction to irregular or turbulent wave conduction in the heart, and thus this medical problem is also a problem of physics and mathematics. In the last century, clinical, experimental, and theoretical studies have shown that dynamical theories play fundamental roles in understanding the mechanisms of the genesis of the normal heart rhythm as well as lethal arrhythmias. In this article, we summarize in detail the nonlinear and stochastic dynamics occurring in the heart and their links to normal cardiac functions and arrhythmias, providing a holistic view through integrating dynamics from the molecular (microscopic) scale, to the organelle (mesoscopic) scale, to the cellular, tissue, and organ (macroscopic) scales. We discuss what existing problems and challenges are waiting to be solved and how multi-scale mathematical modeling and nonlinear dynamics may be helpful for solving these problems.

Mechanisms of ventricular arrhythmias: a dynamical systems-based perspective

Defining the cellular electrophysiological mechanisms for ventricular tachyarrhythmias is difficult, given the wide array of potential mechanisms, ranging from abnormal automaticity to various types of reentry and kk activity. The degree of difficulty is increased further by the fact that any particular mechanism may be influenced by the evolving ionic and anatomic environments associated with many forms of heart disease. Consequently, static measures of a single electrophysiological characteristic are unlikely to be useful in establishing mechanisms. Rather, the dynamics of the electrophysiological triggers and substrates that predispose to arrhythmia development need to be considered. Moreover, the dynamics need to be considered in the context of a system, one that displays certain predictable behaviors, but also one that may contain seemingly stochastic elements. It also is essential to recognize that even the predictable behaviors of this complex nonlinear system are subject to small changes in the state of the system at any given time. Here we briefly review some of the short-, medium-, and long-term alterations of the electrophysiological substrate that accompany myocardial disease and their potential impact on the initiation and maintenance of ventricular arrhythmias. We also provide examples of cases in which small changes in the electrophysiological substrate can result in rather large differences in arrhythmia outcome. These results suggest that an interrogation of cardiac electrical dynamics is required to provide a meaningful assessment of the immediate risk for arrhythmia development and for evaluating the effects of putative antiarrhythmic interventions.

Ventricular fibrillation: dynamics and ion channel determinants

Ventricular fibrillation (VF) is the leading cause of sudden cardiac death. This brief review addresses issues relevant to the dynamics of the rotors responsible for functional reentry and VF. It also makes an attempt to summarize present-day knowledge of the manner in which the dynamic interplay between inward and outward transmembrane currents and the heterogeneous cardiac structure establish a substrate for the initiation and maintenance of rotors and VF. The fragmentary nature of our current understanding of ionic VF mechanisms does not even allow an approach toward a "Theory of VF". Yet some hope is provided by recently obtained insight into the roles played in VF by some of the sarcolemmal ion channels that control the excitation-recovery process. For example, strong evidence supports the idea that the interplay between the rapid-inward sodium current and the inward-rectifier potassium current controls rotor formation, as well as rotor stability and frequency. Solid evidence also exists for an involvement of L-type calcium current in the control of rotor frequency and in determining VF-to-ventricular tachycardia conversion. Less clear, however, is whether or not time dependent outward currents through voltage-gated potassium channels affect the fibrillatory process. Hopefully, taking advantage of currently available approaches of structural, molecular and cellular biology, together with computational and imaging techniques, will afford us the opportunity to further advance knowledge on VF mechanisms.

Intracellular calcium and the mechanism of anodal supernormal excitability in langendorff perfused rabbit ventricles

BACKGROUND

Anodal stimulation hyperpolarizes the cell membrane and increases the intracellular Ca(2+) (Ca(i)) transient. This study tested the hypothesis that the maximum slope of the Ca(i) decline (-(dCa(i)/dt)(max)) corresponds to the timing of anodal dip on the strength-interval curve and the initiation of repetitive responses and ventricular fibrillation (VF) after a premature stimulus (S(2)).

METHODS AND RESULTS

We simultaneously mapped the membrane potential (V(m)) and Ca(i) in 23 rabbit ventricles. A dip in the anodal strength-interval curve was observed. During the anodal dip, ventricles were captured by anodal break excitation directly under the S(2) electrode. The Ca(i) following anodal stimuli is larger than that following cathodal stimuli. The S(1)-S(2) intervals of the anodal dip (203±10 ms) coincided with the -(dCa(i)/dt)(max) (199±10 ms, P=NS). BAPTA-AM (n=3), inhibition of the electrogenic Na(+)-Ca(2+) exchanger current (I(NCX)) by low extracellular Na(+) (n=3), and combined ryanodine and thapsigargin infusion (n=2) eliminated the anodal supernormality. Strong S(2) during the relative refractory period (n=5) induced 29 repetitive responses and 10 VF episodes. The interval between S(2) and the first non-driven beat was coincidental with the time of -(dCa(i)/dt)(max).

CONCLUSIONS

Larger Ca(i) transient and I(NCX) activation induced by anodal stimulation produces anodal supernormality. The time of maximum I(NCX) activation is coincidental to the induction of non-driven beats from the Ca(i) sinkhole after a strong premature stimulation.

