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Biomedical subjects

Alain Karma

Publications and source records attributed to Alain Karma.

At least 19 recordsLinked to original sources
PubMed2006

Inferring the cellular origin of voltage and calcium alternans from the spatial scales of phase reversal during discordant alternans.

Beat-to-beat alternation of the action potential duration (APD) in paced cardiac cells has been linked to the onset of lethal arrhythmias. Both experimental and theoretical studies have shown that alternans at the single cell level can be caused by unstable membrane voltage (V(m)) dynamics linked to steep APD-restitution, or unstable intracellular calcium (Ca) cycling linked to high sensitivity of Ca release from the sarcoplasmic reticulum on sarcoplasmic reticulum Ca load. Identifying which of these two mechanisms is the primary cause of cellular alternans, however, has remained difficult since Ca and V(m) are bidirectionally coupled. Here, we use numerical simulations of a physiologically detailed ionic model to show that the origin of alternans can be inferred by measuring the length scales over which APD and Ca(i) alternans reverse phase during spatially discordant alternans. The main conclusion is that these scales are comparable to a few millimeters and equal when alternans is driven by APD restitution, but differ markedly when alternans is driven predominantly by unstable Ca cycling. In the latter case, APD alternans still reverses phase on a millimeter tissue scale due to electrotonic coupling, while Ca alternans reverses phase on a submillimeter cellular scale. These results show that experimentally accessible measurements of Ca(i) and V(m) in cardiac tissue can be used to shed light on the cellular origin of alternans.

Action Potentials↗
PubMed2006

Interface mobility from interface random walk.

Computational studies aimed at extracting interface mobilities require driving forces orders of magnitude higher than those occurring experimentally. We present a computational methodology that extracts the absolute interface mobility in the zero driving force limit by monitoring the one-dimensional random walk of the mean interface position along the interface normal. The method exploits a fluctuation-dissipation relation similar to the Stokes-Einstein relation, which relates the diffusion coefficient of this Brownian-like random walk to the interface mobility. Atomic-scale simulations of grain boundaries in model crystalline systems validate the theoretical predictions and highlight the profound effect of impurities. The generality of this technique, combined with its inherent spatiotemporal efficiency, should allow computational studies to effectively complement experiments in understanding interface kinetics in diverse material systems.

Journal Article↗
PubMed2006

Spatially discordant alternans in cardiac tissue: role of calcium cycling.

Spatially discordant alternans, where the action potential duration (APD) and intracellular calcium transient (Ca(i)) alternate with opposite phase in different regions of tissue, is known to promote wave break and reentry. However, this phenomenon is not completely understood. It is known that alternans at the cellular level can be caused by dynamical instabilities arising from either membrane voltage (V(m)) attributable to steep APD restitution or to calcium (Ca) cycling. Here, we used a mathematical model of intracellular Ca cycling, coupled with membrane ion currents, to investigate the dynamics of V(m) and Ca(i) transient alternans in an isolated cell, in two electrotonically coupled cells, and in 1D spatially homogeneous tissue. Our main finding is a novel instability mechanism in which the bidirectional coupling of V(m) and Ca(i) can drive the Ca(i) transient of two neighboring cells to be out of phase. This instability is manifested in cardiac tissue by the dynamical formation of spatially discordant alternans. In this case, Ca(i) transient alternans can reverse phase over a length scale of one cell, whereas APD alternans reverses phase over a much longer length scale set by the electrotonic coupling. We analyze this mechanism in detail and show that it is a robust consequence of experimentally established properties of the bidirectional coupling between Ca cycling and V(m) dynamics. Finally, we address the experimental relevance of these findings and suggest physiological conditions under which these patterns can be observed.

Arrhythmias, Cardiac↗
PubMed2006

Orientation selection in dendritic evolution.

