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Alain Karma

Publications and source records attributed to Alain Karma.

7 recordsLinked to original sources
PubMed2002

Eutectic colony formation: a phase-field study.

Eutectic two-phase cells, also known as eutectic colonies, are commonly observed during the solidification of ternary alloys when the composition is close to a binary eutectic valley. In analogy with the solidification cells formed in dilute binary alloys, colony formation is triggered by a morphological instability of a macroscopically planar eutectic solidification front due to the rejection by both solid phases of a ternary impurity that diffuses in the liquid. Here we develop a phase-field model of a binary eutectic with a dilute ternary impurity. We investigate by dynamical simulations both the initial linear regime of this instability, and the subsequent highly nonlinear evolution of the interface that leads to fully developed two-phase cells with a spacing much larger than the lamellar spacing. We find a good overall agreement with our recent linear stability analysis [M. Plapp and A. Karma, Phys. Rev. E 60, 6865 (1999)], which predicts a destabilization of the front by long-wavelength modes that may be stationary or oscillatory. A fine comparison, however, reveals that the assumption commonly attributed to Cahn that lamellae grow perpendicular to the envelope of the solidification front is weakly violated in the phase-field simulations. We show that, even though weak, this violation has an important quantitative effect on the stability properties of the eutectic front. We also investigate the dynamics of fully developed colonies and find that the large-scale envelope of the composite eutectic front does not converge to a steady state, but exhibits cell elimination and tip-splitting events up to the largest times simulated.

Journal Article↗
PubMed2002

Instability and spatiotemporal dynamics of alternans in paced cardiac tissue.

We derive an equation that governs the spatiotemporal dynamics of small amplitude alternans in paced cardiac tissue. We show that a pattern-forming linear instability leads to the spontaneous formation of stationary or traveling waves whose nodes divide the tissue into regions with opposite phase of oscillation of action potential duration. This instability is important because it creates dynamically a heterogeneous electrical substrate for the formation of conduction blocks and the induction of fibrillation if the tissue size exceeds a fraction of the pattern wavelength. We derive an analytical expression for this wavelength as a function of three basic length scales related to dispersion and intercellular electrical coupling.

Action Potentials↗
PubMed2002

Spatiotemporal control of cardiac alternans.

Electrical alternans are believed to be linked to the onset of life-threatening ventricular arrhythmias and sudden cardiac death. Recent studies have shown that alternans can be suppressed temporally by dynamic feedback control of the pacing interval. Here we investigate theoretically whether control can suppress alternans both temporally and spatially in homogeneous tissue paced at a single site. We first carry out ionic model simulations in a one-dimensional cable geometry which show that control is only effective up to a maximum cable length that decreases sharply away from the alternans bifurcation point. We then explain this finding by a linear stability analysis of an amplitude equation that describes the spatiotemporal evolution of alternans. This analysis reveals that control failure above a critical cable length is caused by the formation of standing wave patterns of alternans that are eigenfunctions of a forced Helmholtz equation, and therefore remarkably analogous to sound harmonics in an open pipe. We discuss the implications of these results for using control to suppress alternans in the human ventricles as well as to probe fundamental aspects of alternans morphogenesis. (c) 2002 American Institute of Physics.

Journal Article↗
PubMed1998

Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation.

Wave propagation in ventricular muscle is rendered highly anisotropic by the intramural rotation of the fiber. This rotational anisotropy is especially important because it can produce a twist of electrical vortices, which measures the rate of rotation (in degree/mm) of activation wavefronts in successive planes perpendicular to a line of phase singularity, or filament. This twist can then significantly alter the dynamics of the filament. This paper explores this dynamics via numerical simulation. After a review of the literature, we present modeling tools that include: (i) a simplified ionic model with three membrane currents that approximates well the restitution properties and spiral wave behavior of more complex ionic models of cardiac action potential (Beeler-Reuter and others), and (ii) a semi-implicit algorithm for the fast solution of monodomain cable equations with rotational anisotropy. We then discuss selected results of a simulation study of vortex dynamics in a parallelepipedal slab of ventricular muscle of varying wall thickness (S) and fiber rotation rate (theta(z)). The main finding is that rotational anisotropy generates a sufficiently large twist to destabilize a single transmural filament and cause a transition to a wave turbulent state characterized by a high density of chaotically moving filaments. This instability is manifested by the propagation of localized disturbances along the filament and has no previously known analog in isotropic excitable media. These disturbances correspond to highly twisted and distorted regions of filament, or "twistons," that create vortex rings when colliding with the natural boundaries of the ventricle. Moreover, when sufficiently twisted, these rings expand and create additional filaments by further colliding with boundaries. This instability mechanism is distinct from the commonly invoked patchy failure or wave breakup that is not observed here during the initial instability. For modified Beeler-Reuter-like kinetics with stable reentry in two dimensions, decay into turbulence occurs in the left ventricle in about one second above a critical wall thickness in the range of 4-6 mm that matches experiment. However this decay is suppressed by uniformly decreasing excitability. Specific experiments to test these results, and a method to characterize the filament density during fibrillation are discussed. Results are contrasted with other mechanisms of fibrillation and future prospects are summarized. (c)1998 American Institute of Physics.

Journal Article↗
PubMed1994

Electrical alternans and spiral wave breakup in cardiac tissue.

This paper reports the results of a theoretical investigation of spiral wave breakup in model equations of action potential propagation in cardiac tissue. A general formulation of these equations is described in which arbitrary experimentally determined restitution and dispersion curves can in principle be fitted. Spiral wave behavior is studied in two-dimension as a function of a parameter Re which controls the steepness of the restitution curve at short diastolic intervals. Spiral breakup is found to occur when the minimum period T(min), below which a periodically stimulated tissue exhibits alternans in action potential duration, exceeds by a finite amount the spiral rotation period T(S). At this point, oscillations in action potential duration are of sufficiently large amplitude to cause a spontaneous conduction block to form along the wavefront. The latter occurs closer to the initiation point of reentry (spiral tip) with increasing steepness and, hence, in smaller tissue sizes. Spiral breakup leads to a spatially disorganized wave activity which is always transient, except for tissues larger than some minimum size and within a very narrow range of Re which increases with dispersion.

Journal Article↗