Chaos in the genesis and maintenance of cardiac arrhythmias

Dynamical chaos, an irregular behavior of deterministic systems, has been widely shown in nature. It also has been demonstrated in cardiac myocytes in many studies, including rapid pacing-induced irregular beat-to-beat action potential alterations and slow pacing-induced irregular early afterdepolarizations, etc. Here we review the roles of chaos in the genesis of cardiac arrhythmias, the transition to ventricular fibrillation, and the spontaneous termination of fibrillation, based on evidence from computer simulation of mathematical models and experiments of animal models.

Models of stretch-activated ventricular arrhythmias

One of the most important components of mechanoelectric coupling is stretch-activated channels, sarcolemmal channels that open upon mechanical stimuli. Uncovering the mechanisms by which stretch-activated channels contribute to ventricular arrhythmogenesis under a variety of pathologic conditions is hampered by the lack of experimental methodologies that can record the 3-dimensional electromechanical activity simultaneously at high spatiotemporal resolution. Computer modeling provides such an opportunity. The goal of this review is to illustrate the utility of sophisticated, physiologically realistic, whole heart computer simulations in determining the role of mechanoelectric coupling in ventricular arrhythmogenesis. We first present the various ways by which stretch-activated channels have been modeled and demonstrate how these channels affect cardiac electrophysiologic properties. Next, we use an electrophysiologic model of the rabbit ventricles to understand how so-called commotio cordis, the mechanical impact to the precordial region of the heart, can initiate ventricular tachycardia via the recruitment of stretch-activated channels. Using the same model, we also provide mechanistic insight into the termination of arrhythmias by precordial thump under normal and globally ischemic conditions. Lastly, we use a novel anatomically realistic dynamic 3-dimensional coupled electromechanical model of the rabbit ventricles to gain insight into the role of electromechanical dysfunction in arrhythmogenesis during acute regional ischemia.

Development of an anatomically detailed MRI-derived rabbit ventricular model and assessment of its impact on simulations of electrophysiological function

Recent advances in magnetic resonance (MR) imaging technology have unveiled a wealth of information regarding cardiac histoanatomical complexity. However, methods to faithfully translate this level of fine-scale structural detail into computational whole ventricular models are still in their infancy, and, thus, the relevance of this additional complexity for simulations of cardiac function has yet to be elucidated. Here, we describe the development of a highly detailed finite-element computational model (resolution: approximately 125 microm) of rabbit ventricles constructed from high-resolution MR data (raw data resolution: 43 x 43 x 36 microm), including the processes of segmentation (using a combination of level-set approaches), identification of relevant anatomical features, mesh generation, and myocyte orientation representation (using a rule-based approach). Full access is provided to the completed model and MR data. Simulation results were compared with those from a simplified model built from the same images but excluding finer anatomical features (vessels/endocardial structures). Initial simulations showed that the presence of trabeculations can provide shortcut paths for excitation, causing regional differences in activation after pacing between models. Endocardial structures gave rise to small-scale virtual electrodes upon the application of external field stimulation, which appeared to protect parts of the endocardium in the complex model from strong polarizations, whereas intramural virtual electrodes caused by blood vessels and extracellular cleft spaces appeared to reduce polarization of the epicardium. Postshock, these differences resulted in the genesis of new excitation wavefronts that were not observed in more simplified models. Furthermore, global differences in the stimulus recovery rates of apex/base regions were observed, causing differences in the ensuing arrhythmogenic episodes. In conclusion, structurally simplified models are well suited for a large range of cardiac modeling applications. However, important differences are seen when behavior at microscales is relevant, particularly when examining the effects of external electrical stimulation on tissue electrophysiology and arrhythmia induction. This highlights the utility of histoanatomically detailed models for investigations of cardiac function, in particular for future patient-specific modeling.

The relative role of refractoriness and source-sink relationship in reentry generation during simulated acute ischemia

During acute myocardial ischemia, reentrant episodes may lead to ventricular fibrillation (VF), giving rise to potentially mortal arrhythmias. VF has been traditionally related to dispersion of refractoriness and more recently to the source-sink relationship. Our goal is to theoretically investigate the relative role of dispersion of refractoriness and source-sink mismatch in vulnerability to reentry in the specific situation of regional myocardial acute ischemia. The electrical activity of a regionally ischemic tissue was simulated using a modified version of the Luo-Rudy dynamic model. Ischemic conditions were varied to simulate the time-course of acute ischemia. Our results showed that dispersion of refractoriness increased with the severity of ischemia. However, no correlation between dispersion of refractoriness and the width of the vulnerable window was found. Additionally, in approximately 50% of the reentries, unidirectional block (UDB) took place in cells completely recovered from refractoriness. We examined patterns of activation after premature stimulation and they were intimately related to the source-sink relationship, quantified by the safety factor (SF). Moreover, the isoline where the SF dropped below unity matched the area where propagation failed. It was concluded that the mismatch of the source-sink relationship, rather than solely refractoriness, was the ultimate cause of the UDB leading to reentry. The SF represents a very powerful tool to study the mechanisms responsible for reentry.