Dendritic crystal growth patterns have fascinated scientists for several centuries. Much of the aesthetic appeal of these patterns stems from the hierarchical structure of primary-, secondary-, and higher-order branches, which typically grow along principal crystallographic axes. Atypical growth directions have also been observed. Here, we demonstrate both computationally and experimentally that the range of possible dendrite growth directions, and hence the morphological diversity of the resulting dendritic structures, is much richer than previously anticipated. In particular, we show that primary dendrite growth directions can vary continuously between different crystallographic directions as a function of the composition-dependent anisotropy parameters. The study combines phase-field simulations of equiaxed dendritic growth and directional freezing of Al-Zn alloys. Both simulations and experiments exhibit continuous changes of direction from <100> to <110> for an underlying cubic symmetry. These results have important implications for controlling the microstructure of a wide range of cast alloys that solidify dendritically.

Journal Article↗
PubMed2006

From pulsus to pulseless: the saga of cardiac alternans.

Computer simulations and nonlinear dynamics have provided invaluable tools for illuminating the underlying mechanisms of cardiac arrhythmias. Here, we review how this approach has led to major insights into the mechanisms of spatially discordant alternans, a key arrhythmogenic factor predisposing the heart to re-entry and lethal arrhythmias. During spatially discordant alternans, the action potential duration (APD) alternates out of phase in different regions of the heart, markedly enhancing dispersion of refractoriness so that ectopic beats have a high probability of inducing reentry. We show how, at the cellular level, instabilities in membrane voltage (ie, steep APD restitution slope) and intracellular Ca (Cai) cycling dynamics cause APD and the Cai transient to alternate and how the characteristics of alternans are affected by different "modes" of the bidirectional coupling between voltage and Cai. We illustrate how, at the tissue level, additional factors, such as conduction velocity restitution and ectopic beats, promote spatially discordant alternans. These insights have illuminated the mechanistic basis underlying the clinical association of cardiac alternans (eg, T wave alternans) with arrhythmia risk, which may lead to novel therapeutic approaches to avert sudden cardiac death.

Animals↗
PubMed2006

Turing instability mediated by voltage and calcium diffusion in paced cardiac cells.

In cardiac cells, the coupling between the voltage across the cell membrane (V(m)) and the release of calcium (Ca) from intracellular stores is a crucial ingredient of heart function. Under abnormal conditions and/or rapid pacing, both the action potential duration and the peak Ca concentration in the cell can exhibit well known period-doubling oscillations referred to as "alternans," which have been linked to sudden cardiac death. Fast diffusion of V(m) keeps action potential duration alternans spatially synchronized over the approximately 150-mum-length scale of a cell, but slow diffusion of Ca ions allows Ca alternans within a cell to become spatially asynchronous, as observed in some experiments. This finding raises the question: When are Ca alternans spatially in-phase or out-of-phase on subcellular length scales? This question is investigated by using a spatially distributed model of Ca cycling coupled to V(m). Our main finding is the existence of a Turing-type symmetry breaking instability mediated by V(m) and Ca diffusion that causes Ca alternans to become spontaneously out-of-phase at opposite ends of a cardiac cell. Pattern formation is governed by the interplay of short-range activation of Ca alternans, because of a dynamical instability of Ca cycling, and long-range inhibition of Ca alternans by V(m) alternans through Ca-sensitive membrane ionic currents. These results provide a striking example of a Turing instability in a biological context where the morphogens can be clearly identified, as well as a potential link between dynamical instability on subcellular scales and life-threatening cardiac disorders.

Action Potentials↗
PubMed2006

Control of electrical alternans in canine cardiac purkinje fibers.

Alternation in the duration of consecutive cardiac action potentials (electrical alternans) may precipitate conduction block and the onset of arrhythmias. Consequently, suppression of alternans using properly timed premature stimuli may be antiarrhythmic. To determine the extent to which alternans control can be achieved in cardiac tissue, isolated canine Purkinje fibers were paced from one end using a feedback control method. Spatially uniform control of alternans was possible when alternans amplitude was small. However, control became attenuated spatially as alternans amplitude increased. The amplitude variation along the cable was well described by a theoretically expected standing wave profile that corresponds to the first quantized mode of the one-dimensional Helmholtz equation. These results confirm the wavelike nature of alternans and may have important implications for their control using electrical stimuli.

