People: A. Bilska, K. DeBruin, W. Krassowska Neu

Collaborators: S. Dev (Genetronics), G. Hofmann (Genetronics)

Additional information: Genetronics, Inc. Web Page

Summary:
Our studies of electroporation in drug delivery are based on a model of a single spherical cell electroporated by an external field. This model is a good representation of an “in vitro ” electroporation of a suspension of cells, a technique used widely in biotechnology. Our single cell model reproduced qualitatively and quantitatively experimental data reported by Kinosita et al. for unfertilized sea urchin eggs: flattening of the profile of the rhtransmembrane potential (Vm) near the poles of the cell, saturation of Vm with the increasing shock strength, disappearance of the rest potential, and slow resealing of the cell membrane to preshock conditions.

Next, the single cell model was expanded by accounting explicitly for the ionic composition of the electroporation current. The new, ion-specific model predicts that Vm is symmetric and almost identical to the profile from the non-specific model, but the pore density N has a profound asymmetry with N at the hyperpolarized end of the cell twice the value at the depolarized end. These modeling results may explain experimentally observed preferential uptake of marker molecules at the hyperpolarized end of the cell and suggest that the electroporation process can be maniputated by extracellular concentrations.

The cell model was also used to investigate the effects of pulse frequency on the outcome of electroporation. In this study, a single cell was exposed to a 10 ms train consisting of monophasic or biphasic pulses with frequencies between 200 kHz and 6 MHz. The fraction of the cell membrane occupied by pores was found to decrease with frequency, and the decrease was more gradual for monophasic pulses. This result suggests that varying the frequency of the monophasic pulse train may provide control over the uptake of molecules during electroporation-mediated drug delivery and gene therapy.

Key Publications:

A. O. Bilska, K. A. DeBruin and Wanda Krassowska, Theoretical modeling of the effects of shock duration, frequency, and strength on the degree of electroporation, Bioelectrochem. Bioenerg., 51: 133-143, 2000.

K. A. DeBruin and W. Krassowska, Modeling electroporation in a single cell. I: Effects of field strength and rest potential, Biophys. J., 77: 1213-1224, 1999

K. A. DeBruin and W. Krassowska, Modeling electroporation in a single cell. II: Effects of ionic concentrations, Biophys. J., 77: 1225-1233, 1999

Support: NSF Grant BES-9974185 (GOALI) and Genetronics, Inc.

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People: K. DeBruin, W. Krassowska Neu

Collaborators: J. Neu (UC Berkeley)

Summary:
Electroporation is described mathematically by a partial differential equation (PDE) that governs the distribution of pores as a function of their radius and time. This PDE does not have an analytical solution and, because of the presence of disparate spatial and temporal scales, numerical solutions are hard to obtain. These difficulties limit the application of the PDE only to experimental setups with a uniformly polarized membrane. This project concentrates on developing simplified, macroscopic descriptions of electroporation that would be easy to solve numerically and thus able to model experimental setups with significant spatial dependence, such cells or fibers in an external field.

Initially, we proposed such a macroscopic model on the basis of experimental results published in the literature (Krassowska, 1955). It has a form of an ordinary differential equation (ODE) that describes the dynamics of the pore density N(t). Later, the model was expanded by adding resealing, nonohmic pore conductance, and parameters suitable for cardiac membrane (DeBruin and Krassowska, 1998). Recently, we performed a rigorous, asymptotic reduction of the original PDE to the ODE governing N(t) and confirmed the validity of our macroscopic model (Neu and Krassowska, in press). Given N(t), the precise distribution of pores in the space of their radii can be determined by an asymptotic approximation. Thus, the ODE represents most of the phenomenology contained in the PDE and remains valid under most experimental conditions.

Key Publications:

J. C. Neu and W. Krassowska, Asymptotic model of electroporation, Phys. Rev. E, 59: 3471-3482, 1999

K. A. DeBruin and W. Krassowska, Electroporation and shock-induced transmembrane potential in a cardiac strand during defibrillation strength shocks, Ann. Biomed. Eng., 26: 584-596, 1998

W. Krassowska, Effects of electroporation on transmembrane potential induced by defibrillation shocks, PACE, 18: 1644-1660, 1995

Support: NSF Grant BES-9409026 and NIH Grant HL54071

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People: K. DeBruin, W. Krassowska Neu, K. Skuibine

Collaborators: N. Trayanova (Tulane)

Summary:
Defibrillation shocks, when delivered through internal electrodes, establish transmembrane potentials (Vm) large enough to electroporate the membrane of cardiac cells. The effects of such shocks on the distribution of Vm were investigated using models of a 1-dim. fiber and a 2-dim. sheet of cardiac tissue. The fiber was represented by the core conductor equation and its membrane was represented by the Luo-Rudy model (LRd) and a macroscopic model of electroporation. The goal of this study was to confirm that experimentally observed saturation of the shock-induced Vm with increasing electric field is due to the development of pores. For shocks delivered during the plateau of an action potential, the model reproduced experimental results 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 was qualitatively reproduced even when the sodium and calcium channels were inactivated.

The two-dimensional sheet was modeled as a bidomain with the passive, electroporating membrane. Computer simulations revealed three categories of response to the stimulating current (Is):
(1) Weak Is, below 0.2 A/m, caused essentially no electroporation, and Vm increased linearly with Is.
(2) Strong Is, between 0.2 and 2.5 A/m, electroporated tissue under the physical electrode. In the electroporated region, the growth of Vm was halted and in the region of reversed polarity (virtual electrode), the growth of Vm was accelerated.
(3) Very strong Is, above 2.5 A/m, electroporated tissue under both the physical and the virtual electrodes and the growth of Vm in all electroporated regions was halted.
These results indicate that electroporation of the cardiac membrane plays an important role in the distribution of Vm induced by defibrillation-strength shocks.

Key Publications:

F. Aguel, K. A. DeBruin, W. Krassowska and N. Trayanova, Effects of electroporation on the transmembrane potential distribution in a two-dimensional bidomain model of cardiac tissue, J. Cardiovasc. Electrophysiol., 10: 701-714, 1999

K. A. DeBruin and W. Krassowska, Electroporation and shock-induced transmembrane potential in a cardiac strand during defibrillation strength shocks, Ann. Biomed. Eng., 26: 584-596, 1998

W. Krassowska, Effects of electroporation on transmembrane potential induced by defibrillation shocks, PACE, 18: 1644-1660, 1995

Support: NIH Grant HL54071 and NSF Grant CDR-8622201

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