Effects of hypocalcemia on electrical restitution and ventricular fibrillation

We have shown previously that verapamil reduces the slope of the action potential duration (APD) restitution relation, suppresses APD alternans and converts ventricular fibrillation (VF) into a periodic rhythm. To determine whether these effects result primarily from reduction of the APD restitution slope, as opposed to alteration of calcium dynamics unrelated to restitution, we tested the effects of hypocalcemia ([CaCl2]=31-125 microM) in canine ventricle. At normal [CaCl2] (2.0 mM), the slope of the APD restitution relation was >1, APD alternans occurred during rapid pacing and VF was inducible. During hypocalcemia the slope of the restitution relation remained >1 and the magnitude of APD alternans was unchanged. VF still was inducible and the mean cycle length and the variance of the FFT spectra during VF were not altered significantly. These results suggest that reduction of APD restitution slope, rather than blockade of ICa per se, is responsible for the antifibrillatory effects of verapamil in this model of pacing-induced VF, lending further support to the idea that APD restitution kinetics is a key determinant of VF.

Negative filament tension at high excitability in a model of cardiac tissue

One of the fundamental mechanisms for the onset of turbulence in 3D excitable media is negative filament tension. Thus far, negative tension has always been obtained in media under low excitability. For this reason, its application to normal (nonischemic) cardiac tissue has been questionable, as such cardiac turbulence typically occurs at high excitability. Here, we report expansion of scroll rings (low curvature negative filament tension) in a medium with high excitability by numerical integration of the Luo-Rudy model of cardiac tissue. We discuss the relation between negative tension and the meandering of 2D spiral waves and the possible applications to cardiac modeling.

Intramural foci during long duration fibrillation in the pig ventricle

For more than 50 years, it has been assumed that ventricular fibrillation (VF) is maintained solely by reentry in the working myocardium. This hypothesis has never been tested by recording VF with electrodes spaced sufficiently close to map activation sequences in 3D. We recorded the first 10 minutes of electrically induced VF from the anterior left ventricular (LV) free wall near the insertion of the anterior papillary muscle in 6 pigs. A 3D transmural unipolar electrode array consisting of a 9x9 array of needles with 2-mm spacing and 6 electrodes 2 mm apart on each needle was used for recordings. Automatic analyses were performed to recognize 3D reentry and foci. Our results showed that intramural reentry is present early but not late during VF in the mapped region. The incidence of reentry in working myocardium decreases almost to 0 after 3 minutes of VF. In contrast, intramural foci are present during early VF and, as VF continues, increase in incidence, so that by 10 minutes of VF, 27% of wavefronts arise from intramural foci. These results suggest that, particularly after the first 3 minutes of VF, mechanisms other than local reentry in the working myocardium maintain VF in the anterior LV free wall near the root of the anterior papillary muscle. Intramural foci may play an important role in later VF maintenance. It remains to be determined if these foci arise from Purkinje fibers attributable to abnormal automaticity, afterdepolarizations, or reentry.

Cardiac electrical dynamics: maximizing dynamical heterogeneity

The relationships between key features of the cardiac electrical activity, such as electrical restitution, discordant alternans, wavebreak, and reentry, and the onset of ventricular tachyarrhythmias have been characterized extensively under the condition of constant rapid pacing. However, it is unlikely that this scenario applies directly to the clinical situation, where the induction of ventricular tachycardia (VT) typically is associated with the interruption of normal cardiac rhythm by several premature beats. To address this issue, we have developed a general theory to explain why specific patterns of premature stimuli increase dynamic heterogeneity of repolarization and precipitate conduction block. The theory predicts that conduction block is caused by (1) creation of a spatial gradient in diastolic interval (DI) by waves traveling at slightly different velocities (ie, conduction velocity dispersion) and (2) amplification of the spatial gradient in DI over subsequent action potentials, secondary to a strong dependence of action potential duration on the preceding DI (ie, a steep action potential duration restitution function). Tests of this theory have been conducted in computer models of homogeneous tissue, where increased spatial dispersion of repolarization during premature stimulation can be attributed solely to the development of dynamical heterogeneity, and in a canine model exhibiting spontaneously occurring VT and sudden death. Our results thus far indicate that the probability of inducing ventricular fibrillation (VF) in the animal model is highest for those sequences predicted to cause conduction block in the computer model. An understanding of the mechanisms underlying these observations will help to identify key electrical phenomena in the onset of VT and fibrillation. Drug and electrical therapies can then be improved by targeting these specific phenomena.

Vulnerability to re-entry in simulated two-dimensional cardiac tissue: effects of electrical restitution and stimulation sequence

Ventricular fibrillation is a lethal arrhythmia characterized by multiple wavelets usually starting from a single or figure-of-eight re-entrant circuit. Understanding the factors regulating vulnerability to the re-entry is essential for developing effective therapeutic strategies to prevent ventricular fibrillation. In this study, we investigated how pre-existing tissue heterogeneities and electrical restitution properties affect the initiation of re-entry by premature extrastimuli in two-dimensional cardiac tissue models. We studied two pacing protocols for inducing re-entry following the "sinus" rhythm (S1) beat: (1) a single premature (S2) extrastimulus in heterogeneous tissue; (2) two premature extrastimuli (S2 and S3) in homogeneous tissue. In the first case, the vulnerable window of re-entry is determined by the spatial dimension and extent of the heterogeneity, and is also affected by electrical restitution properties and the location of the premature stimulus. The vulnerable window first increases as the action potential duration (APD) difference between the inside and outside of the heterogeneous region increases, but then decreases as this difference increases further. Steeper APD restitution reduces the vulnerable window of re-entry. In the second case, electrical restitution plays an essential role. When APD restitution is flat, no re-entry can be induced. When APD restitution is steep, re-entry can be induced by an S3 over a range of S1S2 intervals, which is also affected by conduction velocity restitution. When APD restitution is even steeper, the vulnerable window is reduced due to collision of the spiral tips.