Action Potentials↗
PubMed2006

Nonlinear dynamics of paced cardiac cells.

When a cardiac cell is rapidly paced it can exhibit a beat-to-beat alternation in the action potential duration (APD) and the intracellular calcium transient. This dynamical instability at the cellular level has been shown to correlate with the genesis of cardiac arrhythmias and has motivated the application of nonlinear dynamics in cardiology. In this article, we review mathematical approaches to describe the underlying mechanisms for alternans using beat-to-beat iterated maps. We explain the development and properties of these maps, and show that they provide a fruitful framework to understand dynamical instabilities of voltage and calcium in paced cardiac cells.

Action Potentials↗
PubMed2005

Crack path prediction in anisotropic brittle materials.

A force balance condition to predict quasistatic crack paths in anisotropic brittle materials is derived from an analysis of diffuse interface continuum models that describe both short-scale failure and macroscopic linear elasticity. The path is uniquely determined by the directional anisotropy of the fracture energy, independent of details of the failure process. The derivation exploits the gradient dynamics and translation symmetry properties of this class of models to define a generalized energy-momentum tensor whose integral around an arbitrary closed path enclosing the crack tip yields all forces acting on this tip, including Eshelby's configurational forces, cohesive forces, and dissipative forces. Numerical simulations are in very good agreement with analytic predictions.

Journal Article↗
PubMed2005

The dynamics of cardiac fibrillation.

Reentry occurs when the electrical wave propagating through the atria or ventricles breaks locally and forms a rotor (also called a scroll wave or functional reentry). If the waves propagating outward from a rotor develop additional wavebreaks (which may form new rotors), fibrillation results. Tissue heterogeneity, exacerbated by electrical and structural remodeling from cardiac disease, has traditionally been considered the major factor promoting wavebreak and its degeneration to fibrillation. Recently, however, dynamic factors have also been recognized to play a key role. Dynamic factors refer to cellular properties of the cardiac action potential and Ca(i) cycling, which dynamically generate wave instability and wavebreak, even in tissue that is initially completely homogeneous. Although the latter situation can only be created in computer simulations, its relevance to real (heterogeneous) cardiac tissue has been unequivocally demonstrated. Dynamic factors are related to membrane voltage (Vm) and Ca(i). Vm factors include electrical restitution of action potential duration and conduction velocity, short-term cardiac memory, and electrotonic currents. Ca(i) factors are related to dynamic Ca(i) cycling properties. They act synergistically, as well as with tissue heterogeneity, to promote wavebreak and fibrillation. As global properties, rather than local electrophysiological characteristics, dynamic factors represent an attractive target for novel therapies to prevent ventricular fibrillation.

Action Potentials↗
PubMed2005

Low-temperature dynamics of kinks on Ising interfaces.

The anisotropic motion of an interface driven by its intrinsic curvature or by an external field is investigated in the context of the kinetic Ising model in both two and three dimensions. We derive in two dimensions (2D) a continuum evolution equation for the density of kinks by a time-dependent and nonlocal mapping to the asymmetric exclusion process. Whereas kinks execute random walks biased by the external field and pile up vertically on the physical 2D lattice, they execute hard-core biased random walks on a transformed 1D lattice. Their density obeys a nonlinear diffusion equation which can be transformed into the standard expression for the interface velocity, v=M [ (gamma+gamma'') kappa+H] , where M , gamma+gamma", and kappa are the interface mobility, stiffness, and curvature, respectively. In 3D, we obtain the velocity of a curved interface near the 100 orientation from an analysis of the self-similar evolution of 2D shrinking terraces. We show that this velocity is consistent with the one predicted from the 3D tensorial generalization of the law for anisotropic curvature-driven motion. In this generalization, both the interface stiffness tensor and the curvature tensor are singular at the 100 orientation. However, their product, which determines the interface velocity, is smooth. In addition, we illustrate how this kink-based kinetic description provides a useful framework for studying more complex situations by modeling the effect of immobile dilute impurities.

