Wanda Krassowska

Modeling Electroporation in a Single Cell. I. Effects of Field Strength and Rest Potential

This study develops a model for a single cell electroporated by an external electric field and uses it to investigate the effects of shock strength and rest potential on the transmembrane potential V(m) and pore density N around the cell. As compared to the induced potential predicted by resistive-capacitive theory, the model of electroporation predicts a smaller magnitude of V(m) throughout the cell. Both V(m) and N are symmetric about the equator with the same value at both poles of the cell. Larger shocks do not increase the maximum magnitude of V(m) because more pores form to shunt the excess stimulus current across the membrane. In addition, the value of the rest potential does not affect V(m) around the cell because the electroporation current is several orders of magnitude larger than the ionic current that supports the rest potential. Once the field is removed, the shock-induced V(m) discharges within 2 micros, but the pores persist in the membrane for several seconds. Complete resealing to preshock conditions requires approximately 20 s. These results agree qualitatively and quantitatively with the experimental data reported by Kinosita and coworkers for unfertilized sea urchin eggs exposed to large electric fields.

Choosing the optimal monophasic and biphasic waveforms for ventricular defibrillation.

Optimal Monophasic and Biphasic Waveforms. Introduction: The truncated exponential waveform from an implantable cardioverter defibrillator can be described by three quantities: the leading edge voltage, the waveform duration, and the waveform time coastant (τs). The goal of this work was to develop and test a mathematical model of defibrillation that predicts the optimal durations for monophasic and the first phase of biphasic waveforms for different τs values. In 1932, Blair used a parallel resistor‐capacitor network as a model of the cell membrane to develop an equation that describes stimulation using square waves. We extended Blairs model of stimulation, using a resistor‐capacitor network time constant (τm), equal to 2.8 msec, to explicitly account for the waveform shape of a truncated exponential waveform. This extended model predicted that for monophasic waveforms with τs of 1.5 msec, leading edge voltage will be constant for waveforms 2 msec and longer; for τs of 3 msec, leading edge voltage will be constant for waveforms 3 msec and longer; for τs of 6 msec, leading edge voltage will be constant for waveforms 4 msec and longer. We hypothesized that the best phase 1 of a biphasic waveform is the best monophasic waveform. Therefore, the optimal first phase of a biphasic waveform for a given τs is the same as the optimal monophasic waveform.

Current Concepts for Selecting the Location, Size and Shape of Defibrillation Electrodes

Defibrillation would be improved if the shock strength could be decreased. Decreasing shock strength would lessen the chance that the shock itself could damage the heart. With implantable defibrillators, some patients cannot be defibrillated even with the defibrillator at its highest setting; if the shock strength required for defibrillation were sufficiently lowered to bring the required shock voltage into the range of the device, these patients could be defibrillated. Decreasing shock strength requirements would increase the life of the batteries or would allow tbe use of smaller implanted devices. Since the time to charge tbe capacitors would be reduced, it would also shorten the interval until tbe shock was delivered and, hence, decrease the time that tbe patient was without blood flow during fibrillation. Tbe primary variables tbat can be altered to attempt to lower tbe shock strength required for defibrillation include those dealing with the shock waveform, including duration, polarity, and wavesbape, and tbose involving tbe sbock electrodes, including materials of construction, size, shape, and location. This article is concerned witb the last three of tbese variables. It discusses the basic principles, as they are understood today,

Model of Creation and Evolution of Stable Electropores for DNA Delivery

Electroporation, in which electric pulses create transient pores in the cell membrane, is becoming an important technique for gene therapy. To enable entry of supercoiled DNA into cells, the pores should have sufficiently large radii (>10 nm), remain open long enough for the DNA chain to enter the cell (milliseconds), and should not cause membrane rupture. This study presents a model that can predict such macropores. The distinctive features of this model are the coupling of individual pores through membrane tension and the electrical force on the pores, which is applicable to pores of any size. The model is used to explore the process of pore creation and evolution and to determine the number and size of pores as a function of the pulse magnitude and duration. Next, our electroporation model is combined with a heuristic model of DNA uptake and used to predict the dependence of DNA uptake on pulsing parameters. Finally, the model is used to examine the mechanism of a two-pulse protocol, which was proposed specifically for gene delivery. The comparison between experimental results and the model suggests that this model is well-suited for the investigation of electroporation-mediated DNA delivery.