The effects of phase resetting technique in cardiac models

When a brief current pulse is incident on excitable cells in cardiac and other nervous tissue, a change in phase of the cell's response is usually observed. In cardiac tissue, the cells are bared to external stimulation of generally positive currents, which depolarize the cells. In this paper an overview of the application of the phase resetting technique (PRT) in several cardiac models is presented. We discuss the effects of external stimuli in several cardiac cell models and we provide the phase transition curves (PTCs) resulted from the application of PRT with the Zhang et al. sinoatrial node model.

Covariant stringlike dynamics of scroll wave filaments in anisotropic cardiac tissue

It has been hypothesized that stationary scroll wave filaments in cardiac tissue describe a geodesic in a curved space whose metric is the inverse diffusion tensor. Several numerical studies support this hypothesis, but no analytical proof has been provided yet for general anisotropy. In this Letter, we derive dynamic equations for the filament in the case of general anisotropy. These equations are covariant under general spatial coordinate transformations and describe the motion of a stringlike object in a curved space whose metric tensor is the inverse diffusion tensor. Therefore the behavior of scroll wave filaments in excitable media with anisotropy is similar to the one of cosmic strings in a curved universe. Our dynamic equations are valid for thin filaments and for general anisotropy. We show that stationary filaments obey the geodesic equation.

The pinwheel experiment revisited: effects of cellular electrophysiological properties on vulnerability to cardiac reentry

In normal heart, ventricular fibrillation can be induced by a single properly timed strong electrical or mechanical stimulus. A mechanism first proposed by Winfree and coined the “pinwheel experiment” emphasizes the timing and strength of the stimulus in inducing figure-of-eight reentry. However, the effects of cellular electrophysiological properties on vulnerability to reentry in the pinwheel scenario have not been investigated. In this study, we extend Winfree's pinwheel experiment to show how the vulnerability to reentry is affected by the graded action potential responses induced by a strong premature stimulus, action potential duration (APD), and APD restitution in simulated monodomain homogeneous two-dimensional tissue. We find that a larger graded response, longer APD, or steeper APD restitution slope reduces the vulnerable window of reentry. Strong graded responses and long APD promote tip-tip interactions at long coupling intervals, causing the two initiated spiral wave tips to annihilate. Steep APD restitution promotes wave front-wave back interaction, causing conduction block in the central common pathway of figure-of-eight reentry. We derive an analytical treatment that shows good agreement with numerical simulation results.

Calcium transient dynamics and the mechanisms of ventricular vulnerability to single premature electrical stimulation in Langendorff-perfused rabbit ventricles

BACKGROUND

Single strong premature electrical stimulation (S(2)) may induce figure-eight reentry. We hypothesize that Ca current-mediated slow-response action potentials (APs) play a key role in the propagation in the central common pathway (CCP) of the reentry.

METHODS

We simultaneously mapped optical membrane potential (V(m)) and intracellular Ca (Ca(i)) transients in isolated Langendorff-perfused rabbit ventricles. Baseline pacing (S(1)) and a cathodal S(2) (40-80 mA) were given at different epicardial sites with a coupling interval of 135 +/- 20 ms.

RESULTS

In all 6 hearts, S(2) induced graded responses around the S(2) site. These graded responses propagated locally toward the S(1) site and initiated fast APs from recovered tissues. The wavefront then circled around the refractory tissue near the site of S(2). At the side of S(2) opposite to the S(1), the graded responses prolonged AP duration while the Ca(i) continued to decline, resulting in a Ca(i) sinkhole (an area of low Ca(i)). The Ca(i) in the sinkhole then spontaneously increased, followed by a slow V(m) depolarization with a take-off potential of -40 +/- 3.9 mV, which was confirmed with microelectrode recordings in 3 hearts. These slow-response APs then propagated through CCP to complete a figure-eight reentry.

CONCLUSION

We conclude that a strong premature stimulus can induce a Ca(i) sinkhole at the entrance of the CCP. Spontaneous Ca(i) elevation in the Ca(i) sinkhole precedes the V(m) depolarization, leading to Ca current-mediated slow propagation in the CCP. The slow propagation allows more time for tissues at the other side of CCP to recover and be excited to complete figure-eight reentry.

Vulnerability to reentry in a regionally ischemic tissue: a simulation study

Sudden cardiac death is mainly provoked by arrhythmogenic processes. During myocardial ischemia many malignant arrhythmias, such as reentry, take place and can degenerate into ventricular fibrillation. It is thus of great interest to unravel the intricate mechanisms underlying the initiation and maintenance of a reentry. In this computational study, we analyze the probability of reentry during different stages of the acute phase of ischemia. We also aimed at the understanding of the role of its main components: hypoxia, hyperkalemia, and acidosis analyzing the intricate ionic mechanisms responsible for reentry generation. We simulated the electrical activity of a ventricular tissue affected by regional ischemia based on a modified version of the Luo-Rudy model (LRd00). The ischemic conditions were varied to simulate different stages of this pathology. After premature stimulation, we evaluated the vulnerability to reentry. We obtained an unimodal behavior for the vulnerable window as ischemia progressed, peaking at the eighth minute after the onset of ischemia where the vulnerable window yielded 58 ms. Under more severe conditions the vulnerable window decreased and became zero for minute 8.75. The present work provides insight into the mechanisms of reentry generation during ischemia, highlighting the role of acidosis and hypoxia when hyperkalemia is present.