Journal Article↗
PubMed2005

Coupled dynamics of voltage and calcium in paced cardiac cells.

We investigate numerically and analytically the coupled dynamics of transmembrane voltage and intracellular calcium cycling in paced cardiac cells using a detailed physiological model, and its reduction to a three-dimensional discrete map. The results provide a theoretical framework to interpret various experimentally observed modes of instability ranging from electromechanically concordant and discordant alternans to quasiperiodic oscillations of voltage and calcium.

Action Potentials↗
PubMed2005

Modeling wave propagation in realistic heart geometries using the phase-field method.

We present a novel algorithm for modeling electrical wave propagation in anatomical models of the heart. The algorithm uses a phase-field approach that represents the boundaries between the heart muscle and the surrounding medium as a spatially diffuse interface of finite thickness. The chief advantage of this method is to automatically handle the boundary conditions of the voltage in complex geometries without the need to track the location of these boundaries explicitly. The algorithm is shown to converge accurately in nontrivial test geometries with no-flux (zero normal current) boundary conditions as the width of the diffuse interface becomes small compared to the width of the cardiac action potential wavefront. Moreover, the method is illustrated for anatomically realistic models of isolated rabbit and canine ventricles as well as human atria.

Action Potentials↗
PubMed2004

Quantitative phase-field model of alloy solidification.

We present a detailed derivation and thin interface analysis of a phase-field model that can accurately simulate microstructural pattern formation for low-speed directional solidification of a dilute binary alloy. This advance with respect to previous phase-field models is achieved by the addition of a phenomenological "antitrapping" solute current in the mass conservation relation [Phys. Rev. Lett. 87, 115701 (2001)]]. This antitrapping current counterbalances the physical, albeit artificially large, solute trapping effect generated when a mesoscopic interface thickness is used to simulate the interface evolution on experimental length and time scales. Furthermore, it provides additional freedom in the model to suppress other spurious effects that scale with this thickness when the diffusivity is unequal in solid and liquid [SIAM J. Appl. Math. 59, 2086 (1999)]], which include surface diffusion and a curvature correction to the Stefan condition. This freedom can also be exploited to make the kinetic undercooling of the interface arbitrarily small even for mesoscopic values of both the interface thickness and the phase-field relaxation time, as for the solidification of pure melts [Phys. Rev. E 53, R3017 (1996)]]. The performance of the model is demonstrated by calculating accurately within a phase-field approach the Mullins-Sekerka stability spectrum of a planar interface and nonlinear cellular shapes for realistic alloy parameters and growth conditions.

Journal Article↗
PubMed2003

Phase-field approach for faceted solidification.

We extend the phase-field approach to model the solidification of faceted materials. Our approach consists of using an approximate gamma plot with rounded cusps that can approach arbitrarily closely the true gamma plot with sharp cusps that correspond to faceted orientations. The phase-field equations are solved in the thin-interface limit with local equilibrium at the solid-liquid interface [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)]. The convergence of our approach is first demonstrated for equilibrium shapes. The growth of faceted needle crystals in an undercooled melt is then studied as a function of undercooling and the cusp amplitude delta for a gamma plot of the form gamma=gamma0[1+delta(/sin theta/+/cos theta/)]. The phase-field results are consistent with the scaling law Lambda approximately V(-1/2) observed experimentally, where Lambda is the facet length and V is the growth rate. In addition, the variation of V and Lambda with delta is found to be reasonably well predicted by an approximate sharp-interface analytical theory that includes capillary effects and assumes circular and parabolic forms for the front and trailing rough parts of the needle crystal, respectively.

Journal Article↗
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↗