Conclusion

The phases of electroporation, as seen in the section "Example of the Electroporation Process," closely resemble the steps of electropermeabilization identified by Teissie et al.85 based on experimental observations. The "induction step" corresponds to the charging and pore creation phases of the model (Fig. 8), the "expansion step" corresponds to the pore evolution phase (Fig. 9a), the "stabilization step" corresponds to the postshock shrinkage of pores (Fig. 9b), and the "resealing step" corresponds to the resealing of small pores (Fig. 11). The only step not seen in the model is the "memory effect" that describes changes in the membrane and cell behavior persisting on the time scale of hours and that may be related to exchange of charged molecules between the two monolayers of the membrane.86'87 Although some of the model's predictions still must be confirmed experimentally, the resemblance between the model's "phases" and "steps" of electroporation seen in experiments gives us reason to believe that the asymptotic model presented here is sufficiently accurate to provide theoretical support for real-life applications.

In particular, the asymptotic model can be very useful in assessing the effects of defibrillation shocks on cardiac muscle. Since the electroporation process is represented by a set of ODEs for N(t) and rj (t), it can be naturally incorporated into any model of cardiac membrane by adding electroporation current Ip to the transmembrane current. In the past, such simulations were performed for cardiac membrane,59'88 cardiac fibers,60'89 and two- and three-dimensional myocardium.61'62'90 These studies used an older version of the asymptotic model in which pores did not grow: their size was kept constant and equal to rm « 0.8 nm, the minimum-energy radius. The advantage of this simplification is a dramatic reduction in computational cost because only an ODE for N(t) needs to be solved. This simplification is justified in studies that simulate only the increase of membrane conductance and changes in Vm caused by defibrillation shocks. If desired, the exchange of ions through pores can be added by replacing the current-voltage relationship (15) of a pore by the one that accounts for flow of distinct ions.58'73 This extension will allow the use of the model with nongrowing pores to simulate shock-induced changes in ionic concentration that lead to postshock arrhythmias, as well as resealing of pores and restoration of normal concentrations. However, the model with nongrowing pores is not appropriate for the studies of cellular injury associated with the development of large-sized pores or for predicting the electroporation-mediated uptake of macromolecules such as DNA. Such studies need to use the full asymptotic model, which permits pores to expand and shrink, and thus give a complete picture of the electroporation process.

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