Cathodal stimulation in the recovery phase of a propagating planar wave in the rabbit heart reveals four stimulation mechanisms

The stimulation of cardiac tissue in the recovery phase has significant importance in relation to reentry induction. In the theoretical experiment proposed by Winfree, termed the 'pinwheel' experiment, a point stimulus (S2) is applied in the wake of a freely propagating planar wave (S1). Reentry induced from this S1-S2 pinwheel protocol has been observed experimentally in heart preparations. However, in these experiments, which focused on activation outcomes, only mapping of extracellular voltages has been conducted. The lack of transmembrane potential (Vm) distribution data makes it impossible to analyse the underlying stimulation mechanisms which precede the reentry induction. In this work we sought to elucidate the stimulation mechanisms throughout the heart cycle using the pinwheel protocol. We examined the cardiac tissue responses during and immediately after cathodal stimulation in the refractory wake of a propagating planar wave. The voltage-sensitive dye di-4-ANEPPS was utilized to measure Vm directly from quasi two-dimensional preparations of cryoablated Langendorff-perfused rabbit hearts. Four stimulation mechanisms were observed that depended on the Vm magnitude during S2 cathodal stimulation. Make stimulation always occurred during diastolic stimulation. When stimulation was at the beginning of the relative refractory period (RRP), transitional make-break stimulation was detected. During the RRP the excitation was due to the break mechanism. While approaching the effective refractory period (ERP), the tissue response is characterized by a damped wave mediated response. These four stimulation mechanisms were observed in all hearts whether the S1 planar wave propagation was parallel or perpendicular to the fibre direction. This study is the first examination of Vm and the stimulation mechanisms throughout the cardiac cycle using the pinwheel protocol, and the results have implications in the development and improvement of pacing protocols for artificial cardiostimulators.

Theory of action potential wave block at-a-distance in the heart

Propagation failure of an action potential wave at a finite distance from its source (so-called type-II block) may cause spiral wave formation or wave breakup in the heart, phenomena that are believed to underlie lethal and nonlethal heart rhythm disorders. In this study, we develop a sufficient condition for this type of block in a homogeneous, spatially one-dimensional system. Using a topological argument, we find that type-II block of a wave will always occur when launched within a finite range of times if the velocity of the trailing edge of the preceding wave, as measured at the stimulus site, is smaller than the velocity of a wave launched with the minimum diastolic interval (DI) for which propagation is possible. This "blocking condition" is robust, remaining valid even when memory and waveback electrotonic effects are included. The condition suggests that type-II block is greatly facilitated when waves are initiated at irregular intervals in time such that (1) the velocities of consecutive waves are as different as possible and (2) the DIs preceding each wave fall on the steeply sloped portion of the action potential duration restitution curve as often as possible. The set of timing intervals between stimuli that are predicted by the blocking condition to produce block are found to be consistent with these guidelines, and also to agree well with a coupled-maps computer simulation model, for the case of waves launched by four rapidly and irregularly timed stimuli.

Mechanism of origin of conduction disturbances in aging human atrial bundles: experimental and model study

BACKGROUND

Aging is associated with a significant increase in atrial tachyarrhythmias, especially atrial fibrillation. A macroscopic repolarization gradient created artificially by a stimulus at one site before a premature stimulus from a second site is widely considered to be part of the experimental protocol necessary for the initiation of such arrhythmias in the laboratory. How such gradients occur naturally in aging atrial tissue is unknown.

OBJECTIVE

The objective of this study was to determine if the pattern of cellular connectivity in aging human atrial bundles produces a mechanism for variable early premature responses.

METHODS

Extracellular and intracellular potentials were recorded after control and premature stimuli at a single site in aging human atrial bundles. We also measured cellular geometry, the distribution of connexins, and the distribution of collagenous septa. A model of the atrial bundles was constructed based on the morphological results. Action potential propagation and the sodium current were analyzed after premature stimuli in the model.

RESULTS

Similar extracellular potential waveform responses occurred after early premature stimuli in the aging bundles and in the model. Variable premature conduction patterns were accounted for by the single model of aging atrial structure. A major feature of the model results was that the conduction events and the magnitude of the sodium current at multiple sites were very sensitive to small changes in the location and the timing of premature stimuli.

CONCLUSION

In aging human atrial bundles stimulated from only a single site, premature stimuli induce variable arrhythmogenic conduction responses. The generation of these responses is greatly enhanced by remodeling of cellular connectivity during aging. The results provide insight into sodium current structural interactions as a general mechanism of arrhythmogenic atrial responses to premature stimuli.

Simulated interactions between a Class III antiarrhythmic drug and a figure 8 reentry

Ventricular Fibrillation is responsible for a majority of sudden cardiac death, but little is known about how ventricular tachycardia (VT) degenerates into ventricular fibrillation. Several clinical studies focused only on preventing VT with a class III antiarrhythmic drug resulted in many deaths. Our simulations investigate the interactions between an antiarrhythmic drug likely to suppress a VT and a Figure 8 reentry. A parameter AAR is introduced to increase the action potential duration and therefore simulate various Class III drugs. Simulations are ran under several conditions (phases of the reentry, values of AAR, durations). They show that a VT can be suppressed whatever the phase of the reentry but it strongly depends on the duration of the effect. It confirms that a drug which can suppress a reentry can also worsen it. It also shows a great variety of activation patterns and thus the complexity of antiarrhythmic drugs effects. Simulations also demonstrate that suppressing VT is an increasing function of AAR.