Transmural activations and stimulus potentials in three-dimensional anisotropic canine myocardium.

Epicardial and endocardial pacing are widely used, yet little is known about the three-dimensional distribution of potentials generated by the pacing stimulus or the spread of activation from these pacing sites. In six open-chest dogs, simultaneous recordings were made from 120 transmural electrodes in 40 plunge electrodes within a 35 X 20 X 5-mm portion of the right ventricular outflow tract during epicardial and endocardial pacing at a strength of twice diastolic threshold and at 1 mA. The magnitude of extracellular potentials generated by the stimulus and the activation times were compared in regions proximal (less than 10-12 mm) and distal to the pacing site. Local fiber orientation was histologically determined at each recording electrode. For endocardial pacing, endocardial potentials were larger than epicardial potentials only in the proximal region (p less than 0.001); while in the distal region, epicardial potentials were larger (p less than 0.001), and endocardial activation occurred earlier than epicardial activation for both regions (p less than 0.001). For epicardial pacing, epicardial potentials were larger than endocardial potentials in both regions (p less than 0.001), and epicardial activation occurred earlier only in the proximal region (p less than 0.02), while endocardial activation occurred before epicardial activation in the distal region (p less than 0.01). In planes of recording electrodes parallel to the epicardium and endocardium, the initial isochrones were elliptical with the major axes of the ellipses along the mean fiber orientation between the pacing site and recording plane rather than along the local fiber orientation in the recording plane. Thus, the ellipses in each plane rotated with respect to each other so that in three dimensions the activation front was helicoid, yet the twist of the helix was less than that of the corresponding transmural rotation of fibers. For pacing from the right ventricular outflow tract, we conclude that beyond 10-12 mm from endocardial and epicardial pacing sites epicardial stimulus potentials in both cases are larger than endocardial potentials because of resistivity differences inside and outside the heart wall and activation in both cases is primarily endocardial to epicardial because of rapid endocardial conduction, and we conclude that the initial spread of activation is helicoid and determined by transmural fiber direction.

Extracellular field required for excitation in three-dimensional anisotropic canine myocardium.

It is not known how well potential gradient, current density, and energy correlate with excitation by extracellular stimulation in the in situ heart. Additionally, the influence of fiber orientation and stimulus polarity on the extracellular thresholds for stimulation expressed in terms of these factors has not been assessed. To answer these questions for myocardium in electrical diastole, extracellular excitation thresholds were determined from measurements of stimulus potentials and activation patterns recorded from 120 transmural electrodes in a 35 X 20 X 5-mm region of the right ventricular outflow tract in six open-chest dogs. Extracellular potential gradients, current densities, energies, and their components longitudinal and transverse to the local fiber orientation at each recording site were calculated from the stimulus potentials produced by 3-msec constant-current stimuli. The resulting values in regions directly excited by the stimulus field were compared with the values in regions not directly excited but activated by the spread of wavefronts conducting away from the directly excited region. Magnitudes of 3.66 mA/cm2 for current density, 9.7 microJ/cm3 for energy, and 804 mV/cm for potential gradient yielded minimum misclassifications of 8%, 13%, and 17%, respectively, of sites directly and not directly excited. A linear bivariate combination of the longitudinal (l) and transverse (t) components of the potential gradient yielded 7% misclassification (threshold ratio t/l of 2.88), and linear combination of corresponding current density components yielded 8% misclassification (threshold ratio t/l of 1.04). Anodal and cathodal thresholds were not significantly different (p = 0.39). Potential gradient, current density, and energy strength-duration curves were constructed for pulse durations (D) of 0.2-20 msec. The best fit hyperbolic curve for current density magnitude (Jm) was Jm = 3.97/D + 3.15, where Jm is in mA/cm2, and D is in msec. Thus, for stimulation during electrical diastole 1) both current density magnitude and longitudinal and transverse components of the potential gradient are closely correlated with excitation, 2) the extracellular potential gradient along cardiac cells has a lower threshold than across cells, while current density thresholds along and across cells are similar, 3) anodal and cathodal thresholds are approximately equal for stimuli greater than or equal to 5 mA, and 4) the extracellular potential gradient, current density, and energy excitation thresholds can be expressed by strength-duration equations.