Ultrasonically Actuated Silicon Microprobes for Cardiac Signal Recording

In this paper, we report on ultrasonically actuated silicon thin microprobes that successfully penetrated canine cardiac tissue in vitro, and recorded the electrophysiological signals from multiple sites simultaneously within the heart wall. The penetration force--maximum force encountered by the probe during penetration--is found to reduce with increasing ultrasonic driving voltage, on both excised canine right ventricular muscle and chicken breast muscle. The rate of force decrease varies with tissue type and microprobe dimension. With ultrasonic actuation, the silicon microprobes are inserted into isolated perfused canine heart without breakage or significant buckling, under 10Vpp actuating voltage. Recordings were obtained from isolated perfused canine heart during pacing, following the induction of ventricular tachycardia, and during the transition from ventricular tachycardia to ventricular fibrillation. Local conduction velocity of 0.60 +/- 0.03 m/s was observed from the multichannel recordings from the canine right ventricular wall under epicardial pacing. The application of the ultrasonic microprobes in cardiac electrophysiology study can provide information for reconstruction of electrical wave propagation within the heart, which is important to understanding the mechanisms of cardiac arrhythmias.

Vulnerable window for conduction block in a one-dimensional cable of cardiac cells, 2: multiple extrasystoles

Unidirectional conduction block of premature extrasystoles can lead to initiation of cardiac reentry, causing lethal arrhythmias including ventricular fibrillation. Multiple extrasystoles are often more effective at inducing unidirectional conduction block and reentry than a single extrasystole. Since the substrate for conduction block is spatial dispersion of refractoriness, in this study we investigate how the first extrasystole modulates this dispersion to influence the "vulnerable window" for conduction block by subsequent extrasystoles, particularly in relation to action potential duration restitution and conduction velocity restitution properties. Using a kinematic model to represent wavefront-waveback interactions and simulations with the Luo-Rudy model in a one-dimensional cable of cardiac cells, we show that in homogeneous tissue, a premature extrasystole can create a large dispersion of refractoriness leading to conduction block of a subsequent extrasystole. In heterogeneous tissue, however, a premature extrasystole can either reduce or enhance the dispersion of refractoriness depending on its propagation direction with respect to the previous beat. With multiple extrasystoles at random coupling intervals, vulnerability to conduction block is proportional to their number. In general, steep action potential duration restitution and broad conduction velocity restitution promote dispersion of refractoriness in response to multiple extrasystoles, and thus enhance vulnerability to conduction block. These restitution properties also promote spatially discordant alternans, a setting which is particularly prone to conduction block. The equivalent dispersion of refractoriness created dynamically in homogeneous tissue by spatially discordant alternans is more likely to cause conduction block than a comparable degree of preexisting dispersion in heterogeneous tissue.

Vulnerable window for conduction block in a one-dimensional cable of cardiac cells, 1: single extrasystoles

Spatial dispersion of refractoriness, which is amplified by genetic diseases, drugs, and electrical and structural remodeling during heart disease, is recognized as a major factor increasing the risk of lethal arrhythmias and sudden cardiac death. Dispersion forms the substrate for unidirectional conduction block, which is required for the initiation of reentry by extrasystoles or rapid pacing. In this study, we examine theoretically and numerically how preexisting gradients in refractoriness control the vulnerable window for unidirectional conduction block by a single premature extrasystole. Using a kinematic model to represent wavefront-waveback interactions, we first analytically derived the relationship (under simplified conditions) between the vulnerable window and various electrophysiological parameters such as action potential duration gradients, refractoriness barriers, conduction velocity restitution, etc. We then compared these findings to numerical simulations using the kinematic model or the Luo-Rudy action potential model in a one-dimensional cable of cardiac cells. The results from all three methods agreed well. We show that a critical gradient in action potential duration for conduction block can be analytically derived, and once this critical gradient is exceeded, the vulnerable window increases proportionately with the refractory barrier and is modulated by conduction velocity restitution and gap junctional conductance. Moreover, the critical gradient for conduction block is higher for an extrasystole traveling in the opposite direction from the sinus beat than for one traveling in the same direction (e.g., an epicardial extrasystole versus an endocardial extrasystole).

Effects of n-3 polyunsaturated fatty acid on upper limit of vulnerability shocks

BACKGROUND

Ventricular fibrillation (VF) can be induced when a strong shock is delivered during the vulnerable period of a cardiac cycle. VF, however, cannot be induced if the shock strength is increased to the "upper limit of vulnerability" (ULV) level. Docosahexaenoic acid (DHA) has been shown to prevent the occurrence of VF after coronary occlusion. However, its effects on the ULV have not been verified. We tested the hypothesis that ULV shock strength is decreased after DHA administration.

METHODS

In 10 pigs, 10 S1s (square, 5-ms) were delivered from the RV apex electrode at 300 ms cycle length. Shocks (S2, biphasic) were delivered from the RV-SVC electrodes after the last S1. The ULV was determined using an up/down protocol. In group 1 (n = 5), after the control ULV was determined at the beginning of the study, a solution containing 1.0 gm of DHA was infused intravenously within 90 min. The ULV (DHA-ULV) was determined again after the end of infusion. In group 2 (n = 5), the vehicle for DHA was infused instead of DHA to confirm that the vehicle did not have an effect on the ULV.

RESULTS

DHA-ULV (412 +/- 58 V, 12 +/- 3 J) was significantly decreased (P < 0.04) compared to the control ULV (478 +/- 32 V, 16 +/- 3 J). The ULV before (483 +/- 28 V, 16 +/- 1 J) and after (463 +/- 28 V, 15 +/- 2 J) the vehicle infusion was not different (P = 0.4). There was no change in the systolic blood pressure as well as heart rate in both groups.

CONCLUSION

DHA significantly decreases the ULV (13% by voltage and 25% by energy), suggesting that DHA can help to prevent VF induced by a strong stimulus delivered during the vulnerable period.

Our search for the porcine mother rotor

A single stationary mother rotor, located in the fastest activating region and giving rise to activation fronts that propagate throughout the remainder of the myocardium, has been hypothesized to be responsible for the maintenance of ventricular fibrillation (VF). Others have reported a mother rotor in guinea pigs and rabbits. We wanted to see if a mother rotor exists in a larger heart, that is, pigs. Epicardial mapping studies have demonstrated that VF wavefronts in pigs tend to propagate from the posterior basal LV to the anterior LV and on to the anterior RV, raising the possibility of a mother rotor in the posterior LV. However, no sustained reentry consistent with a mother rotor was found on the posterior LV epicardium, even though an intramural mapping study showed that the fastest activating transmural layer was near the epicardium. Many wavefronts in the posterior LV entered the mapped region from the posterior boundary of the mapping array, adjacent to the posterior descending coronary artery, raising the possibility that a mother rotor is located in the right ventricle or septum. Since a previous study has shown that the RV activates more slowly than the LV during VF, the more likely site for a mother rotor was the septum. However, we then performed a study in which we recorded from the right side of the septum and found that reentry was uncommon there also and that the activation rate was slower than the posterobasal LV. Many of the VF wavefronts in the septum passed from the posterior septum toward the anterior septum. This fact coupled with the fact that many wavefronts passed from the posterior LV free wall toward the anterior LV free wall point to the region where the posterior free wall intersects with the septum, the region where the posterior papillary muscle is located, as the possible site of a mother rotor. Indeed, a recent abstract by others reports that, after propranolol, a stable reentrant circuit is present on the endocardium at the insertion of the posterior papillary muscle into the LV free wall in pigs.

Mechanoelectrical excitation by fluid jets in monolayers of cultured cardiac myocytes

Although the prevailing view of mechanoelectric feedback (MEF) in the heart is in terms of longitudinal cell stretch, other mechanical forces are considerable during the cardiac cycle, including intramyocardial pressure and shear stress. Their contribution to MEF is largely unknown. In this study, mechanical stimuli in the form of localized fluid jet pulses were applied to neonatal rat ventricular cells cultured as confluent monolayers. Such pulses result in pressure and shear stresses (but not longitudinal stretch) in the monolayer at the point of impingement. The goal was to determine whether these mechanical stimuli can trigger excitation, initiate a propagated wave, and induce reentry. Cells were stained with the voltage-sensitive dye RH237, and multi-site optical mapping was used to record the spread of electrical activity in isotropic and anisotropic monolayers. Pulses (10 ms) with velocities ranging from 0.3 to 1.8 m/s were applied from a 0.4-mm diameter nozzle located 1 mm above the cell monolayer. Fluid jet pulses resulted in circular wavefronts that propagated radially from the stimulus site. The likelihood for mechanical stimulation was quantified as an average stimulus success rate (ASSR). ASSR increased with jet amplitude and time waited between stimuli and decreased with the application of gadolinium and streptomycin, blockers of stretch-activated channels, but not with nifedipine, a blocker of the L-type Ca channel. Absence of cellular injury was confirmed by smooth propagation maps and propidium iodide stains. In rare instances, the mechanical pulse resulted in the induction of reentrant activity. We conclude that mechanical stimuli other than stretch can evoke action potentials, propagated activity, and reentrant arrhythmia in two-dimensional sheets of cardiac cells.

New approaches for identifying antiarrhythmic drug targets

Sudden cardiac death, secondary to ventricular fibrillation (VF), remains the leading cause of death in many developed countries. Substantial experimental and theoretical support exists for the idea that VF is caused by spiral wave re-entry. The initiation and subsequent break-up of spiral waves have been linked to electrical alternans, a phenomenon typically associated with a steeply sloped restitution relationship. Interventions that reduce the slope of the restitution relationship have been shown to prevent the induction of VF and to terminate existing VF in experimental models. These results suggest that electrical restitution may be a promising new target for antiarrhythmic therapies.