Periodic Conductivity as a Mechanism for Cardiac Stimulation and Defibrillation

This study examines the distribution of the transmembrane potential in the periodic strand of cardiac muscle established by configurations of sources similar to those arising during extracellular stimulation and defibrillation, during intracellular stimulation, and during propagation of action potential. The closed-form solution indicates that during extracellular stimulation with large current and during defibrillation, the periodic component of the transmembrane potential is very important. We postulate that this periodic component causes the depolarization or defibrillation in cardiac muscle, which is different from the depolarization mechanism for a continuous fiber. On the other hand, during propagation and intracellular stimulation, the periodic component only slightly modifies the monotonic decrease of the transmembrane potential, which suggests that the mechanism of propagation in discrete structures may be similar to that of the continuous fiber.

The restitution portrait: A new method for investigating rate-dependent restitution

Introduction: Electrical restitution, relating action potential duration (APD) to diastolic interval (DI), was believed to determine the stability of heart rhythm. However, recent studies demonstrate that stability also depends on long‐term APD changes caused by memory. This study presents a new method for investigation of rate‐ and memory‐dependent aspects of restitution and for assessment of mapping models of APD.

Modeling electroporation in a single cell. II. Effects Of ionic concentrations.

This study expands a previously developed model of a single cell electroporated by an external electric field by explicitly accounting for the ionic composition of the electroporation current. The previous model with non-specific electroporation current predicts that both the transmembrane potential V(m) and the pore density N are symmetric about the equator, with the same values at either end of the cell. The new, ion-specific case predicts that V(m) is symmetric and almost identical to the profile from the non-specific case, but N has a profound asymmetry with the pore density at the hyperpolarized end of the cell twice the value at the depolarized end. These modeling results agree with the experimentally observed preferential uptake of marker molecules at the hyperpolarized end of the cell as reported in the literature. This study also investigates the changes in intracellular ionic concentrations induced around an electroporated single cell. For all ion species, the concentrations near the membrane vary significantly, which may explain the electrical disturbances observed experimentally after large electric shocks are delivered to excitable cells and tissues.

Electroporation and Shock-Induced Transmembrane Potential in a Cardiac Fiber During Defibrillation Strength Shocks

AbstractExperimental studies have shown that the magnitude of the shock-induced transmembrane potential (Vm) saturates with increasing electric field strength. This study uses a mathematical model to investigate the effects of electroporation and membrane kinetics on Vm in a cardiac fiber. The model consists of the core conductor equation for a one-dimensional fiber, where excitability is represented by the Luo–Rudy dynamic model (1994–1995) and electroporation is described by a membrane conductance that increases exponentially with Vm squared. For shocks delivered during the plateau of an action potential, the model reproduces the experimentally observed saturation of Vm with a root mean square error of 4.27% and a correlation coefficient of 0.9992. For shocks delivered during diastole, the saturation of Vm is qualitatively reproduced even when the sodium and calcium channels are inactivated. Quantitative replication of the response to diastolic shocks is hindered by the choice of electroporation parameters (optimized for shocks delivered during the plateau) and differences in the membrane kinetics between model and experiment. The complex behavior of Vm during large shocks is due to a combination of electroporation, electrotonus, propagation, and active membrane kinetics. The modeling results imply that the experimentally observed saturation of Vm is due to electroporation of the lipid bilayer.