Measuring curvature and velocity vector fields for waves of cardiac excitation in 2-D media

Excitable media theory predicts the effect of electrical wavefront morphology on the dynamics of propagation in cardiac tissue. It specifies that a convex wavefront propagates slower and a concave wavefront propagates faster than a planar wavefront. Because of this, wavefront curvature is thought to be an important functional mechanism of cardiac arrhythmias. However, the curvature of wavefronts during an arrhythmia are generally unknown. We introduce a robust, automated method to measure the curvature vector field of discretely characterized, arbitrarily shaped, two-dimensional (2-D) wavefronts. The method relies on generating a smooth, continuous parameterization of the shape of a wave using cubic smoothing splines fitted to an isopotential at a specified level, which we choose to be -30 mV. Twice differentiating the parametric form provides local curvature vectors along the wavefront and waveback. Local conduction velocities are computed as the wave speed along lines normal to the parametric form. In this way, the curvature and velocity vector field for wavefronts and wavebacks can be measured. We applied the method to data sampled from a 2-D numerical model and several examples are provided to illustrate its usefulness for studying the dynamics of cardiac propagation in 2-D media.

Molecular mechanisms and global dynamics of fibrillation: an integrative approach to the underlying basis of vortex-like reentry

Art Winfree's scientific legacy has been particularly important to our laboratory whose major goal is to understand the mechanisms of ventricular fibrillation (VF). Here, we take an integrative approach to review recent studies on the manner in which nonlinear electrical waves organize to result in VF. We describe the contribution of specific potassium channel proteins and of the myocardial fiber structure to such organization. The discussion centers on data derived from a model of stable VF in the Langendorff-perfused guinea pig heart that demonstrates distinct patterns of organization in the left (LV) and right (RV) ventricles. Analysis of optical mapping data reveals that VF excitation frequencies are distributed throughout the ventricles in clearly demarcated domains. The highest frequency domains are found on the anterior wall of the LV at a location where sustained reentrant activity is present. The optical data suggest that a high frequency rotor that remains stationary in the LV is the mechanism that sustains VF in this model. Computer simulations predict that the inward rectifying potassium current (IK1) is an essential determinant of rotor stability and frequency, and patch-clamp results strongly suggest that the outward component of IK1 of cells in the LV is significantly larger than in the RV. Additional computer simulations and analytical procedures predict that the filaments of the reentrant activity (scroll waves) adopt a non-random configuration depending on fiber organization within the ventricular wall. Using the minimal principle we have concluded that filaments align with the trajectory of least resistance (i.e. the geodesic) between their endpoints. Overall, the data discussed have opened new and potentially exciting avenues of research on the possible role played by inward rectifier channels in the mechanism of VF, as well as the organization of its reentrant sources in three-dimensional cardiac muscle. Such an integrative approach may lead us toward an understanding of the molecular and structural basis of VF and hopefully to new preventative approaches.

Dynamics of virtual electrode-induced scroll-wave reentry in a 3D bidomain model

Functional reentry in the heart can be caused by a wave front of excitation rotating around its edge. Previous simulations on the basis of monodomain cable equations predicted the existence of self-sustained, vortex-like wave fronts (scroll waves) rotating around a filament in three dimensions. In our simulations, we used the more accurate bidomain model with modified Beeler-Reuter ionic kinetics to study the dynamics of scroll-wave filaments in a 16 x 8 x 1.5-mm slab of ventricular tissue with straight fibers. Wave fronts were identified as the areas with inward current. Their edges represented the filaments. Both transmural and intramural reentries with I- and U-shaped filaments, respectively, were obtained by the S1-S2 point stimulation protocol through the virtual electrode-induced phase singularity mechanism. The filaments meandered along elongated trajectories and tended to attach to the tissue boundaries exposed to air (no current flow) rather than to the bath (zero extracellular potential). They completely detached from electroporated (zero transmembrane potential) boundaries. In our simulations, the presence of the bath led to generation of only U-shaped filaments, which survived for the 1.5-mm-thick slab but not for the slabs of 0.5- or 3-mm thicknesses. Thus boundary conditions may be another determinant of the type and dynamics of reentry.

Apparent discontinuities in the phase-resetting response of cardiac pacemakers

Injection of a brief stimulus pulse resets the spontaneous periodic activity of a sinoatrial node cell: a stimulus delivered early in the cycle generally delays the time of occurrence of the next action potential, while the same stimulus delivered later causes an advance. We investigate resetting in two models, one with a slow upstroke velocity and the other with a fast upstroke velocity, representing central and peripheral nodal cells, respectively. We first formulate each of these models as a classic Hodgkin-Huxley type of model and then as a model representing a population of single channels. In the Hodgkin-Huxley-type model of the slow-upstroke cell the transition from delay to advance is steep but continuous. In the corresponding single-channel model, due to the channel noise then present, repeated resetting runs at a fixed stimulus timing within the transitional range of coupling intervals lead to responses that span a range of advances and delays. In contrast, in the fast-upstroke model the transition from advance to delay is very abrupt in both classes of model, as it is in experiments on some cardiac preparations ("all-or-none" depolarization). We reduce the fast-upstroke model from the original seven-dimensional system to a three-dimensional system. The abrupt transition occurs in this reduced model when a stimulus transports the state point to one side or the other of the stable manifold of the trajectory corresponding to the eigendirection associated with the smaller of two positive eigenvalues. This stable manifold is close to the slow manifold, and so canard trajectories are seen. Our results demonstrate that the resetting response is fundamentally continuous, but extremely delicate, and thus suggest one way in which one can account for experimental discontinuities in the resetting response of a nonlinear oscillator.